Common Core: Mathematics

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Standards for Mathematical Practice

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Standards for Mathematical Practice

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Make sense of problems and persevere in solving them
MP1
Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution.

They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.
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Reason abstractly and quantitatively
MP2
Mathematically proficient students make sense of quantities and their relationships in problem situations.

They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.
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Construct viable arguments and critique the reasoning of others
MP3
Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments.

They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.
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Model with mathematics
MP4
Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace.

In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.
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Use appropriate tools strategically
MP5
Mathematically proficient students consider the available tools when solving a mathematical problem.

These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.
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Attend to precision
MP6
Mathematically proficient students try to communicate precisely to others.

They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.
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Look for and make use of structure
MP7
Mathematically proficient students look closely to discern a pattern or structure.

Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x² + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)² as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.
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Look for and express regularity in repeated reasoning
MP8
Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts.

Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x² + x + 1), and (x – 1)(x³ + x² + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.
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Kindergarten

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Counting & Cardinality
Contar y usar números cardinales

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Know number names and the count sequence
I can count and know my numbers
Sé los números y puedo contar
K.CC.A.1
Count to 100 by ones and by tens.
K.CC.A.1
I can count to 100 by ones and tens.

K.CC.A.1
Puedo contar hasta el 100 de 1 en 1 y de 10 en 10.

K.CC.A.2
Count forward beginning from a given number within the known sequence (instead of having to begin at 1).
K.CC.A.2
I can count forward starting at any number I have learned.

K.CC.A.2
Puedo contar hacia adelante a partir de cualquier número que he aprendido.

K.CC.A.3
Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20 (with 0 representing a count of no objects).
K.CC.A.3
I can write numbers from 0 to 20.

I can write a number to tell about a group of 0 to 20 things.

K.CC.A.3
Puedo escribir los números del 0 al 20.

Puedo escribir un número para describir un grupo de 0 a 20 cosas.

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Count to tell the number of objects
I can count to tell the number of things
Puedo contar para decir cuántos hay
K.CC.B.4
Understand the relationship between numbers and quantities; connect counting to cardinality.
K.CC.B.4
I can understand how number names go with counting things in the right order.

K.CC.B.4
Entiendo cómo los nombres de números se relacionan al contar de las cosas en orden correcto.

K.CC.B.4.a
When counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object.
K.CC.B.4.a
I can name the number for each thing in a group as I count them.

K.CC.B.4.a
Puedo decir el número de itens en un grupo cuando los cuento.

K.CC.B.4.b
Understand that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the order in which they were counted.
K.CC.B.4.b
I can understand that the last thing I count tells the number of things in a group.

I can understand that things in a group can be moved around and the total number will be the same.

K.CC.B.4.b
Entieno que el último número que cuento es el total de cosas en un grupo.

Entiendo que los objetos en un grupo se pueden mover y el número total será el mismo.

K.CC.B.4.c
Understand that each successive number name refers to a quantity that is one larger.
K.CC.B.4.c
I can understand that the next number I say when I count means that there is one more.

K.CC.B.4.c
Entiendo que cuando digo el próximo número que significa que hay uno más.

K.CC.B.5
Count to answer “how many?” questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1–20, count out that many objects.
K.CC.B.5
I can count up to 20 to tell how many things are in a line, a box or a circle.

I can count up to 10 to tell how many things are in a group.

I can count out a group of things when someone gives me any number from 1 to 20.

K.CC.B.5
Puedo contar al 20 para contar cuántas cosas están en una línea, un rectángulo o un círculo.

Puedo contar al 10 para contar cuántas cosas están en un grupo.

Puedo contar de un grupo de cosas cuando alguien me da un número del 1 al 20.

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Compare numbers
I can compare numbers
Puedo comparar los números
K.CC.C.6
Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies (include groups with up to ten objects).
K.CC.C.6
I can use matching or counting to tell if a group of objects in one group is bigger, smaller or the same as a group of objects in another group.

K.CC.C.6
Puedo relacionar o contar para saber si un grupo de objetos en un grupo es igual, más grande, o más pequeña de un grupo de objetos en otro grupo.

K.CC.C.7
Compare two numbers between 1 and 10 presented as written numerals.
K.CC.C.7
I can compare two written numbers between 1 and 10.

K.CC.C.7
Puedo comparar dos números escritos entre 1 y 10.

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Geometry
La geometría

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Identify and describe shapes
I can name and tell about shapes
Puedo identificar y describir las formas geométricas
K.G.A.1
Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as above, below, beside, in front of, behind, and next to.
K.G.A.1
I can name and tell about shapes I see around me.

I can tell where I see shapes by using words like: above, below, beside, in front of, behind and next to.

K.G.A.1
Puedo identificar y hablar sobre las formas geométricas que veo a mi alrededor.

Puedo decir donde veo formas al usar palabras como: arriba, abajo, al lado, delante y detrás.

K.G.A.2
Correctly name shapes regardless of their orientations or overall size.
K.G.A.2
I can name shapes no matter how big they are or which way they are turned.

K.G.A.2
Puedo identificar formas no importa su tamaño o posición.

K.G.A.3
Identify shapes as two-dimensional (lying in a plane, “flat”) or three- dimensional (“solid”).
K.G.A.3
I can tell if a shape is two-dimensional (flat) or three-dimensional (solid).

K.G.A.3
Puedo decir si una forma es bidimensional (plano) o tridimensional (sólido).

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Analyze, compare, create, and compose shapes
I can think about, compare and make different shapes
Puedo comparar, hacer y pensar en formas geométricas diferentes
K.G.B.4
Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts (e.g., number of sides and vertices/“corners”) and other attributes (e.g., having sides of equal length).
K.G.B.4
I can think about and compare two-dimensional and three-dimensional shapes.

K.G.B.4
Puedo pensar y comparar figuras bidimensionales y tridimensionales.

K.G.B.5
Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes.
K.G.B.5
I can make shapes by drawing them or by using things like sticks and clay.

K.G.B.5
Puedo hacer formas al dibujar o usar cosas como palos y barro.

K.G.B.6
Compose simple shapes to form larger shapes.

For example, “Can you join these two triangles with full sides touching to make a rectangle?”
K.G.B.6
I can use simple shapes to make larger shapes.

K.G.B.6
Puedo usar formas simples para hacer formas más grandes.

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Measurement & Data
La medición y los datos

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Describe and compare measurable attributes
I can tell about and compare things that can be measured
Puedo decir y comparar cosas que pueden ser medidas
K.MD.A.1
Describe measurable attributes of objects, such as length or weight. Describe several measurable attributes of a single object.
K.MD.A.1
I can show and tell about the parts of a thing that I can measure.

K.MD.A.1
Puedo mostrar y hablar de las partes de una cosa que puedo medir.

K.MD.A.2
Directly compare two objects with a measurable attribute in common, to see which object has “more of”/“less of” the attribute, and describe the difference.

For example, directly compare the heights of two children and describe one child as taller/shorter.
K.MD.A.2
I can compare two things that are measured using the same tool by using words like longer and shorter.

K.MD.A.2
Puedo comparar dos cosas que se miden utilizando la misma herramienta al usar palabras como más largo y más corto.

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Classify objects and count the number of objects in each category
I can sort things and put them into groups
Puedo clasificar objetos y ponerlos en grupos
K.MD.B.3
Classify objects into given categories; count the numbers of objects in each category and sort the categories by count (limit category counts to be less than or equal to 10).
K.MD.B.3
I can put things into groups by looking at how they are the same.

I can count the things that I put into groups and then sort them by how many.

K.MD.B.3
Puedo clasificar las cosas en grupos al observar cómo son iguales.

Puedo contar las cosas que he puesto en grupos y luego ordenarlas por el número.

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Number & Operations in Base Ten
Los números y las operaciones de base diez

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Work with numbers 11–19 to gain foundations for place value
I can work with bigger numbers to understand place value
Puedo usar números mayores para entender el valor posicional
K.NBT.A.1
Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or drawings, and record each composition or decomposition by a drawing or equation (e.g., 18 = 10 + 8); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones.
K.NBT.A.1
I can make and take apart numbers from 11 to 19 by telling how many tens and ones are in the number.

I can show how many tens and ones in numbers from 11 to 19 by drawing a picture or writing a number sentence.

K.NBT.A.1
Puedo combinar y separar partes de números del 11 al 19 a través de decir cuántas decenas y unidades hay.

Puedo mostrar cuántas decenas y unidades hay en los números del 11 al 19 al hacer un dibujo o escribir una frase numérica.

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Operations & Algebraic Thinking
Las operaciones y el raciocinio algebraico

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Understand addition and understand subtraction
I can understand addition and subtraction
Entiendo la suma y la resta
K.OA.A.1
Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations. (Drawings need not show details, but should show the mathematics in the problem.)
K.OA.A.1
I can use what makes sense to me to show that I know how to add.

I can use what makes sense to me to show that I know how to subtract.

K.OA.A.1
Puedo usar lo que tiene sentido para mí para demostrar que yo sé sumar.

Puedo usar lo que tiene sentido para mí para demostrar que yo sé restar.

K.OA.A.2
Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.
K.OA.A.2
I can use objects or drawings to show that I can solve addition word problems up to 10.

I can use objects or drawings to show that I can solve subtraction word problems up to 10.

K.OA.A.2
Puedo usar objetos o dibujos para demostrar que puedo resolver problemas de adición al 10.

Puedo usar objetos o dibujos para demostrar que puedo resolver problemas de resta al 10.

K.OA.A.3
Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).
K.OA.A.3
I can take apart any number from 1 to 10 to show that I understand that number. (5 = 2 + 3)

K.OA.A.3
Puedo simplificar cualquier número de 1 a 10 para mostrar que entiendo ese número. (5 = 2 + 3)

K.OA.A.4
For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.
K.OA.A.4
I can take any number from 1 to 9 and show what I need to add to it to make 10.

K.OA.A.4
Puedo combinar cualquier conjunto de dos números de 1 a 9 para demonstrar qué necesita para sumar al 10.

K.OA.A.5
Fluently add and subtract within 5.
K.OA.A.5
I can add numbers within 5.

I can subtract numbers within 5.

K.OA.A.5
Puedo sumar números dentro del 5.

Puedo restar números dentro del 5.

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Grade 1

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Geometry
La geometría

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Reason with shapes and their attributes
I can understand shapes better by using what I notice about them
Entiendo las formas mejor usando lo que noto en ellas
1.G.A.1
Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.
1.G.A.1
I can understand and tell about the parts that make different shapes unique.

I can build and draw shapes that have certain parts.

1.G.A.1
Entiendo y puedo hablar sobre las partes que hacen únicas las diferentes formas.

Puedo construir y dibujar formas que tienen ciertas partes.

1.G.A.2
Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape. (Students do not need to learn formal names such as “right rectangular prism.”)
1.G.A.2
I can create two-dimensional shapes. (rectangles, squares, trapezoids, triangles, half-circles and quarter-circles)

I can create three-dimensional shapes. (cubes, right rectangular prisms, right circular cones and righ circular cylinders)

I can use two- and three-dimensional shapes to create new shapes.

1.G.A.2
Puedo crear formas bidimensionales. (rectángulos, cuadrados, trapecios, triángulos, semicírculos y cuartos de círculo)

Puedo crear formas tridimensionales. (cubos, prismas rectangulares rectos, conos circulares rectos y cilindros circulares rectos)

Puedo usar formas de 2 y 3 dimensiones para crear nuevas formas.

1.G.A.3
Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.
1.G.A.3
I can understand that "halves" means two equal parts and "fourths" or "quarters" means four equal parts.

I can break circles and rectangles into equal parts and use the words whole, halves, fourths, and quarters to talk about them.

I can understand that breaking circles or rectangles into more equal parts means that the parts will be smaller.

1.G.A.3
Entiendo que mitades se refiere a dos partes iguales y que cuartos significa cuatro partes iguales.

Puedo descomponer círculos y rectángulos en partes iguales y usar palabras como entero, mitades y cuartos para describirlos.

Entiendo que la descomposición de un círculo o rectángulo en más partes iguales significa que las partes serán más pequeñas.

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Measurement & Data
La medición y los datos

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Measure lengths indirectly and by iterating length units
I can understand length
Entiendo la longitud
1.MD.A.1
Order three objects by length; compare the lengths of two objects indirectly by using a third object.
1.MD.A.1
I can put three objects in order from longest to shortest and compare their lengths.

1.MD.A.1
Puedo poner tres objetos en orden del más largo al más corto y comparar sus longitudes.

1.MD.A.2
Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps. Limit to contexts where the object being measured is spanned by a whole number of length units with no gaps or overlaps.
1.MD.A.2
I can tell the length of an object using whole numbers.

I can show that I understand how to measure something by using a smaller object as a measurement tool.

1.MD.A.2
Puedo decir la longitud de un objeto utilizando números enteros.

Puedo demostrar que entiendo cómo medir algo mediante el uso de un objeto pequeño como una herramienta de medición.

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Tell and write time
I can tell time
Puedo decir la hora
1.MD.B.3
Tell and write time in hours and half-hours using analog and digital clocks.
1.MD.B.3
I can tell and write time in hours and half-hours using any kind of clock.

1.MD.B.3
Puedo decir y escribir la hora a la hora o a la media hora utilizando cualquier tipo de reloj.

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Represent and interpret data
I can understand how information is shared using numbers
Entiendo cómo se comparte la información usando los números
1.MD.C.4
Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.
1.MD.C.4
I can organize , show and explain number information in a way that makes sense.

I can ask and answer questions about number information that is organized.

1.MD.C.4
Puedo organizar, mostrar y explicar la información numérica de una manera que tiene sentido.

Puedo hacer y contestar preguntas acerca de la información numérica organizada.

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Number & Operations in Base Ten
Los números y las operaciones de base diez

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Extend the counting sequence
I can count up
Puedo contar hasta los próximos números
1.NBT.A.1
Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.
1.NBT.A.1
I can count up to 120 starting at any number under 120.

I can read and write my numbers to show how many objects are in a group.(up to 120)

1.NBT.A.1
Puedo contar hasta 120 a partir de cualquier número menos de 120.

Puedo leer y escribir los números para mostrar cuántos objetos hay en un grupo (hasta 120).

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Understand place value
I can understand place value
Entiendo el valor posicional
1.NBT.B.2
Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:
1.NBT.B.2
I can tell how many tens and how many ones are in a number.

1.NBT.B.2
Puedo decir cuántas decenas y cuántas unidades hay en un número.

1.NBT.B.2.a
10 can be thought of as a bundle of ten ones — called a “ten.”
1.NBT.B.2.a
I can show that I know what a "ten" is.

1.NBT.B.2.a
Yo puedo demostrar que yo sé lo que es una decena.

1.NBT.B.2.b
The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.
1.NBT.B.2.b
I can show that any number between 11 and 19 is a group of "ten" and a certain number of ones.

1.NBT.B.2.b
Puedo demostrar que cualquier número entre 11 y 19 es un grupo de diez y un cierto número de unidades.

1.NBT.B.2.c
The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).
1.NBT.B.2.c
I can show that I understand the numbers I use when I count by tens, have a certain number of tens and 0 ones.

1.NBT.B.2.c
Puedo demostrar que cuando cuento de 10 en 10, estoy usando un cierto número de decenas y ningunas unidades.

1.NBT.B.3
Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.
1.NBT.B.3
I can compare two-digit numbers using <, =, and > because I understand tens and ones.

1.NBT.B.3
Puedo comparar números de dos dígitos utilizando <, = y > porque entiendo qué significa decenas y unidades.

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Use place value understanding and properties of operations to add and subtract
I can use what I know about place value to help me add and subtract
Puedo usar lo que sé sobre el valor posicional para ayudarme a sumar y restar
1.NBT.C.4
Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.
1.NBT.C.4
I can use math strategies to help me solve and explain addition problems within 100.

I can use objects and pictures to help me solve and explain addition problems within 100.

I can understand that adding two-digit numbers means I add the ones and then the tens.

I can understand that when I add two-digit numbers, sometimes I have to make a group of ten from the ones. (regroup)

1.NBT.C.4
Puedo usar estrategias matemáticas para ayudarme a resolver y explicar problemas de suma hasta 100.

Puedo usar objetos e imágenes para ayudarme a resolver y explicar problemas de suma hasta 100.

Entiendo que la adición de números de dos dígitos significa primero sumo las unidades y después las decenas.

Entiendo que cuando sumo números de dos dígitos, a veces se necesita hacer un grupo de diez de las unidades (reagrupar).

1.NBT.C.5
Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.
1.NBT.C.5
I can find 10 more or 10 less in my head.

1.NBT.C.5
Puedo determinar 10 más o 10 menos de un número en mi mente.

1.NBT.C.6
Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
1.NBT.C.6
I can use different strategies to subtract multiples of 10 (10-90) from numbers under 100 , write the matching number sentence and explain my strategy.

1.NBT.C.6
Puedo usar diferentes estrategias para restar múltiplos de 10 (10-90) de los números menos de 100, escribir la frase numérica correspondiente y explicar mi estrategia.

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Operations & Algebraic Thinking
Las operaciones matemáticas y el raciocinio algebraico

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Represent and solve problems involving addition and subtraction
I can write and solve problems using addition and subtraction
Puedo escribir y resolver problemas usando la suma y la resta
1.OA.A.1
Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. (See Glossary, Table 1.)
1.OA.A.1
I can use different strategies for addition to solve word problems. (within 20)

I can use different strategies for subtraction to solve word problems. (within 20)

1.OA.A.1
Puedo usar diferentes estrategias de la suma para resolver problemas verbales (hasta 20).

Puedo usar diferentes estrategias de la resta para resolver problemas verbales (de 20 o menos).

1.OA.A.2
Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.
1.OA.A.2
I can use solve word problems where I have to add 3 whole numbers.

1.OA.A.2
Puedo resolver problemas verbales cuando hay que sumar tres números enteros.

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Understand and apply properties of operations and the relationship between addition and subtraction
I can understand and use what I know about addition and subtraction
Entiendo y uso lo que sé acerca de la suma y la resta
1.OA.B.3
Apply properties of operations as strategies to add and subtract. (Students need not use formal terms for these properties.)

Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)
1.OA.B.3
I can use fact families to help me solve addition problems. (commutative)

I can use addition facts I know well to help me solve problems where there are more than two numbers. (associative)

1.OA.B.3
Puedo usar las operaciones relacionadas para ayudarme a resolver problemas de suma (propiedad conmutativa).

Puedo usar las operaciones de suma que sé bien para ayudarme a resolver problemas en los que hay más de dos números (propiedad asociativa).

1.OA.B.4
Understand subtraction as an unknown-addend problem.

For example, subtract 10 – 8 by finding the number that makes 10 when added to 8.
1.OA.B.4
I can use what I know about addition facts to help me answer subtraction fact problems.

1.OA.B.4
Puedo usar lo que sé acerca de las operaciones de suma para ayudarme a resolver problemas de resta.

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Add and subtract within 20
I can add and subtract any numbers from 0 to 20
Puedo sumar y restar cualquier número de 0 a 20
1.OA.C.5
Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).
1.OA.C.5
I can understand how counting up is like adding and counting down is like subtracting.

1.OA.C.5
Entiendo que contar hacia adelante es sumar y contar hacia atrás es restar.

1.OA.C.6
Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).
1.OA.C.6
I can add facts within 20.

I can subtract facts within 20.

1.OA.C.6
Puedo sumar números hasta 20.

Puedo restar números de 20 y menos.

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Work with addition and subtraction equations
I can work with addition and subtraction number sentences
Puedo trabajar con frases numéricas para sumar y restar
1.OA.D.7
Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false.

For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.
1.OA.D.7
I can tell if addition or subtraction number sentences are true because I understand what an equal sign means.

1.OA.D.7
Puedo decir si una frase numérica (de suma o resta) es verdadera porque entiendo qué significa el signo igual.

1.OA.D.8
Determine the unknown whole number in an addition or subtraction equation relating three whole numbers.

For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = _ – 3, 6 + 6 = _.
1.OA.D.8
I can figure out what a missing number is in an addition or subtraction problem.

1.OA.D.8
Puedo determinar el número que falta en un problema de suma o resta.

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Grade 2

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Geometry
La geometría

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Reason with shapes and their attributes
I can understand shapes better by using what I notice about them
Entiendo las formas mejor usando lo que noto en ellas
2.G.A.1
Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. (Sizes are compared directly or visually, not compared by measuring.) Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.
2.G.A.1
I can name and draw shapes. (I know triangles, quadrilaterals, pentagons, hexagons and cubes.)

2.G.A.1
Puedo identificar y dibujar formas. (Yo sé cuáles son los triángulos, cuadriláteros, pentágonos, hexágonos y cubos.)

2.G.A.2
Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.
2.G.A.2
I can find the area of a rectangle by breaking it into equal sized squares.

2.G.A.2
Puedo encontrar el área de un rectángulo al dividirlo en cuadrados de tamaño igual.

2.G.A.3
Partition circles and rectangles into two, three, or four equal shares, describe the shares using the wordshalves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.
2.G.A.3
I can divide shapes into equal parts and describe the parts with words like halves or thirds.

I can understand that equal parts of a shape may look different depending on how I divide the shape.

2.G.A.3
Puedo dividir las formas en partes iguales y describir las partes usando palabras como mitades o tercios.

Puedo entender que partes iguales de una forma pueden parecer diferentes dependiendo de cómo divido la forma.

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Measurement & Data
La medición y los datos

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Measure and estimate lengths in standard units
I can measure and estimate lengths of objects
Puedo medir y estimar longitudes de los objetos
2.MD.A.1
Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.
2.MD.A.1
I can use different tools to measure objects.

2.MD.A.1
Puedo utilizar diferentes herramientas para medir objetos.

2.MD.A.2
Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.
2.MD.A.2
I can use two different units to measure the same object and tell how the measurements compare.

2.MD.A.2
Puedo usar dos unidades diferentes para medir el mismo objeto y decir cómo se comparan las mediciones.

2.MD.A.3
Estimate lengths using units of inches, feet, centimeters, and meters.
2.MD.A.3
I can estimate the lengths of objects using inches, feet, centimeters and meters.

2.MD.A.3
Puedo calcular las longitudes de los objetos utilizando pulgadas, pies, centímetros y metros.

2.MD.A.4
Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.
2.MD.A.4
I can tell the difference in the lengths of two different objects.

2.MD.A.4
Puedo decir la diferencia en las longitudes de dos objetos diferentes.

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Relate addition and subtraction to length
I can use what I know about addition and subtraction to understand length
Puedo usar lo que sé acerca de la suma y la resta para entender longitud
2.MD.B.5
Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.
2.MD.B.5
I can use addition and subtraction to solve measurement problems.

2.MD.B.5
Puedo usar la suma y la resta para resolver problemas de medición.

2.MD.B.6
Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, ..., and represent whole-number sums and differences within 100 on a number line diagram.
2.MD.B.6
I can make and use a number line.

2.MD.B.6
Puedo hacer y usar una recta numérica.

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Work with time and money
I can count money
Entiendo decir la hora
2.MD.C.7
Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.
2.MD.C.7
I can tell time to five minutes.

I can use a.m. and p.m. in the right ways.

2.MD.C.7
Puedo decir la hora correctamente a los cinco minutos.

Puedo usar a.m. y p.m. de forma correcta.

2.MD.C.8
Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately.

Example: If you have 2 dimes and 3 pennies, how many cents do you have?
2.MD.C.8
I can count money to help me solve word problems.

2.MD.C.8
Puedo contar con dinero para ayudarme a resolver problemas verbales.

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Represent and interpret data
I can understand how information is shared using numbers
Entiendo cómo se comparte la información usando los números
2.MD.D.9
Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.
2.MD.D.9
I can make a table to organize information about measurement.

I can show measurements with a line plot.

2.MD.D.9
Puedo hacer una tabla para organizar la información sobre la medición.

Puedo mostrar las mediciones usando un esquema lineal.

2.MD.D.10
Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put- together, take-apart, and compare problems (See Glossary, Table 1.) using information presented in a bar graph.
2.MD.D.10
I can draw a picture graph to share number information.

I can draw a bar graph to share number information.

I can solve problems using information from a bar graph.

2.MD.D.10
Puedo dibujar una gráfica de dibujos para representar información numérica.

Puedo dibujar una gráfica de barras para representar información numérica.

Puedo resolver problemas usando información de un gráfico de barras.

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Number & Operations in Base Ten
Los números y las operaciones de base diez

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Understand place value
I can understand place value
Entiendo el valor posicional
2.NBT.A.1
Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:
2.NBT.A.1
I can understand and use hundreds, tens and ones.

2.NBT.A.1
Entiendo y usar centenas, decenas y unidades.

2.NBT.A.1.a
100 can be thought of as a bundle of ten tens — called a “hundred.”
2.NBT.A.1.a
I can show that I understand that a bundle of ten "tens" is called a "hundred".

2.NBT.A.1.a
Entiendo que un grupo de diez decenas se llama una centena.

2.NBT.A.1.b
The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).
2.NBT.A.1.b
I can show that I understand the numbers I use when I count by hundreds, have a certain number of hundreds, 0 tens and 0 ones.

2.NBT.A.1.b
Entiendo que cuando cuento por centenas, hay un cierto número de centenas, 0 decenas y 0 unidades.

2.NBT.A.2
Count within 1000; skip-count by 5s, 10s, and 100s.
2.NBT.A.2
I can count to 1,000 by 1s, 5s, 10s and 100s.

2.NBT.A.2
Puedo contar hasta 1000 de 1 en 1, 5 en 5, 10 en 10 y de 100 en 100.

2.NBT.A.3
Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.
2.NBT.A.3
I can read and write numbers to 1,000 in different ways.

2.NBT.A.3
Puedo leer y escribir los números hasta 1000 de formas diferentes.

2.NBT.A.4
Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.
2.NBT.A.4
I can compare three-digit numbers using <, =, and > because I understand hundreds, tens and ones.

2.NBT.A.4
Puedo comparar números de tres dígitos utilizando <, = y > porque entiendo centenas, decenas y unidades.

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Use place value understanding and properties of operations to add and subtract
I can use what I know about place value to help me add and subtract
Puedo usar lo que sé sobre el valor posicional para ayudarme a sumar y restar
2.NBT.B.5
Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.
2.NBT.B.5
I can add two-digit numbers.

I can subtract two-digit numbers.

2.NBT.B.5
Puedo sumar números de dos dígitos.

Puedo restar números de dos dígitos.

2.NBT.B.6
Add up to four two-digit numbers using strategies based on place value and properties of operations.
2.NBT.B.6
I can add up to four 2-digit numbers.

2.NBT.B.6
Puedo sumar hasta cuatro números de 2 dígitos.

2.NBT.B.7
Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three- digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.
2.NBT.B.7
I can use strategies to add numbers within 1000 and know when to regroup.

I can use strategies to subtract numbers within 1000 and know when to borrow.

2.NBT.B.7
Puedo usar estrategias para sumar números hasta 1000 y sé cuándo hay que reagrupar.

Puedo usar estrategias para restar números de 100 o menos y sé cuándo hay que pedir prestado.

2.NBT.B.8
Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given number 100–900.
2.NBT.B.8
I can add and subtract 10 or 100 to any number from 100 to 900 in my head.

2.NBT.B.8
Puedo sumar 10 o 100 a (y restarlos de) un número de 1 - 900 en mi mente.

2.NBT.B.9
Explain why addition and subtraction strategies work, using place value and the properties of operations. (Explanations may be supported by drawings or objects.)
2.NBT.B.9
I can explain why adding and subtracting strategies work using what I know about place value.

2.NBT.B.9
Puedo explicar por qué estrategias de la suma y resta sirven al usar lo que sé sobre el valor posicional.

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Operations & Algebraic Thinking
Las operaciones matemáticas y el raciocinio algebraico

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Represent and solve problems involving addition and subtraction
I can write and solve problems using addition and subtraction
Puedo escribir y resolver problemas usando la suma y la resta
2.OA.A.1
Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. (See Glossary, Table 1.)
2.OA.A.1
I can use strategies to solve addition word problems. (within 100)

I can use strategies to solve subtraction word problems. (within 100)

2.OA.A.1
Puedo usar estrategias para resolver problemas de adición (hasta 100).

Puedo usar estrategias para resolver problemas de resta (de 100 o menos).

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Add and subtract within 20
I can add and subtract any numbers from 0 to 20 in my mind
Puedo sumar y restar cualquier número de 0 a 20 en mi mente
2.OA.B.2
Fluently add and subtract within 20 using mental strategies. (See standard 1.OA.C.6 for a list of mental strategies.) By end of Grade 2, know from memory all sums of two one-digit numbers.
2.OA.B.2
I know my addition facts.

I know my subtraction facts.

2.OA.B.2
Sé las operaciones de suma.

Sé las operaciones de resta.

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Work with equal groups of objects to gain foundations for multiplication
I can work with equal groups of objects to help me start to understand multiplication
Puedo trabajar con grupos iguales de objetos para ayudarme a comenzar a entender la multiplicación
2.OA.C.3
Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.
2.OA.C.3
I can group objects to tell if a number is odd or even.

I can write a number sentence to show how adding two of the same number will equal an even number.

2.OA.C.3
Puedo agrupar objetos para saber si un número es par o impar.

Puedo escribir una frase numérica para mostrar cómo la suma de dos del mismo número será igual a un número par.

2.OA.C.4
Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.
2.OA.C.4
I can use addition to help me figure out how many objects are in an array.

I can write a number sentence to show the total number of objects are in an array.

2.OA.C.4
Puedo usar la suma para ayudarme a calcular cuántos objetos hay en un conjunto.

Puedo escribir una frase numérica para mostrar el número total de objetos que hay en un conjunto.

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Grade 3

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Geometry
La geometría

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Reason with shapes and their attributes
I can understand shapes better by using what I notice about them
Entiendo mejor las formas al usar lo que sé y observo
3.G.A.1
Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.
3.G.A.1
I can place shapes into categories depending upon their attributes (parts).

I can name a category of many shapes by looking at their attributes (parts).

I can recognize and draw quadrilaterals (shapes with four sides) including rhombuses, rectangles and squares.

3.G.A.1
Puedo colocar formas en categorías dependiendo de sus atributos (partes).

Puedo identificar una categoría de muchas formas examinado sus atributos (partes).

Puedo reconocer y dibujar cuadriláteros (figuras con cuatro lados), incluyendo rombos, rectángulos y cuadrados.

3.G.A.2
Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole.

For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape.
3.G.A.2
I can divide shapes into parts with equal areas and show those areas as fractions.

3.G.A.2
Puedo dividir formas en partes con áreas iguales y mostrar esas áreas como fracciones.

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Measurement & Data
La medición y los datos

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Solve problems involving measurement and estimation
I can solve problems that involve measurement and estimation
Puedo resolver problemas que requieren la medición y los cálculos aproximados
3.MD.A.1
Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram.
3.MD.A.1
I can tell and write time to the nearest minute.

I can measure time in minutes.

I can solve telling time word problems by adding and subtracting minutes.

3.MD.A.1
Puedo decir y escribir la hora al minuto más próximo.

Puedo calcular el número de minutos.

Puedo resolver problemas verbales sobre la hora sumando y restando minutos.

3.MD.A.2
Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). (Excludes compound units such as cm³ and finding the geometric volume of a container.) Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem. (Excludes multiplicative comparison problems (problems involving notions of “times as much”; see Glossary, Table 2).)
3.MD.A.2
I can measure liquids and solids with grams (g), kilograms (kg) and liters (l).

I can use addition, subtraction, multiplication and division to solve word problems about mass or volume.

3.MD.A.2
Puedo medir sólidos por gramos (g) y kilogramos (kg) y líquidos por litros (l).

Puedo usar la suma, resta, multiplicación y división para resolver problemas sobre la masa o volumen.

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Represent and interpret data
I can understand how information is shared using numbers
Entiendo cómo se usan los números para expresar información
3.MD.B.3
Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step “how many more” and “how many less” problems using information presented in scaled bar graphs.

For example, draw a bar graph in which each square in the bar graph might represent 5 pets.
3.MD.B.3
I can make a picture or bar graph to show data and solve problems using the information from the graphs.

3.MD.B.3
Puedo hacer una imagen o gráfica de barras para mostrar datos y resolver problemas utilizando los datos de una gráfica.

3.MD.B.4
Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units—whole numbers, halves, or quarters.
3.MD.B.4
I can create a line plot from measurement data, where the measured objects have been measured to the nearest whole number, half or quarter.

3.MD.B.4
Puedo crear un diagrama de puntos utilizando datos de medición, donde los objetos se han medido con una precisión de número entero, la mitad o un cuarto.

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Geometric measurement: understand concepts of area and relate area to multiplication and to addition
I can understand area
Entiendo el concepto de area matemática (superficie)
3.MD.C.5
Recognize area as an attribute of plane figures and understand concepts of area measurement.
3.MD.C.5
I can understand that one way to measure plane shapes is by the area they have.

3.MD.C.5
Entiendo que una manera de medir figuras planas es calcular el área que tienen.

3.MD.C.5.a
A square with side length 1 unit, called “a unit square,” is said to have “one square unit” of area, and can be used to measure area.
3.MD.C.5.a
I can understand that a "unit square" is a square with side lengths of 1 unit and it is used to measure the area of plane shapes.

3.MD.C.5.a
Entiendo que un "cuadrado de unidad" es un cuadrado con lados midiendo 1 y se utiliza para medir el área de figuras planas.

3.MD.C.5.b
A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units.
3.MD.C.5.b
I can cover a plane shape with square units to measure its area.

3.MD.C.5.b
Puedo cubrir una forma plana con unidades cuadradas para medir su área.

3.MD.C.6
Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units).
3.MD.C.6
I can measure areas by counting unit squares (square cm, square m, square in, square ft).

3.MD.C.6
Puedo medir áreas contando cuadrados de la unidad (cm cuadrados, cuadrado m, plaza de, pies cuadrados).

3.MD.C.7
Relate area to the operations of multiplication and addition.
3.MD.C.7
I can understand area by thinking about multiplication and addition.

3.MD.C.7
Entiendo zona pensando en la multiplicación y la adición.

3.MD.C.7.a
Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths.
3.MD.C.7.a
I can find the area of a rectangle using square tiles and also by multiplying the two side lengths.

3.MD.C.7.a
Puedo encontrar el área de un rectángulo con baldosas cuadradas y también multiplicando las longitudes de dos lados.

3.MD.C.7.b
Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning.
3.MD.C.7.b
I can solve real world problems about area using multiplication.

3.MD.C.7.b
Puedo resolver problemas del mundo real sobre el área de uso de la multiplicación.

3.MD.C.7.c
Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning.
3.MD.C.7.c
I can use models to show that the area of a rectangle can be found by using the distributive property (side lengths a and b+c is the sum of a x b and a x c).

3.MD.C.7.c
Puedo usar modelos para mostrar que el área de un rectángulo se puede encontrar utilizando la propiedad distributiva (lado longitudes a y b + c es la suma de axb y axc).

3.MD.C.7.d
Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems.
3.MD.C.7.d
I can find the area of a shape by breaking it down into smaller shapes and then adding those areas to find the total area.

3.MD.C.7.d
Puedo encontrar el área de una figura por lo descomponen en formas más pequeñas y luego añadiendo esas áreas para hallar el área total.

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Geometric measurement: recognize perimeter
I can understand perimeter
Entiendo qué significa un perímetro
3.MD.D.8
Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.
3.MD.D.8
I can solve real world math problems using what I know about how to find the perimeter of shapes.

3.MD.D.8
Puedo resolver problemas de matemáticas del mundo real usando lo que sé sobre cómo encontrar el perímetro de figuras.

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Number & Operations in Base Ten
Los números y las operaciones de base diez

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Use place value understanding and properties of operations to perform multi-digit arithmetic
I can use what I know about place value and operations (+,-,x,÷) to solve problems with larger numbers
Puedo usar lo que sé sobre el valor posicional y las operaciones (+,-,x,÷) para resolver problemas con números mayores
3.NBT.A.1
Use place value understanding to round whole numbers to the nearest 10 or 100.
3.NBT.A.1
I can use place value to help me round numbers to the nearest 10 or 100.

3.NBT.A.1
Puedo usar el valor posicional para ayudarme a redondar números a la próxima decena o centena (o más cercana).

3.NBT.A.2
Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.
3.NBT.A.2
I can quickly and easily add and subtract numbers within 1000.

3.NBT.A.2
Puedo sumar y restar números al 1000 rápidamente y facilmente.

3.NBT.A.3
Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations.
3.NBT.A.3
I can multiply any one digit whole number by a multiple of 10 (6 x 90, 4 x 30).

3.NBT.A.3
Puedo multiplicar cualquier número entero de un dígito por un múltiplo de 10 (6 x 90, 4 x 30).

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Number & Operations - Fractions

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Develop understanding of fractions as numbers
I can understand fractions
Entiendo las fracciones
3.NF.A.1
Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
3.NF.A.1
I can show and understand that fractions represent equal parts of a whole, where the top number is the part and the bottom number is the total number of parts in the whole.

3.NF.A.1
Puedo demostrar y comprender que las fracciones representan partes iguales de un todo, donde el número de arriba es la parte y el número inferior es el número total de partes en el todo.

3.NF.A.2
Understand a fraction as a number on the number line; represent fractions on a number line diagram.
3.NF.A.2
I can understand a fraction as a number on the number line by showing fractions on a number line diagram.

3.NF.A.2
Entiendo una fracción como un número en la recta numérica al mostrar fracciones en una recta numérica.

3.NF.A.2.a
Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.
3.NF.A.2.a
I can label fractions on a number line because I know the space between any two numbers on the number line can be thought of as a whole.

3.NF.A.2.a
Puedo etiquetar fracciones en una recta numérica porque sé que el espacio entre dos números en la recta numérica puede ser pensado como un todo.

3.NF.A.2.b
Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.
3.NF.A.2.b
I can show a fraction on a number line by marking off equal parts between two whole numbers.

3.NF.A.2.b
Puedo mostrar una fracción en una recta numérica marcando partes iguales entre dos números enteros.

3.NF.A.3
Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
3.NF.A.3
I can understand how some different fractions can actually be equal.

I can compare fractions by reasoning about their size.

3.NF.A.3
Entiendo cómo algunas fracciones diferentes en realidad pueden ser iguales.

Puedo comparar fracciones por medio de pensar en su tamaño.

3.NF.A.3.a
Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.
3.NF.A.3.a
I can understand two fractions as equivalent (equal) if they are the same size or at the same point on a number line.

3.NF.A.3.a
Entiendo dos fracciones como equivalentes (igual) si son del mismo tamaño o se colocarían en el mismo punto de una recta numérica.

3.NF.A.3.b
Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction model.
3.NF.A.3.b
I can recognize and write simple equivalent (equal) fractions and explain why they are equal using words or models.

3.NF.A.3.b
Puedo reconocer y escribir fracciones equivalentes simples (iguales) y explicar por qué son iguales usando palabras o modelos.

3.NF.A.3.c
Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.

Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.
3.NF.A.3.c
I can show whole numbers as fractions. (3 = 3/1)

I can recognize fractions that are equal to one whole. (1 = 4/4)

3.NF.A.3.c
Puedo mostrar los números enteros como fracciones. (3 = 3/1)

Puedo reconocer fracciones que son iguales a un todo. (1 = 4/4)

3.NF.A.3.d
Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
3.NF.A.3.d
I can compare two fractions with the same numerator (top number) or the same denominator (bottom number) by reasoning about their size.

I can understand that comparing two fractions is only reasonable if they refer to the same whole.

I can compare fractions with the symbols >, =, < and prove my comparison by using models.

3.NF.A.3.d
Puedo comparar dos fracciones con el mismo numerador (número superior) o el mismo denominador (número inferior) al utilizar su tamaño.

Entiendo que la comparación de dos fracciones es razonable si se refieren a un mismo todo.

Puedo comparar fracciones con los símbolos >, =, < y demostrar mi comparación mediante el uso de modelos.

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Operations & Algebraic Thinking
Las operaciones matemáticas y el raciocinio algebraico

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Represent and solve problems involving multiplication and division
I can write and solve problems using multiplication and division
Puedo escribir y resolver problemas por medio de multiplicar y dividir
3.OA.A.1
Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each.

For example, describe a context in which a total number of objects can be expressed as 5 × 7.
3.OA.A.1
I can understand multiplication by thinking about groups of objects.

3.OA.A.1
Entiendo la multiplicación al pensar en grupos de objetos.

3.OA.A.2
Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each.

For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.
3.OA.A.2
I can understand division by thinking about how one group can be divided into smaller groups.

3.OA.A.2
Entiendo la división al pensar en cómo un grupo se puede dividir en grupos más pequeños.

3.OA.A.3
Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. (See Glossary, Table 2.)
3.OA.A.3
I can use what I know about multiplication and division to solve word problems.

3.OA.A.3
Puedo usar lo que sé acerca de la multiplicación y la división para resolver problemas verbales.

3.OA.A.4
Determine the unknown whole number in a multiplication or division equation relating three whole numbers.

For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = _ ÷ 3, 6 × 6 = ?.
3.OA.A.4
I can find the missing number in a multiplication or division equation.

3.OA.A.4
Puedo encontrar el número que falta en una ecuación de multiplicación o división.

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Understand properties of multiplication and the relationship between multiplication and division
I can understand multiplication and how it is related to division
Entiendo lo que es la multiplicación y cómo se relaciona a la división
3.OA.B.5
Apply properties of operations as strategies to multiply and divide. (Students need not use formal terms for these properties.)

Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)
3.OA.B.5
I can use the Commutative property of multiplication. (I know that if 6 x 4 = 24, then 4 x 6 = 24.)

I can use the Associative property of multiplication. (To figure out 3 x 5 x 2, I can multiply 3 x 5 = 15, then 15 x 2 = 30 OR multiply 5 x 2 = 10, then 3 x 10 = 30.)

I can use the Distributive property of multiplication. (To figure out 8 x 7, I can think of 8 x (5 + 2) which means (8 x 5) + (8 x 2) = 40 + 16 = 56.)

3.OA.B.5
Puedo usar la propiedad conmutativa de la multiplicación (Yo sé que 6 x 4 = 24, por eso 4 x 6 = 24).

Puedo usar la propiedad asociativa de la multiplicación (Para resolver 3 x 5 x 2, puedo multiplicar 3 x 5 = 15, entonces 15 x 2 = 30 O puedo multiplicar 5 x 2 = 10, entonces 3 x 10 = 30).

Puedo usar la propiedad distributiva de la multiplicación (Para resolver 8 x 7, que se me ocurre de 8 x (5 + 2) que significa (8 x 5) + (8 x 2), entonces 40 + 16 = 56).

3.OA.B.6
Understand division as an unknown-factor problem.

For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.
3.OA.B.6
I can find the answer to a division problem by thinking of the missing factor in a multiplication problem. (I can figure out 32 ÷ 8 because I know that 8 x 4 = 32.)

3.OA.B.6
Puedo encontrar la respuesta a un problema de división pensando en el factor faltante en un problema de multiplicación (Puedo imaginar 32 ÷ 8 porque sé que 8 x 4 = 32).

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Multiply and divide within 100
I can multiply and divide numbers within 100
Puedo multiplicar y dividir números al 100
3.OA.C.7
Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.
3.OA.C.7
I can multiply and divide within 100 easily and quickly because I know how multiplication and division are related.

3.OA.C.7
Puedo multiplicar y dividir dentro de 100 con facilidad y rapidez, porque sé cómo la multiplicación y la división están relacionadas.

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Solve problems involving the four operations, and identify and explain patterns in arithmetic
I can use number patterns to help me solve problems
Puedo usar patrones numéricos para ayudarme a resolver problemas
3.OA.D.8
Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. (This standard is limited to problems posed with whole numbers and having whole-number answers; students should know how to perform operations in the conventional order when there are no parentheses to specify a particular order (Order of Operations).)
3.OA.D.8
I can solve two-step word problems that involve addition, subtraction, multiplication and division.

I can solve two-step word problems by writing an equation with a letter in place of the number I don't know.

I can use mental math to figure out if the answers to two-step word problems are reasonable.

3.OA.D.8
Puedo resolver problemas verbales de dos pasos que requieren la suma, resta, multiplicación y/o división.

Puedo resolver problemas verbales de dos pasos al escribir una ecuación con una letra en lugar del número de no sé.

Puedo usar el cálculo mental para determinar si las respuestas a los problemas de dos pasos son razonables.

3.OA.D.9
Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations.

For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.
3.OA.D.9
I can find patterns in addition and multiplication tables and explain them using what I know about how numbers work.

3.OA.D.9
Puedo encontrar patrones en las tablas de suma y de multiplicación y explicarlos utilizando lo que sé sobre cómo funcionan los números.

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Grade 4

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Geometry
La geometría

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Draw and identify lines and angles, and classify shapes by properties of their lines and angles
I can use geometry to help me understand math
Puedo utilizar la geometría para ayudarme a entender las matemáticas
4.G.A.1
Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.
4.G.A.1
I can identify and draw points, lines, line segments, rays, angles and perpendicular & parallel lines.

4.G.A.1
Puedo identificar y dibujar puntos, líneas, segmentos de líneas, rayos, ángulos y líneas perpendiculares y paralelas.

4.G.A.2
Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.
4.G.A.2
I can classify two-dimensional shapes based on what I know about their geometrical attributes.

I can recognize and identify right triangles.

4.G.A.2
Puedo clasificar formas de dos dimensiones usando lo que sé acerca de sus atributos geométricos.

Puedo reconocer e identificar triángulos rectos.

4.G.A.3
Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.
4.G.A.3
I can recognize, identify and draw lines of symmetry.

4.G.A.3
Puedo reconocer, identificar y dibujar lineas de simetría.

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Measurement & Data
La medición y los datos

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Solve problems involving measurement and conversion of measurements
I can solve problems involving measurement and conversion of measurements
Puedo resolver problemas de medición y la conversión de mediciones
4.MD.A.1
Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two- column table.

For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36), ...
4.MD.A.1
I can show that I know the relative size of measurement units within one system of units (including km, m, cm; kg, g; lb, oz; l, ml; hr, min, sec).

I can show the measurements in a larger unit in terms of smaller units and record these in a table.

4.MD.A.1
Puedo demostrar que sé que el tamaño relativo de las unidades de medida dentro de un sistema de unidades (incluyendo km, m, cm; kg, g; lb, oz; l, ml; hr, min, seg).

Puedo mostrar las medidas de una unidad mayor en términos de unidades menores, y anotarlas en una tabla.

4.MD.A.2
Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.
4.MD.A.2
I can use the four operations (+, -, x, ÷) to solve word problems involving measurement.

I can solve measurement problems involving simple fractions and decimals.

I can solve problems that ask me to express measurements given in a larger unit in terms of a smaller unit.

I can show measurement quantities using diagrams that involve a measurement scale (e.g., a number line).

4.MD.A.2
Puedo usar las cuatro operaciones (+, -, x, ÷) para resolver problemas que involucran la medición.

Puedo resolver problemas de medición que involucran fracciones y decimales simples.

Puedo resolver problemas que me piden expresar las mediciones dadas en una unidad mayor en términos de una unidad menor.

Puedo mostrar cantidades de medición utilizando diagramas que involucran una escala de medición (ej: una recta numérica).

4.MD.A.3
Apply the area and perimeter formulas for rectangles in real world and mathematical problems.

For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.
4.MD.A.3
I can use what I know about area and perimeter to solve real world problems involving rectangles.

4.MD.A.3
Puedo usar lo que sé sobre el área y perímetro para resolver problemas del mundo real que involucran rectángulos.

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Represent and interpret data
I can represent and interpret data
Puedo expresar e interpretar datos
4.MD.B.4
Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition and subtraction of fractions by using information presented in line plots.

For example, from a line plot find and interpret the difference in length between the longest and shortest specimens in an insect collection.
4.MD.B.4
I can make a line plot to show a data set of measurements involving fractions.

I can solve problems involving addition and subtraction of fractions by using information shown in line plots.

4.MD.B.4
Puedo hacer una gráfica de línea para mostrar un conjunto de datos de mediciones que involucran fracciones.

Puedo resolver problemas de suma y resta de fracciones mediante el uso de la información que se muestra en gráficos de líneas.

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Geometric measurement: understand concepts of angle and measure angles
I can understand the concept of measurement in geometry with regards to angles
Puedo entender el concepto de medición con respecto a los ángulos geométricos
4.MD.C.5
Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:
4.MD.C.5
I can recognize angles as geometric shapes where two rays share a common endpoint.

I can understand concepts of angle measurement.

4.MD.C.5
Puedo reconocer ángulos como formas geométricas donde dos rayos comparten un punto final común.

Entiendo los conceptos de la medición de ángulos.

4.MD.C.5.a
An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a “one-degree angle,” and can be used to measure angles.
4.MD.C.5.a
I can understand that angles are measured with reference to a 360°circle, with its center at the common endpoint of the rays.

4.MD.C.5.a
Entiendo que los ángulos se miden con referencia a un círculo de 360°, con su centro en el punto final común de los rayos.

4.MD.C.5.b
An angle that turns through n one-degree angles is said to have an angle measure of n degrees.
4.MD.C.5.b
I can understand that an angle that turns through n one-degree angles is said to have an angle measurement of n degrees.

4.MD.C.5.b
Entiendo que un ángulo que gira por n ángulos de un grado se dice que tiene una medición de ángulo de n grados.

4.MD.C.6
Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.
4.MD.C.6
I can use a protractor to measure and sketch angles in whole-number degrees.

4.MD.C.6
Puedo utilizar un transportador para medir y dibujar ángulos en grados de números enteros.

4.MD.C.7
Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.
4.MD.C.7
I can solve real-world and mathematical addition and subtraction problems to find unknown angles.

4.MD.C.7
Puedo resolver problemas del mundo real y de suma y resta matemáticos para encontrar ángulos desconocidos.

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Number & Operations in Base Ten
Los números y las operaciones de base diez

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Generalize place value understanding for multi-digit whole numbers
I can use place value to help me understand larger numbers
Puedo usar el valor posicional para ayudarma a entender los números mayores
4.NBT.A.1
Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right.

For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division.
4.NBT.A.1
I can recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right.

4.NBT.A.1
Puedo reconocer que en un número entero de varios dígitos, un dígito en un solo lugar representa diez veces más de lo que representa en el lugar a su derecha.

4.NBT.A.2
Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.
4.NBT.A.2
I can read and write larger whole numbers using numerals, words and in expanded form.

I can compare two larger numbers by using what I know about the values in each place. symbols to show the comparison.

I can compare two larger numbers and use the symbols >, = and < to show the comparison.

4.NBT.A.2
Puedo leer y escribir números enteros mayores usando números, palabras y forma expandida.

Puedo comparar dos números mayores usando lo que sé de los valores de cada lugar y símbolos para mostrar la comparación.

Puedo comparar dos números mayores y usar los símbolos >, = y < para mostrar la comparación.

4.NBT.A.3
Use place value understanding to round multi-digit whole numbers to any place.
4.NBT.A.3
I can round larger whole numbers to any place.

4.NBT.A.3
Puedo redondear números enteros mayores a cualquier valor posicional.

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Use place value understanding and properties of operations to perform multi-digit arithmetic
I can use what I know about place value and operations (+,-,x,÷) to solve problems with larger numbers
Puedo usar lo que sé sobre el valor posicional y operaciones (+, -, x, ÷) para resolver problemas con números mayores
4.NBT.B.4
Fluently add and subtract multi-digit whole numbers using the standard algorithm.
4.NBT.B.4
I can add and subtract larger numbers.

4.NBT.B.4
Puedo sumar y restar números mayores.

4.NBT.B.5
Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
4.NBT.B.5
I can multiply a whole number up to four digits by a one-digit whole number.

I can multiply two two-digit numbers.

I can illustrate and explain how to multiply larger numbers by using equations, arrays or models.

4.NBT.B.5
Puedo multiplicar un número entero de hasta cuatro dígitos por un número entero de un dígito.

Puedo multiplicar dos números de dos dígitos.

Puedo ilustrar y explicar cómo multiplicar números mayores mediante el uso de ecuaciones, matrices o modelos.

4.NBT.B.6
Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
4.NBT.B.6
I can find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors.

I can illustrate and explain how to divide larger numbers by using equations, arrays or models.

4.NBT.B.6
Puedo encontrar cocientes de números enteros y restos de hasta dividendos de cuatro dígitos y divisores de un dígito.

Puedo ilustrar y explicar cómo dividir números mayores mediante el uso de ecuaciones, matrices o modelos.

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Number & Operations - Fractions
Números y operaciones - Las fracciones

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Extend understanding of fraction equivalence and ordering
I can improve my understanding of fractions
Puedo mejorar mi entendimiento de las fracciones
4.NF.A.1
Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.
4.NF.A.1
I can explain (and show models for) why multiplying a numerator and a denominator by the same number does not change the value of a fraction.

I can recognize and generate equivalent fractions based on my knowledge of numerators and denominators.

4.NF.A.1
Puedo explicar (y mostrar modelos) por qué multiplicar un numerador y un denominador por el mismo número no cambia el valor de una fracción.

Puedo reconocer y generar fracciones equivalentes basadas en mi conocimiento de numeradores y denominadores.

4.NF.A.2
Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
4.NF.A.2
I can compare two fractions with different numerators and different denominators by creating common denominators or numerators or by comparing them to a benchmark fraction like one-half.

I can recognize that comparisons of fractions are valid only when the two fractions refer to the same whole.

I can compare fractions using the symbols >, = and <, and justify the comparison by using models.

4.NF.A.2
Puedo comparar dos fracciones con numeradores y denominadores distintos mediante la creación de denominadores o numeradores comunes o comparándolas con una fracción de referencia como un medio.

Puedo reconocer que las comparaciones de fracciones son válidas solamente cuando las dos fracciones se refieren al mismo número entero.

Puedo comparar fracciones utilizando los símbolos >, = y <, y justificar la comparación mediante el uso de modelos.

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Build fractions from unit fractions
I can build fractions from unit fractions
Puedo mejorar mi entendimiento de las fracciones
4.NF.B.3
Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
4.NF.B.3
I can understand a fraction a/b, with a > 1, as a sum of fractions 1/b.

4.NF.B.3
Entiendo una fracción a / b, con a > 1, como una suma de fracciones 1 / b.

4.NF.B.3.a
Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.
4.NF.B.3.a
I can understand addition and subtraction of fractions as joining and separating parts referring to the same whole.

4.NF.B.3.a
Entiendo la suma y la resta de fracciones como la unión y separación de partes que se refieren al mismo número entero.

4.NF.B.3.b
Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model.

Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.
4.NF.B.3.b
I can decompose a fraction into a sum of fractions with the same denominator in more than one way and justify my work using models.

4.NF.B.3.b
Puedo descomponer una fracción en una suma de fracciones con el mismo denominador en más de una forma y justificar mi trabajo utilizando modelos.

4.NF.B.3.c
Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.
4.NF.B.3.c
I can add and subtract mixed numbers with like denominators.

4.NF.B.3.c
Puedo sumar y restar números mixtos con denominadores comunes.

4.NF.B.3.d
Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.
4.NF.B.3.d
I can solve word problems involving addition and subtraction of fractions that refer to the same whole and that have like denominators.

4.NF.B.3.d
Puedo resolver problemas verbales de suma y resta de fracciones que se refieren al mismo número entero y que tienen denominadores comunes.

4.NF.B.4
Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
4.NF.B.4
I can apply my understanding of multiplication to multiply a fraction by a whole number.

4.NF.B.4
Puedo aplicar mi entendimiento de la multiplicación para multiplicar una fracción por un número entero.

4.NF.B.4.a
Understand a fraction a/b as a multiple of 1/b.

For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).
4.NF.B.4.a
I can understand a fraction a/b as a multiple of 1/b (e.g., I know that 5/4 is the product of 5 x (1/4).)

4.NF.B.4.a
Entiendo una fracción a/b como un múltiplo de 1/b (ej: sé que 5/4 es el producto de 5 por 1/4 ).

4.NF.B.4.b
Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number.

For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)
4.NF.B.4.b
I can understand a multiple of a/b as a multiple of 1/b and use that knowledge to multiply a fraction by a whole number (e.g., n x (a/b) = (n x a)/b).

4.NF.B.4.b
Entiendo un múltiplo de a/b como un múltiplo de 1/b, y uso ese conocimiento para multiplicar una fracción por un número entero (ej: nx (a/b) = (n x a) /b).

4.NF.B.4.c
Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem.

For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?
4.NF.B.4.c
I can solve word problems involving multiplication of a fraction by a whole number.

4.NF.B.4.c
Puedo resolver problemas verbales que involucren la multiplicación de una fracción por un número entero.

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Understand decimal notation for fractions, and compare decimal fractions
I can understand how fractions and decimals are related
Entender cómo se relacionan las fracciones y los decimales
4.NF.C.5
Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100. (Students who can generate equivalent fractions can develop strategies for adding fractions with unlike denominators in general. But addition and subtraction with unlike denominators in general is not a requirement at this grade.)

For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.
4.NF.C.5
I can show a fraction with a denominator of 10 as an equivalent fraction with a denominator of 100 in order to add the two fractions.

4.NF.C.5
Puedo mostrar una fracción con un denominador de 10 como una fracción equivalente con un denominador de 100 con el fin de sumar las dos fracciones.

4.NF.C.6
Use decimal notation for fractions with denominators 10 or 100.

For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.
4.NF.C.6
I can use decimals to show fractions with denominators of 10 and 100.

4.NF.C.6
Puedo usar decimales para mostrar fracciones con denominadores de 10 y 100.

4.NF.C.7
Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model.
4.NF.C.7
I can compare two decimals to hundredths by reasoning about their size and realizing that the comparison is only true if the two decimals refer to the same whole.

I can compare decimals using the symbols >, = and <, and justify the comparison by using models.

4.NF.C.7
Puedo comparar dos decimales hasta las centésimas razonando sobre su tamaño y sabiendo que la comparación es cierto solamente si los dos decimales se refieren al mismo número entero..

Puedo comparar decimales usando los símbolos >, = y <, y justificar la comparación mediante el uso de modelos.

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Operations & Algebraic Thinking
Las operaciones matemáticas y el raciocinio algebraico

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Use the four operations with whole numbers to solve problems
I can use the four operations (+, -, x, ÷) to help me solve problems
Puedo usar las cuatro operaciones (+,-,x,÷) para ayudarme a resolver problemas
4.OA.A.1
Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.
4.OA.A.1
I can understand that multiplication equations can be seen as comparisons of groups (e.g., 24 = 4 x 6 can be thought of as 4 groups of 6 or 6 groups of 4).

4.OA.A.1
Entiendo que las ecuaciones de multiplicación pueden ser vistos como comparaciones de grupos (ej: 24 = 4 x 6 puede ser pensado como 4 grupos de 6 o 6 grupos de 4).

4.OA.A.2
Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison. (See Glossary, Table 2.)
4.OA.A.2
I can multiply or divide to solve word problems by using drawings or writing equations and solving for a missing number.

4.OA.A.2
Puedo multiplicar o dividir para resolver problemas verbales mediante el uso de dibujos o escribir ecuaciones y encontrar el número que falta.

4.OA.A.3
Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
4.OA.A.3
I can use what I know about addition, subtraction, multiplication and division to solve multi-step word problems involving whole numbers.

I can represent word problems by using equations with a letter standing for the unknown number.

I can determine how reasonable my answers to word problems are by using estimation, mental math and rounding.

4.OA.A.3
Puedo usar lo que sé acerca de la suma, resta, multiplicación y división para resolver problemas verbales de múltiples pasos que se usan números enteros.

Puedo representar problemas verbales mediante el uso de ecuaciones con una variable (letra) para representar el número desconocido.

Puedo determinar mis respuestas a problemas verbales son razonables por el uso de la estimación, el cálculo mental y el redondeo.

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Gain familiarity with factors and multiples
I can become familiar with factors and multiples
Puedo familiarizarme con los factores y múltiplos
4.OA.B.4
Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1–100 is prime or composite.
4.OA.B.4
I can find all factor pairs for a whole number from 1 to 100.

I can recognize a whole number as a multiple of each of its factors.

I can determine whether a whole number from 1 to 100 is a multiple of a given one-digit number.

I can determine whether a given whole number up to 100 is a prime or composite number.

4.OA.B.4
Puedo encontrar todos los pares de factores de un número entero de 1 a 100.

Puedo reconocer un número entero como un múltiplo de cada uno de sus factores.

Puedo determinar si un número entero de 1 a 100 es un múltiplo de un número de una cifra determinada.

Puedo determinar si un número entero hasta el 100 es un número primo o compuesto.

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Generate and analyze patterns
I can create and analyze patterns
Puedo crear y analizar los patrones
4.OA.C.5
Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself.

For example, given the rule “Add 3” and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.
4.OA.C.5
I can create a number or shape pattern that follows a given rule.

I can notice and point out different features of a pattern once it is created by a rule.

4.OA.C.5
Puedo crear patrón numérico o de forma geométrica que sigue una regla dada.

Puedo notar y destacar diferentes características de un patrón, una vez que se crea una regla.

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Grade 5

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Geometry

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Graph points on the coordinate plane to solve real-world and mathematical problems
I can graph points on the coordinate plane to solve real-world and mathematical problems
5.G.A.1
Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate).
5.G.A.1
I can graph ordered pairs of numbers on a coordinate plane using what I have learned about the x-axis and coordinate and the y-axis and coordinate.

I can understand a coordinate plane and ordered pairs of number coordinates on that plane.

5.G.A.2
Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.
5.G.A.2
I can represent real-world and mathematical problems by graphing points in the first quadrant of a coordinate plane.

I can understand coordinate values in the context of a real-world or mathematical problem.

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Classify two-dimensional figures into categories based on their properties
I can classify 2-dimensional shapes into categories based on their properties
5.G.B.3
Understand that attributes belonging to a category of two- dimensional figures also belong to all subcategories of that category.

For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.
5.G.B.3
I can understand how attributes of 2-dimensional shapes in a category also belong to all subcategories of those shapes.

5.G.B.4
Classify two-dimensional figures in a hierarchy based on properties.
5.G.B.4
I can classify 2-dimensional shapes based on their properties.

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Measurement & Data

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Convert like measurement units within a given measurement system
I can convert like measurement units within a given measurement system
5.MD.A.1
Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.
5.MD.A.1
I can convert different-sized measurements within the same measurement system.

I can use measurement conversions to solve real-world problems.

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Represent and interpret data
I can represent and interpret data
5.MD.B.2
Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots.

For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally.
5.MD.B.2
I can make a line plot to show a data set of measurements involving fractions.

I can use addition, subtraction, multiplication and division of fractions to solve problems involving information presented on a line plot.

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Geometric measurement: understand concepts of volume
I can understand the concept of measurement in geometry with regards to volume
5.MD.C.3
Recognize volume as an attribute of solid figures and understand concepts of volume measurement.
5.MD.C.3
I can recognize volume as a characteristic of solid figures and understand how it can be measured.

5.MD.C.3.a
A cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume, and can be used to measure volume.
5.MD.C.3.a
I can understand a "unit cube" as a cube with side lengths of 1 unit and can use it to measure volume.

5.MD.C.3.b
A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.
5.MD.C.3.b
I can understand that a solid figure filled with a number of unit cubes is said to have a volume of that many cubes.

5.MD.C.4
Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.
5.MD.C.4
I can measure volume by counting unit cubes.

5.MD.C.5
Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.
5.MD.C.5
I can solve real world problems involving volume by thinking about multiplication of addition.

5.MD.C.5.a
Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication.
5.MD.C.5.a
I can use unit cubes to find the volume of a right rectangular prism with whole number side lengths and prove that it is the same as multiplying the edge lengths (V = l x w x h).

5.MD.C.5.b
Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems.
5.MD.C.5.b
I can solve real-world and mathematical problems involving volume of an object using the formulas V = l x w x h and V = b x h.

I can find the volumes of solid figures made up of two right rectangular prisms by adding the volumes of both.

5.MD.C.5.c
Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.
5.MD.C.5.c
I can solve real-world problems using what I know about adding the volumes of two right rectangular prisms.

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Number & Operations in Base Ten

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Understand the place value system
I can understand the place value system
5.NBT.A.1
Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.
5.NBT.A.1
I can understand and explain the value of digits in a larger number.

5.NBT.A.2
Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
5.NBT.A.2
I can explain patterns of zeros in an answer when multiplying a number by powers of 10.

I can explain patterns pf decimal placement when a decimal is multiplied or divided by a power of 10.

I can use whole-number exponents to show powers of 10.

5.NBT.A.3
Read, write, and compare decimals to thousandths.
5.NBT.A.3
I can read, write, and compare decimals to thousandths.

5.NBT.A.3.a
Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000).
5.NBT.A.3.a
I can read and write decimals to thousandths using base-ten numbers, number names and expanded form.

5.NBT.A.3.b
Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.
5.NBT.A.3.b
I can compare two decimals to thousandths using the >, =, and < symbols correctly.

5.NBT.A.4
Use place value understanding to round decimals to any place.
5.NBT.A.4
I can use place value understanding to round decimals to any place.

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Perform operations with multi-digit whole numbers and with decimals to hundredths
I can solve math equations with larger whole numbers and decimals to the hundredths
5.NBT.B.5
Fluently multiply multi-digit whole numbers using the standard algorithm.
5.NBT.B.5
I can easily multiply larger whole numbers.

5.NBT.B.6
Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
5.NBT.B.6
I can divide four-digit numbers (dividends) by two-digit numbers (divisors).

I can illustrate and explain a division problem using equations, arrays and/or models.

5.NBT.B.7
Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
5.NBT.B.7
I can add, subtract, multiply, and divide decimals to hundredths using what I have learned about place value.

I can relate the strategies I use to add, subtract, multiply and divide decimals to hundredths to a written problem and explain why I chose the strategies to help me solve the problem.

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Number & Operations - Fractions

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Use equivalent fractions as a strategy to add and subtract fractions
I can use equivalent (equal) fractions as a strategy to add and subtract fractions
5.NF.A.1
Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators.

For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)
5.NF.A.1
I can add and subtract fractions with unlike denominators.

5.NF.A.2
Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers.

For example, recognize as incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.
5.NF.A.2
I can solve word problems that involve addition and subtraction of fractions.

I can use number sense and fractions that I know to estimate the reasonableness of answers to fraction problems.

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Apply and extend previous understandings of multiplication and division
I can use and increase my understanding of multiplication and division
5.NF.B.3
Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem.

For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?
5.NF.B.3
I can understand that fractions are really division problems.

I can solve word problems where I need to divide whole numbers leading to answers that are fractions or mixed numbers.

5.NF.B.4
Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
5.NF.B.4
I can use what I know about multiplication to multiply fractions or whole numbers by a fraction.

5.NF.B.4.a
Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b.

For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)
5.NF.B.4.a
I can understand and show with models that multiplying a fraction by a whole number is the same as finding the product of the numerator and whole number and then dividing it by the denominator.

5.NF.B.4.b
Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.
5.NF.B.4.b
I can use unit squares to find the area of a rectangle with fractional side lengths and prove that it is the same as multiplying the side lengths (A = l x w).

5.NF.B.5
Interpret multiplication as scaling (resizing), by:
5.NF.B.5
I can think of multiplication as the scaling of a number (similar to a scale on a map.)

5.NF.B.5.a
Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.
5.NF.B.5.a
I can mentally compare the size of a product to the size of one of the factors by thinking about the other factor in the problem.

5.NF.B.5.b
Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n × a)/(n × b) to the effect of multiplying a/b by 1.
5.NF.B.5.b
I can explain why multiplying a number by a fraction greater than 1 will result in a bigger number than the number I started with.

I can explain why multiplying a number by a fraction less than 1 will result in a smaller number than the number I started with.

I can relate the notion of equivalent fractions to the effect of multiplying a fraction by 1.

5.NF.B.6
Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
5.NF.B.6
I can solve real world problems that involve multiplication of fractions and mixed numbers.

5.NF.B.7
Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. (Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the relationship between multiplication and division. But division by a fraction is not a requirement at this grade.)
5.NF.B.7
I can use what I know about division to divide fractions by whole numbers or whole numbers by fractions.

5.NF.B.7.a
Interpret division of a unit fraction by a non-zero whole number, and compute such quotients.

For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.
5.NF.B.7.a
I can divide a fraction by a whole number (not 0) correctly.

5.NF.B.7.b
Interpret division of a whole number by a unit fraction, and compute such quotients.

For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.
5.NF.B.7.b
I can divide a whole number by a fraction correctly.

5.NF.B.7.c
Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem.

For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?
5.NF.B.7.c
I can use what I know about division problems involving fractions to solve real world problems.

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Operations & Algebraic Thinking

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Write and interpret numerical expressions
I can understand and write number sentences with one or more numbers and operations
5.OA.A.1
Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
5.OA.A.1
I can write and figure out number sentences that have parentheses, brackets and/or braces.

5.OA.A.2
Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them.

For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.
5.OA.A.2
I can correctly write number sentences using mathematic symbols and the order of operations correctly.

I can understand number sentences and estimate their answers without actually calculating them.

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Analyze patterns and relationships
I can study number patterns and figure out their relationships
5.OA.B.3
Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane.

For example, given the rule “Add 3” and the starting number 0, and given the rule “Add 6” and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.
5.OA.B.3
I can create two number patterns using two given rules.

I can identify relationships between two number patterns.

I can form ordered pairs using the relationship between two number patterns and graph them on a coordinate plane.

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Grade 6

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Expressions & Equations

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Apply and extend previous understandings of arithmetic to algebraic expressions
I can apply my understanding of arithmetic to algebraic expressions (number sentences that contain unknowns)
6.EE.A.1
Write and evaluate numerical expressions involving whole-number exponents.
6.EE.A.1
I can write and figure out numerical expressions that have whole-number exponents.

6.EE.A.2
Write, read, and evaluate expressions in which letters stand for numbers.
6.EE.A.2
I can write, read and figure out expressions in which letters stand for numbers.

6.EE.A.2.a
Write expressions that record operations with numbers and with letters standing for numbers.

For example, express the calculation “Subtract y from 5” as 5 – y.
6.EE.A.2.a
I can write expressions with numbers and with letters standing for numbers.

6.EE.A.2.b
Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity.

For example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.
6.EE.A.2.b
I can name the parts of an expression using mathematical words (sum, term, product, factor, quotient, coefficient.)

I can look at one or more parts of an expression in different ways. (Ex: 8 + 7 can be seen as the addition sentence or as the number 15.)

6.EE.A.2.c
Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole- number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations).

For example, use the formulas V = s³ and A = 6 s² to find the volume and surface area of a cube with sides of length s = 1/2.
6.EE.A.2.c
I can figure out different answers to expressions when given specific values for the variable.

I can solve real-world math problems involving expressions that arise from formulas.

I can solve math problems including those with exponents, in the usual order (when no parentheses are there to give a particular order).

6.EE.A.3
Apply the properties of operations to generate equivalent expressions.

For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.
6.EE.A.3
I can apply what I know about the properties of operations (associative, commutative and distributive) to create equivalent (or equal) expressions.

6.EE.A.4
Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them).

For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.
6.EE.A.4
I can recognize when two expressions are equivalent.

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Reason about and solve one-variable equations and inequalities
I can think about and solve one-variable equations and inequalities
6.EE.B.5
Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
6.EE.B.5
I can understand that solving an equation or inequality means that I find out which values can make the equation or inequality true.

I can try different numbers in place of a variable to figure out which makes the equation or inequality true.

6.EE.B.6
Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
6.EE.B.6
I can use variables to represent numbers and write expressions to solve real-world problems.

I can understand that a variable can stand for an unknown number or any number in a given set of numbers.

6.EE.B.7
Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.
6.EE.B.7
I can solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q (where p, q and x are all nonnegative rational numbers).

6.EE.B.8
Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.
6.EE.B.8
I can write an inequality (x > c or x < c) to stand for a limitation or condition in a real-world or mathematical problem that has infinitely many solutions.

I can show the answers to problems involving inequalities on number line diagrams.

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Represent and analyze quantitative relationships between dependent and independent variables
I can write and analyze numerical relationships between dependent and independent variables
6.EE.C.9
Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.

For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.
6.EE.C.9
I can use variables that change in relationship to one another to represent two quantities in a real world problem.

I can write an equation to show one quantity (the dependent variable) in terms of the other quantity (the independent variable).

I can use graphs and tables to show the relationship between dependent and independent variables.

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Geometry

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Solve real-world and mathematical problems involving area, surface area, and volume
I can solve real-world and mathematical problems involving area, surface area and volume
6.G.A.1
Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
6.G.A.1
I can put together and take apart shapes to help me find the area of right triangles, other triangles, special quadrilaterals and polygons.

I can apply what I know about taking apart and putting together shapes to find the area of objects or places in real world situations.

6.G.A.2
Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.
6.G.A.2
I can use unit cubes to find the volume of any right rectangular prism.

I can understand that the mathematical formula (V = l w h or V = b h) will give me the same result as using unit cubes to figure out the volume.

I can use the mathematical formulas V=l w h or V= b h to determine the volume of real world objects.

6.G.A.3
Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.
6.G.A.3
I can draw polygons in the coordinate plane when I am given the coordinates for the vertices.

I can use coordinates to find the length of a side of a polygon joining points with the same first coordinate or the same second coordinate.

I can apply what I have learned about polygons on coordinate planes to real-world and mathematical situations.

6.G.A.4
Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.
6.G.A.4
I can represent and figure out the surface area of a three dimensional shape by using nets made up of rectangles and triangles.

I can apply my skills involving finding surface area with nets in real-world and mathematical problems.

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The Number System

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Apply and extend previous understandings of multiplication and division to divide fractions by fractions
I can apply what I have learned about multiplication and division to the division of fractions
6.NS.A.1
Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.

For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?
6.NS.A.1
I can divide two fractions.

I can solve word problems involving the division of fractions by fractions.

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Compute fluently with multi-digit numbers and find common factors and multiples
I can quickly and easily compute with large numbers and find common factors and multiples
6.NS.B.2
Fluently divide multi-digit numbers using the standard algorithm.
6.NS.B.2
I can easily divide multi-digit numbers.

6.NS.B.3
Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
6.NS.B.3
I can easily add, subtract, multiply and divide multi-digit numbers involving decimals.

6.NS.B.4
Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor.

For example, express 36 + 8 as 4 (9 + 2).
6.NS.B.4
I can find the greatest common factor of two whole numbers less than or equal to 100.

I can find the least common multiple of two whole numbers less than or equal to 12.

I can use the distributive property to show the sum of two whole numbers (1-100) in different ways. (Ex: show 36 + 8 as 4(9+2)).

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Apply and extend previous understandings of numbers to the system of rational numbers
I can apply my understanding of numbers to rational numbers (any numbers that can be made by dividing one integer with another)
6.NS.C.5
Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.
6.NS.C.5
I can understand that positive and negative numbers are used to describe amounts having opposite values.

I can use positive and negative numbers to show amounts in real-world situations and explain what the number 0 means in those situations.

6.NS.C.6
Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.
6.NS.C.6
I can understand that a rational number is a point on a number line.

I can extend number line diagrams to show positive and negative numbers on the line.

I can extend coordinate axes to show positive and negative numbers in the plane.

6.NS.C.6.a
Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., –(–3) = 3, and that 0 is its own opposite.
6.NS.C.6.a
I can recognize opposite signs of numbers as showing places on opposite sides of 0 on the number line.

I can recognize that the opposite of the opposite of a number is actually the number itself. (Ex: -(-3)=3)

I can recognize that 0 is its own opposite.

6.NS.C.6.b
Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
6.NS.C.6.b
I can understand that the signs (- or +) of numbers in ordered pairs indicate locations in quadrants of the coordinate plane.

I can recognize two ordered pairs with differing signs as reflections of each other across one or both axes.

6.NS.C.6.c
Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.
6.NS.C.6.c
I can find and place integers and other rational numbers on a number line diagram.

I can find and place ordered pairs on a coordinate plane.

6.NS.C.7
Understand ordering and absolute value of rational numbers.
6.NS.C.7
I can order rational numbers.

I can understand absolute value of rational numbers.

6.NS.C.7.a
Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.

For example, interpret –3 > –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right.
6.NS.C.7.a
I can understand statements of inequality (ex: -3 > -7) and explain their positions and distances apart on a number line.

6.NS.C.7.b
Write, interpret, and explain statements of order for rational numbers in real-world contexts.

For example, write –3 °C > –7 °C to express the fact that –3 °C is warmer than –7 °C.
6.NS.C.7.b
I can write, understand and explain how the order of rational numbers applies in real-world situations. (Ex: -3 °C > -7°C to show that -3 °C is warmer than -7°C)

6.NS.C.7.c
Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation.

For example, for an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars.
6.NS.C.7.c
I can understand the absolute value of a number as its distance from 0 on the number line.

I can understand absolute values as they apply to real-world situations. (Ex: for an account balance of -30 dollars, write (-30) =30 to describe the size of the debt in dollars.)

6.NS.C.7.d
Distinguish comparisons of absolute value from statements about order.

For example, recognize that an account balance less than –30 dollars represents a debt greater than 30 dollars.
6.NS.C.7.d
I can tell the difference between comparisons of absolute value from statements of order. (Ex: An account balance less than -30 dollars is a debt greater than 30 dollars.)

6.NS.C.8
Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.
6.NS.C.8
I can graph points in all four quadrants of the coordinate plane to help me solve real-world and mathematical problems.

I can use what I know about coordinates and absolute values to figure out the distance between points with the same first coordinate or the same second coordinate.

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Ratios & Proportional Relationships

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Understand ratio concepts and use ratio reasoning to solve problems
I can understand ratios and can use that understanding to solve problems
6.RP.A.1
Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.

For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”
6.RP.A.1
I can use what I know about ratios to describe the relationship between two quantities.

6.RP.A.2
Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship.

For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.”
6.RP.A.2
I can understand how to find a rate when given a specific ratio. (Ex: We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.)

6.RP.A.3
Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
6.RP.A.3
I can use reasoning to solve word problems involving rate and ratios.

6.RP.A.3.a
Make tables of equivalent ratios relating quantities with whole- number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.
6.RP.A.3.a
I can make tables of equivalent ratios, find missing values in the tables and use the tables to compare ratios.

I can plot ratios on a coordinate plane.

6.RP.A.3.b
Solve unit rate problems including those involving unit pricing and constant speed.

For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?
6.RP.A.3.b
I can solve unit rate problems. (Ex: If it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were the lawns being mowed?)

6.RP.A.3.c
Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.
6.RP.A.3.c
I can find a percent of a quantity as a rate per 100. (Ex: 30% of a quantity means 30/100 times the quantity).

I can solve problems involving finding the whole if I am given a part and the percent.

6.RP.A.3.d
Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.
6.RP.A.3.d
I can use what I know about ratios to convert units of measurement.

I can change units of measurement correctly when multiplying or dividing quantities.

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Statistics & Probability

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Develop understanding of statistical variability
I can develop an understanding of the variables involved in statistics
6.SP.A.1
Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers.

For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages.
6.SP.A.1
I can recognize a statistical question as one that expects variability in the data related to the question.

6.SP.A.2
Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.
6.SP.A.2
I can understand that a set of data collected to answer a statistical question has a distribution that can be described by its center, spread and overall shape when plotted on a graph.

6.SP.A.3
Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.
6.SP.A.3
I can understand that a set of numerical data has a measure of center (median and/or mean) that summarizes all of its values with a single number.

I can understand that in a set of numerical data, the measure of variation describes how its values vary with a single number.

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Summarize and describe distributions
I can summarize and describe distributions
6.SP.B.4
Display numerical data in plots on a number line, including dot plots, histograms, and box plots.
6.SP.B.4
I can understand that a distribution of a variable is the description of the relative number of times each possible outcome will occur.

I can show numerical data in plots on a number line (including dot plots, histograms and box plots).

6.SP.B.5
Summarize numerical data sets in relation to their context, such as by:
6.SP.B.5
I can summarize sets of numerical data in relation to their circumstances.

6.SP.B.5.a
Reporting the number of observations.
6.SP.B.5.a
I can summarize data by stating the number of observations.

6.SP.B.5.b
Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.
6.SP.B.5.b
I can summarize data by describing the characteristics of what is being investigated, including how it was measured.

6.SP.B.5.c
Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.
6.SP.B.5.c
I can summarize data by giving numerical measures of center and variability.

I can summarize data by describing the overall pattern of the data and noticing unusual deviations from the overall pattern.

I can summarize data by explaining how the distribution of the data on a graph relates to the choice of measures of center and variability.

6.SP.B.5.d
Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.
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Grade 7

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Expressions & Equations

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Use properties of operations to generate equivalent expressions
7.EE.A.1
Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
7.EE.A.2
Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.

For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.”
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Solve real-life and mathematical problems using numerical and algebraic expressions and equations
7.EE.B.3
Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies.

For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.
7.EE.B.4
Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
7.EE.B.4.a
Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.

For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?
7.EE.B.4.b
Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem.

For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.
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Geometry

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Draw, construct, and describe geometrical figures and describe the relationships between them
7.G.A.1
Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
7.G.A.2
Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
7.G.A.3
Describe the two-dimensional figures that result from slicing three- dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.
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Solve real-life and mathematical problems involving angle measure, area, surface area, and volume
7.G.B.4
Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
7.G.B.5
Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.
7.G.B.6
Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
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The Number System

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Apply and extend previous understandings of operations with fractions
7.NS.A.1
Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.
7.NS.A.1.a
Describe situations in which opposite quantities combine to make 0.

For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.
7.NS.A.1.b
Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.
7.NS.A.1.c
Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.
7.NS.A.1.d
Apply properties of operations as strategies to add and subtract rational numbers.
7.NS.A.2
Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
7.NS.A.2.a
Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.
7.NS.A.2.b
Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then –(p/q) = (–p)/q = p/(–q). Interpret quotients of rational numbers by describing real-world contexts.
7.NS.A.2.c
Apply properties of operations as strategies to multiply and divide rational numbers.
7.NS.A.2.d
Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.
7.NS.A.3
Solve real-world and mathematical problems involving the four operations with rational numbers. (Computations with rational numbers extend the rules for manipulating fractions to complex fractions.)
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Ratios & Proportional Relationships

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Analyze proportional relationships and use them to solve real-world and mathematical problems
7.RP.A.1
Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units.

For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction ½/¼ miles per hour, equivalently 2 miles per hour.
7.RP.A.2
Recognize and represent proportional relationships between quantities.
7.RP.A.2.a
Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
7.RP.A.2.b
Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
7.RP.A.2.c
Represent proportional relationships by equations.

For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.
7.RP.A.2.d
Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.
7.RP.A.3
Use proportional relationships to solve multistep ratio and percent problems.

Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.
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Statistics & Probability

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Use random sampling to draw inferences about a population
7.SP.A.1
Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.
7.SP.A.2
Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions.

For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.
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Draw informal comparative inferences about two populations
7.SP.B.3
Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability.

For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.
7.SP.B.4
Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.

For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book.
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Investigate chance processes and develop, use, and evaluate probability models
7.SP.C.5
Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.
7.SP.C.6
Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability.

For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.
7.SP.C.7
Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.
7.SP.C.7.a
Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.

For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.
7.SP.C.7.b
Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.

For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?
7.SP.C.8
Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
7.SP.C.8.a
Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.
7.SP.C.8.b
Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., “rolling double sixes”), identify the outcomes in the sample space which compose the event.
7.SP.C.8.c
Design and use a simulation to generate frequencies for compound events.

For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?
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Grade 8

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Expressions & Equations

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Work with radicals and integer exponents
8.EE.A.1
Know and apply the properties of integer exponents to generate equivalent numerical expressions.

For example, 3² × 3⁻⁵ = 3⁻³ = 1/3³ = 1/27.
8.EE.A.2
Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
8.EE.A.3
Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.

For example, estimate the population of the United States as 3 × 10⁸ and the population of the world as 7 × 10⁹, and determine that the world population is more than 20 times larger.
8.EE.A.4
Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.
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Understand the connections between proportional relationships, lines, and linear equations
8.EE.B.5
Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.

For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.
8.EE.B.6
Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
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Analyze and solve linear equations and pairs of simultaneous linear equations
8.EE.C.7
Solve linear equations in one variable.
8.EE.C.7.a
Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).
8.EE.C.7.b
Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.
8.EE.C.8
Analyze and solve pairs of simultaneous linear equations.
8.EE.C.8.a
Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.
8.EE.C.8.b
Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.

For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.
8.EE.C.8.c
Solve real-world and mathematical problems leading to two linear equations in two variables.

For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.
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Functions

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Define, evaluate, and compare functions
8.F.A.1
Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. (Function notation is not required in Grade 8.)
8.F.A.2
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.
8.F.A.3
Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.

For example, the function A = s² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.
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Use functions to model relationships between quantities
8.F.B.4
Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
8.F.B.5
Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
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Geometry

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Understand congruence and similarity using physical models, transparencies, or geometry software
8.G.A.1
Verify experimentally the properties of rotations, reflections, and translations:
8.G.A.1.a
Lines are taken to lines, and line segments to line segments of the same length.
8.G.A.1.b
Angles are taken to angles of the same measure.
8.G.A.1.c
Parallel lines are taken to parallel lines.
8.G.A.2
Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
8.G.A.3
Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
8.G.A.4
Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two- dimensional figures, describe a sequence that exhibits the similarity between them.
8.G.A.5
Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.

For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.
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Understand and apply the Pythagorean Theorem
8.G.B.6
Explain a proof of the Pythagorean Theorem and its converse.
8.G.B.7
Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
8.G.B.8
Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
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Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres
8.G.C.9
Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
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The Number System

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Know that there are numbers that are not rational, and approximate them by rational numbers
8.NS.A.1
Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
8.NS.A.2
Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π²).

For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
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Statistics & Probability

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Investigate patterns of association in bivariate data
8.SP.A.1
Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.
8.SP.A.2
Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.
8.SP.A.3
Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.

For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.
8.SP.A.4
Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables.

For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?
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High School: Algebra

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Arithmetic with Polynomials & Rational Expressions

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Perform arithmetic operations on polynomials
HSA-APR.A.1
Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
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Understand the relationship between zeros and factors of polynomials
HSA-APR.B.2
Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).
HSA-APR.B.3
Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.
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Use polynomial identities to solve problems
HSA-APR.C.4
Prove polynomial identities and use them to describe numerical relationships.

For example, the polynomial identity (x² + y²)² = (x² – y²)² + (2xy)² can be used to generate Pythagorean triples.
HSA-APR.C.5
(+) Know and apply the Binomial Theorem for the expansion of (x + y)ⁿ in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle. (The Binomial Theorem can be proved by mathematical induction or by combinatorial argument.)

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
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Rewrite rational expressions
HSA-APR.D.6
Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.
HSA-APR.D.7
(+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
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Creating Equations

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Create equations that describe numbers or relationships
HSA-CED.A.1
Create equations and inequalities in one variable and use them to solve problems.

Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
HSA-CED.A.2
Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
HSA-CED.A.3
Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non- viable options in a modeling context.

For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
HSA-CED.A.4
Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

For example, rearrange Ohm’s law V = IR to highlight resistance R.
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Reasoning with Equations & Inequalities

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Understand solving equations as a process of reasoning and explain the reasoning
HSA-REI.A.1
Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
HSA-REI.A.2
Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
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Solve equations and inequalities in one variable
HSA-REI.B.3
Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
HSA-REI.B.4
Solve quadratic equations in one variable.
HSA-REI.B.4.a
Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)² = q that has the same solutions. Derive the quadratic formula from this form.
HSA-REI.B.4.b
Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.
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Solve systems of equations
HSA-REI.C.5
Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
HSA-REI.C.6
Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
HSA-REI.C.7
Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.

For example, find the points of intersection between the line y = –3x and the circle x² + y² = 3.
HSA-REI.C.8
(+) Represent a system of linear equations as a single matrix equation in a vector variable.

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
HSA-REI.C.9
(+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
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Represent and solve equations and inequalities graphically
HSA-REI.D.10
Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
HSA-REI.D.11
Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
HSA-REI.D.12
Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
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Seeing Structure in Expressions

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Interpret the structure of expressions
HSA-SSE.A.1
Interpret expressions that represent a quantity in terms of its context.
HSA-SSE.A.1.a
Interpret parts of an expression, such as terms, factors, and coefficients.
HSA-SSE.A.1.b
Interpret complicated expressions by viewing one or more of their parts as a single entity.

For example, interpret P(1+r)ⁿ as the product of P and a factor not depending on P.
HSA-SSE.A.2
Use the structure of an expression to identify ways to rewrite it.

For example, see x⁴ – y⁴ as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²).
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Write expressions in equivalent forms to solve problems
HSA-SSE.B.3
Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
HSA-SSE.B.3.a
Factor a quadratic expression to reveal the zeros of the function it defines.
HSA-SSE.B.3.b
Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
HSA-SSE.B.3.c
Use the properties of exponents to transform expressions for exponential functions.

For example the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.
HSA-SSE.B.4
Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.

For example, calculate mortgage payments.
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High School: Functions

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Building Functions

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Build a function that models a relationship between two quantities
HSF-BF.A.1
Write a function that describes a relationship between two quantities.
HSF-BF.A.1.a
Determine an explicit expression, a recursive process, or steps for calculation from a context.
HSF-BF.A.1.b
Combine standard function types using arithmetic operations.

For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
HSF-BF.A.1.c
(+) Compose functions.

For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
HSF-BF.A.2
Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
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Build new functions from existing functions
HSF-BF.B.3
Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
HSF-BF.B.4
Find inverse functions.
HSF-BF.B.4.a
Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse.

For example, f(x) = 2 x³ or f(x) = (x+1)/(x–1) for x ≠ 1.
HSF-BF.B.4.b
(+) Verify by composition that one function is the inverse of another.

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
HSF-BF.B.4.c
(+) Read values of an inverse function from a graph or a table, given that the function has an inverse.

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
HSF-BF.B.4.d
(+) Produce an invertible function from a non-invertible function by restricting the domain.

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
HSF-BF.B.5
(+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
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Interpreting Functions

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Understand the concept of a function and use function notation
HSF-IF.A.1
Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
HSF-IF.A.2
Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
HSF-IF.A.3
Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.
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Interpret functions that arise in applications in terms of the context
HSF-IF.B.4
For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.

Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
HSF-IF.B.5
Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.
HSF-IF.B.6
Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
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Analyze functions using different representations
HSF-IF.C.7
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
HSF-IF.C.7.a
Graph linear and quadratic functions and show intercepts, maxima, and minima.
HSF-IF.C.7.b
Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
HSF-IF.C.7.c
Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
HSF-IF.C.7.d
(+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
HSF-IF.C.7.e
Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
HSF-IF.C.8
Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
HSF-IF.C.8.a
Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
HSF-IF.C.8.b
Use the properties of exponents to interpret expressions for exponential functions.

For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.
HSF-IF.C.9
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.
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Linear, Quadratic & Exponential Models

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Construct and compare linear, quadratic, and exponential models and solve problems
HSF-LE.A.1
Distinguish between situations that can be modeled with linear functions and with exponential functions.
HSF-LE.A.1.a
Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
HSF-LE.A.1.b
Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
HSF-LE.A.1.c
Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
HSF-LE.A.2
Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
HSF-LE.A.3
Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
HSF-LE.A.4
For exponential models, express as a logarithm the solution to abct = d where a , c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.
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Interpret expressions for functions in terms of the situation they model
HSF-LE.B.5
Interpret the parameters in a linear or exponential function in terms of a context.
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Trigonometric Functions

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Extend the domain of trigonometric functions using the unit circle
HSF-TF.A.1
Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
HSF-TF.A.2
Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
HSF-TF.A.3
(+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π-x, π+x, and 2π–x in terms of their values for x, where x is any real number.

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
HSF-TF.A.4
(+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
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Model periodic phenomena with trigonometric functions
HSF-TF.B.5
Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
HSF-TF.B.6
(+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
HSF-TF.B.7
(+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
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Prove and apply trigonometric identities
HSF-TF.C.8
Prove the Pythagorean identity sin²(θ) + cos²(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
HSF-TF.C.9
(+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
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High School: Geometry

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Circles

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Understand and apply theorems about circles
HSG-C.A.1
Prove that all circles are similar.
HSG-C.A.2
Identify and describe relationships among inscribed angles, radii, and chords.

Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
HSG-C.A.3
Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
HSG-C.A.4
(+) Construct a tangent line from a point outside a given circle to the circle.

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
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Find arc lengths and areas of sectors of circles
HSG-C.B.5
Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
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Congruence

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Experiment with transformations in the plane
HSG-CO.A.1
Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
HSG-CO.A.2
Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
HSG-CO.A.3
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
HSG-CO.A.4
Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
HSG-CO.A.5
Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
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Understand congruence in terms of rigid motions
HSG-CO.B.6
Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
HSG-CO.B.7
Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
HSG-CO.B.8
Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
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Prove geometric theorems
HSG-CO.C.9
Prove theorems about lines and angles.

Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.
HSG-CO.C.10
Prove theorems about triangles.

Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
HSG-CO.C.11
Prove theorems about parallelograms.

Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
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Make geometric constructions
HSG-CO.D.12
Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).

Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
HSG-CO.D.13
Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
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Expressing Geometric Properties with Equations

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Translate between the geometric description and the equation for a conic section
HSG-GPE.A.1
Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.
HSG-GPE.A.2
Derive the equation of a parabola given a focus and directrix.
HSG-GPE.A.3
(+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
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Use coordinates to prove simple geometric theorems algebraically
HSG-GPE.B.4
Use coordinates to prove simple geometric theorems algebraically.

For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).
HSG-GPE.B.5
Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
HSG-GPE.B.6
Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
HSG-GPE.B.7
Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.
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Geometric Measurement & Dimension

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Explain volume formulas and use them to solve problems
HSG-GMD.A.1
Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.

Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
HSG-GMD.A.2
(+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
HSG-GMD.A.3
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
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Visualize relationships between two-dimensional and three-dimensional objects
HSG-GMD.B.4
Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
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Modeling with Geometry

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Apply geometric concepts in modeling situations
HSG-MG.A.1
Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).
HSG-MG.A.2
Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).
HSG-MG.A.3
Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).
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Similarity, Right Triangles & Trigonometry

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Understand similarity in terms of similarity transformations
HSG-SRT.A.1
Verify experimentally the properties of dilations given by a center and a scale factor:
HSG-SRT.A.1.a
A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
HSG-SRT.A.1.b
The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
HSG-SRT.A.2
Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
HSG-SRT.A.3
Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
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Prove theorems involving similarity
HSG-SRT.B.4
Prove theorems about triangles.

Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
HSG-SRT.B.5
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
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Define trigonometric ratios and solve problems involving right triangles
HSG-SRT.C.6
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
HSG-SRT.C.7
Explain and use the relationship between the sine and cosine of complementary angles.
HSG-SRT.C.8
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
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Apply trigonometry to general triangles
HSG-SRT.D.9
(+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
HSG-SRT.D.10
(+) Prove the Laws of Sines and Cosines and use them to solve problems.

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
HSG-SRT.D.11
(+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
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High School: Number & Quantity

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Quantities

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Reason quantitatively and use units to solve problems
HSN-Q.A.1
Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
HSN-Q.A.2
Define appropriate quantities for the purpose of descriptive modeling.
HSN-Q.A.3
Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
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The Complex Number System

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Perform arithmetic operations with complex numbers
HSN-CN.A.1
Know there is a complex number i such that = –1, and every complex number has the form a + bi with a and b real.
HSN-CN.A.2
Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
HSN-CN.A.3
(+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
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Represent complex numbers and their operations on the complex plane
HSN-CN.B.4
(+) Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
HSN-CN.B.5
(+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.

For example, (–1 + √3 i)³ = 8 because (–1 + √3 i) has modulus 2 and argument 120°.

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
HSN-CN.B.6
(+) Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
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Use complex numbers in polynomial identities and equations
HSN-CN.C.7
Solve quadratic equations with real coefficients that have complex solutions.
HSN-CN.C.8
(+) Extend polynomial identities to the complex numbers.

For example, rewrite x² + 4 as (x + 2i)(x – 2i).

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
HSN-CN.C.9
(+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
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The Real Number System

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Extend the properties of exponents to rational exponents
HSN-RN.A.1
Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.

For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3)3 to hold, so (51/3)3 must equal 5.
HSN-RN.A.2
Rewrite expressions involving radicals and rational exponents using the properties of exponents.
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Use properties of rational and irrational numbers
HSN-RN.B.3
Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
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Vector & Matrix Quantities

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Represent and model with vector quantities
HSN-VM.A.1
(+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
HSN-VM.A.2
(+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
HSN-VM.A.3
(+) Solve problems involving velocity and other quantities that can be represented by vectors.

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
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Perform operations on vectors
HSN-VM.B.4
(+) Add and subtract vectors.

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
HSN-VM.B.4.a
Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.
HSN-VM.B.4.b
Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
HSN-VM.B.4.c
Understand vector subtraction v – w as v + (–w), where –w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.
HSN-VM.B.5
(+) Multiply a vector by a scalar.

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
HSN-VM.B.5.a
Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(vᵪ, vᵧ) = (cvᵪ, cvᵧ).
HSN-VM.B.5.b
Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).
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Perform operations on matrices and use matrices in applications
HSN-VM.C.6
(+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
HSN-VM.C.7
(+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
HSN-VM.C.8
(+) Add, subtract, and multiply matrices of appropriate dimensions.

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
HSN-VM.C.9
(+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
HSN-VM.C.10
(+) Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
HSN-VM.C.11
(+) Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
HSN-VM.C.12
(+) Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
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High School: Statistics & Probability

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Conditional Probability & the Rules of Probability

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Understand independence and conditional probability and use them to interpret data
HSS-CP.A.1
Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).
HSS-CP.A.2
Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.
HSS-CP.A.3
Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.
HSS-CP.A.4
Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities.

For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.
HSS-CP.A.5
Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.

For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.
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Use the rules of probability to compute probabilities of compound events
HSS-CP.B.6
Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model.
HSS-CP.B.7
Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.
HSS-CP.B.8
(+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
HSS-CP.B.9
(+) Use permutations and combinations to compute probabilities of compound events and solve problems.

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
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Interpreting Categorical & Quantitative Data

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Summarize, represent, and interpret data on a single count or measurement variable
HSS-ID.A.1
Represent data with plots on the real number line (dot plots, histograms, and box plots).
HSS-ID.A.2
Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
HSS-ID.A.3
Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).
HSS-ID.A.4
Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.
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Summarize, represent, and interpret data on two categorical and quantitative variables
HSS-ID.B.5
Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
HSS-ID.B.6
Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
HSS-ID.B.6.a
Fit a function to the data; use functions fitted to data to solve problems in the context of the data.

Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.
HSS-ID.B.6.b
Informally assess the fit of a function by plotting and analyzing residuals.
HSS-ID.B.6.c
Fit a linear function for a scatter plot that suggests a linear association.
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Interpret linear models
HSS-ID.C.7
Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
HSS-ID.C.8
Compute (using technology) and interpret the correlation coefficient of a linear fit.
HSS-ID.C.9
Distinguish between correlation and causation.
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Making Inferences & Justifying Conclusions

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Understand and evaluate random processes underlying statistical experiments
HSS-IC.A.1
Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
HSS-IC.A.2
Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.

For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model?
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Make inferences and justify conclusions from sample surveys, experiments, and observational studies
HSS-IC.B.3
Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.
HSS-IC.B.4
Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.
HSS-IC.B.5
Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.
HSS-IC.B.6
Evaluate reports based on data.
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Using Probability to Make Decisions

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Calculate expected values and use them to solve problems
HSS-MD.A.1
(+) Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
HSS-MD.A.2
(+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
HSS-MD.A.3
(+) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value.

For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes.

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
HSS-MD.A.4
(+) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value.

For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households?

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
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Use probability to evaluate outcomes of decisions
HSS-MD.B.5
(+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
HSS-MD.B.5.a
Find the expected payoff for a game of chance.

For example, find the expected winnings from a state lottery ticket or a game at a fast- food restaurant.
HSS-MD.B.5.b
Evaluate and compare strategies on the basis of expected values.

For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.
HSS-MD.B.6
(+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.
HSS-MD.B.7
(+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

Note: (+) standards are included to increase coherence but are not necessarily expected to be addressed on a high stakes assessments.