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3.MD.7a 
3 
Measurement & Data 
Find the area of a rectangle with wholenumber side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths. 


3.MD.7b 
3 
Measurement & Data 
Multiply side lengths to find areas of rectangles with wholenumber side lengths in the context of solving real world and mathematical problems, and represent wholenumber products as rectangular areas in mathematical reasoning. 


3.MD.7c 
3 
Measurement & Data 
Use tiling to show in a concrete case that the area of a rectangle with wholenumber 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.7d 
3 
Measurement & Data 
Recognize area as additive. Find areas of rectilinear figures by decomposing them into nonoverlapping rectangles and adding the areas of the nonoverlapping parts, applying this technique to solve real world problems. 


3.MD.8 
3 
Measurement & Data 
Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures. 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 area or with the same area and different perimeter. 


3.G.1 
3 
Geometry 
Reason with shapes and their attributes. 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.2 
3 
Geometry 
Reason with shapes and their attributes. 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 is 1/4 of the area of the shape. 


4.OA.1 
4 
Operations & Algebraic Thinking 
Use the four operations with whole numbers to solve problems. Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 x 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.2 
4 
Operations & Algebraic Thinking 
Use the four operations with whole numbers to solve problems. 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. 


4.OA.3 
4 
Operations & Algebraic Thinking 
Use the four operations with whole numbers to solve problems. Solve multistep word problems posed with whole numbers and having wholenumber 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.4 
4 
Operations & Algebraic Thinking 
Gain familiarity with factors and multiples. Find all factor pairs for a whole number in the range 1100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1100 is a multiple of a given onedigit number. Determine whether a given whole number in the range 1100 is prime or composite. 


4.OA.5 
4 
Operations & Algebraic Thinking 
Generate and analyze patterns. 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.NBT.1 
4 
Number & Operations in Base Ten 
Generalize place value understanding for multidigit whole numbers. Recognize that in a multidigit whole number, a digit in one place represents ten times what it represents in the place to its right. For example, recognize that 700 Ã· 70 = 10 by applying concepts of place value and division. (Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.). 


4.NBT.2 
4 
Number & Operations in Base Ten 
Generalize place value understanding for multidigit whole numbers. Read and write multidigit whole numbers using baseten numerals, number names, and expanded form. Compare two multidigit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. (Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.). 


4.NBT.3 
4 
Number & Operations in Base Ten 
Generalize place value understanding for multidigit whole numbers. Use place value understanding to round multidigit whole numbers to any place. (Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.). 


4.NBT.4 
4 
Number & Operations in Base Ten 
Use place value understanding and properties of operations to perform multidigit arithmetic. Fluently add and subtract multidigit whole numbers using the standard algorithm. (Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000. A range of algorithms may be used.). 


4.NBT.5 
4 
Number & Operations in Base Ten 
Use place value understanding and properties of operations to perform multidigit arithmetic. Multiply a whole number of up to four digits by a onedigit whole number, and multiply two twodigit 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. (Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000. A range of algorithms may be used.). 


4.NBT.6 
4 
Number & Operations in Base Ten 
Use place value understanding and properties of operations to perform multidigit arithmetic. Find wholenumber quotients and remainders with up to fourdigit dividends and onedigit 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. (Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000. A range of algorithms may be used.). 


4.NF.1 
4 
Number & Operations  Fractions 
Extend understanding of fraction equivalence and ordering. 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. (Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.). 


4.NF.2 
4 
Number & Operations  Fractions 
Extend understanding of fraction equivalence and ordering. 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. (Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.). 


4.NF.3 
4 
Number & Operations  Fractions 
Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. Understand a fraction a/b with a > 1 as a sum of fractions 1/b. (Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.). 


4.NF.3a 
4 
Number & Operations  Fractions 
Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. 


4.NF.3b 
4 
Number & Operations  Fractions 
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.3c 
4 
Number & Operations  Fractions 
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.3d 
4 
Number & Operations  Fractions 
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.4 
4 
Number & Operations  Fractions 
Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. (Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.). 


4.NF.4a 
4 
Number & Operations  Fractions 
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.4b 
4 
Number & Operations  Fractions 
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.4c 
4 
Number & Operations  Fractions 
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.5 
4 
Number & Operations  Fractions 
Understand decimal notation for fractions, and compare decimal fractions. 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. For example, express 3/10 as 30/100 and add 3/10 + 4/100 = 34/100. (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.) (Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.). 


4.NF.6 
4 
Number & Operations  Fractions 
Understand decimal notation for fractions, and compare decimal fractions. 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. (Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.). 


4.NF.7 
4 
Number & Operations  Fractions 
Understand decimal notation for fractions, and compare decimal fractions. Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons comparisons are valid only when 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. (Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.). 


4.MD.1 
4 
Measurement & Data 
Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit. 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 twocolumn 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), â€¦. 


4.MD.2 
4 
Measurement & Data 
Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit. 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.3 
4 
Measurement & Data 
Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit. 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.4 
4 
Measurement & Data 
Represent and interpret data. 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.5 
4 
Measurement & Data 
Geometric measurement: understand concepts of angle and measure angles. Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:
 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 "onedegree angle," and can be used to measure angles.
 b. An angle that turns through n onedegree angles is said to have an angle measure of n degrees. 


4.MD.6 
4 
Measurement & Data 
Geometric measurement: understand concepts of angle and measure angles. Measure angles in wholenumber degrees using a protractor. Sketch angles of specified measure. 


4.MD.7 
4 
Measurement & Data 
Geometric measurement: understand concepts of angle and measure angles. Recognize angle measure as additive. When an angle is decomposed into nonoverlapping 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.G.1 
4 
Geometry 
Draw and identify lines and angles, and classify shapes by properties of their lines and angles. Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in twodimensional figures. 


4.G.2 
4 
Geometry 
Draw and identify lines and angles, and classify shapes by properties of their lines and angles. Classify twodimensional 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.3 
4 
Geometry 
Draw and identify lines and angles, and classify shapes by properties of their lines and angles. Recognize a line of symmetry for a twodimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify linesymmetric figures and draw lines of symmetry. 


5.OA.1 
5 
Operations & Algebraic Thinking 
Write and interpret numerical expressions. Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. 


5.OA.2 
5 
Operations & Algebraic Thinking 
Write and interpret numerical expressions. 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.3 
5 
Operations & Algebraic Thinking 
Analyze patterns and relationships. 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.NBT.1 
5 
Number & Operations in Base Ten 
Understand the place value system. Recognize that in a multidigit 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.2 
5 
Number & Operations in Base Ten 
Understand the place value system. 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.3 
5 
Number & Operations in Base Ten 
Understand the place value system. Read, write, and compare decimals to thousandths. 


5.NBT.3a 
5 
Number & Operations in Base Ten 
Read and write decimals to thousandths using baseten 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.3b 
5 
Number & Operations in Base Ten 
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.4 
5 
Number & Operations in Base Ten 
Understand the place value system. Use place value understanding to round decimals to any place. 


5.NBT.5 
5 
Number & Operations in Base Ten 
Perform operations with multidigit whole numbers and with decimals to hundredths. Fluently multiply multidigit whole numbers using the standard algorithm. 


5.NBT.6 
5 
Number & Operations in Base Ten 
Perform operations with multidigit whole numbers and with decimals to hundredths. Find wholenumber quotients of whole numbers with up to fourdigit dividends and twodigit 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.7 
5 
Number & Operations in Base Ten 
Perform operations with multidigit whole numbers and with decimals to hundredths. 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.NF.1 
5 
Number & Operations  Fractions 
Use equivalent fractions as a strategy to add and subtract fractions. 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.2 
5 
Number & Operations  Fractions 
Use equivalent fractions as a strategy to add and subtract fractions. 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 an incorrect result 2/5 + 1/2 = 3/7 by observing that 3/7 < 1/2. 


5.NF.3 
5 
Number & Operations  Fractions 
Apply and extend previous understandings of multiplication and division to multiply and divide fractions. Interpret a fraction as division of the numerator by the denominator (a/b = a Ã· 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 50pound 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.4 
5 
Number & Operations  Fractions 
Apply and extend previous understandings of multiplication and division to multiply and divide fractions. Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. 


5.NF.4a 
5 
Number & Operations  Fractions 
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.4b 
5 
Number & Operations  Fractions 
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.5 
5 
Number & Operations  Fractions 
Apply and extend previous understandings of multiplication and division to multiply and divide fractions. Interpret multiplication as scaling (resizing) by:
 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.
 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.6 
5 
Number & Operations  Fractions 
Apply and extend previous understandings of multiplication and division to multiply and divide fractions. 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.7 
5 
Number & Operations  Fractions 
Apply and extend previous understandings of multiplication and division to multiply and divide fractions. 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 of a fraction by a fraction is not a requirement at this grade.). 


5.NF.7a 
5 
Number & Operations  Fractions 
Interpret division of a unit fraction by a nonzero whole number, and compute such quotients. For example, create a story context for (1/3) Ã· 4 and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) Ã· 4 = 1/12 because (1/12) × 4 = 1/3. 


5.NF.7b 
5 
Number & Operations  Fractions 
Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 Ã· (1/5) and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 Ã· (1/5) = 20 because 20 × (1/5) = 4. 


5.NF.7c 
5 
Number & Operations  Fractions 
Solve realworld problems involving division of unit fractions by nonzero 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 1/3cup servings are in 2 cups of raisins?. 


5.MD.1 
5 
Measurement & Data 
Convert like measurement units within a given measurement system. Convert among differentsized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multistep real world problems. 


5.MD.2 
5 
Measurement & Data 
Represent and interpret data. 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.3 
5 
Measurement & Data 
Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition. Recognize volume as an attribute of solid figures and understand concepts of volume measurement.
 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.
 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.4 
5 
Measurement & Data 
Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition. Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units. 


5.MD.5 
5 
Measurement & Data 
Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition. Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume. 


5.MD.5a 
5 
Measurement & Data 
Find the volume of a right rectangular prism with wholenumber 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 wholenumber products as volumes, e.g., to represent the associative property of multiplication. 


5.MD.5b 
5 
Measurement & Data 
Apply the formulas V =(l)(w)(h) and V = (b)(h) for rectangular prisms to find volumes of right rectangular prisms with wholenumber edge lengths in the context of solving real world and mathematical problems. 


5.MD.5c 
5 
Measurement & Data 
Recognize volume as additive. Find volumes of solid figures composed of two nonoverlapping right rectangular prisms by adding the volumes of the nonoverlapping parts, applying this technique to solve real world problems. 


5.G.1 
5 
Geometry 
Graph points on the coordinate plane to solve realworld and mathematical problems. 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., xaxis and xcoordinate, yaxis and ycoordinate). 


5.G.2 
5 
Geometry 
Graph points on the coordinate plane to solve realworld and mathematical problems. 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.3 
5 
Geometry 
Classify twodimensional figures into categories based on their properties. Understand that attributes belonging to a category of twodimensional 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.4 
5 
Geometry 
Classify twodimensional figures into categories based on their properties. Classify twodimensional figures in a hierarchy based on properties. 


6.RP.1 
6 
Ratios & Proportional Relationships 
Understand ratio concepts and use ratio reasoning to solve problems. 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.2 
6 
Ratios & Proportional Relationships 
Understand ratio concepts and use ratio reasoning to solve problems. Understand the concept of a unit rate a/b associated with a ratio a:b with b ? 0 (b not equal to zero), 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. (Expectations for unit rates in this grade are limited to noncomplex fractions.). 


6.RP.3 
6 
Ratios & Proportional Relationships 
Understand ratio concepts and use ratio reasoning to solve problems. Use ratio and rate reasoning to solve realworld and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. 


6.RP.3a 
6 
Ratios & Proportional Relationships 
Make tables of equivalent ratios relating quantities with wholenumber measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios. 


6.RP.3b 
6 
Ratios & Proportional Relationships 
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.3c 
6 
Ratios & Proportional Relationships 
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.3d 
6 
Ratios & Proportional Relationships 
Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. 


6.NS.1 
6 
The Number System 
Apply and extend previous understandings of multiplication and division to divide fractions by fractions. 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) Ã· (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) Ã· (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) Ã· (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4cup 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.2 
6 
The Number System 
Compute fluently with multidigit numbers and find common factors and multiples. Fluently divide multidigit numbers using the standard algorithm. 


6.NS.3 
6 
The Number System 
Compute fluently with multidigit numbers and find common factors and multiples. Fluently add, subtract, multiply, and divide multidigit decimals using the standard algorithm for each operation. 


6.NS.4 
6 
The Number System 
Compute fluently with multidigit numbers and find common factors and multiples. 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 1100 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.5 
6 
The Number System 
Apply and extend previous understandings of numbers to the system of rational numbers. 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, debits/credits, positive/negative electric charge); use positive and negative numbers to represent quantities in realworld contexts, explaining the meaning of 0 in each situation. 


6.NS.6 
6 
The Number System 
Apply and extend previous understandings of numbers to the system of rational numbers. 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.6a 
6 
The Number System 
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.6b 
6 
The Number System 
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.6c 
6 
The Number System 
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.7 
6 
The Number System 
Apply and extend previous understandings of numbers to the system of rational numbers. Understand ordering and absolute value of rational numbers. 


6.NS.7a 
6 
The Number System 
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.7b 
6 
The Number System 
Write, interpret, and explain statements of order for rational numbers in realworld contexts. For example, write 3Â°C > 7Â°C to express the fact that 3Â°C is warmer than 7Â°C. 


6.NS.7c 
6 
The Number System 
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 realworld situation. For example, for an account balance of 30 dollars, write 30 = 30 to describe the size of the debt in dollars. 


6.NS.7d 
6 
The Number System 
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.8 
6 
The Number System 
Apply and extend previous understandings of numbers to the system of rational numbers. Solve realworld 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. 

