# Think Through Math CCSS 6th Grade Pathway

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Think Through Math CCSS 6th Grade Pathway Topic Objective Common Core Kentucky Solving and Solve word problems, 7.EE.B.4a: Solve real-life and mathematical Modeling Two- both algebraically and problems using numerical and algebraic Step Problems arithmetically, leading expressions and equations. Use variables to to equations of the represent quantities in a real-world or form px + q = r. mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. 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? Solving Apply the distributive 7.EE.B.4a: Solve real-life and mathematical Equations with property when solving problems using numerical and algebraic the Distributive equations. expressions and equations. Use variables to Property represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. 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?

Solving Use the distributive 7.EE.B.4a: Solve real-life and mathematical Equations with property to solve problems using numerical and algebraic the Distributive problems. expressions and equations. Use variables to Property in represent quantities in a real-world or Context mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. 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? Interpreting Identify a unit rate 7.RP.A.2b: Analyze proportional relationships and Unit Rates on from a graph, use them to solve real-world and mathematical Graphs especially dealing with problems. Recognize and represent proportional ratios of fractions. relationships between quantities. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. Proportion Recognize and 7.RP.A.2a: Analyze proportional relationships and Concepts represent use them to solve real-world and mathematical proportional problems. Recognize and represent proportional relationships in relationships between quantities. Decide whether different ways, two quantities are in a proportional relationship, including graphs. 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.2c: Analyze proportional relationships and use them to solve real-world and mathematical problems. Recognize and represent proportional

relationships between quantities. 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. Proportional Represent 7.RP.A.2a: Analyze proportional relationships and Relationships in proportional use them to solve real-world and mathematical Tables and relationships in tables problems. Recognize and represent proportional Equations and equations. relationships between quantities. 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.2c: Analyze proportional relationships and use them to solve real-world and mathematical problems. Recognize and represent proportional relationships between quantities. 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. Interpreting Explain what a point 7.RP.A.2d: Analyze proportional relationships and Points on on the graph of a use them to solve real-world and mathematical Graphs of proportional problems. Recognize and represent proportional Proportional relationship means in relationships between quantities. Explain what a Relationships context, especially (0, point (x, y) on the graph of a proportional 0) and (1, r) where r is relationship means in terms of the situation, with the unit rate. special attention to the points (0, 0) and (1, r) where r is the unit rate. 7.RP.A.1: Analyze proportional relationships and use them to solve real-world and mathematical

problems. 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 1/2÷1/4 miles per hour, equivalently 2 miles per hour. Using Analyze proportional 7.G.A.1: Draw, construct, and describe geometrical Proportions to relationships and use figures and describe the relationships between Solve Problems them to solve real- them. Solve problems involving scale drawings of world, multistep ratio geometric figures, including computing actual problems. lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. 7.RP.A.3: Analyze proportional relationships and use them to solve real-world and mathematical problems. 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. Proportions in Solve problems 7.G.A.1: Draw, construct, and describe geometrical Scale Drawings involving scale figures and describe the relationships between drawings of geometric them. Solve problems involving scale drawings of figures, including geometric figures, including computing actual computing actual lengths and areas from a scale drawing and lengths and areas reproducing a scale drawing at a different scale. from a scale drawing 7.RP.A.3: Analyze proportional relationships and and reproducing a use them to solve real-world and mathematical scale drawing at a problems. Use proportional relationships to solve different scale. multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. Introduction of Recognize and apply 7.G.A.1: Draw, construct, and describe geometrical

Similar Figures properties of similar figures and describe the relationships between figures. them. 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.RP.A.2c: Analyze proportional relationships and use them to solve real-world and mathematical problems. Recognize and represent proportional relationships between quantities. 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. Use Similar Use proportional 7.G.A.1: Draw, construct, and describe geometrical Figures to Solve reasoning and the figures and describe the relationships between Problems properties of similar them. Solve problems involving scale drawings of figures to find the geometric figures, including computing actual missing side length in lengths and areas from a scale drawing and a figure. reproducing a scale drawing at a different scale. 7.RP.A.2c: Analyze proportional relationships and use them to solve real-world and mathematical problems. Recognize and represent proportional relationships between quantities. 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. Similarity Generalize the critical 7.G.A.1: Draw, construct, and describe geometrical attributes of figures and describe the relationships between similarity. them. 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. Fraction, Recognize that 7.EE.B.3: Solve real-life and mathematical MA-6-NPO-S-E1: Students will: estimate and Decimal, and fractions, decimals problems using numerical and algebraic mentally compute to solve real-world and/or Percent (that have a finite or expressions and equations. Solve multi-step real- mathematical problems with whole numbers, Equivalents repeating decimal life and mathematical problems posed with fractions, decimals and percents, checking for representation), and positive and negative rational numbers in any reasonable and appropriate computational percents are different form (whole numbers, fractions, and decimals), results representations of using tools strategically. Apply properties of MA-6-NPO-S-NS1: Students will: continue to rational numbers. operations to calculate with numbers in any form; develop number sense using fractions, convert between forms as appropriate; and assess decimals and percents, including percents the reasonableness of answers using mental greater than 100% and improper fractions computation and estimation strategies. For MA-7-NPO-S-NO2: Students will: extend example: If a woman making $25 an hour gets a concepts and application of operations with 10% raise, she will make an additional 1/10 of her fractions and decimals to include percents salary an hour, or $2.50, for a new salary of MA-6-NPO-S-NS5: Students will: compare, $27.50. If you want to place a towel bar 9 3/4 order and convert between whole numbers, inches long in the center of a door that is 27 1/2 fractions, decimals and percents using inches wide, you will need to place the bar about 9 concrete materials, drawings or pictures and inches from each edge; this estimate can be used mathematical symbols (e.g., , "greater than or equal to", =, "not equal to", order on a number line) MA-5-NPO-S-NS5: Students will: explore, investigate, compare, relate and apply relationships among whole numbers, fractions, decimals and percents MA-7-NPO-S-E1: Students will: estimate and mentally compute to solve real-world and/or mathematical problems with fractions, decimals, percents and integers, checking for reasonable and appropriate computational results MA-7-NPO-S-NS5: Students will: compare, order and determine equivalent relationships among fractions, decimals and percents

Percent and Use proportional 7.RP.A.3: Analyze proportional relationships and Percent Change relationships to solve use them to solve real-world and mathematical multistep percent and problems. Use proportional relationships to solve percent change multistep ratio and percent problems. Examples: problems. simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. Percent and Use proportional 7.RP.A.3: Analyze proportional relationships and Percent Error relationships to solve use them to solve real-world and mathematical multistep percent problems. Use proportional relationships to solve change and percent multistep ratio and percent problems. Examples: error problems. simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. Simple Interest Solve problems using 7.RP.A.3: Analyze proportional relationships and MA-8-NPO-S-RP1: Students will: use simple interest use them to solve real-world and mathematical percentages and proportions in problem formulas. problems. Use proportional relationships to solve solving, including consumer applications (e.g., multistep ratio and percent problems. Examples: simple interest, percentages of increase and simple interest, tax, markups and markdowns, decrease, discounts, unit pricing, sale prices) gratuities and commissions, fees, percent increase MA-7-AT-S-VEO2: Students will: substitute and decrease, percent error. values for variables to evaluate algebraic expressions MA-8-AT-S-VEO2: Students will: given a formula, substitute appropriate elements from a real-world or mathematical situation MA-7-NPO-S-RP1: Students will: compute percentages and use percentages in proportional reasoning MA-7-AT-S-EI2: Students will: solve problems involving formulas Adding and Construct expressions 7.NS.A.1d: Apply and extend previous Subtracting and equations to understandings of addition and subtraction to add Rational model real-world and subtract rational numbers; represent addition Numbers I situations. and subtraction on a horizontal or vertical number

line diagram. Apply properties of operations as strategies to add and subtract rational numbers. 7.NS.A.1b: 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. 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.1a: 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. 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. Adding and Construct multiple 7.NS.A.1c: Apply and extend previous Subtracting expressions to show a understandings of addition and subtraction to add Rational difference between and subtract rational numbers; represent addition Numbers II two rational numbers. and subtraction on a horizontal or vertical number line diagram. 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. Multiplying and Learn rules for 7.NS.A.2c: Apply and extend previous Dividing multiplying and understandings of operations with fractions. Apply Rational dividing with signed and extend previous understandings of

Numbers rational numbers. Use multiplication and division and of fractions to the distributive multiply and divide rational numbers. Apply property to show why properties of operations as strategies to multiply the product of a and divide rational numbers. positive number and a 7.NS.A.2a: Apply and extend previous negative number is understandings of operations with fractions. Apply negative. and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. 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.2d: Apply and extend previous understandings of operations with fractions. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. 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.2b: Apply and extend previous understandings of operations with fractions. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. 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. Writing and Interpret signed 7.NS.A.3: Apply and extend previous Interpreting rational numbers in understandings of operations with fractions. Solve Expressions context and write real-world and mathematical problems involving with Rational expressions to model the four operations with rational numbers. Numbers situations. Use 7.EE.B.3: Solve real-life and mathematical multiplication and problems using numerical and algebraic division of signed expressions and equations. Solve multi-step real- rational numbers in life and mathematical problems posed with solving problems. 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. Operations Solve real-world and 7.NS.A.3 with Rational mathematical Apply and extend previous understandings of Numbers I problems involving operations with fractions. Solve real-world and the four operations mathematical problems involving the four with rational operations with rational numbers. numbers. Operations Solve real-world and 7.EE.B.3: Solve real-life and mathematical with Rational mathematical problems using numerical and algebraic Numbers II problems involving expressions and equations. Solve multi-step real-

the four operations life and mathematical problems posed with with rational positive and negative rational numbers in any numbers. 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. Circumference Find the 7.G.B.4: Solve real-life and mathematical problems MA-8-M-U-3: Students will understand that: circumference of a involving angle measure, area, surface area, and measurements are determined by using circle in real-world volume. Know the formulas for the area and appropriate techniques, tools, formulas and and mathematical circumference of a circle and use them to solve degree of accuracy needed for the situation. problems. problems; give an informal derivation of the MA-8-M-S-MPA5: Students will: determine the relationship between the circumference and area area and circumference of circles of a circle. MA-7-M-S-MPA6: Students will: explain how measurements and measurement formulas are related or different (e.g., perimeter and area of rectangles) MA-6-M-U-3: Students will understand that: measurements are determined by using appropriate techniques, tools, formulas and degree of accuracy needed for the situation. MA-6-M-S-MPA1: Students will: find perimeter of regular and irregular polygons in metric and U.S. customary units MA-7-M-S-MPA3: Students will: estimate and

find circle measurements in standard units (radius, diameter, circumference, area) and relationships among them MA-7-M-U-3: Students will understand that: measurements are determined by using appropriate techniques, tools, formulas and degree of accuracy needed for the situation. Area of Circles Find the area of a 7.G.B.4: Solve real-life and mathematical problems MA-8-M-U-3: Students will understand that: circle, and given the involving angle measure, area, surface area, and measurements are determined by using area, find its diameter volume. Know the formulas for the area and appropriate techniques, tools, formulas and or radius. circumference of a circle and use them to solve degree of accuracy needed for the situation. problems; give an informal derivation of the MA-8-M-S-MPA9: Students will: explain how relationship between the circumference and area measurements and measurement formulas of a circle. are related or different (perimeter and area; rate, time and distance; circumference and area of a circle) MA-7-M-U-3: Students will understand that: measurements are determined by using appropriate techniques, tools, formulas and degree of accuracy needed for the situation. MA-8-M-S-MPA5: Students will: determine the area and circumference of circles MA-7-M-S-MPA3: Students will: estimate and find circle measurements in standard units (radius, diameter, circumference, area) and relationships among them Area of Find the area of a 7.G.B.6: Solve real-life and mathematical problems MA-7-M-S-MPA4: Students will: develop and Complex figure that is made up involving angle measure, area, surface area, and use the formulas for area of a triangle, a Composite of more than one volume. Solve real-world and mathematical parallelogram and a trapezoid and relate each Figures shape. problems involving area, volume and surface area to the formula for the area of a rectangle (b x of two- and three-dimensional objects composed h) of triangles, quadrilaterals, polygons, cubes, and MA-7-M-S-MPA3: Students will: estimate and right prisms. find circle measurements in standard units 6.G.A.1: Solve real-world and mathematical (radius, diameter, circumference, area) and

problems involving area, surface area, and relationships among them volume. Find the area of right triangles, other MA-7-M-U-3: Students will understand that: triangles, special quadrilaterals, and polygons by measurements are determined by using composing into rectangles or decomposing into appropriate techniques, tools, formulas and triangles and other shapes; apply these techniques degree of accuracy needed for the situation. in the context of solving real-world and mathematical problems. Surface Area Find the surface area 7.G.B.6: Solve real-life and mathematical problems MA-8-G-S-SR3: Students will: compare and Volume of of a rectangular prism. involving angle measure, area, surface area, and properties of three-dimensional figures Rectangular Use the surface area volume. Solve real-world and mathematical (spheres, cones, cylinders, prisms, pyramids); Prisms of a cube to find one problems involving area, volume and surface area apply these properties and figures to solve of its dimensions. of two- and three-dimensional objects composed real-world problems of triangles, quadrilaterals, polygons, cubes, and MA-8-M-U-3: Students will understand that: right prisms. measurements are determined by using 6.G.A.4: Solve real-world and mathematical appropriate techniques, tools, formulas and problems involving area, surface area, and degree of accuracy needed for the situation. volume. Represent three-dimensional figures MA-8-AT-S-VEO2: Students will: given a using nets made up of rectangles and triangles, formula, substitute appropriate elements from and use the nets to find the surface area of these a real-world or mathematical situation figures. Apply these techniques in the context of MA-8-M-S-MPA7: Students will: develop and solving real-world and mathematical problems. apply formulas for volume and surface area of 6.G.A.2: Solve real-world and mathematical cubes, cylinders and right rectangular prisms; problems involving area, surface area, and investigate relationships between and among volume. Find the volume of a right rectangular them prism with fractional edge lengths by packing it MA-8-G-U-1: Students will understand that: with unit cubes of the appropriate unit fraction characteristics and properties of two- edge lengths, and show that the volume is the dimensional figures and three-dimensional same as would be found by multiplying the edge objects describe the world and are used to lengths of the prism. Apply the formulas V = l w h develop mathematical arguments about and V = b h to find volumes of right rectangular geometric relationships and to evaluate the prisms with fractional edge lengths in the context arguments of others. of solving real-world and mathematical problems. MA-8-AT-S-EI2: Students will: solve problems using formulas MA-7-G-S-SR4: Students will: describe, provide

examples of and identify elements (e.g., vertices, angles, faces, edges, congruent parts) of common three-dimensional figures (spheres, cones, cylinders, prisms and pyramids) MA-HS-M-S-MPA3: Students will: determine the surface area and volume of right rectangular prisms, pyramids, cylinders, cones and spheres in realistic problems MA-HS-M-U-3: Students will understand that: measurements are determined by using appropriate techniques, tools, formulas and degree of accuracy needed for the situation. MA-HS-G-S-SR9: Students will: classify and apply properties of three-dimensional geometric figures MA-7-AT-S-EI2: Students will: solve problems involving formulas MA-6-G-S-SR5: Students will: describe, provide examples of and identify properties (e.g., vertices, angles, faces, edges, congruent parts) of common three-dimensional figures (spheres, cones, cylinders, prisms and pyramids) Surface Area of Find the surface area 7.G.B.6: Solve real-life and mathematical problems MA-8-G-S-SR3: Students will: compare Cylinders of a cylinder. involving angle measure, area, surface area, and properties of three-dimensional figures volume. Solve real-world and mathematical (spheres, cones, cylinders, prisms, pyramids); problems involving area, volume and surface area apply these properties and figures to solve of two- and three-dimensional objects composed real-world problems of triangles, quadrilaterals, polygons, cubes, and MA-8-AT-S-VEO2: Students will: given a right prisms. formula, substitute appropriate elements from 7.G.B.4: Solve real-life and mathematical problems a real-world or mathematical situation involving angle measure, area, surface area, and MA-8-AT-S-EI2: Students will: solve problems volume. Know the formulas for the area and using formulas

circumference of a circle and use them to solve MA-7-G-S-SR4: Students will: describe, provide problems; give an informal derivation of the examples of and identify elements (e.g., relationship between the circumference and area vertices, angles, faces, edges, congruent parts) of a circle. of common three-dimensional figures (spheres, cones, cylinders, prisms and pyramids) MA-HS-M-S-MPA3: Students will: determine the surface area and volume of right rectangular prisms, pyramids, cylinders, cones and spheres in realistic problems MA-HS-M-U-3: Students will understand that: measurements are determined by using appropriate techniques, tools, formulas and degree of accuracy needed for the situation. MA-HS-G-S-SR9: Students will: classify and apply properties of three-dimensional geometric figures MA-7-AT-S-EI2: Students will: solve problems involving formulas Surface Area of Find the surface area 7.G.B.6: Solve real-life and mathematical problems MA-8-G-S-SR3: Students will: compare Pyramids of a pyramid. Use the involving angle measure, area, surface area, and properties of three-dimensional figures pyramid’s base and volume. Solve real-world and mathematical (spheres, cones, cylinders, prisms, pyramids); surface-area problems involving area, volume and surface area apply these properties and figures to solve dimensions to find its of two- and three-dimensional objects composed real-world problems slant height. of triangles, quadrilaterals, polygons, cubes, and MA-8-AT-S-VEO2: Students will: given a right prisms. formula, substitute appropriate elements from 6.G.A.4: Solve real-world and mathematical a real-world or mathematical situation problems involving area, surface area, and MA-8-M-S-MPA7: Students will: develop and volume. Represent three-dimensional figures apply formulas for volume and surface area of using nets made up of rectangles and triangles, cubes, cylinders and right rectangular prisms; and use the nets to find the surface area of these investigate relationships between and among figures. Apply these techniques in the context of them solving real-world and mathematical problems. MA-8-AT-S-EI2: Students will: solve problems using formulas

MA-7-G-S-SR4: Students will: describe, provide examples of and identify elements (e.g., vertices, angles, faces, edges, congruent parts) of common three-dimensional figures (spheres, cones, cylinders, prisms and pyramids) MA-HS-M-S-MPA3: Students will: determine the surface area and volume of right rectangular prisms, pyramids, cylinders, cones and spheres in realistic problems MA-HS-M-U-3: Students will understand that: measurements are determined by using appropriate techniques, tools, formulas and degree of accuracy needed for the situation. MA-HS-G-S-SR9: Students will: classify and apply properties of three-dimensional geometric figures MA-7-AT-S-EI2: Students will: solve problems involving formulas Surface Area of Find the surface area 7.G.B.4: Solve real-life and mathematical problems MA-8-G-S-SR3: Students will: compare Cones of a cone. involving angle measure, area, surface area, and properties of three-dimensional figures volume. Know the formulas for the area and (spheres, cones, cylinders, prisms, pyramids); circumference of a circle and use them to solve apply these properties and figures to solve problems; give an informal derivation of the real-world problems relationship between the circumference and area MA-8-AT-S-VEO2: Students will: given a of a circle. formula, substitute appropriate elements from 7.G.B.6: Solve real-life and mathematical problems a real-world or mathematical situation involving angle measure, area, surface area, and MA-8-M-S-MPA7: Students will: develop and volume. Solve real-world and mathematical apply formulas for volume and surface area of problems involving area, volume and surface area cubes, cylinders and right rectangular prisms; of two- and three-dimensional objects composed investigate relationships between and among of triangles, quadrilaterals, polygons, cubes, and them right prisms. MA-8-AT-S-EI2: Students will: solve problems using formulas

MA-7-G-S-SR4: Students will: describe, provide examples of and identify elements (e.g., vertices, angles, faces, edges, congruent parts) of common three-dimensional figures (spheres, cones, cylinders, prisms and pyramids) MA-HS-M-S-MPA3: Students will: determine the surface area and volume of right rectangular prisms, pyramids, cylinders, cones and spheres in realistic problems MA-HS-M-U-3: Students will understand that: measurements are determined by using appropriate techniques, tools, formulas and degree of accuracy needed for the situation. MA-HS-G-S-SR9: Students will: classify and apply properties of three-dimensional geometric figures MA-7-AT-S-EI2: Students will: solve problems involving formulas Surface Area of Find the surface area 7.G.B.6: Solve real-life and mathematical problems MA-8-M-U-3: Students will understand that: Spheres of a sphere, and, given involving angle measure, area, surface area, and measurements are determined by using the surface area, find volume. Solve real-world and mathematical appropriate techniques, tools, formulas and its diameter or radius. problems involving area, volume and surface area degree of accuracy needed for the situation. of two- and three-dimensional objects composed MA-8-AT-S-VEO2: Students will: given a of triangles, quadrilaterals, polygons, cubes, and formula, substitute appropriate elements from right prisms. a real-world or mathematical situation MA-8-AT-S-EI2: Students will: solve problems using formulas MA-HS-M-S-MPA3: Students will: determine the surface area and volume of right rectangular prisms, pyramids, cylinders, cones and spheres in realistic problems MA-HS-M-U-3: Students will understand that: measurements are determined by using

appropriate techniques, tools, formulas and degree of accuracy needed for the situation. Surface Area of Find the surface area 7.G.B.4: Solve real-life and mathematical problems MA-8-AT-S-VEO2: Students will: given a Composite of a shape that is involving angle measure, area, surface area, and formula, substitute appropriate elements from Solids made up of two or volume. Know the formulas for the area and a real-world or mathematical situation more three- circumference of a circle and use them to solve MA-8-M-S-MPA7: Students will: develop and dimensional shapes. problems; give an informal derivation of the apply formulas for volume and surface area of relationship between the circumference and area cubes, cylinders and right rectangular prisms; of a circle. investigate relationships between and among 7.G.B.6: Solve real-life and mathematical problems them involving angle measure, area, surface area, and MA-HS-M-S-MPA3: Students will: determine volume. Solve real-world and mathematical the surface area and volume of right problems involving area, volume and surface area rectangular prisms, pyramids, cylinders, cones of two- and three-dimensional objects composed and spheres in realistic problems of triangles, quadrilaterals, polygons, cubes, and right prisms. Angle Pairs Find the measures of 7.G.B.5: Solve real-life and mathematical problems MA-HS-G-S-SR3: Students will: analyze and complementary, involving angle measure, area, surface area, and apply angle relationships (e.g., linear pairs, supplementary, volume. Use facts about supplementary, vertical, complementary, supplementary, vertical, and adjacent complementary, vertical, and adjacent angles in a corresponding and alternate interior angles) in angles. multi-step problem to write and solve simple real-world or mathematical situations equations for an unknown angle in a figure. MA-7-G-S-SR2: Students will: identify characteristics of angles (e.g., adjacent, vertical, corresponding, interior, exterior) Angles in a Find the measure of 7.G.B.5: Solve real-life and mathematical problems MA-HS-G-S-SR3: Students will: analyze and Polygon an angle, and find the involving angle measure, area, surface area, and apply angle relationships (e.g., linear pairs, sum of interior angles volume. Use facts about supplementary, vertical, complementary, supplementary, in polygons. complementary, vertical, and adjacent angles in a corresponding and alternate interior angles) in multi-step problem to write and solve simple real-world or mathematical situations equations for an unknown angle in a figure. MA-HS-G-S-SR1: Students will: identify and 8.G.A.5: Understand congruence and similarity apply the definitions, properties and theorems using physical models, transparencies, or about line segments, rays and angles and use geometry software. Use informal arguments to them to prove theorems in Euclidean establish facts about the angle sum and exterior geometry, solve problems and perform basic

angle of triangles, about the angles created when geometric constructions using a straight edge parallel lines are cut by a transversal, and the and a compass angle-angle criterion for similarity of triangles. For MA-8-G-S-SR2: Students will: identify and example, arrange three copies of the same compare properties of two-dimensional triangle so that the sum of the three angles figures (circles; triangles: acute, right, obtuse, appears to form a line, and give an argument in scalene, isosceles, equilateral; quadrilaterals: terms of transversals why this is so. square, rectangle, rhombus, parallelogram, trapezoid; regular/irregular polygons); apply these properties and figures to solve real- world problems MA-6-G-S-SR4: Students will: identify, describe and provide examples and properties of two- dimensional figures (circles, triangles [acute, right, obtuse, scalene, isosceles, equilateral], quadrilaterals, regular polygons); apply these properties and figures to solve real-world problems MA-7-M-S-MPA2: Students will: estimate and find angle measures and segment measures MA-6-G-S-SR1: Students will: formulate and use the rules for the sum of angle measures in a triangle (180°) and in a quadrilateral (360°) MA-7-G-S-SR2: Students will: identify characteristics of angles (e.g., adjacent, vertical, corresponding, interior, exterior) Sampling Use data from random HSS-ID.A.2: Summarize, represent, and interpret samples to make data on a single count or measurement variable. inferences about Use statistics appropriate to the shape of the data populations and to distribution to compare center (median, mean) compare key and spread (interquartile range, standard characteristics deviation) of two or more different data sets. between two 7.SP.A.2: Use random sampling to draw inferences populations. about a population. 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. 7.SP.B.4: Draw informal comparative inferences about two populations. 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. Comparing Use data displays to 7.SP.B.3: Draw informal comparative inferences Data compare the measure about two populations. Informally assess the of both center and degree of visual overlap of two numerical data spread of populations. 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. Simple Find the probability of 7.SP.C.5: Investigate chance processes and MA-8-DAP-S-CD2: Students will: make Probability simple events. develop, use, and evaluate probability models. predictions, draw conclusions and verify Understand that the probability of a chance event results from statistical data and probability is a number between 0 and 1 that expresses the experiments, making use of technology as

likelihood of the event occurring. Larger numbers appropriate indicate greater likelihood. A probability near 0 MA-8-DAP-S-P7: Students will: determine indicates an unlikely event, a probability around theoretical (mathematical) probabilities (e.g., 1/2 indicates an event that is neither unlikely nor express probability as a ratio, decimal, likely, and a probability near 1 indicates a likely percent, area model as appropriate for a given event. situation) 7.SP.C.7b: Investigate chance processes and MA-6-DAP-S-P3: Students will: make develop, use, and evaluate probability models. predictions, draw conclusions and verify Develop a probability model and use it to find results from statistical data and probability probabilities of events. Compare probabilities experiments from a model to observed frequencies; if the MA-5-DAP-U-6: Students will understand that: agreement is not good, explain possible sources of probability can be used to make decisions or the discrepancy. Develop a probability model predictions, or to draw conclusions. (which may not be uniform) by observing MA-8-DAP-U-6: Students will understand that: frequencies in data generated from a chance probability can be used to make decisions or process. For example, find the approximate predictions or to draw conclusions. probability that a spinning penny will land heads MA-6-DAP-U-6: Students will understand that: up or that a tossed paper cup will land open-end probability can be used to make decisions or down. Do the outcomes for the spinning penny predictions or to draw conclusions. appear to be equally likely based on the observed MA-5-DAP-S-P1: Students will: determine the frequencies? possible outcomes of simple probability experiments that are conducted by using manipulatives MA-6-DAP-S-CD1: Students will: make predictions, draw conclusions and verify results from statistical data and probability experiments MA-6-DAP-S-P4: Students will: determine simple probabilities based on the results of an experiment and make inferences based on the data MA-5-DAP-S-P2: Students will: determine the likelihood of an event and represent that likelihood in numerical terms

MA-5-DAP-S-P3: Students will: examine events and describe their probability as likely or unlikely Compound Find the probability of 7.SP.C.5: Investigate chance processes and MA-HS-DAP-S-P13: Students will: determine Probability compound events. develop, use, and evaluate probability models. probabilities involving replacement and non- Understand that the probability of a chance event replacement is a number between 0 and 1 that expresses the MA-HS-DAP-S-P6: Students will: compute the likelihood of the event occurring. Larger numbers probability of a compound event indicate greater likelihood. A probability near 0 MA-HS-DAP-S-P5: Students will: apply the indicates an unlikely event, a probability around concepts of conditional probability and 1/2 indicates an event that is neither unlikely nor independent events and be able to compute likely, and a probability near 1 indicates a likely those probabilities event. 7.SP.C.7a: Investigate chance processes and develop, use, and evaluate probability models. 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. 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.8a: Investigate chance processes and develop, use, and evaluate probability models. Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. 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.8b: Investigate chance processes and develop, use, and evaluate probability models. Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. 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. Making Use probability to 7.SP.C.6: Investigate chance processes and MA-8-DAP-S-CD2: Students will: make Predictions make predictions. develop, use, and evaluate probability models. predictions, draw conclusions and verify Approximate the probability of a chance event by results from statistical data and probability collecting data on the chance process that experiments, making use of technology as produces it and observing its long-run relative appropriate frequency, and predict the approximate relative MA-8-DAP-S-P1: Students will: make frequency given the probability. For example, predictions, draw conclusions and verify when rolling a number cube 600 times, predict results from probability experiments or that a 3 or 6 would be rolled roughly 200 times, simulations, making use of technology as but probably not exactly 200 times. appropriate MA-8-DAP-S-P2: Students will: analyze situations, such as games of chance, board games or grading scales and make predictions using knowledge of probability MA-HS-DAP-U-6: Students will understand that: probability can be used to make decisions or predictions or to draw conclusions. MA-8-DAP-U-4: Students will understand that: inferences and predictions from data are used to make critical and informed decisions. MA-8-DAP-U-6: Students will understand that: probability can be used to make decisions or predictions or to draw conclusions.

MA-HS-DAP-U-5: Students will understand that: inferences and predictions from data are used to make critical and informed decisions. MA-HS-DAP-S-P10: Students will: determine and compare theoretical and experimental probabilities MA-HS-DAP-S-P12: Students will: make predictions and draw inferences from probabilities. and apply probability concepts to practical situations to make informed decisions Simulations of Identify and use 7.SP.C.8c: Investigate chance processes and Simple and simulations to develop, use, and evaluate probability models. Compound represent simple and Find probabilities of compound events using Events compound events. organized lists, tables, tree diagrams, and simulation. 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? Solving Word Rewrite expressions to 7.EE.A.2: Use properties of operations to generate Problems with help identify how equivalent expressions. Understand that rewriting Algebra quantities in a an expression in different forms in a problem problem situation are context can shed light on the problem and how related. 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.” Common Apply properties of 7.EE.A.1 MA-HS-AT-S-VEO8: Students will: factor Factors in operations as Use properties of operations to generate polynomials by removing the greatest Polynomials strategies to add, equivalent expressions. Apply properties of common factor subtract, factor, and operations as strategies to add, subtract, factor, MA-HS-AT-S-VEO6: Students will: add, expand expressions and expand linear expressions with rational subtract and multiply polynomials

with rational coefficients. MA-HS-AT-S-VEO11: Students will: add, coefficients. subtract, multiply, divide and simplify rational expressions MA-HS-AT-S-VEO9: Students will: factor quadratic polynomials Concept of Write and graph 7.EE.B.4b: Solve real-life and mathematical Inequalities II complex inequalities problems using numerical and algebraic and interpret the expressions and equations. Use variables to solution set in the represent quantities in a real-world or context of a real- mathematical problem, and construct simple world problem. equations and inequalities to solve problems by reasoning about the quantities. 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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