Focus Standards

7.RP.A.2.a Recognize and represent proportional relationships between quantities. 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.

Overview

Students are reintroduced to ratio value, equivalent ratios, rate, and unit rate through collaborative work. Students examine situations to determine if they are describing a proportional relationship. Their analysis is applied to relationships in tables, graphs, and verbal descriptions.

Lessons

• Lesson 1: An Experience in Relationships as Measuring Rate *Priority Lesson*

• Lesson 2: Identify Proportional Relationships *Priority Lesson*

• Lesson 3: Identifying Proportional And Non-Proportional Relationships In Tables *Priority Lesson*

• Lesson 4: Identifying Proportional And Non-Proportional Relationships In Tables (continued)

• Lesson 5: Identify Proportional and Non-Proportional Relationships In Graphs *Priority Lesson*

• Lesson 6: Identify Proportional and Non-Proportional Relationships In Graphs (continued)

Focus Standards

7.RP.A.2.Recognize and represent proportional relationships between quantities.

• b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

• 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.

• 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.EE.B.4a 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. 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 p

Overview

Students identify the constant of proportionality by finding unit rate in collection of equivalent ratios. Students relate the equation of a proportional relationship to ratio tables and to graphs and interpret the points on the graph within the context. In Topic B, students learn that the unit rate of a collection of equivalent ratios is called the constant of proportionality and can be used to represent proportional relationships with equations of the form y = kx, where k is the constant of proportionality (7.RP.2b, 7.RP.2c, 7.EE.4a). Students relate the equation of a proportional relationship to ratio tables and to graphs and interpret the points on the graph within the context of the situation (7.RP.2d)

Lessons

• Lesson 7: Unit Rate as the Constant of Proportionality *Priority Lesson*

• Lesson 8: Representing Proportional Relationships with Equations *Priority Lesson*

• Lesson 9: Representing Proportional Relationships with Equations (Continued)

• Lesson 10: Interpreting Graphs of Proportional Relationships *Priority Lesson*

Focus Standards

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.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

Overview

In Topic C, students extend their reasoning about ratios and proportional relationships to compute unit rates for ratios and rates specified by rational numbers, such as a speed of ½ mile per ¼ hour (7.RP.1). Students apply their experience in the first two topics and their new understanding of unit rates for ratios and rates involving fractions to solve multistep ratio word problems (7.RP.3, 7.EE.4a).

Lessons

• Lesson 11: Ratios of Fractions and Their Unit Rates *Priority Lesson*

• Lesson 12: Ratios of Fractions and Their Unit Rates (Continued)

• Lesson 13: Finding Equivalent Ratios Given the Total Quantity *Priority Lesson*

• Lesson 14: Multi-Step Ratio Problems *Priority Lesson*

• Lesson 15: Equations of Graphs of Proportional Relationships Involving Fractions

Focus Standards

7.RP.A.2b Recognize and represent proportional relationships between quantities.

• b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

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.

Overview

Students determine if drawings are a reduction or enlargement of a two-dimensional picture. Students calculate the actual lengths and areas of objects, create their own two-dimensional scale drawings, and produce new scale drawings using a different scale factor.

Lessons

• Lesson 16: Relating Scale Drawings to Ratios and Rates *Priority Lesson*

• Lesson 17: The Unit Rate as the Scale Factor *Priority Lesson*

• Lesson 18: Computing Actual Lengths from a Scale Drawing *Priority Lesson*

• Lesson 19: Computing Actual Areas from a Scale Drawing (Continued)

• Lesson 20: An Exercise in Creating a Scale Drawing *Priority Lesson*

• Lesson 21: An Exercise in Changing Scales *Priority Lesson*

• Lesson 22: An Exercise in Changing Scales (Continued)

Focus Standards

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.

• 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.

• 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.

• 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.

• d. Apply properties of operations as strategies to add and subtract rational numbers.

Overview

In Topic A, students return to the number line to model the addition and subtraction of integers (7.NS.A.1). They use the number line and the Integer Game to demonstrate that an integer added to its opposite equals zero, representing the additive inverse (7.NS.A.1a, 7.NS.A.1b). Their findings are formalized as students develop rules for adding and subtracting integers, and they recognize that subtracting a number is the same as adding its opposite (7.NS.A.1c). Real-life situations are represented by the sums and differences of signed numbers. Students extend integer rules to include the rational numbers and use properties of operations to perform rational number calculations without the use of a calculator (7.NS.A.1d).

Lessons

• Lesson 1: Opposite Quantities Combine to Make Zero *Priority Lesson*

• Lesson 2: Using the Number Line to Model the Addition of Integers *Priority Lesson*

• Lesson 3: Understanding Addition of Integers

• Lesson 4: Efficiently Adding Integers and Other Rational Numbers *Priority Lesson*

• Lesson 5: Understanding Subtraction of Integers and Other Rational Numbers *Priority Lesson*

• Lesson 6: The Distance Between Two Rational Numbers *Priority Lesson*

• Lesson 7: Addition and Subtraction of Rational Numbers *Priority Lesson*

• Lesson 8: Applying Properties of Operations to Add and Subtract Rational Numbers *Priority Lesson*

• Lesson 9: Applying Properties of Operations to Add and Subtract Rational Numbers (Continued)

Focus Standards

7.NS.A.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

• 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.

• 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.

• c. Apply properties of operations as strategies to multiply and divide rational numbers.

• d. Convert a rational number to a decimal number using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats

Overview

Students develop the rules for multiplying and dividing signed numbers in Topic B. They use the properties of operations and their previous understanding of multiplication as repeated addition to represent the multiplication of a negative number as repeated subtraction (7.NS.A.2a). Students make analogies to the Integer Game to understand that the product of two negative numbers is a positive number. From earlier grades, they recognize division as the inverse process of multiplication. Thus, signed number rules for division are consistent with those for multiplication, provided a divisor is not zero (7.NS.A.2b). Students represent the division of two integers as a fraction, extending product and quotient rules to all rational numbers. They realize that any rational number in fractional form can be represented as a decimal that either terminates in 0s or repeats (7.NS.A.2d). Students recognize that the context of a situation often determines the most appropriate form of a rational number, and they use long division, place value, and equivalent fractions to fluently convert between these fraction and decimal forms. Topic B concludes with students multiplying and dividing rational numbers using the properties of operations (7.NS.A.2c).

Lessons

• Lesson 10: Understanding Multiplication of Integers *Priority Lesson*

• Lesson 11: Develop Rules for Multiplying Signed Numbers *Priority Lesson*

• Lesson 12: Division of Integers *Priority Lesson*

• Lesson 13: Converting Between Fractions and Decimals Using Equivalent Fractions *Priority Lesson*

• Lesson 14: Converting Rational Numbers to Decimals Using Long Division *Priority Lesson*

• Lesson 15: Multiplication and Division of Rational Numbers

• Lesson 16: Applying Properties of Operations to Multiply and Divide Rational Numbers

Focus Standards

7.NS.A.3 Solve real-world and mathematical problems involving the four operations with rational numbers.

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”.

7.EE.B.4a Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about quantities.

• 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?

Overview

In Topic C, students problem-solve with rational numbers and draw upon their work from Grade 6 with expressions and equations (6.EE.A.2, 6.EE.A.3, 6.EE.A.4, 6.EE.B.5, 6.EE.B.6, 6.EE.B.7). They perform operations with rational numbers (7.NS.A.3), incorporating them into algebraic expressions and equations. They represent and evaluate expressions in multiple forms, demonstrating how quantities are related (7.EE.A.2). The Integer Game is revisited as students discover “if-then” statements, relating changes in player’s hands (who have the same card-value totals) to changes in both sides of a number sentence. Students translate word problems into algebraic equations and become proficient at solving equations of the form px + q = r and p(x + q) = r, where p, q, and r, are specific rational numbers (7.EE.B.4a). As they become fluent in generating algebraic solutions, students identify the operations, inverse operations, and order of steps, comparing these to an arithmetic solution. Use of algebra to represent contextual problems continues in Module 3. Students problem-solve with rational numbers and draw upon their prior work with expressions and equations. Students perform operations with rational numbers incorporating them into algebraic expressions and equations. They represent and evaluate expressions in multiple forms, demonstrating how quantities are related

Lessons

• Lesson 17: Comparing Tape Diagram Solutions to Algebraic Solutions *Priority Lessons*

• Lesson 18: Writing, Evaluating, and Finding Equivalent Expressions with Rational Numbers *Priority Lessons*

• Lesson 19: Writing, Evaluating, and Finding Equivalent Expressions with Rational Numbers (Continued)

• Lesson 20: Investments—Performing Operations with Rational Numbers *Priority Lessons*

• Lesson 21: If–Then Moves with Integer Number Cards

• Lesson 22: Solving Equations Using Algebra *Priority Lessons*

• Lesson 23: Solving Equations Using Algebra (Continued)

Focus Standards

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.”

Overview

To begin this module, students will generate equivalent expressions using the fact that addition and multiplication can be done in any order with any grouping and will extend this understanding to subtraction (adding the inverse) and division (multiplying by the multiplicative inverse) (7.EE.A.1). They extend the properties of operations with numbers (learned in earlier grades) and recognize how the same properties hold true for letters that represent numbers.

An area model is used as a tool for students to rewrite products as sums and sums as products and can provide a visual representation leading students to recognize the repeated use of the distributive property in factoring and expanding linear expressions (7.EE.A.1). Students examine situations where more than one form of an expression may be used to represent the same context, and they see how looking at each form can bring a new perspective (and thus deeper understanding) to the problem. Students recognize and use the identity properties and the existence of inverses to efficiently write equivalent expressions in standard form (2x + (-2x) + 3 = 0 + 3 = 3)(7.EE.A.2).

Lessons

• Lesson 1: Generating Equivalent Expressions *Priority Lesson*

• Lesson 2: Generating Equivalent Expressions (Continued) *Priority Lesson*

• Lesson 3: Writing Products as Sums and Sums as Products *Priority Lesson*

• Lesson 4: Writing Products as Sums and Sums as Products (Continued)

• Lesson 5: Using the Identity and Inverse to Write Equivalent Expressions *Priority Lesson*

• Lesson 6: Collecting Rational Number Like Terms

Focus Standards

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.

• 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?

• b. Solve word problems leading to inequalities of the form 𝑝𝑥 + 𝑞 > 𝑟 and 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

7.G.B.5 Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and use them to solve simple equations for an unknown angle in a figure

Overview

In Topic B, students use linear equations and inequalities to solve problems. They continue to use bar diagrams from earlier grades where they see fit but will quickly discover that some problems would more reasonably be solved algebraically (as in the case of large numbers). Guiding students to arrive at this realization on their own develops the need for algebra. This algebraic approach builds upon work in Grade 6 with equations (6.EE.B.6, 6.EE.B.7) to now include multi-step equations and inequalities containing rational numbers (7.EE.B.3, 7.EE.B.4). Students solve problems involving consecutive numbers, total cost, age comparisons, distance/rate/time, area and perimeter, and missing angle measures. Solving equations with a variable is all about numbers, and students are challenged with the goal of finding the number that makes the equation true. When given in context, students recognize that a value exists, and it is simply their job to discover what that value is. Even the angles in each diagram have a precise value, which can be checked with a protractor to ensure students that the value they find does indeed create a true number sentence

Lessons

• Lesson 7: Understanding Equations *Priority Lesson*

• Lesson 8: Using If-Then Moves in Solving Equations *Priority Lesson*

• Lesson 9: Using If-Then Moves in Solving Equations (Continued)

• Lesson 10: Angle Problems and Solving Equations *Priority Lesson*

• Lesson 11: Angle Problems and Solving Equations (Continued)

• Lesson 12: Properties of Inequalities *Priority Lesson*

• Lesson 13: Inequalities *Priority Lesson*

• Lesson 14: Solving Inequalities *Priority Lesson*

• Lesson 15: Graphing Solutions to Inequalities

Focus Standards

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.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.

Overview

In Topic C, Students continue work with geometry as they use equations and expressions to study area, perimeter, surface area, and volume. This final topic begins by modeling a circle with a bicycle tire and comparing its perimeter (one rotation of the tire) to the length across (measured with a string) to allow students to discover the most famous ratio of all, pi. Activities in comparing circumference to diameter are staged precisely for students to recognize that this symbol has a distinct value and can be approximated by 22/7 or 3.14 to give students an intuitive sense of the relationship that exists. In addition to representing this value with the pi symbol, the fraction and decimal approximations allow for students to continue to practice their work with rational number operations. All problems are crafted in such a way to allow students to practice skills in reducing within a problem, such as using 22/7 for finding circumference with a given diameter length of 14 cm, and recognize what value would be best to approximate a solution. This understanding allows students to accurately assess work for reasonableness of answers. After discovering and understanding the value of this special ratio, students will continue to use pi as they solve problems of area and circumference (7.G.B.4).

In this topic, students derive the formula for area of a circle by dividing a circle of radius r into pieces of pi and rearranging the pieces so that they are lined up, alternating direction, and form a shape that resembles a rectangle. This “rectangle” has a length that is 1/2 the circumference and a width of r. Students determine that the area of this rectangle (reconfigured from a circle of the same area) is the product of its length and its width: 1/2(C)(r) = 1/2 2(pi)(r)(r) = pi(r)2 (7.G.B.4). The precise definitions for diameter, circumference, pi, and circular region or disk will be developed during this topic with significant time being devoted to student understanding of each term.

Lessons

• Lesson 16: The Most Famous Ratio of All *Priority Lesson*

• Lesson 17: The Area of a Circle *Priority Lesson*

• Lesson 18: More Problems on Area and Circumference *Priority Lesson*

• Lesson 19: Unknown Area Problems on the Coordinate Plane

• Lesson 20: Composite Area Problems *Priority Lesson*

• Lesson 21: Surface Area *Priority Lesson*

• Lesson 22: Surface Area (Continued)

• Lesson 23: The Volume of a Right Prism *Priority Lesson*

• Lesson 24: The Volume of a Right Prism (Continued)

• Lesson 25: Volume and Surface Area *Priority Lesson*

• Lesson 26: Volume and Surface Area (Continued)

Focus Standards

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 1 2 1 4 ⁄ miles per hour, equivalently 2 miles per hour.

7.RP.A.2 Recognize and represent proportional relationships between quantities.

• 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.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.

Overview

Topic A builds on students’ conceptual understanding of percent from Grade 6 (6.RP.3c), and relates 100% to “the whole.” Students represent percents as decimals and fractions and extend their understanding from Grade 6 to include percents greater than 100%, such as 225%, and percents less than 1%, such as 1/2% or 0.5%. They understand that, for instance, 225% means 225/100 , or equivalently, 2.25/1 = 2.25 (7.RP.A.1). Students use complex fractions to represent non-whole number percents.

In Grade 7, they write equations to solve multi-step percent problems and relate their conceptual understanding to the representation: Quantity = Percent × Whole (7.RP.A.2c). Students solve percent increase and decrease problems with and without equations (7.RP.A.3). For instance, given a multi-step word problem where there is an increase of 20% and “the whole” equals $200, students recognize that $200 can be multiplied by 120% or 1.2 to get an answer of $240. They use visual models, such as a double number line diagram, to justify their answers. In this case, 100% aligns to $200 in the diagram and intervals of fifths are used (since 20% = 1/5) to partition both number line segments to create a scale indicating that 120% aligns to $240. Topic A concludes with students representing 1% of a quantity using a ratio, and then using that ratio to find the amounts of other percents. While representing 1% of a quantity and using it to find the amount of other percents is a strategy that will always work when solving a problem, students recognize that when the percent is a factor of 100, they can use mental math and proportional reasoning to find the amount of other percents.

Lessons

• Lesson 1: Percent *Priority Lesson*

• Lesson 2: Part of a Whole as a Percent *Priority Lesson*

• Lesson 3: Comparing Quantities with Percent *Priority Lesson*

• Lesson 4: Percent Increase and Decrease *Priority Lesson*

• Lesson 5: Finding One Hundred Percent Given Another Percent

• Lesson 6: Fluency with Percents

Focus Standards

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 1 2 1 4 ⁄ miles per hour, equivalently 2 miles per hour.

7.RP.A.2 Recognize and represent proportional relationships between quantities.

• 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.

• b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

• 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.

• 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.

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.

Overview

In Topic B, students create algebraic representations and apply their understanding of percent from Topic A to interpret and solve multi-step word problems related to markups or markdowns, simple interest, sales tax, commissions, fees, and percent error (7.RP.A.3, 7.EE.B.3). They apply their understanding of proportional relationships from Module 1, creating an equation, graph, or table to model a tax or commission rate that is represented as a percent (7.RP.A.1, 7.RP.A.2). Students solve problems related to changing percents and use their understanding of percent and proportional relationships to solve the following: A soccer league has 300 players, 60% of whom are boys. If some of the boys switch to baseball, leaving only 52% of the soccer players as boys, how many players remain in the soccer league? Students determine that, initially, 100% = 60% = 40% of the players are girls and 40% of 300 equals 120. Then, after some boys switched to baseball, 100% - 52% of the soccer players are girls, so 0.48p = 120, or p = 120/0.48. Therefore, there are now 250 players in the soccer league.

Lessons

• Lesson 7: Markup and Markdown Problems *Priority Lesson*

• Lesson 8: Percent Error Problems *Priority Lesson*

• Lesson 9: Problem Solving When the Percent Changes

• Lesson 10: Simple Interest *Priority Lesson*

• Lesson 11: Tax, Commissions, Fees, and Other Real-World Percent Problems

Focus Standards

7.RP.A.2 Recognize and represent proportional relationships between quantities. b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

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

Overview

Students revisit scale drawings in Topic C to solve problems in which the scale factor is represented by a percent (7.RP.A.2b, 7.G.A.1). They understand from their work in Module 1, for example, that if they have two drawings where if Drawing 2 is a scale model of Drawing 1 under a scale factor of 80%, then Drawing 1 is also a scale model of Drawing 2, and that scale factor is determined using inverse operations. Since 80% = 4/5, the scale factor is found by taking the complex fraction (1)/(4/5), or 5/4, and multiplying it by 100%, resulting in a scale factor of 125%. As in Module 1, students construct scale drawings, finding scale lengths and areas given the actual quantities and the scale factor (and vice-versa); however, in this module the scale factor is represented as a percent. Students are encouraged to develop multiple methods for making scale drawings. Students may find the multiplicative relationship between figures; they may also find a multiplicative relationship among lengths within the same figure.

Lessons

• Lesson 12: The Scale Factor as a Percent for a Scale Drawing *Priority Lesson*

• Lesson 13: Changing Scales *Priority Lesson*

• Lesson 14: Computing Actual Lengths from a Scale Drawing *Priority Lesson*

• Lesson 15: Solving Area Problems Using Scale Drawings *Priority Lesson*

Focus Standards

7.RP.A.2 Recognize and represent proportional relationships between quantities.

• 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.3 Use proportional relationships to solve multi-step ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.

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.

Overview

The problem-solving material in Topic D provides students with further applications of percent and exposure to problems involving population, mixtures, and counting, in preparation for later topics in middle school and high school mathematics and science. Students apply their understanding of percent (7.RP.A.2c, 7.RP.A.3, 7.EE.B.3) to solve word problems in which they determine, for instance, when given two different sets of 3-letter passwords and the percent of 3-letter passwords that meet a certain criteria, which set is the correct set. Or, given a 12-gallon mixture that is 40% pure juice, students determine how many gallons of pure juice must be added to create a 5-gallon mixture that is 20% pure juice by writing and solving the equation 0.2(5) + j = 0.4(12), where j is the amount of pure juice added to the original mixture.

Lessons

• Lesson 16: Population Problems *Priority Lesson*

• Lesson 17: Mixture Problems

• Lesson 18: Counting Problems *Priority Lesson*

Focus Standards

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 is 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.

• 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.

• 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?

Overview

In Topics A and B, students learn to interpret the probability of an event as the proportion of the time that the event will occur when a chance experiment is repeated many times (7.SP.C.5). They learn to compute or estimate probabilities using a variety of methods, including collecting data, using tree diagrams, and using simulations.

In Topic A, students begin a study of basic probability concepts (7.SP.C.5). They are introduced to the idea of a chance experiment and how probability is a measure of how likely it is that an event will occur. Working with spinners and other chance experiments, students estimate probabilities of outcomes (7.SP.C.6)

Lessons

• Lesson 1: Chance Experiments *Priority Lesson*

• Lesson 2: Estimating Probabilities by Collecting Data

• Lesson 3: Chance Experiments with Equally Likely Outcomes *Priority Lesson*

• Lesson 4: Calculating Probabilities for Chance Experiments with Equally Likely Outcomes *Priority Lesson*

• Lesson 5: Chance Experiments with Outcomes That Are Not Equally Likely *Priority Lesson*

• Lesson 6: Using Tree Diagrams Representing a Sample Space to Calculate Probabilities *Priority Lesson*

• Lesson 7: Calculating Probabilities of Compound Events *Priority Lesson*

Focus Standards

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.

• 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.

• 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.

• 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?

Overview

In Topic B, students move to comparing probabilities from simulations to computed probabilities that are based on theoretical models (7.SP.C.6, 7.SP.C.7). They calculate probabilities of compound events using lists, tables, tree diagrams, and simulations (7.SP.C.8). They learn to use probabilities to make decisions and to determine whether or not a given probability model is plausible (7.SP.C.7).

Lessons

• Lesson 8: The Difference Between Theoretical Probabilities and Estimated Probabilities *Priority Lesson*

• Lesson 9: Comparing Estimated Probabilities to Probabilities Predicted by a Model

• Lesson 10: Conducting a Simulation to Estimate the Probability of an Event *Priority Lesson*

• Lesson 11: Conducting a Simulation to Estimate the Probability of an Event (Continued) *Priority Lesson*

• Lesson 12: Applying Probability to Make Informed Decisions

Focus Standards

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.

Overview

In Topics C and D, students focus on using random sampling to draw informal inferences about a population (7.SP.A.1, 7.SP.A.2). In Topic C, they investigate sampling from a population (7.SP.A.2). They learn to estimate a population mean using numerical data from a random sample (7.SP.A.2). They also learn how to estimate a population proportion using categorical data from a random sample.

Lessons

• Lesson 13: Populations, Samples, and Generalizing from a Sample to a Population *Priority Lesson*

• Lesson 14: Selecting a Sample

• Lesson 15: Random Sampling *Priority Lesson*

• Lesson 16: Methods for Selecting a Random Sample *Priority Lesson*

• Lesson 17: Sampling Variability *Priority Lesson*

• Lesson 18: Sampling Variability and the Effect of Sample Size *Priority Lesson*

• Lesson 19: Understanding Variability When Estimating a Population Proportion

• Lesson 20: Estimating a Population Proportion *Priority Lesson*

Focus Standards

7.SP.B.3 Informally assess the degree of visual overlap of two numerical data distributions with similar variability, 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.

Overview

In Topic D, students learn to compare two populations with similar variability. They learn to consider sampling variability when deciding if there is evidence that the means or the proportions of two populations are actually different (7.SP.B.3, 7.SP.B.4).

Lessons

• Lesson 21: Why Worry About Sampling Variability?

• Lesson 22: Comparing Two or More Populations with Sample Data *Priority Lesson*

• Lesson 23: Comparing Two or More Populations with Sample Data

Focus Standards

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.

Overview

In Topic A, students solve for unknown angles. The supporting work for unknown angles began in Grade 4, Module 4 (4.MD.C.5–7), where all of the key terms in this Topic were first defined, including: adjacent, vertical, complementary, and supplementary angles, angles on a line, and angles at a point. In Grade 4, students used those definitions as a basis to solve for unknown angles by using a combination of reasoning (through simple number sentences and equations), and measurement (using a protractor). For example, students learned to solve for a missing angle in a pair of supplementary angles where one angle measurement is known.

In Grade 7, Module 3, students studied how expressions and equations are an efficient way to solve problems. Two lessons were dedicated to applying the properties of equality to isolate the variable in the context of missing angle problems. The diagrams in those lessons were drawn to scale to help students more easily make the connection between the variable and what it actually represents. Now in Module 6, the most challenging examples of unknown angle problems (both diagram-based and verbal) require students to use a synthesis of angle relationships and algebra. The problems are multi-step, requiring students to identify several layers of angle relationships and to fit them with an appropriate equation to solve. Unknown angle problems show students how to look for, and make use of, structure (MP.7). In this case, they use angle relationships to find the measurement of an angle.

Lessons

• Lesson 1: Complementary and Supplementary Angles *Priority Lesson*

• Lesson 2: Solving for Unknown Angles Using Equations *Priority Lesson*

• Lesson 3: Solving for Unknown Angles Using Equations (Continued...)

• Lesson 4: Solving for Unknown Angles Using Equations (Continued) *Priority Lesson*

Focus Standards

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.

Overview

In Topic B, students work extensively with a ruler, compass, and protractor to construct geometric shapes, mainly triangles (7.G.A.2). The use of a compass is new (e.g., how to hold it, and to how to create equal segment lengths). Students use the tools to build triangles, provided given conditions, such side length and the measurement of the included angle (MP.5). Students also explore how changes in arrangement and measurement affect a triangle, culminating in a list of conditions that determine a unique triangle. Students understand two new concepts about unique triangles. They learn that under a condition that determines a unique triangle: (1) a triangle can be drawn and (2) any two triangles drawn under the condition will be identical. It is important to note that there is no mention of congruence in the CCSS until Grade 8, after a study of rigid motions. Rather, the focus of Topic B is developing students’ intuitive understanding of the structure of a triangle. This includes students noticing the conditions that determine a unique triangle, more than one triangle, or no triangle (7.G.A.2). Understanding what makes triangles unique requires understanding what makes them identical.

Lessons

• Lesson 5: Identical Triangles *Priority Lesson*

• Lesson 6: Drawing Geometric Shapes

• Lesson 7: Drawing Parallelograms

• Lesson 8: Drawing Triangles *Priority Lesson*

• Lesson 9: Conditions for Unique Triangle―Three Sides Two Sides and Included Angle *Priority Lesson*

• Lesson 10: Conditions for a Unique Triangle―Two Angles and a Given Side *Priority Lesson*

• Lesson 11: Conditions on Measurements That Determine a Triangle *Priority Lesson*

• Lesson 12: Unique Triangles—Two Sides and a Non-Included Angle *Priority Lesson*

• Lesson 13: Checking for Identical Triangles *Priority Lesson*

• Lesson 14: Checking for Identical Triangles (Continued)

• Lesson 15: Using Unique Triangles to Solve Real-World and Mathematical Problems

Focus Standards

Overview

Topic C introduces the idea of a slice (or cross section) of a three-dimensional figure. Students explore the two-dimensional figures that result from taking slices of right rectangular prisms and right rectangular pyramids parallel to the base, parallel to a lateral face, and slices that are not parallel to the base nor lateral face, but are skewed slices (7.G.A.3). The goal of the first three lessons is to get students to consider three-dimensional figures from a new perspective. One way students do this is by experimenting with an interactive website which requires students to choose how to position a three-dimensional figure so that a slice yields a particular result (e.g., how a cube should be sliced to get a pentagonal cross section).

Lessons

• Lesson 16: Slicing a Right Rectangular Prism with a Plane *Priority Lesson*

• Lesson 17: Slicing a Right Rectangular Pyramid with a Plane *Priority Lesson*

• Lesson 18: Slicing on an Angle

• Lesson 19: Understanding Three-Dimensional Figures *Priority Lesson*

Focus Standards

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.

Overview

Similar to Topic A, the subjects of area, surface area, and volume in Topics D and E are not new to students, but provide opportunities for students to expand their knowledge by working with challenging applications. In Grade 6, students verified that the volume of a right rectangular prism is the same whether it is found by packing it with unit cubes or by multiplying the edge lengths of the prism (6.G.A.2). In Grade 7, the volume formula V = bh, where b represents the area of the base, will be tested on a set of three-dimensional figures that extends beyond right rectangular prisms to right prisms in general. In Grade 6, students practiced composing and decomposing two-dimensional shapes into shapes they could work with to determine area (6.G.A.1). Now, they learn to apply this skill to volume as well. The most challenging problems in these topics are not pure area or pure volume questions, but problems that incorporate a broader mathematical knowledge such as rates, ratios, and unit conversion. It is this use of multiple skills and contexts that distinguishes real-world problems from purely mathematical ones (7.G.B.6).

Lessons

• Lesson 20: Real-World Area Problems *Priority lesson*

• Lesson 21: Mathematical Area Problems *Priority lesson*

• Lesson 22: Area Problems with Circular Regions

• Lesson 23: Surface Area: 3D *Priority lesson*

• Lesson 24: Surface Area: 3D (Continued)

Focus Standards

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.

Overview

Students explore volume of three-dimensional figures. Students use the volume formula 𝑉 = 𝐵ℎ, where 𝐵 represents the area of the base, to test on a set of three-dimensional figures that extends beyond right rectangular prisms to right prisms in general.

Lessons

• Lesson 25: Volume of Right Prisms *Priority lesson*

• Lesson 26: Volume of Composite Three-Dimensional Objects *Priority lesson*

• Lesson 27: Real-World Volume Problems