solving equations using the distributive...

– grade 7 • Teacher Guide

Solving Equations Using the Distributive Property

LAUNCH (7 MIN) _____________________________________________________________ Before

• What does “per-dozen price” mean? • If you know the price of one bagel, how do you find the price of one dozen bagels?

During • What is the unknown in this situation? • Why can b + 0.05 represent the price of one of the shaky baker’s bagels?

After • Why couldn’t the baker use the expression 12b + 0.05 to represent the per-dozen price?

PART 1 (10 MIN) _____________________________________________________________ During the Intro

• What can you do to get rid of the parentheses when you have an expression that multiplies a number by a sum or difference?

Before solving the Example • What is your goal when solving an equation? How do you achieve this goal? • What is the first step you would use to solve the given equation?

After showing the provided solution • Why can you divide on both sides first, without applying the Distributive Property? • Which of the two solution methods do you prefer? Why? • How do you check your solution?

Sara Says (Screen 2) Use the Sara Says button to point out that there is more than one way to solve this problem. However, distributing may be easier than dividing each side by －8.

PART 2 (8 MIN) _______________________________________________________________ During the Intro

• Why is it a good strategy to analyze an equation before you solve it?

While solving the problem • How can you tell how many operations are needed to solve an equation?

After solving the problem • Did you have to solve each equation to determine its correct bin? Explain.

PART 3 (7 MIN) _______________________________________________________________ Before solving the problem

• What is the total cost, in words, for 1 person?

Sara Says (Screen 1) Use the Sara Says button to point out that the Algebra Tiles tool is impractical in this situation because of the size of the numbers.

While solving the problem • Which quantity should the variable represent? • Which operations act on the variable to give the total cost? Explain.

CLOSE AND CHECK (8 MIN) ______________________________________________

• Think about the airline ticket problem from Example 3. You wrote (167 + 19)n where $167 is the initial cost for each ticket, $19 is the cost for insurance on each ticket, and n = the number of tickets the family buys. Describe what the expression (167 + 19)n means in terms of the problem and how it is related to 167n + 19n.

• How is writing an equation useful when you work with a complex problem?


Solving Equations Using the Distributive Property

LESSON OBJECTIVES 1. 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.

2. Solve word problems leading to equations of the form p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of this form fluently.

FOCUS QUESTION When is it useful to model a situation in two different ways?

MATH BACKGROUND Earlier in this topic, students reviewed writing and solving one-step equations and then learned how to write and solve two-step equations.

In this lesson, students encounter equations of the form p(x + q) = r, where p, q, and r are rational numbers. Students see that there are two methods for solving such an equation: they can use the Distributive Property to write an equivalent two-step equation, or they can divide each side by p to create an equivalent one-step equation. Using either method, students come to realize that solving the equation means writing simpler equivalent equations until the variable is isolated on one side of the equal sign.

In the next lesson, students will consolidate their knowledge of equations by using them to solve a wide variety of real-world problems.

LAUNCH (7 MIN) ____________________________________________________

Objective: Write an expression of the form p(x + q) to represent a quantity.

Author Intent Students are given a real-world problem and write an algebraic expression to represent it. The problem leads to an expression of the form p(x + q). Students will encounter expressions of this form in the equations they solve later in the lesson.

Instructional Design You can use the Words to Expression organizer to help students translate the given information into an expression. Point out that the problem asks for an expression, not an equation. That means students’ answers should not contain an equal sign.

Questions for Understanding Before

• What does “per-dozen price” mean? [It means the price of one dozen (twelve) of the bagels.]

• If you know the price of one bagel, how do you find the price of one dozen bagels? [You multiply the price of one bagel by 12.]

During • What is the unknown in this situation? [the price per bagel charged by the

baker’s rival]

• Why can b + 0.05 represent the price of one of the shaky baker’s bagels? [If b is the price in dollars of one of the rival’s bagels, then b + 0.05 is the price in

Solving Equations Using the Distributive Property continued


dollars of one of the shaky baker’s bagels. This is because the shaky baker wants to charge 5¢, or $.05, more per bagel than his rival.]

After • Why couldn’t the shaky baker use the expression 12b + 0.05 to represent the

per-dozen price? [That expression would mean the baker is charging 5¢ more than his rival for one dozen bagels, rather than 5¢ more for each bagel.]

Solution Notes Students may write the expression 12(b + 0.05) and then use the Distributive Property to rewrite it as 12b + 0.6. Either form of the expression is acceptable.

Some students may represent 5¢ as 5 rather than 0.05, so that the expression they write is 12(b + 5). Explain that this approach is acceptable, but then the variable b represents the cost of one of the rival’s bagels in cents rather than dollars. Evaluating the expression will also give a per-dozen price in cents. Use this opportunity to emphasize that when students evaluate expressions or solve equations, they must interpret the solution in the same way they interpreted the problem when they wrote the expression or equation.

Connect Your Learning Move to the Connect Your Learning screen. Start a conversation about when students last saw expressions with parentheses similar to the expression in the Launch. (Students expanded such expressions in the previous topic, and also wrote such expressions when factoring.) Ask if they remember what properties they used to write equivalent expressions. Discuss how they converted between expressions written as the product of two factors and expressions written as a sum or difference of two terms.

PART 1 (10 MIN) _____________________________________________________

Objective: Solve equations of the form p(x + q) = r, where p, q, and r are specific rational numbers.

Author Intent The Intro reviews the Distributive Property. In the Example, students solve an equation containing an expression that multiplies a number by a sum. They may apply the Distributive Property to transform the equation into a two-step equation, or they may first use division on both sides to transform the equation into a one-step equation. The Example reinforces the idea that many equations can be solved in multiple ways as long as number properties and properties of equality are correctly applied.

Questions for Understanding During the Intro

• What can you do to get rid of the parentheses when you have an expression that multiplies a number by a sum or difference? [Sample answer: I can use the Distributive Property to multiply the number outside the parentheses by each number inside the parentheses.]

Before solving the Example • What is your goal when solving an equation? How do you achieve this goal?

[Sample answer: My goal is to isolate the variable on one side of the equation. I can achieve this goal by applying number properties and properties of equality.]

• What is the first step you would use to solve the given equation? [Sample answer: I would use the Distributive Property to expand the left side. Sample answer: I would divide each side by －8.]



After showing the provided solution • Why can you divide on both sides first, without applying the Distributive

Property? [The Division Property of Equality allows you to divide each side of an equation by the same nonzero number.]

• Which of the two solution methods do you prefer? Why? [Sample answer: I prefer dividing first because it takes fewer steps to get to the solution. Sample answer: I prefer using the Distributive Property because the numerical computations are easier.]

• How do you check your solution? [You substitute the solution for the variable in the original equation and then simplify to make sure you get a true equation.]

Sara Says (Screen 2) Use the Sara Says button to point out that there is more than one way to solve this problem. However, distributing may be easier than dividing each side by －8.

Solution Notes The animated solution is an efficient use of class time and reinforces the idea that there is often more than one way to solve a problem. Point out that in both solution methods, students write equivalent equations using properties that keep the equation balanced.

Differentiated Instruction To help students understand how to use division first (instead of the Distributive Property), you can draw a box around the expression in parentheses to help them consider this expression as one quantity. Once they divide on both sides, they will be able to see that this quantity is part of the new one-step equation.

Error Prevention If students mistakenly expand the expression －8(x + 0.3) as －8x + 0.3, draw arrows from －8 to each term in parentheses to remind students that they must distribute －8 to both terms inside the parentheses.

Got It Notes Unlike the equation in the Example, this equation contains a fraction, and the coefficient of the variable inside the parentheses is not 1. The provided solution shows how to solve the equation using the Distributive Property, but students may also solve it by multiplying each side by

43 to eliminate the fraction on the left side.

If you show answer choices, consider the following possible student errors:

Students who choose B may have multiplied each side of the equation by 34 instead

of 43 .

Students who choose C may have correctly gotten to the equation 6b － 3 = －2, but then may have mistakenly divided the term －3 by the coefficient 6 to get the solution.



PART 2 (8 MIN) ______________________________________________________ Objective: Determine how many operations would be needed to solve an equation by analyzing the equation.

Author Intent Students examine a variety of equations and identify the number of operations needed to solve each one. The equations involve rational numbers and include one-step equations, two-step equations, and equations that require more than two steps to solve. Students learn to analyze an equation before solving it so that they can choose an appropriate strategy for isolating the variable.

Instructional Design Show the Intro to motivate why you might want to analyze an equation before solving it. Move to the Example. Call on students to drag each equation from the tile bank to one of the three bins. When all equations have been placed, click on the Check button to check the answers. Any incorrect answer will snap back to the tile bank. Use this as an opportunity to have students explain why the tile belongs in another bin.

Questions for Understanding During the Intro

• Why is it a good strategy to analyze an equation before you solve it? [Sample answer: It helps me know how many operations I should expect to use to solve it.]

While solving the problem • How can you tell how many operations are needed to solve an equation?

[Sample answer: You can count the number of operations that appear in the equation. An equation containing one operation will take one operation to solve, an equation containing two operations will take two operations to solve, and so on.]

After solving the problem • Did you have to solve each equation to determine its correct bin? Explain. [No; I

only had to identify the number of operations that are needed to solve the equation.]

Differentiated Instruction Some students may find it helpful to solve each equation first and then count the number of operations they used. After using this method, these students should examine the equations in each bin to identify what they have in common. They may then be able to see a way to determine the number of operations needed without solving.

Error Prevention Some students may think that they can use two steps to solve the equations with parentheses. Point out that even if they divide both sides first rather than using the Distributive Property, they will need to divide again (after adding or subtracting) because the coefficient of the variable is not 1. This means that they will need more than two operations no matter which method they use.



Got It Notes If you show answer choices, consider the following possible student errors:

Students who choose A, B, or D may think the fractional coefficient in equation I means that more than one operation is required to solve the equation. Convince these students that only one operation is needed by having them multiply each side of equation I by

54 .

PART 3 (7 MIN) ______________________________________________________

Objective: Write and solve equations of the form p(x + q) = r, where p, q, and r are specific rational numbers for real-world situations.

Author Intent Students answer a real-world problem by writing and solving an equation containing an expression that multiplies a number by a sum. Students need to pull together all the skills they have learned so far in the lesson. You may want to show or mimic the provided solution, which puts the expression that multiplies a number by a sum on the right side of the equation.

Instructional Design A Know-Need-Plan organizer is embedded in this Example. Use the organizer to help students identify the information they are given, the information they need to find, and a plan for solving the problem. Ask three students to come to the whiteboard, and have each student fill in one of the three boxes in the organizer.

Questions for Understanding Before solving the problem

• What is the total cost, in words, for 1 person? [the sum of the cost of 1 ticket and the $19 travel insurance]

Sara Says (Screen 1) Use the Sara Says button to point out that the Algebra Tiles tool is impractical in this situation because of the size of the numbers. This is a good opportunity to discuss the limitations of tools and explain that students need to learn how to solve equations without them.

While solving the problem • Which quantity should the variable represent? [The variable should represent

the number of tickets purchased.]

• Which operations act on the variable to give the total cost? Explain. [multiplication and addition; the number of tickets is multiplied by the sum of the ticket price and the cost of travel insurance]

Solution Notes Some students may choose to divide both sides of the equation by 6 instead of distributing the 6 to start. Accept this method as another way to represent the problem, but show the provided solution to compare.

Differentiated Instruction For struggling students: You may wish to use the Know-Need-Plan organizer to help students organize the information they are given, identify what they need to find, and plan an approach for solving the problem.



For advanced students: You may want to encourage some students who start with the equation in the provided solution to rewrite the equation to put the total cost on the right.

Got It Notes As in the Example, here students write and solve an equation to solve a real-life problem. However, the solution to the equation is not the solution to the problem. The Got It helps reinforce the fact that an equation’s solution must always be interpreted in the context of the problem situation.

If you show answer choices, consider the following possible student errors:

Students who choose A did not account for the cost of the first ounce for each invitation.

Students who choose D have switched the cost of the first ounce and the cost of each additional ounce when writing their equation.

CLOSE AND CHECK (8 MIN) _____________________________________

Focus Question Sample Answer Often the equation you write to model a situation can be rewritten as a simpler, equivalent equation. Both the original equation and the rewritten, simpler equation model the situation.

Focus Question Notes More than one equation can represent the same situation. For example, you can write an expression with parentheses so that you can immediately see the two factors. Then you can use the Distributive Property to write the equation in a different way. The new equation has no parentheses, and you see the problem as a sum (or difference) of two terms.

Essential Question Connection This lesson addresses the Essential Question about when it is useful to model a relationship with an equation and how rewriting an equation helps you think about the relationship in a new way. Use the following questions to help students connect this lesson to the Essential Question.

• Think about the airline ticket problem from Example 3. You wrote (167 + 19)n where $167 is the initial cost for each ticket, $19 is the cost for insurance on each ticket, and n = the number of tickets the family buys. Describe what the expression (167 + 19)n means in terms of the problem and how it is related to 167n + 19n. [Sample answer: (167 + 19)n represents the total cost for each ticket times the number of tickets. You can distribute n and write the equivalent expression 167n + 19n, which represents the initial cost for all the tickets plus the total cost for insurance.]

• How is writing an equation useful when you work with a complex problem? [Sample answer: If you can write an equation that models a complicated problem, you represent the situation symbolically, and sometimes it becomes easier to understand. Then you can write simpler equivalent equations using the properties that you know until you eventually solve the problem.]

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