# Lesson 16

Rewriting Equations for Perspectives

These materials, when encountered before Algebra 1, Unit 4, Lesson 16 support success in that lesson.

## 16.1: No Bad Apples (10 minutes)

### Warm-up

In this warm-up activity students weigh 2 options for purchasing apples with different deals. Either choice can be argued, but they should be supported with some mathematical reasoning. One option is a slightly better deal, but a small savings may not be worth compromising other variables for some students.

Students should construct viable arguments and critique the reasoning of others (MP3) when they explain their reasoning and discuss the reasoning of others.

In the other activities for this lesson, students consider the cost of buying several of the same items. This warm-up engages student thinking about purchases.

### Student Facing

Which option would you select? Use mathematical reasoning to explain your selection.

Option A: Each apple costs $0.97 and are on sale with a “Buy 2, Get 1 Free” offer.

Option B: Bags of 6 apples are on sale “2 for $7.50” but you must buy 2 bags.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

The purpose of the discussion is to compare methods for using mathematical reasoning to select a better option. Select students to share their preferred option and the mathematical reasoning. After each choice is shared, ask if there are any other ways students thought about the choice or other options selected. If all students choose the same option, ask if there are situations for which it might make sense for someone to choose the other option.

## 16.2: A Charity Shopping Trip (20 minutes)

### Activity

In this activity students match descriptions of situations to linear equations that could represent the situation. In the associated Algebra 1 lesson students begin by writing an equation to represent a situation then write the inverse based on the situation. Students are supported by matching equations to the situation in this activity so they can focus on the inverse as a new skill in the Algebra 1 lesson.

In this partner activity, students take turns matching descriptions of situations to an equation representing the situation. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3).

### Launch

Arrange students in groups of 2. Tell students that for each description in column A, one partner finds an equation in column B that represents the situation and explains why they think it is a match. The partner's job is to listen and make sure they agree. If they don't agree, the partners discuss until they come to an agreement. For the next description in column A, the students swap roles. If necessary, demonstrate this protocol before students start working.

### Student Facing

A person has collected a lot of money for providing clothing to those in need. They go to a store to buy several clothing items with the money collected.

Match each description in column A with an equation from column B that represents the situation. Be prepared to explain your reasoning.

- Take turns with your partner to match a description of a situation with an equation that represents the situation.
- For each match that you find, explain to your partner how you know it’s a match.
- For each match that your partner finds, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.

- A store charges $6 for each shirt sold. A person buys \(x\) shirts and pays \(y\) dollars for the total.
- A store charges $6 for each pair of shorts sold. They also offer a $3 coupon to be used on the entire order. A person buys \(x\) pairs of shorts and pays \(y\) dollars for the total after using the coupon.
- A store charges $6 for 3 pairs of socks. A person buys \(x\) pairs of socks and pays \(y\) dollars for the total.
- A store charges $6 for each pair of shoes sold and also charges $3 to lace up all of the shoes in the entire order. A person buys \(x\) pairs of shoes and pays \(y\) for the total including lacing up all the shoes.
- A store charges $3 for 6 handkerchiefs. A person buys \(x\) handkerchiefs and pays \(y\) for the total.
- A store charges $3 for each pair of gloves sold. They also offer a $6 coupon to be used on the entire order when there are more than 4 pairs of gloves purchased. A person buys \(x\) pairs of gloves (with \(x > 4\)) and pays \(y\) dollars for the total after using the coupon.

- \(y = 6x\)
- \(y = \frac{6x}{3}\)
- \(y = \frac{3x}{6}\)
- \(y = 3x - 6\)
- \(y = 6x - 3\)
- \(y = 6x + 3\)

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

Much discussion takes place between partners. Invite students to share how they connected the representations of the situation. Ask students,

- “How did you determine what would be the coefficient of \(x\) in the equations?” (Because each situation involved \(x\) items being purchased, the cost for each item is multiplied by \(x\).)
- “The equations are written from the store’s perspective (‘How much should we charge this person for all of these items?’). From the buyer’s perspective, it may be better to think of it as something like, ‘I have $120 to spend on shirts. How many shirts can I get?’ How can you use the equation you matched with the shirts equation to answer this question?” (I can substitute $120 for \(y\) in the equation to get \(120 = 6x\). Then I can solve that equation to know that the person can buy 20 shirts.)
- “The equations \(y = 6x - 3\) and \(y = 6x + 3\) are somewhat similar. What in the situation determines the difference between them?” (Since \(y\) represents the total cost to the customer, 3 is subtracted when there is a discount or coupon being used, and 3 is added when there is an extra expense.)
- “Describe any difficulties you experienced and how you resolved them.” (I originally thought about the situation with the socks as being $2 for each pair of socks, but I didn’t see any equations involving the number 2. I then realized that the equation was written to include the information about $6 for 3 pairs.)

## 16.3: Isolate the $x$ (10 minutes)

### Activity

In this activity students rearrange equations to isolate a variable. In the associated Algebra 1 lesson students rewrite equations to find the inverse function. This activity allows students to focus on the mechanics of the rearrangement so they can consider the solutions in broader context for the Algebra 1 lesson.

### Launch

If the question from the previous activity synthesis was not asked about perspective, present the question to students now: “The equations are written from the store’s perspective (‘How much should we charge this person for all of these items?’). From the buyer’s perspective, it may be better to think of it as something like, ‘I have $120 to spend on shirts. How many shirts can I get?’ How can you use the equation you matched with the shirts equation to answer this question?” (I can substitute $120 for \(y\) in the equation to get \(120 = 6x\). Then I can solve that equation to know that the person can buy 20 shirts.)

Tell students that “Each equation is written from the store’s perspective to answer ‘How much should the total (\(T\)) be for this customer buying \(x\) items?’ From the customer’s perspective, it is more useful to know how many items can be purchased with the money they have.”

### Student Facing

Rearrange the equations so that one side of the equation is only \(x\). Be prepared to explain or show your reasoning.

- \(T = x - 2\)
- \(T = 2x\)
- \(T = 2x - 1\)
- \(T = \frac{x}{2}\)
- \(T = 2(x-1)\)
- \(T = \frac{x-1}{2}\)

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

The purpose of the discussion is to share the methods students use to rearrange the equation. Select students to share the solutions, including any intermediate they used to arrive at the solution. After each solution shared, ask if there are other methods students used to arrive at the solution. For example, with the equation \(T = 2(x-1)\) some students may distribute the 2 first and then work to isolate \(x\) while others may immediately divide by 2 then solve for \(x\).

If there is additional time, these questions can help students connect their solutions to the context:

- “One of these equations represents a situation in which each customer gets 1 item free then must purchase additional items for $2 each. Which equation matches that situation?” (\(T = 2(x-1)\))
- “Using the equation \(T = \frac{x}{2}\), how much does each item cost?” (50 cents since the equation can also be written as \(T = \frac{1}{2}x\) which means it costs half of a dollar for each item.)
- “The equation \(T = x - 2\) might represent a situation in which each item costs $1 and the customer has a $2 coupon for the total order. In terms of the situation, explain the meaning of the addition in the rewritten equation \(x = T+2\).” (The number of items the person can purchase is equal to the number of dollars they have plus 2 additional items they can get from the coupon.)