# Lesson 26

Solving Problems with Systems of Equations

## 26.1: Are We There Yet? (5 minutes)

### Warm-up

The purpose of this warm-up is to get students to reason about representing a context about distance as an expression. For students who use the equation $$d = rt$$ to choose their answer, encourage them to explain how each part of the expression matches the context (MP3).

### Launch

Give students 2 minutes of quiet work time followed by a whole-class discussion.

### Student Facing

A car is driving towards home at 0.5 miles per minute. If the car is 4 miles from home at $$t=0$$, which of the following can represent the distance that the car has left to drive?

• $$0.5t$$

• $$4+0.5t$$

• $$4 - 0.5t$$

• $$4\boldcdot (0.5t)$$

### Activity Synthesis

Poll students on which expression they chose. Ask students who chose any expression, right or wrong, to explain their reasoning. After each explanation ask the rest of the class if they agree or disagree and how the context is represented in the expression.

## 26.2: Cycling, Fundraising, Working, and ___? (35 minutes)

### Activity

In this activity, students reason about situations involving two different relationships between the same two quantities. Then they invent their own problem of the same type. While students are encouraged by the language of the activity to use a system of equations to solve the problems, they may elect to use a different representation to explain their thinking (MP1, MP3).

As students work through the first three problems, notice the ways students reason about the problems with and without systems of equations. Identify some groups with particularly compelling or clear reasoning to share later.

### Launch

Arrange students in groups of 2. Provide tools for creating a visual display.

Once students have completed the first three problems, select previously identified groups to share their solutions. Bring out why these solutions are particularly good (i.e., clarity, efficiency), and discuss the connections between them, particularly the connections between groups that did and did not use systems of equations where possible. Next, have students begin the second part of the activity and write their own problem to trade with another group.

Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts. After students have solved the first 2–3 problems, check-in with either select groups of students or the whole class. Invite students to share the strategies they have used so far, as well as any questions they have, before continuing.
Supports accessibility for: Organization; Attention
Representing, Conversing: MLR8 Discussion Supports. Use this routine to help students consider audience when preparing a visual display of their work. When students create their displays for the problems they invent, invite them to consider how to display their strategies so that another student can interpret them. For example, students may wish to add notes or details to their equations to help communicate their thinking. When students receive the problem created by another group, provide 2–3 minutes of quiet think time for students to read and interpret the display before they begin their small-group discussion. During whole-group discussion, ask students to share the kinds of details that were most helpful when interpreting another group's display.
Design Principle(s): Cultivate conversation; Maximize meta-awareness

### Student Facing

Solve each problem. Explain or show your reasoning.

1. Two friends live 7 miles apart. One Saturday, the two friends set out on their bikes at 8 am and started riding towards each other. One rides at 0.2 miles per minute, and the other rides at 0.15 miles per minute. At what time will the two friends meet?
2. Students are selling grapefruits and nuts for a fundraiser. The grapefruits cost $1 each and a bag of nuts cost$10 each. They sold 100 items and made $307. How many grapefruits did they sell? 3. Jada earns$7 per hour mowing her neighbors’ lawns. Andre gets paid $5 per hour for the first hour of babysitting and$8 per hour for any additional hours he babysits. What is the number of hours they both can work so that they get paid the same amount?
4. Pause here so your teacher can review your work. Then, invent another problem that is like one of these, but with different numbers. Solve your problem.
5. Create a visual display that includes:
• The new problem you wrote, without the solution.

• Enough work space for someone to show a solution.

6. Trade your display with another group, and solve each other's new problem. Make sure that you explain your solution carefully. Be prepared to share this solution with the class.

7. When the group that got the problem you invented shares their solution, check that their answer is correct.

### Student Facing

#### Are you ready for more?

On a different Saturday, two friends set out on bikes at 8:00 am and met up at 8:30 am. (The same two friends who live 7 miles apart.) If one was riding at 10 miles per hour, how fast was the other riding?