Lesson 16
Solving Problems with Systems of Equations
16.1: Are We There Yet? (5 minutes)
Warmup
The purpose of this warmup 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 wholeclass 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?

\(0.5t\)

\(4+0.5t\)

\(4  0.5t\)

\(4\boldcdot (0.5t)\)
Student Response
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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.
16.2: 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.
Supports accessibility for: Organization; Attention
Design Principle(s): Cultivate conversation; Maximize metaawareness
Student Facing
Solve each problem. Explain or show your reasoning.
 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?
 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?
 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?
 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.
 Create a visual display that includes:

The new problem you wrote, without the solution.

Enough work space for someone to show a solution.


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.

When the group that got the problem you invented shares their solution, check that their answer is correct.
Student Response
For access, consult one of our IM Certified Partners.
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?
Student Response
For access, consult one of our IM Certified Partners.
Anticipated Misconceptions
If students struggle to write a system of equations, ask them to identify any unknown quantities in the problem and assign variables to them. Then ask them if there are ways to describe the relationships between the variables. If students still struggle to think about the relationships, ask them about some possible values for each of the variables including some that make sense (such as 20 grapefruits) and some that do not (such as 1,000 grapefruits). Have students explain why some values are not possible to help them understand the relationships between variables.
Activity Synthesis
Most of the discussions happen within and between groups, but the last question requires a wholeclass discussion. Have each group share the peergenerated question they were assigned and the solution. Though the group that wrote the question will be responsible for confirming the answer, encourage all to listen to the reasoning each group used.
Alternatively, after groups have checked the work of the group that solved their problem, have students complete a gallery walk to see all the created problem. Ask students to look for situations similar to theirs and to identify the most common solution methods used. After the gallery, select a few groups to share a problem and how they solved it.