Unit 4, Lesson 16
Solving Problems with Systems of Equations
Grade 8
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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 to choose their answer, encourage them to explain how each part of the expression matches the context (MP3).
Give students 2 minutes of quiet work time, and follow that with a whole-class discussion.
A car is driving toward home at 0.5 miles per minute. If the car is 4 miles from home at , which of these expressions can represent the distance that the car has left to drive?
Invite students to share their selections and 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.
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. Although 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 in which students reason about the problems with and without systems of equations. Identify some groups with particularly compelling or clear reasoning, and ask them to share later.
Arrange students in groups of 2. Provide tools for creating a visual display.
After students have completed the first three problems, select previously identified groups to share their solutions. Bring out why these solutions are particularly good (for example, greater clarity or 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, in which they write their own problem to trade with another group.
Solve each problem. Explain or show your reasoning.
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). To help students understand the relationships between variables, have them explain why some values are not possible.
Most of the discussions happen within and between groups, but the last question requires a whole-class discussion. Have each group share the peer-generated question that they were assigned and its solution. Although the group that wrote the question will be responsible for confirming the answer, encourage all students to listen to the reasoning that 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 problems. Ask students to look for situations similar to theirs and to identify the most common solution methods used. After the gallery walk, select a few groups to share a problem and how they solved it.