Lesson 16

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). 

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

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.  

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.

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.

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.

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 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.