Lesson 16

The purpose of this warm-up is for students to remember that unit price can be used to figure out which price option is a better deal and also how to compute unit price. When students explain their reasoning, they may engage in constructing arguments and critiquing the reasoning of their classmates (MP3). The question, “Which one would you choose?” is purposefully asked because there is not one correct answer. While there is a choice that is a better deal, that is not the question. In defending their reasoning, students may have other reasons for their choice based on how they make sense of the context. For example, students might reason that a 5-gallon container is easier to store, or that 3 1-gallon containers are easier to share, or they might reject both options because they don't like honey.

As students work, listen for reasoning about the problem in different ways to share in the whole-class discussion. Choose at least one student who reasoned by finding the unit cost.

Arrange students in groups of 2. Give 1 minute of quiet think time followed by 1 minute to discuss their responses with a partner. Follow with a whole-class discussion.

Students may choose the 5 pound jug because they assume that larger quantities are a better deal. Ask if they can be more precise in their reasoning and defend it mathematically.

Poll the class to find who selected each of the options. Ask selected students to share their reasoning. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate ___’s reasoning in a different way?”
• “Does anyone want to add on to _____’s reasoning?”
• “Do you agree or disagree? Why?”

This activity helps students recall what they know about solving problems in a ratio context, and then extends their thinking to consider writing two equations that relate the two quantities in the ratio and representing them with graphs.

For students who struggle to write an equation that relates the two quantities, help them to represent the situation in a concrete way, like with a tape diagram or a discrete diagram. You can also draw their attention to the completed table: “What can you do to each number in the \(r\) column to get the number in the \(y\) column?” Expressing the relationship in words can be a helpful step to expressing it with an equation. 

The discussion should focus on how all the representations (tables, equations, graphs) capture the same information but look at it in a different way.

Some guiding questions:

• “Why is it possible to write two different equations to describe the same situation?” (You can choose either quantity to be the dependent variable.)
• “What do you notice about the numbers that multiply by the independent variable in each equation?” (They are reciprocals, they represent the two unit rates for the situation.)
• “Are there other equations we could have written that describe this situation?” (Yes, for example, we can write equations to describe the relationship between the amount of yellow paint and the total number of cups: \(y=\frac35 t\) and \(t=\frac53 y\))
• “What do you notice about the similarities and differences between the graphs?” (Both have points that appear to lie on a line, both lines go up as you move to the right, the line for Lin’s equation slants up more, the coordinates of the points are reversed.)