This activity is an opportunity to talk about subtracting fractions from 1, multiplying by fractions, and the distributive property, using the number line as a representation for sense-making. In the last question, students are asked to express regularity in repeated reasoning (MP8).

Monitor for students who write different equivalent expressions to help them reason. In the synthesis, draw connections between these different expressions.

The goal of this discussion is to generalize the work students did in these problems to help them approach equivalent expressions. Display this diagram, or create a similar one using the numbers from the problems.

Here are some questions for discussion:

• “How is this diagram related to the problems you solved?” (This diagram represents the last question. The whole trip is  miles long, and the driver has gone of the way. The expressions on the longer part represent the distance remaining.)
• “Are the three expressions on the longer part equivalent? Explain your reasoning.” (Yes. The first expression shows the distance traveled, , being subtracted from the whole, . The second expression uses the distributive property to rewrite the first expression. The third expression combines like terms from the earlier expressions.)

This activity is an opportunity to use the properties that come into play in the associated Algebra 1 lesson to practice creating and identifying equivalent expressions. The expressions with exponents allow for review of performing operations in the conventional order (exponent before multiplication), as well as the convention for raising a number to the 0 power. Identifying moments to use these conventions are opportunities to look for and make use of structure (MP7).

The goal is to clarify how to connect expressions written in various ways. Display the correct matches and encourage students to ask questions about any matches that they had a hard time figuring out. Spend a little time analyzing the expressions that use exponents, reminding students that the conventional order is to evaluate the exponent first and then multiply. Students may need help understanding how to evaluate a fraction raised to a power.