In this activity, students find patterns in situations to connect to the distributive property (MP7). These patterns build understanding of the equations and  . Students should see that the expressions are all representations of the same thing. Students learn that multiplying a number by and adding that product to the original number is the same as multiplying by 1.5. This idea will be extended to percentages later.

Monitor for students who use these different strategies to make sense of the problem:

• Create a drawing or diagram.
• Write addition expressions, like and .
• Write multiplication expressions, like and .
• Identify the table as representing a proportional relationship and the constant of proportionality as (or equivalent).

Plan to have students present in this order to support moving them from concrete to more abstract representations.

As students calculate the total distance for a variety of initial distances and then the variable , they make use of repeated reasoning (MP8).

The purpose of this discussion is to show where Kiran’s expression comes from. Invite previously selected students to share how they computed the total distance. Sequence the discussion of the strategies in the order listed in the Activity Narrative. If possible, record and display the students’ work for all to see.

Connect the different responses to the learning goals by asking questions, such as:

• “How does ‘half that length again’ show up in each method?”
• “Why do the different approaches lead to the same outcome?”
• “Did anyone solve the problem the same way but would explain it differently?”

The key takeaway is that both and represent the situation. The expressions are equivalent due to the distributive property.

In this partner activity, students take turns matching situations, tape diagrams, and equations. They interpret wording like “ more than that” and figure out how that relationship can be expressed using different mathematical representations. Students may initially think that the wording means ; however, the tape diagram will help them realize that this is not true.

As students analyze different representations, they practice reasoning quantitatively and abstractly (MP2). In making connections across representations, they practice looking for and making use of structure (MP7)

The purpose of this discussion is to make connections between the numbers in the description and the numbers in the equation. Select 2–3 groups to share one of their sets of cards and how they matched the description with a tape diagram and equation. Discuss as many different sets of cards as the time allows.

Consider asking:

• “Which matches were tricky? Explain why.”
• “Did you need to make adjustments in your matches? What might have caused an error? What adjustments were made?”
• “How is the number in the equation related to the number in the description?”
• “What is different between a fractional amount more than the original and a fractional amount less than the original?”

This activity gives students more practice connecting descriptions, tape diagrams, and equations for situations that involve an increase or decrease expressed as a fraction of the initial value. First students match descriptions and tape diagrams. Then they write equations that represent the situations. Finally, they apply their understanding to create their own situations for the tape diagrams that did not have a match.

Monitor for different ways students write the equations, including addition or subtraction and multiplication.

The purpose of this discussion is to emphasize the connection between the numbers in the description and the numbers in the equation. First, ask students to share the equations they wrote for each diagram. To discuss the structure of the equations used, consider asking:

• “How is the number in the equation related to the number in the description?”
• “What is different between a fractional amount more than the original versus less than the original?”

If time permits, invite a few students to share the stories they created for the leftover diagrams.