Lesson 1

The purpose of this warm-up is to elicit the idea that there are different ways to represent a value, which will be useful when students apply the distributive property to write expressions in a later activity. While students may notice and wonder many things about these images, the similarities and differences between the two tape diagrams are the important discussion points.

Arrange students in groups of 2. Tell students that they will look at an image, and their job is to think of at least one thing they notice and at least one thing they wonder. Display the image for all to see. Ask students to give a signal when they have noticed or wondered about something. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After each response, ask the class if they agree or disagree and to explain alternative ways of thinking, referring back to the images each time. If the total length of the first diagram does not come up during the conversation, ask students to discuss this idea.

In this activity, students find patterns in situations to connect to the distributive property (MP7). These patterns build understanding of the equations \(x + 0.5x = (1 + 0.5)x = 1.5x\) or \(x + \frac12x = (1 + \frac12)x = 1\frac12x\). Students should see that the expressions are all representations of the same thing. Students learn that multiplying a number by \(\frac12\) and adding that product to the original numbers is the same as multiplying by 1.5. This idea is extended to percents in the following activity.

As students work on the task, monitor the reasoning they come up with for the question comparing Mai and Kiran's representations (MP3). Identify students who agree with both Mai and Kiran; these students should be chosen to share during the discussion.

Demonstrate, or ask students to demonstrate, walking a certain distance and then walking “half as much again”. Ask students how they could draw a tape diagram to represent the first situation. For example, they could draw something like this.

Give students 5 minutes of quiet work time followed by partner and then whole-class discussion.

Students might struggle coming up with an expression in terms of \(x\). Ask students to describe how you would calculate it in words, then see if they can use that to write the expression.

When comparing Kiran and Mai's equations, even though there is no coefficient written in front of the \(x\)  in Mai's equation, \(x\) is equivalent to \(1x\).

Ask selected students to share who they agree with (Mai or Kiran) and have them explain how they know. Highlight instances when students are reasoning with the distributive property, and encourage them to identify it by name. Suggested questions:

• “Are Mai and Kiran's equations equivalent?”
• “What is the same and what is different about the two equations?”
• “How does each equation represent the “one and a half” relationship between the quantities?”
• “How else could you write an equation to represent the relationship?” (\(y=(1+0.5)x\), \(y = \frac32 x\), \(y = 1x + 0.5x\)) Students should see that the equations express the same relationship and understand why.

The purpose of this activity is to connect various representations of proportional relationships including images, equations, and descriptions. Students first match descriptions of a proportional relationship, involving a variable \(x\), to tape diagrams that represent the situations. Then they create equations that describe the proportion using only variables. Finally, they apply their understanding to create their own scenarios and describe a proportional relationship, which is already represented in the form of a tape diagram.

The focus of the discussion following the activity is on the numbers used to go from between significant quantities in a proportional relationship. For example, what numbers are relevant as we go from the quantity of blueberries eaten by Han to the quantity eaten by Mai?

Arrange students in groups of 2. Give students 1--2 minutes of quiet think time followed by partner discussion on the first problem. Then give students 3--5 minutes to complete the problems. Follow with whole-class discussion.

Students may match the equation \(y = \frac23 x\) with the situation “Mai biked \(x\) miles, and Han biked \(\frac23\) more than that.” Ask them who biked farther, Han or Mai? The equation they choose must result in Han biking a greater distance than Mai.

After students complete the task, ask students to share a few of their arguments for the matches they came up with (not all need to be shared). Questions you could ask them as they share:

• What is different about the numbers in the description and the numbers in the equation?
• 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?

In this activity, students receive a set of cards that have 3 different representations of proportional relationships (table, equation, description). The purpose of this activity is for students to interpret wording like “y is \(\frac13\) more than x” and figure out how that relationship can be expressed using different mathematical representations. Students may initially think that the wording means \(y=\frac13 x\), however the numbers in the table representation will help them realize that this is not true. They are engaging in this reasoning here in preparation for upcoming work on percent increase and decrease.

They will work with their partner to match the different representations with each other (each set should contain a table, equation and description) and explain their reasoning (MP3).

There are 8 groups of 3 to match up. Here is an example of one of the groups of 3:

Demonstrate how to set up and do the matching activity. Choose a student to be your partner. Mix up the rest of the cards and place them face-up. Point out that each card contains a description, table, or equation. Select one of each style of card and then explain to your partner why you think the cards do or do not match. Demonstrate productive ways to agree or disagree (e.g., by explaining your mathematical thinking, asking clarifying questions, etc.).

Arrange students in groups of 2. Give each group pre-printed cut-up slips for matching. Place two copies of uncut blackline masters in envelopes to serve as answer keys.

Students may match the equation \(y = \frac23 x\) with the situation “Elena biked \(x\) miles, and Noah biked \(\frac23\) more than that.” Ask them who biked farther, Noah or Elena? The equation they choose must result in Noah biking a greater distance than Elena.

After students complete the task, ask students to share a few of their arguments for the matches they came up with (not all need to be shared). Questions you could ask them as they share:

• What is different about the numbers in the description and the numbers in the equation?
• 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?