Lesson 17

In this warm-up, students are asked to reason which group of blocks is the bluest and explain how they arrived at that decision. The goal is to prompt students to visualize and articulate different ways they can use ratios, equivalent ratios and proportions to support their reasoning.

Students in groups of 2. Tell students you will show them five groups of blocks. Their job is to determine which group of blocks is the bluest. Display the image for all to see. Give students 2 minutes of quiet think time. Encourage students who have one way of supporting their decision to think about another way while they wait.

Ask students to share which group of blocks is the bluest and their reasoning. Record and display student explanations for all to see. To involve more students in the conversation, consider asking some of the following questions:

• Did anyone choose the same group of blocks but would explain it differently?
• Does anyone want to add an observation to the way ____ saw the blocks?
• Do you agree or disagree? Why? Ask students to order the groups of blocks from less blue to bluest after deciding on the bluest group of blocks.

In this activity, students choose from different lists of things to define their own proportional and nonproportional relationships. Some of the things on the list will be familiar and others will be unfamiliar. This is a significant change from previous activities where students were always given two quantities and they had to decide if they were proportional or not. This new step gives students the opportunity to think about what quantities are related to some of the items on the lists, which is an important step of modeling with mathematics (MP4).

This activity and the next go together. Students use the work from this activity to make a visual display of their work in the next activity.

Arrange students in groups of 2.

The names of things in the task may be unfamiliar to both English Language Learners and fluent English speakers. Before students start, take some time to ensure they know the meaning of their chosen things or ask them to do some research on their meanings. You might ask each pair of students to choose a different unfamiliar word, spend five minutes to research it, and prepare a drawing or explanation to share with the class.

It is best to approve of students’ choices before they work. For example, if students choose “legs” and “earthworms,” that will not make for a very interesting relationship.

As students work, pay attention to the numbers they use in their tables. Students can be haphazard when choosing values, and their numbers may end up being unfriendly. You can ask them questions that encourage them to reason about what numbers would be friendlier. Also, depending on the things chosen, they may need to consider scales for their axes. Watch out for scales like {1, 2, 3, . . . } for the number of legs on a centipede!

Creating a relationship that is not proportional may present too significant a challenge for struggling learners. An accommodation would be to change their task to creating only a proportional relationship, or even assigning two quantities that are straightforward like “starfish legs vs. number of starfish.”

Ask groups to trade their work with another group to give feedback about their analysis. 

In this activity, students make a visual display of their scenarios from the previous activity (after sharing their rough draft with another group and getting some feedback).

When the posters are complete and displayed around the room, students view each others' work and use sentence starters to give feedback and to critique the reasoning of others (MP3).

Keep students in the same groups. Tell them that they should incorporate the feedback they received when making their posters.

When the posters are complete, hang them around the room. Provide students with these sentence starters, and give them an opportunity to view their classmates’ work and write their responses. This provides a structured way for students to critique the reasoning of others (MP3).

The most surprising combination of things was ____________________ because ______________________________________________________________.

The group _______________________________ should check their work where they ___________________________________________________________________.

I really liked when the group ____________________________ did this __________________________________ because _______________________________________.