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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.
Which group of blocks is the bluest?
First image of 5 groups of yellow and blue blocks. Figure A has 3 blue blocks on bottom row and 1 yellow square on top row.
Second image of 5 groups of yellow and blue blocks. Figure B has 3 blue blocks on bottom row and 2 yellow blocks on top row.
Third image of 5 groups of yellow and blue blocks. Figure C has 4 blue blocks on bottom row and 2 yellow blocks on top row.
Fourth image of 5 groups of yellow and blue blocks. Figure D has 5 blue blocks on bottom row and 2 yellow blocks on top row.
Fifth image of 5 groups of yellow and blue blocks. Figure E has 4 blue blocks on bottom row and 3 yellow blocks on top row.
Order the groups of blocks from least blue to bluest.
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:
Creating and Representing Situations Handout
In this activity, students create their own proportional and nonproportional relationships. This is a significant change from previous activities where students were always given two quantities and asked to decide if they were proportional or not. This new step gives students the opportunity to think about how quantities are related, which is an important step of modeling with mathematics (MP4).
Next, students make a visual display of their scenarios. The posters are displayed around the room so students can view each other's work, give feedback, and critique the reasoning of others (MP3).
Arrange students in groups of 2. Explain that students are going to come up with relationships that represent things in their daily lives.
Give students 1–2 minutes to brainstorm situations that involve proportional relationships. Invite students to share their ideas. Record the general gist of each idea that is shared and display for all to see. For example, if a student says, “At my favorite arcade, it takes 4 tokens per game of pinball,” you could record this idea as “how many tokens for an arcade game.”
Give students 8–10 minutes of partner work time to invent a relationship and answer the questions. If students struggle to think of situations that involve proportional relationships, consider providing them with copies of the first page of the blackline master, which lists some examples of proportional relationships that they have seen during this unit. Make sure students understand that the examples on the blackline master are just samples and they can still choose to write a different type of relationship.
Make up a situation where there is a proportional relationship between two quantities.
Make a table with at least 5 pairs of numbers relating the two quantities.
If you have time, make up another situation where there is a relationship between two quantities, but the relationship is not proportional.
Make a table with at least 5 pairs of numbers relating the two quantities.
Give students 5 minutes to trade their work with another group to give feedback about their analysis. Tell them that they should incorporate the feedback they received when making their posters.
Ask students to create a visual display of the relationship they came up with that includes all four of the different types of representations (sentences, table, graph, equation). When the posters are complete, display 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).
Conclude with a brief whole-class discussion where students can share similarities they noticed about the different relationships and representations.
The constant of proportionality for a proportional relationship can often be easily identified in a graph, a table, and an equation that represents it. Here is an example of all three representations for the same relationship. The constant of proportionality is circled:
Three representations of the linear function y = 7 fourths times x. Horizontal axis scale is 0 to 8 by 1’s. Vertical axis scale is 0 to 10 by 1’s. The graph of the function has the points (1 comma the fraction 7 over 4) and (4 comma 7). The table has rows (x comma y), (0 comma 0), (1 comma the fraction 7 over 4), (3 comma the fraction 21 over 4), (4 comma 7).
On the other hand, some relationships are not proportional. If the graph of a relationship is not a straight line through the origin, if the equation cannot be expressed in the form
Help us improve by sharing suggestions or reporting issues.
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. Prompt them to reason about what numbers would be more conducive to the situation or easier to calculate.
Some students may struggle to think of a nonproportional situation. Consider providing them with copies of the second page of the blackline master, which lists some examples of nonproportional relationships that they have seen during this unit. However, it is not necessary for every group to come up with a nonproportional relationship, as long as there are a few examples for students to contrast during the discussion.