Four Representations

5 min

Teacher Prep
Setup
Students in groups of 2. Display the image for all to see. Give students 2 minutes of quiet think time followed by a whole-class discussion.

Narrative

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.

Launch

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.

Student Task

  1. Which group of blocks is the bluest?

    A
    First image of 5 groups of yellow and blue blocks.
    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.

    B
    Second image of 5 groups of yellow and blue blocks.
    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.

    C
    Third image of 5 groups of yellow and blue blocks.
    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.

    D
    Fourth image of 5 groups of yellow and blue blocks.
    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.

    E
    Fifth image of 5 groups of yellow and blue blocks.
    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.

  2. Order the groups of blocks from least blue to bluest.

Sample Response

  1. A or D: A if looking at the amount of blue per yellow, D if looking at the total amount of blue or the difference between blue and yellow
  2. Sample responses:
    1. E, B, C, D, A when ordering by the amount of blue blocks per yellow
    2. A or B, C or E, D when ordering by total amount of blue blocks
    3. B or E, A or C, D when ordering by difference between blue and yellow blocks
Activity Synthesis (Teacher Notes)

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.
MLR8 Discussion Supports. Display sentence frames to to support whole-class discussion: “I noticed _____, so I . . . .” “_____’s idea reminds me of . . . .”, and “The next time I put things in order from least to greatest, I will . . . .”
Advances: Speaking, Conversing, Representing
Standards
Building On
  • 6.RP.3·Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
  • 6.RP.A.3·Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
Addressing
  • 7.RP.2·Recognize and represent proportional relationships between quantities.
  • 7.RP.A.2·Recognize and represent proportional relationships between quantities.

30 min