Sequencing Multiple Strategies in a Proportional Reasoning Discussion
Mr. Alvarez teaches seventh-grade mathematics to 27 students, including four multilingual learners and three students with IEPs for written expression. He launches a proportional reasoning task: A recipe uses 3 cups of flour to make 4 dozen cookies. How many cups are needed for 10 dozen? Groups use different strategies. One group sets up 3/4 = x/10 and solves by cross-multiplying. A second finds the unit rate, 3/4 cup per dozen, and multiplies by 10. A third builds a ratio table: 4 dozen→3 cups, 8 dozen→6 cups, 2 dozen→1.5 cups, total 7.5 cups. A fourth says, “Half of 4 dozen is 2 dozen, so half of 3 cups is 1.5; 6 + 1.5 = 7.5.” One group finishes quickly with 7.5 but cannot explain why. Another group is stuck because they added 3 cups to 10 dozen. Mr. Alvarez has 20 minutes for a whole-class discussion and wants to use the different strategies to build a shared understanding of constant of proportionality.
1. Which strategies would you select and sequence for the whole-class discussion, and why? Include how you would use the stuck group's error.
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Sequence from contextual and additive reasoning to the unit rate and then to the equation. Start with the ratio-table group and the halving group because their reasoning is grounded in the recipe and accessible to multilingual learners. Ask them to explain what stays constant: for every 4 dozen, 3 cups; for every 1 dozen, 3/4 cup. Then bring in the unit-rate group to make the constant multiplier explicit: 10 dozen × 3/4 cup per dozen = 7.5 cups. Finally, connect the equation group's 3/4 = x/10 to the same constant: cross-multiplying is a shortcut for finding the missing value in a proportion, but the proportion itself represents equal ratios.
The stuck group's error-adding 3 cups to 10 dozen-is productive if handled carefully. Write “3 cups + 10 dozen” on the board and ask the class whether these quantities can be combined. The units differ, so the sum is meaningless. This moment reinforces that proportional reasoning compares quantities with a constant ratio, not arbitrary addition. Ask the quick group to justify 7.5 with a diagram rather than just repeat the answer.
2. Describe specific discourse moves to support the multilingual learners and students with IEPs for written expression during the discussion.
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Before the discussion, pre-teach key terms-dozen, per, ratio, constant-with a visual word wall and a labeled recipe diagram. Provide sentence frames such as “For every ___ dozen, we need ___ cups, so for 1 dozen we need ___” and “The constant multiplier is ___ because ___.” These frames give multilingual learners a way to participate orally without drafting full sentences from scratch.
During the discussion, use wait time after each question, partner retell before whole-class sharing, and revoicing: “So you are saying that for each dozen, the amount goes up by three-fourths of a cup?” Allow students with written-expression IEPs to explain with a labeled ratio table or diagram instead of a written paragraph. Keep the ratio table visible and color-code the “×3/4” relationship so students can track the constant multiplier.
Finally, ask the quick group to show their thinking on the board with a diagram, not just an answer. This holds them accountable for reasoning and gives visual support to students who are still building meaning.
3. What evidence from an exit ticket would show that the discussion achieved its learning goal? Describe what you would look for and how you would respond to incomplete understanding.
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Exit ticket: “A different recipe uses 5 cups of flour for 6 dozen cookies. How many cups are needed for 9 dozen? Show a ratio table or equation and explain the constant multiplier.”
Strong evidence: the student finds the unit rate 5/6 cup per dozen and multiplies by 9 to get 7.5 cups, or builds a ratio table that preserves the 5:6 relationship. The explanation should identify the constant multiplier-5/6 cup per dozen-and connect the table, equation, and answer. Look for language such as “for each dozen” and an estimate that 9 dozen should need more than 5 cups but less than 10.
If a student only cross-multiplies without explanation, ask them to say what the proportion means in the recipe. If a student adds 5 cups and 9 dozen, reteach with a two-column table and unit labels. If a student gets 7.5 but cannot explain, have them build the ratio table for 6, 3, and 9 dozen and mark the constant multiplier. The exit ticket should reveal whether students can transfer the constant of proportionality to a new context, not just repeat the original answer.