Planning a Three-Part Lesson on Rates
Mr. Iyer is planning a 60-minute Grade 7 lesson on rates using a flyer task: '12 granola bars for $4.50, or 20 for $7.20 - which is the better deal?' His class of 29 includes two students with language IEPs, one student working at a Grade 4 level in number, and five students who finish rich tasks in about three minutes. In the previous unit, 14 students solved a similar problem by subtracting ('$7.20 − $4.50 = $2.70, only $2.70 for 8 more bars, so the second is better') and reached the wrong conclusion. He wants a three-part lesson with a whole-class consolidation, and there is no additional adult support.
1. Anticipate the two most likely student strategies and the misconception behind the subtraction. Then explain how you would choose which two pieces of student work to sequence at the end.
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Strategies: (a) unit rate - $4.50 ÷ 12 = $0.375 per bar against $7.20 ÷ 20 = $0.36 per bar; (b) scaling to a common number - 12 bars scaled ×5 is 60 bars for $22.50, 20 bars scaled ×3 is 60 bars for $21.60.
Misconception: additive reasoning. The subtraction compares the extra cost of the extra bars and treats that incremental rate as the deal's rate. It is only valid if the base is identical, which it is not. The numbers in the previous task happened to punish it, and 14 students still kept the habit, so the error is stable rather than careless.
Sequencing: I would choose one sample with sound structure but incomplete communication - typically a scaling table with no conclusion written - and one complete, efficient unit-rate solution. Starting with the flawed-sample-as-foil risks the class memorising 'don't subtract' rather than seeing why a common basis is needed. Starting with the complete unit-rate answer and then asking whether it agrees with the scaling table gives the class two representations that must be reconciled. The subtraction, if it appears, is placed last as the thing the two agreed answers refute.
2. How would you change the numbers so the additive strategy fails loudly without making the task easier? Explain why the change works.
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Change the second deal to 20 bars for $7.80. The additive move now reads: '$7.80 − $4.50 = $3.30 for 8 more bars, so the second is the better deal.' Unit rates say the opposite - 37.5¢ per bar versus 39¢ per bar - so the first deal wins.
The design point is that the additive conclusion and the correct conclusion now diverge. In the original numbers, the subtraction happened to point the same way as the unit rate, so a student could be right for the wrong reason and never find out. With $7.80, the reasoning is exposed rather than merely wrong.
This does not make the task easier: the arithmetic is comparable, both quantities still require a step beyond the whole-number facts students own, and the numbers are deliberately close enough that estimation cannot settle it. 'Close but not equal' is the condition that forces a common basis. I would keep a pair of numbers where the additive answer is correct as a contrast item later, so students learn that the method fails selectively, not always.
3. Describe your plan for the student working at a Grade 4 level and for the five fast finishers, so that the lesson stays one lesson rather than two.
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Same structure, different numbers. The Grade 4-level student gets the same two-deal comparison with friendly values - 2 bars for $1.00 against 4 for $1.80 - a table to record per-bar and scaled costs, and concrete materials if wanted. The mathematical demand is identical (build a common basis), only the arithmetic load is reduced. That student is then a legitimate presenter in the consolidation: the table is the representation the class needs to see, and presenting it is not a consolation prize.
Fast finishers get an extension that is the same content, deeper. Two options: 'How much should 20 bars cost to be exactly as good a deal as 12 for $4.50?' (answer $7.50, which requires inverting the unit rate), or 'Invent a third deal that beats both, and prove it.' Both keep them inside ratio reasoning rather than moving them on to a new topic, and both produce work that can be used in the consolidation.
What keeps it one lesson: every student is answering a question about the same underlying structure - a common basis for comparison - and the consolidation is organised around representations (table, unit rate, scaling), not around ability groups.