Fourteen Students, Sixteen Days
Class VII has 36 students. Before starting the chapter on simple interest, the teacher gives a four-item proportion quiz. Item 3 reads: '18 workers build a wall in 12 days. How many days will 24 workers take, working at the same rate?' Twenty-one students answer 9 days. Fourteen answer 16 days: they compute 24/18 = 4/3 and multiply 12 by it. One student answers 6 days, having subtracted. The same 14 solve '3 pens cost ₹24, so 7 pens cost ₹____' correctly, and they got a recipe-scaling item right when the number of people doubled. In the discussion several say the rule is 'multiply across', and one asks why the answer gets smaller when there are more workers. The unit test is in ten days, and the next textbook chapter depends on the same reasoning. The school has one working blackboard and no ready-made posters.
1. Diagnose the mental model behind the answer 16 days. Why do the pen and recipe items succeed while the worker item fails, and what does that tell you about the teaching the class has received?
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Sixteen days comes from applying a direct-proportion schema - 'more of one, more of the other' - which is the correct schema for the pen item and for a recipe scaled up. The students are not weak at arithmetic; they have one rule and no way to decide when it applies.
The diagnostic value lies in the split performance. If they held a genuine misconception about inverse variation, they would also fail the pen item. Failing only the worker item means the schema is intact and the selection step is missing. The student who asks why the answer gets smaller is asking exactly the right question, and that question should be treated as the lesson's opening, not as a distraction.
The one 6-day answer is a different thing entirely - a subtraction guess, not a competing model - and should be handled separately rather than grouped with the fourteen.
The teaching implication is sharp: 'multiply across' and cross-multiplication both produce correct answers on direct items without ever forcing a decision about direction, so the class's prior success is not evidence of understanding. What is absent from their notebooks is a representation in which the inverse relation is visible - two quantities whose product stays constant. The ten days should be spent on representation, not on more worked examples of the same type.
2. Plan three 40-minute periods before the unit test, naming the representations and the numbers you will use. What evidence would convince you the fourteen have changed?
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Period 1 - make the constant visible (person-days). Start from the 18 × 12 = 216 idea: 216 worker-days is the fixed amount of wall. Build a two-row ratio table on the board with workers 9, 18, 24, 36 and days left blank, filling days only by dividing 216, then checking by doubling and halving (36 workers must take 6 days). Introduce the double number line alongside so the same relation appears in two forms.
Period 2 - same direction or opposite? Six paired situations side by side (pens/rupees, workers/days, taps/filling time, pages/day, speed/time for a fixed distance, candles/price). Students classify each and must justify from the situation, not the numbers, then test their classification against one calculated example. Include a reversed-numbers item - '24 workers, 18 days, so 18 workers take ___' - to break the habit of reacting to the first number seen.
Period 3 - mixed set and written explanation. Eight items, at least two of which are not proportional (including 'a 12-year-old is 150 cm; how tall at 24?'), plus: 'Explain to a Class V child why more workers means fewer days, using a picture.'
Evidence of change: the proportion who draw a ratio table or number line unprompted; performance on the reversed-numbers item; whether explanations mention the fixed total work rather than the rule; and a one-week delayed item to check retention.
3. What do you do with the 21 students who are already correct, without simply handing out more worksheets?
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Their correctness is partly untested - some of the 21 may be applying the same unselective rule successfully by luck, since the quiz had only one inverse item. So the first step is to find out, using the reversed-numbers item and one non-proportional item; expect a few of the 21 to fall over, and treat that as useful information rather than a problem.
Then use them without turning them into a second teacher:
- Method comparison: they receive the same mixed set but must solve each item two ways (ratio table and equation) and write which was faster and why.
- Error analysis: give them three anonymised wrong solutions - including the 4/3 × 12 one and the subtraction one - and ask them to name the mistake and write the question they would ask the student. This is harder than solving and directly builds the reasoning the unit test rewards.
- Explaining, but late: pair a few with the struggling group only after they have written their own explanation, because students who explain a rule before articulating it tend to restate 'multiply across' louder rather than differently.
Keep the whole class together for the opening 10 minutes of each period so that the fourteen are not publicly labelled, and so that the 21 hear the reasoning, not just the answer.