The gap of one: ratio misconceptions in Class 7
You teach mathematics to 48 Class 7 students in a Patna government school. A five-item diagnostic on ratio and proportion gives four distinct results. Twenty-two students say 2:3 and 4:5 are equivalent 'because the gap is one in both'. Seventeen cross-multiply correctly but cannot explain why the method works, and none of these 17 can solve 'a recipe for 4 people uses 6 cups of rice; how many cups for 10 people'. Six students divide 6 by 4 and then add 1. Three students are correct on every item. The unit test is six periods away. The headmaster has asked that the textbook chapter be finished on schedule, including the unitary-method exercise, because a block education officer's review is due next month. There are no ratio tiles or measuring jars, but the school has a kitchen, an overhead water tank with a tap, and a 1-litre bottle.
1. Name the dominant misconception and explain why the 17 procedurally correct students are also a concern.
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Dominant misconception: additive reasoning replacing multiplicative reasoning. The 22 students are treating a ratio as a pair of numbers with a fixed difference rather than a fixed quotient. The test of this is that 2:3 and 4:5 both 'go up by one', so they look equivalent. The 6 students who divide and then add are the same misconception in another form.
Why the 17 are equally at risk: they have a procedure without a concept. Cross-multiplication works as a check on an equivalence you already understand; it does not tell you whether two quantities are in proportion. That is why all 17 fail the recipe item – they have no unit-rate idea to attach the numbers to. In Class 8 they will meet direct and inverse proportion, and this group will guess which operation to use.
What the data also shows: the three secure students are not a teaching resource to over-use. Use them as recorders and peer checkers for five minutes, not as substitute teachers, or their own progress stalls.
2. Design a 25-minute intervention using only the kitchen, the water tap and the 1-litre bottle that targets the 22 and the 17 together. State the task, the key questions, and the evidence you will collect.
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Task (concrete first, symbol later). In groups, mix squash: 2 cups concentrate to 3 cups water. Taste. Then make a second jug at 4 cups to 6 cups, and a third at 6 cups to 9 cups. Taste all three. Ask: which two taste the same, and how do you know without tasting?
Key questions, in this order:
- 'If I double both amounts, does the taste change? Why not?'
- 'Now make a jug at 4 cups to 5 cups – will it taste the same as the first? Predict before mixing.' This is the deliberate confrontation: the gap is one in both 2:3 and 4:5? No – in 2:3 the gap is 1, in 4:5 it is also 1. Students predict 'same', taste, and find it stronger. The prediction failure is the learning moment.
- 'What number stays the same when the taste stays the same?' Push towards the quotient.
Evidence to collect: each student writes two equivalent ratios and one non-equivalent ratio with a one-line reason. Sort into secure, additive, procedural. Reteach only the additive group next period.
Water tank variant for the recipe item: 1 litre fills 4 bottles; how many for 10?
3. You have six periods, a syllabus deadline, and the review. Allocate the periods and state explicitly what you will drop and why.
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Allocation:
- Period 1: concrete intervention above (squash, water tank).
- Period 2: pictorial – bar models for 2:3 and 4:6; students draw before calculating.
- Period 3: unit rate as the meaning behind the unitary method, still with bottles and cups.
- Period 4: unitary method on paper, but every answer checked against an estimate.
- Period 5: mixed word problems, including one pair contrasting direct and inverse situations.
- Period 6: short assessment plus feedback, not new content.
What I drop: the 20-item drill exercise becomes 6 selected items; the mnemonic table of 'types of proportion' goes; cross-multiplication is taught as a check, not an introduction.
Justification for the headmaster: the review inspects the register and samples student work, not how many exercises were ticked. Drilling 20 identical items with a wrong concept consolidates the error and creates a bigger remediation cost in Class 8. Finishing the chapter means the students can use it, and the bar-model pages in their notebooks are visible evidence of coverage.