Correct Answers, Missing Pictures
In a high school in Cooch Behar, a teacher handles both mathematics and science for Class VII, 40 students. In the last test, 34 of the 40 computed (3/4) × (2/3) correctly as 1/2 using the rule 'multiply numerators, multiply denominators'. When she gave a 1 m × 1 m chart ruled into thirds and asked them to shade what (3/4) of (2/3) of a metre means, only 9 shaded the overlapping region; 22 shaded three-quarters of the whole chart. In science, 27 of 40 agreed that 'a 200 g ball falls faster than a 50 g ball because it is heavier'. She dropped both from the same height and they landed together; a boy said, 'Sir released the heavy one late.' She planned to show a YouTube video, but the projector is broken. She has a stopwatch, a metre scale, and her own phone, which can record slow motion.
1. Diagnose precisely what the fraction data show. Why is 34 out of 40 the more alarming figure, not the reassuring one?
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The two items measure different things and the gap between them is the diagnosis.
- 34/40 correct on the algorithm shows the rule has been memorised and applied reliably. It is procedural knowledge.
- 9/40 correct on the shaded chart shows that 'of' as multiplication has no meaning for most of the class. The 22 who shaded three-quarters of the whole chart are not careless; they read '(3/4) of (2/3)' as 'three-quarters, of a thing that is two-thirds' - the notation, not the operation, is what they interpret.
34 out of 40 is the worrying number because a class that computes correctly without a model will keep producing plausible wrong answers and cannot check itself: nothing in the answer 1/2 feels wrong. Procedural success hides conceptual failure and removes the usual warning sign (wrong answers). The teacher should now treat the algorithm as already taught and spend the next lessons on the area model, going from model to symbol, not symbol to model.
2. Design a three-step lesson sequence using the chart and everyday materials to build the meaning of fraction multiplication.
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Step 1 - Fold before shading (concrete). Each student takes a paper strip 24 cm long. Fold into thirds, shade two-thirds. Then fold the same strip into quarters across and shade three-quarters of the already-shaded part in a second colour. The doubly shaded part is the answer. Folding forces the overlap to be visible; shading on a pre-ruled chart, as in the test, does not.
Step 2 - Name the overlap in two ways (representational). On the 1 m × 1 m chart drawn as 3 columns and 4 rows, ask: 'how many small boxes in all? how many are double-shaded?' Students write 6/12, then reduce to 1/2, and check against the rule. The point is that the rule is now a shortcut for something already seen, not a magic chant.
Step 3 - Reverse the direction (symbolic check). Give 1/2 and ask for three different fraction pairs whose product is 1/2, each with a sketch. This is where 34/40 procedural students are revealed: they can multiply but cannot generate. Exit ticket: one sketch, one sentence - 'multiplying by a fraction less than one makes the result smaller because...'
3. How should the teacher handle the falling-bodies demonstration after the boy's comment, without the projector?
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The boy's comment is not resistance; it is a correct instinct about experimental control - he has identified a confound (release timing), which is exactly the reasoning the lesson should build on.
Plan:
- Predict first. Record every student's prediction and reason on the board before any drop. The 27 who said 'heavier falls faster' must be committed in writing.
- Repeat and vary. Drop the 200 g and 50 g balls three times each, from 1 m and then 2 m, letting two students release simultaneously on a countdown. Have the class time the fall with the stopwatch - no stopwatch is precise enough, which is itself a lesson.
- Slow motion. Record with her phone at the highest frame rate and play back frame by frame on the phone, passed round in four groups. The projector is unnecessary; a phone screen is adequate for 8–10 students at a time.
- Control explicitly. Then crumple paper of equal mass, or drop a flat sheet against a crumpled ball, to show that air resistance can produce the 'heavy falls faster' effect. This preserves the students' observation instead of dismissing it.
- Explain, don't correct. Students write a sentence saying why the heavier ball seemed faster, and what would be true in a vacuum.