Profit per cent and the wrong base
A Class VII mathematics teacher in a Gaya school has 36 students and five 40-minute periods before the unit test on 'Comparing Quantities'. A diagnostic item - 'A shopkeeper buys a fan for ₹800 and sells it for ₹1,000. Find the profit per cent' - got these responses: 21 students wrote 25%, 8 wrote 20%, 4 left it blank, and 3 wrote ₹200. On a separate item, 19 of 36 found the 'area' of a 10 cm by 6 cm rectangle as 32. In class, the same students correctly recite 'profit per cent = profit/CP × 100' when the teacher asks for the formula, and several shout it out before the question is finished. The teacher must also complete the chapter's exercise 8.3 and submit a remediation record for a district inspection.
1. Interpret the 25%, 20%, ₹200 and blank responses. What does the gap between reciting the formula and applying it tell you?
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Each response is diagnostic, not just wrong.
- 25% (21 students): the student computed 200/1000 × 100. The arithmetic is correct; the base is wrong. The student has chosen the selling price because it is the larger, later number in the story. This is a base-identification error, not a division error.
- 20% (8 students): correct. They used 200/800 × 100.
- ₹200 (3 students): the student found the profit but stopped before converting it to a percentage. The idea of 'per cent' as a comparison to 100 has not been applied.
- Blank (4 students): no strategy at all, possibly because they cannot identify the two given quantities as cost and selling price.
The gap between reciting and applying shows inert verbal knowledge. The class can chant a string but has no mental model of 'base': the quantity that the percentage is taken of. When the formula is chanted, CP appears as a symbol; when the problem is read, CP and SP are both rupee amounts and the students guess. More chanting will not fix this. The teaching target is the concept of base, taught through contrast: a profit problem (base = CP) next to a discount problem (base = marked price/SP). The 25% answer should be treated as a useful classroom resource, not simply marked wrong.
2. Plan the five periods using concrete-to-abstract sequencing and a diagnostic re-check. Include what happens to exercise 8.3.
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Period 1 - concrete shop. Each pair gets a card: cost ₹800. They add a 10%, 20% and 25% mark-up using a calculator or mental method and record CP, profit, SP in a table. The teacher asks only one question: 'Which number did you take the percentage of?' The answer is written on the board as a class rule: profit per cent is always of CP.
Period 2 - pictorial bar model. Draw CP as a bar of ₹800, profit as an extra part of ₹200. Shade the part and the whole. Students label 'part' and 'base'. Repeat with a loss. The bar model gives a visual anchor before the formula.
Period 3 - cognitive conflict. Two problems side by side: (a) cost ₹800, sold ₹1,000; (b) marked ₹1,000, discount 20%. Students sort which base each uses. The pair that looks identical but has different bases is the point. Students write one sentence explaining the difference.
Period 4 - error analysis. Give the class the 25% script anonymously and ask pairs to find the mistake and rewrite the solution. Then three mixed problems, one with a loss.
Period 5 - mixed practice and exit ticket. Six problems; the exit ticket is the original fan problem with different numbers. The 8 students who still use SP go to a small reteach group while the others start exercise 8.3.
Exercise 8.3 is set as guided homework after Period 3, not used as class time, so the five periods are protected for the concept. The remediation record for inspection is built from the exit tickets: number of students correct on the base item before and after, with two sample scripts.
3. Nineteen of 36 students gave 32 for the area of a 10 cm by 6 cm rectangle. How do you handle this without derailing the percentage unit?
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Do not ignore it and do not spend a whole period on it. The 32 is not carelessness. It is the perimeter (2 × (10 + 6)), which means 19 students are applying the boundary procedure to an area question. The confusion is between linear units and square units, and it will reappear in mensuration and in graph work.
Ten-minute anchor task, next period. Give grid paper. Students draw the 10 by 6 rectangle, walk the boundary with a finger and count the squares inside. Write two labels on the board: 'distance around = 32 cm' and 'squares covered = 60 square cm'. Ask: 'Which one would you use to buy paint for the floor? Which one for a fence?' The contrast is functional, not verbal.
Follow-up. Put one area/perimeter item in the next two warm-ups and one in the unit test, so the idea is revisited without a separate unit. Tell the class explicitly that this will be tested, because otherwise the 19 students will file it as optional.
Coordination. Mention it to the teacher who handles geometry so the same grid representation is used. The percentage unit is not derailed: ten minutes, one anchor chart, two warm-up items. The alternative - leaving it until the mensuration chapter - means 19 students carry a wrong rule for weeks and the error becomes harder to unlearn.