Solving a problem in stages — Mathematics, 11–13
The same stages, now with fractions of a whole you cannot see: interpret, compare strategies, evaluate the result — and pose problems of your own.
Interpret, carry out, evaluate
You already know the three stages: interpret the problem, choose and carry out a strategy, evaluate the result in the situation. In year 6 the problems often hide the whole behind a fraction. In a class of 28 pupils, 3/7 come by bike. Interpret: the whole is 28 pupils, split into 7 equal parts. You can also pose the problem yourself, from the situation: how many come by bike? (28 ÷ 7 × 3 = 12) And how many do not? (28 − 12 = 16)
Two strategies, the same result
A box holds 36 balls; 2/3 are red, and 1/4 of the red ones have stripes. How many red balls have stripes? Strategy A, a diagram: 36 in 3 groups of 12; 2 groups make 24 red; 24 in 4 groups of 6, so 6. Strategy B, calculation: 36 ÷ 3 × 2 = 24, then 24 ÷ 4 = 6. Both are correct. B is quicker when the numbers are big, A shows why it works. Simulating, working backwards, trial and error (a spreadsheet helps), a simpler problem or particular cases are other strategies. Comparing them shows which is correct, how they differ and which is more efficient.
A case: the book still to be read
Rita read 2/5 of a book on Monday and 1/3 on Tuesday. She still has 56 pages to read. How long is the book? Interpret: the whole is the book, and 56 pages is the part left. Strategy: a bar of 15 equal parts (15 is the smallest number that holds fifths and thirds). 2/5 is 6 parts, 1/3 is 5 parts: 6 + 5 = 11 read, so 4 parts are left. 4 parts = 56, so 1 part = 14, and 15 × 14 = 210. Evaluate: 2/5 of 210 = 84, 1/3 of 210 = 70, and 84 + 70 + 56 = 210 ✓.
Adding fractions as if they were loose numbers
To find out how much Rita has read, Rui added 2/5 + 1/3 and wrote 3/8, adding numerators and denominators. It looks tidy, but the evaluate stage catches it: she had 2/5 of the book and read more, so the total must be more than 2/5. Yet 3/8 = 0.375 is less than 2/5 = 0.4. Adding cannot make a result smaller. The right way goes through 15 parts: 6/15 + 5/15 = 11/15. Never stop at the calculation: ask whether the result fits the situation.
Where you see it: planning a journey
On a 240 km journey you have done 3/5 of the way. From this situation you can pose several problems. How many kilometres have you done? 240 ÷ 5 × 3 = 144. How many are left? 240 − 144 = 96. At an average of 80 km/h, how long is left? 96 ÷ 80 = 1.2 h, that is 1 h 12 min (0.2 h = 12 min). This is how a navigation app or a calculator helps: you pose the question, the technology calculates, and you evaluate whether the result makes sense.
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