Generalising from repeated reasoning
Recognise and use repeated reasoning to generalise: extend patterns in times tables and equivalent fractions, derive unknown facts from known facts efficiently, describe general rules
What a learner can do afterwards
- Notice that all fractions equivalent to 1/2 have a numerator that is half the denominator
- Use the pattern 3×4=12, 3×40=120, 3×400=1200 and explain the generalisation
- Derive 8×7 from 8×5=40 plus 8×2=16 and describe the strategy as a general approach
1 · Read
You already know lots of times-table facts. Patterns hide inside them. Look at 3 × 4 = 12, 3 × 40 = 120, and 3 × 400 = 1200. When one factor gets 10 times bigger, the answer gets 10 times bigger too. One extra zero in gives one extra zero out. Say this rule aloud and use it for facts you have not learned by heart.
Mo needs 8×7 but only remembers 8×5 and 8×2. He splits it: 8×5=40 and 8×2=16. Then he adds them: 40+16=56. So 8×7=56, and Mo used the same splitting trick to solve a fact he had not memorized.
Equivalent fractions follow a rule too. 3 out of 6 shaded is the same amount as 1 out of 2. Look at the numbers: 3 is half of 6. Any fraction equal to 1/2 has a top number that is exactly half the bottom number, so 6/12 works and 5/11 does not.
Spot the pattern, say the rule in your own words, then use it to work out any new fact.
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2 · Watch
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Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
18 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.