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 tables facts, and a pattern hides inside them. Look at 3×4=12, 3×40=120, 3×400=1200. Every time one number gets 10 times bigger, the answer gets 10 times bigger too. One extra zero going in means one extra zero coming out. That is a rule you can say out loud and use on facts you never 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.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 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.