Reasoning with Equivalences
Recognise and use repeated reasoning to generalise: extend patterns in equivalent fractions and percentage conversions, derive unknown facts from known facts, describe general rules for sequences and predict terms
What a learner can do afterwards
- Notice that multiplying any number by 25 can be done by multiplying by 100 then dividing by 4, and explain why
- Describe the general rule for a sequence and predict the 20th term
- Generalise: to find 10% divide by 10, to find 5% halve 10%, and use this to find 35% of any number
1 · Read
When two forms match exactly, state the rule in words and check why the match holds, then use it for fractions, sequences, or percentages. For example, 100 divided by 4 is 25, so x100 then ÷4 matches x25.
Mia's sequence starts at 5 and adds 3 each time: 5, 8, 11, 14, and so on. To find the 20th term, she does not need to write out 20 numbers. She uses the rule: start at 5, then add 3 a total of 19 times. That gives 5 + (3 × 19) = 62.
To find 10% of 40, divide by 10: that's 4. Since 5% is half of 10%, 5% of 40 is 2. Adding 30% (three lots of 10%) and 5% gives 35%: 12 + 2 = 14. So 35% of 40 is 14.
Multiplying by 25 is the same as multiplying by 100, then dividing by 4. That works because 25 is 100 divided by 4, so the two steps together do exactly what ×25 does. Try it: 8 × 25 = (8 × 100) ÷ 4 = 800 ÷ 4 = 200.
100 divided by 4 is 25. So x100 then ÷4 matches x25. For 6 x 25: 6 x 100 = 600, then 600 ÷ 4 = 150. If the rule holds, the match is exact.
Find a rule that works once, explain why it works, then use it again for tricky multiplications, sequences, and percentages.
Percent and sequence items follow OpenMiddle-style and arithmetic-sequence methods with unknown license terms; three readings and two percent items are original in-repo writing.
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.
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.