Generalising with repeated reasoning
Recognise and use repeated reasoning to generalise: describe algebraic rules for nth terms, use properties of operations to simplify, and verify generalisations with specific cases
What a learner can do afterwards
- Explain that the interior angle sum of an n-sided polygon is (n−2) × 180° based on the pattern for triangles, quadrilaterals, pentagons
- Predict the 20th term of a linear sequence by identifying and applying the general rule
- Generalise that dividing by a non-zero whole number n always gives a denominator of n in the fraction
1 · Read
A pattern that works in a few examples often works for every case. Finding a rule like that is called generalising. You use a letter, like n, to stand for any number, so your rule works even for cases you have not tried.
Maya's sequence is 5, 9, 13, 17, and it goes up by 4 each time. Instead of counting all the way to the 20th term, she writes a rule: multiply the position number by 4, then add 1. For the 20th term: 4 × 20 + 1 = 81. She checks the rule on a term she already knows, term 3: 4 × 3 + 1 = 13. It matches, so the rule works.
Dividing a whole number by a non-zero whole number splits it into equal parts. The number you divide by goes on the bottom, and you cannot divide by 0. So 3 divided by 5 is 3/5, and 9 divided by 4 is 9/4. Dividing a unit fraction by a whole number multiplies the bottom. So 1/4 divided by 2 is 1/8.
To go from an angle sum back to sides, undo the rule. For 1080: divide by 180 to get 6, then add 2 to get 8 sides.
Repeated reasoning turns a pattern from a few examples into a rule for any case, then you check the rule against numbers you already know.
Polygon-sum items follow the studied interior-angle method and sequence items follow the arithmetic-sequence method; howto and textbook licenses unknown, recorded null per item. Rewritten readings were written for this dataset.
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Where this leads
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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.