From Examples to a General Argument
Move from checking a pattern on a few cases to an argument that covers every case, using a letter to stand for any number.
What a learner can do afterwards
- Say why testing five cases does not settle a claim about all numbers
- Turn a spotted pattern into a statement with a letter in it and check the statement against a fresh case
- Write a short argument for a claim such as the sum of two even numbers being even
1 · Read
Checking five cases never settles a claim about all numbers. Five examples cover five numbers, while the claim covers infinitely many. One untested number could break the pattern, so examples can suggest but never prove.
Turn a spotted pattern into a statement with a letter in it. Seeing 3 + 5 = 8 and 7 + 9 = 16 suggests a rule: the sum of two odd numbers is even. Or a figure pattern becomes 3n + 1 sticks for the nth figure. Then test it on a fresh case: n = 10 gives 31.
A real argument covers every case at once. An even number is 2 times a whole number, so write two evens as 2a and 2b. Their sum is 2a + 2b, which equals 2(a + b). A multiple of 2 is even, for every a and b. Done for all of them.
Checking one fresh case supports a rule but never proves it. If Priya turns a pattern into 2n and n = 6 gives 12, the rule survives one test with the rest untested. Support is not proof. Proof leaves no case out.
Examples suggest, letters state, and only an argument covering every case proves.
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.