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💡 The Big Idea
In math, a proof is a logical argument that shows something is always true — not just sometimes, not just for the numbers you tried, but for every possible case.
Checking examples can help you believe something is true, but it can never prove it's true for ALL numbers. Only a proof can do that!
🧪 Checking Examples
"2 + 4 = 6 ✓
3 + 5 = 8 ✓
7 + 1 = 8 ✓
So maybe an odd + odd = even?"
❌ This doesn't prove it's ALWAYS true. What about numbers you didn't try?
✅ A Real Proof
"Every odd number has 1 left over when you pair up its dots. If you add two odd numbers, the two leftover dots make a pair — so there's nothing left over. That means the result is always even."
✅ This works for ALL odd numbers, forever!
1 Always, Sometimes, or Never? 🤔
Read each claim. Circle whether it is Always True, Sometimes True, or Never True.
Claim A
When you add two even numbers, the result is even.
Always True
Sometimes True
Never True
Claim B
When you multiply a number by 2, the result is bigger than the original.
Always True
Sometimes True
Never True
Claim C
The sum of three consecutive numbers is divisible by 3.
Always True
Sometimes True
Never True
Claim D
A number that ends in 0 is odd.
Always True
Sometimes True
Never True
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⚠️ When Examples Trick You
Claim: "Every number less than 40 that ends in 1, 3, 7, or 9 is prime."
Let's check: 3 ✓, 7 ✓, 11 ✓, 13 ✓, 17 ✓, 19 ✓, 23 ✓, 29 ✓, 31 ✓, 37 ✓ — wow, they all work!
But wait… is this always true? Try 21. Is 21 prime? No! 21 = 3 × 7.
Also 27 = 3 × 9, and 33 = 3 × 11. So the claim is false!
2 Test the Claim 🧪
For each claim, try at least 3 examples. Then decide: do you think the claim is true or false?
Claim E
If you add 1 to any even number, the result is always odd.
+ 1 =
+ 1 =
+ 1 =
I think TRUE
I think FALSE
Claim F
Doubling a number always gives a number bigger than 10.
× 2 =
× 2 =
× 2 =
I think TRUE
I think FALSE
3 Your Turn! ✍️
Complete these sentences about proof.
A proof shows that something is true for .
Checking examples is not enough because .
To show a claim is false, you only need .
Answer Key:
1A) Always True — 1B) Sometimes True (0×2=0, not bigger) — 1C) Always True (e.g. 4+5+6=15) — 1D) Never True —
2E) True — 2F) False (3×2=6 < 10) —
3) all cases / not every case; you might miss one that fails / one counterexample