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💡 Seeing Is Believing (and Proving!)
Sometimes the best way to prove something is to draw a picture. A good picture can show you WHY a formula works — not just THAT it works. These are called visual proofs or proofs without words.
Picture Proof 1: Odd Numbers Build Perfect Squares
1 + 3 + 5 + 7 + … = a perfect square!
Each odd number (orange L-shape) wraps around the previous square to make a bigger square!
The nth odd number is 2n−1, and it adds exactly enough to go from (n−1)² to n².
1 + 3 + 5 + 7 + ⋯ + (2n−1) = n²
1 Continue the Pattern 🔲
Use the L-shape pattern to answer without calculating each addition!
1 + 3 + 5 + 7 + 9 =
That's 5 odd numbers → ²
1 + 3 + 5 + 7 + 9 + 11 =
That's 6 odd numbers → ²
1 + 3 + 5 + … + 19 =
That's odd numbers → ²
1 + 3 + 5 + … + 99 =
That's odd numbers → ²
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Picture Proof 2: The Staircase
1 + 2 + 3 + 4 = ? Make two staircases into a rectangle!
→
Two copies → 4 × 5 rectangle!
The staircase has 1+2+3+4 = 10 blocks (purple).
Flip a copy (orange) and fit them together → a 4×5 = 20 rectangle.
The staircase is half of the rectangle: 20 ÷ 2 = 10 ✓
1 + 2 + 3 + ⋯ + n = n × (n + 1) ÷ 2
2 Use the Staircase Formula 📐
Use the formula n × (n+1) ÷ 2 to find each sum quickly!
1 + 2 + 3 + 4 + 5
n = 5 → 5 × 6 ÷ 2 =
1 + 2 + 3 + … + 10
n = 10 → 10 × ÷ 2 =
1 + 2 + 3 + … + 20
n = 20 → × ÷ 2 =
1 + 2 + 3 + … + 100
n = 100 → × ÷ 2 =
3 Draw Your Own Picture Proof ✏️
Draw the staircase for 1+2+3+4+5 and show how two copies make a 5×6 rectangle.
Draw your staircase here:
Answer Key:
1) 25 (5²), 36 (6²), 100 (10 odd numbers → 10²), 2500 (50 odd numbers → 50²) —
2) 15, 55, 210, 5050