Mission: The Formula Creator — Level 9
Akuuu is stacking balls to make a triangle:
Row 1: ● (1 ball)
Row 2: ●● (2 balls) → Total: 3
Row 3: ●●● (3 balls) → Total: 6
Row 4: ●●●● (4 balls) → Total: 10
👉 She needs to know the total for 100 rows. Must she add 1+2+3+…+100? Or is there a smarter way?
Think: Adding 100 numbers is slow. Can you find a shortcut by looking at how the totals grow?
✏️ Guess a formula for total balls in n rows. Test it on rows 1-4:
- 📐 Triangular numbers: 1, 3, 6, 10, 15… formed by adding rows 1+2+3+…+n
- 🔍 Look for structure: Pair first + last, second + second-last… they all sum to the same value
- 🎯 A general formula works for ANY n, not just specific cases
For rows 1 to 6: 1+2+3+4+5+6
Pair: (1+6) = 7, (2+5) = 7, (3+4) = 7
👉 How many pairs? What is the total? Does this match 1+2+3+4+5+6?
Akuuu tries the pairing trick on rows 1 to 5: 1+2+3+4+5
She pairs: (1+5)=6, (2+4)=6, and 3 is left alone in the middle.
👉 Why does 5 leave a middle number but 6 doesn’t? What does this tell you about odd vs. even counts?
👉 Does the pairing trick still work for odd counts? Test it and explain:
Using the pairing trick, create a formula for the sum 1+2+3+…+n.
👉 Write your formula and prove it works for n=10 and n=11.
A student “proves” the formula n(n+1)/2 is wrong:
“For n=1, my formula gives 1(1+1)/2 = 1. But the sum is just 1. So 1 ≠ 1, therefore the formula fails!”
👉 What is their error? Explain why the formula actually works for n=1.
Akuuu sees stacked squares instead of triangles:
Layer 1: 1² = 1
Layer 2: 1² + 2² = 5
Layer 3: 1² + 2² + 3² = 14
👉 Can you adapt the pairing idea? Find a formula for 1²+2²+3²+…+n². (Hint: The answer involves n(n+1)(2n+1)/6)
🧠 Part A — Transfer
Before solving, look at this pattern:
1 = 1
1+2+1 = 4
1+2+3+2+1 = 9
1+2+3+4+3+2+1 = 16
👉 Predict: What is the pattern in the totals? Can you write a general formula for the sum 1+2+…+(n-1)+n+(n-1)+…+2+1?
Now prove it:
👉 Create a formula and prove it works for all n. Can you connect it to something you already know?
Think: 1+2+3+2+1 looks like a pyramid. Can you split it into two triangles? What does that tell you?
🧠 Part B — Thinking Pause
- I designed a new formula from a pattern
- I proved it works for all cases, not just examples
- I connected this to formulas I already knew
⭐ Mission Reflection
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