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💡 A Formula Is a Recipe
Think of a formula like a recipe. The letter is the ingredient (the number you put in), and the operations (×, +, −, ÷) are the cooking steps.
To use any formula: ① Read it → ② Plug in the number → ③ Calculate step by step.
📖 Formula Recipe #1: n × n (n squared)
- Take your number n
- Multiply it by itself
- That's your answer!
Squaring a number
n × n = n²
Worked Example
Find n² when n = 4:
Step 1: Write the formula → n × n
Step 2: Replace n with 4 → 4 × 4
Step 3: Calculate → 16 ✓
1 Practice n² 🔲
Follow the steps to find n² for each value.
n = 3 → 3 × 3 =
n = 5 → × =
n = 7 → × =
n = 10 → × =
2 Quick Check: True or False? ✅
Circle True or False for each statement.
5² = 25
True / False
8² = 16
True / False
10² = 100
True / False
n × n only works when n is even
True / False
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📖 Formula Recipe #2: n × (n + 1) ÷ 2
- Take your number n
- Add 1 to it → that gives you n + 1
- Multiply n by n + 1
- Divide the result by 2
Adding all numbers from 1 to n
1 + 2 + 3 + ⋯ + n = n × (n + 1) ÷ 2
Worked Example
Find 1 + 2 + 3 + 4 + 5 using the formula (n = 5):
Step 1: n + 1 = 5 + 1 = 6
Step 2: n × (n + 1) = 5 × 6 = 30
Step 3: Divide by 2 → 30 ÷ 2 = 15 ✓
Check: 1+2+3+4+5 = 15 ✓ It works!
3 Practice n × (n + 1) ÷ 2 📐
Follow the recipe steps for each value of n.
n = 4
Step 1: n + 1 = 4 + 1 =
Step 2: n × (n+1) = 4 × =
Step 3: ÷ 2 =
n = 6
Step 1: n + 1 =
Step 2: n × (n+1) = × =
Step 3: ÷ 2 =
n = 8
Step 1: n + 1 =
Step 2: n × (n+1) =
Step 3: ÷ 2 =
n = 10
Step 1: n + 1 =
Step 2: n × (n+1) =
Step 3: ÷ 2 =
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🎨 Connecting to Picture Proofs
Remember the odd numbers formula: the nth odd number is 2n − 1.
There's an amazing fact: if you add the first n odd numbers together, you always get n²!
Let's check by building up step-by-step:
4 Adding Odd Numbers → Perfect Squares ✨
Fill in each row. Notice the pattern in the last column!
| How many? | Which odd numbers? | Sum | Is it n²? |
|---|
| n = 1 |
1 |
1 |
1² = 1 ✓ |
| n = 2 |
1 + 3 |
|
2² = |
| n = 3 |
1 + 3 + 5 |
|
3² = |
| n = 4 |
1 + 3 + 5 + 7 |
|
4² = |
| n = 5 |
1 + 3 + 5 + 7 + 9 |
|
5² = |
| n = 6 |
1 + 3 + 5 + 7 + 9 + 11 |
|
6² = |
| n = 7 |
1 + 3 + 5 + 7 + 9 + 11 + 13 |
|
7² = |
| n = 8 |
1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 |
|
8² = |
5 Predict Without Adding 🔮
Use the pattern you just found (sum of first n odd numbers = n²) to jump straight to the answer.
Sum of the first 10 odd numbers
= 10² =
Sum of the first 12 odd numbers
= ² =
Sum of the first 7 odd numbers
=
Sum of the first 20 odd numbers
=
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6 Use the Shortcut! ⚡
Now that you know the pattern, use n² to find these sums instantly!
1 + 3 + 5 + … + 17
That's the first 9 odd numbers
(since 2×9−1 = 17)
n = 9 → 9² =
1 + 3 + 5 + … + 39
2 × n − 1 = 39 → n =
² =
1 + 2 + 3 + … + 20
Use n × (n+1) ÷ 2
20 × 21 ÷ 2 =
1 + 2 + 3 + … + 50
Use n × (n+1) ÷ 2
50 × ÷ 2 =
7 Pick the Right Formula 🧠
Decide which formula to use, then solve!
What is 1 + 2 + 3 + … + 12?
Formula:
Answer:
What is 1 + 3 + 5 + … + 23?
How many odd numbers?
Answer:
What is 8²?
Formula: n × n
Answer:
What is the 15th odd number?
Formula: 2 × n − 1
Answer:
8 Real-World Challenge 🌍
Use a formula from this lesson to solve each real-world problem.
A stadium has 12 rows. Row 1 has 1 seat, row 2 has 2 seats, … up to row 12.
Total seats =
A square tile floor is 9 tiles by 9 tiles.
Total tiles = 9² =
A stack of cans has 1 can on top, 3 in the 2nd row, 5 in the 3rd, … through the 6th row.
Total cans =
Which is bigger — 6² or 1 + 2 + … + 11?
Circle: 6² / the sum
Answer Key:
1) 9, 25, 49, 100 —
2) True, False (8²=64), True, False —
3) 5→20→10, 7→42→21, 9→72→36, 11→110→55 —
4) 4, 4; 9, 9; 16, 16; 25, 25; 36, 36; 49, 49; 64, 64 —
5) 100, 144, 49, 400 —
6) 81; n=20→400; 210; 1275 —
7) 78, 12 odd numbers→144, 64, 29 —
8) 78, 81, 36, the sum (66 > 36)