Unit 2 · Topic 01 · Number Play

Number Patterns: Squares and Triangles

Hook

Pixel asked Kabir and Anaya to find the sum of the first 9 odd numbers — 1, 3, 5, 7, 9, 11, 13, 15, and 17.

Kabir reached for paper to add them one at a time.

Anaya said the answer was 81 before he had finished the first row.

Watch the Lesson

The Story

"What is the sum of the first 9 odd numbers?" asked Pixel. Kabir listed them: 1, 3, 5, 7, 9, 11, 13, 15, 17, and started adding carefully. Anaya answered at once: "81."

"How can you know without adding?" Kabir asked. Pixel wrote smaller cases on the board. Sum of the first 2 odd numbers: 1+3=4, which is 2 squared. Sum of the first 3: 1+3+5=9, which is 3 squared. Sum of the first 4: 1+3+5+7=16, which is 4 squared.

"Odd numbers are whole numbers not divisible by 2," said Pixel — 1, 3, 5, 7, 9, and so on. "Watch the pattern: however many odd numbers you add, starting from 1, the total is that count squared."

Kabir tested it. Nine odd numbers meant 9 squared, and 9×9=81. He added all nine terms and got exactly 81. Pixel then set a bigger challenge: "What is the sum of the first 12 odd numbers?"

Anaya used the same rule. Twelve odd numbers, so 12 squared is 144. Kabir wrote out 1+3+5+7+9+11+13+15+17+19+21+23 and checked — the total was 144. "So square numbers grow from odd-number sums," said Anaya, "and that is the square side of our number-pattern story."

1+3=4=2², 1+3+5=9=3², 1+3+5+7=16=4² — each step makes the next square.

So What Just Happened?

Odd numbers are whole numbers not divisible by 2: 1, 3, 5, 7, 9, and so on.

Adding consecutive odd numbers starting from 1 builds square numbers: 1+3=4=2², 1+3+5=9=3², 1+3+5+7=16=4².

The rule is general: the sum of the first n odd numbers equals n squared (n×n). For example, the first 9 odd numbers sum to 9²=81, and the first 12 sum to 12²=144.

You can always check by adding the odd terms directly, but counting how many odd numbers you have and squaring that count is faster.

Nine odd terms means 9 squared: 9 x 9 = 81.

Remember This

Odd numbers are not divisible by 2: 1, 3, 5, 7, 9, and so on.
1+3=4=2² and 1+3+5=9=3² — each new odd term grows the next square number.
The sum of the first n odd numbers always equals n squared.
First 9 odd numbers sum to 9²=81; first 12 odd numbers sum to 12²=144.
Count how many odd numbers you are adding, square that count, then check by adding if you want.

Twelve odd terms means 12 squared: 12 x 12 = 144.

Try It Yourself

Work out 1+3 and 1+3+5, and write each total as a square (2² and 3²).

Find the sum of the first 9 odd numbers using the n-squared rule, then check by adding all nine terms.

Try the video challenge: predict the sum of the first 12 odd numbers as 12 squared, then add 1+3+5+7+9+11+13+15+17+19+21+23 to verify 144.

Word Bank

Odd number: A whole number not divisible by 2, like 1, 3, 5, 7.
Square number: A number formed by multiplying a whole number by itself, like 9=3×3.
Consecutive: Following one after another in order, with nothing skipped.
Pattern: A rule or arrangement that repeats or grows in a predictable way.
Sum: The total when numbers are added together.

Questions

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