Square and triangular numbers
🎯 In this topic you will
- Recognise and extend spatial patterns that represent square numbers and triangular numbers
- Recognise square numbers from 1 to 100
🧠 Key Words
- spatial pattern
- square number
- triangular number
Show Definitions
- spatial pattern: A visual or geometric arrangement of objects, shapes, or dots that follows a predictable structure.
- square number: A number that results from multiplying a whole number by itself, such as 1, 4, 9, 16, and 25.
- triangular number: A number that represents a triangle made of equally spaced dots, found by adding consecutive whole numbers (1, 3, 6, 10, 15, ...).
Patterns of Squares
Look at these examples of patterns of squares.
Exploring Number Patterns
Look along the diagonal of this table square.
Two Special Number Sequences
In this section, you will learn about two number sequences: square numbers and triangular numbers.
💡 Quick Math Tip
Spot the diagonal pattern: The numbers found along the diagonal of a multiplication table are always square numbers, because each one shows a number multiplied by itself.
❓ EXERCISES
1. These patterns of dots show the first four square numbers.
a. Draw a dot pattern for the 5th square number.
b. What is the 10th square number?
👀 Show answer
a. The 5th square number is $5^2 = 25$, so the pattern is a $5 \times 5$ square of dots.
b. $10^2 = 100$
2. Copy and complete the Carroll diagram by writing a number greater than $50$ but less than $100$ in each space.
| Square number | Not a square number | |
|---|---|---|
| Even number | $64$ | $52$ |
| Not an even number | $81$ | $53$ |
👀 Show answer
Example solution shown in the table above. Many correct answers are possible.
3. Look at this number pattern made using counters.
The pattern starts $1, 3, 6, 10$.
a. What are the next two numbers in the sequence?
b. What is the name for this sequence of numbers?
👀 Show answer
a. $15, 21$
b. Triangular numbers
4. Look at this pattern of numbers.
a. Can you see how the pattern continues? Draw the next two rows of the triangle.
b. Find the sum of the numbers in each row, then write down the first eight numbers in the sequence.
c. Describe the sequence.
👀 Show answer
a. Next rows:
$1\quad5\quad10\quad10\quad5\quad1$
$1\quad6\quad15\quad20\quad15\quad6\quad1$
b. Row sums: $1, 2, 4, 8, 16, 32, 64, 128$
c. Each number doubles the previous one (powers of $2$).
5. Calculate these square numbers.
a. $6^2$
b. $8^2$
c. $9^2$
👀 Show answer
a. $36$
b. $64$
c. $81$
🧠 Think like a Mathematician
You can multiply $15$ by itself to give a 3-digit number: $15 \times 15 = 225$.
Investigation Task: Explore which 2-digit numbers multiply by themselves to produce 3-digit answers.
Step 1. Find the smallest 2-digit number that you can multiply by itself to give a 3-digit number.
Step 2. Use a calculator to investigate the largest 2-digit number that you can multiply by itself to give a 3-digit number.
□ □ × □ □ = □ □ □
Success criteria:
- You will show you are specialising when you find numbers that satisfy the given criteria.
- If you explain your results, you will show you are convincing.
Show Answers
- Smallest 2-digit number that squares to a 3-digit number:$10^2 = 100$.
- Largest 2-digit number that squares to a 3-digit number:$31^2 = 961$ because $32^2 = 1024$ is 4-digits.
- General conclusion: All 2-digit numbers from $10$ to $31$ inclusive produce 3-digit results when multiplied by themselves.