Special numbers
🎯 In this topic you will
- Work out square numbers in any position, including identifying terms such as the ninth square number ($9 \times 9 = 81$).
- Use the notation $^2$ to represent a number being squared.
- Find and recall cube numbers up to $5^3$, and explain cubes as repeated multiplication (for example, $5^3 = 5 \times 5 \times 5 = 125$).
🧠 Key Words
- cube number
- square number
Show Definitions
- cube number: A number made by multiplying the same whole number by itself three times, such as $2 \times 2 \times 2 = 8$.
- square number: A number made by multiplying a whole number by itself once, such as $6 \times 6 = 36$.
🔢 Learning About Cube Numbers
In this section you will learn about cube numbers.
❓ EXERCISES
$1.$What is the ninth square number?
👀 Show answer
$2.$Calculate.
$a.$$5^2$
$b.$$10^2$
$c.$$7^2$
👀 Show answer
$a.$$5^2 = 25$
$b.$$10^2 = 100$
$c.$$7^2 = 49$
$3.$Find two square numbers that total $45$ when added together.
👀 Show answer
$4.$Here are three cubes of increasing size.
How many small cubes are in each of the large cubes?
👀 Show answer
$A:$$2 \times 2 \times 2 = 8$
$B:$$4 \times 4 \times 4 = 64$
$C:$$5 \times 5 \times 5 = 125$
$5.$Calculate.
$a.$$5^3$
$b.$$1^3$
$c.$$3^3$
Check your answers to questions $4$ and $5$ with your partner.
👀 Show answer
$a.$$5^3 = 125$
$b.$$1^3 = 1$
$c.$$3^3 = 27$
$6.$Copy this Carroll diagram and write a number less than $100$ in each section.
| Odd | Not odd | |
|---|---|---|
| Cube number | ||
| Not a cube number |
👀 Show answer
- Cube number + Odd: $27$
- Cube number + Not odd: $64$
- Not a cube number + Odd: $15$
- Not a cube number + Not odd: $20$
$7.$Find two cube numbers that total $152$ when added together.
👀 Show answer
$8.$Classify these expressions into two groups. Explain how you chose the groups.
$2^3$ $3^2$ $2^3 + 1$ half of $4^2$ $3^2 - 1$
Discuss your answer with your partner.
👀 Show answer
- Equal to $8$:$2^3$, half of $4^2$, $3^2 - 1$
- Equal to $9$:$3^2$, $2^3 + 1$
🧠 Think like a Mathematician
Two consecutive squares
$1$ and $4$ are two consecutive square numbers. $1 + 4 = 5$
$4$ and $9$ are two consecutive square numbers. $4 + 9 = 13$
Investigate the sums of two consecutive square numbers.
If you are systematic you should find an interesting pattern.
Adding odd numbers
The first two odd number are $1$ and $3$. Their sum is $4$
The first five odd numbers are $1, 3, 5, 7$ and $9$. What is their sum?
Investigate the sums of consecutive odd numbers starting at $1$.
What do you notice?
👀 show answer
For consecutive squares:
- $1 + 4 = 5$
- $4 + 9 = 13$
- $9 + 16 = 25$
- $16 + 25 = 41$
- $25 + 36 = 61$
If the consecutive squares are $n^2$ and $(n+1)^2$, then $n^2 + (n+1)^2 = 2n^2 + 2n + 1$.
A neat pattern: these sums are always odd numbers.
For adding odd numbers starting at $1$:
- $1 = 1 = 1^2$
- $1 + 3 = 4 = 2^2$
- $1 + 3 + 5 = 9 = 3^2$
- $1 + 3 + 5 + 7 = 16 = 4^2$
- $1 + 3 + 5 + 7 + 9 = 25 = 5^2$
So the sum of the first $n$ odd numbers is always $n^2$.