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Special numbers

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visibility 81update a month agobookmarkshare

🎯 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.

📘 Worked example

Which is larger?

$2^3$ or $3^2$

Explain your answer.

Answer: $3^2 > 2^3$

$2^3 = 2 \times 2 \times 2 = 8$

$3^2 = 3 \times 3 = 9$

Explain your answer by showing the calculations for $2^3$ and $3^2$.

❓ EXERCISES

$1.$What is the ninth square number?

👀 Show answer

The ninth square number is $9^2 = 81$.

$2.$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

One correct pair is $36$ and $9$, because $36 + 9 = 45$ (that is, $6^2 + 3^2 = 45$).

$4.$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

Answers will vary. One correct set is:

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

$152 = 125 + 27 = 5^3 + 3^3$, so the two cube numbers are $125$ and $27$.

$8.$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

One sensible way is to group them by value:

Equal to $8$:$2^3$, half of $4^2$, $3^2 - 1$
Equal to $9$:$3^2$, $2^3 + 1$

Explanation: evaluate each expression and then sort expressions with the same result into the same group.

🧠 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 $n^2$.

📘 What we've learned

We learned how to identify and work out square numbers in sequence positions, for example the ninth square number is $9^2 = 81$.
We used the notation $^2$ to represent a number being squared, and practiced calculating square numbers such as $5^2$, $7^2$, and $10^2$.
We learned about cube numbers and used the notation $^3$ to represent cubing, for example $5^3 = 5 \times 5 \times 5 = 125$.
We counted cubes in 3D cube models using cube numbers, including examples like $2^3 = 8$, $4^3 = 64$, and $5^3 = 125$.
We compared powers such as $2^3$ and $3^2$ by evaluating each expression numerically.
We explored number patterns, including sums of consecutive square numbers and the pattern that the sum of the first $n$ odd numbers is $n^2$.
We classified numbers and expressions (for example using odd/not odd and cube/not cube) by calculating values and grouping them logically.

Special numbers

🎯 In this topic you will

🧠 Key Words

🔢 Learning About Cube Numbers

❓ EXERCISES

🧠 Think like a Mathematician

📘 What we've learned

Related Past Papers

Related Tutorials

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