Topics
1. Number and place value
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AustraliaNSW
Stage 3

1.09 Square and triangular numbers

Lesson
Practice
Lesson

Are you ready?

We've seen that  prime and composite numbers  are two types of numbers that can help us with calculations.

Examples

Example 1

The only factors a prime number has is 1 and itself.

Which of the following numbers are prime?

A
10
B
21
C
19
Worked Solution
Create a strategy

Write the factors of each option.

Apply the idea

Option A: 10 has the following factors:1,2,5,10

Option B: 21 has the following factors:1,3,7,21

Option C: 19 has the following factors:1,19

Among the choices, 19 is a prime number as its factors are 1 and itself. So, the correct answer is Option C.

Idea summary

  • Every whole number greater than 1 is either a prime number or a composite number

  • All even numbers greater than 2 are composite numbers

  • To be a prime number, a number can only have itself and 1 as factors

  • 0 and 1 are not prime or composite numbers

Square numbers

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Examples

Example 2

Write down the next square number after 16.

Worked Solution
Create a strategy

Find the 5th square number.

Apply the idea

16 is the 4th square number since 4\times 4=16.

To find the square number after 16, we need to find the 5th square number which is:5\times 5=25

25 is the next square number after 16.

Idea summary

If we multiply a number by itself, we make a square number. If we use dots to picture this as an array it will make a square shape.

Triangular numbers

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Examples

Example 3

Write down the next 3 triangular numbers.1,\,3,\,6,\,10,\,15,\,⬚,\,⬚,\,⬚

Worked Solution
Create a strategy

Add the the position of the number we are finding to the previous triangular number.

Apply the idea

To find the 6th triangular number we add 6 to the previous triangular number of 15:

\displaystyle 15+6\displaystyle =\displaystyle 21

To find the 7th triangular number we add 7 to the previous triangular number of 21:

\displaystyle 21+7\displaystyle =\displaystyle 28

To find the 8th triangular number we add 8 to the previous triangular number of 28:

\displaystyle 28+8\displaystyle =\displaystyle 36

Here is the complete list of triangular numbers:1,\,3,\,6,\,10,\,15,\,21,\,28,\,36

Idea summary

With triangular numbers, we add 1 more dot to each new row and create a triangle shape with dots.

Dot representations of 16 as a square number and 10 as a triangular number. Ask your teacher for more information.

Outcomes

MA3-4NA

orders, reads and represents integers of any size and describes properties of whole numbers

MA3-5NA

selects and applies appropriate strategies for addition and subtraction with counting numbers of any size

MA3-6NA

selects and applies appropriate strategies for multiplication and division, and applies the order of operations to calculations involving more than one operation

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