We see patterns all around us in the world. From the growth of money in a savings account to the decay of radioactive materials. It can be extremely helpful (and fun) to figure out how to get from one number in a pattern to the next. Knowing how a pattern works can help us make important predictions and plan for the future. A simple pattern (or sequence) is formed when the same number is added or subtracted at each step. Let's take a look at the examples below:

Numbers 2, 4, 6, 8 and 10. From each number is an arrow with plus 2 above it pointing to the next number.

This is an increasing pattern, where 2 is added at every step.

Numbers 17, 14, 11, 8 and 5. From each number is an arrow with minus 3 above it pointing to the next number.

This is a decreasing sequence, where 3 is subtracted at every step.

To find the next number that follows in a pattern, it's as simple as figuring out what the pattern is and applying it to the last number. For example, the next number in the decreasing pattern above would be 5-3=2. We could continue this pattern forever if we wanted to.

A table of values can be a nice way to organize a pattern. Below is a drawing of a pattern of flowers.

Flowers in groups of 1, 2, 3 and 4. Each flower has 8 petals.

A table can be generated to count the number of petals visible at a given time, based on how many flowers are present.

Number of flowers	1	2	3	4
Number of petals	8	16	24	32

Notice that the number of petals is increasing by 8 each time - in particular, the value in the table for Number of petals is always equal to 8 times the value for Number of flowers. Therefore, we could generate a rule for this table to say:

\text{Number of petals}=8 \times \text{Number of flowers}

Or to write it more mathematically:

y=8x

where x represents the number of flowers and y represents the number of petals.

This rule can now be used to predict future results. For example, to calculate the total number of petals when there are 10 flowers present, substitute x=10 into the rule to find y=8\times10=50 petals. So even though there were only 1, 2, 3 and 4 flowers present in the picture above, the rule has determined that there would be 80 petals visible when there are 10 flowers present.

Let's explore some different patterns in the practice questions below.

Examples

Example 1

Nadia knows that she is younger than her father, Glen. The following table shows her dad's age compared to hers:

Nadia's age	Glen's age
1	24
5	28
10	33
20	43
30	53

What is the difference in their ages?

Worked Solution

Create a strategy

Refer to the table and subtract their ages in the same row.

Apply the idea

\displaystyle \text{difference}	\displaystyle =	\displaystyle \text{Glen's age}-\text{Nadia's age}	Write the equation
	\displaystyle =	\displaystyle 24-1	Substitute their ages
	\displaystyle =	\displaystyle 23	Evaluate

The difference in their ages is 23 years.

What will Glen's age be when Nadia is 49 years old?

Worked Solution

Create a strategy

Using the answer in part (a), we can find Glen's age by adding 23 to Nadia's age.

Apply the idea

\displaystyle \text{Glen's age}	\displaystyle =	\displaystyle 49+23	Add 23 to Nadia's age
	\displaystyle =	\displaystyle 72	Evaluate

Glen will be 72 years old when Nadia is 49 years old.

Example 2

A catering company uses the following table to work out how many sandwiches are required to feed a certain number of people.

Fill in the blanks:

Number of People	Sandwiches
1	5
2	10
3	15
4	20
5	25

How many sandwiches are needed for each person?

Worked Solution

Create a strategy

Refer to the table for the number of sandwiches for 1 person.

Apply the idea

For each person, the caterer needs to make 5 sandwiches.

How many sandwiches are needed for 6 people?

Worked Solution

Create a strategy

Multiply the number of people by the number of sandwiches 1 person needs.

Apply the idea

\displaystyle \text{sandwiches}	\displaystyle =	\displaystyle 6\times5	Multiply the number of people by 5
	\displaystyle =	\displaystyle 30	Evaluate

For 6 people, the caterer would need to make 30 sandwiches.

Example 3

Consider the pattern shown on this line graph:

If the pattern continues on, what will the next point marked on the line be?

(3, 7)

(4, 5)

(3, 6)

(4, 6)

Worked Solution

Create a strategy

Every time the x-values are increased by 1, the corresponding y-value goes up also by 1.

Apply the idea

Find the coordinates of each point on the graph.

From the last point (2,5), we can now find the next point using the pattern of increasing the x and y-coordinates by 1.

\displaystyle x	\displaystyle =	\displaystyle 2+1	Add 1 to the last x-value
	\displaystyle =	\displaystyle 3	Evaluate

\displaystyle y	\displaystyle =	\displaystyle 5+1	Add 1 to the last y-value
	\displaystyle =	\displaystyle 6	Evaluate

The next point will be (3,6), option C.

Fill in the table with the values from the graph (the first one is filled in for you):

x\text{-value}	y\text{-value}
0	3
1
	5
3

Worked Solution

Create a strategy

Refer to the points on the line and our answer in part (a).

Apply the idea

x\text{-value}	y\text{-value}
0	3
1	4
2	5
3	6

Choose the three statements that correctly describe this pattern:

The rule is x+3=y.

As x increases, y increases.

The rule is y-3=x.

The rule is y+3=x.

Worked Solution

Create a strategy

Substitute one known point into each of the rules stated.

Apply the idea

Let us use the point (2,5).

For the rule x+3=y:

\displaystyle 2+3	\displaystyle =	\displaystyle 5	Substitute x and y values
\displaystyle 5	\displaystyle =	\displaystyle 5	The rule is correct

For the rule y-3=x:

\displaystyle 5-3	\displaystyle =	\displaystyle 2	Substitute x and y values
\displaystyle 2	\displaystyle =	\displaystyle 2	The rule is correct

For the rule y+3=x:

\displaystyle 5+3	\displaystyle =	\displaystyle 2	Substitute x and y values
\displaystyle 8	\displaystyle \neq	\displaystyle 2	The rule is incorrect

The statement "as x increases y increases" is correct since the pattern shows that every time the x-values increase by 1, the y-values increase by 1.

Options B, C and D are correct.

Idea summary

A simple pattern (or sequence) is formed when the same number is added or subtracted at each step.

Number patterns can be represented in tables, graphs and algebraic equations.

Review: Patterns in number tables

Ideas

Patterns in tables and graphs

Examples

Example 1

Example 2

Example 3

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