๐ฌ Watch the rule build a sequence
A consistent change reveals the pattern rule.
๐ฏ By the end of this lesson...
You should be able to:
- Explain what a number pattern is.
- Recognise increasing and decreasing patterns.
- Continue patterns involving addition and subtraction.
- Continue patterns involving multiplication and division.
- Find missing terms in a sequence.
- Describe a pattern using words and mathematical symbols.
- Recognise square-number and cube-number patterns.
- Recognise simple alternating patterns.
- Explore simple shape patterns.
- Solve practical problems involving patterns.
๐ค Think About This...
Planting trees along a road
A school plants more trees each year.
If the pattern continues, how many trees will be planted in the next stage?
Can you describe the rule without listing every future number?
1๏ธโฃ What Is a Number Pattern?
A number pattern is an ordered list of numbers that follows a particular rule.
Example of a number pattern
This is one example of a number pattern.
Its rule is:
Each number in a pattern is called a term.
๐ก Did You Know?
Patterns are not found only in mathematics.
Recognising patterns helps people predict what comes next, organise information and solve problems.
2๏ธโฃ Increasing Patterns
An increasing pattern becomes larger as the sequence continues.
Example 1: Find the rule and the next two terms
Compare consecutive terms.
The same amount is added each time.
3๏ธโฃ Decreasing Patterns
A decreasing pattern becomes smaller as the sequence continues.
Example 2: Find the rule and the next two terms
๐ต๏ธ Pattern Detective
When you meet a new sequence, investigate it instead of guessing.
Look at the direction
Are the terms increasing, decreasing or changing direction?
Compare consecutive terms
What happens from one term to the next?
Test a possible rule
Try addition or subtraction first, then multiplication or division.
Check every term
One rule should explain the complete sequence.
๐ง StudyNest Method: Finding a Pattern Rule
Step 2: Find how the first term changes into the second.
Step 3: Test the same change on the remaining terms.
Step 4: If addition or subtraction does not work, test multiplication or division.
Step 5: Write the rule clearly in words or symbols.
Step 6: Use the rule to continue the pattern.
4๏ธโฃ Finding Missing Terms
A missing term must follow the same rule as the rest of the pattern.
Example 3
The pattern increases by 4.
Example 4
The rule is subtract 5.
5๏ธโฃ Multiplication Patterns
Some patterns are formed by multiplying by the same number.
Example 5
Example 6
6๏ธโฃ Division Patterns
Some decreasing patterns are formed by repeatedly dividing by the same number.
Example 7
โ๏ธ Addition Patterns and Multiplication Patterns
Addition pattern
Add 3 each time.
The difference between consecutive terms stays the same.
Multiplication pattern
Multiply by 2 each time.
The amount added does not remain the same.
7๏ธโฃ Square-Number Patterns
The square numbers form an important pattern.
Example 8: Find the next two terms
These are:
๐ Spot the Difference Pattern
The differences between consecutive square numbers also follow a pattern.
The differences are consecutive odd numbers:
8๏ธโฃ Cube-Number Patterns
Example 9: Find the next term
These are consecutive cube numbers.
9๏ธโฃ Alternating Patterns
In an alternating pattern, more than one operation may repeat in a regular cycle.
Example 10
Compare each step:
The next step is โ1.
๐ Patterns Made from More Than One Operation
Example 11
Multiplying by 2 alone gives numbers that are one too small.
1๏ธโฃ1๏ธโฃ Shape Patterns
Patterns may also be shown using objects or shapes.
Stage 1
Stage 2
Stage 3
Stage 4
The number of dots follows:
๐ง StudyNest Method: Reading a Shape Pattern
Step 2: Write the totals as a number sequence.
Step 3: Compare consecutive totals.
Step 4: Describe what is added, removed or rearranged at each stage.
Step 5: Use the pattern to predict the next stage.
๐ Real-Life Example: Savings
Example 12
Rutendo saves US$5 in the first week and increases her weekly saving by US$3 each week.
How much will she save in the sixth week?
๐ Real-Life Example: Bus Fare
Example 13
A hypothetical bus fare changes by US$0.50 at each stage:
The next fare in the pattern is:
๐ Real-Life Example: Growing Seating Rows
Example 14
A school hall has 12 chairs in its first row.
Each new row contains 6 more chairs than the previous row.
How many chairs are in the sixth row?
๐ฏ Exam Tips
โ Common Mistakes
Assuming that every pattern uses addition.
Finding a rule from only the first two terms and failing to test it on the remaining terms.
Confusing an addition pattern with a multiplication pattern.
2, 4, 6, 8 adds 2, while 2, 4, 8, 16 multiplies by 2.
Ignoring alternating operations.
Counting objects incorrectly in a shape pattern.
Giving the rule but forgetting to answer the actual question.
๐ฎ Your Turn!
๐ข Foundation Questions
-
Write the next three terms:
5, 10, 15, 20, ... -
Write the next three terms:
40, 35, 30, 25, ... -
State the rule:
3, 6, 9, 12, 15, ... -
State the rule:
28, 24, 20, 16, ... -
Find the missing term:
8, 16, __, 32, 40 -
Find the missing term:
90, 80, __, 60, 50
๐ต Developing Questions
-
Write the next two terms:
4, 12, 36, 108, ... -
Write the next three terms:
128, 64, 32, 16, ... -
State the rule and find the next term:
7, 14, 28, 56, ... -
Write the next three square numbers:
1, 4, 9, 16, 25, ... -
Write the next two cube numbers:
1, 8, 27, 64, ... -
Find the next two terms:
2, 5, 4, 7, 6, 9, ... -
Find the next term:
3, 7, 15, 31, ... - A shape pattern has 4 objects at Stage 1, 7 objects at Stage 2 and 10 objects at Stage 3. How many objects will Stage 5 contain?
๐ด Challenge Questions
-
Find the next two terms:
1, 2, 6, 24, 120, ... -
Find the missing term:
5, 11, 23, __, 95 -
Describe the pattern:
100, 50, 25, 12.5, ... - A farmer plants 12 trees in the first row, 18 trees in the second, 24 in the third and 30 in the fourth. How many trees will be in the sixth row?
- A learner saves US$10 in Week 1 and increases the amount saved by US$4 every week. How much will the learner save in Week 8?
- A tile pattern contains 3 tiles in Stage 1, 6 tiles in Stage 2, 9 tiles in Stage 3 and 12 tiles in Stage 4. How many tiles are needed for Stage 20?
โ Show Answers
- 25, 30, 35
- 20, 15, 10
- Add 3
- Subtract 4
- 24
- 70
-
Multiply by 3:
324, 972 -
Divide by 2:
8, 4, 2 -
Multiply by 2.
Next term = 112 - 36, 49, 64
- 125, 216
-
Operations alternate between +3 and โ1.
Next terms: 8, 11 -
Multiply by 2, then add 1.
Next term = 63 -
The pattern increases by 3.
Stage 4 = 13
Stage 5 = 16 -
The terms are multiplied by consecutive whole numbers:
ร2, ร3, ร4, ร5, ...
120 ร 6 = 720
720 ร 7 = 5040
Next terms: 720, 5040 -
Multiply by 2, then add 1.
23 ร 2 + 1 = 47 - Divide by 2 each time.
-
Add 6 each time.
Fifth row = 36
Sixth row = 42 trees -
The amounts are:
10, 14, 18, 22, 26, 30, 34, 38
Week 8 = US$38 -
The pattern is 3 times the stage number.
Stage 20 = 3 ร 20
= 60 tiles
๐ง Remember This
Each number in a sequence is called a term.
Check addition and subtraction before multiplication and division.
Some patterns alternate between two operations.
Square numbers and cube numbers form special patterns.
Shape patterns can be converted into number sequences.
Always check that your rule works throughout the pattern.
๐ฏ Before Moving On...
- โ Can I explain what a number pattern is?
- โ Can I identify the terms of a sequence?
- โ Can I continue an increasing or decreasing pattern?
- โ Can I recognise multiplication and division patterns?
- โ Can I find missing terms?
- โ Can I recognise square-number and cube-number patterns?
- โ Can I investigate an alternating pattern?
- โ Can I convert a shape pattern into a number sequence?
- โ Can I explain a pattern clearly?
๐ Well Done!
You can now investigate, continue and describe different kinds of number patterns.
Compare the terms โ Test a rule โ Check the whole sequence โ Continue the pattern
๐ Your Progress
Lesson 11 of 18