Numeric and geometric patterns (Grade 8) — Mathematics Grades 8–11

Extending number and shape patterns, and describing the rule that generates them.

Gr 8 Term 1 — Numeric and geometric patternsTerm 1

Practise Numeric and geometric patterns in the app →

What gets asked

Extend a sequence of numbers or a diagram pattern.
Describe the rule of a pattern in words or algebra.
Use a rule to find a term far along a sequence.
Find the position of a given term in a sequence.

You must be able to

Test a rule on every given term, not just the first two.
Write a general rule connecting the position number to the term.
Substitute a position number into a rule to find a term.
Count what actually grows in a diagram pattern, not just the shapes.

Traps that cost marks

Working out a rule from the first two terms only, without checking it against the rest. Test your rule on at least three terms before you trust it.
Forcing a constant difference onto a pattern that does not actually have one. If the difference is not the same each time, look for another pattern, like a changing difference.
Counting the shapes in a diagram pattern instead of counting the matchsticks or dots that grow. Look at what is actually being added at each stage, not just how many shapes there are.
Listing out every term one by one to find a term far along the sequence. Use the general rule and substitute the position number directly. It is faster and more accurate.

Worked example

A pattern of soccer-ball stacks has 3 balls in stack 1, 5 in stack 2, 7 in stack 3. Find stack 10.

The difference between terms is always 2, so the rule has the form $2 n + c$ .
Test $n = 1$ : $2 (1) + c = 3$ , so $c = 1$ . Rule: $2 n + 1$ .
Stack 10: $2 (10) + 1 = 21$ balls.

Extend patterns

How to spot the way a list of numbers is growing, so you can carry it on safely.

sequence: A list of numbers written in a set order, like $4; 7; 10; 13$ .
term: One of the numbers in a sequence. In $4; 7; 10$ the second term is $7$ .
position: The place a term holds. The first term is at position $1$ .
constant difference: The same amount added every time. In $4; 7; 10$ you add $3$ each step.
constant ratio: The same number you multiply by every time. In $2; 6; 18$ that is $3$ .

A sequence is a list of numbers in a set order. Each number in it is a term, and each term has a position. To carry a sequence on, first work out how it grows.

Test for a constant difference first. Take each term away from the one after it. In $5; 9; 13$ every gap comes to $4$ . Check three gaps, not one. Two terms alone can agree by luck.

If the gaps differ, test for a constant ratio. Divide each term by the one before it. In $2; 6; 18$ each term is $3$ times the one before. Keep the minus signs: in $5; - 10; 20$ the constant ratio is $- 2$ .

Some patterns have neither. Do not force a gap that is not there. Study the gaps themselves. In $1; 2; 4; 7; 11$ the gaps run $1; 2; 3; 4$ , growing by one.

A picture pattern grows too. Count what is added, like matchsticks or dots, not the shapes. Write the stage number and the count in a table.

Rules to remember

A constant difference means you add. A constant ratio means you multiply.

Examples

Write the next two terms:

6; 10; 14; 18; \dots

Worked answer

Find the gaps: $10 - 6 = 4$ , then $14 - 10 = 4$ , then $18 - 14 = 4$ .
Three gaps agree, so the constant difference is $4$ .
Add it on twice: $18 + 4 = 22$ , then $22 + 4 = 26$ .

Answer: $22$ and $26$

Three gaps agreed, so the rule was safe.

Write the next two terms:

3; - 6; 12; - 24; \dots

Worked answer

The gaps are not equal, so divide instead.
$- 6 \div 3 = - 2$ , and $12 \div (- 6) = - 2$ .
The constant ratio is $- 2$ , minus sign and all.
$- 24 \times (- 2) = 48$ , then $48 \times (- 2) = - 96$ .

Answer: $48$ and $- 96$

Holding the minus sign is what makes the signs swap over.

Matchstick squares in a row use

4

, then

7

, then

10

sticks. How many sticks for shape

5

Worked answer

Put it in a table: shapes $1; 2; 3$ against sticks $4; 7; 10$ .
Count sticks, not squares.
Each new square adds three: $7 - 4 = 3$ and $10 - 7 = 3$ .
Shape $4$ needs $10 + 3 = 13$ sticks.
Shape $5$ needs $13 + 3 = 16$ sticks.

Answer: $16$ matchsticks

The picture grew by one square, but the count grew by three sticks.

Traps

Working out a rule from the first two terms only, without checking it against the rest. Test your rule on at least three terms before you trust it.
Dropping the minus sign, so $3; - 6; 12$ carries on as $24$ . The ratio here is $- 2$ , so the next term is $12 \times (- 2) = - 24$ .
Forcing a constant difference onto a pattern that does not actually have one. If the difference is not the same each time, look for another pattern, like a changing difference.
Counting the shapes in a diagram pattern instead of counting the matchsticks or dots that grow. Look at what is actually being added at each stage, not just how many shapes there are.

Learn and practise “Extend patterns” in the app →

Find the general rule

Writing one short rule that reaches any place in a pattern, without counting your way there.

general rule: A rule that gives any term straight from its position number.
substitute: To write a number in place of a letter, then work the answer out.

Saying "add $3$ each time" only takes you one step further. It cannot reach the 50th term. A general rule can, because it works straight from the position number, which we call $n$ .

Build the rule from the constant difference. If the difference is $3$ , the rule starts with $3 n$ . At position $1$ that gives $3$ , but the first term of $7; 10; 13$ is $7$ . So add $4$ , and the rule is $3 n + 4$ .

Now test it on every term you were given. Position $2$ : $3 (2) + 4 = 10$ . Position $3$ : $3 (3) + 4 = 13$ . Both match, so the rule is safe to use.

With the rule, jump straight to any position. Substitute $n = 50$ to get $3 (50) + 4 = 154$ . You never list the terms in between.

Going backwards is the same rule, undone. If a term is $61$ , take the $4$ away, then divide by $3$ . The $61$ is a term, so it must never be put in place of $n$ .

Rules to remember

The constant difference tells you what to multiply the position by.

Examples

Find the general rule for

5; 8; 11; 14; \dots

Worked answer

The gaps are equal: $8 - 5 = 3$ , so the difference is $3$ .
That makes the rule start with $3 n$ .
At position $1$ , $3 (1) = 3$ , but the term is $5$ , so add $2$ .
Test it: $3 (2) + 2 = 8$ and $3 (4) + 2 = 14$ .

Answer: $3 n + 2$

Every given term was tested, not only the first one.

A stokvel member pays R150 to join, then R80 a month. After

n

months she has paid

80 n + 150

rands. Find the total after

12

months.

Worked answer

Substitute $n = 12$ into the rule.
$80 (12) = 960$ .
$960 + 150 = 1110$ .

Answer: R1 110

The rule reached month $12$ in one line, with no list of months.

A sequence has the rule

3 n + 4

. Which term is

61

Worked answer

The $61$ is a term, not a position, so undo the rule.
Take away the $4$ : $61 - 4 = 57$ .
Undo the multiply: $57 \div 3 = 19$ .
Check it: $3 (19) + 4 = 61$ .

Answer: the $19$ th term

Undoing the rule in reverse order lands on the position, not on a term.

Traps

Answering "add $3$ each time" when the question asks for a general rule. That only gives the next term. A general rule must reach any position on its own.
Writing a rule that fits the first term and never testing it on the others. Check the rule at every position you were given. One match is not proof.
Listing out every term one by one to find a term far along the sequence. Use the general rule and substitute the position number directly. It is faster and more accurate.
Putting the term value $61$ where the position number belongs. To find a position, undo the rule: $61 - 4 = 57$ , then $57 \div 3 = 19$ .

Learn and practise “Find the general rule” in the app →

Grade 8 Numeric and geometric patterns

What gets asked

You must be able to

Traps that cost marks

Worked example

Extend patterns

Rules to remember

Examples

Traps

Find the general rule

Rules to remember

Examples

Traps

About this material