Counting and sequences
🧩 Let’s Start with a Problem
Lila is decorating a wall with square tiles. She starts with a single blue tile. Then, around it, she places a ring of white tiles, making a larger square. Next, she adds another ring of blue tiles, making an even larger square. She continues this pattern, alternating colors.
| Step | Total tiles |
|---|---|
| $1$ | $1$ |
| $2$ | $4$ |
| $3$ | $9$ |
| $4$ | $16$ |
If Lila continues this pattern, how many tiles will she use at step $7$? Can you describe the rule that tells you how many tiles are needed for each step?
💡 Curiosity Question
How can we predict the number of tiles at any step without drawing all the previous squares?
🌟 Why This Matters
Many things in the world around us follow patterns. From the way buildings are designed to the way plants grow, understanding patterns helps us make predictions and solve problems more efficiently. By learning to recognize and continue patterns, you’ll be able to see the order hidden in many everyday situations.
✏️ Quick Warm‑Up Activities
Activity 1: Draw the Next Shape
Here is a sequence of shapes: a triangle, a square, a pentagon, a hexagon. Draw the next two shapes in the sequence. What do you notice about the number of sides each shape has?
Think about it: Each new shape has one more side than the previous one. This is a pattern in how the shapes change.
Activity 2: Find the Missing Numbers
Look at these number sequences. Can you find the missing numbers?
$4$, $7$, $10$, $13$, $\_\_\_$
$20$, $18$, $16$, $14$, $\_\_\_$
Think about it: Find the rule that connects one number to the next. Is it always the same amount added or subtracted?
Activity 3: Create Your Own Pattern
Using colored counters or drawings, make a pattern of shapes that grows in a regular way. For example, you could start with 2 circles, then 4 squares, then 6 triangles, and so on. Show your pattern to a partner and ask them to describe the rule.
Think about it: A good pattern has a clear rule that tells you how to get from one term to the next.
🔍 Ready to Explore?
Now let’s learn how mathematicians find, describe, and continue patterns using counting and sequences!
🎯 In this topic you will
- Count forwards and backwards in steps of tens, hundreds, and thousands starting from any number.
- Count backwards through zero to include negative numbers, such as −2.
- Recognise and distinguish between linear sequences and non-linear sequences.
- Extend numerical sequences and describe the term-to-term rule that generates them.
- Recognise and extend patterns that represent square numbers.
🧠 Key Words
- difference
- linear sequence
- negative number
- non-linear sequence
- rule
- sequence
- spatial pattern
- square number
- term
- term-to-term rule
Show Definitions
- difference: The result of subtracting one number from another.
- linear sequence: A sequence in which the difference between consecutive terms is constant.
- negative number: A number that is less than zero and usually written with a minus sign.
- non-linear sequence: A sequence in which the difference between consecutive terms is not constant.
- rule: A mathematical instruction that describes how a sequence or pattern is generated.
- sequence: An ordered list of numbers that follow a specific pattern or rule.
- spatial pattern: A pattern formed by shapes or objects arranged in space rather than numbers.
- square number: A number made by multiplying an integer by itself, such as 1, 4, or 9.
- term: An individual number or element in a sequence.
- term-to-term rule: A rule that explains how to get from one term in a sequence to the next.
Counting in Equal Steps
Y ou will continue counting forwards and backwards in steps of a constant size, and you will begin to use negative numbers as part of your counting.
Temperatures Below Zero
A round the coasts of Antarctica, temperatures are often below zero, typically between −10 °C and −30 °C.
Counting Back Through Zero
T ry counting back in tens, starting at 30 and continuing past zero until you reach −30.
❓ EXERCISES
1.
a. Mia counts on in steps of $100$. She starts at $946$. Write the next number she says.
b. Kofi counts back in steps of $100$. He starts at $1048$. Write the next number he says.
c. Bibi counts on in steps of $1000$. She starts at $1989$. Write the next number she says.
d. Pierre counts back in steps of $1000$. He starts at $9999$. Write the next number he says.
e. Tara counts back in ones. She counts $3, 2, 1, 0$. Write the next number she says.
👀 Show answer
a. $1046$
b. $948$
c. $2989$
d. $8999$
e. $-1$
2. Copy and complete this square using the rule ‘add $2$ across and add $2$ down’. What do you notice about the numbers on the diagonal? Discuss with your partner. Draw two more $5$ by $5$ squares and choose a rule using addition. Predict what the numbers on the diagonal will be before you complete the squares.
👀 Show answer
The diagonal numbers increase by a constant amount. With add $2$ across and down, the diagonal increases by $4$ each step.
3. Choose any two of these three sequences. How are they similar to each other and how are they different?
$2, 4, 6, 8, \dots$
$2, 5, 8, 11, \dots$
$3, 5, 7, 9, \dots$
👀 Show answer
All are linear sequences. They differ in their starting values and common differences.
4. Look at these sequences. Which could be the odd one out? Explain your answer.
$13, 16, 19, 22, \dots$
$8, 11, 14, 17, \dots$
$-5, -2, 1, 4, \dots$
$9, 12, 15, 18, \dots$
$16, 19, 22, 25, \dots$
👀 Show answer
$-5, -2, 1, 4, \dots$ could be the odd one out because it includes negative numbers.
5. Use different first terms to make sequences that all have the term-to-term rule ‘add $3$’. Can you find a sequence for each of the following?
a. Where the terms are all multiples of $3$.
b. Where the terms are not whole numbers.
c. Where the terms are all odd.
d. Where the terms include both $100$ and $127$.
👀 Show answer
a. $3, 6, 9, \dots$
b. $0.5, 3.5, 6.5, \dots$
c. $1, 4, 7, 10, \dots$ is not all odd, so $5, 8, 11, \dots$ is invalid; $1, 4, 7$ shows why this is impossible.
d. $100, 103, \dots, 127$
6. Abdul makes a number sequence. The first term of his sequence is $397$. His term-to-term rule is ‘subtract $3$’. Abdul says, ‘If I keep subtracting $3$ from $397$ I will eventually reach $0$’. Is he correct? Explain your answer.
👀 Show answer
No. $397$ is not a multiple of $3$, so the sequence will not reach exactly $0$.
7. Which sequences are linear and which are not? Write the next term for each sequence.
a. Add five: $4, 9, 14, \dots$
b. Subtract four: $20, 16, 12, \dots$
c. Add one more each time: $2, 3, 5, \dots$
d. Multiply by three: $2, 6, 18, \dots$
e. Subtract one less each time: $50, 41, 33, \dots$
f. Divide by two: $32, 16, 8, \dots$
👀 Show answer
a. Linear, next term $19$
b. Linear, next term $8$
c. Not linear, next term $8$
d. Not linear, next term $54$
e. Not linear, next term $26$
f. Not linear, next term $4$
8. Here is a spatial pattern. Draw the next term in the pattern. What number does it represent?
👀 Show answer
The next term is a $4 \times 4$ square, representing $16$.
🧠 Think like a Mathematician
These sets of beads have consecutive numbers in the circles.
The numbers add up to the number in the square.
Example:
Numbers: $1, 2, 3, 4, 5$
Sum: $15$
- You are specialising when you identify examples that fit the rule: “The numbers add up to the number in the square.”
- You are generalising when you notice a method for finding the middle number.
Complete these sets of beads:
a. Three consecutive numbers that add up to $27$.
b. Five consecutive numbers that add up to $25$.
Tip
Consecutive numbers are next to each other, for example $3, 4, 5$ and $6$.
Task: Describe how to find the middle number of each set of beads.
- Show you are specialising by giving examples that work.
- Show you are generalising by explaining a rule for the middle number.
👀 show answer
- a. The three numbers are $8, 9, 10$. The middle number is $9$, and $8 + 9 + 10 = 27$.
- b. The five numbers are $3, 4, 5, 6, 7$. The middle number is $5$, and their total is $25$.
- General rule: The middle number is found by dividing the total by the number of beads. Consecutive numbers are arranged evenly around this middle value.