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.