Algebra: sequences, function machines and formulae

Sequences

A sequence is a list of numbers that follows a rule. In a linear sequence, the rule is to add or subtract the same amount each time: 4, 11, 18, 25… goes up in sevens. Year 6 pupils describe sequences, find missing terms, continue them into negative numbers and decimals, and describe the rule in words.

• 4, 11, 18, 25, 32The difference is +7 every time.
• 3.5, 2.75, 2, 1.25, 0.5The difference is −0.75.
• 2, 4, 8, 10The rule changed partway through. Check every pair, not just the first.

To find a term far along the sequence, such as the 20th, children can use the difference: start at the first term and add the difference 19 times (4 + 19 × 7 = 137). This is the bridge to describing a sequence with a formula later.

Function machines

A function machine takes an input, applies one or more operations, and gives an output. Working forwards is straightforward; working backwards needs the inverse operations in reverse order.

• Input 6 → × 3 → + 5 → output 23Forwards: apply each step in order.
• Output 41 → − 5 → ÷ 3 → input 12Backwards: undo the last step first.
• Output 41 → ÷ 3 → − 5Undoing in the original order gives the wrong input.

Letters for numbers

Algebra uses letters to stand for unknown or changing numbers. In 3n + 5, the letter n stands for a number, and 3n means 3 × n. Year 6 pupils:

• find the value of an expression when given n (if n = 4, 3n + 5 = 17);
• solve simple equations (3n + 5 = 23, so n = 6);
• find pairs of numbers that satisfy an equation with two unknowns (a + b = 10);
• use simple formulae, such as perimeter = 2 × (length + width).

A common error is reading 3n as “thirty-something” or as 3 + n. Writing it out in full (3 × n) for the first few weeks prevents this.

Formulae

A formula is a rule written using letters or words. Children use formulae for perimeter, area and volume, and for real-life rules such as “cost = £3 × number of tickets + £2 booking fee”. The skill is substituting the right values in the right places and calculating in the correct order.

In the SATs

Algebra appears on the reasoning papers. Typical questions ask children to continue or find the rule for a sequence, complete a function machine forwards or backwards, find the value of a letter, list possible pairs of values, or use a formula given in words. Missing-number calculations on the arithmetic paper use the same thinking.

Common mistakes

• Assuming the first difference in a sequence continues without checking.
• Using inverse operations in the wrong order when working backwards.
• Reading 3n as 3 + n.
• Finding one pair of values when the question asks for all possible pairs.
• Substituting values into a formula in the wrong places.

Teaching ideas

Human function machines

One child is the machine, with a secret rule. Classmates give inputs and record outputs until someone can state the rule. Then try two-step rules.

Matchstick patterns

Build growing patterns with matchsticks or cubes (squares in a row, triangles, staircases). Pupils record how many sticks each step needs, find the difference, and predict step 20 before checking.

Think of a number

“I think of a number, double it and add 7. The answer is 31. What was my number?” Pupils solve these, then write their own as function machines and as equations.

Systematic pairs

For a + b = 12 or 2a + b = 15, pupils list solutions in an ordered table so none are missed. It's a habit that pays off in many reasoning questions.

Try it: five quick questions

SATs-style questions to use as a starter or a quick check. Tap to see each answer.

1. Find the next two terms: 48, 41, 34, 27, …

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   20, 13

2. The rule is “× 4 then − 3”. What is the output for 9?

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   33

3. The rule is “× 4 then − 3”. The output is 45. What was the input?

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   12

4. If n = 7, what is 5n − 8?

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   27

5. 2a + b = 10, where a and b are whole numbers greater than 0. List all the pairs.

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   a = 1, b = 8; a = 2, b = 6; a = 3, b = 4; a = 4, b = 2