Function Machines: Free Printable Worksheet for Year 4

Function Machines is a free printable worksheet for Year 4 (ages 8 to 9), in Key Stage 2. Input, rule, output. It comes from the Mini Minds Hub, a growing library of free printables from Jaques of London, the British games and toy company that has been making games for families since 1795.

It is two A4 pages. Page 1 is the activity for children. Page 2 is for the grown-up: it has notes on how to use the sheet, help for children who find it hard, a stretch idea, and the answers. Print it at home or in the classroom. Black and white printing works well.

What is on the sheet

This worksheet includes a table to fill in.

A look at what children will meet:

• Input
• Rule
• Output

Who this is for

Year 4 (ages 8 to 9) in Key Stage 2.

Key Stage 2 covers Years 3 to 6, ages 7 to 11. Children are widening what they know and getting better at explaining their thinking, so activities that ask them to reason, not only recall, are especially useful. Use it at home, in class, on a rainy afternoon or at a family gathering.

What children practise

• Number patterns
• Rules
• Reasoning

Child applies a rule to an input to find an output.

Learning focus and curriculum link

Subject: Maths: Algebra. Key stage: Key Stage 2. It supports the subject at this level through short, focused practice.

This sheet is a practice and play activity. It supports classroom learning and is not a replacement for it.

How to use it

1. Print both pages. Page 1 is for the child. Keep page 2 for the grown-up.
2. Do the sums mentally or on paper. Check your answers by working backwards.
3. Work through the activity together the first time. Talk about each choice.
4. Check against the notes on page 2 and the answers.
5. Come back to it another day. Repeating a familiar activity builds confidence.

If it feels too hard

Use counters to show each rule.

To stretch confident children

Find the input when the output is known.

Make it a game

Make up a rule and ask a partner to guess it from the inputs and outputs.

Tips for parents and teachers

• Print on plain A4 paper. Black and white printing works too.
• Work through it together the first time, then let children try alone.
• Praise effort and thinking, not only correct answers.

A screen-free maths: algebra activity for Year 4

Children spend a lot of time on screens. This is a screen-free maths: algebra activity that needs only paper and a pencil. It works as quiet time after school, a rainy day activity, a holiday boredom buster or a calm moment before bed. Screen-free learning gives eyes a rest and gets children talking, thinking and writing by hand.

Free maths: algebra resources for teachers

Teachers can use this worksheet as a starter, a plenary, an early finisher task, a supply teacher resource, a homework task or a table activity. It needs no preparation beyond printing. The learning focus and skills are listed above so you can match it to your planning, and page 2 gives support and stretch ideas for mixed ability groups.

Maths: Algebra ideas for parents at home

You do not need to be a teacher to use this. Read the sheet through, sit with your child for the first go and let them lead. Short bursts of ten to fifteen minutes work better than long sessions. If you are looking for free maths: algebra worksheets for Year 4, or simple home learning for the weekend or school holidays, this is a good place to start.

Home education and tutoring

Home educators and tutors can print this as part of a wider maths: algebra plan. It pairs well with books, outdoor play and everyday talk. Because the sheet is free and printable, you can use it again for brothers and sisters or for revision a few weeks later.

Why it helps children learn

Hands-on, paper based activities slow learning down in a good way. Children read carefully, think before they answer and see their own progress on the page. Games and challenges add a reason to keep going. That mix of play and practice is why printable activities like this are popular with parents, teachers and home educators.

What children will be able to do

• I can apply a rule to an input.
• I can find the output.
• I can spot the rule from inputs and outputs.

Check it worked

Ask these when the sheet is finished. Listen for the reasoning, not just the answer.

1. What does the machine do to each number?
2. What would come out if you put in 10?
3. How could you work out the rule?

More like this

• Indices and Powers
• Solving Two Step Equations

Why function machines matter

A function machine is a simple picture of a rule. A number goes in, the rule changes it, and a new number comes out. The sheet is laid out as Input, Rule, Output. Children fill in the gaps. The same rule is applied every time, so the pattern can be spotted and checked.

Our own view: this is early algebra thinking. A review for the Nuffield Foundation by Anne Watson of the University of Oxford says that "Algebra is the way we express generalisations about numbers, quantities, relations and functions." A function machine is a small, concrete way to meet the idea of a relation between two numbers, long before any letters appear.

The review also says that students need to understand that addition and subtraction are inverses, and so are multiplication and division. Working backwards from an output to an input uses exactly that idea. The sheet asks children to check answers by working backwards, so the inverse idea is built in.

The Ofsted research review of mathematics, published on gov.uk, adds a practical point. It says that even at key stage 3, a pupil's lack of knowledge of the basics in number will have a detrimental effect on their learning of algebra. Rules such as "add 6" or "multiply by 4" lean on number facts. Our own suggestion: help your child learn those facts well, so they can spend their attention on the pattern.

What the research says children find hard

We did not find a study of Year 4 children using function machines that we could open and check, so we are not quoting any figure for this age. The Nuffield review does cover function machines, mainly with older pupils. We set out below what it says, and what it does not.

• Undoing a rule is not the same as doing it. The review reports that most 12-year-olds could reverse operations but not their order when "undoing" to find unknowns. It also says students who understand inversion might not understand that, when inverting a sequence of operations, the inverse operations cannot just be carried out in any order.
• The machine has to be provided. The review reports that Ryan and Williams (2007) found that function machines can be used by most students age 12 to 14 to solve linear equations, but only when provided. Few students chose to introduce them as a method. That study was about secondary pupils and equations, not about Year 4.
• One study saw a benefit. The review describes a UK study that tested pupils aged 8 to 13. It says that in the same study, students who were then taught using a function machine approach reached a better understanding of what an equation is. The review does not tell us how large the gain was.
• The method has limits. The review says this approach only makes sense when an input-output model is appropriate, and that function machines do not extend beyond one-sided equations.
• Weak basics cause errors. The Ofsted review says that pupils who are not able to quickly and easily recall maths facts struggle with calculations due to their working memory being overloaded.

None of these studies tested this sheet. Those studies were not specifically about Year 4 children. Our own view: they still suggest that knowing number facts matters.

What Year 4 children are expected to learn

The national curriculum for England, in its Year 4 programme of study for multiplication and division, says pupils should be taught to:

• "recall multiplication and division facts for multiplication tables up to 12 × 12"
• "recognise and use factor pairs and commutativity in mental calculations"

In addition and subtraction, Year 4 pupils are taught to "estimate and use inverse operations to check answers to a calculation" and to "solve addition and subtraction two-step problems in contexts, deciding which operations and methods to use and why". The non-statutory notes give an example of derived facts: 600 ÷ 3 = 200 can be derived from 2 x 3 = 6.

The Year 6 programme of study includes algebra content. There, pupils are taught to "use simple formulae", "generate and describe linear number sequences" and "express missing number problems algebraically". So a Year 4 function machine is preparation, not a formal Year 4 requirement. The sheet supports the number facts and inverse thinking that come first.

Why a game or activity can help

We want to be straight about this. We did not find a study of a function machine game for 8 to 9 year olds that we could open. So we are not going to claim it raises scores.

What we can offer is reasoning from the sources above. Our suggestion: a function machine gives children many small goes at applying one rule, which gives repeated practice with the number facts that Ofsted says pupils need. Working backwards to check uses inverse operations, which the curriculum asks of Year 4 pupils. The game on the adult page, where one person makes up a rule and a partner guesses it from inputs and outputs, asks the guesser to describe a pattern in words. That is the kind of generalising the Nuffield review links to algebra. It is a plausible link, not a proven result.

The sheet is practice. It is not a lesson and does not replace one.

How to get the most out of Function Machines

1. Say the rule aloud. Before writing anything, read the rule in words: "Whatever goes in, add 5."
2. Do one together. Pick an input, say the output, and write it in the table.
3. Use counters if needed. The page 2 notes suggest counters to show each rule. Put 4 counters in, add 3 more, and count the output.
4. Work backwards. Ask: "The output is 12. What went in?" Use the opposite operation.
5. Mark together. Check against page 2. Ask your child to explain any wrong answer.
6. Swap roles. Our suggestion: let your child make up a rule and test you.
7. Come back another day. Our suggestion: come back to the sheet another day for ten minutes.

Common mistakes, and what to say

These are our own suggestions for conversations, drawn from the Nuffield and Ofsted points above. They are not findings about Year 4 children.

• Using the same operation to go backwards. If the rule is "add 5" and the output is 12, the input is not 17. Say: "Is the machine adding or taking away? To undo it, do the opposite."
• Mixing up input and output. Point to the column headings. Say: "Which number goes in? Which comes out?"
• Doing two steps in the wrong order. For a two-step rule, the Nuffield review notes the order of inverses matters. Our suggestion: write each step on its own line, then undo them from the last step to the first.
• Guessing the rule from one pair. One input and output can fit more than one rule. Say: "Let us test it with a second number."
• Slow tables recall. If the multiplying is the hard part, practise tables separately. The Ofsted review says working memory is overloaded when facts are not to hand.

Why teachers matter to us

We are a games company, and we have been making games for families since 1795. A printable does not replace a teacher, and function machines are a good example. Someone has to choose the rule, notice when a child undoes two steps in the wrong order and ask the next question. A 2011 interim report for the Sutton Trust looked at how much teachers affect pupil achievement. It found that in one year with a very effective maths teacher, pupils gain 40% more learning than with a poorly performing maths teacher. That is a finding about maths teachers.

The same report said that for poor pupils the difference between a good teacher and a bad teacher is a whole year's learning, and that paper qualifications and similar details, such as teaching experience, degrees and certification, explain less than 8% of teacher quality. A University of Bristol paper (CMPO 09/212) linked over 7,000 pupils in England to their teachers and found considerable variability in teacher effectiveness. We are grateful to the maths teachers who turn a rule on a page into understanding.

A caution on the Sutton Trust figures: the report draws on US studies as well as English ones. The 40% compares a very effective teacher (at the 84th percentile) with a poorly performing one (at the 16th percentile). The figure that paper credentials explain less than 8% of teacher quality comes from US research. We use them as pointers, not as a promise about any one child.

Why education matters to us

Rules and patterns are everywhere, from knitting to timetables to recipes that scale up for a crowd. We want children to feel that a rule is something they can discover, test and break. Jaques of London has made things for children to hold and play with for generations. The Mini Minds Hub is our way of carrying that on. It is free, screen-free, and simple enough for a parent at a kitchen table to use alongside a teacher in a classroom.

Function Machines: 20 questions and answers

1. What is a function machine?

It is a way of showing a rule. A number goes in, the rule changes it, and a number comes out. The sheet shows it as Input, Rule, Output.

2. If I put in the same number twice, do I get the same output?

Our own explanation: yes, with a fixed rule, that is what makes it a rule. If a child gets two different outputs for the same input, one of the two sums has a mistake.

3. Is this algebra?

It is early groundwork. The Nuffield review says algebra expresses generalisations about numbers, quantities, relations and functions. A function machine shows a relation between two numbers without using letters.

4. Where does the national curriculum introduce algebra?

The Year 6 programme of study includes algebra content, such as "use simple formulae". Our own view: Year 4 work on multiplication facts, factor pairs and inverse operations comes first.

5. Which Year 4 curriculum aims does this support?

Recalling multiplication and division facts up to 12 × 12, and using inverse operations to check answers. Both appear in the Year 4 programme of study.

6. Why does the sheet say to work backwards?

Working backwards uses the inverse of the rule. The curriculum asks Year 4 pupils to use inverse operations to check answers.

7. What is an inverse?

It is the operation that undoes another. Subtraction undoes addition. Division undoes multiplication. The Nuffield review says these pairs matter for algebra.

8. How do I undo a two-step rule?

Our advice: undo the last step first. If the rule is "multiply by 2, then add 1", the way back is "take away 1, then halve". The Nuffield review notes that the order of inverse operations matters.

9. Do function machines help with equations?

The review reports one UK study where they did seem to help, and another where students aged 12 to 14 used them only when given the machine. They are not a complete method. The review says they do not extend beyond one-sided equations.

10. Does the research say Year 4 children should use them?

No. The studies in the Nuffield review were not specifically about Year 4 children. We did not find a Year 4 study we could open.

11. What if my child is slow at tables?

The Ofsted review says weak recall overloads working memory. Our advice: practise the tables the sheet uses on their own first.

12. What do the counters show?

Page 2 suggests counters to show each rule. A child can put counters in, add or take some away, and count the output. Our suggestion: use a different colour for the counters added by the rule.

13. Can my child use letters for the rule?

Our advice: keep to words and numbers at this stage, such as "the input" and "the output".

14. What happens if a divide rule gives a remainder?

Our advice: choose inputs that divide exactly. Tables facts can help, and the curriculum notes show 600 ÷ 3 = 200 derived from 2 x 3 = 6.

15. How do I make my own function machine?

Choose one rule, such as "add 7". Choose five inputs. Work out the outputs and hide some. Your child fills in the gaps, and you can swap places.

16. How do I stretch a confident child?

The page 2 stretch is to find the input when the output is known. Our extra idea: give two steps in the rule.

17. How do I make it easier?

Use the counters, use small whole numbers, and do the first row together.

18. How does the guess-the-rule game work?

One person thinks of a rule and keeps it secret. The partner gives an input and hears the output. After a few pairs, the partner says the rule. Our suggestion: ask the guesser to explain how they know.

19. What if my child guesses a rule that also fits?

Well done. Test it with a new input. If both rules give the same outputs for every number you try, they may be the same rule written in two ways.

20. Is this a replacement for a maths lesson?

No. It is a short practice sheet. A Sutton Trust interim report found that in one year with a very effective maths teacher, pupils gain 40% more learning than with a poorly performing one.

Sources

1. Watson, A., Key understandings in mathematics learning, Paper 6: Algebraic reasoning, Nuffield Foundation, University of Oxford: nuffieldfoundation.org.
2. Ofsted, Research review series: mathematics (2021): gov.uk.
3. Department for Education, National curriculum in England: mathematics programmes of study: gov.uk.
4. Sutton Trust, Improving the impact of teachers on pupil achievement in the UK, interim findings (2011): dera.ioe.ac.uk.
5. Slater, Davies and Burgess, University of Bristol CMPO Working Paper 09/212 (2009): bristol.ac.uk.

Frequently asked questions

What age is Function Machines for?

It is written for Year 4 (ages 8 to 9), in Key Stage 2.

What will children practise?

Child applies a rule to an input to find an output.

How do I use it?

Do the sums mentally or on paper. Check your answers by working backwards.

What if it feels too hard?

Use counters to show each rule.

Is it free?

Yes. It is free to print for home and classroom use.

Is Function Machines suitable for teachers to use in class?

Yes. Teachers can use it as a starter, plenary, early finisher task, homework or table activity. It needs no preparation beyond printing.

Is this a screen-free activity?

Yes. It is a screen-free printable that needs only paper and a pencil, so it works well for quiet time, rainy days and school holidays.

Can I use it for home education?

Yes. Home educators and tutors can use it on its own or as part of a wider plan. It is free to print again for brothers, sisters or revision.

How long does it take?

Most children finish page 1 in 10 to 20 minutes. The notes on page 2 suggest ways to make it shorter or longer.

Do I need to print in colour?

No. It prints well in black and white on plain A4 paper.

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