Two-Step Function Machines

Overview

In this 45-minute lesson, Year 6 students will explore two-step function machines, building on their prior knowledge of single-step functions from lesson 1. They will learn to apply two arithmetic operations in sequence to an input number, fostering their understanding of algebraic expressions and the order of operations. This lesson is firmly aligned with the National Curriculum for England, focusing on developing pupils' fluency in algebraic manipulation as specified for Key Stage 2.

National Curriculum Links

Mathematics Programme of Study: Number - Algebra
Pupils should be taught to:
- use simple formulae
- generate and describe linear number sequences
- express missing number problems algebraically
Subject Domain: Year 6 - Number: Algebra
Related National Curriculum Statements:
- “Use simple formulae”
- “Generate and describe linear number sequences”
- “Express missing number problems algebraically”

(Reference: Mathematics programmes of study – key stage 2, Department for Education, England)

Learning Objectives

By the end of this lesson, pupils will be able to:

Recognise two-step rule functions combining two arithmetic operations.
Apply two sequential operations accurately to given numbers.
Represent two-step function machines using algebraic expressions involving one variable.
Solve missing number problems involving two-step functions.
Explain reasoning when applying and reversing two-step functions.

Resources

Printed function machine templates (input → two steps → output)
Whiteboards and markers for group work
Visual aids showing two-step function diagrams
Worksheet with differentiated challenges (basic to advanced)
Algebra function cards (e.g. “times 3,” “add 7,” “subtract 4”)
Countdown timer or stopwatch
Mini whiteboards for each pupil

Lesson Structure

Starter (5 minutes)

Activity: Quick Recall and Review from Lesson 1

Recap the single-step function machines learned previously.
Ask 3 volunteers to demonstrate a one-step function on the board (e.g., “input 4, rule: add 5, output 9”).
Write a simple algebraic expression on the board (e.g., x + 5) to introduce algebraic representation.

Purpose: Activate prior learning and create a scaffold for two-step learning.

Introduction (8 minutes)

Teacher Explanation and Demonstration:

Introduce the concept of a two-step function machine: "Now we'll have two operations, one after the other, to transform the input."
Show an example on the board: Input → multiply by 2 → add 3 → Output.
Model working through an input number (e.g., 5):
1. 5 × 2 = 10
2. 10 + 3 = 13
Write an algebraic expression for this: 2x + 3.
Highlight that order matters: the first operation and then the second.

Use questioning to check understanding:

What happens if we add 3 first and then multiply by 2?
How do we write that algebraically?

Main Activity (20 minutes)

Activity 1 (10 minutes): Guided Practice in Pairs

Distribute function machine cards with two-step instructions (e.g., “subtract 4, then times 3”).
Pupils input different numbers and calculate the output.
Pupils then write algebraic expressions (e.g., 3(x - 4)).
Teacher circulates, supporting pupils and probing their understanding.

Activity 2 (10 minutes): Problem Solving on Mini Whiteboards

Give missing number challenges:
- Input: 4 → Output: 17, rules: multiply by 3, then add ?
- Pupils find missing numbers or operations.
Encourage pupils to explain how they arrived at their answers aloud.
Challenge stronger pupils with reverse function problems (finding input from output).

Plenary (7 minutes)

Group Discussion:

Invite 3-4 pupils to share their problem-solving strategies.
Highlight the importance of order in two-step functions.
Together, write a generalised form: ax + b, explaining ‘a’ and ‘b’.

Mini Quiz:

On mini whiteboards, pupils write answers to quick-fire questions, e.g.
- What is output if rule is “×4 then –5” and input is 7?
- What is the algebraic expression for “add 6 then divide by 2”?

Assessment and Feedback

Monitor paired work and mini whiteboard answers for understanding and correct calculations.
Use questioning feedback during plenary to assess grasp of algebraic expressions with two-step functions.
Note misconceptions about operation order or incorrect expressions for individual follow-up.

Differentiation

Group	Support	Challenge
SEN/Lower Attaining	Focus on less complex steps, fewer operation types (add and subtract only). Use concrete objects to model steps visually.	Provide step-by-step guided scaffolding during paired work.
Mainstream	Standard two-step functions with mixed operations.	Use missing number and reverse function challenges.
Higher Attaining	Provide opportunities to write and simplify algebraic expressions (expand brackets, e.g., 2(x + 3)).	Introduce three-step function machines for extension tasks.

Curriculum Links Summary Table

NC Area	Objective	Activity Link
Number – Algebra	Use simple formulae.	Writing algebraic expressions for two-step functions.
Number – Algebra	Generate and describe linear number sequences.	Explore sequences generated by function machines.
Number – Algebra	Express missing number problems algebraically.	Solve missing number problems in function machines.

WOW Element for Teachers

Incorporate 'Function Machine Detective':

Towards the end of the lesson, make the session a detective game—pupils decode mysterious outputs from function machines whose rules are hidden.
Pupils use logical reasoning and algebra to unlock the rules and inputs, making abstract algebra hands-on and exciting.

Technology Boost (optional):
If available, use a simple online function machine simulator to let pupils manipulate inputs and see outcomes dynamically.

Reflection and Next Steps

Pupils will be more confident manipulating algebraic expressions involving linear terms.
The following lesson will deepen understanding by exploring inverse operations and solving equations from function machines.
Teachers should observe pupils’ readiness for formal algebraic solving and adjust further scaffolding if necessary.

End of Lesson 2: Two-Step Function Machines

Teaching Instructions