Evaluating Functions

Overview

This is lesson 5 of 10 in the unit Mastering Algebraic Concepts for Year 11 students. The focus is on developing fluency in evaluating functions using function notation, a key skill underpinning algebraic reasoning and problem solving, aligned to the National Curriculum for England.

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National Curriculum Alignment

Key Stage 4: Years 10-11

Mathematics Programme of Study - Number and Algebra

• Use and interpret algebraic notation, including:
  • using and understanding function notation (f(x)),
  • substituting numerical values into formulae including functions,
  • understanding that variables can represent any number in a specified set,
• Understand and use expressions in different forms to solve problems.
• Work with functions expressed in function notation to solve problems, including linear and quadratic functions.

Specific NC Learning Objectives:

• Solve problems by substituting values into expressions and functions (NC Ref: 1a, 2b (DfE 2013))
• Understand function notation and evaluate functions for given inputs (NC Ref: Number and algebra, KS4)

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Learning Objectives

By the end of this 30-minute lesson, students will be able to:

• Interpret function notation (f(x)) correctly.
• Substitute both integer and fractional values into given functions.
• Evaluate at least 5 different functions accurately.
• Verbally and in writing, explain the step-by-step process of evaluating functions.

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Success Criteria

Students will demonstrate success by:

• Correctly evaluating 5+ functions for given inputs.
• Clearly explaining how substitution into f(x) works using mathematical language.
• Demonstrating understanding independently by applying substitution without prompts.

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Resources Needed

• Whiteboard and markers
• Printed worksheet with 5 varied functions (linear, quadratic, including negative and fractional inputs)
• Mini whiteboard for student use OR digital tablet/laptop for input (if available)
• Stopwatch or timer

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Lesson Structure

1. Starter: Mental Function Warm-Up (5 minutes)

• Teacher writes a simple function on the board, e.g. f(x) = 2x + 3
• Ask the student quickly to evaluate for x=2, then x=0, then x=-1
• Discuss briefly how plugging in the value changes the output
• Success Criterion: Student can mentally substitute without hesitation

2. Direct Teaching & Modelling (8 minutes)

• Explain function notation again: f(x) means “the value of the function f at x”
• Demonstrate substitution step-by-step for f(x) = 3x² - x + 5 at x = 2
• Highlight writing down the substitution clearly, e.g., f(2) = 3(2)² - 2 + 5
• Emphasise following order of operations precisely
• Ask the student to narrate the process back to ensure conceptual understanding
• Success Criterion: Student articulates and records substitution clearly

3. Guided Practice (10 minutes)

• Provide worksheet with 5 functions, a variety of types (linear, quadratic, cubic, fractional inputs, negative inputs)
  Example functions on worksheet:
  1. f(x) = 4x - 7, evaluate f(3)
  2. g(x) = x² + 2x + 1, evaluate g(-1)
  3. h(x) = (1/2)x - 4, evaluate h(6)
  4. p(x) = -3x³ + x, evaluate p(2)
  5. q(x) = 5 - x², evaluate q(0)
• Student works through each, thinking aloud so teacher can check reasoning steps
• Teacher provides immediate feedback and scaffolds if necessary
• Success Criterion: All 5 correctly evaluated with clear working

4. Independent Challenge (5 minutes)

• Pose a ‘wow’ challenge: create your own function rule and evaluate it at two given points, explaining your steps
• Student writes down their function and evaluates
• Teacher listens and praises creativity and precision
• Success Criterion: Clear explanation and correctly evaluated own function

5. Plenary and Reflection (2 minutes)

• Discuss: Why is evaluating functions important? Where else might you see this in maths or real life?
• Student summarises the steps of evaluating functions in their own words
• Teacher highlights progress against success criteria and identifies next steps

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Assessment

• Informal continuous assessment through questioning and observed substitution
• Marking the worksheet for correctness and method
• Peer or self-assessment via reflection discussion
• Teacher notes on student explanations for deeper conceptual insight

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Differentiation & Extension

• Support: Use simpler linear functions with integer inputs, provide partially completed steps
• Extension: Challenge with composite functions or parameterised functions (e.g., f(x) = ax + b where a and b vary)
• Link evaluating functions to real-world contexts (e.g., calculating total cost, physics formulae) for stretch

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Notes for Teachers

• Encourage student to verbalise all algebraic steps to cement understanding.
• Model persistence in revisiting tricky substitutions to build confidence.
• Use mini whiteboards to allow for quick, low pressure trial and error.
• Keep checking understanding by asking “What does f(x) mean?” before substituting each time.

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This lesson supports the mastery approach in algebra and strengthens foundations for GCSE topics such as solving equations, graphing functions, and modelling.