Linear Functions

Linear functions form a foundational concept in mathematics that builds upon basic number operations and introduces students to the relationship between variables. In the Cambridge Primary curriculum, linear functions are introduced at the upper primary level as part of the "Heart of Algebra" unit, preparing students for more advanced algebraic thinking in secondary education. A linear function describes a straight-line relationship between two quantities, where one quantity changes at a constant rate relative to another.

Understanding linear functions is crucial because they appear everywhere in real life—from calculating costs based on quantities purchased, to determining distances traveled at constant speeds, to converting between different units of measurement. These functions help students develop logical thinking and problem-solving skills that extend far beyond mathematics into science, economics, and everyday decision-making.

At the Cambridge Primary level, the focus is on developing intuitive understanding through practical examples, patterns, and simple graphical representations. Students learn to recognize linear relationships in tables, word problems, and real-world situations before progressing to more formal algebraic notation. This foundational knowledge creates a strong platform for future mathematical learning and helps students see mathematics as a tool for understanding and describing the world around them.

Function: A mathematical relationship between two quantities where each input value produces exactly one output value. In primary mathematics, this is often described as a "rule" that transforms one number into another.

Linear function: A function that produces a straight line when graphed, characterized by a constant rate of change. The output changes by the same amount each time the input increases by one unit.

Variable: A symbol (often a letter) that represents a number that can change or vary. In linear functions, we typically use variables to represent the input and output values.

Input: The starting value that goes into a function, also called the independent variable. This is the value we choose or are given first.

Output: The resulting value that comes out of a function after applying the rule, also called the dependent variable. This value depends on what input we choose.

Constant rate of change: The fixed amount by which the output changes when the input increases by one. This is the defining characteristic of linear functions.

Pattern: A regular, predictable arrangement of numbers or shapes. Linear functions create arithmetic patterns where the same number is repeatedly added or subtracted.

Table of values: An organized way of showing the relationship between inputs and outputs, with inputs in one column and corresponding outputs in another column.

Rule: A description (in words or symbols) that explains how to transform the input into the output. For example, "multiply by 3 and add 2."

Graph: A visual representation of a function showing the relationship between inputs (usually on the horizontal axis) and outputs (usually on the vertical axis).

Understanding Function Relationships

A linear function establishes a relationship where one quantity depends on another in a predictable, constant way. At the primary level, students explore these relationships through everyday scenarios. For example, if chocolates cost £2 each, the total cost depends on how many chocolates you buy. If you buy 1 chocolate, you pay £2; if you buy 2 chocolates, you pay £4; if you buy 3 chocolates, you pay £6. This creates a pattern: 2, 4, 6, 8, and so on.

The key characteristic that makes a function linear is that the change is always the same. In our chocolate example, each additional chocolate adds exactly £2 to the total cost. This is called a constant rate of change. If the cost changed by different amounts (£2 for the first chocolate, £3 for the second, £5 for the third), it would not be a linear function.

Recognizing Linear Patterns

Students learn to identify linear functions through number patterns. When numbers increase or decrease by the same amount each time, they form a linear pattern. For example:

• 5, 8, 11, 14, 17... (adding 3 each time)
• 20, 17, 14, 11, 8... (subtracting 3 each time)
• 0, 7, 14, 21, 28... (adding 7 each time)

These patterns can be extended forward or backward by continuing to add or subtract the constant amount. Students practice finding the rule that creates the pattern and using it to predict future values.

Function Tables

Function tables organize the relationship between inputs and outputs systematically. A simple table might look like this:

Students analyze tables to determine the function rule. In this example, the output increases by 2 each time the input increases by 1. To find the rule, students notice that if they multiply the input by 2 and add 3, they get the output. For input 1: (1 × 2) + 3 = 5. For input 2: (2 × 2) + 3 = 7. This rule works for all values.

Function Machines

The function machine is a visual model that helps primary students understand how functions work. Imagine a machine where you put a number in one end, the machine applies a rule, and a different number comes out the other end. If the rule is "multiply by 4," and you put in 3, the machine outputs 12. If you put in 5, it outputs 20.

Function machines can have multiple steps: "multiply by 2, then add 5" or "add 3, then multiply by 4." Students practice working forwards (given the input, find the output) and backwards (given the output, find the input). Working backwards helps develop algebraic thinking and inverse operations.

Real-World Linear Functions

Linear functions appear in countless real-world situations that primary students can understand:

Saving money: If you save £5 each week, after 1 week you have £5, after 2 weeks you have £10, after 3 weeks you have £15. The function rule is "multiply the number of weeks by 5."

Growing patterns: If a plant grows 3 cm each week, its height follows a linear function (though you'd need to add its starting height).

Converting units: Converting between centimeters and meters, or pence and pounds, involves linear functions. To convert centimeters to meters, you divide by 100.

Sharing equally: If you have some sweets to share equally among friends, the number each friend gets depends linearly on the total number of sweets.

These contexts make linear functions meaningful and help students see the practical applications of mathematical thinking.

Graphical Representation (Introduction)

At the upper primary level, students begin exploring how linear functions can be represented as graphs. While detailed graphing is more appropriate for secondary school, primary students can understand that when you plot inputs on a horizontal line and outputs on a vertical line, linear functions create points that form a straight line.

Students might create simple graphs using squared paper, plotting points like (1, 5), (2, 7), (3, 9) from a function table, and observing that these points line up perfectly. This visual representation reinforces the concept of constant change—the line goes up by the same amount each time you move one unit across.

Example 1: Finding the Function Rule from a Pattern

Problem: Look at this sequence of numbers: 4, 9, 14, 19, 24. What is the function rule? What would the 10th number in the sequence be?

Solution:

Step 1: Identify if it's a linear pattern by checking if the change is constant.

• From 4 to 9: increase of 5
• From 9 to 14: increase of 5
• From 14 to 19: increase of 5
• From 19 to 24: increase of 5

The change is constant (+5), so this is a linear pattern.

Step 2: Determine the relationship between position and value.

• Position 1: 4 (which is 1 × 5 - 1 = 4)
• Position 2: 9 (which is 2 × 5 - 1 = 9)
• Position 3: 14 (which is 3 × 5 - 1 = 14)
• Position 4: 19 (which is 4 × 5 - 1 = 19)

The rule is: "Multiply the position number by 5, then subtract 1" or "5 times the position number, minus 1"

Step 3: Apply the rule to find the 10th number.

• 10th number = (10 × 5) - 1 = 50 - 1 = 49

Answer: The function rule is "multiply by 5, then subtract 1." The 10th number is 49.

Example 2: Completing a Function Table and Finding Missing Values

Problem: Complete this function table where the rule is "multiply by 3, then add 4":

Solution:

For Input 2:

• Apply the rule: (2 × 3) + 4
• Calculate: 6 + 4 = 10
• Output is 10

For unknown input with Output 19:

• Work backwards: if (input × 3) + 4 = 19
• First, subtract 4: input × 3 = 15
• Then, divide by 3: input = 5

For Input 5:

• Apply the rule: (5 × 3) + 4
• Calculate: 15 + 4 = 19
• Output is 19

For Input 8:

• Apply the rule: (8 × 3) + 4
• Calculate: 24 + 4 = 28
• Output is 28

Completed table:

Example 3: Real-World Linear Function Problem

Problem: A library charges £2 to join and then £1.50 for each book borrowed. Complete the table showing the total cost for different numbers of books:

Solution:

Step 1: Understand the function rule.

• Fixed cost (joining fee): £2
• Variable cost: £1.50 per book
• Rule: Total cost = £2 + (number of books × £1.50)

For 0 books:

• Total cost = £2 + (0 × £1.50) = £2 + £0 = £2
• You still pay the joining fee even with no books

For 1 book:

• Total cost = £2 + (1 × £1.50) = £2 + £1.50 = £3.50

For 3 books:

• Total cost = £2 + (3 × £1.50) = £2 + £4.50 = £6.50

For 5 books:

• Total cost = £2 + (5 × £1.50) = £2 + £7.50 = £9.50

For total cost of £14:

• £14 = £2 + (number of books × £1.50)
• Subtract the joining fee: £14 - £2 = £12
• Divide by cost per book: £12 ÷ £1.50 = 8 books

Completed table:

Answer: The pattern shows that each book adds £1.50 to the total, starting from a base of £2. To get a total cost of £14, you would need to borrow 8 books.

Question 1: Identifying and Extending Linear Patterns

Typical Question: "Here is a sequence of numbers: 7, 12, 17, 22, 27 a) What is the next number in the sequence? b) What is the rule for this sequence? c) What would the 8th number in the sequence be?"

How to Answer:

Part a): Look at the differences between consecutive numbers.

• 12 - 7 = 5
• 17 - 12 = 5
• 22 - 17 = 5
• 27 - 22 = 5

Since we're adding 5 each time, the next number is 27 + 5 = 32.

Part b): The rule is "add 5 to the previous number" or "start at 7 and add 5 repeatedly". You could also describe it in terms of position: "multiply the position by 5, then add 2" (since 1×5+2=7, 2×5+2=12, etc.).

Part c): To find the 8th number, continue the pattern or use the position rule:

• Using the pattern: 7, 12, 17, 22, 27, 32, 37, 42
• Using the rule: (8 × 5) + 2 = 40 + 2 = 42

Examiner Tip: Always show your working by writing out how you found the difference between numbers. This demonstrates your understanding even if you make a small arithmetic error.

Question 2: Function Machine Problems

Typical Question: "A function machine uses the rule 'multiply by 4, then subtract 3.' a) If the input is 6, what is the output? b) If the output is 29, what was the input? c) Complete the table:

How to Answer:

Part a): Apply the rule step by step.

• Input: 6
• Multiply