Mathematics · Year 7 · The Power of Number · Autumn Term

Negative Numbers and the Number Line

Introducing negative numbers and their application in real-world contexts, using the number line for ordering and operations.

• National Curriculum Attainment Targets
• KS3: Mathematics - Number

About This Topic

Primes and composites are the 'atoms' of the number system. This topic introduces students to the Fundamental Theorem of Arithmetic: the idea that every whole number greater than one is either a prime or can be uniquely represented as a product of primes. By mastering prime factorisation, students gain a powerful tool for simplifying fractions, finding common multiples, and understanding the properties of numbers.

This unit moves beyond simple identification of primes to the application of factor trees and Venn diagrams. These methods provide a visual and logical framework for finding the Highest Common Factor (HCF) and Lowest Common Multiple (LCM), which are essential for algebraic manipulation later. Students grasp this concept faster through structured discussion and peer explanation where they compare different factor tree paths for the same number.

Key Questions

1. Justify the necessity of negative numbers for describing real-world situations.
2. Compare the operations of addition and subtraction with positive and negative integers.
3. Predict the outcome of multiplying two negative numbers.

Learning Objectives

• Calculate the position of positive and negative integers on a number line.
• Compare and order integers, including negative numbers, using a number line.
• Explain the effect of adding and subtracting positive and negative integers using integer counters or a number line.
• Predict the sign of the product when multiplying two negative integers based on observed patterns.

Before You Start

Whole Numbers and the Number Line

Why: Students need a solid understanding of positive whole numbers and their placement on a number line before introducing negative numbers.

Addition and Subtraction of Whole Numbers

Why: The concepts of adding and subtracting positive numbers provide a foundation for understanding how these operations change position on the number line when applied to negative numbers.

Key Vocabulary

Integer	A whole number that can be positive, negative, or zero. Examples include -3, 0, and 5.
Number Line	A visual representation of numbers placed at intervals along a straight line. It is used to order numbers and perform calculations.
Positive Number	A number greater than zero. On a number line, these are typically shown to the right of zero.
Negative Number	A number less than zero. On a number line, these are typically shown to the left of zero.
Opposite Numbers	Two numbers that are the same distance from zero on the number line but in opposite directions. For example, 5 and -5 are opposite numbers.

Watch Out for These Misconceptions

Common Misconception

Thinking that all odd numbers are prime.

What to Teach Instead

Students often conflate 'odd' with 'prime' because many early primes are odd. Use counter-examples like 9, 15, and 21 in a group sorting activity to show that while most primes are odd, not all odd numbers are prime.

Common Misconception

Believing that 1 is a prime number.

What to Teach Instead

This is a common error based on the 'divisible by 1 and itself' definition. Explain that a prime must have exactly two distinct factors. Using a collaborative investigation into factor pairs helps students see that 1 only has one factor, disqualifying it.

Active Learning Ideas

Inquiry Circle: The Sieve of Eratosthenes

In small groups, students work on a large 1-100 grid to systematically cross out multiples. They discuss why certain numbers remain and identify the patterns of primes, specifically looking at why 1 is neither prime nor composite.

30 min·Small Groups

Peer Teaching: Factor Tree Challenge

Pairs are given different large composite numbers. One student creates a factor tree while the other checks the prime status of each branch; then they swap roles to find the HCF of their two numbers using a shared Venn diagram.

25 min·Pairs

Gallery Walk: Prime Art

Students create visual representations of numbers using their prime factors (e.g., using different colours for 2, 3, and 5). They display their work, and the class moves around to identify the original numbers based only on the prime factor 'DNA' shown.

40 min·Whole Class

Real-World Connections

• Temperature readings in weather forecasts often include negative numbers to describe conditions below freezing, such as -5 degrees Celsius in London during winter.
• Financial transactions use negative numbers to represent debt or money owed, for example, a bank statement showing a balance of -£50 indicates an overdraft.
• Elevator floor numbers commonly use negative numbers for basement levels, such as B1 or -1, in large buildings and underground car parks.

Assessment Ideas

Quick Check

Present students with a list of numbers: 5, -3, 0, 8, -10, 2. Ask them to arrange these numbers in ascending order on a mini-whiteboard and hold it up. Observe for common errors in ordering negative numbers.

Exit Ticket

Give each student a card with a scenario: 'A diver descends 20 meters below sea level, then ascends 12 meters.' Ask them to write the calculation using negative numbers (e.g., -20 + 12) and state the final depth.

Discussion Prompt

Pose the question: 'If you add two negative numbers, is the answer always smaller than the original numbers? Explain your reasoning using the number line.' Facilitate a class discussion where students share their justifications.

Frequently Asked Questions

What are the best hands-on strategies for teaching prime numbers?

Using physical manipulatives like multi-link cubes allows students to try and build rectangles with a set number of blocks. If they can only build a single line (a 1 by n rectangle), the number is prime. This tactile approach makes the concept of 'factors' visible and helps students distinguish between prime and composite numbers through physical trial and error.

Why is prime factorisation important for Year 7 students?

It provides a systematic way to break down large numbers. Instead of guessing factors, students learn a logical process that simplifies finding the HCF and LCM, which are crucial for adding fractions with different denominators and simplifying complex ratios.

How do I explain the difference between a factor and a multiple?

Use the 'Factor-Few, Multiple-Many' mnemonic. Factors are the smaller numbers that fit into a target number; multiples are the larger numbers that the target number fits into. Visualising factors as 'building blocks' and multiples as 'towers' helps clarify the distinction.

Is there a limit to how many prime numbers exist?

No, primes are infinite. This was proven by Euclid over 2,000 years ago. Discussing this in class is a great way to introduce the idea of mathematical proof and the concept of infinity in a way that sparks curiosity.