Negative numbers
Ordering positive and negative numbers on the number line, and the 4 operations with directed numbers
About three lessons of 45 to 60 minutes
Golf scores and temperature: when is 'more negative' better?
A golfer who scores -10 is WINNING against a golfer who scores -3, fewer strokes than par is better. But a day that is -10°C is much COLDER, and worse for going outside without a coat, than a day that is -3°C. The maths is identical in both cases, -10 is less than -3, but what 'better' means flips completely depending on the context.
That is exactly why this unit matters: negative numbers behave in one consistent, predictable way on the number line and in every calculation, whatever the real-world story attached to them. Once you can order them and calculate with them accurately, you can apply that same skill to temperature, sea level, bank balances, or a golf scorecard, and let the context (not the maths) decide what 'better' means.
- A temperature of -3°C compared with -10°C-10°C is colder: it is further below 0, so -10 < -3
- A submarine at -50 m compared with -20 m relative to sea level-50 m is deeper, further below sea level, so -50 < -20
- A bank balance of -£40 (overdrawn) compared with -£15-£40 is a bigger debt, even though 40 > 15 as plain numbers
- A golf score of -10 compared with -3-10 is the WINNING score: fewer strokes than par is better in golf
What students will be able to do
Students will order positive and negative integers, decimals and fractions using <, >, = and the number line; find the distance between 2 points on the number line; and apply all 4 operations (addition, subtraction, multiplication and division) to positive and negative integers, including in real-world contexts.
- I can compare 2 positive or negative numbers using <, > or =.
- I can order a set of positive and negative numbers from smallest to largest.
- I can find the distance between 2 points on the number line.
- I can add and subtract positive and negative integers, including subtracting a negative number.
- I can multiply and divide positive and negative integers using the sign rules.
- I can use directed numbers to solve real-world problems involving temperature, depth and money.
Standards this unit teaches
- KS3 Maths: NumberUK National Curriculum (England)Number
Statutory requirement (Department for Education, "National curriculum in England: mathematics programmes of study", updated 28 September 2021, Key stage 3, "Number" strand, https://www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study/national-curriculum-in-england-mathematics-programmes-of-study): pupils should be taught to "order positive and negative integers, decimals and fractions; use the number line as a model for ordering of the real numbers; use the symbols =, ≠, <, >, ≤, ≥"; "use the 4 operations, including formal written methods, applied to integers, decimals, proper and improper fractions, and mixed numbers, all both positive and negative".
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
Words to teach and display
- Integer
- a whole number, positive, negative or 0, with no fraction or decimal part
- Negative number
- a number less than 0, written with a minus sign in front
- Directed number
- a number that has a sign (+ or -) showing its direction from 0, such as -5 or +3
- Magnitude
- the size of a number, ignoring its sign; the magnitude of -8 is 8
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Ordering positive and negative numbers
ConcreteOn the number line, numbers get bigger moving to the right and smaller moving to the left, whatever their sign. -6 sits to the left of 3, so -6 is less than 3, even though 6 is bigger than 3 as a plain number, the minus sign changes everything.
The same idea applies when comparing 2 negative numbers: -8 is less than -3, because -8 is further to the left, further below 0.
The distance between 2 points on the number line is always positive: it is the size of the gap between them, found by taking the difference and ignoring any negative sign (its absolute value).
Compare -8 and -3, then find the distance between them on the number line.
- -8 sits further to the left of 0 than -3, so -8 < -3.
- The distance between them is |-8 - (-3)| = |-5| = 5.
Answer: -8 < -3, and they are 5 apart on the number line.
- Which is greater, -12 or -9, and how do you know from the number line?
- Why isn't -8 less than -3 just because 8 is greater than 3?
2. Adding and subtracting directed numbers
PictorialAdding a positive number moves you RIGHT on the number line; subtracting a positive number moves you LEFT. Subtracting a negative number is where students often get caught out: it moves you RIGHT, the same direction as adding a positive, because taking away a negative undoes a leftward move.
Start at -3. Adding 8 means moving 8 places right, landing on 5: -3 + 8 = 5.
Subtracting a negative number flips direction: 4 - (-6) means starting at 4 and undoing a 6-place leftward move, so you move 6 places RIGHT instead: 4 - (-6) = 4 + 6 = 10.
Calculate 4 - (-6).
- Subtracting a negative number is the same as adding its positive: 4 - (-6) = 4 + 6.
- 4 + 6 = 10.
Answer: 4 - (-6) = 10.
- Why does subtracting a negative number give a bigger answer than the number you started with?
- Calculate -5 + (-3). Which direction do you move, and how far?
3. Multiplying and dividing directed numbers
AbstractThe sign rules for multiplying and dividing are the same: 2 numbers with the SAME sign (both positive or both negative) give a positive answer; 2 numbers with DIFFERENT signs give a negative answer.
Multiplying a negative number by a positive whole number can be pictured as repeated jumps in the negative direction: -3 x 4 means 4 equal jumps of -3 from 0, landing on -12.
Division undoes multiplication, so the same sign rule applies: -12 / -3 = 4 (same sign, positive answer), but -12 / 3 = -4 (different signs, negative answer).
Calculate -3 x 4, then use it to find -12 / -3.
- -3 x 4 means 4 groups of -3, added together: -3 + -3 + -3 + -3 = -12.
- -12 / -3 asks 'how many -3s make -12?' Since -3 x 4 = -12, that means -12 / -3 = 4 (same signs, positive answer).
Answer: -3 x 4 = -12, and -12 / -3 = 4.
- Is the product of 2 negative numbers positive or negative? Give an example.
- Calculate -20 / 4. What sign should the answer have, and why?
Common misconceptions and how to address them
Misconception-8 is greater than -3, because 8 is a bigger number than 3.
Why it happens: Students compare the size (magnitude) of the numbers instead of their actual value, ignoring the negative sign.
How to address it: Always picture (or draw) the number line first: further LEFT always means smaller, whatever the sign. -8 sits to the left of -3, so -8 < -3.
MisconceptionSubtracting a negative number, like 4 - (-6), makes the answer smaller.
Why it happens: Students expect subtraction to always decrease a number, without noticing that subtracting a NEGATIVE reverses that.
How to address it: Subtracting a negative number is the same as adding the positive: 4 - (-6) = 4 + 6 = 10, which is bigger than 4, not smaller.
MisconceptionMultiplying 2 negative numbers gives a negative answer.
Why it happens: Students over-generalise the rule that a negative sign 'makes things negative', applying it every time a negative sign appears rather than checking how many negative signs are involved.
How to address it: Use the sign rule: 2 numbers with the SAME sign multiply or divide to a POSITIVE answer; 2 numbers with DIFFERENT signs multiply or divide to a NEGATIVE answer. -4 x -5 = 20 (same sign, positive), but -4 x 5 = -20 (different signs, negative).
MisconceptionWhen adding a positive and a negative number, like -8 + 3, just add the digits and guess the sign.
Why it happens: Students treat the sign as an afterthought instead of picturing the actual movement on the number line, leading to the wrong VALUE, not just the wrong sign, for example answering -11 or 11 instead of -5.
How to address it: Picture starting at -8 and moving 3 places right (since we are adding a positive): -8 + 3 = -5. The number line shows the correct value and the correct sign at the same time.
Guided practice (with answers)
1. Insert <, > or = to compare: -9 and -4.
Answer: -9 < -4, because -9 is further left (further below 0) on the number line.
2. Order these numbers from smallest to largest: 5, -7, 0, -2, 3.
Answer: -7, -2, 0, 3, 5, reading the number line from left to right.
3. Calculate -6 + 10.
Answer: 4, because starting at -6 and moving 10 places right lands on 4.
4. Calculate 3 - (-7).
Answer: 10, because subtracting a negative is the same as adding: 3 + 7 = 10.
5. Calculate -5 x -6.
Answer: 30, because 2 numbers with the same sign multiply to a positive answer.
6. Calculate -24 / 8.
Answer: -3, because 2 numbers with different signs divide to a negative answer.
Independent practice worksheets
Practise ordering directed numbers, and all 4 operations with positive and negative integers, with computed, never-wrong answer keys.
Differentiation
- Keep a physical or drawn number line visible at all times, and physically move a finger or counter along it for every calculation.
- Start addition and subtraction practice with a number line from -10 to 10 before moving to mental calculation.
- For multiplication and division, write the sign rule as a simple reference: same signs = positive, different signs = negative.
- Use real contexts (temperature, money) for every abstract calculation to keep the sign meaningful.
- Investigate: does the sign rule for multiplication also work for 3 or more negative numbers multiplied together? (An even number of negative factors gives a positive answer, an odd number gives a negative answer.)
- Pose a mixed 4-operation problem, e.g. -3 x (4 - 7) + 10 / -2, and use the order of operations to solve it.
- Explore negative fractions and decimals using the same number-line reasoning, e.g. order -1.5, -2, -0.5.
- Design a real-world scenario (e.g. a submarine dive log) that needs both addition and multiplication of negative numbers to solve.
Assessment: exit ticket
A three-question exit ticket sampling ordering, addition/subtraction, and multiplication of directed numbers.
1. Insert <, > or = to compare: -15 and -6.
Answer: -15 < -6, because -15 is further left on the number line.
2. Calculate -9 - (-4).
Answer: -5, because -9 - (-4) = -9 + 4 = -5.
3. Calculate -7 x 6.
Answer: -42, because 2 numbers with different signs multiply to a negative answer.
Teacher notes and timings
- Rough timing across 3 lessons: Lesson 1 ordering and the number line (section 1), Lesson 2 addition and subtraction (section 2), Lesson 3 multiplication and division plus the exit ticket (section 3).
- This unit assumes the Grade 6 foundational understanding of what a negative number represents (see the linked prior-knowledge unit); it moves straight into calculating with directed numbers rather than re-teaching what a negative number means.
- Language to keep repeating: further left always means smaller, whatever the sign; subtracting a negative reverses direction; same signs multiply/divide to positive, different signs multiply/divide to negative.
- The number line is the single model that unifies all 3 sections: keep returning to it even once students are calculating mentally, since it is the fastest way to catch a sign error.
- Use Student view to project this lesson. Print saves the full teacher unit, including answers and teacher notes; use the linked independent-practice worksheets for student handouts.