This Year 7 negative numbers lesson is written for pupils aged 11-12 and follows the National Curriculum. In short, it shows you how to calculate with negative numbers, factors, multiples, primes, squares and cubes and use priority of operations. Before you start, it helps to revise place value, ordering and rounding, because this lesson builds on it.
Year 7 negative numbers lesson: what you will learn
Numbers are the foundation of KS3 maths. In this chapter you will calculate with negative numbers, use factors, multiples and primes, and apply the priority of operations.
Starter activity
On a winter morning the temperature outside is 2 degrees below zero. By noon it is 5 degrees higher. By midnight it falls by 9 degrees. Draw a number line and show each change. What are the temperatures at noon and at midnight?
Negative numbers
Definition: negative number
A negative number is less than zero, such as \(-3\). On a number line, numbers get smaller as you move left. We use negatives for temperatures, debts and heights below sea level.
Rule: adding and subtracting negatives
Adding a negative is the same as subtracting: \(5+(-3)=5-3=2\). Subtracting a negative is the same as adding: \(5-(-3)=5+3=8\).
Example 1
Work out \(-4+9\), \(-3-5\) and \(2-(-6)\).
\(-4+9=5\) (move 9 right from \(-4\)). \(-3-5=-8\) (move 5 left). \(2-(-6)=2+6=8\).
Example 2
The temperature at night is \(-7^{\circ}\)C and it rises by 12 degrees by midday. Find the midday temperature.
\(-7+12=5^{\circ}\)C.
Multiplying and dividing negatives
Rule: signs when multiplying and dividing
Same signs give a positive answer. Different signs give a negative answer. \((-3)\times(-4)=12\), \((-3)\times4=-12\), \(12\div(-4)=-3\), \((-12)\div(-4)=3\).
Example 3
Work out \((-6)\times5\), \((-8)\div(-2)\) and \((-3)\times(-3)\times(-3)\).
\((-6)\times5=-30\). \((-8)\div(-2)=4\). \((-3)\times(-3)=9\) and \(9\times(-3)=-27\).
Watch out
\(-3^2\) means \(-(3^2)=-9\), but \((-3)^2=9\). Brackets matter when squaring a negative number.
Factors and multiples
Definition: factor and multiple
A factor of a number divides it exactly. A multiple of a number is in its times table. The factors of 12 are 1, 2, 3, 4, 6 and 12. The multiples of 12 are 12, 24, 36 and so on.
Method: finding all the factors of a number
- Find pairs of numbers that multiply to give the number, starting with \(1\times n\).
- Test 2, 3, 4 and so on in order.
- Stop when the pairs repeat, then list the factors in order.
Example 4
List all the factors of 48.
The pairs are \(1\times48\), \(2\times24\), \(3\times16\), \(4\times12\) and \(6\times8\). The factors are 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48.
Prime numbers
Definition: prime number
A prime number has exactly two factors, 1 and itself. The primes up to 30 are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29. The number 1 is not prime. A number with more than two factors is composite.
Example 5
Is 91 a prime number?
Test small primes: \(91=7\times13\). It has other factors, so 91 is not prime.
Prime factorisation
Definition: prime factorisation
Every whole number greater than 1 is either prime or can be written as a product of prime factors in just one way. Writing it like this is prime factorisation. A factor tree helps.
Method: writing a number as a product of primes
- Split the number into any two factors.
- Keep splitting each factor until all are prime.
- Write the primes in order, using powers for repeats.
Example 6
Write 72 as a product of prime factors.
\(72=8\times9=2\times2\times2\times3\times3\). So \(72=2^3\times3^2\).
HCF and LCM
Definition: HCF and LCM
The highest common factor (HCF) of two numbers is the biggest number that divides into both. The lowest common multiple (LCM) is the smallest number that both divide into.
Example 7
Find the HCF and LCM of 12 and 18.
\(12=2^2\times3\) and \(18=2\times3^2\). HCF: the lowest powers shared, \(2\times3=6\). LCM: the highest powers, \(2^2\times3^2=36\).
Example 8
Two buses leave a station together. One returns every 12 minutes and the other every 18 minutes. When do they next leave together?
The LCM of 12 and 18 is 36, so after 36 minutes.
Squares, cubes and roots
Key fact: squares, cubes and roots
Squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. Cubes: 1, 8, 27, 64, 125, 1000. The square root reverses squaring: \(\sqrt{49}=7\). The cube root reverses cubing: \(\sqrt[3]{64}=4\).
Example 9
Work out \(\sqrt{81}+4^3\).
\(9+64=73\).
Priority of operations (BIDMAS)
Rule: BIDMAS
Work in this order: Brackets, Indices, Division and Multiplication, then Addition and Subtraction. Do division and multiplication left to right, and addition and subtraction left to right.
Example 10
Work out \(8+3\times(7-2)^2\div5\).
Brackets: \(7-2=5\). Indices: \(5^2=25\). Then \(3\times25=75\) and \(75\div5=15\). Finally \(8+15=23\).
Common mistakes
- Thinking 1 is a prime number.
- Thinking all odd numbers are prime (9 and 15 are not).
- Treating \(-3^2\) as 9.
- Confusing HCF and LCM: the HCF is never bigger than the numbers, the LCM never smaller.
Watch out
The only even prime is 2. Do not confuse factors (which divide into the number) with multiples (which the number divides into).
Worked example in context
A diver is 12 m below the surface. She swims up 5 m and then dives down 9 m. What is her depth now?
Start at -12. Swim up 5: \(-12+5=-7\). Dive down 9: \(-7-9=-16\). She is 16 m below the surface.
Common mistakes
- Thinking that -9 is greater than -2 because 9 is greater than 2.
- Treating "minus a negative" as another minus.
- Confusing HCF (the greatest factor shared) with LCM (the smallest multiple shared).
Did you know?
Prime numbers are the building blocks of all whole numbers. In 2018 a computer found a prime number with over 23 million digits. Prime numbers also protect your data online: bank cards and websites use huge primes to keep secrets safe.
Starter activity: the number line game
Draw a number line from \(-10\) to \(10\) and put a counter on zero. Roll a dice twice: the first roll tells you how far to move, and a coin decides the direction (heads to the right, tails to the left). After five turns, write down a calculation that describes all your moves, such as \(0+4-6-3+5-2=-2\). This simple game shows that adding moves you to the right, subtracting moves you to the left, and that the final position can be below zero even when you started at zero.
Method: subtracting a negative number
- Rewrite the calculation so that “minus a negative” becomes “plus a positive”: \(a-(-b)=a+b\).
- Start at the first number on a number line.
- Move to the right for addition and to the left for subtraction.
- Read off where you land, and check the size of the answer makes sense.
Example 1
The temperature at midnight is \(-3\) °C. By midday it is 8 °C. By how many degrees has the temperature risen?
The rise is the difference between the two temperatures, which is the later value minus the earlier value: \(8-(-3)\). Subtracting a negative is the same as adding, so this is \(8+3=11\). The temperature has risen by 11 degrees. You can also count on a number line: 3 steps from \(-3\) up to 0, then 8 steps from 0 up to 8, which is 11 steps altogether.
Example 2
Find the highest common factor and the lowest common multiple of 18 and 24 using prime factors.
First write each number as a product of primes: \(18=2\times3^{2}\) and \(24=2^{3}\times3\). For the HCF we take each prime that appears in both numbers, using the lower power: \(2^{1}\times3^{1}=6\). For the LCM we take every prime that appears in either number, using the higher power: \(2^{3}\times3^{2}=72\). A useful check is that HCF × LCM equals the product of the two numbers: \(6\times72=432\) and \(18\times24=432\).
Example 3
Is 119 a prime number?
We only need to test prime divisors up to the square root of 119, which is just under 11, so we try 2, 3, 5 and 7. 119 is odd, so 2 does not divide it. Its digits add to 11, which is not a multiple of 3, so 3 does not divide it. It does not end in 0 or 5, so 5 does not divide it. But \(7\times17=119\), so 119 is not prime. Testing only up to the square root saves a lot of time.
Watch out
Be careful with powers of negative numbers. \((-3)^{2}=(-3)\times(-3)=9\), but \(-3^{2}\) means \(-(3^{2})=-9\), because the power is worked out before the minus sign is applied. Calculators follow this rule too, so always type the brackets when you want to square a negative number.
Did you know?
Negative numbers were accepted surprisingly late. Chinese mathematicians used red and black counting rods for positive and negative numbers more than 2 000 years ago, and the Indian mathematician Brahmagupta wrote rules for them in the year 628, describing them as “debts”. Yet many European mathematicians still called them “absurd numbers” as late as the 1700s.
Otto’s tip
Use a factor tree and keep splitting until every branch ends in a prime. Circle the primes as you go!
How to use this Year 7 negative numbers lesson
First, read the key points so that you know where the Year 7 negative numbers lesson is going. Then work through each definition and worked example, and cover the solution before you read it. After that, practise with the negative numbers, factors, multiples and primes worksheet and check every question with the worksheet answers. When you feel ready, test yourself with the multiple choice questions or the quick quiz. Finally, sit the negative numbers, factors, multiples and primes test and mark it with the test answers.
Also, the starters are a good warm-up, so you can keep the key facts fresh. Moreover, all the pages of this chapter work together, so you can move between them at any time. For example, many pupils read the lesson on Monday, practise on Wednesday and test themselves on Friday. In short, this Year 7 negative numbers lesson gives you everything you need for the chapter.
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