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Ratios are a way of expressing one thing compared to another, if x:y is in the ratio of 1:2 this means y is twice the size of x. There are 8 skills involving ratios you need to learn.

Make sure you are happy with the following before continuing:

• Highest Common Factors

Skill 1: Writing Ratios as Fractions

Ratios can be written as a fraction in a couple of ways.

Example: Red to blue counters are in a bag in the ratio (\textcolor{red}{3} :\textcolor{blue}{2} ) .

There are \dfrac{\textcolor{red}{3}}{\textcolor{blue}{2}} as many red counters as blue counters.

There are \dfrac{\textcolor{blue}{2}}{\textcolor{red}{3}} as many blue counters as red counters.

Alternatively, we can write either part as a fraction of the total i.e. \dfrac{\textcolor{blue}{2}}{\textcolor{black}{5}} of the counters in the bag are blue.

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Skill 2: Simplifying

To simplify a ratio we divide all parts of the ratio by a common factor.

Example: Write the following ratio in its simplest form, \textcolor{red}{15}:\textcolor{blue}{30}:\textcolor{limegreen}{24}

All we do is divide each number by the highest common factor of all three numbers, which is 3 .

We can not simplify this any more, therefore the ratio is in its simplest form .

Skill 3: Scaling Ratios

To scale a ratio we multiply by a common factor.

Example: Meringue is made by mixing cups of egg whites and cups of sugar in the ratio \textcolor{limegreen}{3}:\textcolor{blue}{7} . How many cups of sugar are needed if  \textcolor{limegreen}{12} cups of egg whites are used in the mixture? We know that \textcolor{limegreen}{12} = \textcolor{limegreen}{3} \times \textcolor{black}{4} , so we need to multiply the ratio by \textcolor{black}{4}

So when \textcolor{limegreen}{12} egg whites are used, \textcolor{blue}{28} cups of sugar are needed.

Skill 4: Part to Whole Ratios

Sometimes you may see ratios x:y where y includes x . These are part : whole ratios.

Example: Adam has some apples and oranges in his bag. The ratio of oranges  to fruit in his bag is \textcolor{orange}{2}:\textcolor{blue}{7} .

a) Finding the fraction of a whole – What fraction of Adam’s fruit are oranges?

For every \textcolor{blue}{7} pieces of fruit, \textcolor{orange}2 of these are oranges.

\large{\frac{\text{part}}{\text{whole}} = \frac{2}{7}}

b) Finding the ratio of a part – What is the ratio of oranges to apples?

\textcolor{orange}{2} out of every \textcolor{blue}{7} pieces of fruit are oranges, so \textcolor{blue}{7} - \textcolor{orange}{2} = \textcolor{limegreen}{5} out of every \textcolor{blue}{7} pieces of fruit are apples. For every \textcolor{orange}{2} oranges there are \textcolor{limegreen}{5} apples

\text{\textcolor{Orange}{oranges} : \textcolor{limegreen}{apples}} = \textcolor{orange}{2} : \textcolor{limegreen}{5}

c) Finding the missing amount – Adam has \textcolor{orange}{4} oranges, how many apples does he have?

We need to scale up the ratio \textcolor{orange}{2}:\textcolor{limegreen}{5} , so that the left is equal to \textcolor{orange}{4} . So we multiply the ratio by \textcolor{black}{2} .

So, Adam has \textcolor{limegreen}{10} apples.

Skill 5: Dividing Amounts into Ratio

Being able to split a total amount into a ratio is a key skill needed.

Example: Aaron , Kim and Paul split \textcolor{purple}{£6000} in the ratio of \textcolor{red}{3}:\textcolor{limegreen}{4}:\textcolor{blue}{5} . How much money does Aaron receive?

Step 1: Find the total number of parts in the ratio:

\textcolor{red}{3}+\textcolor{limegreen}{4}+\textcolor{blue}{5} = \textcolor{black}{12} parts.

Step 2: Divide the total amount by the total number of parts in ratio, this finds the value of 1 part .

\textcolor{purple}{£6000} \div \textcolor{black}{12} = \textcolor{orange}{£500} = \, 1 part

Step 3: Multiply the value of one part by the number of parts Aaron has:

\textcolor{orange}{£500} \times \textcolor{red}{3} = £1500

So Aaron gets £1500 .

Skill 6: Difference Between Parts of a Ratio

You may sometimes be given the difference between two parts of the ratio, instead of the total amount.

Example: Josh , James and John share sweets in the ratio \textcolor{orange}{1}:\textcolor{blue}{2}:\textcolor{red}{4} . Josh has \textcolor{limegreen}{9} sweets less than John .  How many sweets does James have?

Step 1: Firstly, work out how many parts of the ratio \textcolor{limegreen}{9} sweets makes up:

\textcolor{limegreen}{9} \, \text{sweets} = \text{John's sweets} - \text{Josh's sweets} = \textcolor{red}{4} \, \text{parts} - \textcolor{orange}{1} \, \text{part} = 3 \, \text{parts} .

Step 2: Then divide to find 1 part:

\textcolor{limegreen}{9} \, \text{sweets} \div 3 = \textcolor{purple}{3} \, \text{sweets}

\textcolor{purple}{3} \text{ sweets } = 1 \text{ part }

Step 3: Multiply by the number of parts James has to find how many sweets he has:

Skill 7: Changing Ratios

You need to be prepared for questions where the ratio changes .

Example: Billy and Claire share marbles in the ratio \textcolor{blue}{5}:\textcolor{limegreen}{3} . Billy gives \textcolor{orange}{4} marbles to Claire and the ratio is now 1:1 . How many sweets did each have initially?

Step 1: At first, Billy had \textcolor{blue}{5}x marbles and Claire had \textcolor{limegreen}{3}x marbles.

Step 2: Billy gives \textcolor{orange}{4} marbles to Claire. Now, Billy has \textcolor{blue}{5}x-\textcolor{orange}{4} marbles and Claire has \textcolor{limegreen}{3}x+\textcolor{orange}{4} .

Step 3: The ratio \textcolor{blue}{5}x-\textcolor{orange}{4} : \textcolor{limegreen}{3}x+\textcolor{orange}{4} is 1:1 .

\dfrac{\textcolor{blue}{5}x-\textcolor{orange}{4}}{\textcolor{limegreen}{3}x+\textcolor{orange}{4}} = \dfrac{1}{1}

Rearranging and solving for x ,

\begin{aligned} (\textcolor{blue}{5}x-\textcolor{orange}{4})& = (\textcolor{limegreen}{3}x+\textcolor{orange}{4}) \\  \textcolor{blue}{5}x &= \textcolor{limegreen}{3}x+\textcolor{orange}{8} \\ 2x &= \textcolor{orange}{8} \\ x &= 4 \end{aligned}

Initially, Billy had \textcolor{blue}{5x = 20} marbles and Claire had \textcolor{limegreen}{3x = 12} .

Skill 8: Reducing Ratios to the form 1 : n

To reduce a ratio to the form 1:n or n:1 , all you have to do is divide the whole ratio by the smallest number.

Example 1: Scaling and Simplifying Decimal Ratios

Write the following ratio in its whole number simplest form.

First, we need to multiply all parts of the ratio until there are only whole numbers left before simplifying.

Example 2: Simplifying Ratios that have Different Units

Change the following ratio into the same unit ratio in its simplest form.

If ratios have different units, we need to convert one of the units to the other, then simplify the ratio to its simplest form .

Ratios Example Questions

Question 1: In a school, the ratio of the number of students with blonde hair to the number of students with brown hair is 4:5 .

a) What fraction of students have blonde hair?

b) If there are 450 students in the school, how many of them have brown hair?

a) In order to work out the fraction of students that have blonde hair, we need to add up the ratio parts. The sum of the ratio is 4 + 5 = 9 .  This means that we are dealing with 9 ths. Since the ratio share for blond students is 4 , this means that the fraction of blonde students is \dfrac{4}{9}

b) We know from the previous question that \dfrac{4}{9} of the students have blonde hair.  Therefore, the fraction of students with brown hair is \dfrac{5}{9} .  If there is a total of 450 students in the school, we need to work out what \dfrac{5}{9} of 450 is:

\dfrac{5}{9} \times 450 = 250 students.

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Question 2: Divide 35 in the ratio 2 : 5 .

In total, there are 7 parts to this ratio (2 + 5 = 7) .  If 7 shares have a value of 35 , then 1 share has a value of 5 (35 \div 7 = 5) .

Since 1 share has a value of 5 , then 2 shares will have a value of 10 (2 \times 5 = 10) .

Since 1 share has a value of 5 , then 5 shares will have a value of 25 (5 \times 5 =  25) .

Question 3: Lucy is tiling her bathroom. She buys white and blue tiles in the ratio of 13 : 2 . Blue tiles cost £2.80 whilst white tiles cost £2.35 . If she buys 16 blue tiles, how much does Lucy spend on tiles in total?

To work out the total cost of the tiles Lucy buys, we need to work out how many white tiles she buys. In order to work out the number of white tiles,  we  need to work out the total number of tiles she buys.

The ratio is 2 parts blue to 13 parts white.  If Lucy buys 16 blue tiles, this is 8 times more than the figure for blue tiles in the ratio (16\div2=8) .

If the number of blue tiles she buys is 8 times more than the blue tiles figure given in the ratio, then the number of white tiles she buys must also be 8 times more than the white tiles figure in the ratio.  Therefore, the number of white tiles she buys is:

13\times8 = 104 white tiles

Now that we know how many tiles of each colour she has bought, we can calculate the total cost of  the tiles.

The cost of 16 blue tiles is:

16\times2.80 = \pounds44.80

The cost of 104 white tiles is:

104\times2.35 =\pounds244.40

Therefore, Lucy’s total spend on tiles is:

44.80+244.40=\pounds289.20

Question 4:  Steve receives £200 each month from his parents as an allowance.  20 \% of this amount is spent on his football magazine subscription, and the rest he spends on football stickers, sweets and fizzy drinks in the ratio of 5 : 2 : 1 .  How much does Steve spend on football stickers?

The first thing we need to do is to deduct the 20 \% spent on the magazine subscription so that we can work out how much of his allowance Steve has left over.  20 \% of £200 can be calculated as follows:

0.2 \times \pounds 200 = \pounds 40

You may prefer to calculate the 20% in your head:

10 \% of £200 is £20 , so 20 \% is

2 \times \pounds 20 = \pounds 40

Then deduct this from £200 :

\pounds200 - \pounds40 = \pounds160

Steve therefore has £160 pounds remaining which he spends on sweets, football stickers and fizzy drinks in the ratio of 5 : 2 : 1 .

By adding up the ratio, we know that we are dealing with eighths.  (We know we are dealing in eighths because 5 + 2 + 1 = 8 ).  We know from the ratio that the share he spends on football stickers is 5 , meaning that Steve spends \frac{5}{8} of the remaining allowance on football stickers.

The actual amount that Steve spends on football stickers can be calculated as follows:

\dfrac{5}{8} \times \pounds160 = \pounds100

Question 5: Alieke, Jon, and Kate tracked the number of books they read last year. Jon read twice as many books as Kate, but Alieke read 4 times as many books as Jon.

a) Write down a ratio of the number of books read by Alieke to the number of books read by Jon to the number of books read by Kate.

b) Alieke read 63 more books than Kate last year. Work out how many books Alieke, Jon, and Kate read in total last year.

a) We are told that Jon reads twice as many books as Kate. As a ratio, this can be written as 2 : 1 .

We are also told that Alieke reads 4 times as many books as Jon.  As a ratio, this can be written as 4 : 1 .

The issue we have now is that in the Jon : Kate ratio, Jon’s share is 2 , while in the Alieke : Jon ratio, Jon’s share is 1 .  By doubling the 4 : 1 ratio for Aleike : Jon to 8 : 2 , Jon’s share is now the same in both ratios.

This means that we can express this is a 3 -way ratio as follows:

\text{Alieke : Jon : Kate }= 8 : 2 : 1

b) We are told that Alieke read 63 more books than Kate last year, and we know from the previous part of the question that the Alieke : Kate reading ratio is 8 : 1 . In this ratio, the share is 8 parts to 1 part, so we can conclude that the difference between the ratio share is 7 parts (8 - 1 = 7) .

If the difference in the ratio share is 7 parts, and the difference in the number of books read is 63 , then we can work out the number of books read that 1 share of the ratio represents:

63 \div 7 = 9\text{ books}

If one share of the ratio represents 7 books read, we can now work out how many books were read in total by the three people.  By adding up the ratio, we know that the total number of shares is 11 (8 + 2 + 1 = 11) , so the total number of books read can be calculated as follows:

11 \times 9\text{ books} = 99\text{ books}

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Understanding ratios: GCSE curriculum

Calculate a ratio, use ratios in problem-solving, common mistakes and tips, ratios in advanced applications, frequently asked questions, how to calculate ratios: gcse maths.

Liam Hindson

• How to calculate ratios in Maths: A GCSE Mathematics guide

• As a fraction : If there are 4 bananas and 8 apples, we can write the ratio of bananas to apples as 1/2. This means there are two times as many apples compared to bananas.
• With a colon : Using the same fruit example, this ratio can also be expressed as 1:2.
• Using the word 'to' : You could also see this comparison written as "1 to 2". This is the least common way to express a ratio.

Types of ratios

• Part-to-part : This ratio compares different parts of a group to each other, like the apples to oranges above. Both of these items are in the same group of fruit.
• Part-to-whole : This compares one part of the group to the entire group. In the fruit example, there are 12 pieces of fruit (4 bananas and 8 apples). There are 4 bananas with a part-to-whole ratio for all the fruit of 4:12. This simplifies to 1:3.
• Rates : A rate is a unique ratio where the quantities being compared have different units. Taking the fruit example further, this could be carbohydrates or calories for bananas and apples.

You're preparing a fruit salad that needs a ratio of apples to oranges of 2:3. If you already have 6 apples, how many oranges do you need?

For every 2 apples, you need 3 oranges. If you have 6 apples, that's 3 groups of 2 apples (6 divided by 2). Therefore, you need 3 groups of 3 oranges, which totals 9 oranges.

• Length of the model - To find the model length, we need to divide the actual length by the scale factor. This is 500 / 100 = 5 meters
• Width of the model = To find the model width, we need to divide the actual width by the scale factor. This is 40 / 100 = 0.4 meters

Simplifying ratios

• Find the GCD: Look for the highest number that divides both parts of the ratio, keeping each part a whole decimal without a remainder.
• Divide: Divide each part of the ratio by the GCD that you have found.

You need to landscape a garden and want a ratio of flowering plants to green plants of 3:2. If there are 45 flowering plants, how many green plants do you need?

20 green plants

30 green plants

40 green plants

There need to be 2 green plants for every 3 flowering plants. If you have 45 flowering plants, which is 15 groups of 3, you will need 15 groups of 2 green plants (15 × 2) = 30 green plants.

• Recipes : You can change the quantity of each ingredient to make a larger or smaller dish while keeping the taste and texture the same.
• Chemistry : To create the desired chemical reaction, you need to mix chemicals with precise ratios to avoid wastage or dangerous outcomes.
• Finance : Ratios are used to analyse financial statements. For example, a ratio called "debt-to-equity" is used to discover the balance between money borrowed by a business and the funds owned by shareholders.
• The ratio is 3 parts grass and 1 part paved. This means a total of 4 parts.
• Each part is the total (16,000) divided by 4 = 4,000 square meters.
• This means that the paved area (1 part) will be 4,000 square meters.

In a classroom, the ratio of students to computers is 3:1. If there are 36 students in the class, how many computers are needed?

12 computers

10 computers

15 computers

There are three times as many students to computers. If there are 36 students, we divide this by 3 per computer. This means there are 12 computers.

Tips for success

• Practice regularly : Irritating as it may sound, practice really does make perfect. Work through from basic to increasingly complex ratios to increase your understanding and confidence.
• Visualise the problem : Draw a diagram or even use a physical object to help you see the relationship shown by the ratio. It's an especially useful technique when you're dealing with complex scenarios.
• Check your work : Always double-check your calculations and the logic of your ratios. It's a good idea to invert the operation and see if you return to the original quantities.

You need to mix paint to create a custom colour. The ratio of red to blue paint is 3:2. If you start with 6 litres of red paint, how many litres of blue paint do you need?

There need to be 2 parts of blue for every 3 parts of red. If you have 6 litres of red paint (2 groups of 3), you need 2 groups of 2 litres of blue paint. The sum of this is 4 litres.

• The Gear ratios in machines determine the speed and torque of moving parts, which are essential for designing efficient mechanical systems.
• Aspect ratios are used in civil engineering to influence the stability and aesthetics of buildings.

Applying advanced ratios

An investor analyses two companies; Company A has a price-to-earnings ratio of 25, while Company B has a ratio of 15. If all other factors are equal, which company's stock represents a better value for money?

Both are equally valuable

A lower price-to-earnings ratio indicates that the stock is undervalued and may represent a better value for money. Company B has a ratio of 15, which means it might offer more earnings per share than Company A, which has a higher ratio of 25.

What is the difference between a ratio and a proportion?

How can i turn a ratio into a decimal, how do i use ratios to make predictions, what examples are there of ratios in everyday life, can ratios help in understanding scales, what are some tips for simplifying ratios.

• Ratio - A comparison of 2+ quantities, showing how many times one value is greater/smaller than the other.
• Proportion - An equation that states two ratios are equal. It helps for solving problems where one part of the ratio is unknown.
• Rate - A specific kind of ratio where the quantities have different units, like speed (miles per hour) or density (people per square mile).
• Scale - The ratio of the dimensions of a model to the dimensions of the original. This is often used in drawings, models and maps.
• Scale factor - A number which scales or multiplies some quantity. It is used to adjust the sizes or amounts in direct proportion to the factor.
• Direct proportion - When two quantities increase or decrease at the same rate. This means their ratio remains constant.
• Inverse proportion - A relationship where one quantity increases as another decreases.
• Simplify - Reducing a ratio to its smallest form by dividing both terms by their greatest common divisor.
• Part-to-part - A comparison between two distinct groups in a larger set, focusing on the relationship between these two groups alone.
• Part-to-whole - A comparison where one part is compared to the total, combining all parts into a total.

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Ratio problem solving for 9-1 GCSE with answers

Subject: Mathematics

Age range: 14-16

Resource type: Worksheet/Activity

Last updated

27 September 2017

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clareturnertutor

A good set of ratio questions that require problem-solving. Thank you for sharing.

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Nice selection of questions, thank you.

This is an excellent worksheet for the most able students because it focuses on the harder questions that initially cause them problems that are reasonably easy to overcome.

Lovely selection of questions, thank you.

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Ratio: Problem Solving Textbook Answers

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The resources on this page will hopefully help you teach AO2 and AO3 of the new GCSE specification - problem solving and reasoning.

This brief lesson is designed to lead students into thinking about how to solve mathematical problems. It features ideas of strategies to use, clear steps to follow and plenty of opportunities for discussion.

The PixiMaths problem solving booklets are aimed at "crossover" marks (questions that will be on both higher and foundation) so will be accessed by most students. The booklets are collated Edexcel exam questions; you may well recognise them from elsewhere. Each booklet has 70 marks worth of questions and will probably last two lessons, including time to go through answers with your students. There is one for each area of the new GCSE specification and they are designed to complement the PixiMaths year 11 SOL.

These problem solving starter packs are great to support students with problem solving skills. I've used them this year for two out of four lessons each week, then used Numeracy Ninjas as starters for the other two lessons.  When I first introduced the booklets, I encouraged my students to use scaffolds like those mentioned here , then gradually weaned them off the scaffolds. I give students some time to work independently, then time to discuss with their peers, then we go through it as a class. The levels correspond very roughly to the new GCSE grades.

Some of my favourite websites have plenty of other excellent resources to support you and your students in these assessment objectives.

@TessMaths has written some great stuff for BBC Bitesize.

There are some intersting though-provoking problems at Open Middle.

I'm sure you've seen it before, but if not, check it out now! Nrich is where it's at if your want to provide enrichment and problem solving in your lessons.

MathsBot  by @StudyMaths has everything, and if you scroll to the bottom of the homepage you'll find puzzles and problem solving too.

I may be a little biased because I love Edexcel, but these question packs are really useful.

The UKMT has a mentoring scheme that provides fantastic problem solving resources , all complete with answers.

I have only recently been shown Maths Problem Solving and it is awesome - there are links to problem solving resources for all areas of maths, as well as plenty of general problem solving too. Definitely worth exploring!

Ratio Problem Solving Worksheet

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Help your students prepare for their Maths GCSE with this free ratio word problem  worksheet of 44 questions and answers

• Section 1 of the ratio word problem worksheet contains 36 skills-based ratio word problem questions, in 3 groups to support differentiation
• Section 2 contains 4 applied ratio word problem questions with a mix of worded problems and deeper problem solving questions
• Section 3 contains 4 foundation and higher level GCSE exam style ratio word problem questions 
• Answers and a mark scheme for all ratio word problem questions are provided
• Questions follow variation theory with plenty of opportunities for students to work independently at their own level
• All questions created by fully qualified expert secondary maths teachers
• Suitable for GCSE maths revision for AQA, OCR and Edexcel exam boards
• Free downloadable and printable resources

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Raise maths attainment across your school with hundreds of flexible and easy to use GCSE maths worksheets and lessons designed by teachers for teachers.

Ratio problem solving at a glance

Ratios can be written as part to part or part to whole. This varied interpretation is necessary when solving worded problems.

Dividing a quantity by a given ratio:

The quantity is divided by the total sum of the parts of the ratio and then multiplied by each required part within the ratio.

Fraction word problems (and percents):

For the part to part ratio A:B, the fraction of the ratio that is A is equal to A (the numerator) divided by A+B (the denominator); the part to whole ratio A:C would be expressed as the fraction A divided by C. To convert the fraction to a percentage, we determine the equivalent fraction where the denominator is equal to 100.

Simplifying and equivalent ratios:

We can simplify ratios by dividing each part of the ratio by a common factor. Writing ratios in their simplest form makes it easier to visualise the relationship between the quantities in the ratios. It also makes it easier to use the ratios. Just as we can simplify ratios by dividing each part by a common factor, we can find other equivalent ratios by multiplying each part by a common number. We can use equivalent ratios to solve problems.

Ratio and proportion word problems:

Ratio word problems may ask us to write a ratio, simplify a ratio, divide a quantity into a ratio or use a ratio to find quantities. We can use a bar model to represent a given ratio and this can help us visualise the ratio problem more easily. 

Rate word problems:

Writing a ratio in the form 1:n allows us to compare two quantities and determine their unit rate. This could be a currency conversion, the price per litre of fuel, or the speed of an object (compound measures).

  Looking forward, students can then progress to additional ratio and proportion worksheets , for example a speed, distance, time worksheet or a   direct proportion worksheet .

For more teaching and learning support on Ratio and Proportion our GCSE maths lessons provide step by step support for all GCSE maths concepts.

You will find free worksheets with real life problems and exam style questions for each topic. Each math worksheet is accompanied by an answer key. 

 When you have students who require more intensive support our one to one GCSE maths revision programme will match them with the most appropriate tutor. This way we can provide individual students with personalised programmes of study while you continue to teach the rest of your class as a whole group. 

Our maths interventions are currently only available for GCSE students, and are not suitable for A level students.

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GCSE Maths: Algebra and Negative Numbers

BBC Teach > Secondary resources > Maths GCSE > The Maths Show

MATT: Welcome to The Maths Show with me Matt Parker. Today we’re doing algebra, specifically negative numbers. Now negative numbers can be quite tricky, in fact for a long time mathematicians refused to admit that they even existed.

MATT: It wasn’t until the 3rd century in China, and the 7th century in India, that mathematicians started to use them.

MATT: In Europe Greeks considered any solution that had a negative value, to be false, absurd and deeply uncool.

MATT: In Europe they didn’t start using negative numbers until the 17th century. Having somehow remained positive through the entire dark ages…

MATT: So in order to be better at negative numbers than this guy was, I present to you the top negative five things about negative numbers.

MATT: In at negative five, it’s correctly adding and subtracting negative numbers. I mean, this was always going to be in the top five because people get it wrong and the secret is, just draw a number line…

MATT: Mark positive in one direction, negative in the other. You start with the first number and then you move either up or down the line,

MATT: so negative four, subtract two… start at negative four and go two in the negative direction.

MATT: Or if you wanna add five, go five in the positive direction. Just always use a number line…

MATT: Hopping in at negative four, it’s two negatives make a positive… you’ve got to remember that two negatives give you a positive. This happens when you’re multiplying and dividing.

MATT: So negative four times negative three… four by three gives you 12… and because you’re multiplying two negatives the answer is a positive.

MATT: This also occurs if you’re subtracting a negative number. So seven minus negative three, the two negatives combine to give you a positive. Seven plus three is 10… two negatives make a positive.

MATT: And at negative three, when two negatives don’t make a positive. So serious we’ve got a lighting change. Now sometimes people will see something like negative one sub... We don’t need it the whole time do we? Can we? Thank you…

MATT: People will see this and go, oh look two negatives, it must be a positive… but we’re not multiplying or dividing, we’re not subtracting a negative… you can just work this out on a number line.

MATT: You start at negative one, you subtract four, so you go four in the negative direction. You end up at negative five.

MATT: Sometimes two negatives don’t make a positive, and I don’t think we need the lights a second time... thank you.

MATT: In at negative two it’s expanding brackets. Just because you see brackets doesn’t mean you can forget everything you know about negative numbers. Pay close attention to where the negatives go.

MATT: The first bit is reasonably straight forward, three times two x, six x, and three times negative four, is negative 12. For the second section you need to multiply everything inside the brackets by negative two. Negative two times x, negative two x… and then negative two times negative four gives you positive eight.

MATT: Make sure you follow every single step, and then simplify it down and you’re done. Pay very close attention to negatives when you’re simplifying expressions.

MATT: Finally, in at negative one, it’s negative numbers… hate your haircut and think you’re going to f… who wrote this one?

MATT: No, it’s actually squaring negative numbers.

MATT: Because, negative three squared is very different to

MATT: negative, three squared.

MATT: One is negative three times negative three,

MATT: and the other is the negative of three times three.

MATT: You get different answers.

MATT: So if you’re substituting into x squared, it’s the first one. It’s the negative number times itself, it will give you a positive answer…

MATT: And that’s it, that’s our top negative five things about negative numbers. If you’ve paid close attention you are now officially better at negative numbers than any ancient Greek mathematician.

MATT: And don’t forget the number one golden rule about negative numbers, always stay positive!... That doesn’t sound right?

Video summary

Negative numbers are made easy in this handy guide for maths GCSE students from Matt Parker, who breaks down his top five problems with negative numbers.

The mathematician and comedian uses this video to explain the sometimes confusing topic of negative numbers.

A mixture of graphics and jokes help break down negative numbers into Matt’s “Top Five” categories of negative number errors.

Matt also explains the history of negative numbers, including how Europeans didn’t use them until long after Chinese and Indian mathematicians.

He also reinforces the importance of order of operations when dealing with an equation and the use of acronyms such as BIDMAS and BODMAS in revision.

While the videos have been designed as revision tools for GCSE level students, they could also be used in part to introduce topics for earlier Key Stages.

This short film is from the BBC series, The Maths Show.

Teacher Notes

During the video:

• As Matt sets up the various examples using brackets and powering negative numbers, pause before he answers and test your students on the questions. See if they can set up number lines correctly to solve equations that require adding and subtracting negative numbers.
• Use Always/Sometimes/Never to elicit discussion of negative number ‘rules’ during the pausing of the video.
• Students could use whiteboards or think pair share to encourage collaboration of checking methods that some students may already utilize.
• A calculator emulator or similar would be useful to look at calculator skills.

After the video:

• Look at other past exam questions that highlight the use of negative number skills. Use ‘spot the mistake’ questions to ensure use of appropriate checking skills. Explore different ways of creating number lines to use in a timely manner with sufficient but not excessive information.
• Use a matching activity or similar to substitute negative values into formulae - ensure a mix of the four operations. For students that need negative number reinforcement before approaching algebra, a similar activity with sums alone will help students.
• Students could create an odd one out activity where either two answers are the same and one is different, or two answers are correct, and one is wrong (breaking one of the top -5 rules).
• Ask students to create a poster or display using their own examples for Matt’s “Top -5 mistakes with negative numbers”, including how to check that they haven’t made the errors. Students could start with a simpler substitution expression then slowly build it using various positive and negative terms.
• Ask your pupils if they find any parts of negative numbers tricky that Matt didn’t mention?

Suitable for teaching maths at GCSE in England, Wales and Northern Ireland and National 4/5 or Higher in Scotland.

More from The Maths Show

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Matt offers tips on identifying different graph types and their uses, as well as tips to avoid common mistakes when working with formulae.

Matt simplifies the use of lines of best fit for students struggling to achieve a passing mark on the maths GCSE.

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Matt demonstrates how to handle probabilities using probability and frequency trees.

Matt breaks down key GCSE angle problems into easy steps, including internal angles of polygons and bearings.

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Matt offers advice for teachers and students on test-taking techniques specifically for the maths GCSE.

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A guide to converting fractions, decimals and percentages, applied to real-world problems.