- National Curriculum Tool
- Year 6 - Ratio and Proportion
Year 6 - Ratio and Proportion
New Curriculum
- solve problems involving the relative sizes of two quantities where missing values can be found by using integer multiplication and division facts
- solve problems involving the calculation of percentages [for example, of measures, and such as 15% of 360] and use percentages for comparison
- solve problems involving similar shapes where the scale factor is known or can be found
- solve problems involving unequal sharing and grouping using knowledge of fractions and multiples
Non-Statutory Guidance
Pupils recognise proportionality in contexts when the relations between quantities are in the same ratio (e.g. similar shapes, recipes).
Pupils link percentages or 360° to calculating angles of pie charts.
Pupils should consolidate their understanding of ratio when comparing quantities, sizes and scale drawings by solving a variety of problems. They might use the notation a:b to record their work.
Pupils solve problems involving unequal quantities e.g. ’for every egg you need three spoonfuls of flour’, ‘⅗ of the class are boys’. These problems are the foundation for later formal approaches to ratio and proportion.
Links and Resources
Pupils should make rich connections across mathematical ideas to develop fluency, mathematical reasoning and competence in solving increasingly sophisticated problems. They should also apply their mathematical knowledge to science and other subjects.
– National Curriculum, September 2013, page 10)
Connections within Mathematics
Making connections to other topics within this year group
Number – Fractions
- use common factors to simplify fractions; use common multiples to express fractions in the same denomination
- associate a fraction with division and calculate decimal fraction equivalents (e.g. 0.375) for a simple fraction (e.g. ⅜)
- recall and use equivalences between simple fractions, decimals and percentages, including in different contexts
When working on ratio and proportion and/or fractions, including decimals and percentages, there are opportunities to make connections between them, for example:
In the same way that fractions describe the relationship between parts of a whole and the whole of which they are a part, proportion also expresses the relationship between parts and wholes. Ratio, on the other hand, describes the relationships between parts. When working with ratio, the whole can be inferred by understanding the total number of parts. In turn, once the total number of parts is known, any number of those parts can be expressed as a fraction or proportion of the whole. It is helpful to work through an example: imagine 10 counters of which 4 are red and 6 are blue. When comparing the red with the blue, we see that there are 4 red for every 6 blue; the ratio is 4:6. This can be simplified to 2:3. Proportion is usually represented as a fraction. Four out of ten counters are red, so, 4/10 are red. Six out of ten counters are blue, so, 6/10 are blue. These fractions can be simplified to ⅖ and ⅗. You can practise this with the children using counters, interlocking links or cubes or a collection of classroom items, for example, 8 pencils and 4 rubbers. 8⁄12of the collection are pencils, this fraction can be simplified to ⅔. 4⁄12 are rubbers, this can be simplified to ⅓. Simplifying fractions covers the first objective in the above list.
The fractions made can also be converted to decimals and percentages, for example 0.4 or 40% of the counters are red and 0.6 or 60% of the counters are blue. 0.3 of 30% of the collection are pencils and 0.7 or 70% are rubbers.
Using the ITP Fractions gives opportunities to visually make comparisons between fractions, decimals, ratio and proportion.
Ratio and proportion is another ITP worth exploring with the children. In this ITP you can demonstrate comparisons between different ratios and proportions with volumes of liquids.
Number-Multiplication and division
When working on ratio and proportion and/or multiplication and division there are opportunities to make connections between them, for example:
Scaling an object or an amount down by multiplying by a fraction, for example, if sketching a tree or building in the school’s grounds, the height would need to be scaled down. You could ask the children to make an estimate of the height of the object by walking away from it until, when they bend down they can see its top from between their legs. They put a marker at this point and then measure from the marker to the base of the object. This will give a reasonable estimate of its height because the child will be looking at the top of the tree at an angle of approximately 45 degrees if viewed in this way. Therefore, the height of the tree will be the same as the distance from it. To make a sketch, this measurement needs to be scaled down. For example if it was 10m tall and they wanted to make a scale drawing showing the tree as 1m high, they would need to draw the tree at a scale of 1:10 (or multiply the height of the tree by a scale factor of 1/10).
Children could draw classroom objects to scale. For example, the height of a table measuring 50cm could be scaled down to (multiplied by scale factor) 1/5 to make the table10cm in the drawing.
Statistics
- interpret and construct pie charts and line graphs and use these to solve problems
When working on ratio and proportion and/or statistics there are opportunities to make connections between them, for example:
Using the ITP Data Handling is an effective way to explore comparisons using a pie chart. In the example below, the children could estimate the percentages of people of different ages and compare them. They could present the information as proportions using fraction and/or decimals. They could compare data as ratios, for example the ratio of those under 20 to those between 20 and 50 is 1:2.
Making connections to this topic in adjacent year groups
Year 5
Number-Fractions (including decimals and percentages)
- read and write decimal numbers as fractions [for example, 0.71 = 71⁄100 ]
Non statutory guidance
- Pupils should be taught throughout that percentages, decimals and fractions are different ways of expressing proportions.
Key Stage 3
Number
- interpret fractions and percentages as operators
Ratio, proportion and rates of change
- use scale factors, scale diagrams and maps
- express one quantity as a fraction of another, where the fraction is less than 1 and greater than 1
- use ratio notation, including reduction to simplest form
- divide a given quantity into two parts in a given part:part or part:whole ratio; express the division of a quantity into two parts as a ratio
- understand that a multiplicative relationship between two quantities can be expressed as a ratio or a fraction
Cross-curricular and real life connections
Learners will encounter ratio and proportion:
Within the science curriculum there are opportunities to work with ratio and proportion. For example pupils should select the most appropriate ways to answer science questions using different types of scientific enquiry, including observing changes over different periods of time, noticing patterns, grouping and classifying things, carrying out comparative and fair tests and finding things out using a wide range of secondary sources of information. The children could, for example, construct pie charts or use ratio and proportion to compare groupings and classifications or the results of tests that they carry out.
Within the geography curriculum there are opportunities to connect with ratio and proportion, for example in the introduction of the Key Stage 2 Programme of Study it states that pupils should extend their knowledge and understanding beyond the local area to include the United Kingdom and Europe, North and South America. This will include the location and characteristics of a range of the world’s most significant human and physical features. Children could, for example, find and compare distances between countries or cities, compare population statistics, temperatures, lengths of rivers, heights of mountains. The results of any comparisons could be displayed in a pie chart.
There are also opportunities to work with ratio and proportion, linked to history, for example, ‘pupils should continue to develop a chronologically secure knowledge and understanding of British, local and world history, establishing clear narratives within and across the periods they study. The children could, for example, represent the lengths of the different periods in history and the rules of different British monarchs using pie charts’ (History Programme of Study). This would enable them to make comparisons using proportion as fractions or percentages.
| Programme of Study statements | Activities | |||
|---|---|---|---|---|
| A | B | C | D | |
| solve problems involving the relative sizes of two quantities where missing values can be found by using integer multiplication and division facts | ||||
| solve problems involving the calculation of percentages [for example, of measures, and such as 15% of 360] and the use of percentages for comparison | ||||
| solve problems involving similar shapes where the scale factor is known or can be found | ||||
| solve problems involving unequal sharing and grouping using knowledge of fractions and multiples | ||||
Activity set A
You could give the children, for example, one green and three yellow counters, the children could identify, for example, the ratio of green to yellow, the proportion of green. They could then work out other numbers of counters that would give the same ratio, for example two green and six yellow, three green and nine yellow,. This would help them to understand that the numbers of counters might change but the ratio does not change, if the relationship between the different parts remains the same. You could ask them to explore what they need to do to find a formula for ‘g’ green and ‘y’ yellow so that they can work out any number of these counters that give the same ratio and proportion.
Give the children some blue and white interlocking cubes and ask them to show you different ratios and proportions, such as:
- a 5:2 ratio of blue to white
- a 2:5 ratio of blue to white
- blue is 2/5 of white
- blue is 5/2 or 2½ of white
- the proportion of cards from a normal pack of 52 that is red
- the proportion of cards from a normal pack of 52 that are aces
You could give the children a selection of recipes from the internet and ask them to work out the ratio of, for example, flour:sugar:margarine in a sponge cake, or the proportion of a ready meal that should be eaten by 2 people if the meal is intended to serve 6 people. You could then ask them to rewrite a list of ingredients for a recipe, originally written for 4 so that it will serve 6 or 12 people.
You could give the children problems such as:
- Tom and Nafisat win £750 between them. They agree to divide the money in the ratio 2:3. How much do they each receive?
- A necklace is made using gold and silver beads in the ratio 3:5. If there are 80 beads in the necklace:
- How many are gold?
- How many are silver?
- To make a tasty chocolate milkshake James needs one part chocolate sauce to six parts milk. Each part is 150ml
- If he used 4 parts of chocolate sauce how much milk would he need?
- If he had 420 ml milk, how much chocolate sauce?
Activity set B
The children could construct a pie chart and then make up and solve problems from it. You could set scenarios such as: the local health authority are surveying the eating habits of school children and want to know how many of the 360 children in a local school have school dinners, packed lunches or go home. If appropriate the children could find out this information or could make up the data. They could then construct a pie chart using a protractor with every degree representing one child. They could then find the numbers, fractions or percentages of children having each type of lunch.
Set problems such as this for the children to solve:
- Tammy was saving for a laptop. The laptop she wanted cost £360. She has saved 60% of the amount. How much more money does she need?
Encourage the children to use effective methods to find 60%, such as find 50% by halving and then 10% by dividing £360 by ten and adding the two amounts together.
- In the sale a coat has been reduced by 20%. It now costs £56. What was its original price?
The children could use the bar model to help them solve this:
Each section of the bar is worth £14 so the original cost of the coat must be £70.
You could give the children the Nrich activity ‘Rod Ratios’ or, as a challenge, ‘Weekly Problem 27’
Activity set C
- Children could look at photographs of themselves or famous buildings and discuss why they are smaller than the actual children or buildings. Establish that they have been scaled down. Discuss where else they might see scaled down images, for example, maps, models, architects plans.
- The children could measure the lengths/heights and widths of objects around the classroom, scale these measurements down by an amount they choose and then sketch the object to that size on paper. Other children could estimate by how much these have been scaled down.
- The children could look at maps and work out distances from one place to another using the given scale.
- You could discuss when objects might need to be scaled up – explain that this is called ‘enlarged’. A good example would be looking at very small objects under a magnifying glass or microscope. If you have any available the children could use the equipment to see what different objects, such as an apple pip or the head of a pin, would look like if scaled up by the magnification on the apparatus.
Activity set D
- The children could look at paintings such as ‘Tiger in a storm’, by Henri Rousseau. They could explore mixing blue and yellow paints to get the colours that Rousseau has achieved. What ratios of blues and yellows have they made? What are these as proportions or fractions of the total paint mix? They could paint a jungle scene using their mixed paints.
- See The Art of Mathematics from Issues 8-63 of our (now archived, as PDF documents) Primary Magazine for other ideas of how to link ratio and proportion to art
Examples of what children should be able to do, in relation to each (boxed) Programme of Study statement
solve problems involving the relative sizes of two quantities where missing values can be found by using integer multiplication and division facts
Answer problems such as:
- Here is a recipe for pasta sauce.
Pasta sauce
300 g tomatoes
120 g onions
75 g mushrooms
Sam makes the pasta sauce using 900 g of tomatoes. What weight of onions should he use? What weight of mushrooms?
- A recipe for 3 portions requires 150 g flour and 120 g sugar. Desi’s solution to a problem says that for 2 portions he needs 80 g flour and 100 g sugar. What might Desi have done wrong? Work out the correct answer.
-
This map has a scale of 1 cm to 6 km.
The road from Ridlington to Carborough measured on the map is 6.6 cm long. What is the length of the road in kilometres?
solve problems involving the calculation of percentages (e.g. of measures) such as 15% of 360 and the use of percentages for comparison
Find simple percentages of amounts and compare them. For example:
- A class contains 12 boys and 18 girls. What percentage of the class are girls? What percentage are boys?
- 25% of the apples in a basket are red. The rest are green. There are 21 red apples. How many green apples are there?
solve problems involving similar shapes where the scale factor is known or can be found
- Solve simple problems involving direct proportion by scaling quantities up or down, for example:
Two rulers cost 80 pence. How much do three rulers cost?
- Use the vocabulary of ratio and proportion to describe the relationships between two quantities solving problems such as:
Two letters have a total weight of 120 grams. One letter weighs twice as much as the other. Write the weight of the heavier letter.
The distance from A to B is three times as far as from B to C. The distance from A to C is 60 centimetres. Calculate the distance from A to B.
solve problems involving unequal sharing and grouping using knowledge of fractions and multiples
Relate fractions to multiplication and division (e.g. 6 ÷ 2 = ½ of 6 = 6 × ½), simplify fractions by cancelling common factors, find fractions of whole-number quantities and solve problems such as:
- What fraction is 18 of 12
- What fraction is 500ml of 400ml?
- What is 14⁄35 in its simplest form? ⅖
- What ⅓ × 15? What about 15 × ⅓? How did you work it out?
- What is two thirds of 66?
- What is three quarters of 500?