Ratio and proportion
Ratio notation and simplifying, sharing a quantity in a given ratio, ratio as a fraction, and scale factors, maps and direct proportion
About four lessons of 45 to 60 minutes
Which paint mix makes the stronger red?
A decorator mixes 2 parts red paint to 3 parts white to make Mix A, and 3 parts red to 5 parts white to make Mix B. Mix B has MORE red parts than Mix A (3 versus 2), so it looks like Mix B should be redder. It isn't.
The trick is that the two mixes also have different total amounts of paint: Mix A makes 5 parts of paint altogether, Mix B makes 8. To compare them fairly you need the FRACTION of each mix that is red: Mix A is 2 out of 5 parts red, 2/5 = 40%. Mix B is 3 out of 8 parts red, 3/8 = 37.5%. Mix A is actually the stronger red, even with fewer red parts, because ratio is about the relationship between quantities, not the raw numbers. This unit is about reading that relationship correctly: writing ratios, sharing amounts by ratio, turning a ratio into a fraction, and scaling amounts up or down proportionally.
- Paint mixed 2 parts red to 3 parts white2/5 of the mix is red, a stronger red than the '3' in 3 : 5 first suggests
- Squash mixed 1 part cordial to 4 parts waterthe ratio 1 : 4 means 1/5 of the drink is cordial, not 1/4
- A map with a scale of 1 cm to 25 kmevery centimetre measured on the map stands for 25 real kilometres
- Splitting a $150 prize between two winners in the ratio 2 : 3the ratio decides how the whole prize divides into shares
What students will be able to do
Students will write and simplify ratios to their lowest terms and in the unit form 1 : n, divide a quantity into two or three parts in a given ratio and reason from a known part to the total, express one quantity as a fraction of another (both less than and greater than 1) and convert between a ratio and a fraction, and use scale factors, scale diagrams, maps and the unitary method to solve direct proportion problems.
- I can simplify a ratio to its lowest terms by dividing every term by their highest common factor.
- I can write a ratio in the unit form 1 : n.
- I can share a quantity into two or three parts in a given ratio, and find a missing part or the total when only one part is known.
- I can express one quantity as a fraction of another, whether the fraction is less than 1 or greater than 1.
- I can convert between a ratio and a fraction that describe the same multiplicative relationship.
- I can use a scale factor, a map scale, or the unitary method to solve a direct proportion problem.
Standards this unit teaches
- KS3 Maths: Ratio, proportion and rates of changeUK National Curriculum (England)Ratio, proportion and rates of change
Statutory requirement (Department for Education, "National curriculum in England: mathematics programmes of study", updated 28 September 2021, Key stage 3, "Ratio, proportion and rates of change" 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 "use ratio notation, including reduction to simplest form"; "divide a given quantity into 2 parts in a given part:part or part:whole ratio; express the division of a quantity into 2 parts as a ratio"; "express 1 quantity as a fraction of another, where the fraction is less than 1 and greater than 1"; "understand that a multiplicative relationship between 2 quantities can be expressed as a ratio or a fraction"; "use scale factors, scale diagrams and maps".
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
- Grade 6 ratios & rates teaching unitthe same-age unit on simplifying ratios and unit rates; this unit builds further into sharing, fraction conversions and scale
- Ratio in the glossarya quick refresher on what a ratio compares
- Simplifying fractions in the glossarythe same highest-common-factor technique simplifies a ratio
- Fraction in the glossaryneeded to move between a ratio and a fraction
Words to teach and display
- Ratio
- a comparison of two (or more) quantities, written a : b, showing how much of one there is for a given amount of the other
- Simplest form
- a ratio written so its terms share no common factor larger than 1, found by dividing every term by their highest common factor
- Part:part ratio
- a ratio comparing two parts of a whole to each other, such as boys to girls in a class
- Part:whole ratio
- a ratio comparing one part to the whole amount, such as boys to the whole class
- Unitary method
- finding the value of ONE unit first, then multiplying to find any other amount, the method behind scaling recipes and reading a map scale
- Scale factor
- the number a length is multiplied by to enlarge or reduce it to a similar shape
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. Ratio notation and simplifying
ConcreteA ratio compares two (or more) quantities using the ':' symbol. Mix A's paint, 2 parts red to 3 parts white, is written 2 : 3. Just like a fraction, a ratio can be simplified: if every term shares a common factor, dividing all of them by it gives an equivalent ratio in simplest form.
To simplify a ratio, find the highest common factor (HCF) of all its terms and divide every term by it. 18 : 24 has HCF 6 (18 = 6 x 3, 24 = 6 x 4), so 18 : 24 simplifies to 3 : 4. The two ratios describe exactly the same relationship, just with smaller numbers.
A ratio can also be written in the unit form 1 : n, which is useful for comparing rates directly. Dividing BOTH terms of 8 : 20 by the first term, 8, gives 1 : 2.5 (since 20 / 8 = 2.5). This form always makes the first quantity exactly 1, so n on its own tells you the full multiplicative relationship.
Write the ratio 18 : 24 in its simplest form.
- Find the highest common factor of 18 and 24: HCF(18, 24) = 6.
- Divide both terms by 6: 18 / 6 = 3, and 24 / 6 = 4.
Answer: 18 : 24 simplifies to 3 : 4.
- What is the highest common factor of 12 and 18, and how do you use it to simplify 12 : 18?
- Why does dividing every term of a ratio by the same number keep the ratio equivalent?
3. Ratio as a fraction: comparisons above and below 1
AbstractA ratio and a fraction can describe the exact same relationship. A 75 km journey has a 45 km motorway section and a 30 km country-road section. What fraction of the WHOLE journey is the motorway? What fraction of the country-road distance is the motorway distance?
The motorway is 45 out of the 75 total km, so as a fraction of the whole journey that is 45/75, which simplifies (divide both by 15) to 3/5, a proper fraction less than 1, exactly as you would expect for a part of a whole.
But the motorway compared only to the country-road section is 45/30, which simplifies (also divide by 15) to 3/2. That fraction is GREATER than 1, because the motorway section is actually longer than the country-road section, 1.5 times as long. A fraction comparing two separate quantities is not restricted to being a part of one whole, so it can be more than 1.
A 75 km journey has a 45 km motorway section and a 30 km country-road section. What fraction of the whole journey is the motorway, and what fraction of the country-road distance is the motorway distance?
- Fraction of the whole journey: 45/75. Both terms share the factor 15: 45/75 = 3/5.
- Fraction of the country-road distance: 45/30. Both terms again share the factor 15: 45/30 = 3/2.
Answer: The motorway is 3/5 of the whole journey, and 3/2 (one and a half times) the length of the country-road section.
- Can a fraction comparing two quantities ever be greater than 1? Give an example.
- If two amounts are in the ratio 2 : 7, what fraction of their total is the first amount?
4. Scale factors, maps and direct proportion
AbstractA scale factor is the number every length is multiplied by to enlarge or reduce a shape, drawing, or map. Map scales and recipes both use the SAME underlying idea, the unitary method: find the amount for ONE unit first, then scale up (or down) to any other amount.
A map scale of 1 cm to 25 km means every centimetre measured on the map represents 25 real kilometres. Two towns 6 cm apart on the map are 6 x 25 = 150 km apart in real life. Going the other way, if you know two real places are 150 km apart, the map distance is 150 / 25 = 6 cm.
The unitary method works the same way for recipes and rates: find how much ONE person, ONE hour, or ONE unit needs, then multiply by however many you actually have. It always starts with a division (to find the value of one) followed by a multiplication (to scale to the amount you need).
A recipe for 4 people uses 200 g of pasta. How much pasta is needed for 10 people?
- Find the amount for ONE person (the unitary method): 200 g / 4 = 50 g per person.
- Scale up to 10 people: 10 x 50 g = 500 g.
Answer: 500 g of pasta is needed for 10 people.
- A photo 5 cm wide is enlarged to 20 cm wide. What scale factor was used?
- On a map with a scale of 1 cm to 25 km, how far apart in real life are two towns that are 6 cm apart on the map?
Common misconceptions and how to address them
MisconceptionA ratio like 3 : 5 means '3 out of 5 total', the same as the fraction 3/5 of the whole.
Why it happens: Students confuse a part:part ratio with a part:whole fraction. In a 3 : 5 ratio there are 3 + 5 = 8 total parts, so the first part is actually 3/8 of the whole, not 3/5.
How to address it: Always add the ratio terms first to find the TOTAL number of parts before turning a ratio into a fraction of the whole. A 3 : 5 ratio gives the fractions 3/8 and 5/8 of the whole, never 3/5.
MisconceptionTo share an amount in the ratio a : b, divide the total by a and by b separately.
Why it happens: Students skip finding the value of one share and instead divide the total by each ratio term on its own, giving two numbers that do not add back up to the original total.
How to address it: Add the ratio terms first to find the total number of shares, divide the total amount by that sum to find the value of ONE share, then multiply by each ratio term. Checking that the parts add back to the total catches this mistake every time.
MisconceptionSimplifying a ratio means subtracting the same number from both terms, the way you balance an equation.
Why it happens: Simplifying is confused with the 'do the same to both sides' habit from solving equations, where subtracting is a valid balancing move.
How to address it: A ratio is simplified by DIVIDING every term by their highest common factor, never by subtracting. Subtracting changes the actual proportion: 10 : 20 is the same relationship as 1 : 2, but subtracting 5 from each term gives 5 : 15, which simplifies to 1 : 3, a completely different ratio.
MisconceptionA scale factor smaller than 1 does not make sense, or scale factors are always whole numbers greater than 1.
Why it happens: Students have usually only met enlargements that make a shape bigger, using whole-number scale factors.
How to address it: A scale factor between 0 and 1 shrinks a shape (a reduction); for example, a scale factor of 0.5 halves every length. Reductions and enlargements use exactly the same scale-factor idea, just with the factor below or above 1.
MisconceptionWhen comparing two quantities as a fraction, the bigger quantity always goes on the bottom, so the fraction is always less than 1.
Why it happens: Students over-generalise from earlier fraction work, where a fraction was always a part of one whole and so always less than 1.
How to address it: A fraction comparing two separate quantities keeps whichever quantity is named FIRST as the numerator, whether or not it is the bigger one. Comparing 45 km to 30 km gives 45/30 = 3/2, a fraction properly greater than 1.
Guided practice (with answers)
1. Simplify the ratio 16 : 24.
Answer: 2 : 3, because the highest common factor of 16 and 24 is 8, and 16 / 8 = 2, 24 / 8 = 3.
2. Write 5 : 8 in the form 1 : n.
Answer: 1 : 1.6, because 8 / 5 = 1.6.
3. Share 60 sweets in the ratio 2 : 3 : 5.
Answer: 12, 18 and 30 sweets, because 2 + 3 + 5 = 10 shares, 60 / 10 = 6 sweets per share, so 2 x 6 = 12, 3 x 6 = 18 and 5 x 6 = 30 (and 12 + 18 + 30 = 60).
4. A garden is split into a lawn and a vegetable patch in the ratio 5 : 3. The vegetable patch is 15 m². What is the total area of the garden?
Answer: 40 m², because the vegetable patch's 3 shares equal 15 m², so 1 share is 15 / 3 = 5 m², and the whole garden is 5 + 3 = 8 shares: 8 x 5 = 40 m².
5. Two numbers are in the ratio 4 : 9. Write the first number as a fraction of the second, and as a fraction of their total.
Answer: 4/9 of the second number, and 4/13 of the total, because 4 + 9 = 13.
6. A model car has a scale factor of 1 : 18 compared with the real car. The model is 25 cm long. How long is the real car?
Answer: 450 cm (4.5 m), because the real car is 18 times the model: 25 x 18 = 450.
Independent practice worksheets
Practise ratio notation, sharing a quantity in a ratio, ratio-to-fraction conversions, and scale/proportion problems with computed, never-wrong answer keys.
Differentiation
- Give the total number of shares (a + b) already added up for early sharing questions, so the first step is a single division.
- Use counters or coloured cubes to physically build a ratio (e.g. 2 red cubes and 3 blue cubes) before moving to numbers alone.
- For ratio-to-fraction conversions, always write the total number of parts underneath the ratio first, before writing any fraction.
- Start scale-factor work with whole-number factors only (x2, x3) before introducing decimal scale factors below 1.
- Introduce ratios with non-integer terms (e.g. 1.5 : 4) and ask students to rewrite them with whole-number terms.
- Pose a three-part ratio sharing problem where only the TOTAL and the ratio are known, working backward to find every part.
- Investigate: does simplifying a ratio a : b ever change the value of the fraction a/b? (No, a/b has the same value before and after simplifying; only how it is WRITTEN changes.)
- Combine scale drawing with area: if a shape's sides are enlarged by scale factor k, what happens to its area? Test the conjecture (area scales by k²) with a real rectangle.
Assessment: exit ticket
A three-question exit ticket sampling simplifying a ratio, sharing a quantity, and the unitary method.
1. Simplify the ratio 21 : 35.
Answer: 3 : 5, because the highest common factor of 21 and 35 is 7, and 21 / 7 = 3, 35 / 7 = 5.
2. Share $150 in the ratio 2 : 3. How much is each share?
Answer: $60 and $90, because 2 + 3 = 5 shares, $150 / 5 = $30 per share, so 2 x $30 = $60 and 3 x $30 = $90 (and $60 + $90 = $150).
3. A recipe for 6 people needs 300 g of rice. Using the unitary method, how much rice is needed for 9 people?
Answer: 450 g, because 1 person needs 300 / 6 = 50 g, and 9 people need 9 x 50 = 450 g.
Teacher notes and timings
- Rough timing across four lessons: Lesson 1 ratio notation and simplifying (section 1), Lesson 2 sharing a quantity in a ratio (section 2), Lesson 3 ratio as a fraction (section 3), Lesson 4 scale factors, maps and proportion (section 4) plus the exit ticket.
- This is the site's first teaching unit anchored to the real UK National Curriculum's Key Stage 3 mathematics programme of study (the KS2 'Roman Empire and its impact on Britain' history unit was the first UK NC citation overall). The KS3 programme of study is one continuous specification for Years 7 to 9, not split by year the way KS1/KS2 are; ratio and proportion is placed here at Year 7 as the conventional, curriculum-coherent starting point, following straight on from Year 6's introduction to ratio.
- Scope note: this unit deliberately covers the 'core' ratio and proportion skills (notation, sharing, fraction conversion, scale factors and the unitary method) and leaves direct/inverse proportion in full algebraic form (y = kx, y = k/x) for a later unit, matching how the KS3 programme of study lists that as a separate, more advanced bullet within the same strand.
- Language to keep repeating: the TOTAL number of shares is always the SUM of the ratio terms; the unitary method always starts by finding the value of ONE.
- This unit assumes comfort with equivalent fractions and finding a highest common factor (see the linked glossary entries). Revisit those first if either foundation is shaky.
- Present mode and print both work: build the bar models live with the class for sections 1-3, then print the four linked worksheets for independent practice.