Ratios and rates
Ratio as a part-to-part comparison, equivalent ratios by scaling, unit rates, and sharing in a given ratio
About four to five lessons of 45 to 60 minutes
Why does one jug of cordial taste right and the next taste wrong?
You make a jug of cordial with 2 cups of syrup and 3 cups of water. It tastes perfect. Your friend makes theirs with 2 cups of syrup and 6 cups of water, and it tastes weak and watery. Same amount of syrup, totally different drink. The secret is the ratio, the comparison of syrup to water, and it is what controls the taste.
Ratios and rates are the maths of mixing, pricing, speed and scaling a recipe up or down. Today you will learn to read and write a ratio, keep a mixture tasting the same by scaling it, work out a unit rate such as the price for one item, and share an amount in a given ratio.
- 2 cups syrup to 3 cups waterthe ratio is 2 to 3, written 2:3, for every 2 of syrup there are 3 of water
- A smoothie of 2 mango to 3 bananadouble it to 4 mango and 6 banana and it tastes exactly the same
- 3 drinks for $6the unit rate is $2 per drink, the price for a single one
- 20 sweets shared 2 to 35 equal parts, so one child gets 8 and the other gets 12
What students will be able to do
Students will describe a relationship between two quantities using ratio language and notation, generate equivalent ratios by scaling both parts by the same number, find and use a unit rate, and share a quantity in a given ratio, while keeping ratio (part to part) distinct from fraction (part to whole).
- I can write a ratio and describe it with for every language.
- I can tell that the order of a ratio matters.
- I can make equivalent ratios by multiplying both parts by the same number.
- I can find a unit rate, such as the price of one item.
- I can share an amount in a given ratio.
Standards this unit teaches
- 6.RP.A.1Common Core (US)Understand the concept of a ratio
Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, the ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.
- 6.RP.A.2Common Core (US)Understand unit rates
Understand the concept of a unit rate a/b associated with a ratio a:b with b not equal to 0, and use rate language in the context of a ratio relationship. For example, this recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.
- 6.RP.A.3Common Core (US)Solve ratio and rate problems
Use ratio and rate reasoning to solve real-world and mathematical problems, e.g. by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
- AC9M7N08Australian Curriculum v9 (ACARA)Ratios (Year 7)
Recognise, represent and solve problems involving ratios. Australia introduces ratios at Year 7, so this US Grade 6 unit runs about a year ahead of the ACARA placement.
- AC9M8M05Australian Curriculum v9 (ACARA)Rates (Year 8)
Recognise and use rates to solve problems involving the comparison of two related quantities of different units of measure. In Australia rates follow ratios and are met at Year 8, a year after the ratio work above.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
- Multiplicationscaling a ratio up means multiplying both parts by the same number
- Divisiona unit rate comes from dividing to find the amount for one
- Fractionsequivalent ratios work like equivalent fractions, and a ratio can become a fraction of the whole
- Grade 6 percentages teaching unita percent is a rate per hundred, a close cousin of the rates here
Words to teach and display
- Ratio
- a comparison of two amounts, such as 2 to 3, written 2:3
- Term
- one of the numbers in a ratio, so 2:3 has the terms 2 and 3
- Equivalent ratios
- ratios that describe the same relationship, such as 2:3 and 4:6
- Rate
- a comparison of two quantities measured in different units, such as km per hour
- Unit rate
- a rate for exactly one unit, such as the price for one item
- Scaling
- multiplying or dividing both parts of a ratio by the same number to keep it equivalent
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. A ratio compares two amounts
ConcreteMix the cordial in front of the class: 2 cups of syrup and 3 cups of water. A ratio is a way of comparing those two amounts. We say the ratio of syrup to water is 2 to 3, and write it 2:3. Read it as for every 2 cups of syrup there are 3 cups of water. Draw the two amounts as bars side by side and the comparison is plain to see.
A ratio compares one part with another part, not a part with a whole. Here the two parts are the syrup and the water. The colon is read as to, so 2:3 is two to three.
The total of the two parts is 2 + 3 = 5 cups of cordial, but the ratio itself is the comparison 2:3, the recipe that keeps the taste right.
- What does the ratio 2:3 tell you about the cordial?
- How do you read the symbol 2:3 out loud?
- How many cups of cordial does one batch make in total?
2. Order matters, and ratio is not a fraction of the whole
PictorialA ratio must be written in the order you name the amounts. Syrup to water is 2:3, but water to syrup is 3:2, a different statement. Swapping the order describes a different mixture, so keep the order matching the words.
Be careful not to confuse a ratio (part to part) with a fraction (part to whole). In the 2:3 cordial, syrup is not 2/3 of the drink. The whole is 5 parts, so syrup is 2/5 of the drink and water is 3/5. The ratio compares the two parts to each other, the fraction compares one part to the whole.
A different mix makes the point: a squash of 1 part concentrate to 4 parts water is the ratio 1:4, and the concentrate is 1/5 of the whole drink, not 1/4.
- If the ratio of red to blue counters is 3:5, what is the ratio of blue to red?
- In the 2:3 cordial, what fraction of the whole drink is water?
3. Equivalent ratios: scaling a mixture
PictorialTo make more cordial without changing the taste, scale the recipe. Double 2:3 by multiplying both parts by 2, giving 4:6. That is twice as much drink, but the same taste, because syrup and water grew in step. 2:3 and 4:6 are equivalent ratios.
The rule is exactly like equivalent fractions: multiply (or divide) both parts by the same number and the ratio stays equivalent. Tripling 2:3 gives 6:9. Each of these keeps for every 2 syrup, 3 water true, just with bigger batches.
This is also how you simplify. A mix given as 6:9 can be divided by 3 on both parts to reach its simplest form, 2:3. Simplest form is the smallest whole-number version of the ratio.
Write two ratios equivalent to 2:3, then give 10:15 in its simplest form.
- Multiply both parts by 2: 2:3 becomes 4:6.
- Multiply both parts by 5: 2:3 becomes 10:15.
- To simplify 10:15, divide both parts by 5, giving 2:3.
Answer: 4:6 and 10:15 are equivalent to 2:3, and 10:15 in simplest form is 2:3.
- What do you multiply both parts of 2:3 by to get 8:12?
- Write 12:18 in its simplest form.
4. Rates and unit rates
PictorialA rate compares two quantities measured in different units, such as dollars and drinks, or kilometres and hours. A unit rate is the rate for exactly one unit. If 3 drinks cost $6, divide to find the cost of one: 6 divided by 3 is $2, so the unit rate is $2 per drink.
Rate language uses the word per, meaning for each one. Miles per hour, cost per item, words per minute, all are rates. The unit rate is powerful because once you know the cost of one, you can find the cost of any number by multiplying.
A recipe ratio can also give a rate. With 3 cups of flour to 4 cups of sugar, there is 3/4 of a cup of flour for each single cup of sugar. Dividing the two parts turns the ratio into a per-one rate.
A car travels 150 km in 3 hours at a steady speed. Find the unit rate (its speed).
- A unit rate here is the distance for one hour, so divide the distance by the time.
- 150 km divided by 3 hours is 50.
- Attach the units: 50 kilometres per hour.
Answer: The unit rate is 50 km per hour.
- If 4 pens cost $12, what is the cost of one pen?
- What word in a rate tells you it is for each single unit?
Common misconceptions and how to address them
MisconceptionA ratio is treated as a fraction of the whole, so in a 2:3 mix the syrup is called 2/3 of the drink.
Why it happens: Both use two numbers, and the top-and-bottom look of a fraction gets mapped onto the two ratio terms.
How to address it: A ratio compares part to part, a fraction compares part to whole. Add the parts to find the whole: 2 + 3 = 5, so syrup is 2/5 of the drink. Use the bar model to show all five parts.
MisconceptionThe order of a ratio does not matter, so 2:3 and 3:2 are read as the same.
Why it happens: Students focus on the pair of numbers and overlook which quantity each one describes.
How to address it: Tie each term to its label. Syrup to water 2:3 is a different mixture from water to syrup 3:2. Always write the ratio in the order the words are said.
MisconceptionEquivalent ratios are made by adding the same number to both parts, so 2:3 becomes 3:4.
Why it happens: Adding feels like keeping things fair, and it mirrors a common (wrong) move with fractions.
How to address it: Scaling multiplies both parts by the same number, it does not add. 2:3 doubled is 4:6, and you can check the taste is unchanged. Adding 1 to each part changes the mixture, so 2:3 and 3:4 are not equivalent.
MisconceptionA rate is left without its units, so the answer to a speed problem is just 50 rather than 50 km per hour.
Why it happens: The number feels like the whole answer, and the units get dropped along the way.
How to address it: A rate always carries two units joined by per. Insist that every rate answer names both quantities: dollars per drink, kilometres per hour, words per minute.
MisconceptionWhen sharing in a ratio, the amount is divided by one of the terms instead of by the total number of parts.
Why it happens: Students grab a number from the ratio (the 2 or the 3) rather than adding them to find how many equal parts there are.
How to address it: Always add the terms first to get the number of equal parts, then divide the amount by that total. For 2:3 sharing 20, divide by 5, not by 2 or 3.
MisconceptionBigger numbers in a ratio are read as a stronger mixture, so 10:15 is thought to be more syrupy than 2:3.
Why it happens: Larger terms look like more, so equivalence between scaled ratios is missed.
How to address it: Simplify 10:15 by dividing both parts by 5 to reach 2:3, the same ratio. Scaling changes the batch size, not the taste.
Guided practice (with answers)
1. Write the ratio of circles to squares if there are 4 circles and 6 squares, then simplify it.
Answer: 4:6, which simplifies to 2:3 by dividing both parts by 2.
2. Give two ratios equivalent to 3:5.
Answer: 6:10 and 9:15 (multiply both parts by 2 and by 3).
3. In a 1:4 squash, what fraction of the drink is concentrate?
Answer: 1/5, because the whole has 1 + 4 = 5 parts.
4. If 5 apples cost $10, what is the unit rate?
Answer: $2 per apple, because 10 divided by 5 is 2.
5. Share 24 marbles in the ratio 1:2.
Answer: 8 and 16. There are 3 parts, 24 divided by 3 is 8, so 1 x 8 = 8 and 2 x 8 = 16.
6. A runner covers 100 m in 20 seconds at a steady pace. Find the unit rate.
Answer: 5 metres per second, because 100 divided by 20 is 5.
Independent practice worksheets
ChalkBee does not yet have a dedicated ratio and rate worksheet generator, so set the closely related fraction, division and word-problem worksheets, whose answer keys are computed in code and never wrong. These rehearse the exact skills ratios rest on: equivalence and simplifying, dividing to find a unit rate, and reasoning through a multi-step problem.
Differentiation
- Keep the bar models in view so a ratio stays a picture of two parts, not two bare numbers.
- Start with small ratios such as 1:2 and 2:3 and real mixing before scaling or sharing.
- For sharing, always write the three steps in order: add the parts, divide, multiply.
- Use a ratio table with columns doubling and tripling so equivalence is seen as repeated scaling.
- Compare two mixtures given as different ratios and decide which is stronger by scaling to a common part.
- Work backwards: if the larger share in a 2:3 split is 12, find the whole amount.
- Link rates to best-buy problems, comparing price per 100 g for two package sizes.
- Explore three-part ratios, such as sharing in the ratio 1:2:3.
Assessment: exit ticket
A three-question exit ticket for the last five minutes, sampling ratio language and simplifying, a unit rate, and sharing in a ratio.
1. Simplify the ratio 6:9.
Answer: 2:3, by dividing both parts by 3.
2. If 4 muffins cost $8, what is the cost of one muffin?
Answer: $2 per muffin, because 8 divided by 4 is 2.
3. Share 25 in the ratio 2:3.
Answer: 10 and 15. Five parts of 5, so 2 x 5 = 10 and 3 x 5 = 15.
Teacher notes and timings
- Rough timing across four to five lessons: Lesson 1 what a ratio is (section 1), Lesson 2 order and ratio versus fraction (section 2), Lesson 3 equivalent ratios (section 3), Lesson 4 rates and unit rates (section 4), Lesson 5 sharing in a ratio plus the exit ticket (section 5 and assessment).
- Language to keep saying: for every, part to part versus part to whole, scale both parts by the same number, per means for each one, add the parts before you divide. These pre-empt most of the misconceptions.
- The comparison bar models put the two parts of a ratio side by side to a shared scale, so their proportions are exact. The part-whole bars in the sharing section show the whole split into the two shares. Describe them as the ratio made visible, not as measured cups.
- Keep ratio distinct from fraction throughout. The single most common error is reading 2:3 as 2/3 of the whole. Return to adding the parts to find the whole every time it slips.
- Curriculum note and a US and AU alignment: the US teaches ratios, unit rates and rate reasoning together in Grade 6 (6.RP.A.1, 6.RP.A.2, 6.RP.A.3). ACARA introduces ratios a year later at Year 7 (AC9M7N08) and rates later still at Year 8 (AC9M8M05), so in an Australian setting the ratio work here suits late Year 6 or Year 7 and the rate work is an early look at Year 8.
- Present mode and print both work: use the Print button for a clean teacher copy or a student handout, and project the page to teach straight from the diagrams.