What is a ratio?

Classroom resources

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Students will be able to

• I can write a ratio of two quantities in a : b form, in the order asked.
• I can tell part to part from part to whole.
• I can write a part-to-whole ratio as a fraction.

Timing

Predict and Run 8 min · Investigate 10 min · Modify 12 min · Practise and Make 10 min

Before class

Do the Run board yourself first. Students need fractions as parts of a whole (Equivalent fractions).

The one idea: a ratio compares quantities; the order in which they are named is the order of the numbers. The common mistakes are writing part to part when part to whole was asked (Pip), reversing the order, and using the whole (5) as a part.

Use the same bag all lesson so students see 2 : 3 and 2 : 5 about the same objects.

Running the lesson

1. Predict: students who chose 3 : 2 reversed the order; 2 : 5 compared to the whole.
2. Run: ask students to say both ratios aloud, e.g. 'three to five' and 'three to eight'.
3. Investigate: have students hold up the whole bag and a part to show which is which.
4. Modify: ask which ratio matches the fraction 3/8.
5. Make: collect ratios; check each adds to 10.

The hard part (on purpose)

Telling part to part from part to whole. Ask 'is the second number a part, or the whole bag?' every time before writing.

When students get stuck

Answers to the checks

1. red to blue: A bag has 2 red counters and 3 blue counters. What is the ratio of red to blue?
   2 : 3. Red comes first, so 2 comes first. 3 : 2 is blue to red; 2 : 5 is red to all the counters.
2. build the bag: Make 3 red blocks for every 5 blue. What is the ratio of red to the whole?
   
   Red to blue is 3 : 5. In all there are 3 + 5 = 8 blocks, so red to the whole is 3 : 8.
3. Whole bag: A bag has 2 red and 3 blue counters. The ratio of red to all the counters is $2 : ?$
   5
   2 + 3 = 5 counters in all, so red to all is 2 : 5.
4. Find the mistake: Pip finds the ratio of apples to all the fruit. Click the line where the working goes wrong.
   line 3
   Line 3: 4 : 3 is apples to pears. Apples to all the fruit is 4 : 7, the 7 from line 2.
5. Which way round?: A drink has 2 cups of blue cordial and 5 cups of water. What is the ratio of water to cordial?
   5 : 2
   Water first: 5, then cordial: 2. The ratio of water to cordial is 5 : 2.
6. Dogs and cats: Show 4 dogs for every 3 cats. What is the ratio of dogs to the whole?
   
   Dogs to cats is 4 : 3, and 4 + 3 = 7, so dogs to all the animals is 4 : 7.
7. Add the parts: At a school the ratio of walkers to bus riders is $3 : 2$. How many parts are there altogether?
   5
   3 + 2 = 5 parts.
8. As a fraction: A shelter has 3 dogs and 5 cats. What fraction of the animals are dogs?
   3/8
   Dogs to all the animals is 3 : 8, so $\tfrac{3}{8}$ of the animals are dogs.
9. Strong or weak?: A drink is made with juice and water in the ratio $1 : 4$. Is it strong or weak?
   Weak: only 1 part juice in every 5 parts
   1 part juice to 4 parts water: 1 of every 5 parts is juice, so $\tfrac{1}{5}$ juice.
10. Pencils to pens: A box has 5 pens and 8 pencils. The ratio of pencils to pens is $8 : ?$
    5
    Pencils first: 8, then pens: 5. 8 : 5.
11. Pens to all: In the same box, the ratio of pens to all the items is $5 : ?$
    13
    5 + 8 = 13 items, so pens to all the items is 5 : 13.
12. Wins to losses: A team won 7 games and lost 3. The ratio of wins to losses is $7 : ?$
    3
    Wins first: 7. Losses: 3. 7 : 3.
13. Review: estimating: Review: estimate $6.8 \times 31$.
    210
    6.8 ≈ 7 and 31 ≈ 30, so 7 × 30 = 210.
14. a three-colour mix: Make a three-colour ratio with 10 parts in all, and 3 parts of the first colour.
    
    For example 3 : 2 : 5 or 3 : 3 : 4. The whole is the parts added: 3 + 2 + 5 = 10.

Solutions

One possible solution for each task. Other correct programs are fine.

change the bag › Dogs and cats

Open task: any program that does what the step asks.

change the bag › Add the parts

Open task: any program that does what the step asks.

change the bag › As a fraction

Open task: any program that does what the step asks.

change the bag › Strong or weak?

Open task: any program that does what the step asks.

a three-colour mix

Open task: any program that does what the step asks.

Stretch and support

Stretch: A ratio of boys to girls is 3 : 4. What fraction of the group are girls? What if 2 more boys join, no more girls? Is the ratio still 3 : 4?

Support: Draw one block for each item and colour them; count before writing.

Coach prompt (optional)

For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.

You are a patient maths coach for a Year 7 student in Australia. Help me understand ONE idea only, from the lesson "What is a ratio?".

The idea: Ratio as a relationship between quantities: I can Students recognise that a ratio expresses a multiplicative relationship between two quantities and can represent ratios using diagrams and notation.

What understanding looks like:
- Writes a given ratio correctly in a:b notation.
- Distinguishes between part-part and part-whole interpretations of a ratio.
- Represents a ratio accurately using a diagram.

Mistakes students often make here:
- A student writes 4 : 3 for apples to all the fruit (Pip's slip).
- A student writes the ratio the wrong way round.
- A student says 3 : 5 and 3/5 are the same.

How to coach me (rules from the Waypoints Kit coach):
- Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question.
- Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit).
- If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go.
- Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back.
- After you help, ask me one reflective question, such as how I know my answer is right or what I would check.
- Praise my effort and method, not my ability.
- You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so.
- Use simple words.
- Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "What is a ratio?".

Exit ticket

A jar has 4 black and 5 white beads. Write the ratio of black to white, and of black to all the beads.

4 : 5 and 4 : 9.

What each step is for

• red to blue: Committing to a ratio shows whether they write the order asked for (2 : 3), reverse it (3 : 2) or compare to the whole (2 : 5).
• build the bag: Building 3 red for every 5 blue shows the ratio as rows of equal blocks, and the whole as the blocks added together.
• part to part, part to whole: Part to whole adds the parts, so the second number is the total; Pip writes part to part when asked for part to whole. The order matters.
• change the bag: Changing one number at a time, then turning part to whole into a fraction, links ratio to the fractions students know.
• ratios: Writing a ratio in the order asked, part to whole, and a review of estimation.
• a three-colour mix: A ratio can have three parts, and the whole is all of them added; making 10 parts shows many ratios give the same total.

Where it leads

Next: Equivalent ratios, scaling a ratio up and down.

Where it leads: sharing in a ratio, rates, scale drawings, and proportion in Years 8 and 9.