At a glance. One lesson of about 60 minutes. Students cut two bars into equal groups, read the simplest form as the number of groups, put lengths in one unit first, and write 1 : n.
Calculator: no calculator. Every number divides cleanly and the highest common factor is found by listing. The skill is seeing the groups, not computing.
What students will learn
Know
- Dividing or multiplying both parts by the same number keeps a ratio.
- The simplest form uses the highest common factor of the parts.
- Both amounts go into the same unit before a ratio is simplified.
Understand
- Equal ratios are one comparison in different numbers.
- Adding or subtracting the same amount from both parts changes the ratio, unless the parts are equal.
- A ratio of 1 : n lets two ratios be compared by their n.
Do
- Simplify a ratio of two or three whole numbers.
- Find a missing number in equal ratios.
- Convert units, then simplify; write a ratio as 1 : n.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: equal ratios, highest common factor and simplest form (questions 1 to 7).
- Criterion B. Investigating patterns: find every pair in the figure whose simplest form is 1 : 3, then describe the pattern in the numbers.
- Criterion C. Communicating: explaining in words why taking the same amount off both parts changes a ratio (questions 3 and 9).
- Criterion D. Applying mathematics in real-life contexts: squash mixes, ropes and ribbons, paths and hedges (the hook and question 5).
The one thing to do with the figure
Ask for a prediction before the sliders move. “4 cups to 12 cups and 1 cup to 3 cups. Is it the same drink? What do you divide by?” Take answers, then set 4 and 12 to settle it.
Then set 7 and 13 and ask why no lines appear. Name it: the only group that fits both is a group of 1, so the ratio is already in its simplest form.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook and the figure, with predictions first. Include a pair with no common factor. |
| 10 min | Key idea; practise saying the groups aloud: “4 is 1 group of 4, 12 is 3 groups of 4”. |
| 10 min | Both worked examples; students finish the faded ribbon example in pairs. |
| 5 min | Question 1, the check. If many choose A, B or D, go back to the figure's cut lines. |
| 15 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Exit ticket, question 9, on paper or on screen. |
The answers
| Warm-up | 8; 10; 5 : 12. |
|---|---|
| Worked example 1 | 3 : 4. |
| Worked example 2 | 5 : 2. |
| Faded example | 3 : 10. |
| 1. Simplest form | C, 2 : 3. A, B and D are equal to 24 : 36 but each still has a common factor above 1. |
| 2. Missing number in equal ratios | 5. 15 ÷ 5 = 3 and 25 ÷ 5 = 5. |
| 3. Spotting equal ratios | A. B adds 2 to both; C and D double only one number correctly. |
| 4. Simplify with the highest common factor | 4. 56 ÷ 14 = 4 and 42 ÷ 14 = 3, so the ratio is 4 : 3. |
| 5. Convert units, then simplify | 5. 400 : 250 = 8 : 5. |
| 6. The form 1 : n | 2.5. 5 ÷ 2 = 2.5. |
| 7. Three-part ratio, simplest form | 5. 12 : 18 : 30 = 2 : 3 : 5. |
| 8. Working backwards from the simplest form | 63. 36 : 63 = 4 : 7 (both divided by 9). |
| 9. Why subtraction breaks a ratio | A. B, C and D all keep subtraction as a valid method. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1; 8, 28 or 14 in question 4 | Divided by a common factor but not the highest one and stopped, or gave the highest common factor itself as the answer. | Ask: can both numbers still be divided by something? Keep going until the answer is no. |
| 13, 22 or 125 in question 2 | Subtracted to find the missing number, or multiplied the second part when it should be divided. | Ask what the first number was divided by, and write it as an arrow: 15 ÷ 5 = 3, so 25 ÷ 5. |
| B, C or D in question 3 | Added a number to both parts, or doubled only one. | Ask students to double the first ratio and check whether the second is the result. |
| 125, 250 or 50 in question 5 | Did not convert, stopped before simplifying, or gave the highest common factor itself. | Write 4 m = 400 cm first, then the ratio, then the highest common factor. |
| 5, 3 or 0.4 in question 6 | Left the second part alone, subtracted, or divided the wrong way. | Divide the first part by itself to get 1 and ask what must happen to the second part. |
| 15, 10 or 30 in question 7 | Divided all three by a factor of only two of them, by a smaller common factor, or left the last number as it was. | Ask: does this number go into all three? Check each one. |
| 39, 28 or 9 in question 8 | Added the same amount to both parts, multiplied by the first number instead of the scale factor, or gave the scale factor itself. | Draw the multiplication: 4 × ? = 36, then the same on the 7. |
| B, C or D in question 9 | Subtraction taken as valid, or the mistake missed. | Simplify 12 : 18 by dividing, then compare it with 6 : 12. |
Questions worth asking
- Why must you divide both parts by the same number? So the comparison is kept. Listen for “the groups are the same size” or “it is the same ratio in smaller numbers”.
- How do you know a ratio is fully simplified? No number above 1 divides both. Listen for “the highest common factor is 1”.
- Is 2 : 3 the same as 2/3? Linked, but not the same. 2 : 3 says the first amount is 2/3 of the second amount. It is not 2/3 of the whole: the whole is 2 + 3 = 5 parts, so the first part is 2/5 of it. Link back to lesson 1.
- Can a ratio have a simplest form with a decimal, such as 1 : 1.25? Yes. It is a way of comparing: for every 1 of the first there are 1.25 of the second.
Differentiation
A way in for anyone stuck
- Give students strips of squared paper to cut into equal groups; every question can be done with strips.
- Start with pairs that have an obvious common factor: 10 : 15, 6 : 8, 12 : 18.
- Write the factor pairs of each number in a list before looking for the highest one.
Stretch
- Question 7, then: find a ratio of three different whole numbers in simplest form in which two of the parts are even. (For example 4 : 6 : 9: 4 and 6 share the factor 2, but 9 does not, so no number above 1 divides all three.)
- Write the ratio 3 : 8 in the form 1 : n and in the form n : 1. Which would you use to compare it with 2 : 5, and why? (1 : 2.67 to 2 decimal places, or exactly 1 : 2 2/3, and 0.375 : 1; compare the n values.)
Students learning in English as an additional language
“Simplest form” means “the smallest whole numbers that say the same thing”. Show 6 : 8 and 3 : 4 side by side with groups, and say “the same, in smaller numbers”.
| Word | Plain meaning | Example |
|---|---|---|
| equal ratios | ratios that say the same thing in different numbers | 4 : 12 and 1 : 3 |
| simplest form | the ratio with the smallest whole numbers | 2 : 3, not 4 : 6 |
| common factor | a number that divides both parts exactly | 6 divides 12 and 18 |
| highest common factor | the largest number that divides both parts exactly | 6 for 12 and 18 |
| same unit | both amounts measured in the same way | 500 cm and 80 cm |
Sentence starters for explaining aloud or in writing:
- I divided both parts by … because …
- The highest common factor of … and … is …, so the simplest form is …
- I changed … into … first because the units were different.
Two things not to say
“Cancel it down.” Cancel is a word without a reason: say “divide both parts by the same number”.
“Whatever you do to the top, do to the bottom.” It sounds the same for adding as for dividing, and adding breaks a ratio.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- SimplifyWrite 18 : 27 in its simplest form.
2 : 3. - Missing number6 : 15 = 2 : ?
5.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA cup of paint mix holds 150 ml of blue and 1 litre of white. Write blue to white in simplest form.
3 : 20. 1 litre is 1000 ml, and 150 : 1000 = 3 : 20. - Earlier work (Lesson 1)A bag has 4 red, 6 blue and 2 green beads. Write the ratio of blue to all the beads, and simplify it.
6 : 12 = 1 : 2.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningFind a ratio that simplifies to 3 : 8 and whose two parts add to 44.
12 : 32. The parts 3 + 8 = 11, and 44 = 11 × 4. - SpottingIs 7 : 21 in simplest form? Write it as 1 : n.
No. 7 divides both: 1 : 3.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both sliders work with the keyboard (tab to a slider, then the arrow keys) and by touch. Under the figure, “Show me the steps” gives the same states in words.
No device? Use squared paper strips and cut them into equal groups.