At a glance. One lesson of about 60 minutes. Students build fraction multiplication from an area model, simplify before multiplying, and use it for fractions of amounts and mixed numbers.
Calculator: no calculator. Nothing here needs one. Cancelling first keeps numbers small, which is the point of the second figure.
What students will learn
Know
- a/b × c/d = (a × c)/(b × d).
- Simplify before multiplying by dividing a top and a bottom by a shared factor.
- A mixed number is changed to an improper fraction before multiplying.
Understand
- Multiplying by a fraction is taking a part of a quantity, so it can make the quantity smaller.
- The area model shows why the bottoms multiply.
- Any top can cancel with any bottom, because multiplication can be done in any order.
Do
- Multiply fractions, whole numbers and mixed numbers.
- Take a fraction of an amount.
- Estimate to check that the answer is sensible.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: multiplication of fractions and mixed numbers (questions 1 to 8).
- Criterion B. Investigating patterns: the what-if question and the first figure. Ask students to predict the product from the numbers of columns and rows before the figure shows it.
- Criterion C. Communicating: explaining why multiplying by a fraction less than 1 gives a smaller answer (question 9).
- Criterion D. Applying mathematics in real-life contexts: a school plot, a water tank and a recipe (the hook and questions 3 and 5).
The one thing to do with the figure
Ask for a prediction before the first slide. “Half of a third of a plot: more or less than a third?” Let students draw it on paper before you show the figure.
Then change the second fraction to 2/3, then 3/4, and ask students to say the count of overlap cells aloud. In the second figure, ask which top and bottom share a factor with the screen turned away or the figure covered, then show it.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook, prediction and the first figure. Include two fractions whose product simplifies. |
| 10 min | Key idea; the second figure. Students find the shared factors with the figure covered, then check against it. |
| 10 min | Worked examples; students finish the faded example on their own. |
| 5 min | Question 1, the check. If many choose A or D, go back to the first figure. |
| 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 | 2/5; 24; 1/2. |
|---|---|
| Worked examples | 3/8 × 4/9 = 1/6; 1 1/3 × 2 1/4 = 3. |
| Faded example | 4/9 × 15/16 = 5/12. |
| 1. Multiplying two fractions | B, 3/10. A adds both; C cross-multiplies; D multiplies the tops and adds the bottoms. |
| 2. Unit fraction of a unit fraction | 10. 1/2 × 1/5 = 1/10. |
| 3. A fraction of an amount | 21. 3/4 of 28 = 28 ÷ 4 × 3. |
| 4. Simplify before multiplying | 3. 5/6 × 9/10 = 3/4. |
| 5. A mixed number times a fraction | 1.5. 5/2 × 3/5 = 3/2. |
| 6. A fraction of a fraction of an amount | 4. 1/3 × 1/4 = 1/12, and 1/12 of 48 = 4. |
| 7. Working back from a fraction of a number | 20. The number is 30. |
| 8. Greatest whole number that fits | 23. 69/7 < 10 < 72/7. |
| 9. Does multiplying make it bigger? | B. A thinks multiplying always enlarges; C mixes up the rules for adding; D adds tops and bottoms. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A or D in question 1; 7, 5 or 2 in question 2 | Added the bottoms, added the tops, or used only one of the two bottoms. | Use the area model: how many cells in the whole? The bottoms give that. |
| C in question 1 | Cross-multiplied tops with bottoms. | Ask what each number in the area model counts: columns and rows. |
| 7, 31 or 112 in question 3 | Found one quarter, added instead of taking a part, or multiplied by the bottom. | Ask: is 3/4 of the tank more or less than the whole tank? Then check the answer. |
| 45, 15 or 9 in question 4 | Did not simplify at all, or stopped after dividing by one common factor (3 or 5). | Ask if the top and bottom of the answer still share a factor. |
| 1.2, 0.3 or 3.1 in question 5 | Multiplied only the whole number or only the fraction part of the mixed number, or added 2.5 + 0.6. | Convert 2 1/2 to 5/2 first and use the area model. |
| 16 or 12 in question 6 | Took only one of the two fractions. | Ask what is one third of one quarter of the whole bag: 1/12. |
| 12, 30 or 27 in question 7 | Took 2/3 of 18 (the wrong quantity), gave the number itself, or divided 18 by 2/3. | Make a bar model: 5 equal parts, 3 of them are 18. |
| 24, 70 or 10 in question 8 | Rounded up, mixed up top and bottom, or stopped at a value that works but is not the greatest. | Test 23 and 24 in the inequality. |
| A, C or D in question 9 | Thought multiplying always enlarges, or applied the wrong rule. | Ask for half of 3, then half of 1/2 on the number line. |
Questions worth asking
- Why does the bottom multiply? The number of equal cells in the whole is columns times rows. Listen for “cut each part again”.
- Can the product of two fractions be bigger than both? Yes, if both are greater than 1: 3/2 × 5/4 = 15/8. If one is below 1 the product is smaller than the other, and for two fractions below 1 it is smaller than both.
- Without working it out, is 7/8 × 45 more or less than 45? Is it more or less than 40? Less than 45, because 7/8 is less than 1. Less than 40 as well: 7/8 × 45 = 39 3/8. Listen for comparing 7/8 with 1 and for taking away a part.
- Why can you cancel a top with a bottom of the other fraction? The product is the same number whatever the order, and the factors can be divided by the same amount top and bottom. Accept the area model.
- What is 1/2 of 1/2 of 1/2? 1/8. Each half cuts the whole again.
Differentiation
A way in for anyone stuck
- Give squared paper: draw a 6 by 6 whole and shade columns and rows with two colours.
- Start with unit fractions: 1/2 of 1/3, 1/3 of 1/4, 1/2 of 1/7.
- Say “of” for × until the area model is secure.
Stretch
- Is 3/4 × 3/4 more or less than 3/4? Than 1/2? Explain with the area model. (9/16: less than 3/4, more than 1/2 = 8/16.)
- What is (1 − 1/2) × (1 − 1/3) × (1 − 1/4) × (1 − 1/5)? (1/5: each cancels.)
Students learning in English as an additional language
“Of” means multiply when it comes after a fraction: “half of a third”. Write “of = ×” on the word wall and point at the overlap in the picture.
| Word | Plain meaning | Example |
|---|---|---|
| product | the result of multiplying | the product of 1/2 and 1/3 is 1/6 |
| of | multiplied by, when a fraction comes before it | half of 12 is 1/2 × 12 |
| cancel | divide a top and a bottom by the same number | in 4/9 × 3/8, cancel 4 and 8 |
| area model | a picture where a fraction of a fraction is an overlap | columns for one, rows for the other |
Sentence starters for explaining aloud or in writing:
- The whole is cut into … columns and … rows, so there are … cells.
- I can cancel … and … because both divide by …
- The answer is smaller than … because I took a part of a part.
Two things not to say
“Multiply across.” Say what is being multiplied: tops together, bottoms together.
“Multiplying makes bigger.” Say “multiplying a positive number by something greater than 1 makes it bigger, by 1 keeps it, and by something between 0 and 1 makes it smaller”.
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.
- Multiply2/3 × 3/5.
2/5. The 3s cancel. - Of an amountWhat is 5/8 of 24?
15. 24 ÷ 8 = 3, and 5 × 3 = 15.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA cake is cut so that 3/4 is left after lunch, and Idris eats 2/3 of what is left. What fraction of the whole cake does he eat?
1/2. 3/4 × 2/3 = 6/12. - Earlier work (Lesson 4 of this unit)Work out 1/2 + 1/3 and then multiply the answer by 6/5.
5/6, then 5/6 × 6/5 = 1.
3Challenge
Reasoning, generalising and working backwards.
- ReasoningIs 1/4 of 3/5 equal to 3/5 of 1/4? Explain using the area model.
Yes, both 3/20. Swapping rows and columns keeps the overlap. - Working backA fraction multiplied by 3/4 gives 9/20. What is the fraction?
3/5. Multiplying by 3/4 multiplies the top by 3 and the bottom by 4, so the top was 9 ÷ 3 = 3 and the bottom 20 ÷ 4 = 5.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a slider, then the arrow keys) and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Use squared paper and two colours of pencil, one for columns and one for rows.