At a glance. One lesson of about 60 minutes. Students count how many fit, group parts to find the reciprocal, divide with whole and mixed numbers and use the order of operations.
Calculator: no calculator. Nothing here needs one. The numbers are chosen so that cancelling gives whole-number or simple answers.
What students will learn
Know
- a ÷ c/d = a × d/c.
- The reciprocal of c/d is d/c.
- Mixed numbers become improper fractions before dividing; brackets, then multiply and divide, then add and subtract.
Understand
- Dividing by a fraction asks how many of it fit.
- Dividing by a fraction less than 1 makes a bigger number.
- Only the fraction you divide by is turned over.
Do
- Divide whole numbers, fractions and mixed numbers by fractions.
- Use the reciprocal.
- Work out an expression with several operations.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: division of fractions and mixed numbers, and the order of operations (questions 1 to 8).
- Criterion B. Investigating patterns: the first figure. Ask students to find what happens to the number of copies as the copy gets smaller, and to describe the pattern in words.
- Criterion C. Communicating: explaining in words why dividing by a half doubles (question 9), using the word reciprocal.
- Criterion D. Applying mathematics in real-life contexts: ribbon, juice and a rectangle (the hook and questions 2, 6 and 8).
The one thing to do with the figure
Ask for a prediction before the first slide. “3 metres of ribbon cut into quarters: more or fewer than 3 pieces?” Collect the votes and read the figure. Ask what happens to the count as the piece gets shorter.
Then keep the copy the same and change the length, so students see the count scale. In the second figure, set the wholes to 1 and the fraction to 3/4 and ask where the remainder goes: 1 part left is 1/3 of a group.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook, prediction and the first figure. Include a copy that does not fit a whole number of times. |
| 10 min | Key idea; the second figure, to see the reciprocal as parts per whole. |
| 10 min | Worked examples and the order-of-operations section; students finish the faded example. |
| 5 min | Question 1, the check. If many choose C, go back to the warn card. |
| 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 | 1/7; 11; 3 5/6. |
|---|---|
| Worked examples | 3/8 ÷ 9/16 = 2/3; 2 1/4 ÷ 3/4 = 3; 5/6 − 1/2 × 2/3 = 1/2. |
| Faded example | 1/2 + 2/3 × 3/4 = 1. |
| 1. Which fraction is turned over | B, 3/5 × 7/2. A multiplies by 2/7; C turns over the first; D keeps the division with the reciprocal. |
| 2. How many unit fractions fit | 20. 4 ÷ 1/5 = 4 × 5. |
| 3. Reciprocal | 9. The fraction is 4/9. |
| 4. Dividing a fraction by a fraction | 10. 5/6 = 10/12, which is 10 twelfths. |
| 5. Sharing a fraction equally | 8. 3/4 ÷ 6 = 1/8. |
| 6. Missing side of a rectangle | 3. Width = 2/3 m. |
| 7. Order of operations with fractions | 3. 3/4 ÷ 1/2 = 1 1/2. |
| 8. Dividing mixed numbers | 6. 15/2 × 4/5 = 60/10. |
| 9. Why dividing by a half doubles | B. A thinks division always shrinks; C mixes up halving with dividing by a half; D treats dividing as subtracting. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Multiplied, turned over the wrong fraction, or kept the division. | Say aloud which fraction you divide by, and turn only that one over. |
| 0.8, 5 or 4 in question 2 | Multiplied by 1/5, gave the cups per litre, or gave the litres. | Ask: how many cups does 1 litre fill? Then 4 litres? |
| 4, 13 or 36 in question 3 | Read the bottom of the reciprocal, added, or multiplied the two numbers. | Write 9/4 and flip it: 4/9. Ask for the bottom. |
| 60 or 17 in question 4 | Forgot the bottom after turning over, or added. | Write 5/6 × 12/1 and ask for the bottom number: 6. |
| 1 1/12 for 3/4 ÷ 2/3, in the first figure | Measured the leftover in metres, not in copies. | Set the length to 3/4 and the copy to 2/3. Ask what fraction of a whole copy the shaded leftover is: 1/8, not 1/12. |
| 24 or 4 in question 5 | Did not simplify, or did not change the fraction at all. | Ask whether the answer is smaller or bigger than 3/4, and whether it is in simplest form. |
| 6 or 24 in question 6 | Did not simplify, or multiplied instead of dividing. | Area = length × width, so write width = area ÷ length. |
| 4.5 or 1.875 in question 7 | Added first, or multiplied instead of dividing. | Underline the operation that comes first. Ask what divide by 1/2 does. |
| 6.25 or 8.75 in question 8 | Subtracted or added the lengths. | Ask what the question is asking: how many pieces fit? |
| A, C or D in question 9 | Thought division always shrinks, or confused halving with dividing by a half. | Ask: how many halves are in 5? Draw them. |
Questions worth asking
- Why does dividing by 1/4 give four times as many? Each whole holds 4 quarters. Listen for “4 quarters in every whole”.
- Can dividing a number by a fraction give a smaller number? Yes, if the fraction is more than 1: 6 ÷ 3/2 = 4.
- What is the reciprocal of 5? Of 1/5? 1/5; 5. A whole number is a fraction over 1. Zero has no reciprocal.
- Does 3/4 ÷ 1/2 = 1/2 ÷ 3/4? No: 1 1/2 and 2/3. Division is not commutative.
- Write 3/4 ÷ 2/3 over a common denominator. What are you asking? 9/12 ÷ 8/12: how many 8-twelfths fit in 9-twelfths? 9 ÷ 8 = 1 1/8, the same as 3/4 × 3/2. Listen for “same size of part, so divide the counts”.
Differentiation
A way in for anyone stuck
- Give strips of paper marked in halves, thirds and quarters, so students can lay copies end to end.
- Start with whole numbers divided by unit fractions: 2 ÷ 1/4, 3 ÷ 1/2.
- Write the multiplication underneath every division: “÷ 1/4 is × 4”.
Stretch
- How many times does 2/3 go into 7/9? Draw it, and explain the leftover. (7/6 = 1 1/6: one 2/3 fits, and the 1/9 left is 1/6 of a copy.)
- Find two fractions whose quotient is 5/6. How many pairs are there? (Infinitely many: 5/6 ÷ 1, 5/12 ÷ 1/2, ...)
Students learning in English as an additional language
“Reciprocal” is a long word: say “the fraction turned over” as well. “How many fit” and “how many go into” mean the same; say both.
| Word | Plain meaning | Example |
|---|---|---|
| reciprocal | a fraction turned upside down: the top and bottom swapped | the reciprocal of 2/7 is 7/2 |
| divide | find how many times one number fits in another, or share equally | 6 ÷ 2 = 3 |
| quotient | the result of a division | the quotient of 12 ÷ 4 is 3 |
| copy | one piece of the same size | four copies of 1/4 make a whole |
| brackets | symbols that say what to do first | in (1/2 + 1/4) ÷ 3/8 the brackets come first |
Sentence starters for explaining aloud or in writing:
- … fits into … exactly … times, because …
- I turned over … because it is the fraction I divide by.
- The answer is bigger than where I started because …
Two things not to say
“Keep, change, flip.” as a chant with no picture. Say how many fit first, and let the rule come from the counting.
“Dividing makes it smaller.” Only dividing by a number greater than 1 does that.
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.
- Divide4 ÷ 1/3.
12. - Divide2/5 ÷ 1/10.
4. 2/5 = 4/10.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA bottle holds 3/4 of a litre. How many glasses of 1/8 of a litre can be filled?
6. - Earlier work (Lesson 5 of this unit)Work out 3/4 of 2/9, then divide the answer by 1/3.
1/6, then 1/6 ÷ 1/3 = 1/2.
3Challenge
Reasoning, generalising and working backwards.
- ReasoningWithout calculating, which is greater: 5 ÷ 2/3 or 5 × 2/3? Explain.
5 ÷ 2/3, because dividing by a fraction less than 1 makes it bigger. 7 1/2 against 3 1/3. - Working backA number divided by 3/5 is 20. What is the number?
12. 20 × 3/5.
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 strips of paper cut into quarters and ask how many quarters fit in a whole strip, then two strips.