At a glance. One lesson of about 60 minutes. Students fill boxes with parts, turn improper fractions into mixed numbers, and go back along a number line.
Calculator: no calculator. Nothing here needs one. The division facts are small, and the skill is knowing what the remainder is a count of.
What students will learn
Know
- An improper fraction has a top that is at least its bottom.
- A mixed number is a whole number plus a fraction that is less than 1.
- w r/d = (w × d + r)/d.
Understand
- A remainder is a count of parts of the same size as the fraction.
- Each whole number holds as many parts as the bottom number says.
- A mixed number is a sum, not a product.
Do
- Write an improper fraction as a mixed number, and the reverse.
- Place both forms on a number line.
- Explain what the whole-number part and the remainder count.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: improper fractions and mixed numbers (questions 1 to 8).
- Criterion B. Investigating patterns: the boxes figure. Ask students to find which counts of quarters fill exactly whole boxes, and to describe the pattern in the top numbers.
- Criterion C. Communicating: explaining in words why 8/4 is not 5 3/4 (question 9), using whole, part and parts of the same size.
- Criterion D. Applying mathematics in real-life contexts: measuring rice, flour and practice time (the hook and questions 4 and 5).
The one thing to do with the figure
Ask for a prediction before the first slide. “17 quarter-cups: more or less than 4 cups?” Take estimates, then fill the boxes until the answer is read from the figure, not from a rule.
Then change the size of the part and keep the number of parts. Ask what stays the same (the number of parts) and what changes (the number of wholes). In the second figure, ask students to say the count of parts before the figure does.
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 count that fills the boxes exactly. |
| 10 min | Key idea; the second figure. Students say the improper fraction before reading 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 | 3/4; 20; 8 with 2 left over. |
|---|---|
| Worked examples | 47/9 = 5 2/9; 4 5/8 = 37/8. |
| Faded example | 41/7 = 5 6/7. |
| 1. Improper fraction to mixed number | B, 7 3/4. A rounds the whole part up; C swaps the remainder and the bottom; D splits the digits of 31. |
| 2. Placing an improper fraction between whole numbers | B, 5 and 6. 23/4 = 5 3/4. A counts one box too few; C reads the digits of 23; D rounds up. |
| 3. Mixed number to improper fraction | C, 23/7. A adds the whole to the top; B swaps the multiplication; D joins the digits. |
| 4. Mixed number as a count of parts | 11. 2 3/4 = 11/4. |
| 5. Remainder as a count of parts | 2. 38/4 = 9 2/4, which is 9 1/2 hours. |
| 6. Spotting a wrong conversion | C. 3 × 6 + 6 = 24, not 19. 19/6 = 3 1/6. |
| 7. Improper fraction just above a whole number | 19. 3 = 18/6, so 19/6 is the next sixth. |
| 8. Counting fractions in a range | 8. n = 40, 41, ..., 47. |
| 9. Why the sum is not the count | B. A adds the whole and the top; C joins the digits; D multiplies the whole by the top. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A in question 1 | Rounded the whole number up. | Fill the boxes: ask how many boxes are full, not how many boxes are started. |
| A, C or D in question 2 | Counted one box too few, read the digits of 23 as 2 and 3, or rounded up. | Ask how many quarters 5 boxes hold, and 6 boxes, then place 23 between them. |
| A, B or D in question 3 | Added the whole and the top, swapped the numbers in the product, or joined the digits. | Ask how many sevenths are in one whole and in 3 wholes before adding the 2. |
| 8 or 10 in question 4 | Left out the extra parts, or multiplied by the wrong number. | Ask: how many quarters in one cup? So how many in 2 cups? |
| 9 or 36 in question 5 | Gave the whole hours, or the parts inside them. | Write the units on the answer line: whole hours, or quarter-hours left over. |
| A, B or D in question 6 | Picked a correct conversion, or did not check by multiplying back. | Check each by w × d + r. The wrong one does not give the top. |
| 18 in question 7 | Gave a fraction that equals 3, not one that exceeds it. | Write 3 as eighteen sixths on a number line and ask what is the next sixth. |
| 7 or 9 in question 8 | Missed 40/8 or included 48/8. | List the smallest and the largest numerators and count them. |
| A, C or D in question 9 | Added, joined or multiplied the whole and the top. | Ask how many quarters are in 5 wholes before adding the 3. |
Questions worth asking
- What does the remainder count? Parts of the size of the fraction: 2 left over means 2 thirds, say. Listen for the unit.
- Can an improper fraction be a whole number? Yes: 12/4 = 3. The remainder is 0.
- Which is more useful for comparing, 23/7 or 3 2/7? Mixed numbers show quickly between which wholes it sits; improper fractions are easier for calculating. Accept both reasons.
- Why is 3 2/7 not 3 × 2/7? Because there is no multiplication sign: the whole number and the fraction are added.
Differentiation
A way in for anyone stuck
- Give each student a strip of paper cut into 4 equal parts and a pile of more strips so they can fill boxes themselves.
- Start with halves and quarters only, where the remainder is at most 3.
- Write the check in a box: whole × bottom + top should give back the improper top.
Stretch
- A mixed number has a whole-number part of 3, and its improper fraction has a top number of 29. List all the possible bottom numbers. (8 and 9: 29/8 = 3 5/8 and 29/9 = 3 2/9.)
- How many improper fractions with a bottom number of 12 have a whole-number part of 2? (24/12 to 35/12, so 12 of them.)
Students learning in English as an additional language
“Improper” here means “top bigger than bottom”, not “wrong”. “Mixed” means a whole number and a fraction together. Point at the whole number and the fraction when you say each.
| Word | Plain meaning | Example |
|---|---|---|
| improper fraction | a fraction whose top is at least as big as its bottom | 9/4 is an improper fraction |
| mixed number | a whole number and a fraction less than 1, added together | 2 1/4 means 2 + 1/4 |
| whole-number part | how many full wholes there are | in 5 2/9 the whole-number part is 5 |
| remainder | what is left over after dividing as far as whole numbers go | 47 ÷ 9 has a remainder of 2 |
| proper fraction | a fraction less than 1: the top is less than the bottom | 3/8 is a proper fraction |
Sentence starters for explaining aloud or in writing:
- … wholes use up … parts, and … parts are left over.
- The bottom number is … because each whole is cut into … parts.
- I checked by multiplying back: … × … + … = …
Two things not to say
“Bottom times whole plus top.” as a chant with no picture. Say what each part counts first: parts in the wholes, then extra parts.
“Improper fractions are wrong.” They are a perfectly good way to write a number bigger than 1.
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.
- ConvertWrite 17/5 as a mixed number.
3 2/5. - Convert backWrite 6 1/2 as an improper fraction.
13/2.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA ribbon is 4 3/8 metres long. How many eighth-metre pieces would that be if you cut it all?
35. 4 × 8 = 32, and 3 more makes 35. - Earlier work (Lesson 1 of this unit)Write 30/45 in simplest form, then say whether it is more or less than 1.
2/3, less than 1. Divide by 15, the greatest common factor.
3Challenge
Reasoning, generalising and working backwards.
- ReasoningWhich is greater, 29/6 or 4 3/4? Explain without converting both to decimals.
29/6 = 4 5/6, and 5/6 is more than 3/4, so 29/6 is greater. - Working backA mixed number has a whole-number part of 4, and the top of its improper fraction is 23 more than its bottom. Find a possible bottom number.
7: 4 2/7 = 30/7, and 30 − 7 = 23. Also 6: 4 5/6 = 29/6, and 29 − 6 = 23.
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: draw boxes six squares wide and shade parts of the right size.