At a glance. One lesson of about 60 minutes. Students recut fractions into same-size parts to add and subtract, then extend to mixed numbers.
Calculator: no calculator. Nothing here needs one. The skill is deciding what to count, and the numbers stay small.
What students will learn
Know
- To add or subtract fractions, write them over a common denominator.
- The bottom number stays the same when you add or subtract.
- A sum above 1 can be written as a mixed number.
Understand
- The bottom number is the size of the part, so only the count changes.
- With equal bottoms, adding the bottoms halves the answer; with any bottoms, the tops-plus-bottoms fraction lies between the two fractions, so it is never the sum.
- A mixed-number subtraction may need a whole cut into parts.
Do
- Add and subtract fractions with different bottoms.
- Add and subtract mixed numbers.
- Use a known total to find a missing fraction.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: addition and subtraction of fractions and mixed numbers (questions 1 to 8).
- Criterion B. Investigating patterns: the second figure. Ask students to find how the wrong answer relates to the right one when the bottoms match (always half), then test 1/2 and 1/3 on the first figure: is it still half?
- Criterion C. Communicating: explaining in words why the bottom number is not added (the warn card and question 9).
- Criterion D. Applying mathematics in real-life contexts: sharing cake and pizza and cutting planks (the hook and questions 5 and 7).
The one thing to do with the figure
Ask for a prediction before the first press. “1/2 + 1/3: more or less than half a cake?” The claim 2/5 is less than 1/2 is the whole lesson: a sum cannot be smaller than one of its parts. Set the sliders to 1/2 and 1/3 only after the prediction.
Then, in the first figure, switch to Subtract and ask students to say which fraction is taken from which. In the second figure, make the sum cross 1 and ask what the second bar 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 an add that crosses 1 and a subtract. |
| 10 min | Key idea; the second figure. Students predict the wrong answer before reading it. |
| 10 min | Worked examples including the mixed-number subtraction; students finish the faded example. |
| 5 min | Question 1, the check. If many choose A, go back to the second 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 | 8/12 and 9/12; 8/5; 2/3. |
|---|---|
| Worked examples | 2/3 + 3/10 = 29/30; 1 2/5 + 2 3/4 = 4 3/20; 5 1/3 − 2 3/4 = 2 7/12. |
| Faded example | 5/6 − 3/8 = 11/24. |
| 1. Adding fractions with different bottoms | B, 13/20. A adds tops and bottoms; C multiplies; D converts 2/5 incorrectly. |
| 2. Adding to more than one whole | 2. 9/7 = 1 2/7. |
| 3. Adding with different bottoms | 19. 9/12 + 10/12 = 19/12. |
| 4. Adding mixed numbers | 53. 32/12 + 21/12 = 53/12 = 4 5/12. |
| 5. Subtracting mixed numbers in context | 6. 13/4 − 7/4 = 6/4 = 1 1/2 m. |
| 6. Finding a missing fraction | 3. 4/8 + 3/8 = 7/8. |
| 7. Three fractions of a whole | 3. 12 − (4 + 3 + 2) = 3. |
| 8. Two different unit fractions | 6. 1/3 + 1/6 = 2/6 + 1/6 = 3/6 = 1/2. |
| 9. Adding with one bottom a multiple of the other | B, 9/10. A adds tops and bottoms; C stops after converting; D keeps one top and adds the bottoms. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A in question 1; 8 in question 3 | Added tops and bottoms, or tops only, without a common denominator. | Show the second figure: ask how much of the cake you have when the parts are different sizes. |
| 9 or 1 in question 2 | Gave the improper top, or the whole-number part. | Ask what each number in 1 2/7 counts. |
| 14 in question 3 | Converted only one fraction. | Ask for both fractions over 12 on separate lines. |
| 17, 41 or 36 in question 4 | Added only the fractions, lost the whole carried out of the fractions, or left the fractions out. | Ask for 2 2/3 and 1 3/4 as twelfths on separate lines, then add the twelfths. |
| 10 in question 5 | Subtracted the smaller fraction from the larger without regrouping. | Ask: can you take 3 quarters from 1 quarter? Convert to improper fractions. |
| 4, 6 or 7 in question 6 | Reported 1/2 as 4/8 and stopped, subtracted tops directly (7 − 1), or reported the total. | Write the unknown as a box over 8 and write 1/2 as 4/8: 4 is what you already have. |
| 9 or 4 in question 7 | Gave what was eaten (4 + 3 + 2), or stopped at the first friend's share. | Ask what the question asks: left or eaten? Write all three shares, then both totals. |
| 4, 3 or 2 in question 8 | Added unit fractions to see if they reach a half, gave the bottom of the fraction already given, or used the total's bottom. | Subtract: 1/2 − 1/3 over 6. |
| A, C or D in question 9 | A added tops and bottoms; C stopped after converting; D kept one top and added the bottoms (3/15). | Write the conversion on one line and the sum on the next. |
Questions worth asking
- Why does the bottom number not change when you add? Because it names the size of the part, and the parts are not being changed. Listen for “same size parts”.
- Can the sum of two fractions be less than one of them? No, not with positive fractions. This is a quick check for the tops-and-bottoms slip.
- Is a common denominator always the product of the two bottoms? No: the lowest common multiple is often smaller (4 and 6 give 12, not 24). Both work.
- Why might you regroup a whole when subtracting mixed numbers? The fraction part may be too small to take from. Cutting a whole into parts gives enough.
Differentiation
A way in for anyone stuck
- Give fraction strips for halves, thirds, quarters, sixths and twelfths so students can lay them end to end.
- Start with one bottom a multiple of the other: 1/2 + 1/4, 2/3 + 1/6.
- For mixed numbers, convert to improper fractions first and convert back at the end.
Stretch
- Find three different unit fractions that add to 1. Find four. (1/2 + 1/3 + 1/6; 1/2 + 1/4 + 1/6 + 1/12.)
- Two fractions are each in simplest form. Is their sum always in simplest form? Try 1/6 + 1/3. (No: 1/6 + 1/3 = 3/6 = 1/2.)
- The mean of two fractions is their sum divided by 2. Find the mean of 1/3 and 1/2 and show that it lies between them. (5/12: 1/3 + 1/2 = 5/6, and half of 5/6 is 5/12; 4/12 < 5/12 < 6/12.)
Students learning in English as an additional language
“Common” here means “shared by both”, not “usual”. Say “the same size of part” each time you say “common denominator”.
| Word | Plain meaning | Example |
|---|---|---|
| common denominator | a bottom number that both fractions can be written over | 12 for quarters and sixths |
| sum | the result of adding | the sum of 1/2 and 1/3 is 5/6 |
| difference | the result of subtracting; how much more one is than the other | the difference between 3/4 and 1/4 is 1/2 |
| regroup | cut one whole into parts so there are enough to take away | 5 1/3 can be regrouped as 4 4/3 |
| simplest form | a fraction where only 1 divides both top and bottom | 3/6 simplifies to 1/2 |
Sentence starters for explaining aloud or in writing:
- Both fractions are written over … so the parts are the same size.
- The answer is …/…, which simplifies to … because … divides both.
- I know the answer cannot be … because it is less than …
Two things not to say
“Find a common denominator by multiplying the bottoms.” It works but is not always the lowest. Ask for the lowest common multiple.
“Borrow from the whole.” Say “cut one whole into twelfths” so it is clear what is happening.
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.
- Add2/5 + 1/10.
1/2. 4/10 + 1/10 = 5/10. - Subtract7/8 − 1/4.
5/8. 7/8 − 2/8.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA recipe uses 2/3 cup of milk and 1/4 cup of cream. How much liquid is that? How much more milk is there than cream?
11/12 cup; 5/12 cup. Over 12: 8/12 and 3/12. - Earlier work (Lesson 2 of this unit)Write 19/12 as a mixed number, then add 1/4 to it.
1 7/12; 1 10/12 = 1 5/6.
3Challenge
Reasoning, generalising and working backwards.
- ReasoningShow that 1/n − 1/(n + 1) = 1/(n(n + 1)) for n = 2, 3 and 4. What do you notice about the answers?
1/6, 1/12, 1/20: the bottom is the product of consecutive numbers. - Working backA fraction is added to 3/8 to make 1 1/4. What is the fraction?
7/8. 1 1/4 = 10/8, and 10/8 − 3/8 = 7/8.
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 12 squares long, cut into halves, thirds, quarters and sixths.