Add: same-size parts first
2/3 + 1/4
= 8/12 + 3/12
= 11/12
Fractions: the four operations Why can't you just add the tops and add the bottoms?
Foundation and Higher · AQA · 45 minutes
Students estimate 2/3 + 1/4 on a number line and catch the classic wrong answer, then see each operation as a picture: same-size parts to add, "of" to multiply, "how many fit" to divide. Slides, a vote that shows who's stuck, practice and homework.
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The idea in one minute
A fraction is a size, not two separate whole numbers. To add or take away, the parts must be the same size first: cut both bars into the same number of parts (a common denominator), then count. To multiply, read × as "of": 3/4 of 2/3 is part of a part, so the tops multiply and the bottoms multiply. To divide, ask how many fit: 3 ÷ 3/4 is how many three-quarters fit in 3, which is 4, the same as 3 × 4/3. Mixed numbers become improper fractions first, and an estimate catches most mistakes.
The lesson starts with an estimate on a number line, before any rule, so the classic error (2/3 + 1/4 = 3/7) shows itself: adding made it smaller. Each operation then gets its picture and its rule together, and students practise choosing which method a calculation needs. The core is for Foundation and Higher; the stretch works backwards from an answer.
By the end, students can
- add and subtract fractions by making the parts the same size first
- multiply fractions, reading × as "of"
- divide by a fraction by asking how many fit, then multiplying by the second fraction flipped
- work with mixed numbers by making them improper fractions first
- estimate an answer to check it is the right size
Key words
- numerator
- denominator
- common denominator
- equivalent fraction
- mixed number
- improper fraction
- reciprocal
Before this lesson
Students should find equivalent fractions, simplify a fraction and know their times tables. Common denominators and the four operations are taught here.
The lesson, step by step
30 minutes in total. Each step says what changes for this length.
3 min 1 · Hook · students think
Where is 2/3 + 1/4?
Students estimate a sum on a number line, then judge a classic wrong answer.
For this length: Skip the number line estimate; vote on Sam's answer only
Say“Don't work it out exactly. Put 2/3 + 1/4 on the number line where you think it goes. Then decide whether Sam can be right.”
Ask- Is 2/3 + 1/4 more or less than 2/3?
- Is 3/7 more or less than a half?
Watch forStudents who work out 3/7 and then place it: ask whether adding a quarter can make 2/3 smaller.
Don't work it out exactly: put it where you think it goes.
- Without working it out exactly, drag the marker to where 2/3 + 1/4 is on the number line.They drag a marker on a number line from 0 to 2.
For the teacher
Show the number line with 1/4 and 2/3 marked. Students drag a marker to where they think 2/3 + 1/4 goes: an estimate, not an exact value.
Then the vote: Sam says 2/3 + 1/4 = 3/7. Before anyone calculates, compare 3/7 with 2/3. 3/7 is less than a half, so adding made the answer smaller: it can't be right.
Leave the question open: so how do you add fractions? That is the next step.
Give the estimate first (Quizzes), then the vote on Sam's answer. Keep the two dots on the board for the next step.
Shorter: Skip the number line estimate; vote on Sam's answer only (−2 min).
10 min 2 · Explain · you explain
Four operations, four pictures
Each operation as a picture and its rule: same-size parts, "of", and "how many fit".
For this length: Skip the divide-fractions-and-mixed slide; do it in Your turn
Say“Every rule today comes from a picture. If you forget a rule, draw the picture.”
Ask- The bars are cut into thirds and quarters. Into how many parts could we cut both?
- Is 3/4 of 2/3 more or less than 2/3?
- How many three-quarters fit in 3? Will the answer be more or less than 3?
Watch forChanging the bottom without changing the top (2/3 = 2/12): point at the bar, 2 of 12 parts is not two thirds.
Subtract mixed numbers: improper first
2 1/3 − 1 3/4
= 7/3 − 7/4
= 28/12 − 21/12 = 7/12
Multiply: × means "of"
3/4 × 2/3
= (3 × 2)/(4 × 3)
= 6/12 = 1/2
Divide: how many fit?
3 ÷ 3/4: how many 3/4 in 3?
= 3 × 4/3
= 4
Divide fractions, and multiply mixed numbers
4/5 ÷ 2/3 = 4/5 × 3/2 = 12/10 = 6/5 = 1 1/5
2 1/2 × 1 1/3 = 5/2 × 4/3 = 20/6 = 10/3 = 3 1/3
For the teacher
Add: the bars for 2/3 and 1/4 are cut into thirds and quarters, so their parts are different sizes and can't be counted together. Cut both into twelfths (count along the axis, 0 to 12): 2/3 is 8 of them, 1/4 is 3. The bottom bar puts them together: 11 of 12, so 11/12, just under 1, as the estimate said. Whatever you do to the bottom, do to the top.
Subtract: the same idea. With mixed numbers, make them improper first: 2 1/3 − 1 3/4 = 7/3 − 7/4 = 28/12 − 21/12 = 7/12. Taking 1 from 2 and 3/4 from 1/3 goes wrong, because 3/4 doesn't come out of 1/3.
Multiply: 3/4 × 2/3 means 3/4 of 2/3. Shade 2/3 of the bar (8 twelfths), then cut that 2/3 into quarters and take 3 of them (6 twelfths): 6/12 = 1/2. Tops multiply, bottoms multiply, and no common denominator is needed. Multiplying by a fraction less than 1 makes it smaller.
Divide: 3 ÷ 3/4 asks how many three-quarters fit in 3. Hop along the number line in 3/4s: 4 hops. That is 3 × 4/3. So to divide by a fraction, multiply by it flipped: keep the first, flip the second. Dividing by a fraction less than 1 makes it bigger.
Mixed numbers: always improper first for × and ÷. 2 1/2 × 1 1/3 = 5/2 × 4/3 = 20/6 = 10/3 = 3 1/3; 2 × 1 and 1/2 × 1/3 would give only 2 1/6, too small: 2 1/2 × 1 1/3 is more than 2 1/2 × 1, which is already 2 1/2.
Ask for an estimate before every example. On the divide slide, count the hops aloud with the class.
Shorter: Skip the divide-fractions-and-mixed slide; do it in Your turn (−4 min).
Longer: Why flipping works: 1 ÷ 1/4 = 4, so dividing by 1/4 is multiplying by 4 (+5 min).
5 min 3 · Your turn · students write
Worked, then yours
Each worked example beside a twin that students solve themselves.
For this length: Do the add and divide pairs only
Say“Watch me do one, then you do its twin. Estimate first, every time.”
Ask- Before you start: which method does this one need?
Watch forMixed numbers multiplied part by part: make them improper first.
Add
3/5 + 1/3
= 9/15 + 5/15
= 14/15
Then theirs
2/7 + 1/2
Subtract mixed numbers
3 1/4 − 1 2/3
= 13/4 − 5/3 = 39/12 − 20/12
= 19/12 = 1 7/12
Then theirs
2 1/5 − 1 1/2
Multiply mixed numbers
1 1/2 × 2 2/3
= 3/2 × 8/3 = 24/6
= 4
Then theirs
2 1/4 × 1 1/3
Divide
2/3 ÷ 4/9
= 2/3 × 9/4 = 18/12
= 3/2 = 1 1/2
Then theirs
3/4 ÷ 5/8
For the teacher
Work each example aloud, saying the estimate first (3/5 + 1/3 is a bit less than 1). Then students solve its twin while you watch who makes the parts the same size, and who flips the second fraction.
The mixed-number pairs catch part-by-part working: 1 1/2 × 2 2/3 is 4, not 2 1/3.
Open the answers slide once most have finished their twins.
Shorter: Do the add and divide pairs only (−4 min).
8 min 4 · Who's stuck? · students vote
Vote, then practise
One question shows who's stuck; then practice in levels, and a stretch.
For this length: After the vote, questions 1–6 only
Say“Choose an answer on your own. Nobody talks until everyone has chosen.”
Ask- Dividing by 2/3: will the answer be more or less than 3/4?
- Which fraction do you flip, and why that one?
Watch forMixing up the rules for dividing: ask which fraction tells you the size of each piece.
Estimate first, then work it out. Give fractions in their simplest form. Show your working on paper. The stretch is optional.
The question that shows who's stuck
- Work out 3/4 ÷ 2/3.
In your Classroom, as they answer (an example) - A9
- B11
- C7
- D3
Each wrong answer is a different mistake. While they answer, your Classroom shows who chose which, so you know in seconds whom to help with what.
- 2/5 + 1/3 = ___
- Work out 3 1/5 − 1 2/3. Type the answer as a fraction or a mixed number, such as 7/4 or 1 3/4.
- The bar is cut into 12 equal parts. Shade 3/4 of 2/3 of the bar.They shade parts of a bar cut into 12 equal parts.
- Drag the marker to the answer of 2 ÷ 4/5 on the number line. (How many four-fifths fit in 2?)They drag a marker on a number line from 0 to 4.
- Work out 5/6 ÷ 2/3, a line at a time. Start each line with x =, and simplify until the last line is x = the answer in its simplest form (a mixed number such as 1 1/4 is fine, written with a space).They solve x = 5/6 / 2/3 line by line, each line checked as they write it.
- Work out 2 1/2 × 1 1/5, a line at a time. Start each line with x =, make the mixed numbers improper fractions first, and end with x = the answer in its simplest form.They solve x = 2 1/2 × 1 1/5 line by line, each line checked as they write it.
- Which method does each calculation need? Sort them.They sort 7 items into Same-size parts first and Multiply tops and bottoms and Flip the second, then multiply.
- Without working it out: is 5/8 ÷ 1/4 more or less than 5/8?
- A ribbon is 3 3/4 m long. It is cut into pieces each 5/8 m long. How many pieces are there?
- Ben spends 1/4 of his money on a ticket, then 2/5 of what is left on food. What fraction of his money is left? ___
- Stretch: 3/4 × ___ = 5/6. Give the missing fraction.
Support · For dividing: keep the first fraction, flip the second, multiply. Then check the size: dividing by a fraction less than 1 gives more than you started with.
For the teacher
Everyone chooses before anyone explains. Then ask each group to say how they got their answer, and check every answer for size: how many two-thirds fit in three-quarters?
Questions 1–4 build fluency (adding, taking away mixed numbers, a bar to shade, a number line); 5–6 are solved line by line, each line checked; 7–8 ask for reasons; 9–10 are in context. The stretch works backwards.
Students show their working on paper or the board; the quiz keeps their final answers.
If the votes split, hop 2/3 along a number line up to 3/4 together before they practise, and ask how many times it fits.
Shorter: After the vote, questions 1–6 only (−5 min).
Longer: The stretch, then: make up a division whose answer is 2 (+10 min).
4 min 5 · Exit · students show
Why not 2/5?
A picture and a sentence show who knows why the parts must match.
Say“Draw a picture that shows why 1/2 + 1/3 is not 2/5, and write the right answer.”
Watch forBars of different lengths: the wholes must be the same size.
A picture and a sentence are enough.
- Use a picture to explain why 1/2 + 1/3 is not 2/5, and give the right answer.
For the teacher
Students draw two bars of the same length, one in halves and one in thirds, then cut both into the same parts and count.
Then one sentence saying why 2/5 can't be right.
The pictures show who understands common denominators: start the next lesson with two of them. Set the homework.
Longer: Which is bigger, 2/3 ÷ 1/2 or 2/3 × 1/2? Say why without working them out (+5 min).
One to one: before every calculation, ask the student for a rough size (more or less than 1? than the first fraction?), and draw the bar or number line together whenever the answer surprises them.
Homework: all four operations
Estimate each answer first. Give fractions in their simplest form, as a mixed number where the answer is more than 1. Show your working.
- Work out 3/8 + 5/12.
- Work out 4 1/3 − 2 3/4.
- Work out 2/5 of 3/7. Draw a bar to show it.
- Work out 1 3/5 × 2 1/2.
- How many 3/4-litre bottles can be filled from 4 1/2 litres of juice?
- Ali says 2 1/2 × 2 1/2 = 4 1/4. Explain his mistake and give the right answer.
Print the worksheet with working space and homework. Teacher answers come with your Library copy.
Teaching fractions: the four operations: common questions
Why can't you add the numerators and the denominators?
Because the parts are different sizes. Thirds and quarters can't be counted together until both are cut into the same parts (twelfths). Adding the tops and the bottoms also gives answers that are too small: 1/2 + 1/2 would be 2/4, which is still a half.
How do you explain dividing fractions so it makes sense?
Ask how many fit. 3 ÷ 3/4 asks how many three-quarters fit in 3: hop along a number line and count 4. That is the same as 3 × 4/3, which is why you multiply by the second fraction flipped. It also shows why dividing by a fraction less than 1 gives a bigger answer.
Should students turn mixed numbers into improper fractions?
For multiplying and dividing, always: multiplying the whole numbers and the fractions separately is a common lost mark. For adding and subtracting, improper fractions are the safest method, especially when the fraction parts can't be taken away (2 1/3 − 1 3/4).
Is this fractions lesson suitable for Foundation GCSE?
Yes. All four operations with fractions and mixed numbers are on both tiers, and the core of the lesson is for both. The context questions and the stretch, which works backwards from an answer, suit Higher students and confident Foundation ones.
Can I use the lesson without creating an account?
Yes. Read the lesson, print the worksheet and open its slides without signing in; nothing you draw there is kept. A free teacher account keeps the whole lesson in your Library to change and teach, with the quizzes, the answers and the answer slides.
Curriculum reference: AQA GCSE Mathematics 8300: N2 (the four operations with fractions, including mixed numbers), N8 (calculating exactly with fractions) and N1 (ordering fractions on a number line).
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