Adding and subtracting fractions

Classroom resources

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Students will be able to

• I can add and subtract fractions with different denominators using a common denominator.
• I can add and subtract mixed numerals and decimals.
• I can explain why the tops and the bottoms are not simply added.

Timing

Predict and Run 8 min · Investigate 15 min · Modify 10 min · Practise and Make 12 min

Before class

Do the Run board yourself first. Students need equivalent fractions (Equivalent fractions) and the LCM (HCF and LCM).

The one idea: parts can be counted together only when they are the same size, so cut both fractions into the same parts first. The common mistake is adding the tops and the bottoms (Ella's mistake); a quick check is that the sum must be bigger than each fraction.

Decimals are the same idea: line up the places so tenths are added to tenths.

Running the lesson

1. Predict: students who chose 2/7 added across. Ask: is 2/7 bigger than 1/3?
2. Run: ask which values of k work, and what they have in common.
3. Investigate: for the decimals task, write 2.7 as 2.70 and ask why that is allowed.
4. Modify: for the edge case, ask why only one fraction needs to change.
5. Make: collect pairs; then challenge for one where neither bottom divides the other.

The hard part (on purpose)

Finding a common denominator without guessing. Ask students to list a few multiples of each bottom number and look for the first match, instead of telling them the LCM.

When students get stuck

Answers to the checks

1. a third and a quarter: What is $\tfrac{1}{3} + \tfrac{1}{4}$?
   7/12: 1/3 = 4/12 and 1/4 = 3/12, and 4 + 3 = 7 twelfths. 2/7 is less than 1/3 on its own, so it cannot be the sum.
2. cut them the same: Change $k$, the number of parts in each whole, until both fractions split evenly. Then read the sum. Was your prediction right?
   
   12 works (and 24): both are multiples of 3 and 4. 12 is the lowest.
3. Same parts, then count: Work out $\tfrac{2}{5} + \tfrac{1}{3}$. Give your answer in simplest form.
   11/15
   2/5 = 6/15 and 1/3 = 5/15, so 6/15 + 5/15 = 11/15.
4. Find the mistake: Ella adds $\tfrac{1}{2} + \tfrac{1}{3}$. Click the line where her method goes wrong.
   line 2
   Line 2: halves and thirds are different sizes, so they cannot be counted together. 1/2 = 3/6 and 1/3 = 2/6: 5/6. (2/5 is less than 1/2 on its own.)
5. Decimals: Work out $3.45 + 2.7$.
   6.15
   Line up the points: 3.45 + 2.70 = 6.15. Tenths with tenths, like fractions with the same bottom.
6. Subtract: Find $\tfrac{3}{4} - \tfrac{1}{6}$ with the lowest common denominator.
   
   12: 3/4 = 9/12 and 1/6 = 2/12, so 9/12 − 2/12 = 7/12.
7. Without the board: Work out $\tfrac{5}{6} - \tfrac{1}{4}$ in simplest form.
   7/12
   5/6 = 10/12 and 1/4 = 3/12, so 7/12.
8. A mixed numeral: Work out $1\tfrac{1}{2} + \tfrac{2}{3}$.
   2 1/6
   1 1/2 = 9/6 and 2/3 = 4/6, so 13/6 = 2 1/6.
9. One already fits: Edge case: what is the lowest common denominator for $\tfrac{1}{4} + \tfrac{3}{8}$?
   8, since 8 is already a multiple of 4
   4 divides 8, so only the 1/4 changes: 2/8 + 3/8 = 5/8.
10. Add: Work out $\tfrac{3}{4} + \tfrac{1}{6}$ in simplest form.
    11/12
    9/12 + 2/12 = 11/12.
11. Mixed numerals: Work out $3\tfrac{1}{3} - 1\tfrac{1}{2}$.
    1 5/6
    3 1/3 = 20/6 and 1 1/2 = 9/6, so 11/6 = 1 5/6.
12. Decimals in context: A plank is 2.4 m long. 0.65 m is cut off. How long is the rest, in metres?
    1.75
    2.40 − 0.65 = 1.75 m.
13. Review: LCM: Review: find the LCM of 6 and 10.
    30
    6 = 2 × 3 and 10 = 2 × 5: LCM = 2 × 3 × 5 = 30.
14. a sum of one: Make two fractions with different bottoms that add to exactly 1.
    
    For example 1/2 + 2/4, 1/3 + 4/6 or 1/4 + 6/8. Neither bottom a multiple of the other: 2/4 + 3/6 (6/12 + 6/12).

Solutions

One possible solution for each task. Other correct programs are fine.

one change at a time › Subtract

Open task: any program that does what the step asks.

one change at a time › Without the board

Open task: any program that does what the step asks.

one change at a time › A mixed numeral

Open task: any program that does what the step asks.

one change at a time › One already fits

Open task: any program that does what the step asks.

a sum of one

Open task: any program that does what the step asks.

Stretch and support

Stretch: Find three different fractions with different bottoms that add to 1.

Support: Use the board for every sum; fraction walls and a list of multiples help find the common denominator.

Coach prompt (optional)

For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.

You are a patient maths coach for a Year 7 student in Australia. Help me understand ONE idea only, from the lesson "Adding and subtracting fractions".

The idea: Addition and subtraction of fractions and decimals: I can Students add and subtract fractions (including mixed numerals) and decimals fluently, using common denominators and place value alignment.

What understanding looks like:
- Correctly adds and subtracts fractions with unlike denominators.
- Correctly adds and subtracts decimals using place value alignment.
- Does not add numerators and denominators directly when adding fractions.

Mistakes students often make here:
- A student adds the tops and the bottoms (Ella's mistake).
- A student changes the bottom but not the top (1/3 = 1/12).
- A student subtracting mixed numerals takes the wholes and the parts separately and gets stuck at 1/3 − 1/2.

How to coach me (rules from the Waypoints Kit coach):
- Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question.
- Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit).
- If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go.
- Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back.
- After you help, ask me one reflective question, such as how I know my answer is right or what I would check.
- Praise my effort and method, not my ability.
- You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so.
- Use simple words.
- Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "Adding and subtracting fractions".

Exit ticket

Work out 2/3 + 1/5 and 1 1/4 − 2/3.

10/15 + 3/15 = 13/15. 15/12 − 8/12 = 7/12.

What each step is for

• a third and a quarter: Committing to a sum for 1/3 + 1/4 shows whether they add the tops and the bottoms (2/7), which the bars then contradict.
• cut them the same: Changing the number of parts until both bars split evenly shows that a common multiple of 3 and 4 makes the parts countable.
• common denominators: A common denominator makes the parts the same size; Ella adds tops and bottoms as if the parts already matched. Mixed numerals and decimals follow the same rule: line up like with like.
• one change at a time: One change at a time: subtracting on the board with the lowest common denominator, subtracting without the board, a mixed numeral answer, then the case where one denominator is a multiple of the other.
• add and subtract: Fractions, a mixed numeral and decimals in context, mixed; adding across is the planted slip; the review keeps the LCM fresh.
• a sum of one: Choosing two different fractions that add to exactly 1, and showing it with a common denominator, works the method backwards.

Where it leads

Next: Multiplying and dividing fractions (node 2), where the rule is different: no common denominator needed.

Where it leads: algebraic fractions (Year 10) and probability with fractions.