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# Adding and subtracting fractions

Number Foundations · Fractions and decimals

<!-- Speaker note: Students cut two fraction bars into the same number of parts to add and subtract fractions with different denominators, use the LCM as the lowest common denominator, and extend the idea to mixed numerals and decimals (AC9M7N06, node 1). About 45 minutes (Predict and Run 8 min · Investigate 15 min · Modify 10 min · Practise and Make 12 min). -->

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## Today

**We are learning to** parts can be counted together only when they are the same size. Cut both fractions into the same parts first: 1/3 + 1/4 = 4/12 + 3/12 = 7/12. The lowest common multiple of the bottoms is the smallest choice.

- 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.

<!-- Speaker note: 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. -->

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## Why it matters

> A recipe needs 1/3 of a cup of milk and 1/4 of a cup of cream. Will it all fit in a half-cup measure? You cannot tell until the parts are the same size.

<!-- Speaker note: You can add 2 apples and 3 apples, but 2 apples and 3 pears do not make 5 of anything. Fractions are the same: $\tfrac{2}{7} + \tfrac{3}{7} = \tfrac{5}{7}$ (sevenths and sevenths), but thirds and quarters are different sizes. To add or subtract fractions with different bottoms, cut both into the **same parts** first, using equivalent fractions. The new bottom is a **common denominator**: a common multiple of the two bottoms, and the lowest common multiple is the smallest. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Predict

**What is $\tfrac{1}{3} + \tfrac{1}{4}$?**

- $\tfrac{2}{7}$
- $\tfrac{1}{7}$
- $\tfrac{7}{12}$
- $\tfrac{2}{12}$

<!-- Speaker note: What happens: 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. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: students who chose 2/7 added across. Ask: is 2/7 bigger than 1/3? -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Fnumber%2Fadding-fractions%2Fstep-2&code=widget%3A+fracadd%0Aa%3A+1%0Ab%3A+3%0Ac%3A+1%0Ad%3A+4%0Aslider+k%3A+1%2C+24%2C+7%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/number/adding-fractions/?full=step-2#step-2)

<!-- Speaker note: The board says when a bar does not split evenly into $k$ parts. -->

<!-- Speaker note: Teacher: ask which values of k work, and what they have in common. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## What happened?

* 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.

<!-- Speaker note: Reveal after running. Ask who predicted it, and what made them think so. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Same parts, then count

Here is the method, then a mixed numeral and a decimal.

Now try one: $\tfrac{2}{5} + \tfrac{1}{3}$.

$$\tfrac{1}{3} + \tfrac{1}{4} = \tfrac{4}{12} + \tfrac{3}{12}$$

1) $= \tfrac{7}{12}$ — Same parts: add the tops, keep the bottom.
2) $2\tfrac{1}{2} - \tfrac{3}{4} = \tfrac{10}{4} - \tfrac{3}{4} = \tfrac{7}{4} = 1\tfrac{3}{4}$ — A mixed numeral: write it as quarters first.
3) $3.45 + 2.70 = 6.15$ — Decimals: line up the points (the same parts: tenths, hundredths).

<!-- Speaker note: Model this live on the board before revealing anything: write each line and say why, pausing to ask the class what comes next. Then click to reveal the lines and compare. Script. Start: write 1 ÷ 3 + 1 ÷ 4 = 4 ÷ 12 + 3 ÷ 12 and say "The LCM of 3 and 4 is 12: write both in twelfths." Line 1: write = 7 ÷ 12 and say "Same parts: add the tops, keep the bottom." Line 2: write 21 ÷ 2 - 3 ÷ 4 = 10 ÷ 4 - 3 ÷ 4 = 7 ÷ 4 = 13 ÷ 4 and say "A mixed numeral: write it as quarters first." Line 3: write 3.45 + 2.70 = 6.15 and say "Decimals: line up the points (the same parts: tenths, hundredths)." -->

<!-- Speaker note: Teacher: for the decimals task, write 2.7 as 2.70 and ask why that is allowed. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Same parts, then count

**Work out $\tfrac{2}{5} + \tfrac{1}{3}$. Give your answer in simplest form.**

* Answer: 11/15

<!-- Speaker note: 2/5 = 6/15 and 1/3 = 5/15, so 6/15 + 5/15 = 11/15. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Find the mistake

**Ella adds $\tfrac{1}{2} + \tfrac{1}{3}$. Click the line where her method goes wrong.**

```
 1  Work out 1/2 + 1/3.
 2  Add the tops and add the bottoms.
 3  1 + 1 = 2 and 2 + 3 = 5
 4  So 1/2 + 1/3 = 2/5.
```

* Answer: line 2

<!-- Speaker note: 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.) -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Decimals

**Work out $3.45 + 2.7$.**

* Answer: 6.15

<!-- Speaker note: Line up the points: 3.45 + 2.70 = 6.15. Tenths with tenths, like fractions with the same bottom. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## One change at a time



- **Subtract:** Each task changes one thing.
- **Without the board:** 
- **A mixed numeral:** 
- **One already fits:** 

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/number/adding-fractions/?full=step-4#step-4)

<!-- Speaker note: What this step is for: 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. Hints, in order: 24 works. Is there a smaller number that 4 and 6 both divide? For students who need more support: Use the board for every sum; fraction walls and a list of multiples help find the common denominator. For early finishers: Find three different fractions with different bottoms that add to 1. 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: A student adds the tops and the bottoms (Ella's mistake). Ask: Is your answer bigger than each fraction you added? Fix: Use the bars: change both into the same parts, then count. / A student changes the bottom but not the top (1/3 = 1/12). Ask: What did you multiply the bottom by? Did the top get the same? Fix: Equivalent fractions: the same multiplication on the top and the bottom. / A student subtracting mixed numerals takes the wholes and the parts separately and gets stuck at 1/3 − 1/2. Ask: What if you write both as improper fractions first? Fix: Convert to improper fractions with a common denominator, then subtract. -->

<!-- Speaker note: Teacher: for the edge case, ask why only one fraction needs to change. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Add

**Work out $\tfrac{3}{4} + \tfrac{1}{6}$ in simplest form.**

* Answer: 11/12

<!-- Speaker note: 9/12 + 2/12 = 11/12. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Mixed numerals

**Work out $3\tfrac{1}{3} - 1\tfrac{1}{2}$.**

* Answer: 1 5/6

<!-- Speaker note: 3 1/3 = 20/6 and 1 1/2 = 9/6, so 11/6 = 1 5/6. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Decimals in context

**A plank is 2.4 m long. 0.65 m is cut off. How long is the rest, in metres?**

* Answer: 1.75

<!-- Speaker note: 2.40 − 0.65 = 1.75 m. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Review: LCM

**Review: find the LCM of 6 and 10.**

* Answer: 30

<!-- Speaker note: 6 = 2 × 3 and 10 = 2 × 5: LCM = 2 × 3 × 5 = 30. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Make: a sum of one

Then find a pair where neither bottom is a multiple of the other.

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/number/adding-fractions/?full=step-6#step-6)

<!-- Speaker note: What this step is for: Choosing two different fractions that add to exactly 1, and showing it with a common denominator, works the method backwards. Hints, in order: Try 1/2 and something else, or 1/3 and something else. / Choose k so both bars split, then count the parts. -->

<!-- Speaker note: Teacher: collect pairs; then challenge for one where neither bottom divides the other. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## 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.

<!-- Speaker note: Students answer on paper or a mini whiteboard before the reveal. Anyone who cannot answer needs the main idea again next lesson. Where this 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. -->