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# Calculating with negative fractions and decimals

Number Foundations · Fractions and decimals

<!-- Speaker note: Students carry the integer sign rules to negative fractions and decimals (sign first, then size) and choose efficient strategies, reordering products and splitting numbers with the distributive law (AC9M8N04, node 2). About 45 minutes (Predict and Run 8 min · Investigate 12 min · Modify 13 min · Practise and Make 12 min). -->

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## Today

**We are learning to** the sign rules for integers work for every fraction and decimal: work out the sign, then the size. Choose an efficient order: multiplying can be done in any order (25 × 4 first), and a hard product can be split (6 × 2.5 = 6 × 2 + 6 × 0.5).

- I can add, subtract, multiply and divide negative fractions and decimals.
- I can work out the sign first, then the size.
- I can choose an efficient order or split a number to make a calculation easier, and say why it works.

<!-- Speaker note: Before class: Do the reasoning task yourself first. Students need Multiplying and dividing integers, Adding and subtracting integers, and Equivalent fractions. The one idea: the sign rules do not change for fractions and decimals; and the order of multiplying, or a split, can make the size easy. The common mistake is the addition rule for multiplying (two negatives make a negative); it is planted in Leo's working and again in Practise. The strategy tasks use the reasoning check: students write their own steps (not marked); the answer and the reason for the key step are marked, then they compare with a model. -->

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## Why it matters

> A temperature falls half a degree an hour for three hours. A share loses 2.4 dollars over 6 days. Negative fractions and decimals are everywhere money and weather are measured.

<!-- Speaker note: You know the sign rules for integers: the same signs give a positive, different signs a negative, for multiplying and dividing. They work just the same for fractions and decimals: $-0.5 \times (-4) = 2$, and $-\tfrac{3}{4} \times \tfrac{2}{5} = -\tfrac{3}{10}$. This lesson also looks for **efficient** ways to calculate: choosing the order, or splitting a number, so the arithmetic is easy. -->

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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 $-3 \times (-\tfrac{1}{2})$?**

- $1\tfrac{1}{2}$
- $-3\tfrac{1}{2}$
- $\tfrac{1}{6}$
- $-1\tfrac{1}{2}$

<!-- Speaker note: What happens: 1 1/2. Same signs give a positive, and 3 × 1/2 = 1 1/2. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: take a show of hands. Students who chose −1 1/2 used the addition rule. -->

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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%2Frational-operations%2Fstep-2&code=widget%3A+numberline%0Arange%3A+-4%2C+4%0Afractions%3A+2%0Aproduct%3A+a+%C3%97+b%0Aslider+a%3A+-4%2C+4%2C+3%0Aslider+b%3A+-2%2C+2%2C+0.5%2C+0.5%0A&fresh=1}}

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

<!-- Speaker note: Then try $-3 \times \tfrac{1}{2}$ and $3 \times (-\tfrac{1}{2})$. Which sign rule does each follow? -->

<!-- Speaker note: Teacher: ask whether it matters that the jump is a half, not a whole number. -->

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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?

* 1 1/2. Same signs give a positive, and 3 × 1/2 = 1 1/2.

<!-- 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>

## Sign, then size, then strategy

Here is how to work with negative fractions and decimals, and two shortcuts.

Try one: $-\tfrac{3}{4} \times \tfrac{2}{5}$.

$$-\tfrac{3}{4} \times \tfrac{2}{5}$$

1) $= -\tfrac{6}{20} = -\tfrac{3}{10}$ — Then size: multiply tops and bottoms, then simplify.
2) $25 \times (-3.7) \times 4 = 100 \times (-3.7)$ — Any order: pair 25 and 4 first (commutative and associative).
3) $= -370$ — × 100 is easy.
4) $6 \times (-2.5) = 6 \times (-2) + 6 \times (-0.5)$ — Split a hard number (distributive): $-12 + (-3)$.
5) $= -15$ — Then add.

<!-- 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 -3 ÷ 4 × 2 ÷ 5 and say "Sign first: different signs, so negative." Line 1: write = -6 ÷ 20 = -3 ÷ 10 and say "Then size: multiply tops and bottoms, then simplify." Line 2: write 25 × (-3.7) × 4 = 100 × (-3.7) and say "Any order: pair 25 and 4 first (commutative and associative)." Line 3: write = -370 and say "× 100 is easy." Line 4: write 6 × (-2.5) = 6 × (-2) + 6 × (-0.5) and say "Split a hard number (distributive): -12 + (-3)." Line 5: write = -15 and say "Then add." -->

<!-- Speaker note: Teacher: collect different ways to do 25 × (−3.7) × 4 and ask which is least work. -->

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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>

## Sign, then size, then strategy

**Work out $-\tfrac{3}{4} \times \tfrac{2}{5}$. Simplify.**

* Answer: -3/10

<!-- Speaker note: Different signs: negative. 3 × 2 = 6 and 4 × 5 = 20: −6/20 = −3/10. -->

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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

**Leo works out $-0.4 \times (-0.5)$. Click the line where his working goes wrong.**

```
 1  Work out −0.4 × (−0.5).
 2  4 × 5 = 20, so 0.4 × 0.5 = 0.20.
 3  Both numbers are negative, so the answer is negative.
 4  −0.4 × (−0.5) = −0.2.
```

* Answer: line 3

<!-- Speaker note: Line 3: for multiplying, the same signs give a positive. −0.4 × (−0.5) = 0.2. -->

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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>

## Choose the order

**Work out $25 \times (-3.7) \times 4$ the easy way. Write your steps your own way.**

* Answer: $25 \times 4 = 100$ (multiplying can be done in any order, so pair the friendly numbers first); $100 \times (-3.7) = -370$ (different signs give a negative; × 100 moves the digits two places)

<!-- Speaker note: 25 × 4 = 100, then 100 × (−3.7) = −370. -->

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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



- **Add on the line:** Each task changes one thing.
- **Divide decimals:** 
- **Divide fractions:** 
- **Split a decimal:** 
- **A power:** 

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

<!-- Speaker note: What this step is for: One change at a time: adding negative fractions on the line, dividing decimals, dividing fractions, splitting with the distributive law, then the edge case of a power of a negative decimal. Hints, in order: Start at −1 1/4 and move 2 1/2 right. For students who need more support: Use the number line board for every addition and for multiplying by a whole number of jumps. For early finishers: Work out (−1/2)⁵ and (−0.2)³. When is a power of a negative number positive? When students get stuck: A student uses the addition rule for multiplying (Leo's mistake). Ask: What does −3 × (−2) give? Does a decimal change that? Fix: Same signs give a positive for multiplying and dividing, whatever the numbers. / A student gets the size wrong with decimals. Ask: Can you work with whole numbers first (0.3 × 40 = 3 × 4)? Fix: Count the decimal places, or scale: 2.4 ÷ 0.6 = 24 ÷ 6. / A student works strictly left to right. Ask: Which two numbers are friendly together? Fix: Multiplication can be done in any order: pair 25 and 4. -->

<!-- Speaker note: Teacher: for the fraction division, recall multiplying by the reciprocal if it is new. -->

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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 negatives

**Work out $-1.5 + (-2.75)$.**

* Answer: -4.25

<!-- Speaker note: Both move left: −4.25. -->

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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>

## Multiply a decimal

**Work out $-0.3 \times 40$.**

* Answer: -12

<!-- Speaker note: 3 × 4 = 12, different signs: −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>

## Subtract a negative

**Work out $\tfrac{5}{6} - (-\tfrac{1}{3})$. Simplify.**

* Answer: 7/6

<!-- Speaker note: 5/6 + 2/6 = 7/6 (or 1 1/6). -->

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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>

## Review: simplest form

**Review: write $\tfrac{15}{35}$ in its simplest form.**

* Answer: 3/7

<!-- Speaker note: ÷ 5: 3/7. -->

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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>

## Two negatives that make −1

Then: two negative fractions whose product is $\tfrac{1}{4}$?

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

<!-- Speaker note: What this step is for: Choosing two negative fractions that add to −1 reverses the lesson: from the answer back to the parts, using quarters. Hints, in order: Both must be between −1 and 0. / Try −1/4 and −3/4. -->

<!-- Speaker note: Teacher: ask for the product question too. -->

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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 −3/5 × 10/9 and −4 × (−2.25). Show one strategy.**

* −2/3. 9 (−4 × −2 = 8, −4 × −0.25 = 1, so 8 + 1 = 9).

<!-- 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: This completes AC9M8N04 (both nodes). Where it leads: substituting negative fractions into formulas, gradient from two points, and financial calculations (Year 8 and 9). -->