Calculating with negative fractions and decimals

Classroom resources

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Students will be able to

• 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.

Timing

Predict and Run 8 min · Investigate 12 min · Modify 13 min · Practise and Make 12 min

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.

Running the lesson

1. Predict: take a show of hands. Students who chose −1 1/2 used the addition rule.
2. Run: ask whether it matters that the jump is a half, not a whole number.
3. Investigate: collect different ways to do 25 × (−3.7) × 4 and ask which is least work.
4. Modify: for the fraction division, recall multiplying by the reciprocal if it is new.
5. Make: ask for the product question too.

When students get stuck

Answers to the checks

1. a negative half: What is $-3 \times (-\tfrac{1}{2})$?
   1 1/2. Same signs give a positive, and 3 × 1/2 = 1 1/2.
2. jumps of a half: Set the board to $-3 \times (-\tfrac{1}{2})$. Was your prediction right?
   
3. Sign, then size, then strategy: Work out $-\tfrac{3}{4} \times \tfrac{2}{5}$. Simplify.
   -3/10
   Different signs: negative. 3 × 2 = 6 and 4 × 5 = 20: −6/20 = −3/10.
4. Find the mistake: Leo works out $-0.4 \times (-0.5)$. Click the line where his working goes wrong.
   line 3
   Line 3: for multiplying, the same signs give a positive. −0.4 × (−0.5) = 0.2.
5. Choose the order: Work out $25 \times (-3.7) \times 4$ the easy way. Write your steps your own way.
   $25 \times 4 = 100$ (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)
   25 × 4 = 100, then 100 × (−3.7) = −370.
6. Add on the line: Set the board to $-1\tfrac{1}{4} + 2\tfrac{1}{2}$. Where do you land?
   
   −1 1/4 + 2 1/2 = 1 1/4.
7. Divide decimals: Work out $-2.4 \div 0.6$.
   -4
   24 ÷ 6 = 4, different signs: −4.
8. Divide fractions: Work out $-\tfrac{2}{3} \div (-\tfrac{4}{9})$. Simplify.
   3/2
   Same signs: positive. 2/3 × 9/4 = 18/12 = 3/2 (or 1 1/2).
9. Split a decimal: Work out $8 \times (-1.25)$ by splitting $-1.25$. Write your steps.
   $8 \times (-1) = -8$ and $8 \times (-0.25) = -2$ (split −1.25 into −1 and −0.25 (distributive law)); $-8 + (-2) = -10$ (add the two parts)
   8 × (−1) + 8 × (−0.25) = −8 + (−2) = −10.
10. A power: Edge case: what is $(-0.1)^2$?
    $0.01$
    (−0.1) × (−0.1) = 0.01: positive, and smaller than 0.1.
11. Add negatives: Work out $-1.5 + (-2.75)$.
    -4.25
    Both move left: −4.25.
12. Multiply a decimal: Work out $-0.3 \times 40$.
    -12
    3 × 4 = 12, different signs: −12.
13. Subtract a negative: Work out $\tfrac{5}{6} - (-\tfrac{1}{3})$. Simplify.
    7/6
    5/6 + 2/6 = 7/6 (or 1 1/6).
14. Review: simplest form: Review: write $\tfrac{15}{35}$ in its simplest form.
    3/7
    ÷ 5: 3/7.
15. two negatives that make −1: Set $a$ and $b$ to two negative fractions (not whole numbers) that add to $-1$.
    
    −1/4 + (−3/4), −1/2 + (−1/2) or −3/4 + (−1/4).

Solutions

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

one change at a time › Add on the line

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

one change at a time › Divide decimals

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

one change at a time › Divide fractions

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

one change at a time › Split a decimal

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

one change at a time › A power

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

two negatives that make −1

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

Stretch and support

Stretch: Work out (−1/2)⁵ and (−0.2)³. When is a power of a negative number positive?

Support: Use the number line board for every addition and for multiplying by a whole number of jumps.

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 8 student in Australia. Help me understand ONE idea only, from the lesson "Calculating with negative fractions and decimals".

The idea: Efficient strategies for rational number operations: I can Students apply all four operations to rational numbers including negative fractions and decimals, selecting efficient strategies and digital tools appropriately.

What understanding looks like:
- Correctly computes with negative fractions and decimals.
- Selects an efficient strategy and can justify why.
- Does not default to the most laborious method when a simpler one is available.

Mistakes students often make here:
- A student uses the addition rule for multiplying (Leo's mistake).
- A student gets the size wrong with decimals.
- A student works strictly left to right.

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.
- If I seem to be missing the idea that comes before this one (Multiplication and division of integers), check it quickly, then come back.
- 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 "Calculating with negative fractions and decimals".

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).

What each step is for

• a negative half: Committing to −3 × (−1/2) shows whether the integer sign rules carry over to fractions, before the board shows the jumps.
• jumps of a half: Setting −3 jumps of −1/2 on the line shows the same picture as with integers: the sign rules do not care whether the jump is a whole number.
• sign, size, strategy: Sign first, then size; and choose an order or a split that makes the size easy. Leo's mistake uses 'two negatives make a negative' from adding.
• one change at a time: 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.
• mixed rational calculations: Adding, multiplying and subtracting with negative fractions and decimals, mixed; the addition-rule slip planted again; the review keeps simplifying fresh.
• two negatives that make −1: Choosing two negative fractions that add to −1 reverses the lesson: from the answer back to the parts, using quarters.

Where it 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).