--- title: "Calculating with negative fractions and decimals" subtitle: "Number Foundations · Fractions and decimals · Lesson 16" --- # Calculating with negative fractions and decimals **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** 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. :::teacher-note **In one sentence:** 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). **Students should be able to say:** 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. **Width:** the unmarked parts are the floor every student does (predict, run, investigate and make). Typical adds the modify practice; wider adds the real-world contexts. They add contexts, not difficulty. Students work in the online tutorial alongside this sheet. ::: ## Predict Which sign rule applies? Then: what is $3 \times \tfrac{1}{2}$? :::keep-together **What is $-3 \times (-\tfrac{1}{2})$?** ☐ $1\tfrac{1}{2}$ ☐ $-3\tfrac{1}{2}$ ☐ $\tfrac{1}{6}$ ☐ $-1\tfrac{1}{2}$ My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** 1 1/2. Same signs give a positive, and 3 × 1/2 = 1 1/2. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Sign, size, strategy :::keep-together ### Sign, then size, then strategy Here is how to work with negative fractions and decimals, and two shortcuts. 1. $-\tfrac{3}{4} \times \tfrac{2}{5}$ — Sign first: different signs, so negative. 2. $= -\tfrac{6}{20} = -\tfrac{3}{10}$ — Then size: multiply tops and bottoms, then simplify. 3. $25 \times (-3.7) \times 4 = 100 \times (-3.7)$ — Any order: pair 25 and 4 first (commutative and associative). 4. $= -370$ — × 100 is easy. 5. $6 \times (-2.5) = 6 \times (-2) + 6 \times (-0.5)$ — Split a hard number (distributive): $-12 + (-3)$. 6. $= -15$ — Then add. Try one: $-\tfrac{3}{4} \times \tfrac{2}{5}$. **Work out $-\tfrac{3}{4} \times \tfrac{2}{5}$. Simplify.** Answer: {{field-15}} ::: :::teacher-note **Answer:** -3/10 Different signs: negative. 3 × 2 = 6 and 4 × 5 = 20: −6/20 = −3/10. ::: :::keep-together ### 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. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 3 Line 3: for multiplying, the same signs give a positive. −0.4 × (−0.5) = 0.2. ::: :::keep-together ### Choose the order **Work out $25 \times (-3.7) \times 4$ the easy way. Write your steps your own way.** Answer: {{field-15}} ::: :::teacher-note **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) 25 × 4 = 100, then 100 × (−3.7) = −370. ::: :::typical ## One change at a time **Add on the line.** Each task changes one thing. **Divide decimals.** **Divide fractions.** **Split a decimal.** **A power.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Mixed rational calculations :::keep-together ### Add negatives **Work out $-1.5 + (-2.75)$.** Answer: {{field-15}} ::: :::teacher-note **Answer:** -4.25 Both move left: −4.25. ::: :::keep-together ### Multiply a decimal **Work out $-0.3 \times 40$.** Answer: {{field-15}} ::: :::teacher-note **Answer:** -12 3 × 4 = 12, different signs: −12. ::: :::keep-together ### Subtract a negative **Work out $\tfrac{5}{6} - (-\tfrac{1}{3})$. Simplify.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 7/6 5/6 + 2/6 = 7/6 (or 1 1/6). ::: :::keep-together ### Review: simplest form **Review: write $\tfrac{15}{35}$ in its simplest form.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 3/7 ÷ 5: 3/7. ::: ## Two negatives that make −1 Then: two negative fractions whose product is $\tfrac{1}{4}$? Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **Work out −3/5 × 10/9 and −4 × (−2.25). Show one strategy.** {{lines-3}} ::: :::teacher-note **Answer:** −2/3. 9 (−4 × −2 = 8, −4 × −0.25 = 1, so 8 + 1 = 9). :::