--- title: "Recurring decimals" subtitle: "Irrational numbers and index laws · Recurring decimals · Lesson 1" --- # Recurring decimals **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** Type 1 ÷ 7 into a calculator and the screen fills up. Does it ever stop? Long division can tell you, and it does not need a bigger screen. :::teacher-note **In one sentence:** Students divide fractions out on a long-division board, watch the remainders, and see why decimals either stop or repeat, writing recurring decimals with bars and dots (AC9M8N03, node 1). **Students should be able to say:** I can tell whether a fraction's decimal terminates or recurs. I can write a recurring decimal with a bar or dots. I can explain why a remainder coming back means the digits repeat. **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 Try dividing 1 by 3 on paper for a few steps. What remainder do you keep getting? :::keep-together **Does the decimal for 1/3 ever stop?** ☐ No: the 3s go on forever ☐ Yes, after about 10 digits ☐ Yes: 0.33 My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** No. Every step of 1 ÷ 3 leaves a remainder of 1, so every digit is 3, forever. 0.33 is an approximation. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Why it repeats :::keep-together ### Notation Here is the routine. 1. $1 \div 6$ — Bring down a 0: 10 ÷ 6 = 1 remainder 4. 2. $40 \div 6 = 6 \text{ r } 4$ — Remainder 4 again. 3. $\tfrac{1}{6} = 0.1\overline{6}$ — The 6 repeats forever; the 1 does not. 4. $\tfrac{1}{8} = 0.125$ — A remainder of 0: it stops. **How should 0.272727… be written with a bar?** ☐ A bar over 2 only ☐ A bar over 27: 0.27 with the bar over both digits ☐ A bar over the last 7 only ☐ 0.27 with no bar ::: :::teacher-note **Answer:** A bar over 27: 0.27 with the bar over both digits The repeating block is 27, so the bar goes over both digits. ::: :::keep-together ### Find the mistake **Zoe decides whether 1/7 is a rational number (a fraction). Click the line where her reasoning goes wrong.** ``` 1 Zoe divides 1 by 7 on a calculator. 2 The screen shows 0.1428571429. 3 The digits go on past the end of the screen. 4 A decimal that never stops is irrational, so 1/7 is irrational. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 4 Line 4: 1/7 is a fraction, so it is rational. Its decimal never stops, but it repeats (142857). Only a decimal that never stops and never repeats is irrational. ::: :::keep-together ### Eleven **Divide 1 by 11 (on paper, or on the Run board above). How many digits are in its repeating block?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 2 1/11 = 0.090909…: the block 09 repeats. ::: :::typical ## Stop, repeat, delay **Stops after 3 places.** **A delay.** **Thirteenths.** **0.999….** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Recurring decimals :::keep-together ### Stops or repeats **Which fraction has a terminating decimal?** ☐ 2/7 ☐ 1/3 ☐ 3/8 ☐ 5/6 ::: :::teacher-note **Answer:** 3/8 3/8 = 0.375: it stops. ::: :::keep-together ### Sevenths **How many digits are in the repeating block of 1/7?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 6 1/7 = 0.142857142857…: 6 digits. ::: :::keep-together ### Dot notation **How is 0.1666… written with a dot?** ☐ 0.17 ☐ 0.16 with a dot over the 1 ☐ 0.16 with dots over the 1 and the 6 ☐ 0.16 with a dot over the 6 ::: :::teacher-note **Answer:** 0.16 with a dot over the 6 Only the 6 repeats: a dot over the 6. ::: :::keep-together ### Review: fractions to decimals **Review: write 3/4 as a decimal.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 0.75 3 ÷ 4 = 0.75. ::: ## Make: a two-digit block Then write your fraction's decimal with a bar over the repeating block. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **Does 5/6 terminate or recur? Write it with a dot.** {{lines-3}} ::: :::teacher-note **Answer:** It recurs: 5/6 = 0.8333… = 0.83 with a dot over the 3. :::