--- title: "Which decimals stop?" subtitle: "Irrational numbers and index laws · Recurring decimals · Lesson 2" --- # Which decimals stop? **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** Without dividing anything, you can tell that 7/40 stops and 7/30 goes on forever. The secret is in the denominator's prime factors. :::teacher-note **In one sentence:** Students predict whether a fraction's decimal terminates from the prime factors of its simplified denominator, and check on the long-division board (AC9M8N03, node 2). **Students should be able to say:** I can predict whether a fraction terminates without dividing. I can explain the rule using 10 = 2 × 5. I can simplify a fraction before using the rule. **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 Break 50 into primes. :::keep-together **Without dividing: does 11/50 terminate or recur?** ☐ Terminate ☐ Recur ☐ You cannot tell without dividing My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** It terminates: 50 = 2 × 5 × 5, only 2s and 5s. 11/50 = 22/100 = 0.22. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Why 2s and 5s :::keep-together ### Why it works **Why do only 2s and 5s in the denominator give a terminating decimal?** ☐ Because 2 + 5 = 7 ☐ 2 and 5 are the smallest primes ☐ 10 = 2 × 5, so only those can multiply up to 10, 100, 1000… ☐ Because even numbers always terminate ::: :::teacher-note **Answer:** 10 = 2 × 5, so only those can multiply up to 10, 100, 1000… A terminating decimal is a fraction over a power of 10, and 10 = 2 × 5. ::: :::keep-together ### Find the mistake **Max decides whether 3/12 terminates. Click the line where his reasoning goes wrong.** ``` 1 Max wants to know if 3/12 terminates. 2 12 = 2 × 2 × 3. 3 12 has a prime factor of 3, so 3/12 recurs. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 3 Line 3: simplify first. 3/12 = 1/4, and 4 = 2 × 2, so it terminates: 0.25. ::: :::keep-together ### Sort them **Terminate or recur?** Groups: Terminates, Recurs - 1/16: {{field-15}} - 3/20: {{field-15}} - 1/12: {{field-15}} - 2/15: {{field-15}} - 7/25: {{field-15}} - 6/15: {{field-15}} ::: :::teacher-note **Answer:** 1/16 → Terminates; 3/20 → Terminates; 1/12 → Recurs; 2/15 → Recurs; 7/25 → Terminates; 6/15 → Terminates Simplify, then check the denominator for primes other than 2 and 5. ::: :::typical ## Find one that stops **Fifteenths.** **Twenty-fourths.** **Eightieths.** **Thirtieths.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Which stop :::keep-together ### Simplify first **Does 7/56 terminate or recur?** ☐ Terminate: 7/56 = 1/8 ☐ Recur: 56 has a 7 ☐ Recur: 56 is too big ☐ You cannot tell ::: :::teacher-note **Answer:** Terminate: 7/56 = 1/8 7/56 = 1/8 = 0.125. ::: :::keep-together ### Places **1/16 terminates. How many decimal places does it have?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 4 16 = 2⁴, so 1/16 = 625/10 000 = 0.0625: four places. ::: :::keep-together ### Always stops **Which denominator always gives a terminating decimal, whatever the numerator?** ☐ 12 ☐ 21 ☐ 30 ☐ 50 ::: :::teacher-note **Answer:** 50 50 = 2 × 5 × 5: only 2s and 5s. ::: :::keep-together ### Review: block length **Review: how many digits are in the repeating block of 1/9?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 1 1/9 = 0.111…: one digit. ::: ## Make: sixtieths Then explain, using primes, why your fraction stops and 1/60 does not. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **Does 9/24 terminate? Explain.** {{lines-3}} ::: :::teacher-note **Answer:** Yes: 9/24 = 3/8, and 8 = 2 × 2 × 2, only 2s, so it terminates (0.375). :::