--- title: "When brackets are needed" subtitle: "Algebra Foundations · Expressions · Lesson 5" --- # When brackets are needed **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** "Add 4, then double it" and "double it, then add 4" sound alike. They give different answers, and different expressions. :::teacher-note **In one sentence:** Students decide where brackets belong when a sentence applies a later step to a whole result, and see that brackets change the value (AC9M7A02, node 2). **Students should be able to say:** I can say which step is done first, and what is then done to its whole result. I can put brackets around the part that is done first, and leave them out when they are not needed. I can show with a number that 2(n + 3) and 2n + 3 are different. **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 Dan with 4 stickers: 4 and 3 more is 7, doubled is 14. Which expression gives 14? :::keep-together **Dan has $n$ stickers. He gets 3 more, then doubles the lot. Which expression is the number he ends with?** ☐ $n + 6$ ☐ $2n + 3$ ☐ $5n$ ☐ $2(n + 3)$ My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** 2(n + 3). He adds 3 first (n + 3), then doubles all of it. 2n + 3 would double only his own stickers and add 3 afterwards. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Where do the brackets go? :::keep-together ### Add 3, then double Here is how to decide where the brackets go. 1. $\text{add 3 to } n, \text{ then double the result}$ — Find the step done first, and what is done to its result. 2. $n + 3$ — The first step: add 3 to $n$. 3. $\text{double the result}$ — "The result" is the whole of $n + 3$, not just $n$. 4. $2(n + 3)$ — Brackets keep $n + 3$ together, so it is worked out first and then doubled. 5. $n = 4: \; 2(4 + 3) = 2 \times 7 = 14$ — Check: add 3 to 4 to get 7, then double it. 6. $2n + 3$ — Compare: this doubles $n$ only and then adds 3. For $n = 4$ it gives 11, not 14. **Write an expression for "add 4 to $n$, then multiply the result by 5".** Answer: {{field-15}} ::: :::teacher-note **Answer:** 5(n + 4) 5(n + 4): the brackets keep n + 4 together. Check with n = 1: 1 + 4 = 5, then × 5 = 25, and 5(1 + 4) = 25. ::: :::keep-together ### Find the mistake **Which expression means "subtract 3 from $n$, then divide the result by 2"?** ☐ $\frac{n}{2} - 3$ ☐ $\frac{n - 3}{2}$ ☐ $n - \frac{3}{2}$ ::: :::teacher-note **Answer:** $\frac{n - 3}{2}$ The whole of n − 3 is divided by 2. The fraction bar works like brackets. Check with n = 11: 11 − 3 = 8, and 8 ÷ 2 = 4. ::: :::keep-together ### Your turn **Jo writes "add 2 to $x$, then multiply by 6". Click the line where she goes wrong.** ``` 1 Write "add 2 to x, then multiply by 6" as an expression. 2 Add 2 to x is x + 2. 3 Then multiply by 6, so I write 6x + 2. 4 So the expression is 6x + 2. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 3 Line 3: 6x + 2 multiplies only x, then adds 2. To multiply the whole x + 2 by 6 she needs brackets: 6(x + 2). Check with x = 1: 1 + 2 = 3, × 6 = 18, but 6 × 1 + 2 = 8. ::: :::typical ## New sentences **Subtract, then multiply.** The board does what the words say. Write your expression, then check it with a number. **No brackets.** **Two letters.** **Halving.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Brackets or not? :::keep-together ### Books **Books cost 8 dollars each. Write an expression for the cost of $n$ books after taking 5 dollars off the total.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 8n - 5 8n − 5: the books cost 8n, then 5 dollars is taken off. ::: :::keep-together ### A group **A group of $n$ adults and 3 children each pay 6 dollars. Write an expression for the total.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 6(n + 3) 6(n + 3): the group has n + 3 people and each pays 6 dollars. ::: :::keep-together ### Which is bigger? **When $n = 5$, which is bigger, $2(n + 3)$ or $2n + 3$?** ☐ $2n + 3$, which is 13 ☐ They are the same ☐ $2(n + 3)$, which is 16 ::: :::teacher-note **Answer:** $2(n + 3)$, which is 16 2(5 + 3) = 2 × 8 = 16 and 2 × 5 + 3 = 13. Brackets change the answer. ::: :::keep-together ### Review **Review: find $4(n - 1)$ when $n = 6$.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 20 4 × (6 − 1) = 4 × 5 = 20. ::: ## Make: find the rule Say the steps in words: what does the rule do first, and what next? Stretch: write a rule of your own with brackets and ask a partner to find it from the board. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **Write an expression for: (a) add 6 to n, then multiply by 3; (b) multiply n by 3, then add 6. Which one gives 15 when n = 3?** {{lines-3}} ::: :::teacher-note **Answer:** (a) 3(n + 6); (b) 3n + 6. (b): 3 × 3 + 6 = 15; (a) gives 27. :::