--- title: "An expression for any case" subtitle: "Algebra Foundations · Expressions · Lesson 6" --- # An expression for any case **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** A party for 5 people, 50 people or 500: one expression gives the cost for all of them, if you say clearly what the letter stands for. :::teacher-note **In one sentence:** Students say what a letter stands for, write an expression for a situation with a fee and a rate, test it with a case they can check, and use it for any number (AC9M7A02, node 3). **Students should be able to say:** I can say what the letter in an expression stands for, a number that changes. I can write an expression for a situation with a once-only amount and an amount for each one. I can test my expression with a case I can check in words, then use it for a big case. **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 If $n$ is the number of students, what does 1 student cost? What do 10 cost? :::keep-together **A school trip costs 12 dollars per student plus a 50 dollar bus. You want one expression for the total cost. What should the letter stand for?** ☐ The bus ☐ The total cost ☐ The 12 dollars ☐ The number of students My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** The number of students. It is the quantity that changes from trip to trip. The cost is worked out from it (12n + 50), and 12 and 50 are constants. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## From a situation to an expression :::keep-together ### A hall Here is how to build an expression for a situation. 1. $\text{Hall: 60 dollars, plus 5 dollars per guest}$ — A hall costs 60 dollars to hire, and then 5 dollars for each guest. 2. $\text{Let } g = \text{number of guests}$ — Say what the letter stands for. The number of guests changes, so $g$ stands for that. 3. $5g$ — Each guest costs 5 dollars, so $g$ guests cost $5 \times g$. 4. $5g + 60$ — Add the hire, which is paid once. 5. $g = 10: \; 5(10) + 60 = 110$ — Test with a case you can check in words: 10 guests cost 50 + 60 = 110 dollars. 6. $g = 40: \; 5(40) + 60 = 260$ — Now the same expression gives any case, with no counting. **A phone plan costs 15 dollars a month plus 4 dollars per GB. Let $g$ be the number of GB used. Write an expression for the monthly cost.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 4g + 15 4g + 15: 4 dollars for each of g GB, plus 15 dollars once. ::: :::keep-together ### Find the mistake **Zoe starts with "Let $n$ be the cost" for tickets that cost 9 dollars each. Why is that not a good start?** ☐ 9 should be the letter ☐ The cost changes with the number of tickets, so the letter should stand for the number of tickets ☐ A letter can never stand for a cost ::: :::teacher-note **Answer:** The cost changes with the number of tickets, so the letter should stand for the number of tickets The cost depends on how many tickets there are. Let n be the number of tickets, then the cost is 9n. ::: :::keep-together ### Your turn **Pia writes the cost of $a$ apples at 3 dollars each. Click the line where her thinking goes wrong.** ``` 1 Apples cost 3 dollars each. Write the cost of a apples. 2 Cost = 3a. I read 3a as "3 apples". 3 So the cost of a apples is 3a. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 2 Line 2: 3a means 3 × a: 3 dollars times the number of apples. It does not mean "3 apples". The letter a stands for how many apples, a number. ::: :::typical ## New situations **A taxi.** The board follows the situation and shows the result. Write your expression, then check it with a number. **A tank.** **Two groups.** **A big case.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Build and use :::keep-together ### A plumber **A plumber charges 60 dollars call-out plus 40 dollars per hour. Let $h$ be the number of hours. Write an expression for the cost.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 40h + 60 40h + 60: 40 dollars for each hour, plus 60 dollars once. ::: :::keep-together ### Using it **Using $40h + 60$, what does a job of 3.5 hours cost, in dollars?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 200 40 × 3.5 + 60 = 140 + 60 = 200 dollars. ::: :::keep-together ### The letter **Cupcakes cost 3 dollars each. What should the letter $c$ stand for in the expression $3c$ for the total cost?** ☐ The cost ☐ The number of cupcakes ☐ Cupcakes ::: :::teacher-note **Answer:** The number of cupcakes c is a number: how many cupcakes. 3c is then 3 dollars times that number. ::: :::keep-together ### Review one **Review: find $3(n + 4)$ when $n = 5$.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 27 3 × (5 + 4) = 3 × 9 = 27. ::: :::keep-together ### Review two **Review: write an expression for "add 5 to $n$, then multiply the result by 2".** Answer: {{field-15}} ::: :::teacher-note **Answer:** 2(n + 5) 2(n + 5): add 5 first, then double the whole result. ::: ## Make: a situation of your own Make up a situation with a fee paid once and a price for each one: a club, a game, a delivery. Write "Let ___ be the number of ___." Then write an expression for the total, and test it with one case. Stretch: swap with a partner and see if they can write your expression from the story alone. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **A gym costs 30 dollars to join plus 15 dollars a week. Write what your letter stands for, an expression for the total, and the cost after 8 weeks.** {{lines-3}} ::: :::teacher-note **Answer:** Let w be the number of weeks. Total = 15w + 30. After 8 weeks: 15 × 8 + 30 = 150 dollars. :::