--- title: "Letters that vary" subtitle: "Algebra Foundations · Expressions · Lesson 1" --- # Letters that vary **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** Recipes, phone bills and the perimeter of a garden bed all follow a formula. The letters are what changes; the numbers are what stays. :::teacher-note **In one sentence:** Students tell the variables in a formula from the constants, and see that in an equation a letter can be one unknown number (AC9M7A01, node 1). **Students should be able to say:** I can say which letters in a formula are variables and which numbers are constants. I can say what a constant means in a real formula, such as a call-out fee. I can tell a variable (can take many values) from an unknown (one value makes the equation true). **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 $$P = 2l + 2w$$ Think of two different rectangles: a long thin one and a square. What is the same about how you find their perimeters? :::keep-together **$P = 2l + 2w$ gives the perimeter of a rectangle. When you draw a different rectangle, which parts of the formula change?** ☐ Only $P$ changes ☐ $l$, $w$ and $P$ change; the 2s stay the same ☐ The 2s change as well ☐ Nothing: each letter stands for one fixed number My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** l, w and P change, because different rectangles have different lengths, widths and perimeters. The 2s stay the same for every rectangle: there are always two lengths and two widths. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Variables, constants and unknowns :::keep-together ### Reading a formula Here is how to read the parts of a formula. 1. $P = 2l + 2w$ — The perimeter of a rectangle: twice the length plus twice the width. 2. $P, \; l, \; w$ — These letters are **variables**: quantities that change from one rectangle to the next. 3. $2$ — The 2s are **constants**: they are the same for every rectangle. 4. $l = 5, \; w = 3$ — For one particular rectangle, each variable has a value. 5. $3x + 2 = 14$ — In an equation, a letter can be an **unknown**: one particular number. 6. $x = 4$ — Only one value makes this equation true. **A plumber charges $C = 5h + 20$ dollars for a job of $h$ hours. Which are the variables?** ☐ 5 and 20 ☐ $C$ and $h$ ☐ Only $h$ ::: :::teacher-note **Answer:** $C$ and $h$ C and h change from job to job. 5 (dollars an hour) and 20 (the call-out fee) are constants: the same for every job. ::: :::keep-together ### Find the mistake **Kim works out the perimeter. Click the line where her thinking goes wrong.** ``` 1 P = 2l + 2w 2 l is the 12th letter of the alphabet, so l = 12. 3 w is the 23rd letter, so w = 23. 4 P = 2 × 12 + 2 × 23 = 70. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 2 Line 2: a letter stands for a quantity, here the length, not its place in the alphabet. l can be any length: you need to measure or be told it. ::: :::keep-together ### Unknown or variable? **In the equation $3x + 2 = 14$, what does $x$ stand for?** ☐ An unknown: one number makes it true ☐ The letter x, not a number ☐ A variable that can be any number ::: :::teacher-note **Answer:** An unknown: one number makes it true In an equation, the letter is an unknown: only x = 4 makes 3x + 2 = 14 true. In a formula, a letter is a variable that can take many values. ::: :::typical ## Other formulas **A plumber.** Each task loads a new formula. Change the variables and watch what stays the same. **A car trip.** **Area.** **A letter that never changes.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Read the formula :::keep-together ### What the constant means **Water in a tank: $V = 3t + 10$ litres after $t$ minutes. What does the 10 mean?** ☐ The tank fills in 10 minutes ☐ 10 litres in the tank at the start ☐ 10 litres added every minute ::: :::teacher-note **Answer:** 10 litres in the tank at the start When t = 0, V = 10: the 10 litres were there at the start. 3 litres are added each minute. ::: :::keep-together ### Variables **Which letters are variables in $T = 12n + 6$ (the cost of $n$ concert tickets)?** ☐ $T$ and $n$ ☐ 12 and 6 ☐ Only $T$ ::: :::teacher-note **Answer:** $T$ and $n$ T and n change with the order. 12 (dollars a ticket) and 6 (the booking fee) stay the same. ::: :::keep-together ### Unknown **In $2y + 1 = 9$, what is $y$?** ☐ An unknown: $y = 4$ makes it true ☐ A variable: any value works ☐ A constant ::: :::teacher-note **Answer:** An unknown: $y = 4$ makes it true It is an equation, so y is an unknown: 2 × 4 + 1 = 9, and no other value works. ::: ## Make: a formula of your own Name the variables and the constant in your formula. Stretch: write a formula for the perimeter of a triangle with sides $a$, $b$ and $c$. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **A taxi charges T = 2d + 4 dollars for d km. Name the variables and the constants, and say what the 4 means.** {{lines-3}} ::: :::teacher-note **Answer:** Variables: T and d. Constants: 2 and 4. The 4 is the fee to get in, paid even for 0 km. :::