--- title: "Finding an unknown" subtitle: "Algebra Foundations · Expressions · Lesson 3" --- # Finding an unknown **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** You know how much fence you have and how long the garden bed must be. How wide can it be? A formula answers that too, worked backwards. :::teacher-note **In one sentence:** Students substitute the known values into a formula, solve the equation that is left for the unknown, and check by substituting back (AC9M7A01, node 3). **Students should be able to say:** I can substitute the values I know into a formula to make an equation. I can solve the equation by undoing each operation, last first. I can check my answer by substituting it back into the formula. **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 $$P = 2l + 2w$$ Which letters do you know? Which one is unknown? :::keep-together **A rectangle has perimeter $P = 20$ cm and length $l = 6$ cm. Using $P = 2l + 2w$, what is its width $w$?** ☐ 4 ☐ 7 ☐ 8 ☐ 14 My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** 4. 20 = 2 × 6 + 2w, so 20 = 12 + 2w. Then 2w = 8 and w = 4. (8 forgets to halve; 14 takes the length away only once.) ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Substitute, then solve :::keep-together ### Area Here is how to find an unknown in a formula. 1. $P = 2l + 2w$ — Write the formula. 2. $P = 20, \; l = 6$ — Write what you know. The unknown is $w$. 3. $20 = 2 \times 6 + 2w$ — Substitute. Now it is an equation with one unknown. 4. $20 = 12 + 2w$ — Work out what you can. 5. $8 = 2w$ — Undo the + 12: take 12 from both sides. 6. $w = 4$ — Undo the × 2: divide both sides by 2. 7. $2 \times 6 + 2 \times 4 = 20 \;\checkmark$ — Check by substituting back into the formula. **A rectangle has area $A = 24$ cm² and length $l = 6$ cm. Using $A = lw$, find its width $w$.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 4 24 = 6w, so w = 24 ÷ 6 = 4. Check: 6 × 4 = 24. ::: :::keep-together ### A plumber **A plumber charges $C = 5h + 20$ dollars. A job cost 45 dollars. How many hours, $h$?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 5 45 = 5h + 20, so 5h = 25 and h = 5. Check: 5 × 5 + 20 = 45. ::: :::keep-together ### Find the mistake **Mia finds the width of a rectangle. Click the line where she goes wrong.** ``` 1 P = 2l + 2w, with P = 20 and l = 6. 2 20 = 2 × 6 + 2w 3 20 = 12 + 2w 4 8 = 2w 5 w = 16 ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 5 Line 5: 2w = 8 means 2 × w = 8. To undo × 2, divide by 2: w = 4. Mia multiplied again. Check: 2 × 6 + 2 × 16 = 44, not 20. ::: :::typical ## Other unknowns **A car trip.** Each task loads a new formula. Write the equation first, solve it, then set the board to check. **A different unknown.** **Going down.** **A box.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Find the unknown :::keep-together ### Tickets **Concert tickets cost $T = 12n + 6$ dollars. An order cost 66 dollars. How many tickets?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 5 66 = 12n + 6, so 12n = 60 and n = 5. ::: :::keep-together ### Substitute or solve? **Using $T = 12n + 6$, what do 3 tickets cost?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 42 T = 12 × 3 + 6 = 42 dollars. Here you know n, so you just substitute. ::: :::keep-together ### A bike ride **Average speed is $s = d / t$. A cyclist rides at 20 km/h for $t = 3$ hours. How far, $d$, in km?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 60 20 = d ÷ 3, so d = 20 × 3 = 60 km. ::: :::keep-together ### Review **Review: solve $4x - 3 = 17$.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 5 4x = 20, so x = 5. Check: 4 × 5 − 3 = 17. ::: ## Make: a garden bed The bed must use exactly 24 m of edging, and be twice as long as it is wide. Stretch: with the same 24 m, which rectangle gives the biggest area? 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 T = 15w + 30 dollars for w weeks. Someone paid 120 dollars. How many weeks? Check your answer.** {{lines-3}} ::: :::teacher-note **Answer:** 120 = 15w + 30, so 15w = 90 and w = 6. Check: 15 × 6 + 30 = 120. :::