--- title: "Between the squares" subtitle: "Number Foundations · Squares, powers and primes · Lesson 5" --- # Between the squares **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** A square vegetable patch needs 43 m² of soil. No whole number squares to 43, but a builder still needs to know the side, to the nearest 10 cm. :::teacher-note **In one sentence:** Students bound a square root between two whole numbers using the perfect squares either side, estimate it to one decimal place by squaring tenths on a grid board, and solve area and perimeter problems with roots (AC9M7N01, nodes 3 and 4). **Students should be able to say:** I can say which two whole numbers a square root is between, with a reason. I can estimate a square root to one decimal place by squaring decimals. I can solve problems with the side, area and perimeter of a square. **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 Which perfect squares are near 43? :::keep-together **$\sqrt{43}$ is between which two whole numbers?** ☐ 6 and 7 ☐ 3 and 4 ☐ 40 and 50 ☐ 21 and 22 My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** 6 and 7. 6² = 36 and 7² = 49, and 43 is between them. 21 and 22 come from halving: 21 × 21 = 441. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## The squares either side :::keep-together ### Bound it, then get closer Here is the method for $\sqrt{43}$. 1. $36 < 43 < 49$ — Find the perfect squares either side of 43. 2. $6 < \sqrt{43} < 7$ — Take their roots: √43 is between 6 and 7. 3. $6.5^2 = 42.25 \quad 6.6^2 = 43.56$ — Square some tenths. 6.5 is under and 6.6 is over. 4. $\sqrt{43} \approx 6.6$ — 43.56 is closer to 43 (0.56 over) than 42.25 (0.75 under). Now try one: between which whole numbers is $\sqrt{70}$? **$\sqrt{70}$ is between 8 and which whole number?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 9 8² = 64 and 9² = 81, and 70 is between them, so √70 is between 8 and 9. ::: :::keep-together ### Find the mistake **Noah estimates $\sqrt{50}$. Click the line where his reasoning goes wrong.** ``` 1 Estimate the square root of 50. 2 The square root is half the number. 3 50 ÷ 2 = 25 4 So √50 is about 25. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 2 Line 2: the square root is not half (25 × 25 = 625). 7² = 49 and 8² = 64, so √50 is just over 7. ::: :::keep-together ### With reasons **A square garden bed has area 90 m². Between which two whole numbers of metres is its side? Write each step with its reason.** Answer: {{field-15}} ::: :::teacher-note **Answer:** $9^2 = 81$ and $10^2 = 100$ (these are the perfect squares either side of 90); $81 < 90 < 100$, so $9 < \sqrt{90} < 10$ (a bigger square has a bigger side) 81 < 90 < 100, so the side is between 9 m and 10 m (about 9.5 m: 9.5² = 90.25). ::: :::typical ## One change at a time **Twenty.** Each task changes one thing. **A bigger root.** **Backwards: from the perimeter.** **Perfect or not.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Roots between :::keep-together ### To one decimal place **Estimate $\sqrt{30}$ to one decimal place. (5² = 25 and 6² = 36.)** Answer: {{field-15}} ::: :::teacher-note **Answer:** 5.5 5.4² = 29.16 and 5.5² = 30.25: 30.25 is closer to 30, so √30 ≈ 5.5. ::: :::keep-together ### Between **$\sqrt{150}$ is between which two whole numbers?** ☐ 15 and 16 ☐ 75 and 76 ☐ 11 and 12 ☐ 12 and 13 ::: :::teacher-note **Answer:** 12 and 13 12² = 144 and 13² = 169. ::: :::keep-together ### Two steps **A square floor needs 64 square tiles, each 1 m by 1 m. How much skirting board goes round the edge, in metres?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 32 Side = √64 = 8 m. Perimeter = 4 × 8 = 32 m. ::: :::keep-together ### Review: squares **Review: work out $12^2$.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 144 12 × 12 = 144. ::: ## Make: an area in the thirties Then find the smallest and the biggest side (in tenths) that work. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **Between which two whole numbers is √55? Give a reason.** {{lines-3}} ::: :::teacher-note **Answer:** 7 and 8, because 7² = 49 and 8² = 64, and 55 is between them. :::