--- title: "Irrational numbers on the line" subtitle: "Irrational numbers and index laws · Irrational numbers · Lesson 4" --- # Irrational numbers on the line **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** A carpenter needs a brace across a 1 m by 2 m gate: √5 metres. Where is √5 on a tape measure, and how close is close enough? :::teacher-note **In one sentence:** Students place square roots and π on the number line using the perfect squares either side and decimal squares, and decide when an approximation is enough (AC9M8N01, node 2). **Students should be able to say:** I can find the two whole numbers a square root is between. I can place a root or π to the nearest tenth. I can tell an exact value from an approximation. **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 :::keep-together **Between which two whole numbers is √40?** ☐ 39 and 41 ☐ 20 and 21 ☐ 6 and 7 ☐ 4 and 5 My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** 6 and 7: 6² = 36 and 7² = 49, and 40 is between them. (Halving 40 gives 20, but 20² = 400.) ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Squares either side :::keep-together ### Squares either side **√50 is between two whole numbers. What is the smaller one?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 7 7² = 49 and 8² = 64, so √50 is between 7 and 8 (just over 7). ::: :::keep-together ### Find the mistake **Ben places √50 on a number line. Click the line where his reasoning goes wrong.** ``` 1 Ben places √50 on a number line. 2 Half of 50 is 25, so √50 is about 25. 3 He marks it between 25 and 26. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 2 Line 2: a square root is the number that multiplies by itself to give 50, not half of it. 7² = 49, so √50 is just over 7. ::: :::keep-together ### Real numbers **Which is true of the number line?** ☐ Every point is a real number, and there are rational and irrational numbers between any two points ☐ Irrational numbers sit between the whole numbers only ☐ Only rational numbers have a place on it ☐ √2 has no exact point ::: :::teacher-note **Answer:** Every point is a real number, and there are rational and irrational numbers between any two points The real numbers fill the whole line, rationals and irrationals mixed together. ::: :::typical ## More roots, and π **Root 10.** **π.** **In order.** **Good enough.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Placing roots :::keep-together ### Bounds **√20 is between two whole numbers. What is the smaller one?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 4 16 < 20 < 25, so √20 is between 4 and 5. ::: :::keep-together ### Compare **Which is bigger: √8 or 2.8?** ☐ They are equal ☐ √8 ☐ 2.8 ::: :::teacher-note **Answer:** √8 2.8² = 7.84 < 8, so √8 > 2.8 (√8 = 2.828…). ::: :::keep-together ### Round **Write √2 rounded to 2 decimal places.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 1.41 √2 = 1.41421…, so 1.41. ::: :::keep-together ### Review: irrational **Review: which number is irrational?** ☐ √4 ☐ 0.5 ☐ √12 ☐ 1/3 ::: :::teacher-note **Answer:** √12 12 is not a perfect square. ::: ## Make: root seven Then mark √7 on a sketch of the number line between 2 and 3. 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 √70? Which is it nearer?** {{lines-3}} ::: :::teacher-note **Answer:** 64 < 70 < 81, so between 8 and 9, nearer 8 (√70 ≈ 8.37). :::