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# Irrational numbers on the line

Irrational numbers and index laws · Irrational numbers

<!-- Speaker note: 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). About 35 minutes (Predict and Run 8 min · Investigate 8 min · Modify 10 min · Practise and Make 9 min). -->

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## Today

**We are learning to** every point on the number line is a real number, rational or irrational. To place √30, find the perfect squares either side: 25 < 30 < 36, so √30 is between 5 and 6, nearer 5. A rounded value like 5.48 is close, not exact.

- 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.

<!-- Speaker note: Before class: Do the Run board yourself first. Students need the previous lesson and square numbers to 144. The one idea: perfect squares either side bound a root (25 < 30 < 36 so 5 < √30 < 6), and decimal squares narrow it. The common mistake is halving (Ben's mistake). The Run and Modify boards are number lines with a point to drag to the nearest tenth; the Make board zooms and counts wrong presses. Wider: the real numbers are dense: between any two there are rationals and irrationals. -->

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## Why it matters

> 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?

<!-- Speaker note: The **real numbers** are every point on the number line: the rational numbers and the irrational ones together. Between any two numbers, however close, there are both. To place a square root, find the **perfect squares** either side: $$25 < 30 < 36 \quad\Rightarrow\quad 5 < \sqrt{30} < 6$$ 30 is a little nearer 25 than 36, so $\sqrt{30}$ is a bit under 5.5 (check: $5.5^2 = 30.25$). To get closer, square decimals: $5.4^2 = 29.16$ and $5.5^2 = 30.25$; then $5.47^2 = 29.92$ and $5.48^2 = 30.03$, so $\sqrt{30} \approx 5.48$. A rounded value is an **approximation**; the exact value is $\sqrt{30}$. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Predict

**Between which two whole numbers is √40?**

- 39 and 41
- 20 and 21
- 6 and 7
- 4 and 5

<!-- Speaker note: What happens: 6 and 7: 6² = 36 and 7² = 49, and 40 is between them. (Halving 40 gives 20, but 20² = 400.) Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: ask what 15² is. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Freal-numbers%2Firrationals-on-the-line%2Fstep-2&code=widget%3A+numberline%0Arange%3A+0%2C+5%0Asnap%3A+0.1%0Apoint+P%3A+0%2C+drag%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/real-numbers/irrationals-on-the-line/?full=step-2#step-2)

<!-- Speaker note:  -->

<!-- Speaker note: Teacher: ask how they know √5 is nearer 2 than 3. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## What happened?

* 6 and 7: 6² = 36 and 7² = 49, and 40 is between them. (Halving 40 gives 20, but 20² = 400.)

<!-- Speaker note: Reveal after running. Ask who predicted it, and what made them think so. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Squares either side

**√50 is between two whole numbers. What is the smaller one?**

* Answer: 7

<!-- Speaker note: 7² = 49 and 8² = 64, so √50 is between 7 and 8 (just over 7). -->

<!-- Speaker note: Teacher: ask Ben to check 25 × 25. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## 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.
```

* Answer: line 2

<!-- Speaker note: 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. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## 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

* Answer: Every point is a real number, and there are rational and irrational numbers between any two points

<!-- Speaker note: The real numbers fill the whole line, rationals and irrationals mixed together. -->

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## More roots, and π



- **Root 10:** 
- **π:** 
- **In order:** 
- **Good enough:** 

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/real-numbers/irrationals-on-the-line/?full=step-4#step-4)

<!-- Speaker note: What this step is for: Placing √10 and π, ordering a root among decimals, and deciding when an approximation is good enough. Hints, in order: 9 < 10 < 16. / 3.1² = 9.61 and 3.2² = 10.24. For students who need more support: A list of squares 1 to 144, and a number line from 0 to 12 to mark them on. For early finishers: Find a rational number and an irrational number between 1.41 and 1.42. The hard part, on purpose: Using squares, not halves. Ask students to write the perfect squares either side before they guess. When students get stuck: A student halves the number. Ask: What times itself gives 50? Fix: Find the perfect squares either side. / A student cannot list perfect squares. Ask: What is 6 × 6? 7 × 7? Fix: Write the squares to 12² down the side of the page. / A student treats 1.41 as exactly √2. Ask: What is 1.41²? Fix: 1.9881: close, not 2. 1.41 is an approximation. -->

<!-- Speaker note: Teacher: ask when 1.41 is good enough. -->

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## Bounds

**√20 is between two whole numbers. What is the smaller one?**

* Answer: 4

<!-- Speaker note: 16 < 20 < 25, so √20 is between 4 and 5. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Compare

**Which is bigger: √8 or 2.8?**

- They are equal
- √8
- 2.8

* Answer: √8

<!-- Speaker note: 2.8² = 7.84 < 8, so √8 > 2.8 (√8 = 2.828…). -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Round

**Write √2 rounded to 2 decimal places.**

* Answer: 1.41

<!-- Speaker note: √2 = 1.41421…, so 1.41. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Review: irrational

**Review: which number is irrational?**

- √4
- 0.5
- √12
- 1/3

* Answer: √12

<!-- Speaker note: 12 is not a perfect square. -->

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## Make: root seven

Then mark √7 on a sketch of the number line between 2 and 3.

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/real-numbers/irrationals-on-the-line/?full=step-6#step-6)

<!-- Speaker note: What this step is for: Finding √7 to 2 places without a wrong press makes them check each square before they choose. Hints, in order: 4 < 7 < 9. / 2.6² = 6.76 and 2.7² = 7.29. -->

<!-- Speaker note: Teacher: ask for the squares they checked. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Exit ticket

**Between which two whole numbers is √70? Which is it nearer?**

* 64 &lt; 70 &lt; 81, so between 8 and 9, nearer 8 (√70 ≈ 8.37).

<!-- Speaker note: Students answer on paper or a mini whiteboard before the reveal. Anyone who cannot answer needs the main idea again next lesson. Where this leads: This completes AC9M8N01. Where it leads: exact answers with Pythagoras, and surds (Year 10). -->