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# Between the squares

Number Foundations · Squares, powers and primes

<!-- Speaker note: 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). About 40 minutes (Predict and Run 8 min · Investigate 12 min · Modify 10 min · Practise and Make 10 min). -->

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## Today

**We are learning to** 43 is not a perfect square, but it sits between 36 and 49, so √43 is between 6 and 7. Closer to 49 or to 36? Square a decimal to find out: 6.6² ≈ 43.6.

- 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.

<!-- Speaker note: Before class: Do the Run board yourself first. Students need the perfect squares to 144 (previous lesson) and multiplying decimals (calculators are fine for squaring tenths). The one idea: find the perfect squares either side, take their roots, and the answer lies between. The scaffold matters: without it students guess, or halve (Noah's mistake). The reasoning check never marks the free working; only the reason for the key step is chosen and marked. -->

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## Why it matters

> 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.

<!-- Speaker note: Most numbers are not perfect squares, but they still have square roots. $\sqrt{43}$ is the side of a square with area 43: no whole number, but somewhere between two. The method: find the perfect squares either side. $36 < 43 < 49$, and $\sqrt{36} = 6$, $\sqrt{49} = 7$, so $\sqrt{43}$ is between 6 and 7. Then try decimals to get closer. -->

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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

**$\sqrt{43}$ is between which two whole numbers?**

- 6 and 7
- 3 and 4
- 40 and 50
- 21 and 22

<!-- Speaker 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. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: students who chose 21 and 22 halved. Ask them to square 21. -->

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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%2Fnumber%2Fbetween-squares%2Fstep-2&code=widget%3A+square%0Aslider+s%3A+1%2C+9%2C+6%2C+0.1%0Atarget%3A+43%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/number/between-squares/?full=step-2#step-2)

<!-- Speaker note: The board says how far your square is from 43, over or under. -->

<!-- Speaker note: Teacher: ask whether 6.6 is a little over or under, and how they decided between 6.5 and 6.6. -->

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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 43 is between them. 21 and 22 come from halving: 21 × 21 = 441.

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

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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>

## Bound it, then get closer

Here is the method for $\sqrt{43}$.

Now try one: between which whole numbers is $\sqrt{70}$?

$$36 < 43 < 49$$

1) $6 < \sqrt{43} < 7$ — Take their roots: √43 is between 6 and 7.
2) $6.5^2 = 42.25 \quad 6.6^2 = 43.56$ — Square some tenths. 6.5 is under and 6.6 is over.
3) $\sqrt{43} \approx 6.6$ — 43.56 is closer to 43 (0.56 over) than 42.25 (0.75 under).

<!-- Speaker note: Model this live on the board before revealing anything: write each line and say why, pausing to ask the class what comes next. Then click to reveal the lines and compare. Script. Start: write 36 < 43 < 49 and say "Find the perfect squares either side of 43." Line 1: write 6 < 43 < 7 and say "Take their roots: √43 is between 6 and 7." Line 2: write 6.5^2 = 42.25 6.6^2 = 43.56 and say "Square some tenths. 6.5 is under and 6.6 is over." Line 3: write 43 6.6 and say "43.56 is closer to 43 (0.56 over) than 42.25 (0.75 under)." -->

<!-- Speaker note: Teacher: chant the routine: squares either side, roots, between. -->

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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>

## Bound it, then get closer

**$\sqrt{70}$ is between 8 and which whole number?**

* Answer: 9

<!-- Speaker note: 8² = 64 and 9² = 81, and 70 is between them, so √70 is between 8 and 9. -->

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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

**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.
```

* Answer: line 2

<!-- Speaker note: Line 2: the square root is not half (25 × 25 = 625). 7² = 49 and 8² = 64, 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>

## 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: $9^2 = 81$ and $10^2 = 100$ (these are the perfect squares either side of 90); $81 &lt; 90 &lt; 100$, so $9 &lt; \sqrt{90} &lt; 10$ (a bigger square has a bigger side)

<!-- Speaker note: 81 < 90 < 100, so the side is between 9 m and 10 m (about 9.5 m: 9.5² = 90.25). -->

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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>

## One change at a time



- **Twenty:** Each task changes one thing.
- **A bigger root:** 
- **Backwards: from the perimeter:** 
- **Perfect or not:** 

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/number/between-squares/?full=step-4#step-4)

<!-- Speaker note: What this step is for: One change at a time: a new target on the board, a bigger root, an area from a perimeter (backwards), then which numbers are perfect squares at all. Hints, in order: 4² = 16 and 5² = 25, so start between 4 and 5. / Compare how far 4.4² and 4.5² are from 20. For students who need more support: Keep the list of squares to 144 visible and use a number line marked with 36, 43 and 49. For early finishers: Between which two whole numbers is √1000? Estimate it to one decimal place, then check with a calculator. The hard part, on purpose: Choosing between two neighbouring decimals (6.5 or 6.6). Ask which square is closer to the target and by how much, and let them compare the two distances. When students get stuck: A student halves the number (Noah's mistake). Ask: Square your answer. Do you get the number back? Fix: The root times itself gives the number; use the squares either side. / A student finds the interval but cannot get closer. Ask: Square 6.5. Is it over or under 43? Fix: Square tenths and compare the distances. / A student mixes up side and area in a problem. Ask: Is this number a length or an area? Fix: Root for area to side; square for side to area. -->

<!-- Speaker note: Teacher: in the perimeter task, ask which is known first: the side or the area? -->

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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>

## To one decimal place

**Estimate $\sqrt{30}$ to one decimal place. (5² = 25 and 6² = 36.)**

* Answer: 5.5

<!-- Speaker note: 5.4² = 29.16 and 5.5² = 30.25: 30.25 is closer to 30, so √30 ≈ 5.5. -->

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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>

## Between

**$\sqrt{150}$ is between which two whole numbers?**

- 15 and 16
- 75 and 76
- 11 and 12
- 12 and 13

* Answer: 12 and 13

<!-- Speaker note: 12² = 144 and 13² = 169. -->

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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>

## 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: 32

<!-- Speaker note: Side = √64 = 8 m. Perimeter = 4 × 8 = 32 m. -->

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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: squares

**Review: work out $12^2$.**

* Answer: 144

<!-- Speaker note: 12 × 12 = 144. -->

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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>

## Make: an area in the thirties

Then find the smallest and the biggest side (in tenths) that work.

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/number/between-squares/?full=step-6#step-6)

<!-- Speaker note: What this step is for: Choosing a side whose square lands between 30 and 40 turns the bounding argument round: the side must be between √30 and √40. Hints, in order: 5² = 25 is too small and 6² = 36 is inside. Are there others? / Step in tenths: 5.5² = 30.25. -->

<!-- Speaker note: Teacher: collect sides that work and order them; the ends approach √30 and √40. -->

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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 √55? Give a reason.**

* 7 and 8, because 7² = 49 and 8² = 64, and 55 is between them.

<!-- 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: Next: exponent notation, prime factors and HCF and LCM (AC9M7N02). Where it leads: Pythagoras (Year 8) gives lengths like √20, and estimating them first checks a calculator answer. -->