---
marp: true
theme: default
html: true
---

# Squares and square roots

Number Foundations · Squares, powers and primes

<!-- Speaker note: Students grow a square on a grid to see a square number as an area, learn the perfect squares to 144, and use the square root as the inverse, the side of a square with a given area (AC9M7N01, nodes 1 and 2). About 40 minutes (Predict and Run 8 min · Investigate 12 min · Modify 10 min · Practise and Make 10 min). -->

---

## Today

**We are learning to** a square number is the area of a square: 6² = 6 × 6 = 36. The square root undoes it: √36 = 6, the side of a square with area 36.

- I can work out the square of a number and list the perfect squares to 144.
- I can find the square root of a perfect square and explain what it means.
- I can find the side and the perimeter of a square from its area.

<!-- Speaker note: Before class: Do the Run board yourself first. Students need times tables and area as side × side (Year 6). The one idea: squaring is a number times itself (the area of a square), and the square root goes back (the side). The common mistakes are squaring as doubling (5² = 10) and rooting as halving (√36 = 18); both are planted. Recall of the perfect squares to 144 is the hinge for the next lesson (roots between whole numbers). -->

---

## Why it matters

> A square patio uses 49 tiles. How long is each side? Squares and square roots go back and forth between a side and an area.

<!-- Speaker note: Multiply a number by itself and you get its **square**: $6 \times 6 = 36$, written $6^2$ ("6 squared"). The name comes from the shape: a square with side 6 covers 36 unit squares. Numbers like 1, 4, 9, 16, 25 and 36 are **perfect squares**. The **square root** goes the other way: $\sqrt{36} = 6$, because $6 \times 6 = 36$. It is the side of a square with that area. -->

---

<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

**What is $5^2$?**

- 7
- 10
- 25
- 52

<!-- Speaker note: What happens: 25. 5² means 5 × 5: a square with side 5 has 5 rows of 5 unit squares. 10 is 5 × 2, double, not squared. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: take a show of hands. Students who chose 10 doubled. -->

---

<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%2Fsquares-and-roots%2Fstep-2&code=widget%3A+square%0Aslider+s%3A+1%2C+12%2C+3%0A&fresh=1}}

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

<!-- Speaker note: $s$ is the side. The board counts the unit squares: the area. -->

<!-- Speaker note: Teacher: ask how the board knew the area. Count one row, then the rows. -->

<style scoped>pre, marp-pre { font-size: 0.5em; line-height: 1.25; } section { font-size: 25px; }</style>

---

<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?

* 25. 5² means 5 × 5: a square with side 5 has 5 rows of 5 unit squares. 10 is 5 × 2, double, not squared.

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

---

<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>

## Undo the square

The square root asks: what number, times itself, makes this?

Now try one: $\sqrt{81}$.

$$6^2 = 6 \times 6 = 36$$

1) $\sqrt{36} = 6$ — The square root goes back: 6 × 6 = 36, so the side is 6.
2) $1,\ 4,\ 9,\ 16,\ 25,\ 36,\ 49,\ 64,\ 81,\ 100,\ 121,\ 144$ — The perfect squares from 1² to 12². Learn them.

<!-- 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 6^2 = 6 × 6 = 36 and say "Squaring: multiply the number by itself (a square with side 6)." Line 1: write 36 = 6 and say "The square root goes back: 6 × 6 = 36, so the side is 6." Line 2: write 1,4,9,16,25,36,49,64,81,100,121,144 and say "The perfect squares from 1² to 12². Learn them." -->

<!-- Speaker note: Teacher: write 6² = 36 and √36 = 6 on the board as a pair that undo each other. -->

---

<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>

## Undo the square

**Work out $\sqrt{81}$.**

* Answer: 9

<!-- Speaker note: 9 × 9 = 81, so √81 = 9. -->

---

<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

**Ava finds the area of a square with side 5 cm. Click the line where her working goes wrong.**

```
 1  A square has side 5 cm. Find its area.
 2  Area = side squared.
 3  5² = 5 × 2 = 10
 4  So the area is 10 cm².
```

* Answer: line 3

<!-- Speaker note: Line 3: 5² means 5 × 5 = 25, not 5 × 2. The area is 25 cm². -->

---

<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>

## Why it works

**Why is $\sqrt{64} = 8$?**

- Because $8 \times 8 = 64$
- Because 64 is even
- Because $64 \div 8 = 8$ and $64 \div 2 = 32$

* Answer: Because $8 \times 8 = 64$

<!-- Speaker note: The square root of 64 is the number that times itself makes 64: 8 × 8 = 64. -->

---

<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



- **The biggest in the list:** Each task changes one thing.
- **A root:** 
- **A side from an area:** 
- **Two steps: the perimeter:** 
- **Not a perfect square:** 

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

<!-- Speaker note: What this step is for: One change at a time: the biggest square in the list, a root, a side from an area in context, a perimeter (two steps), then a number that is not a perfect square. Hints, in order: Try sides of 10, 11 and 12. For students who need more support: Keep the list of squares to 144 visible, and use the grid board for every root. For early finishers: The last digits of square numbers are 0, 1, 4, 5, 6 or 9. Use this to show that 2023 and 1458 cannot be perfect squares. The hard part, on purpose: Telling squaring from doubling under time pressure. Ask students to draw the square for 5² before they answer, rather than correcting the 10. When students get stuck: A student writes 5² = 10 (Ava's mistake). Ask: Draw a square with side 5. How many unit squares? Fix: Squared means times itself: 5 × 5. / A student halves for a root (√36 = 18). Ask: Does 18 × 18 make 36? Fix: The root times itself must give the number back. / A student gives the side when the question asks for a perimeter. Ask: How many sides go round the garden? Fix: Find the side with the root, then multiply by 4. -->

<!-- Speaker note: Teacher: before the perimeter task, ask what the 49 is: an area or a length? -->

---

<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>

## Square

**Work out $9^2$.**

* Answer: 81

<!-- Speaker note: 9 × 9 = 81. -->

---

<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>

## Root

**Work out $\sqrt{100}$.**

* Answer: 10

<!-- Speaker note: 10 × 10 = 100. -->

---

<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>

## A side

**A square rug has area 9 m². How long is each side, in metres?**

* Answer: 3

<!-- Speaker note: √9 = 3, so each side is 3 m. -->

---

<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: integers

**Review: work out $-4 \times 3$.**

* Answer: -12

<!-- Speaker note: 4 × 3 = 12, and the signs differ, so −12. -->

---

<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: a bigger square

Then write its square root, and check by multiplying.

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

<!-- Speaker note: What this step is for: Choosing a perfect square past 100 and naming its root extends the list beyond 12² by the same reasoning. Hints, in order: Try a side of 13 or 14. / The area is side × side. -->

<!-- Speaker note: Teacher: collect squares past 144 and their roots; look at the last digits. -->

---

<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

**A square has area 81 cm². Find its side and its perimeter.**

* Side √81 = 9 cm; perimeter 4 × 9 = 36 cm.

<!-- 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: Between the squares (nodes 3 and 4), then exponent notation (AC9M7N02). Where it leads: Pythagoras (Year 8) uses squares and roots on every side of a right-angled triangle. -->