Between the squares

Classroom resources

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Students will be able to

• 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.

Timing

Predict and Run 8 min · Investigate 12 min · Modify 10 min · Practise and Make 10 min

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.

Running the lesson

1. Predict: students who chose 21 and 22 halved. Ask them to square 21.
2. Run: ask whether 6.6 is a little over or under, and how they decided between 6.5 and 6.6.
3. Investigate: chant the routine: squares either side, roots, between.
4. Modify: in the perimeter task, ask which is known first: the side or the area?
5. Make: collect sides that work and order them; the ends approach √30 and √40.

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

Answers to the checks

1. where is √43?: $\sqrt{43}$ is between which two whole numbers?
   6 and 7. 6² = 36 and 7² = 49, and 43 is between them. 21 and 22 come from halving: 21 × 21 = 441.
2. as close as you can: The dashed square has area 43. Change the side (in tenths) to make your square as close to 43 as you can. Was your prediction right?
   
   6.5² = 42.25 (0.75 under) and 6.6² = 43.56 (0.56 over), so 6.6 is closer: √43 ≈ 6.6.
3. Bound it, then get closer: $\sqrt{70}$ is between 8 and which whole number?
   9
   8² = 64 and 9² = 81, and 70 is between them, so √70 is between 8 and 9.
4. Find the mistake: Noah estimates $\sqrt{50}$. Click the line where his reasoning goes wrong.
   line 2
   Line 2: the square root is not half (25 × 25 = 625). 7² = 49 and 8² = 64, so √50 is just over 7.
5. 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.
   $9^2 = 81$ 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).
6. Twenty: The dashed square has area 20 m². Make your square as close to it as you can.
   
   4.4² = 19.36 (0.64 under), 4.5² = 20.25 (0.25 over): √20 ≈ 4.5.
7. A bigger root: $\sqrt{200}$ is between which two whole numbers?
   14 and 15
   14² = 196 and 15² = 225, and 200 is between them.
8. Backwards: from the perimeter: A square has perimeter 36 cm. What is its area, in cm²?
   81
   Side = 36 ÷ 4 = 9 cm. Area = 9² = 81 cm².
9. Perfect or not: Edge case: which of these is a perfect square?
   400
   20² = 400. A perfect square needs a whole-number root.
10. To one decimal place: Estimate $\sqrt{30}$ to one decimal place. (5² = 25 and 6² = 36.)
    5.5
    5.4² = 29.16 and 5.5² = 30.25: 30.25 is closer to 30, so √30 ≈ 5.5.
11. Between: $\sqrt{150}$ is between which two whole numbers?
    12 and 13
    12² = 144 and 13² = 169.
12. 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?
    32
    Side = √64 = 8 m. Perimeter = 4 × 8 = 32 m.
13. Review: squares: Review: work out $12^2$.
    144
    12 × 12 = 144.
14. an area in the thirties: Make a square whose area is between 30 and 40.
    
    Any side from 5.5 to 6.3 works: √30 ≈ 5.48 and √40 ≈ 6.32.

Solutions

One possible solution for each task. Other correct programs are fine.

one change at a time › Twenty

Open task: any program that does what the step asks.

one change at a time › A bigger root

Open task: any program that does what the step asks.

one change at a time › Backwards: from the perimeter

Open task: any program that does what the step asks.

one change at a time › Perfect or not

Open task: any program that does what the step asks.

an area in the thirties

Open task: any program that does what the step asks.

Stretch and support

Stretch: Between which two whole numbers is √1000? Estimate it to one decimal place, then check with a calculator.

Support: Keep the list of squares to 144 visible and use a number line marked with 36, 43 and 49.

Coach prompt (optional)

For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.

You are a patient maths coach for a Year 7 student in Australia. Help me understand the ideas in ONE lesson only, from the lesson "Between the squares".

The ideas:
- Locating non-perfect square roots: I can identify which two consecutive integers a square root falls between for numbers that are not perfect squares.
- Applying square roots to problems: I can Students use the relationship between perfect square numbers and square roots to solve practical problems, including finding the side length of a square given its area.

What understanding looks like:
- For any given number, correctly identifies the two consecutive perfect squares it lies between.
- States the corresponding square root interval (e.g. '√43 is between 6 and 7').
- Explains the reasoning, not just the answer.
- Given the area of a square, correctly determines the side length using square root reasoning.
- Solves a two-step problem involving squares and square roots in a practical context.
- Checks that their answer is reasonable (e.g. side length must be positive).

Mistakes students often make here:
- A student halves the number (Noah's mistake).
- A student finds the interval but cannot get closer.
- A student mixes up side and area in a problem.

How to coach me (rules from the Waypoints Kit coach):
- Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question.
- Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit).
- If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go.
- If I seem to be missing the idea that comes before this one (Square root as inverse operation), check it quickly, then come back.
- Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back.
- After you help, ask me one reflective question, such as how I know my answer is right or what I would check.
- Praise my effort and method, not my ability.
- You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so.
- Use simple words.
- Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "Between the squares".

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.

What each step is for

• where is √43?: Committing to an interval for √43 shows whether they use the squares either side, or halve, or think of 43's digits.
• as close as you can: Stepping the side in tenths until the square is as close to the target as possible estimates a root to one decimal place, and shows it is between the whole numbers found first.
• the squares either side: Bounding a root by the perfect squares either side gives an interval with reasons; Noah's halving is the squaring-as-doubling mistake run backwards.
• one change at a time: 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.
• roots between: An estimate to one decimal place, an interval, and a two-step problem, mixed; halving is the planted slip; the review keeps squares fresh.
• an area in the thirties: Choosing a side whose square lands between 30 and 40 turns the bounding argument round: the side must be between √30 and √40.

Where it 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.