009i - Estimating Square Roots

Lesson 9 — Estimating and Calculating Square Roots

Strand: Number | Descriptor: AC9M7N01 | Duration: 45 minutes

Learning Intentions

• To estimate the square root of a non-perfect square by locating it between consecutive integers.
• To use prime factorisation to evaluate the square root of a larger perfect square.

Success Criteria

I can:

1. State which two consecutive whole numbers a given square root lies between.
2. Judge which of those two it is closer to, and justify the judgement.
3. Use prime factorisation to find the square root of a large perfect square.
4. Check the reasonableness of a calculator square root.

Warmup

(6 minutes — recall race, mini whiteboards)

1. Rapid-fire the squares: through .
2. Now the roots: , , , , .
3. Which two whole numbers does lie between? How do you know?

Answer to 3: and , and , so .

Activities

Activity 1 — Explicit Instruction: Trapping a Root between Integers (12 min)

Method, modelled explicitly:

1. Identify the nearest perfect square below the number.
2. Identify the nearest perfect square above the number.
3. The root lies between their roots.
4. Decide which end it is closer to by comparing distances.

I do: Estimate .

Which end? and . Since is closer to , is closer to . Estimate: about .

(Calculator check: .)

We do: , , .

A caution to raise: being closer to the lower square does not guarantee the root is below the midpoint, because squares are not evenly spaced. Students should treat these as estimates and check with a calculator where precision matters.

You do: Between which consecutive integers do these lie? , , , , .

(Answers: –; –; –; –; –.)

Activity 2 — Exact Roots via Prime Factorisation (10 min)

Connects Lesson 6 to Lesson 8. This gives an exact method for large perfect squares without a calculator.

I do: Find .

Check: ✓

Why it works: halving each even exponent splits the factorisation into two identical halves — exactly what a square root does.

We do: and .

You do: , , .

Activity 3 — Inquiry: how Good Can an Estimate Be? (8 min)

Pairs, calculators available for checking only at the end.

Estimate to one decimal place without a calculator. Then check.

Socratic scaffolding:

Prompt	Purpose
Understand: what is the unknown?	A number that, squared, gives about .
What squares do you already know near ?	; . So the root is between and .
Can you narrow it?	— just above . So the root is just below .
Devise a plan	Test .
Carry it out	. So .
Narrow further	; . Much closer to . Try .
Compute	. Try . So .
Looking back	Calculator gives — the estimate holds to one decimal place.

Discussion: Which method — trapping between integers, or prime factorisation — suits which kind of problem? (Prime factorisation is exact but only works for perfect squares; trapping works for any number but gives an approximation.)

Checks for Understanding

(5 minutes — exit ticket)

1. Between which two consecutive whole numbers does lie? Which is it closer to?
2. Use prime factorisation to find .
3. A calculator shows . Is this reasonable? Justify without recalculating.
4. Estimate to one decimal place.

Answers: 1. and (), closer to ; 2. , so ; 3. Yes — , so the root must lie between and ; 4. , ; is much closer to , and , so about .

Common Misconceptions

Misconception	How to pre-empt it
(halving).	Sanity check: , nowhere near . Always ask “what times itself?”
Assuming every square root is a whole number.	Deliberately mix perfect and non-perfect squares in every practice set.
Linear interpolation treated as exact, e.g. claiming exactly because is midway-ish.	Emphasise “estimate” language; compare with the calculator value and discuss the small error.
Halving the number instead of the exponents in the prime factorisation method.	Model the step "" aloud as “halve the exponent, not the base”.
Applying the prime factorisation method to a non-perfect square and forcing an answer.	If any exponent is odd, there is no whole-number root — return to estimation.
Believing is the same as .	The product rule holds; the sum rule does not. Test both with , .

Enrichment — Competition-Style Problems

E1 (Kangaroo style). How many whole numbers satisfy ?

Answer

means . Counting gives 24 values (or 25 if is allowed).

E2 (AMC Junior style). What is the value of ?

Answer

Equivalently .

E3 (Challenge). . What is ?

Answer

, so . In index form, .

E4 (Challenge). Between and , how many whole numbers lie?

Answer

and . The whole numbers are — 8 numbers.

E5 (Reasoning). Explain why cannot be a whole number, using prime factorisation.

Answer

. Both exponents are odd, so the factorisation cannot be split into two identical halves — hence is not a perfect square.

Homework

1. State the two consecutive whole numbers each root lies between: (a) (b) (c) (d) (e) .
2. For each in Q1, state which whole number it is closer to.
3. Use prime factorisation to evaluate exactly: (a) (b) (c) (d) .
4. Estimate to one decimal place, then check with a calculator: (a) (b) .
5. A square paddock has area . Find its side length exactly using prime factorisation.
6. Reasoning. Anya says . Test her claim and explain the result. Is it a coincidence?
7. Challenge. Find the smallest whole number that must be multiplied by to make it a perfect square.

Answers: Q1/Q2 — (a) –, closer to (b) –, closer to (c) –, closer to (d) –, closer to (e) –, closer to . Q3 — (a) (b) (c) (d) . Q4 — (a) (b) . Q5 — , side m. Q6 — and , so the claim is false; roots do not distribute over addition. Q7 — , so multiply by to give .