007i - Estimating Square Roots

Lesson 7 — Estimating and Locating Square Roots on a Number Line

Strand: Number | Descriptor: AC9M8N01 | Duration: 45 minutes

Learning Intentions

• To estimate the value of the square root of a non-perfect square using known perfect squares.
• To locate irrational square roots accurately on a number line.

Success Criteria

I can:

1. Identify the two consecutive perfect squares that a given number lies between.
2. Estimate a square root to one decimal place using a “trial and refine” method.
3. Locate square roots accurately on a number line, using estimates.
4. Check that an estimate is reasonable by squaring it.

Warmup

(5 minutes — mini whiteboards, rapid recall)

1. State the squares of the whole numbers from to .
2. Which of these is a perfect square: , , ?
3. Between which two whole numbers does lie? How do you know, without a calculator?
4. Is closer to or closer to ? Explain your reasoning.

Teacher note: Question 4 is the hook — most students will guess “halfway,” which is imprecise. Activity 1 formalises a better method.

Activities

Activity 1 — Explicit Instruction: the Bracketing Method (10 min)

I do: Estimate step by step.

Step 1 — bracket between perfect squares.

Step 2 — judge closeness. is closer to than to (; ), so should be a little past the midpoint, closer to than to .

Step 3 — refine by testing decimals.

So , to one decimal place.

We do: Together estimate to one decimal place, following the same three steps.

You do: Estimate and to one decimal place, showing the bracketing perfect squares and one refining test.

(Answers: (since , ); (since , ).)

Activity 2 — Guided Practice: Placing Square Roots on a Number Line (10 min)

Draw a number line from to , marked at every whole number.

I do: Place , , and accurately, using the bracketing method to justify each position before marking it.

We do: Together place and .

You do: On your own number line (0 to 10), place: , , , . For each, write the bracketing perfect squares used.

(Approximate positions: ; ; ; .)

Activity 3 — Applied Task: Estimation in Context (14 min)

Pairs.

Problem 1. A square garden bed has an area of . Estimate the side length of the garden bed to one decimal place, and explain how you would check your estimate using a tape measure and squaring.

Problem 2. Two square offcuts of carpet have areas and . Estimate each side length to one decimal place. Which side length is closer to a whole number of metres?

Problem 3 (harder). A rectangular TV screen has a diagonal length, in inches, equal to . Without a calculator, estimate the diagonal length to the nearest whole inch, and decide whether this screen would be advertised as a “44-inch” or “45-inch” screen.

Socratic scaffolding for Problem 3:

Prompt	Purpose
Understand the problem	We need a whole-number estimate for , accurate enough to choose between two nearby advertised sizes.
Devise a plan	Find two perfect squares that bracket tightly, then refine using a suitable multiple of as a starting trial.
Carry out the plan — find a rough bracket	and , so is between and — too wide to be useful yet. Try closer values: and . Since , is between and .
Judge closeness	; . is closer to , so is closer to .
Carry out — refine	; . So .
Looking back	Since rounds to , the screen would most likely be advertised as a “45-inch” screen. Check: is sensibly between the two bracketing perfect-square roots, and ? Yes.

Answers: Problem 1 — m; check by measuring m and squaring: , close to . Problem 2 — m; m; neither is very close to a whole number, but is slightly closer to a whole number than . Problem 3 — , rounding to a “45-inch” screen.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. Between which two consecutive whole numbers does lie?
2. Estimate to one decimal place, showing a refining calculation.
3. A square field has an area of . Estimate the side length to one decimal place.
4. Explain why must be closer to than to .

Answers: 1. Between and , since and . 2. (since , ). 3. m. 4. and ; while , so is closer to than to , meaning is closer to than to .

Common Misconceptions

Misconception	How to pre-empt it
Assuming a square root is always exactly halfway between its bracketing whole numbers.	Compare the gaps to each perfect square numerically (as in CFU Q4) — the square root is not linearly halfway, since squaring is not a linear operation.
Bracketing with the wrong perfect squares, e.g. placing between and instead of and .	Insist on listing consecutive perfect squares in order and checking the target number falls strictly between adjacent ones.
Believing must give a “nice” decimal, and rounding too early during refinement.	Show that refining requires testing at least two trial decimals and comparing which is closer, not stopping after one guess.
Forgetting to check the estimate by squaring it back.	Make the final squaring check a required last line of working in every estimation problem.
Confusing “closer to the smaller perfect square” with “closer to the smaller whole number” when the target is near the top of its bracket.	Use the numeric gap comparison (as modelled in Activity 1, Step 2) rather than a visual guess.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). Between which two consecutive integers does lie?

Answer

and , and , so lies between and .

E2 (Kangaroo style). Estimate to the nearest whole number.

Answer

: , , so .

: , , so .

Sum .

E3 (Challenge). A student estimates . Use squaring to check this estimate, then improve it to one decimal place.

Answer

— close but slightly high, since . Try (too low) and . So a better one-decimal-place estimate is still rounds correctly, but shows it sits just under .

E4 (Investigation). Without a calculator, decide which is larger: , or .

Answer

They are equal. , and squaring gives , the same as . Since both are positive and have equal squares, they must be equal: .

Homework

1. Between which two consecutive whole numbers does each lie? (a) (b) (c) .
2. Estimate each to one decimal place, showing a refining calculation: (a) (b) .
3. A square rug has an area of . Estimate its side length to one decimal place.
4. Place these values on a single number line from to : , , , .
5. Reasoning. Explain why is much closer to than is to , even though both and sit near the top of their bracketing range.
6. Challenge. A rectangular field has length twice its width, and its diagonal is m. Using the bracketing method, estimate the diagonal to one decimal place, and then estimate the width, given that (width).

Answers: 1(a) Between and . (b) Between and . (c) Between and . 2(a) (since ). (b) (since ). 3. m. 4. Approximate order: , , , . 5. and is only away, an extremely small gap relative to the bracket; and is away too, but the relative closeness depends on the size of the bracketing interval and how squaring compresses values near larger numbers — near , small changes in the root correspond to larger changes in the square, so sits very close to . 6. , , so m; width, so width m.