010i - Problem Solving with Squares and Roots

Lesson 10 — Problem Solving and Consolidation: Squares and Square Roots

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

Learning Intentions

• To use squares and square roots to solve practical problems.
• To consolidate the connection between place value, prime factorisation and perfect squares.

Success Criteria

I can:

1. Choose whether a problem requires squaring or square-rooting, and justify the choice.
2. Solve area-and-side-length problems in context, with correct units.
3. Apply the order of operations to expressions involving squares and roots.
4. Explain my reasoning clearly in words.

Warmup

(6 minutes — “Always, sometimes, never”, pairs)

Decide whether each statement is always, sometimes, or never true. Give a supporting example or counterexample.

1. The square of a whole number is larger than the number itself.
2. The square root of a number is smaller than the number.
3. A perfect square ends in , , or .
4. The square of an odd number is odd.

Answers: 1. Sometimes — true for , but and . 2. Sometimes — true for ; . 3. Never — perfect squares end only in . 4. Always — odd × odd = odd.

Discussion of 3: Have students square through and read off the units digits: . This gives a quick screening test for perfect squares.

Activities

Activity 1 — Order of Operations with Squares and Roots (10 min)

Explicit instruction, then practice.

Remind students that squares and roots sit at the same level as other indices — they are evaluated after brackets, before multiplication and division.

I do:

Critical contrast to display side by side:

You do:

5. versus

Activity 2 — Applied Problems (14 min)

Pairs. Every answer must carry correct units and a one-sentence justification.

Problem 1. A square tile has an area of . What is the perimeter of the tile?

Problem 2. A square rug is laid centrally in a square room of side m, leaving a border of m all the way around. What is the area of the rug? What area of floor is left uncovered?

Problem 3. A school wants to plant seedlings in a square array. How many rows are needed? If they only have seedlings, what is the largest complete square array they can plant, and how many are left over?

Problem 4. A square photograph of area is placed in a square frame whose outer edge is cm beyond the photo on every side. What is the area of the frame’s visible border?

Socratic scaffolding for Problem 4:

Prompt	Purpose
Understand: what is being asked?	The area of the border only, not the whole frame.
Draw a diagram. What do you label?	The photo’s side, and the frame’s outer side.
What is the photo’s side length?	cm.
How much wider is the frame on each side?	cm on the left and cm on the right — so cm in total.
So what is the outer side length?	cm.
Devise a plan for the border area	Whole area minus photo area.
Carry it out	.
Looking back	Is the answer sensible? The border is a thin ring — is plausible against a total of ? Check by decomposing the border into 4 rectangles and 4 corner squares: ✓

Answers: 1 — side cm, perimeter cm. 2 — rug side m, area ; uncovered . 3 — rows; with , the largest square is , leaving seedlings. 4 — .

Checks for Understanding

(6 minutes — exit ticket, collected)

1. Evaluate .
2. A square courtyard has area . Find its perimeter.
3. Explain why is not equal to .
4. Is a perfect square? Use the units-digit test and explain.
5. Reasoning. A square pond of side m sits inside a square garden of side m. Write, in a single expression, the grassed area, then evaluate it.

Answers: 1. ; 2. Side m, perimeter m; 3. but — the square root of a sum is not the sum of the roots; 4. No — it ends in , and perfect squares end only in ; 5. .

Common Misconceptions

Misconception	How to pre-empt it
.	Display the numerical counterexample prominently. Revisit whenever brackets appear.
.	Same treatment: .
Answering the sub-question (giving the side length when the perimeter is asked for).	Build “re-read the question” into the Looking Back step. Require units in every final answer.
Adding only cm (not ) to a border dimension.	Always require a labelled diagram before calculating for border problems.
Forgetting units, or using cm instead of cm².	Insist that every area answer carries squared units and every length answer does not.
Treating “largest square array from 380” as rounded up.	Emphasise rounding down — you cannot plant a partial row. .

Enrichment — Competition-Style Problems

E1 (AMC Junior style). A square has area and perimeter , and numerically. What is the side length?

Answer

Since , . Check: area , perimeter ✓

E2 (Kangaroo style). The area of a square is . What is its perimeter?

Answer

Area , so the side is cm and the perimeter is cm. (A deliberate two-step trap — students often stop at .)

E3 (Challenge). How many squares of any size are there on a standard chessboard?

Answer

There are squares of size , of size , and so on down to of size .

E4 (Challenge). A number is both a perfect square and a perfect cube. What is the smallest such number greater than ?

Answer

It must be a perfect sixth power: . Indeed .

E5 (Investigation). Two squares have areas and . A third square has area equal to their sum. Find its side length. What do you notice about the three side lengths? (This previews Pythagoras.)

Answer

, so the side is cm. The sides are , , — a Pythagorean triple. The two smaller squares’ areas sum exactly to the largest.

Homework

1. Evaluate: (a) (b) (c) (d) .
2. A square swimming pool has an area of . Find (a) its side length (b) its perimeter.
3. A square picture of side cm is mounted on a square card with a cm border all around. Find the area of the card and the area of the visible border.
4. A farmer plants trees in a square grid. How many rows are there? If trees die, can he still form a complete square grid? Explain.
5. State whether each could be a perfect square, using the units-digit test only: (a) (b) (c) (d) .
6. Reasoning. Explain why doubling the side length of a square multiplies its area by , not by . Use a diagram.
7. Challenge. The difference between the areas of two squares is , and their side lengths are consecutive whole numbers. Find both side lengths.

Answers: Q1 — (a) (b) (c) (d) . Q2 — (a) m (b) m. Q3 — card side cm, area ; border . Q4 — rows; with trees, is the largest complete grid, leaving . Q5 — (a) possible (ends in 4; in fact ) (b) no (ends in 7) (c) no (ends in 3) (d) possible (ends in 6; in fact ). Q7 — has no whole-number solution, since is even and is always odd — a good “no solution” discussion. Teacher variant: use , giving sides and .