008i - Perfect Squares and Square Roots

Lesson 8 — Perfect Squares and Square Roots

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

Learning Intentions

• To understand the relationship between perfect square numbers and square roots.
• To recall the perfect squares up to and their roots.

Success Criteria

I can:

1. Explain what makes a number a perfect square, using an area model.
2. Evaluate squares up to from memory.
3. Evaluate the square root of a perfect square, e.g. .
4. Describe squaring and square-rooting as inverse operations.

Warmup

(6 minutes — square tiles or grid paper)

Give pairs square tiles.

1. Build a square using tiles. What are its dimensions?
2. Build a square using tiles. And ?
3. Try to build a square using tiles. What happens?
4. List the tile counts, from smallest, that do form a perfect square.

Bridging question: If a square has area square units, what is its side length? How did you work it out?

Activities

Activity 1 — Explicit Instruction: Squares and the Area Model (10 min)

Definition. A perfect square is a number that can be written as a natural number multiplied by itself.

Reading: ” squared”, or ” to the power of two”. The name comes directly from the area of a square with side .

Build the reference table with the class — students should learn these:

Noticing prompt: Look at the differences between consecutive squares: What do you notice? (Consecutive odd numbers. Show this geometrically: adding an L-shaped border of tiles turns an square into an square.)

Activity 2 — Explicit Instruction: Square Roots (10 min)

Definition. The square root of a number is the value which, when squared, gives that number.

Present squaring and rooting as an inverse pair, using a function machine:

Note on notation (state briefly, do not labour it at Year 7): denotes the positive square root by convention. So , even though as well.

We do: Evaluate , , , , .

You do: Mixed practice.

Activity 3 — Inquiry: Which Numbers Are Perfect Squares? (10 min)

Pairs. Connects back to Lesson 6.

Here are the prime factorisations of six numbers. Without evaluating them, decide which are perfect squares.

Socratic scaffolding:

Prompt	Purpose
Understand: what does “perfect square” mean here?	The number equals something multiplied by itself.
Try a known case: what is the prime factorisation of ?	. And where .
What about ?	.
What do the exponents in these cases have in common?	They are all even.
Why would that matter?	To split the factorisation into two identical halves, each prime must divide evenly between them.
Test the conjecture on	Exponents and are odd — so not a perfect square. Check: , and , . ✓
State the rule	A number is a perfect square exactly when every exponent in its prime factorisation is even.

Answers: Perfect squares are , , , . Not perfect squares: , .

Extension: What is the smallest number you could multiply by to make it a perfect square? (By , giving .)

Checks for Understanding

(5 minutes — exit ticket)

1. Evaluate and .
2. Is a perfect square? Explain how you know.
3. A square has area . What is its side length?
4. A student writes . What has gone wrong?
5. Between which two consecutive whole numbers does lie? Explain.

Answers: 1. and ; 2. No — and , so falls between consecutive squares; 3. cm; 4. The student doubled instead of multiplying by itself: ; 5. Between and , since .

Common Misconceptions

Misconception	How to pre-empt it
(doubling instead of squaring).	Always read as “seven times seven” before evaluating. Reinforce with the area model — a square clearly holds more than tiles.
(halving instead of rooting).	Ask “what number times itself gives 16?” Never “what is half of 16?”
Believing is always smaller than — always true for , but confused at .	Note ; the root equals the number.
Reading as .	Evaluate both: , but . Roots do not distribute over addition.
Confusing with .	Expand both explicitly: versus .
Thinking every number has a whole-number square root.	Use the “between which squares” question routinely (see Lesson 9).

Enrichment — Competition-Style Problems

E1 (Kangaroo style). What is the sum of the first five perfect squares?

Answer

E2 (AMC Junior style). How many perfect squares are there between and ?

Answer

(excluded if “between” is strict), up to ; is too large. That gives — 12 perfect squares.

E3 (Challenge). A perfect square has exactly factors. What can you say about it?

Answer

It must be the square of a prime: has factors . Examples: .

E4 (Challenge). The difference between two consecutive perfect squares is . What are the two squares?

Answer

The squares are and . Check: ✓

E5 (Investigation). Show that the sum of the first odd numbers is always . Try , , , , and explain with a diagram.

Answer

; ; ; . Geometrically, each odd number is an L-shaped border added to grow an square into an square.

Homework

1. Evaluate: (a) (b) (c) (d) (e) .
2. Evaluate: (a) (b) (c) (d) (e) .
3. Calculate: (a) (b) (c) .
4. A square garden bed has an area of . What length of edging is needed to go all the way around it?
5. State whether each is a perfect square, giving the root if it is: (a) (b) (c) (d) (e) .
6. Reasoning. Explain, using the area model, why is not equal to .
7. Challenge. A number multiplied by itself gives a result ending in . List all possible units digits of the original number.

Answers: Q3 — (a) (b) (c) . Q4 — side m, perimeter m. Q5 — (a) yes, (b) no (c) yes, (d) no (e) yes, . Q6 — , not . Q7 — or (since , ).