016. Squares, Roots and Perfect Squares

Learning Intentions

• To understand what it means to square a number or to take its square root
• To know what a perfect square is
• find the square of a number
• find the square root of a perfect square
• locate square roots between two consecutive whole numbers

Pre-requisite Summary

• Understand multiplication as repeated multiplication of equal factors
• Know that $a\times a=a^2$ for a whole number $a$
• Recall multiplication facts up to at least $12\times 12$
• Understand that inverse operations undo each other
• Be able to compare whole numbers and place them in order on a number line
• Recognise that not every number is a perfect square
• Know nearby square numbers such as $1,4,9,16,25,36,49,64,81,100$

Worked Examples

Worked Example 1

a) Write $6^2$ as repeated multiplication.
b) Evaluate $6^2$.
c) Explain what it means to square a number.

Worked Example 2

a) Find $\sqrt{49}$.
b) Check the answer using multiplication.

Worked Example 3

a) State whether $36$ is a perfect square.
b) State whether $40$ is a perfect square.
c) Justify each answer.

Worked Example 4

a) Find the square of $11$.
b) Find the square of $15$.

Worked Example 5

a) Find $\sqrt{81}$.
b) Find $\sqrt{121}$.
c) Find $\sqrt{144}$.

Worked Example 6

a) Locate $\sqrt{20}$ between two consecutive whole numbers.
b) Locate $\sqrt{70}$ between two consecutive whole numbers.

Problems

Problem 1

a) Write $7^2$ as repeated multiplication.
b) Evaluate $7^2$.
c) Explain what it means to square a number.

Problem 2

a) Find $\sqrt{64}$.
b) Check the answer using multiplication.

Problem 3

a) State whether $49$ is a perfect square.
b) State whether $50$ is a perfect square.
c) Justify each answer.

Problem 4

a) Find the square of $12$.
b) Find the square of $14$.

Problem 5

a) Find $\sqrt{100}$.
b) Find $\sqrt{169}$.
c) Find $\sqrt{196}$.

Problem 6

a) Locate $\sqrt{18}$ between two consecutive whole numbers.
b) Locate $\sqrt{90}$ between two consecutive whole numbers.

Exercises

Understanding and Fluency

1. Find the square:
   a) $3^2$
   b) $5^2$
   c) $9^2$

2. Find the square:
   a) $8^2$
   b) ${10}^2$
   c) ${13}^2$

3. Find the square root:
   a) $\sqrt{16}$
   b) $\sqrt{25}$
   c) $\sqrt{64}$

4. Find the square root:
   a) $\sqrt{36}$
   b) $\sqrt{81}$
   c) $\sqrt{144}$

5. Decide whether each number is a perfect square:
   a) $24$
   b) $49$
   c) $121$

6. Decide whether each number is a perfect square:
   a) $72$
   b) $100$
   c) $225$

7. Locate each square root between two consecutive whole numbers:
   a) $\sqrt{15}$
   b) $\sqrt{27}$
   c) $\sqrt{50}$

8. Locate each square root between two consecutive whole numbers:
   a) $\sqrt{35}$
   b) $\sqrt{80}$
   c) $\sqrt{99}$

Reasoning

9. Explain why $\sqrt{64}=8$.

10. A student says $9^2=18$. Explain the misunderstanding.

11. Explain why $48$ is not a perfect square.

12. Explain why $\sqrt{30}$ lies between $5$ and $6$.

Problem-solving

13. A square garden has side length $7$ m. Find its area.

14. A square floor has area $144{\text{ m}}^2$. Find the length of one side.

15. Find two consecutive whole numbers between which $\sqrt{45}$ lies.

16. A square tile has area $81{\text{ cm}}^2$. Find the side length.

17. Compare $4^2+5^2$ with $6^2$.

18. A square picture frame has side length $12$ cm. Find its area.

Potential Misunderstandings

• A student may think squaring a number means multiplying by $2$ rather than multiplying the number by itself
• A student may think taking a square root means dividing by $2$
• A student may not recognise that square root is the inverse of squaring
• A student may think every whole number is a perfect square
• A student may confuse $a^2$ with $2a$
• A student may identify a nearby perfect square incorrectly when locating a square root between whole numbers
• A student may forget to compare the number with consecutive perfect squares when estimating square roots
• A student may think $\sqrt{49}$ could be $-7$ in this context, rather than using the principal square root convention of $7$