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
• Solve 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 for a whole number
• Recall multiplication facts up to at least
• 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

Worked Examples

Worked Example 1

a) Write as repeated multiplication.

b) Evaluate .

c) Explain what it means to square a number.

Worked Example 2

a) Find .

b) Check the answer Use multiplication.

Worked Example 3

a) State whether is a perfect square.

b) State whether is a perfect square.

c) Justify each answer.

Worked Example 4

a) Find the square of .

b) Find the square of .

Worked Example 5

a) Find .

b) Find .

c) Find .

Worked Example 6

a) Locate between two consecutive whole numbers.

b) Locate between two consecutive whole numbers.

Problems

Problem 1

a) Write as repeated multiplication.

b) Evaluate .

c) Explain what it means to square a number.

Problem 2

a) Find .

b) Check the answer using multiplication.

Problem 3

a) State whether is a perfect square.

b) State whether is a perfect square.

c) Justify each answer.

Problem 4

a) Find the square of .

b) Find the square of .

Problem 5

a) Find .

b) Find .

c) Find .

Problem 6

a) Locate between two consecutive whole numbers.

b) Locate between two consecutive whole numbers.

Exercises

Understanding and Fluency

Exercise 1.

Find the square:

a)

b)

c)

Exercise 2.

Find the square:

a)

b)

c)

Exercise 3.

Find the square root:

a)

b)

c)

Exercise 4.

Find the square root:

a)

b)

c)

Exercise 5.

Decide whether each number is a perfect square:

a)

b)

c)

Exercise 6.

Decide whether each number is a perfect square:

a)

b)

c)

Exercise 7.

Locate each square root between two consecutive whole numbers:

a)

b)

c)

Exercise 8.

Locate each square root between two consecutive whole numbers:

a)

b)

c)

Reasoning

Exercise 9.

Explain why .

Exercise 10.

A student says . Explain the misunderstanding.

Exercise 11.

Explain why is not a perfect square.

Exercise 12.

Explain why lies between and .

Problem-solving

Exercise 13.

A square garden has side length m. Find its area.

Exercise 14.

A square floor has area . Find the length of one side.

Exercise 15.

Find two consecutive whole numbers between which lies.

Exercise 16.

A square tile has area . Find the side length.

Exercise 17.

Compare with .

Exercise 18.

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

Potential Misunderstandings

• A student may think squaring a number means multiplying by rather than multiplying the number by itself
• A student may think taking a square root means dividing by
• 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 with
• 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 could be in this context, rather than using the principal square root convention of