Perfect Squares & Square Roots: Step-by-Step Math Worksheet for 13-Year-Olds (Algebra)

Instructions

This worksheet explores the connection between squaring a number (finding a perfect square) and finding the square root of that number. These two operations are inverses of each other—they undo one another.

1. Review the definitions and the example table in Section 1.
2. Complete the calculation problems in Sections 2 and 3.
3. Apply your knowledge to the real-world scenarios in Section 4.

Section 1: The Inverse Relationship

Definition 1: Squaring a Number When you square a number, you multiply it by itself. The result is called a Perfect Square.

Definition 2: Square Root The square root of a perfect square is the original number that was multiplied by itself. Finding the square root undoes the squaring operation.

The Perfect Square / Square Root Connection

Section 2: Finding Perfect Squares (Squaring)

Calculate the following squares. Show the multiplication steps.

1. 9 squared (9 x 9): Answer: ____

2. 11 squared (11 x 11): Answer: ____

3. 16 squared (16 x 16): Answer: ____

4. 25 squared (25 x 25): Answer: ____

5. If a large parking lot is 30 units wide and 30 units long, what is its total area (the perfect square)? Calculation: ___ Area: ____

Section 3: Finding Square Roots (Inverse Operation)

Determine the positive whole number that was multiplied by itself to get the following perfect squares. (Hint: What number times itself equals the Perfect Square?)

1. Square Root of 49: Answer: ____ (because 7 x 7 = 49)

2. Square Root of 121: Answer: ____

3. Square Root of 225: Answer: ____

4. Square Root of 400: Answer: ____

5. Square Root of 1,600: Answer: ____

Section 4: Real-World Application (Area and Sides)

In geometry, if a shape is a perfect square, its area is a perfect square number. The side length is the square root of the area.

Problem Set A: Finding the Area (Squaring)

Problem Set B: Finding the Side Length (Square Roots)

Section 5: Challenge and Extension

These problems require two steps: finding the square root, and then using that number for another calculation.

1. A square bulletin board has an area of 1,024 square inches. a) What is the length of one side of the bulletin board? (Find the square root). Side Length: ____ b) If you wanted to put trim around the entire perimeter (all four sides), how many total inches of trim would you need? Perimeter Calculation: ___ Perimeter: ____

2. The square root of a certain perfect square (X) is 18. What is the value of X? (Hint: Use the inverse operation to solve). X = ___

Answer Key

Section 1: The Inverse Relationship (Completed Table)

Section 2: Finding Perfect Squares

1. 81
2. 121
3. 256
4. 625
5. Calculation: 30 x 30. Area: 900

Section 3: Finding Square Roots

1. 7
2. 11
3. 15
4. 20
5. 40

Section 4: Real-World Application

Problem Set A

Problem Set B

Section 5: Challenge and Extension

1. a) Side Length: 32 inches (32 x 32 = 1,024) b) Perimeter Calculation: 32 + 32 + 32 + 32 or 4 x 32. Perimeter: 128 inches

2. X = 324 (18 x 18 = 324)