Instructions
This worksheet explores the fundamental relationship between squaring a number (finding a perfect square) and taking the square root. Read the background information, complete the practice chart, and solve the application problems.
- Review the definitions in Section 1.
- Complete the Perfect Squares Chart in Section 2.
- Use your knowledge of inverse operations to solve the real-world problems in Section 4.
Section 1: Understanding the Relationship
A Perfect Square is the result of multiplying an integer by itself (e.g., $4 \ imes 4 = 16$). A Square Root ($\sqrt{}$) is the inverse operation of squaring. It asks: "What number, when multiplied by itself, equals the given number?" (e.g., $\sqrt{16} = 4$ because $4^2 = 16$).
Quick Check (Matching)
Draw a line to connect the term on the left with its description on the right.
- Squaring a number A. The inverse operation of squaring.
- Square Root ($\sqrt{}$) B. Multiplying a number by itself.
- Perfect Square C. A number whose square root is an integer.
Section 2: Building the Perfect Squares Chart
Complete the following chart. This helps you quickly recognize perfect square numbers 1 through 12. Note: We focus on the principal (positive) square root.
| Number (N) | Calculation ($N \ imes N$) | Perfect Square ($N^2$) | Square Root ($\sqrt{N^2}$) |
|---|---|---|---|
| Example: 3 | $3 \ imes 3$ | 9 | $\sqrt{9} = 3$ |
| 4 | |||
| 5 | |||
| 7 | |||
| 10 | |||
| 11 |
Section 3: Inverse Operations Practice
Calculate the result for each problem. (5 points each)
- $8^2$ (8 squared)
Answer: __
- $\sqrt{81}$
Answer: __
- $15^2$
Answer: __
- $\sqrt{400}$
Answer: __
- If $x^2 = 144$, what is $x$?
Answer: __
- If $y = 12$, what is $y^2$?
Answer: __
Section 4: Real-World Applications (Area and Geometry)
Squaring and square roots are essential when dealing with the area of a square, since Area = side $\ imes$ side ($A = s^2$).
Problem 1: The New Garden Plot
A landscaper is designing a square garden. If the length of one side is 13 feet, what is the total area of the garden?
Show your calculation (Hint: Use squaring):
Area: __ square feet.
Problem 2: The Tile Floor
Sarah is installing a square floor in her kitchen. The total area of the floor is 225 square meters. What is the length of one side of the kitchen?
Show your calculation (Hint: Use the square root):
Side Length: __ meters.
Problem 3: Finding the Perimeter
Mr. Chen bought a square piece of land with an area of 64 square kilometers. He plans to build a fence around the entire perimeter of the property. What is the total length of the fence he needs?
Step 1: Find the side length (s).
Step 2: Calculate the perimeter (P = 4s).
Total Fence Length: __ kilometers.
Challenge Zone (Optional Extension)
When a number is not a perfect square, we can estimate its square root.
- The number 50 is not a perfect square. Find the two consecutive integers (whole numbers) that $\sqrt{50}$ lies between.
Hint: Which two perfect squares is 50 between?
$\sqrt{50}$ is between __ and __.
- Estimate the square root of 30, rounding to the nearest tenth. (You may use a calculator to check your estimate, but try to guess first based on the perfect squares chart.)
Estimate of $\sqrt{30}$: __
Answer Key
Section 1: Understanding the Relationship
- Squaring a number (B. Multiplying a number by itself.)
- Square Root ($\sqrt{}$) (A. The inverse operation of squaring.)
- Perfect Square (C. A number whose square root is an integer.)
Section 2: Building the Perfect Squares Chart
| Number (N) | Calculation ($N \ imes N$) | Perfect Square ($N^2$) | Square Root ($\sqrt{N^2}$) |
|---|---|---|---|
| 4 | $4 \ imes 4$ | 16 | $\sqrt{16} = 4$ |
| 5 | $5 \ imes 5$ | 25 | $\sqrt{25} = 5$ |
| 7 | $7 \ imes 7$ | 49 | $\sqrt{49} = 7$ |
| 10 | $10 \ imes 10$ | 100 | $\sqrt{100} = 10$ |
| 11 | $11 \ imes 11$ | 121 | $\sqrt{121} = 11$ |
Section 3: Inverse Operations Practice
- $8^2 = 64$
- $\sqrt{81} = 9$
- $15^2 = 225$
- $\sqrt{400} = 20$
- If $x^2 = 144$, $x = 12$
- If $y = 12$, $y^2 = 144$
Section 4: Real-World Applications (Area and Geometry)
Problem 1: The New Garden Plot Area = $13^2 = 169$ square feet.
Problem 2: The Tile Floor Side Length = $\sqrt{225} = 15$ meters.
Problem 3: Finding the Perimeter Step 1: Side length (s) = $\sqrt{64} = 8$ km. Step 2: Perimeter = $4 \ imes 8 = 32$ km. Total Fence Length: 32 kilometers.
Challenge Zone (Optional Extension)
-
$7^2 = 49$ and $8^2 = 64$. Since 50 is between 49 and 64, $\sqrt{50}$ is between 7 and 8.
-
$5^2 = 25$ and $6^2 = 36$. 30 is closer to 25. Estimate of $\sqrt{30}$: 5.5 (Actual value is approximately 5.48, so 5.5 is a great estimate.)