Instructions
This worksheet explores the relationship between multiplying a number by itself (squaring) and finding the original number (square root). Follow these steps carefully:
- Read the definitions and the key relationship summarized in Section 1.
- Complete the tables in Sections 2 and 3 by performing the required calculations.
- Solve the real-world problems in Section 4, writing your steps clearly.
- Attempt the Challenge Question in Section 5 if you finish the core sections.
Section 1: Defining the Relationship
What is a Perfect Square? A perfect square is the result of multiplying a whole number by itself. Example: $6 \ imes 6 = 36$. Therefore, 36 is a perfect square.
What is a Square Root? The square root of a number is the value that, when multiplied by itself, gives you the original number. We use the radical symbol ( $\sqrt{}$ ) to denote the square root. Example: $\sqrt{36} = 6$. (Because $6 \ imes 6 = 36$)
The Key Relationship (Inverse Operations): Squaring a number and finding the square root are inverse operations. They undo each other.
If $n^2 = P$, then $\sqrt{P} = n$.
Section 2: Squaring Numbers Practice
Find the perfect square by multiplying the given number by itself. Use the formula $n^2 = P$.
| Number ($n$) | Calculation ($n \ imes n$) | Perfect Square ($P$) |
|---|---|---|
| 3 | $3 \ imes 3$ | 9 |
| 5 | ||
| 8 | ||
| 11 | ||
| 15 | ||
| 18 |
Section 3: Finding Square Roots Practice
Find the principal (positive) square root of the perfect square. Use the formula $\sqrt{P} = n$.
| Perfect Square ($P$) | Square Root Symbol ($\sqrt{P}$) | Square Root ($n$) |
|---|---|---|
| 49 | $\sqrt{49}$ | 7 |
| 36 | ||
| 100 | ||
| 144 | ||
| 225 | ||
| 400 |
Section 4: Real-World Applications
Use your knowledge of perfect squares and square roots to solve these problems. Remember that the area of a square is the side length squared ($s^2$).
- The Garden Plot: Ms. Harris is designing a square garden plot with an area of 64 square meters. What is the exact length of one side of the garden?
Calculation: (Show the square root operation needed)
Answer: The side length is
- The New Bedroom: Leo's square bedroom is 12 feet wide. He needs to buy carpeting that covers the entire floor. What is the total area (in square feet) that the carpet must cover?
Calculation: (Show the squaring operation needed)
Answer: The area is
- Floor Tiles: A builder used 289 identical small square tiles to cover a larger square area. If the tiles were laid perfectly in a square grid, how many tiles are along one edge of the larger area?
Hint: The total number of tiles is the perfect square, and the number of tiles on one edge is the square root.
Calculation:
Answer: There are
Section 5: Challenge and Extension
Non-Perfect Squares:
- The number 150 is not a perfect square. Find the two consecutive whole numbers that the square root of 150 falls between.
Step 1: Find the largest perfect square less than 150.
Step 2: Find the smallest perfect square greater than 150.
Step 3: Identify the square roots of those two numbers.
*Answer: $\sqrt{150}$ is between ______ and ______.
Answer Key
Section 2: Squaring Numbers Practice
| Number ($n$) | Calculation ($n \ imes n$) | Perfect Square ($P$) |
|---|---|---|
| 3 | $3 \ imes 3$ | 9 |
| 5 | $5 \ imes 5$ | 25 |
| 8 | $8 \ imes 8$ | 64 |
| 11 | $11 \ imes 11$ | 121 |
| 15 | $15 \ imes 15$ | 225 |
| 18 | $18 \ imes 18$ | 324 |
Section 3: Finding Square Roots Practice
| Perfect Square ($P$) | Square Root Symbol ($\sqrt{P}$) | Square Root ($n$) |
|---|---|---|
| 49 | $\sqrt{49}$ | 7 |
| 36 | $\sqrt{36}$ | 6 |
| 100 | $\sqrt{100}$ | 10 |
| 144 | $\sqrt{144}$ | 12 |
| 225 | $\sqrt{225}$ | 15 |
| 400 | $\sqrt{400}$ | 20 |
Section 4: Real-World Applications
-
The Garden Plot: Calculation: $\sqrt{64} = 8$ Answer: The side length is 8 meters.
-
The New Bedroom: Calculation: $12^2 = 12 \ imes 12 = 144$ Answer: The area is 144 square feet.
-
Floor Tiles: Calculation: $\sqrt{289} = 17$ Answer: There are 17 tiles along one edge.
Section 5: Challenge and Extension
- The perfect square less than 150 is 144 (which is $12^2$).
- The perfect square greater than 150 is 169 (which is $13^2$). *Answer: $\sqrt{150}$ is between 12 and 13.