1. Perfect Numbers- Squares and Cubes
Essential Question:
How can algebraic expressions and equations be used to model, analyze, and solve mathematical situations?
This is targeted to Algebra I students to help them identify perfect numbers. The term “perfect” will be new to some students. Most students have been exposed to perfect numbers that are whole numbers. They will need to differentiate between perfect squares and perfect cubes. They will need to learn about fractions, and negatives. In the future, they will use this skill to simplify radicals.
There are a variety of instructional strategies used in this presentation. The introductions of perfect squares and cubes is presented by giving examples and then giving the rule. The sections on fractions and negative numbers are presented by giving a rule and then several examples. After each section the quiz provides feedback to the student. The instruction in negative numbers and the repetition of rules throughout the presentation help to stimulate recall of prior knowledge. The use of sounds and clipart helps gain and keep the attention of the student.

2. Radicals (also called roots) are directly related to exponents.
Square and Cube Roots
Roots and Radicals
Radicals (also called roots) are directly related to exponents.

3. The simplest types of radicals are square roots and cube roots.
Square and Cube Roots
Roots and Radicals
The simplest types of radicals are square roots and cube roots.
Radicals beyond square roots and cube roots exist, but we will not explore them here.

4. Square and Cube Roots
Roots and Radicals
The rules for radicals that you will learn work for all radicals – not just square roots and cube roots.

5. The symbol used to indicate a root is the radical symbol -
Square and Cube Roots
Roots and Radicals
The symbol used to indicate a root is the radical symbol -

6. Perfect Cubes and Cube Roots
Area
Volume
cube
square

7. What does square root mean?
Square and Cube Roots
Roots and Radicals
What does square root mean?
What does cube root mean?

8. Perfect Cubes and Cube Roots
Remember:
A number is a perfect square if you multiply the same number such as:
1 x 1 unit squares and form them into a square. The side length of the square you form is that number’s square root.

9. 3 x 3 = 9 3 3

10. 3
The square root of 9 is

11. Remember:
A number is a perfect cube if you multiply the same number such as:
1 x 1 x 1 unit cubes and form them into a cube. The side length of the cube you form is that number’s cube root.

12. Cubes 5 6 7 8 1 2 3 4

13. 2 x 2 x 2 = 8 2 2 2

14. 3 x 3 x 3 = 27

15. Remember
If a number is a perfect square, then you can find its exact square root.
A perfect square is simply a number (or expression) that can be written as the square [raised to 2nd power] of another number (or expression).

16. If a number is a perfect cube, then you can find its exact cube root.
Remember…….
If a number is a perfect cube, then you can find its exact cube root.
A perfect cube is simply a number (or expression) that can be written as the cube [raised to 3rd power] of another number (or expression).

17. Examples:

18. Thinking ahead: What if they aren’t perfect?
The End until next week
Thinking ahead: What if they aren’t perfect?

19. Non Perfect Squares and Cubes
Essential Question: How can algebraic expressions and equations be used to model, analyze, and solve mathematical situations?

20. Non-Perfect Squares Here is the list of perfect squares from 1 to 256.
Not every number is a perfect square. If they aren’t, we call them non-perfect squares. To find the square root of a number that is not a perfect square, we use estimation with perfect squares.

21. Estimating Square Roots
To calculate the square root of a non-perfect square
1. Place the values of the adjacent perfect squares on a number line.
2. Interpolate between the points to estimate to the nearest tenth.

22. Approximating Square Roots
You need to estimate its value of non-perfect squares by determining which two perfect squares it falls in between.
Example:
11 is a non-perfect square
11 falls between perfect squares 9 & 16
Therefore, is between and
Since, = 3 and = 4
Then is between 3 and 4

23. Estimating Square Roots
Example:
What are the perfect squares on each side of 27? 25 30 35 36

24. Estimating Square Roots
Example: 5 6 25 30 35 36 27
Estimate = 5-6

25. Find the two consecutive numbers the following non-perfect square fall between. SHOW WORK!
√55 
√23
√5 
√14
√44
and
Between 7 & 8
and
Between 4 & 5
and
Between 2 & 3
and
Between 3 & 4
and
Between 6 & 7

26. Answer the following problem SHOW WORK!
If a square bedroom has an area of 144 square feet, what is the length of one wall?
12 feet 26

27. Check It Out: Example
You want to build a fence around a square garden that is 250 square feet. Calculate the length of one side of the garden and the total length of the fence, to the nearest tenth.
First find the length of a side. Then you can use the length of a side to find the perimeter, the length of the fence.