1. Homework: Cumulative Review

2. Squares & Square Roots

3. Rational and Irrational Numbers Essential Question
How do I distinguish between rational and irrational numbers?

4. Vocabulary
real number
irrational number

5. The set of real numbers is all numbers that can be written on a number line. It consists of the set of rational numbers and the set of irrational numbers.
Irrational numbers
Rational numbers
Real Numbers
Integers
Whole
numbers

6. Recall that rational numbers can be written as the quotient of two integers (a fraction) or as either terminating or repeating decimals. 4 5 2 3
= 3.8
= 0.6
1.44 = 1.2

7. Irrational numbers can be written only as decimals that do not terminate or repeat. They cannot be written as the quotient of two integers. If a whole number is not a perfect square, then its square root is an irrational number. For example, 2 is not a perfect square, so is irrational.
A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits.
Caution!

8. Reals Make a Venn Diagram that displays the following sets of numbers:
Reals, Rationals, Irrationals, Integers, Wholes, and Naturals.
Reals
Rationals
-2.65
Integers -3
-19
Wholes
Irrationals
Naturals
1, 2, 3...

9. Additional Example 1: Classifying Real Numbers
Write all classifications that apply to each number. A. 5
5 is a whole number that is not a perfect square.
irrational, real B.
–12.75
–12.75 is a terminating decimal.
rational, real 16 2
= = 2 4 2 16 C.
whole, integer, rational, real

10. Write all classifications that apply to each number.
Check It Out! Example 1
Write all classifications that apply to each number. A. 9
9 = 3
whole, integer, rational, real B.
–35.9
–35.9 is a terminating decimal.
rational, real 81 3
= = 3 9 3 81 C.
whole, integer, rational, real

11. A fraction with a denominator of 0 is undefined because you cannot divide by zero. So it is not a number at all.

12. Additional Example 2: Determining the Classification of All Numbers
State if each number is rational, irrational, or not a real number. A. 21
irrational 3 3
= 0 B.
rational

13. Additional Example 2: Determining the Classification of All Numbers
State if each number is rational, irrational, or not a real number.
4 0 C.
not a real number

14. State if each number is rational, irrational, or not a real number.
Check It Out! Example 2
State if each number is rational, irrational, or not a real number. A. 23
23 is a whole number that is not a perfect square.
irrational 9 B.
undefined, so not a real number

15. State if each number is rational, irrational, or not a real number.
Check It Out! Example 2
State if each number is rational, irrational, or not a real number. 64 81 8 9 = 64 81 C.
rational

16. Square Number Also called a “perfect square”
A number that is the square of a whole number
Can be represented by arranging objects in a square.

17. Square Numbers

18. Square Numbers
1 x 1 = 1
2 x 2 = 4
3 x 3 = 9
4 x 4 = 16

19. Square Numbers 1 x 1 = 1 2 x 2 = 4 3 x 3 = 9 4 x 4 = 16 Activity:
Calculate the perfect
squares up to 152…

20. Square Numbers 1 x 1 = 1 2 x 2 = 4 3 x 3 = 9 4 x 4 = 16 5 x 5 = 25

21. Activity: Identify the following numbers as perfect squares or not.16 15
146
300
324
729

22. Activity: Identify the following numbers as perfect squares or not.
16 = 4 x 4 15
146
300
324 = 18 x 18
729 = 27 x 27

23. Squares & Square Roots
Square Root

24. Square Numbers
One property of a perfect square is that it can be represented by a square array.
Each small square in the array shown has a side length of 1cm.
The large square has a side length of 4 cm.
4cm
4cm
16 cm2

25. Square Numbers The large square has an area of 4cm x 4cm = 16 cm2.
The number 4 is called the square root of 16.
We write: 4 =
4cm
4cm
16 cm2

26. Square Root
A number which, when multiplied by itself, results in another number.
Ex: 5 is the square root of 25.
5 = 25

27. Finding Square Roots 4 x 9 = 4 x 9 36 = 2 x 3 6 = 6
We can use the following strategy to find a square root of a large number.
4 x 9 = x 9
= x 3 =

28. Finding Square Roots 4 x 9 = 4 9 36 = 2 x 3 6 = 6=
We can factor large perfect squares into smaller perfect squares to simplify.

29. Finding Square Roots 256 = 4 x 64 = 2 x 8 = 16
Activity: Find the square root of 256
256
= x 64
= 2 x 8
= 16

30. Estimating Square Root
Squares & Square Roots
Estimating Square Root

31. Estimating Square Roots
25 = ?

32. Estimating Square Roots
25 = 5

33. Estimating Square Roots
49 = ?

34. Estimating Square Roots
49 = 7

35. Estimating Square Roots
27 = ?

36. Estimating Square Roots
27 = ?
Since 27 is not a perfect square, we
have to use another method to
calculate it’s square root.

37. Estimating Square Roots
Not all numbers are perfect squares.
Not every number has an Integer for a square root.
We have to estimate square roots for numbers between perfect squares.

38. 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.

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

40. Estimating Square Roots
Example:
half 5 6 25 30 35 36 27
Estimate = 5.2

41. Estimating Square Roots
Example:
Estimate: = 5.2
Check: (5.2) (5.2) = 27.04

42. CLASSWORK
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