1. Estimating &
Approximating
Square Roots

2. All of the examples so far have been from perfect squares.
What does it mean to be a perfect square?
The square of an integer is a perfect square.
A perfect square has a whole number square root.

3. You know how to find the square root of a perfect square.
What happens if the number is not a perfect square?
Does it have a square root?
What would the square root look like?

4. Square Perfect
Root Square
1 1
2 4
3 9
4 16
5 25
6 36
7 49
8 64
9 81
Think about the square root of 50.
Where would it be on this chart?
What can you say about the square root of 50?
50 is between the perfect squares 49 and 64 but closer to 49.
So the square root of 50 is between 7 and 8 but closer to 7.

5. Square Perfect
Root Square
1 1
2 4
3 9
4 16
5 25
6 36
7 49
8 64
9 81
When estimating square roots of numbers, you need to determine:
Between which two perfect squares it lies (and therefore which 2 square roots).
Which perfect square it is closer to (and therefore which square root).
Example: 110
Lies between 100 & 121, closer to 100.
So is between 10 & 11, closer to 10.

6. Estimate the following: 30
Square Perfect
Root Square
Estimate the following: 30
200
215
Teacher Instructions:
√30 - Lies between 25 & 36, closer to 25 but since almost right in the middle,  5.5
√ Lies between 196 & 225, closer to 196 so  14
√ Lies between 196 & 225, closer to 225 so  15

7. Approximating a Square Root
Approximate to the nearest integer
<
<
Identify perfect squares closest to 38
Take square root 6
<
< 7
Answer: Because 38 is closer to 36 than to 49, is closer to 6 than to 7. So, to the nearest integer, = 6

8. Approximate to the nearest integer
Identify perfect squares closest to 70
Take square root
Identify nearest integer
<
Approximate to the nearest integer

9. Another way to think about it is to use a number line.
Since 8 is closer to 9 than to 4, √8 is closer to 3 than to 2, so √8 ≈ 2.8 √8

10. Example:
Approximate

11. The square root of 40 falls between which two perfect squares?41
The square root of 40 falls between which two perfect squares? A
9 and 16 B
25 and 36 C
36 and 49 D
49 and 64
Answer: C

12. Which whole number is 40 closest to?42
Which whole number is closest to?
Identify perfect squares closest to 40
Take square root
Identify nearest integer
<
Answer: 6

13. The square root of 110 falls between which two perfect squares?43
The square root of 110 falls between which two perfect squares? A
36 and 49 B
49 and 64 C
64 and 84 D
100 and 121
Answer: D

14. Estimate to the nearest whole number. 11044
Estimate to the nearest whole number.
110
Answer: 10

15. Estimate to the nearest whole number. 21945
Estimate to the nearest whole number.
219
Answer: 15

16. Estimate to the nearest whole number. 9046
Estimate to the nearest whole number. 90
Answer: 9

17. What is the square root of 400?47
What is the square root of 400?
Answer: 20

18. Watch this video!

19. Approximate to the nearest integer.48
Approximate to the nearest integer.
Answer: 5

20. Approximate to the nearest integer. 9649
Approximate to the nearest integer. 96
Answer: 10

21. Approximate to the nearest integer. 16750
Approximate to the nearest integer.
167
Answer: 13

22. Approximate to the nearest integer. 14051
Approximate to the nearest integer.
140
Answer: 12

23. Approximate to the nearest integer. 4052
Approximate to the nearest integer. 40
Answer: 6

24. The expression is a number between 53
3 and 9 B
8 and 9 C
9 and 10 D
46 and 47
From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.
Answer: C

25. Rational & Irrational
Numbers

26. Rational & Irrational Numbers
is rational because the radicand (number under the radical) is a perfect square
If a radicand is not a perfect square, the root is said to be irrational.
Ex:

27. Sort the following numbers.24 25 32 36 40 52 64
100
200
225
625
1225
300
1681
3600
Rational Irrational

28. Rational or Irrational?54
Rational or Irrational? A
Rational B
Irrational
Answer: A

29. Rational or Irrational?55
Rational or Irrational? A
Rational B
Irrational
Answer: B

30. Rational or Irrational?56
Rational or Irrational? A
Rational B
Irrational
Answer: A

31. Rational or Irrational?57
Rational or Irrational? A
Rational B
Irrational
Answer: A

32. Rational or Irrational?58
Rational or Irrational? A
Rational B
Irrational
Answer: B

33. Which is a rational number?59
Which is a rational number? A B p C D
From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.
Answer: C

34. 60
Given the statement: “If x is a rational number, then is irrational.” Which value of x makes the statement false? A B 2 C 3 D 4
Answer: D
From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.