Transcript Approximation of Square Roots

4-6
Square
Roots
4-6 Estimating
Estimating
Square
Roots
Warm Up
Problem of the Day
Lesson Presentation
Course
Course
33
4-6 Estimating Square Roots
Warm Up
Find the two square roots of each
number.
1. 144
12
2. 256
16
Evaluate each expression.
3. 8+ 144 20
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4. 7
289
119
4-6 Estimating Square Roots
Essential Question
How can you estimate the square root of a
number?
Standard: MFANSQ3a. Find and estimate decimal expansions
of an irrational number locating the approximations on a
number line (MGSE8.NS.1,2)
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4-6 Estimating Square Roots
1
12  1
3
2
22  4
32  9
The square of an integer is a
perfect square.
The opposite of squaring a
number is taking the square root.
Course 3
4-6 Estimating Square Roots
Example
• For example
81
asks what number multiplied by itself is equal to
81?
Is there another solution to that problem?
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4-6 Estimating Square Roots
Simplify each square root
100
 16
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4-6 Estimating Square Roots
Squares and roots
• Here is a list that will be helpful:
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12  1
1 1
22  4
4 2
32  9
9 3
42  16
16  4
52  25
25  5
62  36
36  6
7 2  49
49  7
82  64
64  8
92  81
81  9
102  100
100  10
112  121
121  11
122  144
144  12
4-6 Estimating Square Roots
Estimating square roots
• Once we have memorized these squares and
their roots, we can estimate square roots
that are not perfect squares
• For example, what about
?
8
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4-6 Estimating Square Roots
Estimating square roots
• We find the two perfect squares that are before and
after the square root of 8. . .
•
9
4 and
• Look at them on a number line:
2
3
4
2
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5
6
7
8
9
3
4-6 Estimating Square Roots
Estimating square roots
• We can see that
8 is between 2 and 3 but
is closer to 3. We would say that 8 is approximately
3.
2
3
4
2
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5
6
7
8
9
3
4-6 Estimating Square
Roots
TRY THIS:
Estimate to the nearest whole number
27
 78
50
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4-6 Estimating Square Roots
Try to place six color tiles in an array. Since we are trying
to find the square root, the array should be in the form of
a square.
Course 3
4-6 Estimating Square Roots
What is the largest square you can make using six tiles?
Course 3
4-6 Estimating Square Roots
The largest square that we can construct out of six
squares is a 2x2 which uses four tiles. Lets use a different
color for the remaining tiles.
Course 3
4-6 Estimating Square Roots
In order to get to the next square we will need to add
three more tiles. Use another color since were adding tiles
to the six.
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4-6 Estimating Square Roots
Additional Example 1A: Estimating Square Roots of
Numbers
Each square root is between two integers.
Name the integers. Explain your answer.
55
Think: What are perfect
squares close to 55?
72 = 49
49 < 55
82 = 64
64 > 55
55 is between 7 and 8 because 55 is
between 49 and 64.
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4-6 Estimating Square Roots
Additional Example 1B: Estimating Square Roots of
Numbers Continued
Each square root is between two integers.
Name the integers. Explain your answer.
Think: What are perfect
– 90
squares close to 90?
–92 = 81
81 < 90
–102 = 100
100 > 90
– 90 is between –9 and –10 because 90 is
between 81 and 100.
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4-6 Estimating Square Roots
Check It Out: Example 1A
Each square root is between two integers.
Name the integers. Explain your answer.
80
Think: What are perfect
squares close to 80?
82 = 64
64 < 80
92 = 81
81 > 80
80 is between 8 and 9 because 80 is
between 64 and 81.
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4-6 Estimating Square Roots
Check It Out: Example 1B
Each square root is between two integers.
Name the integers.
– 45
Think: What are perfect
squares close to 45?
–62 = 36
36 < 45
–72 = 49
49 > 45
– 45 is between –6 and –7 because 45 is
between 36 and 49.
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4-6 Estimating Square Roots
Additional Example 2: Problem Solving Application
You want to sew a fringe on a square
tablecloth with an area of 500 square
inches. Calculate the length of each side of
the tablecloth and the length of fringe you
will need to the nearest tenth of an inch.
Understand the Problem
First find the length of a side. Then you can
use the length of a side to find the
perimeter, the length of fringe around the
tablecloth.
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4-6 Estimating Square Roots
Additional Example 2 Continued
Make a Plan
The length of a side, in feet, is the number
that you multiply by itself to get 500. To be
accurate, find this number to the nearest
tenth. If you do not know a step-by-step
method for finding 500, use guess and
check.
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4-6 Estimating Square Roots
Additional Example 2 Continued
Solve
Because 500 is between 222 and 232, the square root
of 500 is between 22 and 23.
Guess 22.5
Guess 22.2
Guess 22.4
Guess 22.3
22.52 = 506.25
22.22 = 492.84
22.42 = 501.76
22.32 = 497.29
Too high
Too low
Too high
Too low
Square root is
between 22
and 22.5
Square root is
between 22.2
and 22.5
Square root is
between 22.2
and 22.4
Square root is
between 22.3
and 22.4
2
22.0
22.2
4
3
22.4
1
22.6
The square root is between 22.3 and 22.4.
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4-6 Estimating Square Roots
Additional Example 2 Continued
Solve
The square root is between 22.3 and 22.4. To
round to the nearest tenth, look at the next
decimal place. Consider 22.35.
22.352 = 499.5225
Too low
The square root must be greater than 22.35, so
round up.
To the nearest tenth,
500 is about 22.4.
The length of each side of the table is about
22.4 in.
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4-6 Estimating Square Roots
Additional Example 2 Continued
Solve
The length of a side of the tablecloth is 22.4
inches, to the nearest tenth of an inch. Now
estimate the length around the tablecloth.
4 • 22.4 = 89.6
Perimeter = 4 • side
You will need about 89.6 inches of fringe.
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4-6 Estimating Square Roots
Additional Example 2 Continued
Look Back
The length 90 inches divided by 4 is 22.5
inches. A 22.5-inch square has an area of
506.25 square inches, which is close to 500,
so the answers are reasonable.
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4-6 Estimating Square Roots
Check It Out: Example 2
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.
Understand the Problem
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.
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4-6 Estimating Square Roots
Check It Out: Example 2 Continued
Make a Plan
The length of a side, in feet, is the number
that you multiply by itself to get 250. To be
accurate, find this number to the nearest
tenth. If you do not know a step-by-step
method for finding 250, use guess and
check.
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4-6 Estimating Square Roots
Check It Out: Example 2 Continued
Solve
Because 250 is between 152 and 162, the square
root of 250 is between 15 and 16.
Guess 15.5
Guess 15.9
Guess 15.7
Guess 15.8
15.52 = 240.25
15.92 = 252.81
15.72 = 246.49
15.82 = 249.46
Too Low
Too high
Too Low
Too Low
Square root is
between 15.5
and 16
Square root is
between 15.5
and 15.9
Square root is
between 15.7
and 15.9
Square root is
between 15.8
and 15.9
1
15.5
3
15.7
4
2
15.9
16.1
The square root is between 15.8 and 15.9.
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4-6 Estimating Square Roots
Check It Out: Example 2 Continued
Solve
To round to the nearest tenth, look at the next
decimal place.
Consider 15.85.
15.852 = 251.2225
The square root is lower than 15.85, so round
down.
To the nearest tenth, 250 is about 15.8.
The length of each side of the garden is
about 15.8 ft.
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4-6 Estimating Square Roots
Check It Out: Example 2 Continued
Solve
The length of a side of the garden is 15.8
feet, to the nearest tenth of a foot. Now
estimate the length around the garden.
4 • 15.8 = 63.2
Perimeter = 4 • side
You will need about 63.2 feet of fence.
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4-6 Estimating Square Roots
Check It Out: Example 2 Continued
Look Back
The length 63.2 feet divided by 4 is 15.8
feet. A 15.8 foot square has an area of
249.64 square feet, which is close to 250,
so the answers are reasonable.
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4-6 Estimating Square Roots
Additional Example 3: Using a Calculator to Estimate
the Value of a Square Root
Use a calculator to find
the nearest tenth.
Using a calculator,
Rounded,
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600. Round to
600 ≈ 24.49489742….
600 is 24.5.
4-6 Estimating Square Roots
Check It Out: Example 3
Use a calculator to find
the nearest tenth.
Using a calculator,
Rounded,
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800. Round to
800 ≈ 28.2842712….
800 is 28.3.
4-6 Estimating Square Roots
Lesson Quiz
Each square root is between two integers.
Name the two integers.
1.
27
5 and 6
2. –
456
–22 and –21
Use a calculator to find each value. Round
to the nearest tenth.
3.
89
9.4
4.
1223
35.0
5. A square field has an area of 2000 square
feet. To the nearest foot, how much fencing
would be needed to enclose the field? 179 ft
Course 3