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Mathematics
Quarter 1 – Module 10:
Estimating the Square Root of
a Whole Number to the
Nearest Hundredth
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Mathematics – Grade 7
Quarter 1 – Module 10: Estimating the Square Root of a Whole Number to the Nearest
Hundredth
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Regional Director: Gilbert T. Sadsad
Assistant Regional Director: Jessie L. Amin
Development Team of the Module
Writer:
Jeric C. Dino
Editor:
Lilli Ann O. Hermogenes
Reviewer:
Dioleta B. Borais
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MODULE 10
ESTIMATING THE SQUARE ROOT OF A WHOLE NUMBER
TO THE NEAREST HUNDREDTH
INTRODUCTION
In your previous modules, you were able to differentiate a rational number
from irrational number. You were also able to determine in what two integers the
square root of a given number lie.
But have you ever wondered how to estimate the square root of a number that
is not a perfect square up to the nearest hundredth?
In this module, you are going to learn about: estimating the square root of a
whole number to the nearest hundredth.
VOCABULARY LIST
✓ Estimate – to find something close to a specific value.
✓ Hundredth – the second digit after the decimal point of a decimal number.
PRE – TEST
Estimate the following square roots to the nearest hundredth.
1. √17
3. √75
2. √23
4. √99
To get things started, you may begin by answering Activity 1 to recall your
knowledge on rational and irrational numbers.
ACTIVITY 1: Which one is Irrational?
Identify
whether
each
number
is
rational
or
1. −12
6. 1.234789423412 …
2. 24
7. 𝜋
3. √3
8. −8.7
4.
2
3
5. 0.5555555 …
irrational.
9. √81
10. √12
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Were you able to determine which of those numbers are rational and which
ones are irrational? When is a number rational? When is it irrational?
𝑎
Rational numbers are numbers that can be expressed as a ratio 𝑏 , where
𝑎 𝑎𝑛𝑑 𝑏 are integers, while irrational numbers are numbers that cannot be
expressed as a ratio of two integers. Some of these numbers are commonly
expressed as square roots, or sometimes a non – terminating, non – repeating
decimal.
Now, proceed to Activity 2 to recall your knowledge on estimating the
square root of a whole number to the nearest two integers.
ACTIVITY 2: Place in Between
Determine in what two integers the following numbers lie.
1. √7
4. √82
2. √15
5. √103
3. √57
The numbers listed in Activity 2 are irrational numbers. Their square root is not an
integer or a terminating, repeating decimal. But what if we look for its square root up
to the nearest tenth or even to the nearest hundredth, is there a way to do it without
using calculators?
Here are some of the ways to estimate the square root of a whole number up to
the nearest hundredth:
A. Divide and Average Method
It is difficult and tedious to use guess and check in estimating the square root
of a whole number up to the nearest hundredth. So, this method called the Divide
and Average Method can be used. The following are the steps in estimating the
square root of a whole number using this method.
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STEP 1: Estimate. Find two perfect squares between which the number lies.
STEP 2: Divide. Divide the number by one of those square roots. Use the
square root closest to that number.
STEP 3: Average. Get an improved estimate by getting the average of the
divisor and quotient in Step 2.
STEP 4: Using the result in Step 3, repeat Steps 2 and 3 until we find a
number that is accurate enough (depending on the decimal place
we want to approximate).
(
Illustrative Examples
1. Estimate √10 to the nearest tenth.
Solution:
Step 1: Find two perfect squares between which the number lies.
9 < 10 < 16
→
9 and 16 are perfect squares
√9 < √10 < √16
→
10 is closer to 9
3 < √10 < 4
→
3 will be the first estimate of √10
Step 2: We divide 10 by our first estimate which is 3.
10
3
≈ 3. 33
→
we will
Since we are looking for the nearest tenth,
get the quotient to the nearest
hundredth.
Step 3: Get the average of the estimate (3) and the quotient in Step 2 (3.33).
3+3.33
2
=
6.33
2
= 3.165
→
the average is 3.165 or 3.17
Step 4: Repeat Steps 2 and 3 until the divisor and the quotient in Step 3
agree in tenths place.
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➢ We divide 10 by our new estimate which is 3.17.
10
≈ 3.154
3.17
→
the new quotient is 3.154 or 3.15
Notice that the new quotient (3.15) and divisor (3.17) agree in the tenths
place.
➢ We get the average of 3.15 and 3.17.
3.15 + 3.17 6.32
=
= 3.16 𝑜𝑟 3.2
2
2
Thus, √𝟏𝟎 ≈ 𝟑. 𝟐 to the nearest tenths.
2. Estimate √31 to the nearest hundredth.
Solution:
Step 1: Find two perfect squares between which the number lies.
25 < 31 < 36
→
25 and 36 are perfect squares
√25 < √31 < √36
→
31 is closer to 36
5 < √31 < 6
→
6 will be our first estimate of √31
Step 2: We divide 31 by our first estimate which is 6.
31
6
≈ 5.167
→
since we are looking for the nearest
hundredth,
we will get the quotient to the
nearest
thousandth
Step 3: Get the average of the estimate (6) and the quotient in Step 2 (5.167).
6+5.167
2
=
11.167
2
= 5.5835
→
the average is 5.5835 or 5.584
6
Step 4: Repeat Steps 2 and 3 until the divisor and the quotient in Step 3
agree in hundredths place.
➢ We divide 31 by our second estimate which is 5.584
31
≈ 5.556
5.584
→
the second quotient is 5.556
➢ Get the average of the second estimate (5.584) and the second
quotient (5.556)
5.584 + 5.556 11.14
=
= 5.57
2
2
Here, since our divisor and quotient do not agree yet in the hundredths place,
we repeat Steps 2 and 3.
➢ Divide 31 by our new estimate which is 5.57
31
≈ 5.566
5.57
→
the new quotient is 5.5655 or 5.566 (or 5.57)
Note that our divisor and quotient already agree in the hundredths place.
➢ Get the average of 5.57 and 5.566
5.57 + 5.566 11.136
=
= 5.568 𝑜𝑟 5.57
2
2
Thus, √𝟑𝟏 ≈ 𝟓. 𝟓𝟕to the nearest hundredth.
Checkpoint 1
Estimate the value of the following square roots to the nearest hundredth
using Divide and Average Method.
1. √11
3. √71
2. √46
4. √103
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B. Square Root Estimate Formula
Aside from the Divide and Average Method, this formula in estimating for the
square root of a whole number to the nearest hundredth can also be used:
𝑵 →the number which we want the square root of
𝑺 →the closest perfect square to the given number N
Illustrative Examples
1. Estimate √14 to the nearest hundredth.
Solution:
√𝑵 ≈ √𝑺 +
𝑵−𝑺
𝟐√𝑺
𝑵 = 𝟏𝟒
𝑺 = 𝟏𝟔 → 𝑐𝑙𝑜𝑠𝑒𝑠𝑡 𝑝𝑒𝑟𝑓𝑒𝑐𝑡 𝑠𝑞𝑢𝑎𝑟𝑒 𝑡𝑜 14
14 − 16
√14 ≈ √16 +
2√16
2
≈4−
2 ∙4
1
≈4−
4
≈ 4 − 0.25
≈ 3.75
Hence, √𝟏𝟒 ≈ 𝟑. 𝟕𝟓
2. Estimate √105 to the nearest hundredth.
Solution:
√𝑵 ≈ √𝑺 +
𝑵−𝑺
𝟐√𝑺
𝑵 = 𝟏𝟎𝟓
𝑺 = 𝟏𝟎𝟎 → 𝑐𝑙𝑜𝑠𝑒𝑠𝑡 𝑝𝑒𝑟𝑓𝑒𝑐𝑡 𝑠𝑞𝑢𝑎𝑟𝑒 𝑡𝑜 105
105 − 100
√105 ≈ √100 +
2√100
5
≈ 10 +
2 ∙ 10
1
≈ 10 +
4
≈ 10 + 0.25
≈ 10.25
Hence, √𝟏𝟎𝟓 ≈ 𝟏𝟎. 𝟐𝟓
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Checkpoint 2
Estimate the following square roots to the nearest hundredth using Square
Root Estimate Formula and verify it using the Divide and Average Method.
1. √29
2. √57
3. √90
4. √115
Were you able to perform Checkpoint 2? Which method is more accurate in
estimating the square root of a whole number to the nearest hundredth? Why?
Do you know other ways/methods in estimating the square root of a whole
number to the nearest hundredth? Share it with your classmates!
REMEMBER
Divide and Average Method, and Square Root Estimate Formula are just two
of many ways to estimate the square root of whole number to the nearest
hundredth. But neither of these methods give us the EXACT value of the square
root of irrational numbers. The only value we can determine is its estimate.
ENRICHMENT
Square Root Algorithm Method
This algorithm for square roots resembles the long division algorithm, and it
was taught in schools in days before calculators. Here are the steps and examples
using this method:
Illustrative Example
1. Estimate √8 to the nearest hundredth.
➢ Step 1: Group the given number into pairs.
Since we are estimating to the nearest
hundredth, we are going to add two pairs of
zeros after the decimal point.
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➢ Step 2: Determine the largest integer whose square is less than or
equal to the given number and write that integer above and to the left
of the given number.
12 = 1
𝟐𝟐 = 𝟒
32 = 9
The integer 2 will be used since its square is
less than 8, also because 32 is already more
than 8.
➢ Step 3: Square the first integer estimate, and write it below the given
number.
➢ Step 4: Subtract the result just like in division algorithm. Then bring
down the next two pairs of numbers (00 in this example).
➢ Step 5: Double the units digit of the first integer estimate, and write it to
the left of the result in Step 4. This will become the tens digit of our
second integer estimate.
This is 4 because 2 times 2 equals 4.
This is our second integer estimate.
➢ Step 6: Determine the largest integer to be written as the units digit of
our second integer estimate such that the product of that largest
integer and our second integer estimate is less than or equal to our
next number (400). Then, repeat Steps 3 and 4.
Why 48?
Here, we will try to get as close to 400 as
possible, but not more than 400.
47 x 7 = 329
→ this is still small
48 x 8 = 384
→ this is close to 400
49 x 9 = 441
→ this is already more than
400
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➢ Step 7:Repeat Step 5
48 = 40 + 8
Now, double the units digit 8.
8 x 2 = 16
Then, 40 + 16 = 56
This is our 3rd integer estimate
➢ Step 8: Repeat Step 6, only that we will try to get as close to 1600 as
possible (it can be less than or greater than 1600)
Why 563?
Here, we will try to get as close to 1600 as
possible
562 x 2 = 1124 → this is too small for
1600
563 x 3 = 16689
→ this is close to
1600
Note: Here, we will only take which
value is closest since we are already
looking for the hundredths digit.
Hence, √𝟖 ≈ 𝟐. 𝟖𝟑
Try this!
Estimate √120 to the nearest hundredth using this method. Compare the
value that you have obtained to the result when the other two methods (you have
learned) are used.
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POST TEST
Estimate the following square roots to the nearest hundredth.
1. √27
3. √175
2. √68
4. √390
Assignment
Estimate √120 to the nearest hundredth using this method. Compare the
value that you have obtained to the result when the other two methods (you have
learned) are used.
ANSWER KEY
Pretest
2. 3 and 4
1. 4.12
3. 7 and 8
2. 4.80
4. 9 and 10
3. 8.66
5. 10 and 11
4. 9.95
ACTIVITY 1
CHECKPOINT 1
1. Rational
1. 3.32
2. Rational
2. 6.78
3. Irrational
3. 8.43
4. Rational
4. 10.15
5. Rational
CHECKPOINT 2
6. Irrational
1. 5.39
7. Irrational
2. 7.54
8. Rational
3. 9.49
9. Rational
4. 10.72
10. Irrational
ACTIVITY 2
1. 2 and 3
POST TEST
1. 5.20
12
2. 8.25
ENRICHMENT (TRY THIS!)
3. 13.23
Answer: 10.95
4. 19.75
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REFERENCES
Internet Sources:
•
https://www.showme.com/sh/?h=dQ9fHWq
•
http://www.math.com/school/subject1/lessons/S1U1L9DP.html
•
https://www.homeschoolmath.net/teaching/square-root-algorithm.php
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