Alignment
Domains
Developing the Concept of Square Root:
A New View
What happens when we apply what is known about
how children learn to developing an understanding of
square root? Is there a more meaningful method or
algorithm for estimating square roots than those in
general use? Estimation is one of the basic skill areas
as defined by the National Council of Supervisors of
Mathematics.
Modeling a concept using concrete objects
provides the best basis for building understanding.
What follows exemplifies the value of looking at a
concrete model to develop an algorithm that has
meaning and utility. Such a process for developing
an algorithm can serve as a model even for those not
responsible for teaching this particular subject.
The term “square root” implies that such a
number must have something to do with a square
and the “root,” or source, from which it springs. So,
let’s use the idea of building squares. We will use
uniform objects that embody the square shape such
as centicubes, square pattern blocks, Unifix cubes, or
simply square pieces of card stock.
In the first example, we will find the approximate
square root of four. For this purpose, four congruent
squares will be used. The goal is to build the largest
possible square with these materials.
The largest square that can be formed measures
two units by two units. Three small squares remain,
not enough to finish the next tier to form a 3 x 3
square.
At this point we know that the approximate square
root is greater than two but less than three. What is
to be done with the three that remain unused? They
will be used to arrive at a closer approximation of the
square root in the following manner.
How many remain unused? Three.
How many are needed for the next tier? Five.
That is, 3/5 of the next tier can be constructed with
the available pieces. Therefore, the approximate root
is 2 3/5 or 2.6. This is very close to the calculated
square root: 2.65. The difference between the two is
less than 2%!
The result is that we have a 2 x 2 square.
The number whose square root is to be found forms
the area of the square. The square root is defined as
the measure of its side. This first square has an area
of four square units and the measure of its side is two
linear units.
But results are not always this neat. In this next
example the assignment is to find the square root of
seven. Seven congruent square objects must be used.
To form the fraction, the number left over is used
as the numerator and the number needed for the
next tier becomes the denominator.
Let’s continue to use this method to find the square
root of 13. This requires 13 pieces with square faces.
We quickly determine that a 3 x 3 square using nine
of the 13 pieces is the largest square that can be built.
The square root of 13 lies between three and four.
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© 2006 AIMS Education Foundation
Four pieces are left over. The next tier requires
seven. Therefore, the approximate square root of 13
is 3 4/7 or 3.57. This compares with the calculated
result of 3.60. This time the difference is 0.03 and
the approximation deviates from the actual by less
than 1%!
We are now in a position to generalize this method
by using our experience with concrete objects to guide
our thinking. Suppose we want to find the square root
of 45. What is the largest square that can be built
from 45 pieces? The answer is a 6 x 6 square. The
side such a square has a measure of six units and six
is the whole number of our estimated square root.
How many are left over? Nine pieces. Nine
becomes the numerator of the answer. How many are
required for the next tier? Thirteen pieces. Thirteen
becomes the denominator. The approximate square
root is 6 9/13 or 6.69. The calculated result is 6.71, a
difference of 0.02 or one-third of one percent. As the
numbers get larger, this method becomes more and
more accurate.
Since students readily learn the square numbers
when memorizing multiplication facts, this method
of estimating square roots utilizes student strength. If
the square numbers up to 12 x 12 = 144 are known,
the approximate square root of any number up to 168
can be determined by this method.
Mathematics may be thought of as the search
for an easier way… the pursuit of laziness. To make
this task even easier with larger numbers, we must
address the matter of finding the denominator in the
method used here.
Returning to using concrete objects, we note that
a new tier always uses the same number as are used
in the existing side twice (once at the right and once
at the top) plus one more to fill in the upper right
hand corner. Study this example to note the transition
from a 3 x 3 square to a 4 x 4 square.
If the largest square that has been built has a side
whose measure is 21, the next tier would require
2(21) + 1 = 43 pieces. This would be the denominator
of the fraction.
Algebra students are frequently asked to memorize
square numbers up to and including 25 squared. This
increases the range in which students can use this
method to mentally determine the square root.
The emergence of this algorithm from the concrete
model serves as a reminder that we need to look at
other instances in which algorithms can be derived
from the concrete.
How accurate are these approximations? In the
table below the percent of error occurring with this
approximation is listed for the square roots of the
numbers one to10. Thereafter, the highest percent of
error occurring in any decade is given for numbers
up to 60.
Actual
Approximate
Number Square Root Square Root Difference
3
1
Percent
of Error
1
1
1
0
0
2
1.414
1.333
0.081
5.71
3
1.732
1.667
0.065
3.77
4
2
2
0
0
5
2.236
2.2
0.036
1.61
6
2.449
2.4
0.049
2.02
7
2.646
2.6
0.046
1.72
8
2.828
2.8
0.028
1.0
9
3
3
0
0
10
3.162
3.143
0.019
0.61
12
3.464
3.429
0.035
1.02
20
4.472
4.444
0.028
0.61
30
5.477
5.455
0.022
0.41
42
6.481
6.462
0.019
0.29
56
7.483
7.467
0.016
0.22
For numbers above 60, the approximate result
differs from the actual by less than one-fifth of one
percent!
3
Therefore, to obtain the denominator, the length
of the side of the completed square must be doubled
and one added to the result. In the above example,
we double three and add one to obtain seven.
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© 2006 AIMS Education Foundation
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© 2006 AIMS Education Foundation
Your estimation of the square root and the calculator's calculation of
the square root should be very similar. Explain how you can use this
estimation method to estimate square roots of larger numbers.
1. Using the number of blocks listed on the chart make the largest
perfect square possible.
2. Record the length of the edge of the square and the number of the
remaining blocks.
3. Determine and record how many blocks it would take to fill the next
tier of blocks to make the next larger square.
4. Record the length of the complete square and the fraction of what
you have to make the next larger square and use a calculator to
record it as a decimal.
5. Use a calculator and record the square root of the number of blocks
listed on the chart.
1
1
2
3
16
15
14
13
12
11
10
9
8
7
6
5
4
1
1
2
1
0
3
3
3
13
2
13
1
13
0
No. Edge Length
Tiles
Total Tiles Perfect Sq.
of
of Full
Towards
Needed for Length and
Tiles
Perfect
Next Perfect Next Perfect Fraction
Square
Square
Square
of Next
1
Square
Root
on
Calculator
1.33... 1.41...
1
Decimal
Equivalent
of
Fraction
Use two-centimeter blocks or tiles (hex-a-link, uni-fix cubes, etc.) to discover a way to approximate a square root of a number.
Developing the Concept of Square Root