Some Mental Computation techniques
Mental computation is the process of finding an exact answer to a computation
without the use of any other computational aid.
Here are a few common mental computation techniques
Name
Count on and
Count Back
What
When
Use this technique
Count up or down by place
when the number to
value. For example, 352 – 3 be added or subtracted
would be calculated, “352,
is 1, 2, 3; 10, 20, 30;
351, 350.
100, 200, 300; and so
on.
How
Begin by saying the
larger number. Count
on to add or back to
subtract.
Choose
Compatible
Numbers
Select pairs of compatible
numbers (numbers that are
easy to compute mentally)
to perform the computation.
Use this technique if
one or more pairs of
numbers can be easily
added, subtracted,
multiplied, r divided,
or if they can produce
multiples of 10, 100,
or other numbers that
make calculations
easy.
First look for pairs of
numbers that are easy
to calculate. Perform
these calculations
first. Then look for
other numbe
combinations that can
be calculated easily.
Left to Right
Break apart the numbers
into their place values and
perform you work from
right to left.
Use this technique
when one of the
numbers is a single or
when most digit-bydigit computations are
simple.
Think about each
number in its
expanded form. Do
the calculations for
the largest place
values on down to the
smallest place values.
Now combine your
answers to each of the
smaller computations.
Examples:
5800+3100
1455-200
256+30
715+67+15
41125
63+18+27+12
345+130
2324
22121÷11
Use
Compensation
Substitute a compatible
number for one of the
numbers so that you can
complete the computation
mentally.
Use this technique
when a calculations
can be chosen that is
close to the original
one and that is easy to
do mentally.
Equal
Additions
Since the difference
between two numbers does
not change if the same
number is added to both of
the original numbers, select
an addend that will
transform the subtraction
into a recognizable
difference.
Use this when one of
the numbers in a
subtraction
calculation can be
changed to make the
resulting computation
easy to do mentally.
Change the original
calculation to one that
is easy to do mentally.
(Try not to change
more than one
number.) Keep track
of how you adjusted
the original
calculation, and find
the answer to the
original calculation.
Identify a number that
can be added to one
of the numbers in the
original calculation to
give a new
computation that is
easy to do. Then add
this number to both
numbers in the
original problem.
Examples:
65+38
196
908 - 39
3493÷7
145 – 77
1456 - 397
More mental mathematics – this time for comparing the magnitude of rational numbers.
a
, where a may be any
b
integer, and b may be any integer except the integer 0. The rational numbers may be viewed in
two interrelated ways:
Key Ideas: Every rational number may be expressed in the form
(Equations) The set of rational numbers in the smallest possible set of numbers that are
necessary for solving all possible equations of the form bx+a=0, where a and b are
integers, b≠0. (This point of view dominates high school mathematics or college algebra
courses.)
(Ratios) The set of rational numbers is the set of all possible ratios. Ratios come in two
forms: Part-to-Whole and Part-to-Part. (In elementary curriculum, the ratio view
dominates.) Our focus is on the Part-to-Whole representation.
Part-to-Whole Ratios
2
2
, a reasonable question to ask is, “ of what?” In order to relate
7
7
rational number to a whole number of objects, we must define a context in which a “whole” or
“unit” is clearly identifiable. Is the unit a loaf of bread? Is it a one dollar bill? Is it the distance
to the nearest mile from here to San Francisco? Is it the number of teaspoons in a cup? The
2
understanding of the number
changes with respect to the context. Thus when working with
7
rational numbers, it is quite important to clearly identify the whole (unit) you are considering.
When I write the ratio
Comparing Fractions – How can you determine the largest fraction between two given
fractions?
Using Benchmarks to Estimate
8
9
1
8
7
6
6
11
2
17
11
24
4
9
8
10
4
25
1.
How do you know when a fraction is close to 1?
2.
Which of the fractions above are close to 1? Among these, which are greater than 1?
3.
How do you know when a fraction is close to
4.
Which of the fractions above are close to
5.
How do you know when a fraction is close to 0?
6.
Which of the fractions above are close to 0?
7.
Which fraction in each pair listed below is larger? Do not find a common denominator! Use
1
the benchmarks 1 and to reason through each comparison. Is one fraction more than 1
2
1
1
while the other is less than 1? Is the fraction more than while the other is less than ?
2
2
Explain your thinking for each comparison.
a.
7
8
b.
c.
1
1
? Among these, which are greater than ?
2
2
d.
7
3
or
12
10
1
3
or
10
8
e.
5
1
or
9
3
4
13
or
3
15
f.
9
7
or
4
5
or
5
9
1
?
2
Using Reasoning to Compare Fractions
Many fractions can be compared by simply reasoning from a conceptual knowledge (a geometric
understanding) of fractions rather than by using a process to find a common denominator or by
using cross multiplication. Use the hints given to develop a thinking strategy based on
conceptual knowledge for comparing each pair of fractions.
1.
The fractions
and why?
7
3
and have the same size parts (the same denominator). Which is larger
8
8
7
7
and
both have the same number of parts (the same numerator). Which
9
12
is larger and why?
2.
The fractions
3.
Both of the fractions
larger and why?
4.
Both of the fractions
why?
5.
11
5
and are one fractional part away from being a whole. Which is
12
6
7
5
1
and are one fractional part away from . Which is larger and
12
8
2
Determine which fraction in each pair is larger by reasoning from your conceptual
knowledge of fractions. Explain your thinking for each comparison.
a.
b.
5
5
or
8
6
2
5
or
9
9
c.
23
3
or
24
4
d.
3
3
or
5
7
e.
3
5
or
8
12
f.
2
9
or
3
10