| There are a variety of strategies
that can be used to figure out mentally math and remember facts.
1. Addition “Counting On”
means starting with the first number of the equation and then just
counting forward the number of digits of the next number on a number line.
Example: 7 + 3: Start with “7” and count forward “3”
places on a number line. Or…look at a RULER…again, start with “7” and
then count forward “3” places (inches) and you end up at… You got it…”10”
*Tip: Since it is “addition,” it is generally easier
to start with the LARGER number and then “count on” from there. The
ability in addition to start with either number first is mentioned in the
Commutative Property Lesson. In the “number line” hand out example, it
just so happens that this IS the order anyway!
For SUBTRACTION:
"Counting Back"
Similarly, for facts such as 9 – 1, you can count
back. (Counting back is often harder to master than counting on. AND you
need to make sure you KEEP THE PROBLEM IN ORDER as well.) Refer to the
Number line handout again. Let’s look at another example…suppose we have
the equation: “12-2-5” Make your OWN number line. Make sure you include
“12” (on the right side of “0”). Again, you start at the first number on
the number line, which is – yes, “12.” Then you count BACK (or DOWN,
toward the Left) “2”…where do you end up? Yes, you are at “10”now…but are
you DONE? No….what’s next? We are now at “10” and we need to count back
“5” and where do we end up? You got it…we end up at “5” Super!
2. Doubles
If the numbers you are adding are almost the “same
number” than it may be easier to just “double” the number and then add or
subtract from there!
Let’s take the example: “15 + 14 = ?”
This is almost a double…(15 + 15, right?) and since
we know that “10 + 10 = 20” AND “5 = 5 = 10,” THEN we put the TWO TOGETHER
and we have “30,” RIGHT?...and since “14” is “1” less than “15,” we just
subtract “1” from out total and get “29.”
Let’s try another…
“98 + 101 = ?”
What would we do first using the DOUBLES
method?...Yes..”98” is extremely close to “100” right? OK..now what about
“101?” Isn’t this also closes to “100?” We therefore take “100” and
double it…we’re then at “200.” OK, now comes the fun part. We know that
“98” is “2 less than 100” so we take “2” away from our “double end product
of 200.” This now becomes “198.” OK…now are we done? NO. We must now look
at the 2nd number which was “101” and we know that this is “1”
more than 100. We then must ADD that one to our current end product of
“198” and we NOW get “199.” Wasn’t that EASY?
3. 2-digit Addition &
Subtraction with Mental Math
How to mentally add two two-digit numbers.
·
Add 83 and 35.
·
First add the two tens places (8
+ 3 = 11)
·
Next add the two ones places (3 +
5 = 8)
·
The sum of the ones places is
less than ten so the answer is 118
NOW…try another:
·
Add 95 and 67.
·
First add the two tens places (9
+ 6 = 15)
·
Next add the two ones places (5 +
7 = 12)
·
The sum of the ones places is
more than ten so
o
increase the tens sum by 1 (15 +
1 = 16)
o
decrease the ones sum by 10 (12 -
10 = 2)
·
Combine the tens and ones sums to
give the answer of 162
How to mentally subtract two two-digit numbers..
·
Subtract 35
from 84.
·
First subtract
the two tens place digits (8 - 3 = 5)
·
Notice that
the bottom ones digit is larger than the top ones digit. Decrease the
answer for the tens place by one (5 - 1 = 4) and increase the top ones
place by 10 (4 + 10 = 14).
·
Next subtract
the two ones place digits (14 - 5 = 9)
·
Put the tens
and ones place answers together to make 49.
·
Subtract 62
from 94.
·
First subtract
the two tens place digits (9 - 6 = 3)
·
Next subtract
the two ones place digits (4 - 2 = 2)
·
Combine the
tens and ones place differences to give the answer of 32
4. Using 10 to add 9
The place-value system makes adding 10 to a number easy – just increase
the digit in the tens place by 1. You can use this to help add 9 to a
number. Just add 10 to the number, then subtract 1. Let’s look at an
example:
9 + 9 + 5 = ?
We know that “9” is almost “10” and so we just take the first part of
the addition problem, “9 + 9” and we can use the “double mental math”
using “10.” So we take “10 + 10 = 20.” We then add the “5” on the end.
NOW we are at the product “25.” We know that both “9s” are 1 less each so
we take one away from BOTH nines and get “minus 2” from our product – and
what are we left with? YES… “23!”
You really can look like a MATHEMATICAL WIZARD to ALL!
Fact families
A fact family is a group of related facts using the same numbers. One
example would be 4 + 3 = 7, 3 + 4 = 7, 7 – 3 = 4, and 7 – 4 = 3. Fact
families are a very powerful tool for mastering facts; once you know one
fact in a family, you can work out the other facts in the same family.
Fact families are also useful for solving problems with missing addends,
such as 4 + __ = 7.
REVIEW of “0,1,2” Number Facts…
When zero (which is
really nothing) is added or subtracted to any number the answer is the
other number(s)
Adding one is just like
counting up by one. Subtracting one is just like counting down by one.
Adding two is like
counting up by two. Subtracting two is just like counting down by two.
“3 and 4 Addends” – An “addend” is just an extra
number that you use to get the total “addition” or the “answer. It is
just a term for the missing number.
For example: Adding three numbers (for example 4 + 8
+ 7) involves two steps.
·
Mentally add the first two
numbers. (4 + 8 = 12).
·
Add the other number to this sum.
(12 + 7 = 19).
*Tip: The figuring out of
an “addend” – for example when you take the above problem and change the
missing number of the equation:
___ + 8 + 7 = 19
In this case, the “addend”
is again the “missing number” and you can work the problem out backwards
by SUBTRACTION (8 + 7 = “15) . You have to remember to KEEP THE SAME
PROPERTY ON THE SAME SIDE OF THE EQUAL SIGN. (All the numbers to the LEFT
of the “=” sign are ADDED, just as it says.) And then to COMPLETE the
equation…
___ + 15 = 19
Mentally, you realize that
“15 plus what equals 19” and you get the “addend” or ANSWER of “4.”
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