Mental Math
(Level I)
C. David Pilmer
Nova Scotia School for Adult Learning
Adult Education
2007
This resource is the intellectual property of the Adult Education Division of the Nova Scotia
Department of Labour and Advanced Education.
The following are permitted to use and reproduce this resource for classroom purposes.
• Nova Scotia instructors delivering the Nova Scotia Adult Learning Program
• Canadian public school teachers delivering public school curriculum
• Canadian nonprofit tuition-free adult basic education programs
The following are not permitted to use or reproduce this resource without the written
authorization of the Adult Education Division of the Nova Scotia Department of Labour and
Advanced Education.
• Upgrading programs at post-secondary institutions
• Core programs at post-secondary institutions
• Public or private schools outside of Canada
• Basic adult education programs outside of Canada
Individuals, not including teachers or instructors, are permitted to use this resource for their own
learning. They are not permitted to make multiple copies of the resource for distribution. Nor
are they permitted to use this resource under the direction of a teacher or instructor at a learning
institution.
Table of Contents
Introduction to Mental Math…………………………………………………………………. 1
Addition Strategies………………………………………………………………………….. 5
Black Line Masters for Addition Facts……………………………………………………… 11
Commutative Property Flashcards…………………………………………………... 11
Doubles Flashcards………………………………………………………………….. 12
Nothing Changes Flashcards………………………………………………………… 14
Next Number Flashcards…………………………………………………………….. 16
Doubles Plus One Flashcards…………………………………………………………18
Next Even/Next Odd Flashcards…………………………………………………….. 20
Make Ten Flashcards………………………………………………………………… 22
Last Six Addition Facts Flashcards………………………………………………… 25
Addition Strategies List……………………………………………………………… 27
Multiplication Strategies…………………………………………………………………….. 28
Black Line Masters for Multiplication Facts………………………………………………… 37
Commutative Property Flashcards…………………………………………………... 37
Doubles Flashcards………………………………………………………………….. 38
Nifty Nines Flashcards………………………………………………………………. 40
Fun Fives Flashcards………………………………………………………………… 42
Tricky Zero Flashcards……………………………………………………………… 44
No Change Flashcards………………………………………………………………. 45
Thrilling Threes Flashcards…………………………………………………………. 46
Double and Double Flashcards……………………………………………………… 47
Last Six Multiplication Facts Flashcards……………………………………………. 48
Multiplication Strategies List………………………………………………………... 49
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Introduction to Mental Math
Mental math is the process of calculating the exact numerical answer without the aid of any
external calculating or recording device. Research shows that as adults over 80% of the
mathematics we encounter in our daily lives involves the mental manipulation of numerical
quantities rather than the traditional paper and pencil math so often stressed in schools. Most
learners feel that mental math is important however, they mistakenly believe that written math is
learned in school while mental math is learned outside of school.
Although learners may value mental math, they may not be able to perform even the most
straightforward calculations mentally. Consider that on the Third National Mathematics
Assessment, only 45% of 17 year olds were able to multiple 90 and 70 mentally. The findings
may indicate that little classroom time has been devoted to doing mental math. Another study
regarding performance on mental math of learners in grades 2, 4, 6, and 8 showed the range of
strategies selected by learners was very narrow and that the most popular strategy selected by
grade 4 and grade 8 learners reflected the learned paper/pencil strategy. Many learners were
even unable to propose an alternate strategy when prompted and even surprised that there were
alternate strategies.
Research states that mental math activities should be integrated into daily classroom practices,
rather than being taught as an isolated unit. The exposure should be gradual and continuous. For
the Level I Math program, we recommend that such activities serve as a 5 to 10 minute warm-up
at the beginning of a session. There are three reasons for this approach. The first is that mental
math is pervasive in the real-world and the curriculum. The continued classroom exposure to
mental math is meant to be reflective of the math that learners encounter in the real-world and in
the curriculum. The second reason deals with the memory demands of mental math. Mental
math tends to be easier for individuals whose addition and multiplication facts are firmly
entrenched in their long-term memory. If this is so, the working memory is available to work
flexibility with number and operations. For learners who have not retained the addition and
multiplication facts in their long-term memory, mental math can be very challenging. In these
cases, their working memory is consumed with determining the fact, and they have little time or
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space left to address the question that was asked. When such memory deficits occur, we, as
instructors, need to present the learners with a series of strategies that allow quick retrieval of
pertinent facts. This goes beyond asking learners to go back and memorize addition and
multiplication charts. Many adult learners have been unsuccessful with this approach, so more
innovative strategies are required; strategies that are less taxing on the long-term memory.
Consider the following examples.
(1) Using the Commutative Property
Many learners struggle with 9 × 3 because they attempt to figure out 9 sets of 3,
however, if they know 9 × 3 = 3 × 9 , then the question is more accessible to some.
Using the commutative property cuts the long-term memory requirements for the
addition and multiplication tables almost in half.
(2) Doubles Plus One
Most learners are comfortable with doubles so this can be used to their advantage
when attempting to answer mental math questions like 6 + 7 . The learner can think
of this question as 6 + 6 + 1 or “double 6 plus 1.” If double 6 is 12, then the answer to
6 + 7 is 13.
(3) Make Ten
Most learners are comfortable adding 10 to a whole number. This skill can be used to
their advantage when attempting to answer mental math questions like 9 + 5 . The
learner can take 1 from 5 and add it to the 9. The question now becomes 10 + 4
which equals 14.
(4) Nifty Nines
Multiplying by nines can be difficult for a lot of learners but there is a pattern in the
products that one can exploit to make it far less taxing on long-term memory.
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9 × 9 = 81
8 × 9 = 72
7 × 9 = 63
6 × 9 = 54
5 × 9 = 45
4 × 9 = 36
3 × 9 = 27
Notice that the product’s tens digit is one less than the
first factor.
Ex. 8 × 9 = 72
Also notice that the sum of the digits for any of these
products is 9.
Ex. 8 × 9 = 72
7+2=9
2 × 9 = 18
Although these strategies, and the others that you will be exposed to in this resource, reduce the
demands on the long-term memory, learners will still require time and practice to solidify the
strategies in their long-term memory. If this is accomplished then learners can develop a higher
level of automaticity. A considerable amount of time should be spent discussing the strategies
that the learners decided to employ. This is of particular importance when learners start to
accumulate a greater number of varying strategies. The third reason for the gradual yet
continuous approach to teaching of mental math focuses on learner success. Many of our Level I
students have had little success in mathematics. We need the mental math experience to be
positive so that the learner can feel empowered. The success in mental math can be used to
breed further success in their future mathematical pursuits.
For the Level I program, we are going to limit the scope of mental math to addition and
multiplication of whole numbers. This will include whole numbers which are multiples of 10.
Examples:
4+5
400 + 500
3× 7
3× 70
Some of you may wish to introduce mental estimation questions to your learners. Although
these estimation skills are highly beneficial in the real-world, we were concerned that these
additional demands may be too taxing on our instructors and learners at this time. If you did
choose to include estimation questions, our recommendation is to limit it to numbers that
learners would associate with money.
Examples:
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2.99 + 7.99
4 × 3.99
3
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In terms of using mental math questions in your classroom, we have four suggestions.
(1) The first one was previously recommended. Mental math activities should be gradual
and continuous therefore we suggest that these types of activities could serve as a
warm-up activity at the beginning of each class.
(2) The second suggestion is that strategies and the associated facts are presented after
learners have obtained conceptual understanding of the operations with whole
numbers. This may require that the learners examine patterns, use manipulatives,
and/or work in familiar contexts before the mental math strategies are introduced.
(3) The third suggestion looks at the presentation of the mental math question. Merely
saying the question aloud may not serve the needs of all learners. A visual
component must also be provided either on the board, overhead, or flash cards.
(4) The fourth suggestion focuses on the learner’s method of reporting their answer. We
suggest using small hand-held white boards. Learners can write the solution to the
problem on the board and then turn the board in the instructor’s direction. This
technique allows the instructor to quickly scan the responses and ascertain the level of
student understanding. This method eliminates the possibility of embarrassing an
adult learner whose response is incorrect, but still allows for discussions of correct
solutions.
In closing, mental math activities can be very empowering to adult learners and should be used.
If successfully done, these new skills can ultimately facilitate the development of other math
concepts.
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Addition Strategies
For many adult learners, the addition table is comprised of 100 disconnected facts that seem
impossible to memorize. Many would rather resort to the highly inefficient counting strategy
that has offered them some level of success in the past.
+
0
1
2
3
4
5
6
7
8
9
0
0
1
2
3
4
5
6
7
8
9
1
1
2
3
4
5
6
7
8
9
10
2
2
3
4
5
6
7
8
9
10
11
3
3
4
5
6
7
8
9
10
11
12
4
4
5
6
7
8
9
10
11
12
13
5
5
6
7
8
9
10
11
12
13
14
6
6
7
8
9
10
11
12
13
14
15
7
7
8
9
10
11
12
13
14
15
16
8
8
9
10
11
12
13
14
15
16
17
9
9
10
11
12
13
14
15
16
17
18
When you look at the table above, you can understand why visually it is overwhelming for many
adult learners. You need to convince your learners that these 100 facts can be reduced to 7
strategies, leaving only 6 facts that need to be memorized in isolation of a clear strategy.
Introduce the strategies gradually and in the order supplied in this document. In the first two or
three days, you may be able to introduce the first four strategies. The subsequent strategies will
take much longer to introduce and master. Not only are these new strategies more demanding
but you will have to continually reinforce previously learned strategies. Combining the
flashcards from the different strategies is the easiest way to reinforce old strategies. May sure
that students discuss which strategy is employed to solve each question. Don’t be surprise if
students use different strategies. For example some students will handle 9 + 9 using the doubles
strategy while other will use the make ten strategy (9 + 9 = 10 + 8). In this case, both are
perfectly acceptable. It may be useful to keep a list of the strategy names (Doubles, Nothing
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Changes, Next Number, …) on the board for your learners. This will make it easier for them to
recall which strategy they used. It should be stressed that you don’t overuse the flashcards.
There are other techniques that can be used to practice the various strategies and increase
automaticity. One suggestion is to have two spinners divided into ten equal increments labeled 0
through 9. Both spinners are used simultaneously and the learner must find the sum of the two
random numbers generated by the spinners.
Strategy 1: Commutative Property of Addition (Flip Flop)
Changing the order in which two numbers are added has no effect on the sum.
a+b =b+a
Examples:
5+8=8+5
7+3=3+7
1+9=9+1
This strategy is very useful to learners because the 100 facts can now be viewed as only 55 facts.
The revised addition table makes this far easier to visualize. This visual should probably be
presented to your learners.
+
0
1
2
3
4
5
6
7
8
9
0
0
1
2
3
4
5
6
7
8
9
2
3
4
5
6
7
8
9
10
4
5
6
7
8
9
10
11
6
7
8
9
10
11
12
8
9
10
11
12
13
10
11
12
13
14
12
13
14
15
14
15
16
16
17
1
2
3
4
5
6
7
8
18
9
There are flashcards provided in the black line masters that can be used to practice the
commutative property. The use of the commutative property will be stressed throughout this
resource. One instructor referred to this strategy as the “flip flop” strategy because she found it
easier for her learners to remember.
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Strategy 2: Doubles
Doubles are a good place to start because many adults are already familiar with this concept
although some may not realize this. Here are a few examples where adults may have
encountered doubles.
- playing basketball (scoring)
- counting change (often more efficient to count by two’s)
- gambling (double or nothing)
Whatever the real-world context, adult learners should find this starting point fairly familiar and
straight-forward.
1+1 = 2
2+2 = 4
3+3 = 6
4+4 =8
5 + 5 = 10
6 + 6 = 12
7 + 7 = 14
8 + 8 = 16
9 + 9 = 18
This will be extended to numbers that are multiples of 10.
Examples:
30 + 30 = 60
700 + 700 = 1400
Strategy 3: Nothing Changes Strategy
When we add 0 to something, nothing changes. If you have 6 apples and you add 0 apples, you
still end up with 6 apples. Nothing changes.
0 +1 = 1
0+2 = 2
0+3=3
0+4 = 4
1+ 0 = 1
2+0 = 2
3+0 = 3
4+0 = 4
0+5=5
5+0 = 5
0+6 = 6
6+0 = 6
0+7 = 7
7+0=7
0+8 =8
8+0 =8
0+9 =9
9+0 =9
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Notice that commutative property is used here and in all the strategies that follow.
As with each of the strategies, this will be extended to numbers that are multiples of 10.
0 + 50 = 50
Examples:
600 + 0 = 600
Strategy 4: Next Number Strategy
When you add 1 to a whole number, it means that you just find to next whole number.
1+ 2 = 3
1+ 3 = 4
1+ 4 = 5
2 +1 = 3
3 +1 = 4
4 +1 = 5
and so on…
Extend this to numbers that are multiples of 10.
10 + 80 = 90
Examples:
500 + 100 = 600
Strategy 5: Doubles Plus One Strategy
This strategy really combines strategy 2 and strategy 4. It is used with questions like 2 + 3 ,
where the two numbers being added together differ by one. For this example, the learner should
mentally change 2 + 3 into 2 + 2 + 1 . By doing so, the learner just needs to double 2 and then
go to the next number.
2 + 3 = 2 + 2 +1
= 4 +1
=5
This can now be applied to a variety of sums.
3 + 4 = 3 + 3 +1
=7
5 + 6 = 5 + 5 +1
= 11
4 + 5 = 4 + 4 +1
=9
and so on…
4 + 3 = 3 + 3 +1
=7
5 + 4 = 4 + 4 +1
=9
6 + 5 = 5 + 5 +1
= 11
Extend this to numbers that are multiples of 10.
Examples:
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60 + 70
800 + 700
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Strategy 6: Next Even/Next Odd Strategy
When you add 2 to a whole number you just have to find the next even or odd number depending
on the number you start with. If you start with an even number and add 2, then the sum is the
next even number. If you start with an odd number and add 2, then the sum is the next odd
number.
(Even)
2+4 = 6
2+6 =8
2 + 8 = 10
4+2 = 6
6+2 =8
8 + 2 = 10
2+7 =9
7+2=9
2 + 9 = 11
9 + 2 = 11
(Odd)
2+5= 7
5+2 = 7
Extend this to numbers that are multiples of 10.
Examples:
20 + 70
600 + 200
Strategy 7: Make Ten Strategy
This strategy is introduced in two phases. In the first phase learners use this strategy to add 9 to
a whole number. Once they have mastered this, the second phase is initiated. In this new phase
learners use this strategy to add 8 to a whole number.
This strategy relies on the fact that most learners are comfortable adding 10 to a whole number.
This skill can be used to their advantage when attempting to answer mental math questions like
9 + 7 . The learner can take 1 from 7 and add it to the 9. The question now becomes 10 + 6
which equals 16.
9 + 3 = 12
3 + 9 = 12
9 + 4 = 13
4 + 9 = 13
9 + 5 = 14
5 + 9 = 14
and so on…
Extend this to numbers that are multiples of 10.
Examples:
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90 + 60
700 + 900
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Similarly this same strategy can be used with questions like 8 + 6 . The learner can take 2 from 6
and add it to 8. The question now becomes 10 + 4 which equals 14.
8 + 3 = 11
8 + 4 = 12
8 + 5 = 13
3 + 8 = 11
4 + 8 = 12
5 + 8 = 13
and so on…
Extend this to numbers that are multiples of 10.
Examples:
80 + 70
300 + 800
Last Six Facts
The last six facts don’t fit nicely into any one strategy. Some learners will just memorize these
facts. Others learners will develop new strategies like doubles plus two or use existing strategies
like make ten.
5+3=8
3+5 = 8
6+3=9
3+6 = 9
6 + 4 = 10
4 + 6 = 10
7 + 3 = 10
3 + 7 = 10
7 + 4 = 11
4 + 7 = 11
7 + 5 = 12
5 + 7 = 12
Extend this to numbers that are multiples of 10.
Examples:
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50 + 30
400 + 700
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Commutative Property Flashcards
5+7
can
3+2
can
8+4
can
be written as: be written as: be written as:
6+5
can
0+9
can
5+8
can
be written as: be written as: be written as:
1+7
can
7+2
can
9+4
can
be written as: be written as: be written as:
3+5
can
2+6
can
4+3
can
be written as: be written as: be written as:
7+2
can
8+1
can
6+9
can
be written as: be written as: be written as:
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Doubles Flashcards
1+1
2+2
3+3
4+4
5+5
6+6
7+7
8+8
9+9
10 + 10
20 + 20
30 + 30
40 + 40
50 + 50
60 + 60
70 + 70
80 + 80
90 + 90
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100 + 100 200 + 200 300 + 300
400 + 400 500 + 500 600 + 600
700 + 700 800 + 800 900 + 900
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Nothing Changes Flashcards
0+1
0+2
0+3
0+4
0+5
0+6
0+7
0+8
0+9
1+0
2+0
3+0
4+0
5+0
6+0
7+0
8+0
9+0
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0 + 20
30 + 0
70 + 0
0 + 80
50 + 0
0 + 40
600 + 0
0 + 900
100 + 0
0 + 200
300 + 0
800 + 0
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Next Number Flashcards
1+2
1+3
1+4
1+5
1+6
1+7
1+8
1+9
2+1
3+1
4+1
5+1
6+1
7+1
8+1
9+1
30 + 10
50 + 10
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10 + 40
80 + 10
10 + 70
100 + 300 500 + 100 600 + 100
100 + 200 800 + 100 400 + 100
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Doubles Plus One Flashcards
2+3
3+4
4+5
5+6
6+7
7+8
8+9
3+2
4+3
5+4
6+5
7+6
8+7
9+8
30 + 20
40 + 50
70 + 60
90 + 80
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60 + 50
70 + 80 200 + 300
500 + 400 900 + 800 800 + 700
500 + 600 300 + 400 700 + 600
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Next Even/Next Odd Flashcards
2+4
2+5
2+6
2+7
2+8
2+9
4+2
5+2
6+2
7+2
8+2
9+2
20 + 40
50 + 20
70 + 20
20 + 80
90 + 20
20 + 60
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400 + 200 200 + 900 500 + 200
200 + 700 600 + 200 200 + 800
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Make 10 Flashcards
Phase 1: From 9
9+3
9+4
9+5
9+6
9+7
9+8
3+9
4+9
5+9
6+9
7+9
8+9
90 + 30
40 + 90
90 + 50
90 + 60
70 + 90
90 + 80
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300 + 900 900 + 400 500 + 900
600 + 900 900 + 700 800 + 900
Phase 2: From 8
8+3
8+4
8+5
8+6
8+7
3+8
4+8
5+8
6+8
7+8
80 + 30
80 + 40
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50 + 80
80 + 60
70 + 80
300 + 800 400 + 800 800 + 500
800 + 500 600 + 800 800 + 700
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Last Six Addition Facts Flashcards
5+3
6+3
6+4
7+3
7+4
7+5
3+5
3+6
4+6
3+7
4+7
5+7
50 + 30
30 + 60
60 + 40
70 + 30
40 + 70
70 + 50
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300 + 500 600 + 300 400 + 600
300 + 700 700 + 400 500 + 700
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Addition Strategies List
1. Commutative Property
(Flip Flop)
2. Doubles
3. Nothing Changes
4. Next Number
5. Doubles Plus One
6. Next Even/Next Odd
7. Make Ten
(Last Six Facts)
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Multiplication Strategies
For many adult learners, the multiplication table is comprised of 100 disconnected facts that
seem impossible to memorize. Many would rather resort to the highly inefficient repeated
adding strategy that has offered them some level of success in the past.
×
0
1
2
3
4
5
6
7
8
9
0
0
0
0
0
0
0
0
0
0
0
1
0
1
2
3
4
5
6
7
8
9
2
0
2
4
6
8
10
12
14
16
18
3
0
3
6
9
12
15
18
21
24
27
4
0
4
8
12
16
20
24
28
32
36
5
0
5
10
15
20
25
30
35
40
45
6
0
6
12
18
24
30
36
42
48
54
7
0
7
14
21
28
35
42
49
56
63
8
0
8
16
24
32
40
48
56
64
72
9
0
9
18
27
36
45
54
63
72
81
When you look at the table above, you can understand why visually it is overwhelming for many
adult learners. You need to convince your learners, that these 100 facts can be reduced to 8
strategies, leaving only 6 facts that need to be memorized in isolation of a clear strategy.
It is important to introduce the strategies gradually and in the order supplied in this document.
This is a long process and must proceed at a rate dictated by successful learner performance. Be
prepared that students may use different yet valid strategies on the same question. For example
the question 2 × 9 can be solved using the doubles strategy or the nifty nines strategy. It is
important to initiate discussions so that all learners understand what valid strategy or strategies
can be employed.
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Strategy 1: Commutative Property of Multiplication (Flip Flop)
Changing the order in which two numbers are multiplied has no effect on the product.
a×b = b×a
Examples:
5 × 8=8 × 5
7 × 3=3 × 7
1 × 9=9 × 1
This strategy is very useful to learners because the 100 facts can now be viewed as only 55 facts.
The revised multiplication table makes this far easier to visualize. This visual should be
presented to your learners.
×
0
1
2
3
4
5
6
7
8
9
0
0
0
0
0
0
0
0
0
0
0
1
2
3
4
5
6
7
8
9
4
6
8
10
12
14
16
18
9
12
15
18
21
24
27
16
20
24
28
32
36
25
30
35
40
45
36
42
48
54
49
56
63
64
72
1
2
3
4
5
6
7
8
81
9
There are flash cards for the commutative property found in the black line master section of this
document. One instructor referred to this strategy as the “flip flop” strategy because she found it
easier for her learners to remember.
Strategy 2: Doubles
This should be a fairly easy strategy for two reasons. The first is most adults have some intuitive
understanding of doubles and the second is that this strategy was stressed when learning the
addition strategies.
2 ×1 = 2
2× 2 = 4
2×3 = 6
2× 4 = 8
2 × 5 = 10
2 × 6 = 12
2 × 7 = 14
2 × 8 = 16
2 × 9 = 18
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Extend this to numbers that are multiples of 10. Notice that only one of the numbers is a
multiple of 10.
Examples:
2 × 30
400 × 2
200 × 8
Strategy 3: Nifty Nines
There are three different strategies learners can use to figure out the product of 9 and a whole
number. You can choose the method that best suits your learners. We’re going to call them the
See the Pattern strategy, the Finger strategy, and the Times Ten Less One Set strategy. We’d
advocate the use of the first strategy.
See the Pattern Strategy
9 × 9 = 81
8 × 9 = 72
7 × 9 = 63
6 × 9 = 54
5 × 9 = 45
4 × 9 = 36
3 × 9 = 27
2 × 9 = 18
Notice that the product’s tens digit is one less than the
first factor.
Ex. 8 × 9 = 72
Also notice that the sum of the digits for any of these
products is 9.
Ex. 8 × 9 = 72
7+2=9
This technique will require significant verbal coaching before learners can do it
independently.
The Finger Strategy
Many of you may be familiar with this technique. It’s really exploits the pattern
seen in the previous strategy but doesn’t require that the learners see the pattern
for themselves. The learner holds up ten fingers. If they are asked to multiply 9
by 6, then they turn down their sixth finger (counting from left to right). There
are now five fingers to the left of the down-turned finger and four fingers to the
right of the down-turned finger. The first five fingers represent the tens digit.
The last four fingers represent the ones digit. Therefore 9 × 6 = 54 .
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Times Ten Less One Set Strategy
Many learners are comfortable multiplying 10 by a whole number. This can be
used to their advantage when learning multiplication facts involving 9. If they are
asked to find the product of 9 and 6, then ask them to find the product of 10 and 6,
then reduce the answer by 6.
9 × 6 = 10 × 6 − 6
= 60 − 6
= 54
This may not work well for all students particularly if subtracting mentally is a
challenge for the learner.
9 × 3 = 27
9 × 4 = 36
9 × 5 = 45
9 × 6 = 54
9 × 7 = 63
9 × 8 = 72
9 × 9 = 81
Regardless of the strategy you decide to teach, you should eventually extend to numbers that are
multiples of 10. Notice that only one of the numbers is a multiple of 10.
Examples:
9 × 60
400 × 9
900 × 8
Strategy 4: Fun Fives
There are two different strategies learners can use to figure out the product of 5 and a whole
number. You can choose the method that best suits your learners.
The Five Pattern Strategy
Many adult learners have some level of comfort when it comes to multiplying by
5. Some learners will know that when you multiply five by an even number the
ones digit for the product must always be a 0. These same individuals would
probably know that when you multiply five by an odd number the ones digit for
the product must always be a 5. Another could be related to the fact that many
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adults are comfortable counting by fives due to the “almost rhyming” nature of
the sequence (5, 10, 15, 20, 25, …).
The Clock Strategy
For a small group of adults, however, multiplying by 5 can be problematic. One
can use the learner’s knowledge of analog clocks to simplify the process. When
the minute hand is on 3, you know that it’s 15 minutes after the hour. When the
minute hand is on 7, you know that it’s 35 minutes after the hour. Based on these
two examples, you can illustrate the following two multiplication facts.
5 × 3 = 15
5 × 7 = 35
This connection to analog clocks can allow for quick retrieval of the
multiplication facts of 5.
5 × 3 = 15
5 × 4 = 20
5 × 5 = 25
5 × 6 = 30
5 × 7 = 35
5 × 8 = 40
5 × 9 = 45
Extend this to numbers that are multiples of 10. Notice that only one of the numbers is a
multiple of 10.
Examples:
5 × 60
300 × 5
500 × 8
Strategy 5: Tricky Zero Strategy
Learners often confuse the rule for multiplying by 0 with the rule for adding 0. For the question
5 × 0 or 0 × 5 , learners need to remember that these mean 5 sets of 0, or 0 sets of 5. In either
case, the answer is 0. This can seem counter intuitive for some learners because they start with 5
and end up with nothing. That’s why we’ve called this the “tricky” zero strategy.
0×0 = 0
0 ×1 = 0
0× 2 = 0
0×3 = 0
0× 4 = 0
0×5 = 0
0×6 = 0
0×7 = 0
0×8 = 0
0×9 = 0
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Extend this to numbers that are multiples of 10.
0 × 70
Examples:
500 × 0
Another way to reinforce this concept is to make an analogy to snakes. Ask the learners the
following three questions. How many legs does one snake have? How many legs do three
snakes have? How many legs do five snakes have? You can now connect these questions and
their responses to the following three questions; 1 × 0 = 0 , 3 × 0 = 0 , and 5 × 0 = 0 . If you
choose to explain the process in this manner you may want to call this the Snake Legs Strategy,
rather than the Tricky Zero Strategy.
You may want to introduce strategy 5 and 6 on the same day.
Strategy 6: No Change Strategy
The product of 1 and another number will be the other number. There is no change. For the
question 4 × 1 or 1× 4 , learners should think of it as 4 sets of 1, or 1 set of 4. In either case, the
answer is 4. We started with 4 and ended with 4; there was no change. We didn’t call it the
Nothing Changes Strategy because we used that term in the addition strategies when explaining
the sum of zero and another number.
1×1 = 1
1× 2 = 2
1× 3 = 3
1× 4 = 4
1× 5 = 5
1× 6 = 6
1× 7 = 7
1× 8 = 8
1× 9 = 9
Extend this to numbers that are multiples of 10. Notice that only one of the numbers is a
multiple of 10.
Examples:
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500 × 1
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Strategy 7: Thrilling Threes
There are two different strategies learners can use to figure out the product of 3 and a whole
number. You can choose the method that best suits your learners.
Double and One More Set
This method is used when multiplying 3 by another number. If you have 3× 7 ,
then it can be expressed as 3 sets of 7, or as double 7 plus one more 7. The
second expression, double 7 plus one more 7, gives you 14 + 7 which is 21. This
strategy relies on a typical learner strength, doubling.
3× 7 = 2 × 7 + 7
= 14 + 7
= 21
Use this strategy to calculate each of the following.
3× 3 = 9
3 × 4 = 12
3 × 7 = 21
3 × 8 = 24
3 × 6 = 18
Tic Tac Toe Threes
The double and one more set strategy can be difficult for students who have
difficulties doing mental addition. An alternate method, tic tac toe threes, allows
the learner to quickly generate a 3 by 3 grid that contains the multiplication facts
for 3. This is the only method that requires the learner to generate the facts using
pencil and paper. The learner would be expected to quickly generate the grid
prior to do any mental multiplication activities.
Step 1: Draw a tic tac toe grid
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Step 2: Starting in the lower left hand corner, moving up and then over to the next
column, fill in the numbers 1 to 9.
3
6
9
2
5
8
1
4
7
Step 3: Take all the numbers in the middle row and give them a tens digit of 1.
Take all the numbers in the bottom row and give them a tens digit of 2
3
6
9
12
15
18
21
24
27
If you look at the grid, you will notice that you have all the multiplication
facts for 3.
3×1 = 3
3 × 2 =6
3×3 = 9
3 × 4 = 12
etc.
Regardless of the method you decide to teach, you should eventually extend to numbers that are
multiples of 10. Notice that only one of the numbers is a multiple of 10.
Examples:
3× 40
600 × 3
300 × 7
Strategy 8: Double and Double
Multiplying a number by 4 is the same as doubling the number and then doubling that
new answer. If you have 4 × 6 , then you double 6 and then double that answer.
4 × 6 = 2(2 × 6 )
= 2(12 )
= 24
This strategy works nicely for 4 × 4 , 4 × 6 , and 4 × 7 . It is, however, more challenging
for 4 × 8 because many learners find it difficult to mentally double 16.
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Extend this to numbers that are multiples of 10. Notice that only one of the numbers is a
multiple of 10.
Examples:
4 × 60
800 × 4
400 × 7
Last Six Multiplication Facts
The last six facts don’t fit nicely into any one strategy. Most students will just memorize these.
6 × 6 = 36
6 × 7 = 42
6 × 8 = 48
7 × 7 = 49
7 × 8 = 56
8 × 8 = 64
Extend this to numbers that are multiples of 10. Notice that only one of the numbers is a
multiple of 10.
Examples:
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7 × 60
800 × 6
36
700 × 8
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Commutative Property Flashcards
5×7
can
3×2
can
8×4
can
be written as: be written as: be written as:
6×5
can
0×9
can
5×8
can
be written as: be written as: be written as:
1×7
can
7×2
can
9×4
can
be written as: be written as: be written as:
3×5
can
2×6
can
4×3
can
be written as: be written as: be written as:
7×2
can
8×1
can
6×9
can
be written as: be written as: be written as:
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Doubles Flashcards
2×1
2×2
2×3
2×4
2×5
2×6
2×7
2×8
2×9
1×2
3×2
4×2
5×2
6×2
7×2
8×2
9×2
20 × 2
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2 × 30
4 × 20
50 × 2
2 × 60
70 × 2
8 × 20
20 × 9
200 × 2
3 × 200
400 × 2
5 × 200
2 × 600
200 × 7
2 × 800
900 × 2
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Nifty Nines Flashcards
3×9
4×9
5×9
6×9
7×9
8×9
9×9
9×3
9×4
9×5
9×6
9×7
9×8
30 × 9
9 × 40
5 × 90
90 × 6
70 × 9
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90 × 8
9 × 90
9 × 300
400 × 9
5 × 900
900 × 6
9 × 700
800 × 9
900 × 9
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Fun Fives Flashcards
5×3
5×4
5×5
5×6
5×7
5×8
3×5
4×5
6×5
7×5
8×5
5 × 30
40 × 5
50 × 5
6 × 50
70 × 5
5 × 80
500 × 3
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500 × 4
5 × 500
700 × 5
800 × 5
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Tricky Zero Flashcards
0×0
1×0
0×2
3×0
4×0
0×5
0×6
0×7
8×0
0×9
20 × 0
0 × 40
70 × 0
0 × 90
300 × 0
0 × 500
600 × 0
0 × 800
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No Change Flashcards
1×1
1×2
3×1
4×1
1×5
1×6
7×1
8×1
1×9
1 × 20
40 × 1
70 × 1
1 × 90
300 × 1
1 × 500
600 × 1
1 × 800
100 × 1
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Thrilling Threes Flashcards
3×3
3×4
3×6
3×7
3×8
4×3
6×3
7×3
8×3
3 × 30
40 × 3
3 × 60
70 × 3
30 × 8
300 × 3
600 × 3
300 × 7
3 × 800
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Double and Double Flashcards
4×4
4×6
4×7
4×8
4 × 40
60 ×4
40 × 7
8 × 40
400 × 4
4 × 600
700 × 4
400 × 8
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Last Six Multiplication Facts Flashcards
6×6
6×7
6×8
7×7
7×8
8×8
7×6
8×6
8×7
6 × 60
70 × 6
60 × 8
70 × 7
80 × 7
600 × 6
700 × 7
7 × 800
8 ×600
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Multiplication Strategies List
1. Commutative Property
(Flip Flop)
2. Doubles
3. Nifty Nines: _____________
4. Fun Fives: ______________
5. Tricky Zero (Snake Legs)
6. No Change
7. Thrilling Threes: ________
8. Double and Double
(Last Six Facts)
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