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BASIC FACTS are the facts that are memorized and
then used to figure out the facts that are not
memorized. They are basic because they are the
basis for all the other facts. For each
operation, 100 facts are generally considered to
be basic facts.
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A CURRENT ISSUE in elementary mathematics
education is When teaching basic facts, what
should we be emphasizing? strategies? or
relationships?
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• Relationships are the connections that provide a
  meaningful context for mathematical concepts and
  skills.
• Relationships are the basis for number sense.

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• Strategies are step-by-step procedures, based on
  useful relationships, that will efficiently
  produce a desired result.
• Effective strategies usually are very automatic,
  requiring a minimum of decision-making as they
  are executed.

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• Recall, that the issue is Should we be
  emphasizing strategies or relationships?
• We should emphasize both.
• Relationships and strategies both have an
  important role.

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Now, we will examine some fact strategies for the
hard basic addition facts. Although there are
many strategies for finding answers to hard basic
addition facts, the strategies that are most
successful in resulting in memorization of those
facts all have two common characteristics.
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The two common characteristics of successful
strategies for finding answers to hard basic
facts are1. They are mental strategies (not
pencil/paper or mechanical strategies.2. They
always require the child to use facts that are
already memorized.
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As we consider selected strategies, look for
these two characteristics.1. They are mental
strategies (not pencil/paper or mechanical
strategies.2. They always require the child to
use facts that are already memorized.
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Three successful strategies for figuring out
answers to hard addition facts are
One More Building on Doubles Make TenWe will
examine each of these strategies.
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The easiest strategy for children to use is the
One More strategy.This strategy requires the
lowest level of knowledge and skills. It is also
very easy for children to visualize.
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This fact is one more than a fact that you
already know.
5 6
13
This fact is one more than a fact that you
already know.
5 6 is
5 5
and one more
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Although this strategy is easy, it requires that
the child already know a specific fact before
finding the answer to the target fact. If the
child does not know the required fact, then the
child cannot use this strategy to find the target
fact.
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Is this fact one more than a fact that you
already know?
To use the One More strategy, the target fact
must be one more that a fact that you know.
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Consider this fact.
7 6
What fact do I already need to know to use the
One More Strategy?
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Consider this fact.
7 6
What fact do I already need to know to use the
One More Strategy?
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Although the One More strategy is easy for the
children, it is not easy for a teacher to use
with a group of more than 2 or 3 children.When
using this strategy, it is necessary for the
teacher to be aware, precisely, which facts each
child has already mastered. Otherwise, the
teacher cannot know which facts the child is
ready to figure out.With a full class of
children, it is virtually impossible for a
teacher to keep up with which facts the
individual children have mastered. This strategy
is not recommended for classroom
teachers.However, this strategy is recommended
for individuals of small groups, particularly in
a remedial setting.
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A strategy that many teachers have found to be
effective is the Doubles strategy.Since the
doubles are typically among the facts the
children master early, this strategy requires
children to use the doubles to figure out other
hard basic addition facts.
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For each target fact, the child thinks of a
double that is close to the target fact.
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This fact is close to a double that you already
know.
8 7
What double that you already know is this fact
close to?
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This fact is close to 7 7.
8 7 7 7
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This fact is close to 7 7.
8 7 is 7 7
and 1 more.
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Is this fact close to a double that you already
know?
What double is this fact close to?
6 7
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6 7 is close to 6 6.
6 7 is 6 6
and 1 more.
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The Doubles strategy has been a relatively
successful strategy for the hard basic addition
facts. It is a generally easy strategy for facts
that are within one or two of a known double (for
example, 65, 79, or 87). However, it is not an
easy strategy for facts that are more than two
away from a double (for example, 58, 48, or
96).Consequently the Doubles strategy is often
used together with several other strategies and
children choose the best strategy for each
specific fact. This often results in confusion
at that point in the process where the child must
choose a strategy.
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A strategy that many teachers feel may be most
effective is the Make Ten strategy.Since the
tens are typically among the facts the children
master early, this strategy requires children to
use the tens to figure out other hard basic
addition facts.
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In this strategy, the child will make ten and see
how many are left over. Teachers often use a
device called a Ten-Frame to help children
visualize the process.
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We can find the answer to this fact by making ten.
5 6
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Place the larger number in a ten frame.
5 6
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Use part of the other number to fill the ten
frame.
5 6
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Then you can look at what is left outside the
ten-frame and tell what the answer is.
5 6 10 1
11
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We can find the answer to this fact by making ten.
8 4
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Place the larger number in the ten-frame.
8 4
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Fill the ten-frame with part of the other number.
8 4
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What is left outside tells you the answer. You
have 10 and 2 more.
8 4 12
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What number goes in the ten-frame?
6 9
How many more do you have to move?
How many are left outside?
Whats the answer?
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What number goes in the ten-frame?
8 5
How many more do you have to move?
How many are left outside?
Whats the answer?
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The Make Ten strategy is very successful strategy
for several reasons. It is a universal strategy
(it will always work) for all hard basic addition
facts. The children only need to learn a single
strategy.The Make Ten strategy is directly
related to the base-10 numeration system.
Numbers less than one hundred are always named as
tens and ones.Finally, the Make Ten strategy
lays groundwork for regrouping in addition, that
children will learn to use when adding
multi-digit numbers.
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We will now examine three strategies that have
proven to be effective for figuring out answers
to hard subtraction facts Think of Related
Addition One More Subtract from Ten
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The first of these strategies is the Think of
Related Addition strategy.This strategy
requires the child to think of an addition fact
that is related to the target subtraction
fact.Of course, this will only be helpful if
the child knows that related addition fact.
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Think of a related addition fact.
12 - 7 ?
? 7 12
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Think of a related addition fact.
15 - 8 ?
? 8 15
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Think of a related addition fact.
12 - 5 ?
? 5 12
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The Think of Related Addition strategy is
popular among many teachers because it emphasizes
the very important relationship between addition
and subtraction. It is a universal strategy (it
will always work) for all hard basic subtraction
facts. The greatest drawback of this strategy
is that children who are having difficulty with
15 - 8 7 usually do not know 7 8 15 either.
Children who are having difficulty with 12 - 5
7 usually do not know 7 5 12 either.
Remember that this strategy will only work if
the child already knows the related addition fact.
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The next strategy that we will consider is the
One More strategy.This strategy requires the
child to think of a known subtraction and then
take away one more.
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Do you already know 12 - 3 ?
12
12 - 3 9
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Then, to find 12 - 4, take away one more.
12
12 - 4 12 - 3 minus 1 more
8
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15 - 9 is easy if you already know 15 - 8 ?
15
15 - 8 7
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15 - 9 is 15 - 8 minus one more.
15 - 8 7 minus 1 more 6 So, 15 - 9 6.
15
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If I know 13 - 5 8
13 - 6
Then, to find 13 - 6, take away one more.
So, 13 - 6 7.
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The One More strategy is a very easy strategy for
children. The process is easy to understand, and
it is easy to subtract 1. The strategys
greatest drawback is that children who are trying
to figure out 15 - 8 must already know 15 - 7
8, to find the answer to 15 - 7 the child must
already know 15 - 6 9, etc.A
child is able to work only on facts that are 1
away from facts the child already knows.
Children master facts at different speeds and in
different orders, two children will seldom have
already mastered the same facts. The teacher
must be aware of what facts each child knows in
order to know what facts they should be working
on. This is very difficult for a classroom
teacher to manage.
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A strategy that is a general strategy for hard
basic subtraction facts is the Subtract from Ten
strategy.This strategy requires the child
already to know the subtraction facts having 10
as the subtrahend (10-7, 10-4, 10-6, etc.).
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To find 15 - 8, start with 15.
15 is 10 5
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To find 15 - 8, take 8 from the ten. You can see
that 7 is left.
15 - 8 7
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To find 12 - 7, start with 1.
12 is 10 2
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To find 12 - 7, take 7 from the ten. You can see
that 5 is left.
12 - 7 5
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To find the answer to 13 - 8
13
- 8
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Remember that 13 is 10 and 3.
13
10 3
- 8
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Remember that 13 is 10 and 3.
Take 8 from the 10.
13
10 3
- 8
- 8
2
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Remember that 13 is 10 and 3.
Take 8 from the 10.
13
10 3
- 8
- 8
Combine this with the other 3.
2 3
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Remember that 13 is 10 and 3.
Take 8 from the 10.
13
10 3
- 8
- 8
Combine this with the other 3.
2 3
5
13 - 8 5
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Take 8 from the 10.
Combine this with the other 2.
Altogether, what is the answer?
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The Subtract From Ten strategy is very easy to
teach. It is easy for children to visualize and
understand. Some teachers object to the child
having to do two steps to get the answer instead
of just memorizing the fact. Keep in mind,
however, that this strategy is not to be used
instead of memorizing the facts. This strategy
is an effective vehicle for helping the child
commit the facts to memory.If the child does
not already know the fact, the strategy provides
a quick way for the child to figure out the
correct answer. Without an effective strategy,
the childs only options are to guess or to say
I dont know. It is far better for the child
to have a way to find the answer.
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Next, we will now examine two effective
strategies for figuring out answers to hard
multiplication facts One More Partial
Products
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The first strategy for figuring out hard basic
multiplication facts that we will consider is the
One More strategy.This strategy requires the
child to build on a known multiplication fact.
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To find 6 X 8, Visualize 6 rows of eight.
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Thats 5 eights and one more eight.
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If I know 5 X 8 40, then I just add one more
eight.
40
8
6 X 8 48
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To find 9 X 5, Visualize 9 rows of five.
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Thats 8 fives and one more five.
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If I know 8 X 5 40, then I just add one more
five.
40
5
9 X 5 45
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Do I know 6 X 8 ?
7 X 8 ?
6 X 8 48
7 X 8 is 6 X 8 and one more 8.
7 X 8 48 8 or 56
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The second strategy for figuring out hard basic
multiplication facts that we will consider is the
Partial Products strategy.This strategy
requires the child to break a hard fact that the
child doesnt know into two easier facts that are
known.
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To find 8 X 7, Visualize 8 rows of seven. We
will break 8 X 7 into two easy parts.
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4 X 7 4 X 7
But 4 X 7 28
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4 X 7 28 4 X 7 28
So, 8 X 7 28 28 56
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Can I change this to two easier facts that I know?
6 X 9
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3 X 9 3 X 9
But 4 X 7 27
80
So 6 X 9 27 27
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6 X 9
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Can I change this to two easier facts that I know?
8 X 8
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The Partial Products strategy is a very good
strategy to teach. It is easy for children to
visualize and understand. Not only does it
help the child to figure out hard basic
multiplication facts, but it also provides a
strong background for the use of partial products
in multi-digit multiplication.
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ASSIGNMENT 2Most commercially published
elementary textbook series include the
development of strategies for figuring out the
hard basic facts..With a partner, search for a
strategy that is developed in a textbook series
for the hard basic facts for the operations,
addition, subtraction, and multiplication. You
should expect to find both hard addition facts
and hard subtraction facts taught late in grade
one and early in grade two. Look for the
teaching of hard multiplication facts in grade
three.After finding the textbook development,
with your partner, present to the class the
strategies that are developed and explain to the
class how it is accomplished on the textbook
pages.