**Mental
Math Strategies**

**
Basic Addition Facts
| Basic Multiplication Facts**

**Teaching
for Mastery of Basic Facts**

It is important
that most students have **mastery of basic facts**. It
is equally important that they **make sense** of number
combinations as they are learning these facts. Here are some strategies
to help with this understanding.

**Adding
Zero: **

Model adding zero (with younger students) or review it with older students.
If a child understands that when you add zero you add nothing, he/she
should never get a basic fact with zero wrong. Make sure this understanding
is in place.

**Adding
One (Count up)**

Adding one means saying the larger number, then jumping up one number,
or counting up one number. This happens every time you add one. It never
changes. Never recount the larger number, just say it and count up one.

Example:
6 + 1 = say 6 then 7

44 + 1 = say 44 then 45

**Adding
Two – Count up Two**

Adding two means saying the larger number, then jumping up or counting
up twice. Again this is always correct and never changes.

Example:
9 + 2 = say 9 then 10 then 11

45 + 2 say 45 then 46 then 47

**Commutative
Property: **

You also have to teach or review the commutative property. The answer
will be the same regardless of the order you add the two numbers. 9
+ 2 = 2 + 9 Order doesn’t matter.

The following
printable resources can be used to check student understanding of the
above strategies. If a child does not get them all right you can check
to see which strategy they don’t understand and reteach it.

**Printable
Resources:(in PDF Format)**

**Adding
Ten **

Adding ten means jumping up ten (think of a hundred’s chart).
The ones digit stays the same but the ten’s digit increases by
one. Students must understand this. Using a hundreds board to teach
this works well to build understanding. Have students actually count
up the ten and write down the result. Then affirm with them the pattern
and explain why it works every time.

Example:
5 + 10 = 15

10 + 7
= 17

For older students you can relate this to higher numbers:

Example
23 + 10 = 33

48 + 10 = 58

**Adding
9 **

Adding 9 makes sense if students understand adding ten. It sounds more
difficult than it actually is.

Remind
students of the jump of ten – 5 + 10 = 15. A student would say
(in their head) “5 plus 10 = fifteen”

The five and fifteen are naming the same number of ones.

With the
nines – a student must count down one in the ones.

A student would say “5 + 9 = fourteen”.

It sounds difficult but once they catch on it is really simple.

Work with
lots of examples until the idea is understood:

5 + 10 = fifteen 5 + 9 = fourteen 7 + 10 = 17 7 + 9 = sixteen

**Adding
8**

This works exactly the same only a child must think 2 less. Using the
examples above students would say; 5 + 10 = 15 so 5 +8 = 13, 7 + 10
= 17 so 7 + 8 = 15 (2 less)

**Double
Numbers**

To add double numbers there are a couple of strategies that might help
students.

When you
add a double you are counting by that number once.

For example: 4 + 4 = think of 4,8 … counting by fours

Practice skip counting by each number in turn:

2-4

3-6

4-8 etc. This gets harder with the higher numbers but skip counting
is an important skill for students to have.

Doubles
occur everywhere in life.

For example: an egg carton is 6 + 6

two hands are 5 + 5

16 pack of crayons has 8 + 8

two weeks 7 + 7 =

Do a variety of activities with double numbers and have students determine
and explain which strategies help them remember. Each student should
look at each fact and relate to a visual image or counting by strategy
that works for them.

**Near
Doubles **

To use the near doubles strategy a student first has to master the doubles.
Then, if the double is known, they use that and count up or down one
to find the near double.

Example: 4 + 4 = 8 5 + 4 = 9 (count up one)

Or: 4 + 4 = 8 so 4 + 3 = 7 (count down one)

**Adding
5**

Adding five has a strategy that is helpful but not completely effective
as it is a bit tricky. You can decide if it is helpful or not.

Example:
5 + 7 = (10 + 2) = 12

5 + 8 =
5 + 5 + 3 = 13

Students
who can see the five in 8 should have no difficulty. Students who can’t
visualize numbers will find this hard. Most students can be taught to
do this with some extra work.

**Printable
Resources:(in PDF Format)**

**Let’s
look at where this takes us. **

If you think
about the 10 by 10 grid for addition facts it would look like the one
below. That is a lot to memorize.

The strategies
we have discussed should have eliminated the need to memorize most of
the facts.

On the
chart below, all the facts that can be taught using a strategy, are
covered, leaving only the facts that need to be drilled.

All addition facts with zero

All addition facts with one

All
addition facts with two

All addition facts with nine

All addition facts with ten

All addition facts with eight

All addition facts with double numbers and near doubles

All addition facts with five

**Strategies
for Basic Multiplication Facts**

The basic
number facts are among the tools that students need to be successful
in their mathematics program. In the past, students memorized the facts
once they had been introduced to Multiplication as a faster method of
addition.

**Now
it is recommended that students learn patterns and strategies for as
many facts as possible so that they strengthen their understanding of
the relationships between numbers and the patterns in mathematics. **Then
they begin to memorize. There are many strategies out there. Here are
some that have been successful with many students.

Before
these strategies are taught, students must gain a complete understanding
of the concept of multiplication. They should actually make groups of
things and relate these groups to the number facts. They should skip
count and make arrays to gain a complete understanding of multiplication.

**Multiply
by zero **

If you have zero groups of anything you have nothing. It is fun to teach
this by offering several different groups of zero to students.

“Here,
you can have zero Smarties. How many did you get? zero

Work through
several examples. The idea is that it doesn’t matter how many
numbers are in a set or group, if you have zero sets you have nothing.
So 1 x 0 is 0 one group of zero and 0 x 1 = 0 zero groups of one = zero

Once students understand this they will never have to practice it.

**Commutative
Property (Turn around facts)**

Students may as well learn this right away. If you have 2 groups of
zero or zero groups of two, you have the same amount. Work through several
examples with zero to be sure that students understand. Then, review
this with all the other strategies as all facts have a turn around fact.

**Multiplying
by one**

Again this is a concept that students need only to understand and then
they will always know the one times facts. One times any number means
one group of that number which is the same number.

1 x 6 is one group of six = six

Turn around fact; 6 groups of one = 6 x 1 = 6

If students do lots of examples to gain this understanding, they will
not have to practice this.

**Multiply
by Two **

This is just double numbers, which they should already be familiar with.

For example: 2 x 8 = 8 + 8 = 16

It would take a couple of lessons to work through examples where you
relate the two ideas and give students a chance to practice. Then they
should be able to use this strategy.

**Multiply
by Ten**

A hundreds board works great for this as do base ten rods. Students
need to make groups of tens. They will see the pattern fairly quickly
but they need to see the number pattern of increasing by ten as well
as the “adding zero” factor. Once they explore with groups
of ten then they can use the rule of adding zero to multiply 10 by any
number. Again, they should review the turn around fact as well.

Two groups
of ten = 20 10 groups of 2 = 20

**Multiply
by five**

Counting by fives is a common factor in our society so multiplying by
fives can fit right in here. Use a clock to introduce the five times
table.

We talk about 5 after, ten after, fifteen after – so this is one
group of five, two groups of five, etc.

Have students count by fives and review the zero – five pattern
5, 10, 15, 20 (ends in zero, ends in five).

Work with examples like these to help children find patterns in the
five times table and then remind them of the turn around facts.

**Multiplying by 9**

There are several ways to help students with this but the neatest one
is that there is a nifty pattern to the nines. If students look at some
examples: one group of nine is 9. Two groups of nine is 18, three groups
of nine is 27 they can see that the answer adds up to nine and the tens
digit is one less than the factor the nine is being multiplied by. Correspondingly
the last digit, when added to the factor makes ten.

For example:

4 x 9 – the first digit is one less than 4 (the factor) and the

last digit will add up to 9 if added to the first digit. Also, the factor
4 and the last digit will add up to ten.

**It
is confusing until you try it out several times and then the pattern
appears much more simple. **

Those are
some basic strategies that along with the turn around strategies help
give students a solid base on which to build their multiplication facts.
The Nelson program also teaches students to build new facts from known
facts.

For example:
If a child knows 5 x 3 = 15 they can figure out 6 x 3 = 18 (one more
group of 3)

If a child
knows 6 x 7 = 42 then 7 x 7 = one more group of seven = 49

**Halving
strategies**

This can be used on facts with 5’s and 10’s.

If a child knows 8 x 5 = 40 she can halve and double to find 4 x 10
= 40. (half of 4 and double 5)

Another example; 4 x 5 = 20 half and double 2 x 10 = 20.

**Multiplying
by eleven**

It quickly becomes very obvious that multiplying by 11 follows an easy
pattern. If students do some examples 2 x 11 = 22, 8 x 11 = 88 etc.
they soon see that it is taking the original number and multiplying
it by ten and then itself. Make sure they understand the pattern and
then let them practice with other numbers. Again this pattern never
changes.

**Number
Neighbours**

A child who doesn’t know 7 x 6 = might know 6 x 6 = If so they
can just add one more 6.

A child
may not know 5 x 6 but they might know 5 x 5 so they can just add one
more five.

**Multiplication
Table (lines crossed out represent use of strategies instead of memorization)
**The strategies we have discussed should have eliminated the
need to memorize most of the facts.

On the
chart below, all the facts that can be taught using a strategy, are
covered, leaving only the facts that need to be drilled highlighted
in blue.

**
**

Multiply by zero

Commutative
property (Turn around facts)

Multiply by one

Factor of 2 – double numbers

Multiply by ten

Multiply by Five (clock/counting by 5)

Multiplying by 9 (nifty nines) or (using tens)

Multiplying by 3 or 6 (adding on strategy)