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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)

  • Printable Resources:(in PDF Format)

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.

  • To add fives look for the five in both numbers to make a ten then count on the extra digits.

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)

 

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