Learning the Basic Facts - While flash cards are still a great tool to help
students learn the basic arithmetic facts, students also use strategies to
help them with facts. This page contains some of the strategies students
learn to master their facts.
- Math at Home, or How Parents Can Help Their Children
- Basic Facts Strategies
- Plus 2
- Doubles and Plus Tens
- Adding Near Doubles or 'Neighbor Numbers'
- Adding 'Neighbor in the Middle' Numbers
- Make a Ten
- Adding Hungry Eights and Hungry Nines
- Multiplying by Zero and by One
- Multiplying by Two
- Multiplying by Three
- Multiplying by Four
- Multiplying by Five
- Multiplying by Six
- Multiplying Squares
- The Dreaded 7 x 8
- And the rest....
Math at Home, or How Parents Can Help Their Children
It is critical that students have a working command of the basic
arithmetic facts in fifth grade. The computation work will be
more easily accomplished and less prone to errors if students
have quick rote recall of facts in addition, subtraction,
multiplication, and division. We will have some practice drills
on these skills, but our curriculum does not provide time to
teach these skills since they are covered in previous grades.
In Reading, students must be able to read 98% of the words in a
text in order to comprehend at least 70% of the material.
Similarly, in Math a student must have nearly 100% accuracy in
basic facts in order to solve problems. Math facts that are shown
to be deficient on a timed test will be sent home for practice.
Time for painless practice of basic facts can be found throughout
the day: at school, in the car, during chores like table
setting and clearing/washing dishes and folding laundry, or
during television commercials.
The Math Newsletter, which will be posted in the Newsletter
section of this site, will explain ways to practice the current
unit's skills at home.
Basic Facts Strategies
Students generally know most of the facts, but once they start
working on multiplication and division, their addition and
subtraction facts slip. Since much of our work involves some
mental math facility, students need to keep their addition,
subtraction, multiplication, and division skills sharp.
Students know their +0 and +1 facts.
Using strategies, students actually learn all 100 facts on a 1-10
Hundreds Chart except 4 facts. Because addition is commutative
(the order of the factors does not change the sum) there are
really only 2 facts to learn since 3 + 6 is the same as 6 + 3 for
example. The 'unknown facts' that must be learned are: 3 + 6 and
4 + 7.
Subtraction facts are found by using Fact Families, just like
students found in the primary grades. 7, 8 and 15 are a Fact
Family: 7 + 8 = 15, 8 + 7 = 15, 15 - 7 = 8 and 15 - 8 = 7.
Plus 2
The +2 facts are found by skip counting to the next even or odd
number depending on the other addend. Practice skip counting by
twos using even numbers or odd numbers. Start at a different
point each time, like 39, 41, 43, 45, 47 etc.
Doubles and Plus Tens
Students know the doubles facts from playing board games and they
can add ten by increasing the tens digit by one.
Adding Near Doubles or 'Neighbor Numbers'
Another strategy, Neighbor Numbers, is used when the addends are
consecutive numbers on the number line, like 5 + 6. The strategy
is based on a known doubles fact: 6 = 5 + 1, so 5 + 6 is the same
as 5 + 5 + 1 or 11.
Try laying out pennies or other objects: five in one group and
six in the next group. By moving one penny from the group of six
to the side, students can see the 5 + 5 fact. Repeating this
activity for other Neighbor Number pairs helps students to
internalize the concept.
Adding 'Neighbor in the Middle' Numbers
Another strategy works for numbers separated by one place on a
number line like 6 + 8. I call these facts Neighbor in the Middle
facts since the sum of the two numbers is the Doubles fact for
the Neighbor in the Middle on the number line, or 7 in this case.
So, 6 + 8 = 7 + 7 or 14.
Try laying out pennies or other objects: six in one group and
eight in the next group. By moving one penny from the group of
eight to the group of six, students can see the 7 + 7 fact.
Repeating this activity for other Neighbor in the Middle pairs
helps students to internalize the concept.
Make a Ten
Make Ten is usually an easy strategy for most students. They
learn the combinations that make ten: 1 + 9, 2 + 8, 3 + 7, etc.
Adding Hungry Eights and Hungry Nines
My last strategy is Hungry Eights and Hungry Nines. In lower
grades, this strategy is taught using blocks that link together.
To use Hungry Eights, students have a 'train' of eight blocks
hooked together. Eights are always hungry for two more blocks ...
which will make them a ten. Students take two from the other
addend to put with the eight and make a ten. The remaining
amount from the second addend is added to ten. For example, 8 + 7
would mean that students should take two from the seven. That
makes a ten and five more or fifteen. The same strategy works for
Hungry Nines, which are always hungry for one more to make them a
ten. So, to add 9 + 5, take one from the five to make a ten and
add the remaining four for fourteen.
Multiplying by Zero and by One
Zero times any number means we have that number zero times and
the product will always be zero. This is known as the Zero
Property of Multiplication.
The Identity Property of Multiplication, or Identity Property,
says that any number times one means we have that number one
time, so one times any number is that number.
Multiplying by Two
The twos facts are the same as the Doubles in addition, so
students should already know the twos. Twos facts are always even
numbers, so they are divisible by 2 and end in 0, 2, 4, 6, or 8.
Multiplying by Three
Multiplying by three means three sets of a group. Students can
find threes by building from their twos. Three sets are like
doubles plus one more set. So, for example, 6 x 3 can be three
sets of six -- 6 + 6 = 12, 12 + 6 more = 18 or double six is
twelve, plus one more six is eighteen.
Multiplying by Four
To multiply by four, students have already learned to double the
Doubles fact. A mnemonic story can also help students to skip
count by fours: Four ate twelve sixteens going down Easy Street
when Thirty-Two and Thirty-Six passed him doing Forty. The
diagram that accompanies the story shows a caricature character
Four eating Sixteens. He already has a 3 x 4 array in his tummy,
but one number is missing from a corner -- it is the twelfth
sixteen being eaten. Easy Street is a row of townhouses with
house numbers 20, 24 and 28 for easy skip counting. Two more
caricature characters, Thirty-Two and Thirty-Six are riding
tandem on a scooter by a speed limit sign that reads 40 mph. So
students recall the skip counting multiples: 4, 8, 12, 16, 20,
24, 28, 32, 36, 40.
Numbers that are divisible by four are numbers whose two
rightmost digits form a multiple of four. This is only helpful
about half the time ;-O
Multiplying by Five
Skip counting by fives has been done so often that most students
already know the fives facts. Multiples of five always end in
zero or five with even multiples being the zero in the ones place
and odd multiples ending in the five.
Students can also use the tens facts for even multiples of five --
5 x 6 can be found from 10 x 6 = 60. Since 5 is half of ten, 5 x
6 is half of 10 x 6 or 30. Some students see a pattern in the
even multiples: cut the even factor in half and tack on a zero
for the ten. It's the same idea, but the student cuts in half
before making the ten multiple :-)
Multiplying by Six
Just as we double the twos facts for the fours, we double the
threes facts for the sixes. Numbers that are divisible by six are
even numbers whose digits add up to a multiple of three.
Multiplying Squares
A number times itself is a square number because it can be
represented in a square array, and squares are usually easily
recalled by most fifth graders, like 4 x 4 = 16 and 5 x 5 = 25.
The Dreaded 7 x 8
This fact is considered the most challenging for students to
learn. I tell students about my teacher who shouted 5 - 6 - 7 - 8
in my face when I couldn't immediately supply the answer to 7 x
8. Unfortunately for me, I had been absent the day before when
she taught students to say five, six, seven, eight for 56 = 7 x 8.
Alternately, students can start at the bottom and write the even
digits in a column of ten numbers: 0,2,4,6,8,0,2,4,6,8. Starting
at the top in the tens column, skip the first number (8) and then
list 1,2,3,4,4,5,6,7,8 going down the column. These are the first
ten multiples of eight!
And the rest....
Nine facts use 'tricks' like the hand trick and the pattern
trick. We use these in class.
Tens are easy to skip count and form an easy pattern.
That just leaves the sevens and eights. Most of them are already
known since students know the commutative facts for the factors 1-
6, like 7 x 4 = 4 x 7 and that's a double the doubles fact. So
students need to learn 7 x 6, 7 x 8, and 8 x 6. Students know
many of these 100 multiplication facts already also, and they
should know the math property that applies. They can also use the
Build from Fives strategy. If finding 7 x 8, students can
determine 5 x 8, which is 40, and then determine that 2 more sets
of eight are needed to get from 5 eights to 7 eights, so 2 x 8 =
16. Put the partials together: 40 + 16 = 56, so 7 x 8 = 56.
Once students know a multiplication fact, they can figure out the
related division fact.