Multiplication and Division Strategies

Multiplication and division are introduced to students with materials as they model problems.  They act out sharing 15 cookies among 3 children (division) or distribute sets of 4 marbles to each of 5 friends (multiplication).  Second graders are not expected to memorize the multiplication facts at first.  They are exposed to problem situations in order to focus on concept development.  Multiplication strategies are included here simply as a preview of how they will be addressed in later years and to make parents aware of helpful strategies for learning them.

As with addition, the commutative property is very powerful when learning basic multiplication facts.  Explore the concept that 3 x 4 (or three groups of four) will give the same product as 4 x 3 (or four groups of three).  Also, relate division to multiplying as you move through the strategies. 

x0, x1:  0 x  3 and 3 x 0 can be thought of as zero groups of three and three groups of zero.  Also 1 times any number is always the other factor.  It is always important to model these ideas.  It is easy to assume that because a concept seems very obvious to adults, it may not need to be explored.  Usually it takes very little time to convince children of these particular strategies, but it is wise to begin with a concrete model before moving on.

x2:  These are the addition doubles.  Again, using the commutative principle, 2 x 7 = 7 x 2, although they may be modeled in different ways.

x5:  The pattern children hear when counting by fives makes these facts fun to learn.  Your child may want to count fingers to keep track of the number of fives counted.  This is perfectly good use of available counters!

x9:  Children are very interested in the patterns of the nine facts.  Recognizing them will help them learn these facts easily and build their confidence quickly.  The nine facts are listed in the table.

Ask your child to describe the patterns he or she sees.  One pattern in the products is as the tens place increases by one, the ones place decreases by one.  Also, when the digits in the product are added together, they always make 9.  For example, 1 + 8 = 9 and 2 + 7 = 9.  Your child will recognize the opposites in the products such as 18 and 81, 27 and 72.  A strategy for knowing the digit in the tens place of a product is to think of one less than the factor that is not the 9.  For example, in 6 x 9, 6 is the factor that is not 9.  One less than 6 is 5, therefore 5 is the tens digit of the product.  Since 5 + 4 equals 9, the product must be 54.

Another fun strategy for learning the nines involves a finger trick.  When holding both hands spread palms down, each finger is named 1 through 10.  To show the product 9 x 4, for example, bend finger #4.  Looking on either side of the bent finger, there are 3 fingers to the left of it, and 6 to the right, making the product 36.  This works for the facts 9 x 1 through 9 x 10.  Ask your child to show you this if he or she has learned it in school.  An example of the hand position is illustrated below.

Squares:  These facts include 0 x 0, 1 x 1, 2 x 2, ... 9 x9.  They are called the squares facts because each fact can be shown as a square array.  For example, a 3 x 3 array would look like the table on this page.

It is 3 squares wide and 3 squares long, having 9 squares in all.  You and your child can make a set of the squares facts using graph paper.  Many of the previously learned facts are already a part of the squares facts.  This overlapping provides for a built-in review as your child reviews 0 x0, 1 x 1, 2 x 2, 5 x 5,, and 9 x 9 and learns only five new facts:  3 x 3, 4 x 4, 6 x 6, 7 x 7, and 8 x 8.

Through learning the x0, x1, x2, x5, x9 and squares facts, your child will know a great majority of the multiplication facts.  Keep track of the facts you and your child have been working on by creating a table such as the one below.  You may download a blank table like this one.

Last Facts:  The facts that make holes in the table above are the last facts, of which there are only 20 (really 10).  They are in the table below.

As you are working on these facts, encourage your child to use a known fact to help learn a new fact.  For example, if your child is learning 6 x 8, he or she may think, I know 5 x 8 = 40, so six 8's would be 8 more, or 48.  Or, to figure out 4 x 6, I know 2 x 6 is 12, so 4 x 6 would be twice that, or 24.

Source:  Montgomery County Public Schools. 1996.  Mathematics at home:  A guide for parents, grades K-2.  Rockville, MD:  Author.