Multiplication
Grandchildren  Trixie, Eva and Rose, from left to right, in February, 2012.
I separate multiplication word problems into 3 categories depending on
how they are most easily modeled. Here are examples of each type.
Easy Multiplication: Eva has 3 fish tanks. Each fish tank has 2
fish in it. How many fish does Eva have?
Hard Multiplication: Rose has 2 pet fish. Trixie has 3 times as
many pet fish as Rose has. How many pet fish does Trixie have?
Combinations: Punch, the dog, has 2 different collars and 3
different scarves. How many different outfits does Punch have?
These 3 problems can all be represented by the equation 2 x 3 = □^{1}.
Easy Multiplication
1. What Is The Model For Easy Multiplication?
To model the Easy Multiplication problem above you would:
 Count out 3 objects to represent the 3 fish tanks – perhaps 3 paper plates.
 Count out 2 objects to represent the fish in the first tank. Place them on the first paper plate.
 Count out 2 objects to represent the fish in the second tank. Place them on the second paper plate.
 Count out 2 objects to represent the fish in the third tank. Place them on the third paper plate.
 Join all the “fish” together and count them.
2. How Do You Teach Easy Multiplication?
Once again, the way to introduce children to solving any particular type of word problem is to present them with examples and help them to build suitable models. Here, at 3 years and 8 months, Trixie surprisingly needs very little help.
Hard Multiplication
1. What Is The Model For Hard Multiplication?
To model the Hard Multiplication problem above you would:
 Count out 2 objects to represent Rose's 2 pet fish.
 Count out 2 "fish" for Trixie  she now has the same number of fish as Rose has.
 Count out 2 more "fish" for Trixie  she now has 2 times (or twice) as many fish as Rose has.
 Count out 2 more "fish" for Trixie  she now has 3 times as many fish as Rose has.
 Count Trixie's fish altogether  "1, 2, 3, 4, 5, 6."
2. How Do You Teach Hard Multiplication?
Here is a good example of direct, explicit, instruction. Trixie at 3 years and 9 months does not know how to model a Hard Multiplication problem. It is obvious to me that I should show her. In doing so I am not telling her how to think  I am telling her what the question means.
Combinations
1. What Is The Model For Combination Problems?
How can we model this problem?
Punch, the dog, has 2 different collars and 3 different scarves. How many different outfits does Punch have?
It is easy to begin.
 I might represent the 2 collars with 2 crayons – perhaps one red and one blue.
 I might represent the 3 scarves with 3 toy blocks – one with the letter A, one with the letter B, and one with the letter C.
I can now use the crayons and blocks to model the different possible outfits.
 For example, one outfit can be represented by the red crayon and the block with the letter A.
 Another outfit can be represented by the blue crayon and the block with the letter B.
But,
unfortunately, it is not possible to use the crayons and blocks to
represent all the possible outfits simultaneously. For example,
while there are 3 outfits that involve the red crayon, we can only
assemble these 3 outfits one at a time.
Altogether, there are 6 possible outfits, as shown in the table below.
But with the crayons and blocks you can only model 2 of them at any one
time – one with the red crayon and one with the blue crayon.






Obviously, the inability to represent all 6 outfits at once makes it difficult to count how many there are.
2. How Do You Teach Combination Problems?
Most elementary schools would not consider introducing Combination problems to children. They are considered "too hard." That is a big mistake. As this example shows, young children can learn to model them quite successfully.
Summary and Additional Comments
Click here to download a printable PDF summary, in a new
window, of the models of the 3 types of multiplication word problems.
If you are curious about the relationship between Hard Addition and
Hard Multiplication, click here.
Click here for a
further discussion of classifying multiplication word problems.
Shortcuts
Parents and teachers often make the mistake of overemphasizing the
“facts.” This is especially true in the case of multiplication where
young children typically spend way too much time memorizing
multiplication tables at the expense of experiences that would help
them to learn to reason. The consequences of such overemphasis for
these children include a permanent distaste for arithmetic, the
inability to apply their learning to solving word problems, and a
complete misunderstanding of what mathematics is all about.
In these two examples Rose starts to uses her fingers to make a model
but has a problem grouping her fingers appropriately. Then, in both
cases, on her own, she abandons her model and uses a shortcut 
counting by 2's. While I have no problem with suggesting
shortcuts to children, in this case I played no such
role. Rose's reasoning is entirely her own.