Homeschooling parents know that one of the biggest challenges for any middle-elementary math student is to master the multiplication facts. It can seem like an unending task to memorize so many facts and be able to pull them out of mental storage in any order on demand.
Too often, we are tempted to stress the rote aspect of such memory work, which makes our children lose their focus on what multiplication really means. Before practicing the times table facts, make sure your student gets plenty of practice recognizing and using the common models for multiplication.
To help your children see what multiplication looks like in real life, explore the multitude of Multiplication Models collected at the Natural Math website. Or sign up for their Multiplication Explorers Online Course.
You may want to pick up this poster and use it for ideas as you play the Tell Me a (Math) Story game. Word problems are important for children learning any new topic in math, because they give children a mental “hook” on which to hang the abstract number concepts.
And for extra practice, you can play my free card game…
Multiplication Model Games
Or pick up the free Multiplication & Fraction Printables file, with a full-color multiplication model card deck and much more.
The cards do not include every math fact on the times table, but they should offer enough variety to cement the most common multiplication models in your children’s minds. Each set consists of four cards: the multiplication equation itself and three pictures that help us visualize what multiplication means.
- The set model: cookies per plate.
- The rectangular array model: blocks per row.
- The measurement model: count per unit on a measuring tape or stick.
How To Play
For a quick and easy game using the multiplication cards, play Concentration: Lay all the cards face down on the table, then try to turn up two matching cards — any two cards that belong to the same math fact. For young children, try playing Concentration with the cards face up.
Or try Go Fish, according to your family’s favorite rules. At our house, we match in pairs (not sets of four) and allow a free turn whenever you draw the card you asked for from the fishing pond.
Both of these games help children grow comfortable with the multiplication models.
Let’s Look Closer at the Multiplication Models
My card game is based on three basic models — three mental pictures of multiplication:
(1) Set Model
“_____ sets of ______ objects per set”
This model represents discrete (countable) items collected into groups: apples per basket, pennies per dime, or cookies per plate. The set model clearly relates to the idea of multiplication as repeated addition, so it is the most common way of introducing multiplication in elementary school textbooks.
(2) Measurement Model
“_____ units of _____ measures per unit”
This model represents continuous quantities measured out in parts: inches per foot, cups per recipe, dollars per pound (for buying in bulk), or spaces per jump on a number line. The measurement model can also include the idea of multiplication as scaling, stretching, or shrinking something from its original size, which makes it useful when thinking about fractions.
(3) Rectangular Array Model
“_____ rows of _____ items per row”
In early elementary math, the this model represents an array of discrete items: chairs per row, buttons per column, or soldiers on parade. As students grow, however, the model expands to include continuous rectangular area. At its most mature, this model becomes the basis for many topics in high school math and beyond, including integral calculus. Because of its flexibility, the rectangular model is the most important one for our children to master.
Notice that in each model, the two numbers of a multiplication problem have different roles. The first number is a scale factor, which tells you how many sets, units, or rows you are talking about, while the second number is a this-per-that ratio.
In addition and subtraction, both numbers must represent the same type of thing, but this is almost never true in multiplication and division. Even in rectangular area, where both numbers represent a length measurement, one must be the horizontal length and the other vertical (the x- and y-dimensions in coördinate graphing).
Update: An expanded version of this post is featured in my book Multiplication & Fractions: Math Games for Tough Topics, available now at your favorite online book dealer.
More Tips for Teaching Multiplication
And my Times Table Series:
- Math Facts Are like Learning to Type
- How to Conquer the Times Table, Part 1
- How to Conquer the Times Table, Part 2
- How to Conquer the Times Table, Part 3
- How to Conquer the Times Table, Part 4
- How to Conquer the Times Table, Part 5
[Poster by Maria Droujkova of NaturalMath.com. These games were originally published as part of the Homeschooling with a Profound Understanding of Fundamental Mathematics Series.]
Denise Gaskins' Let's Play Math 
I love the idea of multiple representations. One of the pushbacks I hear from teachers is that once the kids learn the standard algorithm, or a method their parents teach them, the students don’t want to learn another way. Perhaps with something more visual it would help them to get the kids excited.
Hi, Miah! I’ve noticed that problem, too: Once kids think they “know how to do it,” their brain turns off and they just go through the motions. That’s why I like math games that make them do the figuring in their heads. The game is fun, so it makes practice palatable, and the mental calculation (imo) builds deeper understanding.
I created a skip-counting card game which helps children learn to multiply. Both kids and parents love it and the game has been a popular Christmas gift this season. http://highhillhomeschool.blogspot.com/p/highhill-educational-supplies.html