**The Development of Models**

Before you use a model (such as the open number line or the array) in a number

string, students should have been introduced to the model through a context.

Models begin as representations of a context, then representations of strategies,

and then finally become tools for thinking. If you want to learn more about this

sequence, you can read Cathy Fosnot and Martin Dolk’s book on multiplication

and division: Young Mathematicians at Work; Constructing Multiplication and

Division (2002).

**Open number line**

**Array**

This powerful tool for multiplication and division is all around us: fruit boxes,

windows of skyscrapers, tiles on the floor. It takes time for kids to understand that

you can find the area of an array by multiplying columns by rows (Battista et al.,

1998). Kids first move through strategies like counting by ones, making groups,

and skip counting. There are obvious connections to area and volume. The array is

also a great model for multiplication with rational numbers.

You can design your own investigations for kids to develop understanding of the

array, or you can also use an already made unit. Groceries, Stamps, and

Measuring Strips (1)1 is a great introduction to multiplication for kids in grades 2. In

Muffles Truffles(2), kids will work with 2 by 5 arrays of candy, great for kids in

grades 2 -3. For division with the array, try the soda machine problem in The

Teachers’ Lounge(3).

**Ratio table**

The ratio table is a flexible tool that kids can use in understanding rational

numbers, algebra, etc. Once it is a tool for thinking, kids can use it to make moves

that make sense for them, often beginning with doubling or halving and then

gradually making bigger moves. In the two-week unit The Big Dinner(4), kids in

grades 3 are introduced to the ratio table when they plan a big dinner.

1 Van Galen & Fosnot (2007). *Groceries, Stamps, and Measuring Strips; Early Multiplication.* Heinemann. (Two week unit)

2 Cameron(2007). *Muffles Truffles;Multiplication and Division with the Array.* Heinemann. (Two week unit)

3 Fostnot & Natale (2007). *The Teachers’ Lounge; Place Value and Division.* Heinemann. (Two week unit)

4 Fosnot(2007). *The Big Dinner; Multiplication with the Ratio Table*. Heinemann. (Two week unit)