 Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum            Session 9, Part A:
Models for the Multiplication and Division of Fractions

In This Part: Area Model for Multiplication | Try It Yourself | Area Model for Division
The Common Denominator Model for Division | Translating the Process to Decimals

We can apply the area model for the multiplication of fractions to visualize the division of two fractions when each is less than 1. To model division with fractions, we more or less reverse the process used for multiplication. We start with an area we're looking for, and we find one of the missing factors that makes up that area. Note 3

For example, here's how we would use the model to demonstrate the problem 1/4 2/3:

 Shade one square, partitioned vertically, to represent 1/4 (as in the multiplication model, it's shaded purple): Superimpose a square partitioned into thirds, positioned horizontally, onto the fourths square, and draw a bracket to the right of the thirds square to show the size of 2/3:    What you see now is the purple (1/4) area and the size of one of the factors that made that area.

We know from the multiplication model that the product of 2/3 and another factor (the quotient) defines an area equivalent in size to 1/4. To find the quotient, we need to move the top part of the purple area so that it's the same height as the 2/3 factor.

 Subdivide the fourths square to make an eighths square: Move the top two purple pieces into the 2/3 height area (the area within the 2/3 bracket): Now shade the rectangles immediately to the right and immediately above the purple area:      This shows that there are 3 • 2, or 6, purple parts out of 8 • 3, or 24, parts in all. The purple area equals 1/4, and it came from the product of 2/3 multiplied by what? We can see that the other factor is 3/8. Problem A3 A town plans to build a community garden that will cover 2/3 of a square mile. They would like to situate it on a pasture of an old horse farm. One dimension of the garden area will be determined by a fence that is 3/4 of a mile long. Use the area model for division to determine the other dimension of the new garden area. Problem A4 Describe how the area model shows that the quotient of two positive fractions, each less than 1, must be larger than the first fraction.   Session 9: Index | Notes | Solutions | Video