Using interactive figures, students can investigate how changing the height of a rectangle with a fixed width changes its area. As discussed in the Number Standard, understanding multiplication by fractions and decimals can be challenging for middle-grades students if experiences with multiplication by whole numbers have led them to believe that "multiplication makes bigger." In these dynamic figures, the rectangle represents the familiar area model of multiplication; changing the rectangle's height can help students see the effect of multiplying a fixed positive number by numbers greater than 1 and less than 1.
The following links to an updated version of the e-example.
||Understanding Multiplication Using Dynamic Sketches of an Area Model (6.1)
Students are encouraged to build conceptual reasoning for
multiplying decimals by manipulating an area model.
The e-example below contains the original applet which conforms more closely to the pointers in the book.
||Learning about Multiplication Using Dynamic Sketches of an Area Model
Students can learn to visualize the effects of multiplying a fixed positive number by positive numbers greater than 1 and less than 1 with
Your task is to explore the effects of multiplying 3 by numbers greater than 1 and less than 1. To do this, you will use the area model of multiplication. The figure below shows a rectangle with width 3 and height y. The product 3y represents the area of the 3-by-y rectangle. Change the value of y by dragging the red point up and down the vertical axis. Note that as the point is dragged, the area of the rectangle changes simultaneously. Use the area of the 3-by-1 rectangle as a referent (3 square units), and compare it to the area of 3-by-y rectangles when y is greater than 1 and when y is less than 1. What do you observe?
How to Use the Interactive Figure
This interactive figure shows a rectangle with width 3 units and height y units. Change the value of y by dragging the red point up and down the vertical axis.
In the middle grades, students should refine their understandings of the four basic operations as they use those operations with fractions, decimals, percents, and integers. Teachers need to be attentive to the conceptual obstacles that many students encounter as they make the transition from working primarily with whole numbers. Multiplying and dividing fractions and decimals can be challenging for many students for reasons that are largely conceptual rather than procedural. For example, from their experience with whole numbers, many students develop a belief that "multiplication makes bigger and division makes smaller." When students are asked to solve problems in which they need to decide whether to multiply or divide fractions or decimals, this belief can have negative consequences (Greer 1992).
Take Time to Reflect
- What other experiences might challenge students to think about why the statement "multiplication makes bigger" is not always true? For example, how can looking for patterns in organized lists be useful?
- How can the dynamic area model of multiplication be used to help students "see" the distributive property of multiplication over addition in action?
- What sorts of questions might a teacher ask, after students have worked with the dynamic area model of multiplication, to prompt them to think and talk about the density of rational numbers between 0 and 1?
- How can a teacher use the dynamic area model of multiplication and the resulting classroom conversations to assess students' understanding of the multiplication of rational numbers and to plan worthwhile instructional tasks?
Greer, Brian. "Multiplication and Division as Models of Situations." In Handbook of Research on Mathematics Teaching and Learning, edited by Douglas A. Grouws, pp. 276–95. New York: Macmillan Publishing Co., 1992.