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 this tool. 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.
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 a 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?
to Use the Interactive Figure]
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
- 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
- What sorts
of questions might a teacher ask, after students have worked with
the dynamic area model of multiplication, to prompt students to
think and talk about the density of rational numbers between 0
- 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
Greer, Brian. "Multiplication
and Division as Models of Situations." In Handbook of Research on
Mathematics Teaching and Learning, edited by Douglas A. Grouws, pp.
27695. New York: Macmillan Publishing Co., 1992.