Introductory Lesson – Area Model with Multiplication of
Decimals
Standard: MCC5.NBT.7 Add, subtract, multiply, and divide decimals to
hundredths, using concrete models or drawings and strategies based on place
value, properties of operations, and/or the relationship between addition and
subtraction; relate the strategy to a written method and explain the reasoning
used.
EQ: What strategies can I use to multiply and divide decimals efficiently?
How do I use an area model to help me multiply decimals?
Materials: Base –Ten Grid Paper
Activator: Review with students how to use an area model when multiplying with
whole numbers. See example below:
This model was intended to help link the calculations of the standard algorithm
for multi-digit multiplication to the distributive property and reveal the partial
products as areas of smaller rectangles. The students produced area models for
multi-digit whole-number problems such as 57 x 14, shading and labeling four
partial products in the model (500 + 200 + 70 + 28). You also want students to
recognize that the distributive property was at work. Ex. (50 + 7) x (10 + 4).
Lesson Part 1: Using the grid paper, model this problem: 5.7 x 1.4. Explain to
students that indeed, the decimal multiplication problem 5.7 x 1.4 could be
shown with the same base-10 diagram as the whole number multiplication 57 x
14, but with dimensions that are labeled as one-tenth the size of those in the
whole number problem.
Tell students that now that a square of 100 little boxes is 1 whole, a row of 10
squares is .1, and one tiny square is .01. You can do this by showing a key.
Students need to note that the multipliers are the dimensions of the rectangle
and the area shows the product. Further, in order to find the total product, one
sums the areas of the smaller rectangles. In the diagram below, the smaller
rectangles have the following dimensions: (1 x 5), (1 x .7), (.4 x 5), (.4 x .7) and
respective areas of 5, 0.7, 2.0, and 0.28 square units, for a total of 7.98 square
units. Finally, we want students to make the connection that the smaller regions
have the same dimensions and the same areas as the partial products
produced using the distributive property as this calculation shows:
5.7 x 1.4 = (5 + .7) x (1 + .4) = (5 x 1) + (5 x .4) + (.7 x 1) + (.7 x .4)
= 5 + 2.0 + 0.7 + 0.28 = 7.98.
Lesson Part 2: Pass out grid paper and have students set up the new problem:
2.6 x 4.3. Allow students to use two different color crayons; one color for the
width (4.3) and one for the length (2.6). Allow them to work in groups. Give
plenty of grid paper for mistakes.
Once everyone has their grid labeled, then see if they can find the partial
products.
Find and color the product of 4 x 2
Find and color the product of .3 x 2
Find and color the product of .6 x 4
Find and color the product of .6 x .3
This also can look like this:
(4 + .3) x (2 + .6) – Use distributive property to get answer….
Summarizer: Students are given a problem (2.3 x 4.1) and they will need to do it
independently using grid paper.