# Using Diagrams to Represent Multiplication

Warmup

For each of the following products, without using a calculator, choose the best estimate of its value.

Activity #1

Multiplication of Natural Numbers using Unit Squares. • Move the sliders through different values and observe the number of squares.
• Verify by multiplying the two numbers either manually or using a calculator.

Activity #2

Use Diagrams to Multiply Decimals.

Before you get into this activity, I would like us understand how 2.3 x 1.5 is multiplied using diagrams. 2.3 x 1.5 = ( 2 + 0.3) x (1 + 0.5) = (2 x 1) = 2 = (0.3 x 1) = 0.3 = (2 x 0.5) = 1.0 = (0.3 x 0.5) = 0.15

Therefore, the total area = (2 + 0.3) + (1.0 + 0.15) = 2.3 + 1.15 = 3.45

Take note of this breakdown of the numbers, their multiplication, and the addition of the separate parts to give the total. • Use the sliders to choose other pairs of numbers to multiply.​ Try the following numbers;

(1.) 1.4  x  2.8​

(2.) 3.2  x  1.6​

(3.) 4.6  x  3.4​

(4.) 5.5  x  2.9

Activity #3

Explore Different Ways to Multiply Decimals.

In this activity, you are going to label the area diagram below to represent (2.5) · (1.2) in other to find the product. Follow the instructions below carefully. • Decompose each number into its base-ten units (ones, tenths, etc.)
• Write them in the boxes on each sides of the rectangle.
• Label regions A, B, C, and D with their areas.
• Then, find the product that the area diagram represents.
• In the boxes next to each number, use the text tool to enter the letter(s) of the corresponding region(s).

In this next activity, you are going to use the applet below to multiply the numbers (2.5) · (1.2).

• To adjust the values, move the dots on the ends of the segments. You may have to use the Up and Down arrow keys on your computer key board to move them. After you adjust the sliders to 2.5 and 1.2, you will have a rectangle partitioned into 4 regions.
• Check the boxes in the following order; “Show Partial Products” , “Show Sum”, “Show Product”.
• Compare the answer with one you would get in a calculation.
• Click the Reset button to practice with your own examples.

• You may be familiar with different ways to write multiplication calculations. Here are two ways to calculate 24 x 13. Study the two methods and answer the questions that follow.

(1.) In Calculation A, how are each of the partial products obtained? For instance, where does the 12 come from?

(2.) In Calculation B, how are the 72 and 240 obtained?

(3.) Look at the diagrams in the Challenge #1 below. Which diagram corresponds to Calculation A? Which one corresponds to Calculation B?

(4.) How are the partial products in Calculation A and the 72 and 240 in Calculation B related to the numbers in the diagrams?

• Here below are two ways to calculate (2.5) · (1.2). Each number with a box gives the area of one or more regions in the area diagram. Look at them closely and answer the questions that follow.

(1.) In Calculation B, which two numbers are being multiplied to obtain 0.5?

(2.) Which numbers are being multiplied to obtain 2.5?

Challenge #1

Here below are three ways of finding the area of a rectangle that is 24 units by 13 units.

(2.) How are they different?

(3.) If you were to find the area of a rectangle that is 37 units by 19 units, which of the three ways of decomposing the rectangle would you use? Why?

Challenge #2

Complete the calculations in the applet below so that each shows the correct sum.

Challenge #3

Here below is a rectangle that has been partitioned into four smaller rectangles. For each expression, choose the sub-rectangle whose area, in square units, matches the expression.

Quiz Time