Lesson 7: Using Diagrams to Represent Multiplication
Let’s use area diagrams to find products.
Illustrative Math Unit 6.5, Lesson 7 (printable worksheets)
Lesson 7 Summary
The following diagram shows how to use area diagrams and partial products to represent and find products of decimals.
Lesson 7.1 Estimate the Product
For each of the following products, choose the best estimate of its value. Be prepared to explain your reasoning.
- (6.8) · (2.3)
- 74 · (8.1)
- 166 · (0.09)
- (3.4) · (1.9)
Lesson 7.2 Connecting Area Diagrams to Calculations with Whole Numbers
- Here are three ways of finding the area of a rectangle that is 24 units by 13 units.
Discuss with your partner:
a. What do the diagrams have in common? How are they alike?
b. How are they different?
c. 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?
- You may be familiar with different ways to write multiplication calculations. Here are two ways to calculate 24 times 13.
Discuss with your partner:
a. In Calculation A, how are each of the partial products obtained? For instance, where does the 12 come from?
b. In Calculation B, how are the 72 and 240 obtained?
c. Look at the diagrams in the first question. Which diagram corresponds to Calculation A? Which one corresponds to Calculation B?
d. How are the partial products in Calculation A and the 72 and 240 in Calculation B related to the numbers in the diagrams?
- Use the two following methods to find the product of 18 and 14, then compare the values obtained.
a. Calculate numerically.
b. Here is a rectangle that is 18 units by 14 units. Find its area, in square units by decomposing it. Show your reasoning.
- Compare the values of that you obtained using the two methods. If they are not the same, check your work.
Use the applet to verify your answers and explore your own scenarios. In order to adjust the values, move the dots on the ends of the segments.
Lesson 7.3 Connecting Area Diagrams to Calculations with Decimals
- You can use area diagrams to represent products of decimals. This area diagram represents (2.4) · (1.3).
a. Which region represents (0.4) · (0.3)? Label that region with its area of 0.12.
b. Label each of the other regions with their respective areas.
c. Find the value of (2.4) · (1.3). Show your reasoning.
- Here are two ways of calculating 2.4 times 1.3.
Analyze the calculations and discuss with a partner:
a. Which two numbers are being multiplied to get 0.12 in Calculation A? Which numbers are being multiplied to get 0.72 in Calculation B? How are the other numbers in blue calculated?
b. In each calculation, why are the numbers in blue lined up vertically the way they are?
- Find the product of (3.1) · (1.5) by drawing and labeling an area diagram. Show your reasoning.
- Show how to calculate (3.1) · (1.5) using numbers without a diagram. Be prepared to explain your reasoning. If you are stuck, use the examples in a previous question to help you.
- Use the applet to verify your answers and explore your own scenarios. In order to adjust the values, move the dots on the ends of the segments.
Are you ready for more?
How many hectares is the property of your school? How many morgens is that?
Lesson 7.4 Using the Partial Products Method
- Label the area diagram to represent (2.5) · (1.2) and to find that product.
a. Decompose each number into its base-ten units (ones, tenths, etc.) and write them in the boxes on each side of the rectangle.
b. Label Regions A, B, C, and D with their areas. Show your reasoning.
c. Find the product that the area diagram represents. Show your reasoning.
- Here 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.
a. In the boxes next to each number, write the letter(s) of the corresponding region(s).
b. In Calculation B, which two numbers are being multiplied to obtain 0.5? Which two are being multiplied to obtain 2.5?
Lesson 7 Practice Problems
- Here is a rectangle that has been partitioned into four smaller rectangles.
For each expression, choose a sub-rectangle whose area, in square units, matches the expression.
a. 3 · (0.6)
b. (0.4) · 2
c. (0.4) · (0.6)
d. 3 · 2
- Here is an area diagram that represents (3.1) · (1.4).
a. Find the areas of sub-rectangles A and B.
b. What is the area of the 3.1 by 1.4 rectangle?
- Draw an area diagram to find (0.36) · (0.53). Label and organize your work so that it can be followed by others.
- Find each product. Show your reasoning.
a. (2.5) · (1.4)
b. (0.64) · (0.81
- Complete the calculations so that each shows the correct sum or difference.
- Diego bought 12 mini muffins for $4.20.
a. At this rate, how much would Diego pay for 4 mini muffins?
b. How many mini muffins could Diego buy with $3.00? Explain or show your reasoning. If you get stuck, consider using the table.
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