standard_description: Construct a tangent line from a point outside a given circle to the circle.
problem_activity: Optional activity
This activity continues using partial products and area diagrams to compute products of decimals and connect area diagrams with vertical calculations. In these problems, the side lengths and areas of the rectangle are left blank. Students fill in these fields by decomposing the factors by place value and multiplying them. Like many of the activities in this unit, the work involves understanding the structure of the base-ten system used in multiplication calculations (MP7) and making sense of it from arithmetic and geometric perspectives.
Students also make another key connection about several previously developed ideas here. They see that we can write the non-zero digits of decimal factors as whole numbers, use vertical calculations to multiply them, and then attend to the decimal point in the product afterwards. In other words, they notice the algorithm for multiplication can streamline the reasoning processes they have used up to this point.
Launch
Arrange students in groups of 2. Give groups 1–2 minutes to make sense of the first set of questions and area diagram. Listen to their discussions to make sure they understand how to label the appropriate side lengths. If needed, pause the class and have a student or a group that correctly decomposed the side lengths by place value share their reasoning.
Give students 5 minutes of quiet work time for the rest of the task. Follow with a whole-class discussion.
Student Facing
Label the area diagram to represent
\((2.5) \boldcdot (1.2)\)
and to find that product.
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Decompose each number into its base-ten units (ones, tenths, etc.) and write them in the boxes on each side of the rectangle.
Label Regions A, B, C, and D with their areas. Show your reasoning.
Find the product that the area diagram represents. Show your reasoning.
Here are two ways to calculate
\((2.5) \boldcdot (1.2)\)
. Each number with a box gives the area of one or more regions in the area diagram.
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In the boxes next to each number, write the letter(s) of the corresponding region(s).
In Calculation B, which two numbers are being multiplied to obtain 0.5?
Which numbers are being multiplied to obtain 2.5?
Student Response
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Anticipated Misconceptions
If students label side lengths with numbers that are not single-digit multiples of powers of ten, remind them to work with base-ten units as practiced in the previous lesson. Provide them with one label for a length and see if they can then find the other sections.
Activity Synthesis
Display the solution shown here for all to see.
[IMG]
Select a few students to explain how they determined which factor belongs to which side of the rectangle and how they found each partial area. Then, ask how each blue number in the calculations, especially Calculation B, is obtained. If not mentioned in students' explanations, emphasize that the blue 0.5, represented by the area of A + B in the diagram, is the product of 0.2 and 2.5. To highlight this connection, consider shading or coloring the corresponding areas in the diagram and circling these values in the calculations.  Likewise, the blue 2.5, represented by the area of C + D in the diagram, is the product of 1 and 2.5 in the calculation.


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