Unit 1, Lesson 5: Bases and Heights of Parallelograms

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Unit 1, Lesson 5: Bases and Heights of Parallelograms

Lesson Goals

? Write a formula for the area of a parallelogram in terms of a base and

corresponding height, and use the formula to find the area of parallelograms on a

grid.

? Find a base-height pair for parallelograms. ? Understand that there are always two base-height pairs for a given parallelogram. ? Use the terms base and height of a parallelogram to refer to the length of one side

and the perpendicular distance between that side and the opposite side.

? Understand that the base and height of a parallelogram determine its area.

Required Materials

? geometry toolkits

5.1: A Parallelogram and Its Rectangles (10 minutes)

Setup:

Students in groups of 2. Access to geometry toolkits. 2 minutes of quiet think time, followed by partner and whole-class discussions.

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Student task statement

Elena and Tyler were finding the area of this parallelogram:

Here is how Elena did it:

Here is how Tyler did it:

Possible responses

Answers vary. Sample responses:

? Similar: Both parallelograms are

cut off and rearranged to make identical rectangles.

? Different: Cuts were made in

different places. Different shapes were cut off. The rectangles are not in the same place.

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GRADE 6 MATHEMATICS BY How are the two strategies for finding the area of a parallelogram the same? How they are different?

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5.2: The Right Height? (20 minutes)

Setup:

Display image of examples and non-examples for all to see. A minute to notice and wonder, followed by a brief whole-class discussion. Students in the same groups. 4?5 minutes of partner discussion on the first question, followed by another class discussion. 3?4 minutes of quiet think time for the second question and 2?3 minutes of partner discussion. Access to geometry toolkits.

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Student task statement

1. Each parallelogram has a side that is labeled "base."

Study the examples and non-examples of bases and heights of parallelograms. Then, answer the questions that follow.

Examples: The dashed segment in each drawing represents the corresponding height for the given base.

Non-examples: The dashed segment in each drawing does not represent the corresponding height for the given base.

Select all statements that are true about bases and heights in a parallelogram. a. Only a horizontal side of a parallelogram can be a base. b. Any side of a parallelogram can be a base. c. A height can be drawn at any angle to the side chosen as the base.

Possible responses

1. Statements B, D, and F are true.

2. A, C, and D are correct.

Anticipated misconceptions

Students may not yet internalize that any side of parallelogram can be a base (they may think that a base must be the bottom, horizontal side), or that the height needs to be perpendicular to the base. Point out where the right angle symbols are located and how they relate to the height. Students may think a segment showing the height cannot be drawn outside of the parallelogram (as in Parallelogram C).

Students may relate how they think about the side lengths of a rectangle and inaccurately apply it to Parallelogram E.

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d. A base and its corresponding height must be perpendicular to each other. e. A height can only be drawn inside a parallelogram. f. A height can be drawn outside of the parallelogram, as long as it is drawn at a 90-degree angle to

the base. g. A base cannot be extended to meet a height. 2. Five students labeled a base and a corresponding height for each of these parallelograms. Are all drawings correctly labeled? Explain how you know.

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5.3: Finding the Formula for Area of Parallelograms (15 minutes)

Setup:

Students in the same groups. 5?7 minutes of quiet think time, followed by partner and whole-class discussions. Access to geometry toolkits.

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Student task statement

For each parallelogram:

? Identify a base and a corresponding height, and record their lengths in the table that follows. ? Find the area and record it in the right-most column.

In the last row, write an expression using and for the area of any parallelogram.

parallelogram A B C D

any parallelogram

base (units)

height (units)

area (sq units)

Possible responses

A: 6, 4, 24 or 4, 6, 24

B: 5, 3, 15

C: 2, 3, 6

D: 4, 2, 8

Any parallelogram:

Anticipated misconceptions

Finding a height segment outside of the parallelogram may still be a rather unfamiliar idea to students. Have examples from the "The Right Height?" section visible so they can serve as a reference in finding heights.

Students may say that the base of Parallelogram D cannot be determined because, as displayed, it does not have a horizontal side. Remind students that in an earlier activity we learned that any side of a parallelogram could be a base.

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