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# Area of Trapezoids

Lesson 8 of 14

## Objective: SWBAT derive and apply the area formula for a trapezoid.

#### Think About It

*7 min*

Students work independently on today's Think About It problem.

Using a student work sample that decomposes the trapezoid into triangles, we create the expression (1/2 bh) + (1/2 bh) to represent the area of the trapezoid. However, we can't use b for both bases, give that they have different values. This leads into the Intro to New Material Section.

#### Resources

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#### Intro to New Material

*15 min*

To start the Intro to New Material, I review the properties of a trapezoid with my students. We label the bases as b_{1} and b_{2}. Because my students have not had much exposure to subscripts, I explicitly tell them that the numbers on the variables are called subscripts and don’t have any numerical meaning. We are using these to distinguish between the first base and the second base. We also label the height of the trapezoid, h.

Because a trapezoid can be decomposed into two triangles, the area of any trapezoid can be found using the formula:

*Area = ½ hb _{1} + ½ hb_{2}*

To gain this insight, I have my students work with their partners to apply what they've learned about the Distributive Property to re-write the equation. Most students eventually arrive at A = ½ h(b_{1} + b_{2}), which we put into the formula box in Intro to New Material notes.

Once we derive the formula and record it, we return to the Think About It problem and apply the formula to the trapezoid. I ask a student to identify the height. I then ask a student to identify base one (b_{1).} No matter which base value the student gives me, I say, "Huh. I was thinking that (the other value) would be base one. Does it matter?" I want students to identify that it does not matter which base we name as b_{1} and b_{2 }because the values are being added together and addition is commutative.

For Example 2, I have students identify the height and bases. Students identify the bases as the parallel sides. There can be some misconceptions around the height, because of the orientation of the trapezoid. If a student states that 19 is the height, I ask for another student to share out why 19 *can't * be the height. I am looking to hear something like: "19 inches is the diagonal distance between the two bases and isn't the perpendicular distance. A right angle needs to be present between the two bases for it to be considered the height."

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#### Partner Practice

*15 min*

Students work in pairs on the Partner Practice problem set. As they work, I circulate around the room and check in with each pair. I am looking for:

- Are scholars accurately finding the area of the given trapezoid?
- Are scholars correctly identifying and labeling the dimensions of the trapezoid?
- Are scholars correctly showing all steps?
- Are scholars correctly substituting the correct values into the formula?
- Are students correctly applying the concept of area to solve real world problems?
- Are scholars using the correct units?

I am asking:

- How did you know to use this formula for area?
- Where are the bases in this trapezoid? How do you know?
- Why can't (diagonal length value) be the height of the trapezoid?
- How did you know to substitute these values into the formula?
- How did you know this is the correct area?
- How did you know which units to use?

After 10 minutes of partner work time, students complete the Check for Understanding problem independently. I then have students whisper out their answer choice. I call on a student to share his/her work on the document camera. The class offers feedback on the written work.

#### Resources

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#### Independent Practice

*20 min*

Students work on the Independent Practice problem set.

In Problem 1, the height of the trapezoid is shown external to the figure. My students know that a dashed line signifies the dimension of height, but this could be confusing for some students.

I like Problem 2, because if forces students to check all of the possible answers. This is something I suspect we'll see on the Smarter Balanced assessment, and it's something my students were not doing at the start of the year. I've been working to ingrain the idea that there can be more than one correct answer for multiple choice questions.

#### Resources

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#### Closing and Exit Ticket

*8 min*

At the end of Independent Practice time, I bring the class back together to discuss Problem_6. I have students turn to their partners to discuss strategies and answers. Then, I ask for 3 pairs to share out their strategies for getting to the answer. Most students will find the area of each trapezoid, and compare. Some students will notice that the larger trapezoid has 18 more cm worth of base length, and will multiply 18 x 4 (which is 1/2h for both trapezoids).

Students work independently to complete the Exit Ticket to close the lesson.

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- UNIT 1: Number Sense
- UNIT 2: Division with Fractions
- UNIT 3: Integers and Rational Numbers
- UNIT 4: Coordinate Plane
- UNIT 5: Rates and Ratios
- UNIT 6: Unit Rate Applications and Percents
- UNIT 7: Expressions
- UNIT 8: Equations
- UNIT 9: Inequalities
- UNIT 10: Area of Two Dimensional Figures
- UNIT 11: Analyzing Data

- LESSON 1: Area of Rectangles
- LESSON 2: Rectangles with the Same Area and Different Perimeters
- LESSON 3: Rectangles with the Same Perimeters and Different Areas
- LESSON 4: Area of Parallelograms
- LESSON 5: Area of Parallelograms, Applying the Formula
- LESSON 6: Finding the Area of Triangles by Composing Parallelograms
- LESSON 7: Area of Triangles, Applying the Formula
- LESSON 8: Area of Trapezoids
- LESSON 9: Area of Shapes on the Coordinate Grid
- LESSON 10: Draw Shapes When Given an Area
- LESSON 11: Area of Compound Shapes on the Coordinate Grid
- LESSON 12: Area of Compound Shapes
- LESSON 13: Area of Compound Shapes with Unknown Dimensions
- LESSON 14: Area of Compound Shapes, Using Decomposition