Area of Trapezoids

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Objective

SWBAT derive and apply the area formula for a trapezoid.

Big Idea

Formulas for calculating the area of shapes can be derived by decomposition into shapes for which an area formula is already known.

Think About It

7 minutes

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.

Intro to New Material

15 minutes

To start the Intro to New Material, I review the properties of a trapezoid with my students. We label the bases as b1 and b2. 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 hb1 + ½ hb2

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(b1 + b2), 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 (b1). 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 b1 and bbecause 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."

Partner Practice

15 minutes

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.

 

 

Independent Practice

20 minutes

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.  

 

Closing and Exit Ticket

8 minutes

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.