# Area of Triangles and Trapezoids

This lesson is based from the textbook Holt Middle School Math:  Course 2.
Bennet, J.M., Chard, D.J., Jackson, A., Milgram, J., Scheer, J.K., Waits, B.K.  (2004).  Holt middle school math:  Course 2.  Orlando:  A Harcourt Education Company.

Lesson Title:  Finding Areas
Grade Level:  7th grade
Course Title:  Compacted Math
Time Allotted:  1 class period
Number of Students:
24-34 students
Extra Information About Students:
None
Day 4

Goals and Objectives:

According to the NCTM Principles and Standards of Mathematics, the following standards are met in this lesson:
1.   To understand measurable attributes of objects and the units, systems, and processes of measurement.

• Understand both metric and customary systems of measurement.
• Understand relationships among units and convert from one unit to another within the same system.
• Understand, select, and use units of appropriate size and type to measure angles, perimeter, area, surface area, and volume.

2.   To apply appropriate techniques, tools, and formulas to determine measurements.

• Develop and use formulas to determine the circumference of circles and the area of triangles, parallelograms, trapezoids, and circles and develop strategies to find the area of more-complex shapes.

According to the NCSCOS, the following standards are met in this lesson:

1.   Competency Goal 2:  The learner will demonstrate an understanding and use of the properties and relationships in geometry, and standard units of metric and customary measurement.

Goals:

• For the students to understand how to find the areas of triangles and trapezoids.
• For the students to apply this information to their everyday encounters.

Materials Needed and/or Use of Space:

• Calculators
• Rulers
• Graph Paper
• Scissors

Math Fact of the Day:

The line connecting the midpoints of the two nonparallel segments of a trapezoid is called the midsegment.

• Share this with the students at the beginning of class.  It is similar to when an English teacher shares a quote at the beginning of language arts class!!

Problem of the Day:

An isosceles trapezoid has bases 7 in. and 4 in. and heigh 1.5 in.  Find its area by using only the formula for the area of the parallelogram.  Draw a picture, too.

• Give this to the students at the beginning of class and have them turn in their answers with their unit portfolio at the end of the unit.  The answer for this problem is:  (5.5)(1.5)=8.25 square inches
• This problem is an introduction into the day's lesson.
• The students should be able to solve this problem after today's lesson.

Motivational Activity:

The students will learn about how to find the area of triangles and trapezoids today.

Before teaching the lesson, you should review the parts of the triangle, particularly the altitude and base of the triangle.  Other things to review include the following:
• Draw an acute triangle on the board.
• Define the base of the triangle as any side of the triangle and label it.
• Draw the altitude that corresponds to the base.
• Be sure to explain that the altitude is the perpendicular segment from a vetex of the triangle to the opposite base.
• The height of the triangle is the length of the altitude.
After reviewing this, explain to the students some information about the Bermuda Triangle.
• The Bermuda Triangle is a triangular region between Bermuda, Florida, and Puerto Rico.   The Bermuda Triangle is an interesting phenomenon.  More information about this phenomena can be found at http://www.parascope.com/en/bermuda1.htm
• This website may be an interesting resource to explore if there is enough time AND if the students all have access to computers.  A webquest could possibly be made in correspondence with this website.  This would provide the students with a technology component during class.
• In order to find the area of this region, you could use the formula that we will learn today to find the area of a triangle!  This formula is closely related to the formula used to find the area of the parallelogram.

Lesson Procedure:

Yesterday we learned how to find the area of parallelograms.  Today we are going to discuss how to find the areas of triangles and trapezoids.  When looking at a parallelogram, we notice that the diagonal of a parallelogram divides the parallelogram into two congruent triangles.  (Draw on the board if necessary for students to see).  Thus, the area of each triangle is half the area of the parallelogram. The base of a triangle can be any side.  The height of a triangle is the perpendicular distance from the base to the opposite side.

Here is a chart similar to those we looked at yesterday to help you understand how to find the area of a triangle, along with a picture to help you learn.

AREA OF A TRIANGLE
 The area A of a triangle is half the product of its base b and its height h. A=1/2 bh Now that we know what area is and how to find the area of a triangle, let's do an example together!

Example 1:  Find the area of the triangle. We use the formula and substitute for l and w.
A=1/2(bh)
A=1/2(7*4)
A=1/2(28)
A=14 square units

Thus, the area of the rectangle is 14 square units.
• Note:  Our units are in square units.  Make sure to point this out to students and to explain that because we do not have a specific unit of measurement to use, that we assume the measurement to be in units.
A parallelogram can be divided into two congruent triapezoids.  The area of each trapezoid is one-half the area of the parallelogram. The two parallel sides of a trapezoid are its bases.  If we call the longer side b1 and the shorter side b2, then the base of the parallelogram is b1+b2. AREA OF A TRAPEZOID
 The area A of a trapezoid is half its height multiplied by the sum of the lengths of its two bases. A=1/2h(b1+b2) • Note:  Remind the students the difference between b1 and b2 - I think that it will become more clear when doing an example together.

Example 2:  Find the area of the trapezoid. A=1/2h(b1+b2)
A=1/2(5 cm)(2.5 cm + 6 cm)
A=(2.5 cm)(8.5 cm)
A= 21.25 square centimeters.

Thus, the area of our trapezoid is 21.25 square centimeters.

Now that we know how to find the area of a triangle and the area of a trapezoid, let's do an activity utilizing the new concepts that we have just learned.

Activity:
• Have the students get into small groups - probably 3 or 4 depending upon how many students are in the class.  Today, separate the groups based on the letters of their first names; put the first 3 people with A-names together, the next 3 people who have B-names, and continue on until there are groups with 3-4 students in each group.
• Have the students draw a triangle (with the use of a ruler) whose lengths have whole number measures.  Make sure to note that the triangle should fit on the piece of graph paper that was given to them.
• Now have the students cut out the triangle and use the formula to show that any of the three sides can serve as the base of the triangle.
• In order to find the height, students can draw a rectangle around the cutout triangle on another sheet of graph paper to find the height of the triangle for each base.
• This activity is good for those who are visual learners and kinesthetic learners.
• The teacher should provide scaffolding to the students so that they understand the big picture and see what is going on.
• Some questions to ask are:
• Why do we need to make sure that we are using whole numbers for the lengths of our sides?
• What is another way that we can find the height of our triangle?
• How is this particular activity helpful to your learning process?

Extension:

If an extension activity is needed, the students can think and write responses to the following questions:

• Tell how to use the sides of a right triangle to find its area.
• Explain how to find the area of a trapezoid.

Closure:

Review with the students the various formulas that they have learned thus far - circumference, area of a parallelogram, triangle, and trapezoid.  Have students come up to the board and write the formulas and have other students come up to the board and draw the specific diagram that corresponds with the specific formula.

As you can see, we have learned how to find the area of triangles and trapezoids.  As we saw in one of our examples, we utilize this concept in the real world sometimes.  This is an important component that will be of use in your future with whatever you do.  Tomorrow we will discuss finding the area of circles and see how that is different from what we have been discussing the past two days - the areas of parallelograms, triangles, and trapezoids.

Assessment:

• Students will complete the problem of the day.  Their completion of this problem will show the teacher whether or not they understand what they have learned in class that day.  This activity also teaches the students more about mathematical communication.
• The students will also have a worksheet to complete.  If the worksheet is not completed in class, it is to be finished as homework.  This worksheet will count as a homework grade.
• The students are expected to participate in the activity, too.  This will be part of their classwork grade and their behavior/conduct grade.

Evaluation of Lesson Upon Completion:

• What would you do differently?
• What would you do the same?
• Was this a good lesson - why or why not?