Area of Parallelograms and Triangles


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1 Context: Grade 7 Mathematics. Area of Parallelograms and Triangles Objective: The student shall derive the formulas for the area of the triangle and the parallelogram and apply these formulas to determine the area of triangles and quadrilaterals. SOL (2001): 7.7 The student will, given appropriate dimensions, will estimate and find the area of polygons by subdividing them into rectangles and right triangles. Materials/Resources: Laptop computer connected to digital projector, laptop cart or use of computer lab, NCTM Illuminations Shape Cutter applet, Area of Parallelograms and Triangles Guided Notes, and Homework worksheet. Time required: 45 minutes Content and Instructional Strategies: 1. Introduce the lesson by reviewing the properties of special quadrilaterals, including the parallelogram, rhombus, rectangle, square, and trapezoid. Review the concept of area and the formula for the area of a rectangle. The area of a rectangle is equal to the length times the width. Discuss the concept of height of a parallelogram and a triangle. The height must be perpendicular to the side that is specified as the base. 2. Pass out the handout titled Area of Parallelograms and Triangles. 3. Consider Activity  Recognize a pattern. Instruct the students to open the Shape Cutter Applet at Demonstrate how to use the new shape tools to construct a rectangle with a length of 10 units and a width of 6 units. Have the students do the same. How do we find the area of a rectangle? The area of a rectangle is the length times the width. Write this formula on your handout under problem #1. Fill in the dimensions for the length and width and calculate the Area. l=10, w = 6, and A= 60 square units. We could also use the grid to count all 60 squares. 4. Interpret Activity  Pose a conjecture [Problem #1 on page 410]. Teacher demonstrates how to construct a parallelogram with a base of 10 units and a height of six 6 units using the applet. On the projection on the white board, write in the dimensions. Use the cut tool to draw the height from the base to the vertex and cut off the triangle. Use the slide tool to drag the cut triangle to the opposite end to transform the parallelogram into a rectangle. Did we change the area? No. Does this now look like the rectangle in #1? Yes. What is the area of the new quadrilateral? 60 square units. Fill in the area for the parallelogram on the handout (#2) 5. Interpret Activity  Develop an Argument. Instruct the students to construct a new parallelogram with the same base and height measurements, but different interior angles. Use the same procedure to cut the triangle and form a rectangle. What is the area of this rectangle? 60 square units Fill in #3 on the handout. For the rectangle, we would say length times width. As we think about our parallelogram, the height was transformed
2 into the width. The base of the parallelogram was cut and moved. Did the overall base measurement change? No. So, the base of the parallelogram was transformed into the length of the rectangle. So, the area of the parallelogram is base times height. Do the interior angles of the parallelogram have any effect on the parallelogram? No. Area is only affected by the base and the height. Remember that the height must be perpendicular to whichever side we select as the base. It is the perpendicular distance from that base to the other base. What is the formula for the area of a parallelogram? A = bh. Complete the formula for the area of the parallelogram at the bottom of the first page of the handout. 6. Interpret Activity  Develop an Argument. [Problem #4 on page 410.] Instruct the students to complete problem #3 on the handout using the same construction and cutting in the applet. Is it necessary to know the length of the other side of the parallelogram? No. We do not need that measurement to find the area of the parallelogram. Instruct the students to find the area of the parallelogram. A=bh. A=3*4=12 ft. 7. [Problem #2 on page 410]. On the board, draw a parallelogram with a base of 5 ft and height of 5 m. Mark the height outside the parallelogram. Does it matter if the height is drawn outside of the parallelogram? No. Find the area of the parallelogram. A=bh. A=5*5=25 m. Complete #5 on the handout. Students may check their answers using the applet. (Evaluate Activity  Test a Solution.) 8. On the board, draw a parallelogram with a base of 7 and a height of 4 as shown in problem # 6 on the handout. Mark the dimensions of the other sides as shown. Complete #6 on the handout. If the height is 4, how do we know which side is the base? We must use the side which is perpendicular to the height. In this case, the base has a length of 7 units. Students may check their answers using the applet. Students may need to rotate the drawing in order to draw it on the applet. (Evaluate Activity  Test a Solution.) 9. Consider Activity  Recognize a pattern. Teacher demonstrates how using the applet, construct a rectangle, with a base of 6 units and a height of 4 units. Mark the dimensions on the board. Use the cut tool to cut the rectangle into two right triangles. Use the rotate and slide tools to show that the two triangles are congruent. What is the area of the rectangle? A=6*4=24. What is the area of the right triangle? The area of the triangle is one half of the area of the rectangle. Therefore, the area of the of the triangle is onehalf of the base times the height, which is 12 square units. Complete #7 on the handout. 10. Interpret Activity  Develop an Argument. Does this work for all triangles? Instruct the students to use the applet to construct a new parallelogram, with base of 6 and a height of 4. Use the cut tool to divide the parallelogram into two triangles on the diagonal. Use the rotate and slide tools to confirm that the two triangles are congruent. Is it safe to say that the area of triangle is onehalf of the area of the parallelogram? If the area of the parallelogram is base times height, what is the formula for the area of the triangle? The area of the triangle is one half of the base times the height, which is 0.5*6*4 = 12 square units. Complete #8 on the handout.
3 11. Problem #7 on page 410. On the board, draw a triangle with a base of 14 cm and a height of 8 cm. Instruct the students to find the area. A=0.5*14*8 = 56 cm^2. Complete #9 on the handout. Students may wish to check their answers using the applet. (Evaluate Activity  Test a Solution.) 12. Problem #10 on page 410. On the board, draw a triangle with a base of 12 km and a height of 12 km. When I am finding area, does it make any difference if the height is shown outside the triangle? No. Does it matter if my "base" is at the top? It is fine as long as the "height" is perpendicular to the specified base. Instruct the student to find the area of the triangle. A=0.5*12*12 = 72 km^2. Complete #10 on the handout. Students may wish to check their answers using the applet. (Evaluate Activity  Test a Solution.) Plan B: There are varying degrees of failures and modifications which could be implemented to ensure an effective lesson. First, if the laptop cart or lab is unavailable, the teacher could simply perform steps 35, and 910 as a demonstration with a single laptop and the digital projector. The remaining steps would be presented on the whiteboard and students would work them out on paper, and the class would go over these problems together. If the technology is not at all possible on the lesson day, steps 35 and 910 could be demonstrated using large square graph paper and scissors. Students could then use quadrule graph paper to model the other area problems, as necessary. Some students may prefer this tactile method of learning to the visual method provided by the virtual manipulatives. Differentiation and Adaptations: Students who may need extra assistance could work in partners. Students who may be advanced could use computers in the classroom and use a similar method to derive the formula for the area of a trapezoid by dividing a parallelogram into two congruent trapezoids. Advanced students might also apply the Pythagorean theorem to find the area of a right triangle, given the base and hypotenuse. Advanced students might also do the constructions in Geogebra, where they could use the coordinate system to construct the polygons of specified dimensions. The amount of practice could be adjusted to meet the needs of the students. Evaluation: Worksheet Practice 82 #118.
4 Reflection: The design of the technology enhanced lesson plan for the area of parallelograms and triangles truly makes use of technology to enhance the mathematical concept. The lesson is designed for guided discovery. Concept development is important to students, and whenever possible, students should understand the basis of the formulas that they are using. In this way, there is some hope that if the student is not able to recall a particular formula, he or she will be able to deduce it, based on prior knowledge. By using the virtual manipulative in this lesson, students are actually deriving the formula and making sense out of it. This could easily be done as a teacher led demonstration, if resources are not available or if time does not allow for the student led activity described in the lesson plan. While exploring various virtual manipulatives, I found that this particular applet, although the most simplistic, could provide the greatest conceptual understanding. Because it has the grid, students can easily count the squares to find the area, so it reinforces the fundamental understanding of area using an array model, in the same way as it is introduced in the elementary grades. Other manipulatives simply provided drill and practice. By using this particular applet, students must construct their own parallelograms, which will strengthen the basic geometry skills, rather than others, in which students may adjust the lengths of the sides of the parallelogram, and the applet computes the area. I like the flexibility that the applet provides in the classroom. It offers a sort of "arts and crafts" approach to engage the students, with a more technological spin. In this way, the students have quick and easy access to a manipulative, without the cleanup. It also enhances students' spatial skills, which is not something that is commonly enhanced in a middle school mathematics classroom.
5 Problem #2 Screen shots of Shape Cutter Applet in use with guided notes. Problem #3 Problem #4 Problem# 7 Problem #8
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