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TRIANGLES, RECTANGLES, AND PARALLELOGRAMS Author(s): Melfried Olson and Judith Olson Source: The Mathematics Teacher, Vol. 76, No. 2 (February 1983), pp. 112-116 Published by: National Council of Teachers of Mathematics Stable URL: http://www.jstor.org/stable/27963364 . Accessed: 18/07/2014 11:44 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to The Mathematics Teacher. http://www.jstor.org This content downloaded from 129.130.252.222 on Fri, 18 Jul 2014 11:44:07 AM All use subject to JSTOR Terms and Conditions

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Page 1: TRIANGLES, RECTANGLES, AND PARALLELOGRAMS

TRIANGLES, RECTANGLES, AND PARALLELOGRAMSAuthor(s): Melfried Olson and Judith OlsonSource: The Mathematics Teacher, Vol. 76, No. 2 (February 1983), pp. 112-116Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27963364 .

Accessed: 18/07/2014 11:44

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extendaccess to The Mathematics Teacher.

http://www.jstor.org

This content downloaded from 129.130.252.222 on Fri, 18 Jul 2014 11:44:07 AMAll use subject to JSTOR Terms and Conditions

Page 2: TRIANGLES, RECTANGLES, AND PARALLELOGRAMS

arti Wies -OA?

Edited by Evan M. Maletsky, Montclair State College, Upper Montclair, NJ 07043 Christian Hirsch, Western Michigan University, Kalamazoo, MI 49008 Daniel Yates, Mathematics and Science Center, Richmond, VA 23223

TRIANGLES, RECTANGLES, AND PARALLELOGRAMS

By Melfried Olson, University of Wyoming, Laramie, WY 82071 Judith Olson, University of Wyoming, Laramie, WY 82071

Teacher's Guide

Grade level: 7-10

Materials: Scissors, rulers, extra sheets of paper, and a set of worksheets for each student

Objectives: Students will manipulate physical models of geometric figures, en

gage in spatial visualization, and observe

relationships between triangles and paral lelograms and between triangles and rect

angles.

Directions: Depending on the back

ground and level of your students, the

suggested physical manipulation of the cutouts may not be a prerequisite for the

discovery of relationships among the ar eas of different figures. Nevertheless, many students will profit from these phys ical experiences. Alternative pictorial ap proaches are suggested below for each

activity sheet.

Sheet 1. After distributing the materi als, you may wish to use two triangular

cutouts to demonstrate on an overhead

projector how the shapes are to be posi tioned along a common side. The more

capable student can complete exercise 2

by using pairs of tracings of the triangle so that the copies have a common side. Since each side of one copy of the given triangle can be matched in two ways with the

corresponding congruent side of the sec ond copy, students will be able to form six different quadrilaterals (3 [sides] x 2 [matchings]). Of the three quadrilaterals formed that are not parallelograms, two are chevrons and one is a kite. A chevron is a nonconvex quadrilateral with two

pairs of congruent adjacent sides. A kite is the convex counterpart of a chevron.

Getting students to identify all related

parallelograms is frequently a challenge. Focusing their attention on the fact that one side of the given triangle is a diagonal of the related parallelogram is usually a successful approach. If pupils are experi encing difficulty completing exercise 5,

This section is designed to provide mathematical activities suitable for reproduction in worksheet and

transparency form for classroom use. This material may be photoreproduced by classroom teachers for use in their own classes without requesting permission from the National Council of Teachers of Mathematics. Laboratory experiences, discovery activities, and model constructions drawn from the

topics of seventh, eighth, and ninth grades are most welcome for review.

112 Mathematics Teacher

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Page 3: TRIANGLES, RECTANGLES, AND PARALLELOGRAMS

you might encourage them first to make

three copies of the given triangle and then

to form a related parallelogram on one

side of each figure. Students who complete this sheet more

quickly than others could be encouraged to complete exercise 2 using different tri

angles and investigate the conditions on a

triangle for which chevrons occur.

Sheet 2. The primary intent of this

worksheet is to show that, with respect to

area, any triangle can be viewed as one

half of some rectangle. You might encour

age students to cut piece A to fit on piece C. Similarly, ask them to fit pieces E and

G on piece F and fit pieces H and J on

piece /. The more capable student can

complete exercises 7-9 simply by using the given diagrams and visualization, or

possibly area formulas. These students

might be encouraged to find all possible

rectangles for each triangle in exercise 10.

Students who complete this sheet more

quickly than others could be asked to

draw a triangle so that none of its sides are

in a horizontal position and then find a

rectangle whose area is twice the area of

the triangle. It is informative to observe

how students tackle this problem.

Answers

Sheet 1. 2. a. 6; b. 3; c. quadrilaterals or, more specifically, chevrons and kites; d. The area of the triangle is one-half the

area of each quadrilateral. 3. Possible

solutions include these:

4. 3; 5. b. The resulting figure is a triangle similar to the given one. The new triangle has an area four times the original.

Sheet 2. 7. d. Of course, they are all the same in area. 8. b. Pieces A and F have

equal area, as do pieces ? and G. Piece F

is twice as large as piece E. c. They are

equal. 9. b. Pieces A, /, and F have equal area. Piece H will be larger or smaller than

piece / depending on where the point is

chosen. 10. Answers will vary, but make

sure the rectangles are correctly placed. Possible solutions include the following:

Answers for the Calendar Problems?February

The September 1982 issue contained a colorful

twelve-month calendar insert suitable for use on a

bulletin board. The general solution that appeared in

September can be adapted to express each day of the

month using six 2s for February; other answers

follow.

2. 3x4x5x6+1 = 19 19

4x5x6x7+1= 29 x 29 5x6x7x8+1= 41 41

(n) (n + 1) (n + 2) (n + 3) + 1 = ((n + 1) (n + 2) - l)2

4. 1073

6. 4.01 109 miles

10. 73 beats per minute

14. n(n -

1) when is the number of people in the

class

20. 8:10 p.m.

24. 9

25. 1.05 m

28. The second completed problem should have

read

48 (4 + 8) = 43 + 83

Then the pattern is

111 (11 + 1) = II3 + l3 147 x (14 + 7) = 143 + 73 148 (14 + 8) = 143 + 83

Single copies of the two-year calendar that ap

peared in the September 1982 Mathematics Teacher

are available for $2.50 (stock #311). Five copies cost

$5.00 (stock #312). A 20 percent individual member ship discount applies. Use the NCTM Educational

Materials Order Form in the ''New Publications"

section for your order.

COLLEGE ENTRANCE EXAMINATION Indexed Sample Objective Questions

154 SAT-Math Problems 216 Math Ach.-Level 1 & 2 Problems 90 AP Calculus AB & BC Problems Detachable Answers

at $6.50 (Price includes postage.) Send check or money order to:

NATHANIEL B. BATES 277 Nashoba Rd., Concord, Ma. 01742

February 1983 113

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Page 4: TRIANGLES, RECTANGLES, AND PARALLELOGRAMS

TRIANGLES AND PARALLELOGRAMS SHEET 1

1. On a separate sheet of paper, make two copies of the triangle at the right.

2. Cut out the two triangles and then

place them together so that they have a side in common.

a. How many different four-sided figures can be formed in this way? _Make a small sketch of each shape in the space below.

b. How many of these figures are parallelograms?_

c. What might you call the remaining figures?_

d. How does the area of the original triangle above compare to the area of each

quadrilateral you formed?_

3. The area of any triangle can be viewed as one-half the area of some parallelo gram. For each triangle below, draw a parallelogram so that the area of the

triangle is one-half the area of the parallelogram.

?\

4. The parallelograms you drew in 3 are called related parallelograms. How many related parallelograms can be drawn for each triangle?

5. a. Draw all three related parallelograms on the

triangle at the right.

b. What do you notice about the size, shape, and area of the resulting figure?_

From the Mathematics Teacher, February 1983

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Page 5: TRIANGLES, RECTANGLES, AND PARALLELOGRAMS

RECTANGLES SHEET 2

6. Cut out the four rectangles on sheet 3.

7. a. Divide one rectangle vertically into two congruent parts and label them as shown.

b. Divide a second rectangle horizontally into two congruent

parts and label them as shown.

c. Cut apart the four pieces, A, B, C, and D.

d. Which piece is larger in area:

AorB?_ CorD?_ A or C?_ A or D?

D

8. a. Take another rectangle, find the midpoint of one of the

longer sides, draw the segments, and label them as shown at

the right. Cut apart the three pieces.

b. Which piece is larger in area:

A or Fl_ E or Fl_ E or G?_

c. How does the area of E plus the area of G compare to the area

of Fl_

9. a. Take the fourth rectangle; pick any point, other than mid

point, on one of the longer sides; draw the segments; and label them as shown at the right. Cut apart the three pieces.

b. Which piece is larger in area:

A or II_ H or Jl_ / or Fl_

10. For any rectangle, we can find many triangles whose areas are one-half the area

of the rectangle. Let's try reversing the process. For each triangle below, draw a

rectangle so that the area of the triangle is one-half the area of the rectangle.

From the Mathematics Teacher, February 1983

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Page 6: TRIANGLES, RECTANGLES, AND PARALLELOGRAMS

SHEET 3

From the Mathematics Teacher, February 1983

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