View
220
Download
0
Embed Size (px)
Citation preview
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
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
This content downloaded from 129.130.252.222 on Fri, 18 Jul 2014 11:44:07 AMAll use subject to JSTOR Terms and Conditions
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
This content downloaded from 129.130.252.222 on Fri, 18 Jul 2014 11:44:07 AMAll use subject to JSTOR Terms and Conditions
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
This content downloaded from 129.130.252.222 on Fri, 18 Jul 2014 11:44:07 AMAll use subject to JSTOR Terms and Conditions
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
This content downloaded from 129.130.252.222 on Fri, 18 Jul 2014 11:44:07 AMAll use subject to JSTOR Terms and Conditions
SHEET 3
From the Mathematics Teacher, February 1983
This content downloaded from 129.130.252.222 on Fri, 18 Jul 2014 11:44:07 AMAll use subject to JSTOR Terms and Conditions