Area Formulas for Parallelograms, Triangles, and Trapezoids Developed and Published by AIMS Education Foundation This book contains materials developed by the AIMS Education Foundation. AIMS (Activities Integrating Mathematics and Science) began in 1981 with a grant from the National Science Foundation. The non-profit AIMS Education Foundation publishes hands-on instructional materials that build conceptual understanding. The foundation also sponsors a national program of professional development through which educators may gain expertise in teaching math and science. Copyright © 2009 by the AIMS Education Foundation All rights reserved. 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Contact us or visit the AIMS website for complete details. AIMS Education Foundation 1595 S. Chestnut Ave., Fresno, CA 93702 • 888.733.2467 • aimsedu.org ISBN 978-1-60519-009-9 Printed in the United States of America AREA FORMULAS © 2009 AIMS Education Foundation Area Formulas for Parallelograms, Triangles, and Trapezoids Table of Contents Welcome to the AIMS Essential Math Series! .................... 3 BIG IDEA: The area of a parallelogram is found by multiplying base by height. All parallelograms that have the same length base and height have the same area no matter how far they have been skewed. Lesson One: Parallelogram Cut-Ups ........................................... 9 Days 1 and 2 Investigation Parallelogram Cut-Ups............................................................. 10 By measuring the base, height, and sides of parallelograms, students recognize that the height and sides of parallelograms are different lengths. They learn to use the appropriate measures to determine the perimeters of parallelograms. By cutting up a parallelogram and reforming it into a rectangle, students discover the relationship of the two and the similarity of ﬁnding the areas of both. A critical understanding is being able to differentiate between a side and the height. Students should come to recognize that any of the sides can be designated as the base. Comic Parallelogram Cut-Ups .............................................................................. 15 The comic summarizes the relationship of a parallelogram to a rectangle with the same base and height and develops the meaning of the general formula for area. Lesson Two: Areas on Board: Parallelograms .................. 17 Day 3 AREA FORMULAS Investigation Areas on Board: Parallelograms .............................................. 19 By recognizing that every parallelogram can be transformed into an equal area rectangle, students conﬁrm the area formula of base times height. 5 © 2009 AIMS Education Foundation Day 4 Day 5 Animation A Shifting Parallelogram ................................................................ 21 A parallelogram is skewed while keeping its sides the same length resulting in a changed height and area. The parallelogram is then skewed again, this time changing the length of the sides to keep the height constant resulting in a constant area. The visual display accentuates the critical nature of the height. Problem Solving Parallelograms ..................................................................23 These problems provide an assessment of understanding of the area formula as students are asked to ﬁnd various unknown dimensions. BIG IDEA: A triangle is half the area of a parallelogram with the same height and width. The formula A = (b • h)/2 = ½ (b • h) describes this relationship. Lesson Three: Triangle Cut-Ups ....................................................25 Day 6 Investigation Triangle Cut-Ups ......................................................................26 By matching pairs of congruent triangles and forming parallelograms with them, students will recognize that a triangle is half of a parallelogram. This understanding connects with the formula for the area of a triangle as: A = (b • h)/2 = ½ (b • h). Comic Triangle Cut-Ups .......................................................................................29 The comic emphasizes the relationship of two congruent triangles to a parallelogram and develops the formula by showing how it represents this relationship. Lesson Four: Triangles to Parallelograms ............................. 31 Day 7 Investigation Triangles to Parallelograms......................................................32 All triangles can be cut so their pieces can be reformed into parallelograms. A parallelogram will have a base or height that is half the base or height of the triangle from which it was made. The experience provides a visual model of the formula in the form A = ½ b • h = b • ½ h. Comic Triangles to Parallelograms .......................................................................35 The comic emphasizes the relationship of two congruent triangles to a parallelogram and develops the formula by showing how it represents this relationship. AREA FORMULAS 6 © 2009 AIMS Education Foundation Lesson Five: Areas on Board: Triangles .................................. 37 Day 8 Day 9 Investigation Areas on Board: Triangles........................................................39 A geoboard or dot paper provides a grid for counting square area. By looking at triangles with equal areas, students ﬁnd that the triangles have a common base and height. Multiplying the base and height gives the area of a rectangle that is twice the size of the triangle. Animation Triangle Transformations ............................................................... 41 A triangle is transformed into a parallelogram or rectangle in four ways. The vivid visual models reinforce the understanding of the area formula and demonstrate several forms of the formula. Practice: Bigger Triangles .........................................................................................43 Using larger triangles drawn on dot paper, students apply and practice what they learned from the initial investigation. Day 10 Problem Solving Triangles ...........................................................................45 Students use the area formula to solve problems with triangles. BIG IDEA: The area of a trapezoid is a calculated by multiplying the average base by the height. The formula is A = ((b1 + b2)/2) • h = ((b1 + b2) • h) / 2 = ½ ((b1 + b2) • h). Lesson Six: Trapezoid Cut-Ups.......................................................49 Day 11 Investigation Trapezoid Cut-Ups...................................................................50 Cutting out and combining two trapezoids (b1 + b2) • h . into a parallelogram demonstrates the area formula A = 2 Any two congruent trapezoids form a parallelogram that has a base that is the combined lengths of the top and bottom bases of the trapezoid. Dividing the area of the parallelogram by two gives the area of the trapezoid. Comic Trapezoid Cut-Ups ...................................................................................53 The comic demonstrates that two congruent trapezoids form a parallelogram and reinforces how the formula represents this relationship. AREA FORMULAS 7 © 2009 AIMS Education Foundation Lesson Seven: Trapezoids to Parallelograms .....................55 Day 12 Day 13 Investigation Trapezoids to Parallelograms ..................................................56 Cutting a trapezoid in half and rotating it forms a parallelogram of the same area. Calculating the area of the parallelogram, which is half the height of the trapezoid, gives the area of the trapezoid. The transformation of the trapezoid is a visual model of the formula in the form A = ½h • (b1 + b2). Animation Trapezoid Tumbles ........................................................................ 57 The animation demonstrates both doubling the trapezoid to make a parallelogram twice as big as the trapezoid and cutting the trapezoid in half to make a parallelogram equal in size. The dynamic visual reinforces the ideas developed in the investigations and encourages students to seek the commonalities of the formulas to clarify the concepts expressed in the formulas. Problem Solving Trapezoids ........................................................................59 Students apply and practice using a visual model or an area formula to solve trapezoid problems. Day 14 Day 15 Problem Solving Polygon Puzzle ................................................................. 61 A ﬁve-piece puzzle can be formed into two rectangles, two parallelograms, one triangle, and three trapezoids. It provides an opportunity to review and summarize the meaning of the formulas and see their interrelatedness. Assessment Geoboard Designs......................................................................65 Some very interesting and complex shapes are made by combining polygons on dot paper. The problems can be solved visually or by using formulas. Glossary ............................................................................................................................................69 National Standards and Materials.................................................................................................... 71 Using Comics to Teach Math ............................................................................................................72 Using Animations to Teach Math .....................................................................................................73 The Story of Area Formulas ..............................................................................................................75 The AIMS Model of Learning ............................................................................................................79 AREA FORMULAS 8 © 2009 AIMS Education Foundation P ARALLELOGRAM UPS UT C How is ﬁnding the area of a parallelogram different from ﬁnding the area of a rectangle? It is crucial to be able to differentiate among a side, the base, and the height. A parallelogram has four sides. The base is one of these sides. The height is only a side of a parallelogram when the parallelogram is a rectangle. Using the Measuring Pad, students focus on these differences. By cutting up a parallelogram and reforming it into a rectangle, one discovers the relationship of the two and the similarity of ﬁnding area of both. ARALLELOGRAM n n UT PS attiioo tig iigga t t s s e nvve IIn P Materials Scissors Parallelograms Measuring Pad Each group of 4 or 5 students will need 2 of each of the parallelograms. ARALLELOGRAM P 1. What two dimensions did you use to determine the perimeter? U these are some pressing questions. the lengths of each pair of opposite sides 2. How did you use those dimensions to ﬁnd the perimeter? ﬁnd the sum of all four sides CUT UPS 3. What two dimensions do you use to determine the area of the rectangle made from the two pieces of the parallelogram? How is ﬁnding the area of a parallelogram different from ﬁnding the area of a rectangle? find the perimeter by adding up the four sides. C base and height A 1. Make sure Use the Measuring Pad to ﬁnd and record the lengths of B the sides of each parallelogram. (Round to the nearest students align A whole centimeter, if necessary.) C each side to the Determine and record the perimeter of each parallelogram. D ruler as they measure. 2. Make the long side of each parallelogram the base. D B C measure the base... ...and the height. 4. How are the dimensions used for area different than the ones used for perimeter? Base is a side, but height is at right angles to the base. For the rectangle, the base and height are the same as the sides. 5. Does it matter which side is the base? Measure and record the length of the base and the height. Area is the product of base and height. See that students recognize the difference between height and side. No, as long as you measure the height perpendicular from the base you chose. Make the short side of each parallelogram the base. Measure and record the length of the base and the height. 3. cut and arrange the parallelograms into rectangles. Cut a dotted line marking one of the heights of the parallelogram. Cut different lines on each pair of congruent parallelograms. Area = base • height, A = b • h Make a rectangle using the two pieces from each parallelogram. Determine and record the area of each parallelogram by ﬁnding the area of its two pieces making the rectangle. 1. Short Long Perimeter Side (cm) Side (cm) (cm) Parallelogram A 8 12 40 Parallelogram B 9 12 42 Parallelogram C 12 16 56 AREA FORMULAS Base (cm) Height (cm) Area (cm2) Long Base 12 8 96 Short Base 8 12 96 Long Base 12 9 6 8 72 72 16 12 9 12 144 144 2. Short Base Long Base Short Base 12 6. Write a formula to describe how you use these two dimensions to get the area (A) of the parallelogram. AREA FORMULAS 3. 13 © 2009 AIMS Education Foundation Confirm that students recognize that the areas of the parallelogram and the corresponding rectangle are equal. © 2009 AIMS Education Foundation Compare the two sides of the chart so it is recognized that any side can be used as the base but the height is not a side except in the rectangle. ics Students reinforce their understanding of the relationship of a parallelogram to a m Co rectangle when ﬁnding perimeter and area. AREA FORMULAS 9 © 2009 AIMS Education Foundation PARALLELOGRAM UT U PS C cut out the parallelograms along the bold lines. your group will need two of each parallelogram. C A B AREA FORMULAS 10 © 2009 AIMS Education Foundation 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 Measuring Pad 10 11 12 13 14 15 16 17 18 19 20 21 22 centimeters BASE © 2009 AIMS Education Foundation 11 AREA FORMULAS centimeters HEIGHT P ARALLELOGRAM UT C UPS How is ﬁnding the area of a parallelogram different from ﬁnding the area of a rectangle? 1. A find the perimeter by adding up the four sides. D B B A C Determine and record the perimeter of each parallelogram. C D Use the Measuring Pad to ﬁnd and record the lengths of the sides of each parallelogram. (Round to the nearest whole centimeter, if necessary.) measure the base... ...and the height. 2. Make the long side of each parallelogram the base. Measure and record the length of the base and the height. Make the short side of each parallelogram the base. Measure and record the length of the base and the height. cut and arrange the parallelograms into rectangles. 3. Cut a dotted line marking one of the heights of the parallelogram. Cut different lines on each pair of congruent parallelograms. Make a rectangle using the two pieces from each parallelogram. Determine and record the area of each parallelogram by ﬁnding the area of its two pieces making the rectangle. 1. 2. Short Long Perimeter Side (cm) Side (cm) (cm) Height (cm) Area (cm2) 3. Long Base Parallelogram A Short Base Long Base Parallelogram B Short Base Long Base Parallelogram C AREA FORMULAS Base (cm) Short Base 12 © 2009 AIMS Education Foundation P ARALLELOGRAM UT C 1. What two dimensions did you use to determine the perimeter? UPS these are some pressing questions. 2. How did you use those dimensions to ﬁnd the perimeter? 3. What two dimensions do you use to determine the area of the rectangle made from the two pieces of the parallelogram? 4. How are the dimensions used for area different than the ones used for perimeter? 5. Does it matter which side is the base? Explain. 6. Write a formula to describe how you use these two dimensions to get the area (A) of the parallelogram. AREA FORMULAS 13 © 2009 AIMS Education Foundation KEEP GOING We only needed to measure two sides because the sides that are across from each other are the same length. Yeah, we measured the sides. We used this pad thing to measure them. It was pretty far. The perimeter is how far it is around the parallelogram. After measuring, we added up the lengths of the four sides to find the perimeter. Okay, how many sides did you have to measure? After you cut out the parallelograms at the beginning of this activity, you measured some lengths. Let’s talk about that. 4. How do you find the height of a parallelogram? 3. What is the base of a parallelogram? ESSENTIAL MATH SERIES 0 1 2 3 4 5 6 7 8 12 C 12 + 16 + 12 + 16 = 56 Like for parallelogram C, it was 12 plus 16 plus 12 plus 16. that’s 56. 16 9 Measuring Pad I just multiplied the long side and the short side each by 2 and then added them! What else did you do? We call them opposite sides. So, we can say that opposite sides of a parallelogram have equal length. 10 11 12 13 14 15 16 17 18 19 20 21 22 centimeters BASE C That’s right, mark. For every parallelogram, the sides across from each other are not only parallel, but they are also equal in length. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 The measuring pad was like two rulers that were perpendicular to each other. It was easy to put one side of a parallelogram on one of the rulers and measure it. 5. What is the formula for finding the area of a parallelogram? 2. How do you find the perimeter of a parallelogram? 1. What do you know about the sides of a parallelogram? THINGS TO LOOK FOR: Parallelogram Cut-Ups centimeters HEIGHT 1 12 C Well done. Either way is fine for finding the perimeter. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 centimeters BASE C 0 1 2 3 4 5 I get it, the parallelogram is taller when it’s resting on the short side, and it’s shorter when it’s resting on the long side. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 6 7 8 9 Well, yeah, red, that’s pretty obvious. 10 11 12 13 14 15 16 17 18 19 20 21 22 centimeters BASE Like if parallelogram C is resting on the long side, then that’s the base and that’s 16. If you measure the height from that base it’s 9. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 using the pad, we rested one of the sides of the parallelogram on the bottom ruler and measured the height of the perpendicular dotted line. 16 2 • 12 + 2 • 16 = 56 The perimeter for parallelogram C was 2 times 12 plus 2 times 16. centimeters HEIGHT AREA OF PARALLELOGRAMS, TRIANGLES, AND TRAPEZOIDS centimeters 15 HEIGHT AREA FORMULAS C © 2009 AIMS Education Foundation So, when parallelogram C is resting on the short side, the height is 12. Now we know how tall it is! For each base there is a height that goes with it. Hold on a minute, vanessa, that is a very good observation. Yeah, Red, things change when you use a different side for the base. Wait, now the height changed. It’s not 12 anymore? Yeah, and whatever side the parallelogram is resting on, we call that the base. Well, we first had to measure the heights of the parallelograms. The next thing we did was to find the area of each of the parallelograms, right? 2 3 4 16 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 centimeters BASE 0 1 2 3 4 5 A 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 centimeters BASE So, class, how did we do it? How did we figure out the formula for finding the area of any parallelogram? But right now, we can use base times height to help us figure out a formula for the area of any parallelogram. 12 8 If 12 is the base of the rectangle, then the height is 8, right? And the area is base times height, that’s 12 times 8 or 96. 5 0 2 0 1 1 0 2 1 3 4 5 6 7 8 9 10 11 12 13 14 15 16 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 What we found out was that length and width on a rectangle are the same thing as base and height. So, a rectangle has a base and height just like all the rest of the parallelograms. And I wanted to show you how that helped us find a formula for finding the area of any parallelogram. Not really, redmond. You’ll usually think about the area of a rectangle as length times width. And the parallelogram and the rectangle have the same area because they’re both made out of the same pieces. We just cut up the parallelogram and put it back together to make a rectangle. It was fun! Does that mean we have to know two different formulas for the area of a rectangle? We found out that another way to think about the formula for area of a rectangle is that it’s base times height. That’s exactly right, Mark. Yeah, we already knew that the formula for finding the area of a rectangle is length times width. 3 16 16 9 That’s right, juana. You just have to be sure that you measure the height from that base. Is that the formula for the area of a parallelogram? Is it base times height for every parallelogram? The same thing happens when you cut up parallelogram B. 9 C 16 16 So, if base times height tells us the area of the rectangle, then base times height tells us the area of the parallelogram as well! The area of the rectangle is base times height, or 9 times 16, same as the parallelogram. 8 9 9 9 B 8 b But for a parallelogram the area is just base times height. h For every parallelogram, if you measure one of the sides, that’s called the base. And if you then measure the height from that base, then the area is base times height. 8 I think I’ve got it! The area of the rectangle is length times width or base times height! They mean the same thing, right? It is, redmond. B B You can turn the parallelogram into a rectangle and both shapes have the same base and height. They also have the same area. 9 9 Like for parallelogram C, if we cut it up into two pieces, we can move that triangle to the other side and it makes a rectangle. C It’s because we already knew the formula to find the area of a rectangle. C Class, there is an important reason why we included the rectangle along with the other two parallelograms. centimeters HEIGHT centimeters HEIGHT C AREA FORMULAS A A © 2009 AIMS Education Foundation That is an excellent summary, redmond. You’ve got it! And you can use the short side for the base or you can use the long side, right? So, if the area of the rectangle is base times height, then the area of the parallelogram is base times height as well. 4

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