THE ILLINOIS MATHEMATICS TEACHER
EDITORS Marilyn Hasty and Tammy Voepel Southern Illinois University Edwardsville Edwardsville, IL 62026
REVIEWERS
Kris Adams Chip Day Karen Meyer Jean Smith Edna Bazik Lesley Ebel Jackie Murawska Clare Staudacher Susan Beal Dianna Galante Carol Nenne Joe Stickles, Jr. William Beggs Linda Gilmore Jacqueline Palmquist Mary Thomas Carol Benson Linda Hankey James Pelech Bob Urbain Patty Bruzek Pat Herrington Randy Pippen Darlene Whitkanack Dane Camp Alan Holverson Sue Pippen Sue Younker Bill Carroll John Johnson Anne Marie Sherry Mike Carton Robin Levine-Wissing Aurelia Skiba
ICTM GOVERNING BOARD 2012
Don Porzio, President Rich Wyllie, Treasurer Fern Tribbey, Past President Natalie Jakucyn, Board Chair Lannette Jennings, Secretary Ann Hanson, Executive Director
Directors: Mary Modene, Catherine Moushon, Early Childhood (2009-2012) Com. College/Univ. (2010-2013) Polly Hill, Marshall Lassak, Early Childhood (2011-2014) Com. College/Univ. (2011-2014) Cathy Kaduk, George Reese, 5-8 (2010-2013) Director-At-Large (2009-2012) Anita Reid, Natalie Jakucyn, 5-8 (2011-2014) Director-At-Large (2010-2013) John Benson, Gwen Zimmermann, 9-12 (2009-2012) Director-At-Large (2011-2014) Jerrine Roderique, 9-12 (2010-2013)
The Illinois Mathematics Teacher is devoted to teachers of mathematics at ALL levels. The contents of The Illinois Mathematics Teacher may be quoted or reprinted with formal acknowledgement of the source and a letter to the editor that a quote has been used. The activity pages in this issue may be reprinted for classroom use without written permission from The Illinois Mathematics Teacher. (Note: Occasionally copyrighted material is used with permission. Such material is clearly marked and may not be reprinted.) THE ILLINOIS MATHEMATICS TEACHER
Volume 61, No. 1 Fall 2012
Table of Contents
Guidelines for Submitting Manuscripts...... 35
Using Literature to Get to Pi ...... 3 Deborah A. Crocker & Betty B. Long – General Interest
Using the SmartArtTM Organization Chart to Create a Graphic Organizer ...... 6 Dianna Galante – Technology
Mind Mapping Fractions, Decimals and Percents ...... 11 Geoffrey Lewis, Robert Hayes, & Marianne Wysocki – Elementary & Middle School
Playground Equipment ...... 15 Ann Hanson & Peter Insley – High School
Variations on the Witch of Agnesi ...... 20 John Boncek & Sig Harden – High School
Confetti – It’s Not Just for a Party Anymore! ...... 25 Rena Pate – Elementary
Problem Solving in the Primary Classroom ...... 28 Rena Pate – Elementary
101 102 uses for a paperclip ...... 32 Charlotte Schulze-Hewett & Hana Sallouh – High School
Quadrilaterals: Transformations for Developing Student Thinking ...... 36 Colleen Eddy, Kevin Hughes, Vincent Kieftenbeld, & Carole Hayata – High School
Illinois Mathematics Teacher – Fall 2012 ...... 1 From the Editors…
Welcome to the Fall 2012 issue of the Illinois Mathematics Teacher. It has been awhile, but we now have a great selection of articles to share with the members of ICTM.
This issue of the IMT contains general interest articles as well as articles specific to a variety of levels of student learning. Deborah A. Crocker and Betty B. Long have written an article about using literature and technology to have students discover pi. Dianna Galante also addresses technology in her article about graphic organizers. Fractions are the topic of focus in an article by Geoffrey Lewis, Robert Hayes, and Marianne Wysocki. Two articles that may be of interest to high school teachers involve the mathematics associated with a playground swing and the mathematics of two famous curves. Rena Pate provides two articles in this issue that will be of interest to elementary level math teachers which discuss a readily available manipulative and problem solving. Charlotte Schulze-Hewett and Hana Sallouh discuss a manipulative that is readily available and how to implement it in your middle school or high school classroom. Finally, there is an article about using transformations in the high school geometry classroom.
We want to thank all the readers of the IMT. We have enjoyed being the editors of this journal for the past eleven years. As Marilyn retires, we are passing the job of editor to two other members of ICTM. We wish them the best of luck and hope you continue to enjoy the IMT under their direction.
Typically included at the end of each issue is a form for becoming a reviewer for the IMT. This form has not been included in this issue due to page considerations. If you are interested in becoming a reviewer, please email [email protected] and list the subject(s) or grade level(s) you would be interested in reviewing. In order to ensure that the articles are of interest to our readers, we send them to reviewers to get their approval. We would love to have reviewers willing to review in only one area or multiple areas. As postage prices continue to rise, we are trying to conduct most of our communications by email, so please update your information and your email address if you have not reviewed in the past year.
Please consider submitting an article or classroom activity to the IMT. Consider writing about an activity that you use in your classroom and you would like to share with others. Your articles are needed to continue the sharing of ideas and the publishing of this journal on a regular basis.
Thank you for sharing.
Marilyn and Tammy editors
Illinois Mathematics Teacher – Fall 2012 ...... 2 Using Literature to Get to Pi
Deborah A. Crocker – Appalachian State University, [email protected] Betty B. Long – Appalachian State University, [email protected]
Literature is often used as a man. At this point, Radius goes to the springboard into mathematics at the carpenters, Geo of Metry and his brother elementary and middle school levels. Sym. While there, Radius has an idea about Stories can help motivate students to explore “across the middle” and “around the circle” and discover mathematical concepts. Even (p. 15). For the next several pages, in the if the reading level of the book is below story, Radius measures various circular grade level, the story can help students recall items. In table 1 is a list of the items that mathematical concepts and ideas. Sir Radius finds and their measures. Cumference and the Dragon of Pi: A Math Table 1 Adventure is an example of this type of Items measured by Radius in the story book. This activity incorporates literature in this manner and makes use of a handheld Across the Around the graphing device to discover pi using Item Middle Outside Edge multiple representations. Some prior (inches) (inches) knowledge of a handheld graphing device is Wheel 49 154 assumed. Onion Slice 3.5 11 As you read the story to your class, Basket 7 22 they will be introduced to the characters Sir Bowl 14 44 Cumference, his son Radius, and his wife Round of 17.5 55 Lady Di of Ameter. During lunch one day, Cheese Sir Cumference is accidentally turned into a dragon when Radius brings him a bottle Students should measure other circular labeled “Fire Belly” from the doctor’s objects such as tops of soda cans, hula- workroom to cure his stomachache. In the hoops, cookies, or plastic lids and add their doctor’s workroom, Radius looks for a cure measurements to table 1. The data can be to change his father back. He finds a entered on any handheld graphing device container with a poem written on the outside with the capability for scatterplots and linear that says: regression models. In this article, we will Measure the middle and circle show screens captured from the TI-73. The around, final table will be entered into the list editor Divide so a number can be found. on the TI-73. The “Across the Middle” Every circle, great and small – measurements are in L1 and the “Around the The number is the same for all. Outside Edge” measurements from the It’s also the dose, so be clever, literature book are in L2 (see fig. 1). Or a dragon he will stay . . . forever. (p. 13)
Radius realizes he must solve the riddle in the poem involving a circle in order to change his dad back from a dragon into a Fig. 1 Screen shot of L1 and L2
Illinois Mathematics Teacher – Fall 2012 ...... 3 The data in the list editor can be analyzed both graphically and numerically. Numerically, you can have the students create lists L3, L4, L5, and L6 containing L2/L1, L2 + L1, L2 - L1, and L2*L1, respectively. By analyzing the sum, Fig. 4 Screen shots showing how to set up a difference, product and quotient, students stat plot should observe that only the quotient appears to be approximately constant. (see The data in the scatter plot should appear fig. 2). linear as in figure 5.
Fig. 2 Lists showing ratio of “Around the Fig. 5 Scatter plot of data from the story Outside Edge” to “Across the Middle” Have the students estimate the slope Ask the students if they see anything that the of the line and the y-intercept using points numbers in L3 have in common. They on the stat plot and the mathematics they should notice that all of the numbers in the have studied. Then ask them to choose new list are a little more than three. A good manual fit from the stat menu. Students can representation for all of the numbers in L3 position the cursor at a starting place (see + might be the mean. Go back to the home sign on the second screen in figure 6), press screen by pressing 2ND MODE (for QUIT) ENTER and use the arrow keys to create a and press 2ND STAT MATH Mean( ). Be line segment they believe is a good model sure to indicate, by typing, that you want the for the data. mean of L3 and press ENTER (see fig. 3).
Fig. 6 Screen shots of Manual-Fit Options Fig. 3 Screen shots of commands for finding the mean When they have the segment they want, they press ENTER again to see the equation of The students should notice that the mean of their line at the top of the screen (see fig. 7). the numbers in L3 is approximately 3.14. To analyze the data graphically, set up a stat plot (see fig. 4).
Fig. 7 Manual-Fit line with equation
Illinois Mathematics Teacher – Fall 2012 ...... 4 Students should start with the segment they line of best fit generated by the handheld want and adjust the slope by pressing the up graphing device. or down arrow keys on the handheld graphing device and adjust the y-intercept by pressing the left or right arrow keys on the handheld graphing device, as needed, to find a model for the linear data (see fig. 8). The model will be displayed at the top of the screen and can be stored as Y1. Fig. 10 Graph of the line of best fit and the model found using Manual-Fit
After this numerical and graphical analysis of the data from across and around several circles, the students should draw the conclusion that the dosage needed to change Radius’ father back into a man is approximately 3.14 spoonfuls. This is the approximate ratio of “around” to “across” in any circle. In the story, Radius saved his father by computing the correct dosage of Fig. 8 Screen shots of adjustments to the the potion to give him. That dosage turned slope and y-intercept out to be pi spoonfuls. We have approximated pi numerically and graphically What do you notice about the slope? It and understand that it is a ratio that holds should be close to the numerical average true for any size circle. To celebrate, Sir you found in the list L3 earlier. Let’s Cumference, Radius, and Lady Di of Ameter compare the students’ model to the line of joined all the people of the kingdom for a best fit generated by the handheld graphing piece of pie. device. Press STAT CALC LinReg (see fig. 9). NOTE: The teacher could give all of the students a cookie or something round as a treat after this activity.
References
Neuschwander, Cindy. Sir Cumference and the Dragon of Pi: A Math Adventure. New York: Scholastic Inc., 2000.
Fig. 9 Screen shots for getting the line of best fit from the handheld graphing device
Like the graph shown in figure 10, students can look at a graph of their model and the
Illinois Mathematics Teacher – Fall 2012 ...... 5
Using the SmartArtTM Organization Chart to Create a Graphic Organizer
Dianna Galante – Governors State University, [email protected]
One way to increase understanding Create and use representations to and develop problem solving skills in your organize, record, and communicate mathematics class is to develop lessons that mathematical ideas appeal to the visual learner. You can Select, apply, and translate among accomplish this by inserting graphic mathematical representations to solve organizers when preparing worksheets and problems cooperative learning activities. A graphic Use representation to model and organizer can help your students tackle interpret physical, social and difficult concepts by providing a visual tool mathematical phenomena (p. 360) to help arrange key ideas and definitions, organize categories, or analyze One limitation with adding a graphic mathematical processes. By helping organizer to your lesson plan is the amount students present what they know visually, of time and effort needed to create a high- you can increase student interest and quality document. One easy option available motivation while promoting conceptual with Microsoft WordTM is the use of the understanding in your classroom. SmartArt graphic tool on the Insert tab. You You can use an organizer to aid can quickly create a graphic organizer and students with understanding the sequence in drop it into your document like a piece of a process or to group related information. clipart, then easily copy and paste it, move Using graphic organizers can also help it, or resize it. One graphics template useful students with developing reading strategies for mathematics is the Organization Chart in mathematics by supporting their ability to which can provide a picture of a hierarchical break apart difficult text, to group related relationship or create a decision tree. information, or by providing a structure for Another useful template is the Basic Venn what they already know. diagram which can be used to visually display the union and intersection of sets. Supporting Problem Solving Getting Started This ability to create and use a visual representation of a problem supports and To get started, open a new document enhances students’ problem-solving in Microsoft Word and then select the Insert strategies. According to the Principles and tab and SmartArt. Click on the SmartArt Standards for School Mathematics (PSSM) button (Figure 1) near the center of the [National Council of Teachers of toolbar. Your toolbar may appear differently Mathematics (NCTM), 2000], “different depending on the software version you are representations support different ways of currently using. The Diagram Gallery dialog thinking about and manipulating box with a pallet of diagrams will appear mathematical objects” (p. 361). Using a (Figure 2). There are six diagram types variety of representations in instruction available including List, Process, Cycle, should enable students to: Hierarchy, Relationship, Matrix and
Illinois Mathematics Teacher – Fall 2012 ...... 6
Pyramid. Click on the diagram type you layout of the diagram depending on the want to select and then click OK. As an template you selected. Finally, resize or example, select Hierarchy, then move the graphic to fit your document. You Organization. can experiment with the many options and The diagram template will appear with then use the UNDO button to backtrack to guides (Figure 3). At this point you can type the best choice. text, add additional graphics, or change the
Figure 1 SmartArt Icon
Figure 2 Diagram Types
Figure 3 Organization Chart
Illinois Mathematics Teacher – Fall 2012 ...... 7
Classroom Uses
Example 1: In a geometry class, you can students first put each equation in slope- use the Organization Chart to help students intercept form. Next, ask students to identify describe the properties of quadrilaterals. the slope value and place the equation in the Individually or in groups, ask students to appropriate slot on the chart (Figure 5). describe the properties of each type of Put each equation in slope-intercept form quadrilateral by supplying information about then complete the chart: sides, angles and diagonals in the chart (Figure 4). 1. 5. 2. 6. Example 2: In an algebra class, the 3. 7. Organization Chart can chart help students 4. 8. focus on an important property of the slope of a line. From a list of equations, have
Quadrilateral Parallelogram Rectangle Square Rhombus Trapezoid
sides
angles
diagonals
Figure 4 Properties of Quadrilaterals
y = mx + b
m > 0 m < 0 m = 0 undefined
y = 3x+2 y = 1/2x y =‐2x+5 y = ‐x +3 y = 4 y = ‐2 x = 3 x = ‐7
Figure 5 Solution for Example 2
Illinois Mathematics Teacher – Fall 2012 ...... 8
Create a Decision Tree D .01 Example 3: You can use the Organization .50 Chart in a probability and statistics class to A Dc .99 support a problem-solving strategy that uses a decision tree to help clarify a difficult Find the D .02 topic. The accompanying tree diagram B 30 (Figure 7) represents a two-stage experiment probability . .98 where students use the “branches” of the Dc tree to help calculate probabilities. C D .02 Vector Electronics – The circuit boards for .20 new laptop computers are manufactured at Dc .98 three different locations and then shipped to Figure 7 Decision Tree with Probabilities the main plant of Vector Electronics for
final assembly. Plants A, B, and C supply
50%, 30% and 20%, respectively, of the Example 4: You can use a graphic organizer circuit boards used by Vector Electronics. to help students select a mathematical Production has been carefully monitored by representation that best suits their learning the quality control department. It has been style or to help translate between different determined that 1% of the circuit boards representations. The Basic Venn graphic produced by Plant A are defective, whereas organizer can be found by clicking on the 2% of the circuit boards for Plants B and C SmartArt icon then selecting the are defective. If a Vector Electronics Relationship group then Venn. computer is selected at random, and the circuit board is found to be defective, what Senior Class – At Midwest High School is the probability that the circuit board was there are 150 students in the senior class. A manufactured in Plant C? Use the tree total of 30 students are enrolled in Physics diagram to help with the calculation of II, 35 students are enrolled in A. P. Calculus, . and 100 students are enrolled in English IV.
There are 15 students in physics and Solution: Let A, B, and C denote the events calculus, 15 in physics and English, and 20 that the laptop computer selected has a in calculus and English. Five seniors signed circuit board from Plant A, B, or C, up for all three classes. How many seniors respectively. Let D indicate the event that are enrolled in the three classes? How many the laptop has a defective circuit board. The seniors are not taking any of the three complement, DC indicates no defect. classes? Construct a tree diagram (Figure 7) to represent the situation and supply values for Solution 1: Using a symbolic representation, the equation below. students would use a counting formula for
three sets where number of students taking physics number of students taking calculus number of students taking English Then
Illinois Mathematics Teacher – Fall 2012 ...... 9
5 10 10 5 15 5 70 120 the number in the three classes 30 35 100 15 15 20 5 150 120 30 120 the number in the three classes the number of students not in any of the classes 150 120 30 the number of students not in any of the classes Helpful Information and Available Support
Solution 2: Using a graphic organizer as a The Microsoft website offers a multitude of visual aid, students label each section of the support options for use with SmartArt Basic Venn diagram (Figure 8) for the three including additional graphics, tips, sample overlapping sets. ideas, and online tutorials. To visit the website, go to http://office.microsoft.com and select the Help and How-to tab. Physics 30 5 References
10 10 Microsoft Office Online (2009). Retrieved 5 July 17, 2012 from www.office. 5 15 70 microsoft.com. National Council of Teachers of
Calculus English Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.
Figure 8 Basic Venn Diagram with Labeled Regions
Plan to attend the NCTM Regional Conference at the Hyatt Regency Hotel in Chicago on November 28-30, 2012.
Illinois Mathematics Teacher – Fall 2012 ...... 10
Mind Mapping Fractions, Decimals and Percents
Geoffrey Lewis – Sunnybrook School District 171, [email protected] Robert Hayes – Sunnybrook School District 171, [email protected] Marianne Wysocki – Sunnybrook School District 171, [email protected]
Nathan Hale Elementary in Sunnybrook School District 171 was recognized by the state for school improvement on the ISAT test in part due to a 21% increase in fourth grade math scores.
One of the most difficult topics for exception is a series of worksheet after middle school students is fractions. worksheet that seem to provide no increase Students spend long hours on worksheets in standardized test scores and high school that have them practicing the rules for students that continually answer . addition, subtraction, multiplication, and When you ask any high school teacher if division of fractions. Yet when time comes their students can do rational number for testing teachers are at a loss to explain operations in algebra or geometry the why their students perform poorly. The answer is almost always no. What happened same can be said to a lesser extent for the to all the time spent on worksheets? Was student scores on decimals and percent. there any learning? Are we just teaching our The National Council of Teachers of students what key strokes to perform on a Mathematics addresses the teaching of these calculator and hoping they can even topics in the Standards for Grades 6-8 remember this? Read the assigned chapter, (Principles and Standards for school answer the questions at the end of chapter, Mathematics 2000). The specific goals are: sit and take lecture notes, complete the
worksheet, and take a test are all listed as That students be able to work flexibly this is not teaching in How to be an Effective with fractions, decimals and percents. Teacher, (by Harry Wong, pp. 218-219, Select appropriate methods and tools for 1998). Bloom’s taxonomy arranges the computing with fractions, decimals and verbs into six related groups: percents. 1. Knowledge, Understand the meaning and effects of 2. Comprehension, arithmetic operations with fractions, 3. Application, decimals and integers. 4. Analysis, Select appropriate methods and tools for 5. Synthesis, computing with fractions and decimals 6. Evaluation. from among mental computation, Most worksheets involve answering only estimation, calculators or computers, and level 1 type questions. NCTM standards paper and pencil. start at level 2 type questions. No wonder Develop and analyze algorithms for children have trouble remembering or computing with fractions and decimals. learning fractions. NCTM standards talk Develop and use strategies to estimate about communication and connections. the results of computation and judge the Each is very difficult to ask, evaluate, or reasonableness of the results. teach with traditional worksheets. The solution for more effective Is this what is happening in our middle lessons involves both the NCTM standards schools? No! What we see almost without and being an effective teacher. Elementary
Illinois Mathematics Teacher – Fall 2012 ...... 11
teachers have long known that they must method recommended and used by the establish connections for their students to NCTM in the standards. understand fractions. Often these It is not necessary to use large or connections involve making a drawing or complicated fractions. Only the most diagram of the concept or problem. The common should be used, halves, thirds, effective teacher is the one who has the fourths, fifths, eighths, and tenths. These students doing the work. The one doing the provide more than an adequate background work is the one doing the learning. Often for a fundamental understanding of fractions times middle school teachers end up with a decimals and percent. If the students have a high degree of frustration and are worn out firm understanding of the basics of fractions because they are the ones doing the work the rest is easy. and their students are not doing the learning. A few examples are shown, put in Middle school teachers must spend order so that students can also see the more time having their students working on equivalent values: developing concepts and less time on abstract practice. This can be accomplished by mind mapping fractions, decimals and percents as an interrelated activity with a connection to the kinesthetic activity of making the related picture or model. Students must see the marriage of picture, fraction, decimal, and percent. Once the basics are well understood it is easy to progress to more difficult problems. The mind map consists of 5 separate but related parts. A rectangle to show fractions, a dollar sign to show the value in money, a decimal point to show the decimal value, a triangle to show the percent and a 10 by 10 grid to show the appropriate picture.
The first part: rectangle, $, decimal, %, grid are easy for the students to remember. This forms the first part of the connection. The second part of the connection is the pictorial representation of the fraction, decimal and percent. This connected to money, something that all students see and think about each day provide a strong memory basis for the students. The 10 by 10 grid is the picture
Illinois Mathematics Teacher – Fall 2012 ...... 12
Use all the fractions to help with the symbols. Fractions are shown with a connections. You need to use all the rectangle so that the fraction bar fits parallel halves, all the thirds, all the fourths, all to the top and bottom of the rectangle. The the fifths, all the eighths, and all the decimal point follows the decimal with the tenths. Only a sample is shown above. It dollar sign. A triangle is used to show per is necessary to show all of these to show cent because the side of the percent sign the connections. resembles the side of a fraction. Squares to It is now easy for students to find show pictures are drawn in a 10 10 equivalent fractions. They are the ones with format to match money, percent and NCTM the pictures that have shaded the same. The standards. (NCTM, p. 215) The visual comparison of order of fractions decimal connections must be the first established for and percent is also as simple as looking at the students to make the connections the picture. Basic fractions are easy to between fractions, decimals and percent. reduce simply by using the illustration. Other connections are shown in money and Decimal and percent equivalents to fractions picture scale. Yes a few worksheets are can be very visual. Performing addition necessary for practice and review. Yes a and subtraction with and without a common few larger problems are necessary, but this denominator is as easy to do as counting is a question of balance. Now the scale has squares. More importantly pictures aid slipped to more and more problems with less students in making estimates. This is and less understanding of the marriage of especially important if the students are using the basic concepts that bring fractions, a calculator. Is this the right answer? Or decimals and percent together. Calculators have I made a mistake with the decimal or can perform the most difficult computation denominator. The use of money, with the problems, but students must be able to picture also reinforces the concept of interpret the correctness of their results and estimation. Estimation is one of the key they must also be able to make a connection goals of the NCTM. Multiplication and between the use of the calculator and the division of fractions and decimals makes basics problems. Fractions, decimals and sense to students in picture and money form. percent should not be taught as individual No more ½ of $.50 is 1.00! Large posters of activities, but must be taught as different the basics should be placed around the room representations of the same processes for for easy and constant reference. Once the students to have complete understanding. basic fractions and operations have been When students see these connections, mastered most students can easily make the they can choose the method of problem transformation to more complicated solving that makes most sense to them as problems. Not that the problems are any recommended by the NCTM standards. more complicated only that the numbers are Finding 50% of an unknown number is the larger. Is this important for your students to same as finding of it or in decimal form know? Do we really add sevenths and multiplying by .50. The picture provides the eights or multiply 2 4 . Does this really final check of estimation. Have they shaded have a real life connection? Is it tested on in the appropriate amount of the given? State exams or the ACT test? Use your time Likewise, why does dividing by make the to teach what they will need in their real life. number larger? Try dividing by .50 and see Note the physical mind mapping what happens? This method also appeals to connections: There is the brain friendly a variety of learning styles, most importantly connection involving the pictures and
Illinois Mathematics Teacher – Fall 2012 ...... 13
the visual and kinesthetic learners’ needs are understanding that is fun and involved for met. today’s learners. Working math now will This all fits with current research that become a fun activity and not a bore. These says explaining basic concepts behind math sites must be carefully chosen to be of problems improves children’s learning. interest to the students and sound “New research from Vanderbilt University mathematical content. has found students benefit more from being Look how these activities fit together taught the concepts behind math problems with the NCTM standards. Students are rather than the exact procedures to solve the working flexibly, they understand the problems. The finding offers teachers new meaning, they can select from the insights on how best to shape math appropriate method or tool, they can analyze instruction to have the greatest impact on the different algorithms, they can develop student learning. The research by Bethany strategies, and estimate the reasonableness Rittle-Johnson, assistant professor of of the results. Always refer back to the psychology and human development at posters. Students will need to see the Vanderbilt University’s Peabody College connections many times before it all makes and Percival Mathews, a Peabody doctoral sense to them. candidate, is in press at the Journal of New state and national testing Experimental Child Psychology. Teaching standards relate to Bloom’s taxonomy. children the basic concepts behind the math Level one type questions, knowledge, are problems was more useful than teaching rarely asked. Students can simply crunch it children a procedure for solving the out on a calculator. Comprehension, problems-these children gave better applications, analysis, synthesis, and explanation and learned more. …This adds evaluation are the types of questions asked. to the growing body of research illustrating This requires a different approach to the the importance of teaching children concepts teaching of mathematics than the worksheets as well as having them practice solving of the past. Students must evaluate, problems.” (Science Daily, April 2009). The develop, compare and contrast different shapes, the relationships and the pictures all methods of doing the problems in order to add to the student’s understanding of the fully understand the relationships of concepts of fractions, decimals and percents. fractions, decimals and percents. The internet can also provide a valuable Years of teaching experience and tool for both the practice and the research now offer all teachers methods of understanding of fractions, decimals and providing effective instruction. State testing percents. There are many web sites that can evaluates how successful we are at teaching provide a wide variety of interactive standards using research based methods. activities and practice that would not be This article provides a start in the directing possible for the classroom teacher to of successful research based instruction. duplicate. This provides the help in
Illinois Mathematics Teacher – Fall 2012 ...... 14 Playground Equipment
Ann Hanson – Columbia College Chicago, [email protected] Peter Insley – Columbia College Chicago, [email protected]
Abstract similar triangles, trigonometry, Newton’s Second Law and much more. In this paper, the authors analyze the The diagrams and equations below math and science in a common piece of are idealized in order to simplify the playground equipment, a swing. The mathematics. In a real experiment using a authors derive the equations needed to swing, there would be several different describe the motion of a swing. Next, they possibilities, which would greatly perform an actual experiment with a real complicate the mathematics and/or the swing and then give teachers some measurements. Friction introduces a force suggestions on how to use this activity in other than gravity and if we do not assume a their classroom. straight line, then we cannot use similar triangles in our explanations. Here are some of the assumptions being made: 1) there is no air friction, 2) the swing swings in a straight line not in an arc, 3) the weight of the chain is ignored and 4) the length of the swing is the distance from the pivot point to the center of mass. In the diagram below, l is the length of the chain, P is the point where it is attached, E is the equilibrium position and A is the maximum position. Pull the swing out to A. The swing is the projection on the X- axis of a point moving on a circle.
In every community, you can find a Figure 1 children’s playground and Forest Park, IL is no exception. This playground is on the corner of Randolph and Circle in Forest Park and contains a lot of different equipment. One might think that this playground equipment for children is very simple but actually there is a lot of mathematics and physics used in the development of this equipment. The playground has several different pieces of equipment, but the focus for this paper is on the swing set. The mathematics and physics involved in the swings uses
Illinois Mathematics Teacher – Fall 2012 ...... 15 Then the sum of these forces equals the total force ( ) on the swinger.
mg
Where
is the period or the time for one Assuming the swing is not pulled back very vibration (or rotation) far, the force triangle is similar to the
triangle formed by the swing where ( ) is is the displacement from the the distance from the swinger to the pivot equilibrium position (E) and ( ) is the distance the swing is pulled
back. is the time from when the swing is
released
is the amplitude or how far the swing
is pulled back before it is
released
should be 0.1 of or smaller for the best is the speed of the swing at time results. Then or , the swing is the highest speed of the swing force equation. (at 0) When the swing is pulled back, the weight of the swing (mg) and the tension is the acceleration of the swing in the chain ( ) together produce the force ( ) that restores the swing to its equilibrium is the highest acceleration of the position ( 0) - assuming there is no swing (at ) friction and the swing moves in a straight line not an arc.
The force equation for the swing can be Figure 2 found from similar triangles.
mg
First draw the forces acting on the swinger. The triangles are similar and therefore: They are gravity (mg) pulling down and the rope () pulling up along the rope.
Illinois Mathematics Teacher – Fall 2012 ...... 16 Figure 3 we get
4 2 2 0
4 F mg x l 2 Where x is the displacement from the equilibrium position (x = 0). The maximum position ( ) is how far the swing is pulled which is the time for one swing. back and l is the length of the swing from 2 the equilibrium position to the center of mass. The figures are force diagrams of the forces on the swing. If is positive, then 2 2 is negative as are and . is the amplitude or maximum value of . It “comes into play” in the general equations for position, velocity and acceleration. 4 2 Newton’s Second Law is , so
. 0 0 g A (Note: acceleration is defined as ) l 0.25 0 0 0, then assuming a g A trigonometric solution, we get l 0.50 0 g A 2 l 0.75 0 g 0 A 2 2 l 0 g A l
4 2 Note to the teacher. The table serves a summary of all of the possibilities. Please adapt this activity to suit your class. For and by substitution in your class, you may want to use only the time equation and use a pendulum or swing to check it. Or, perhaps you want your class 0 to work with variables, such as, changing
Illinois Mathematics Teacher – Fall 2012 ...... 17 the length of the swing, weight, etc. You We found our percent of error to be 14%, could also take your class to the playground which was far too much. This calculation and have the students take turns swinging on was made by assuming that 2.43 was the . . the swing and timing the person on the correct answer. The % of error is swing and recording this information. If it is . 0.1358 or 13.58% or 14%. not possible to do this activity during class, We decided that the error lay in the it could be given as a homework assignment length of the swing. Also, when we derived or students could hang weights from their the equation we placed the weight vector desks. When this is done the student “mg” (see fig. 2) at the bottom of the swing changes the length of the string and observes where we assumed the weight was the change in period ( ). In this paper, the concentrated. However looking at the actual length of the chain is constant. swing, it is clear that most of the weight is in The next section of this paper, the chains and not in the seat. So the force describes what happened when we went to vector should be placed further up the chain. the actual swing set and tried out our If we substitute 2.43 for in the theories. equation, we find the effective to be When the equation for the period ( ) 1.47m, quite different from our measured of the swing was derived, several distance 1.89m. assumptions were made. Going to a In the next trial, we sat on the swing playground and measuring and can test and our weight of 170 pounds became the these assumptions. The equation was most significant weight in the system. This 2 time we measured from the pivot point at the top to where we estimated the center of where is the period or the time for one mass of the person on the swing was. swing (back and forth), l is the length of the swing from the pivot point at the top to the Trial 2 – with a person on the swing seat at the bottom, and is the gravity 2 constant 9.8 m/s . 1. is about 1.6m
Procedure: . 2. 2 and 2.54 sec 1. Measure . 2. Predict using the equation 3. Measure by timing 10 complete 3. The time for ten swings is 2.66sec swings and dividing by 10 We next found the % of error . . Trial 1 – with no person on the swing 4.7% which is more reasonable. . 1. is about 1.89m We recalculated this time and got 1.75m and we guessed that we were off by 15cm in estimating the center of mass of the . 2. 2 and, therefore rider. . Another assumption that was made 2.76sec. in deriving the equation for was that the
swing moves in a straight line ( in fig. 2). 3. Using a stop watch time ten swings We know the swing actually curves but we of the swing, which is 2.43sec. thought that if we kept the amplitude
Illinois Mathematics Teacher – Fall 2012 ...... 18 (amount of swing) small, we could assume This activity could be used in any that the difference between the arc of the grade from 3rd to calculus. Some teachers circle the swing actually makes and the may need to talk with their science length of the chord of that arc would be colleagues for some extra guidance. small. If we use amplitude of 0.3m and the Regardless, even at the earliest grades, the swing length was 1.89m, then the angle is students could measure “ ” and calculate . 0.159 or 9 degrees. . from the equation 2 or We tested this assumption by 4 . Then they could measure (and find swinging as high as possible. ) and compare the answers. Eighth Trial 3 – with a person on the swing and graders could calculate the percent of high swing uncertainties and discuss the effects of air friction and the straight-line assumption. 1. is 1.7m The students could calculate x, v, and a from 2. 2.62sec the motion results. And calculus students 3. Time ten swings at 2.83sec, which could integrate the acceleration equation to result in an 8% error. find the velocity at various points. This activity can generate a lot of discussion and This time we estimated our amplitude to be questions and definitely show how math and 1.2m so our angle is 1.2/1.89 = 0.635 physics is applied even to children’s radians or 36 degrees. Our conclusion was playground equipment. This should settle that keeping the angle under 10 or 15 the questions of “when am I (or anyone) degrees would give good results but big ever going to use math?” swings could throw the answer off.
Check out the ICTM website at ictm.org. The
latest issue of the ICTM Bulletin is available here as well as information about the officers and directors. You may want to share the scholarship information with your students. Consider nominating a colleague for one of the ICTM awards. All of this and more is on the website – check it out.
Illinois Mathematics Teacher – Fall 2012 ...... 19
Variations on the Witch of Agnesi
John Boncek – Troy University-Montgomery Campus, [email protected] Sig Harden – Troy University-Montgomery Campus, [email protected]
Introduction
At some point in their mathematical If we use , the angle that the line training, most teachers encounter the Witch segment OA makes with the x-axis, as a of Agnesi. The Witch, a plane curve named parameter, the equation of the line OA is and studied in the mid-1700s by the Italian seen to be mathematician Maria Agnesi, is constructed x tan , as follows: 2 y 2,a . 1. Center a circle with radius a at the point 2 0,a . We get a piecewise representation
since the tangent of a right angle is 2. Choose any point A on the line ya 2 undefined. The equation of the circle on and connect it to the origin O with a line which the construction is based is also easily segment. determined, since we know both its center 222 3. Label the point where the line segment 0,a and radius: x ya a. OA intersects the circle as B. To find the coordinates of any point P that lies on the Witch: 4. Let P be the point whose x-coordinate is the same as point A and whose y- 1. We find the x-coordinate of the point A, coordinate is the same as point B. the point where the line OA intersects the line ya 2 . Using the The Witch is the collection of all parameterization we constructed above, points P obtained by moving the point A if , we have x tan 2a , or along the line ya 2 . (Figure 1) 2 xa 2cot . If the line OA is 2 precisely the y-axis, and so x 0 .
2. We then find the y-coordinate of the point B, the point where OA intersects the circle. If , we determine 2 where the line yx tan intersects the 2 circle x22 ya a:
Figure 1. The Witch of Agnesi
Illinois Mathematics Teacher – Fall 2012 ...... 20
222 2a x ya a y 2 222 x yyaacot 1 2a yay221cot 2 0 8a3 y yay22csc 2 0 x22 4a
2 yy csc 2 a 0 The function described by this equation is continuous, with a graph that is Since our geometric understanding of the symmetric about the y-axis and asymptotic problem tells us that y 0 , we must have to the x-axis. Some simple calculus reveals the surprising fact that while the Witch lies entirely above the x-axis, the area of the 2 yacsc 2 0 region bounded by the Witch and the x-axis 2a is a finite value: y csc2 2 8adx3 dx ya 2sin 2a . 22 2 ya1cos2 xa 4 x 1 2a using the double angle identity x 2sin2 1 cos2 . Note that if , Make the change of variables u : 2 2a no computation is necessary. We have ya 2 . dx2 du This allows us to describe the Witch 24aa 221 u parametrically using the following x 1 equations: 2a 4tanau21 4 a 2 . xa 2cot ya1cos2 For those teachers whose students where 0 . Now that we have the enjoy experimenting with this type of curve described parametrically, we can construction, a follow-on exercise can easily eliminate the parameter to describe the be created by replacing the circle used to Witch explicitly as a function of the variable draw the Witch with another geometric x. shape, such as a square or a triangle. Two resulting curves, which our students have ya1cos2 named “Maria’s Bridge” and “Agnesi’s Tent,” while perhaps not as elegant as the 2a y original Witch, are interesting to construct csc2 and study in their own right. 2a y 1cot 2
Illinois Mathematics Teacher – Fall 2012 ...... 21
Maria’s Bridge for tan 2 tan 2, and
To construct Maria’s Bridge, we 2 cot replace the circle used to construct the tan Witch of Agnesi with a square. for tan . Eliminating the 1. Position a square with side length 2a so parameter , we get the following piecewise that the midpoint of its lower edge is at mathematical description of the Bridge: the origin O. 2. Choose any point A on the line ya 2 2 , and connect it to the origin O with a line segment. 2 , 3. Label the point where OA intersects the 2 , square as B. 4. Let P be the point whose x-coordinate is the same as point A and whose y- Among the interesting properties of coordinate is the same as point B. Maria’s Bridge:
Maria’s Bridge is the collection of all It is a continuous curve. points P obtained by moving the point A The function that defines the curve is along the line ya 2 . (Figure 2) differentiable everywhere except at . Its graph is symmetric with respect to the -axis. Its graph is asymptotic to the -axis. Its graph has no points of inflection. The region bounded by the Bridge and the -axis has infinite area. Given that the area bounded by the Witch and the -axis is finite, this result is perhaps surprising. Some elementary calculus establishes the result.
Figure 2. Maria’s Bridge 2 2 2 . Using , the angle OA makes with the horizontal, as a parameter, we have The last integral in this expression 2 cot diverges to infinity, since tan lim → for 0 tan 2, lim 2 ln ∞. → 2 cot 2
Illinois Mathematics Teacher – Fall 2012 ...... 22
Agnesi’s Tent 2 tan 2 tan Agnesi’s Tent is constructed by replacing the circle in the Witch of Agnesi for . When using this with a select isosceles triangle. parameterization, we will also agree that the point 0, 2 lies on the curve. Note that 1. Let ,0 , ,0 , and be the three vertices of an 0, 2 2 tan 2 sec isosceles triangle. Draw its sides. lim lim 2 2 tan sec 2. Choose any point on the line 2 → → and connect it with a line segment to the origin . and 3. Label the point where the line segment intersects the triangle as . 2 tan 2 sec lim lim 2 , 4. Let be the point whose -coordinate is → 2 tan → sec the same as point and whose -
coordinate is the same as point . so from the perspective of continuous
functions, the point is precisely where we Agnesi’s Tent is the collection of all want it to be. points obtained by moving the point When we eliminate the parameter , along the line 2 . (Figure 3) we get the following description of the Tent:
2 , 0 2 ,0
Agnesi’s Tent has the following properties:
It is a continuous curve. Figure 3. Agnesi’s Tent The function that defines the curve is differentiable everywhere except at If we again use , the angle 0. makes with the horizontal as a parameter, Its graph is symmetric with respect to we can describe the tent as the - axis. Its graph is asymptotic with respect to 2 cot the -axis. 2 tan Its graph has no points of inflection. 2 tan The area of the region bounded by the
Tent and the -axis is infinite. In fact, for 0 , and the area of the region bounded just by the portion of the Tent that lies in the 2 cot right half plane and the - axis is itself infinite.
Illinois Mathematics Teacher – Fall 2012 ...... 23
naturally when solving geometric problems. 2 lim2 ln ∞. → 2 3. We have presented only two curves that can be obtained from the Witch of Some remarks Agnesi. What happens if you replace the circle with a rectangle? Change the 1. Note that this exercise can be adapted triangle from isosceles to equilateral? If for use in a wide variety of classes, you use an inverted parabola? Let your ranging from the most elementary students experiment and see what they courses through calculus classes. discover.
2. Many of our beginning students struggle with the idea of piecewise defined References functions. These constructions illustrate that piecewise functions arise quite G. B. Thomas and R. L. Finney. Calculus and Analytic Geometry, Sixth Edition. Addison-Wesley, 1984.
Marilyn Hasty and Tammy Voepel have asked us, the incoming editors, to introduce ourselves and mention some important upcoming changes to the journal.
Daniel Jordan: I earned my PhD in mathematics from Indiana University, Bloomington, and am now an associate professor at Columbia College Chicago. I have published and presented on both mathematics research and on educating students, including my recent book, Exploring Discrete Mathematics with Maple.
Christopher Shaw: I joined Columbia College as an assistant professor in 2010, after receiving my PhD and subsequently teaching at the University of Maryland. My pure mathematical research is in logic, and I recently created a general education mathematics course at Columbia for students in the liberal arts.
We have been charged by the ICTM leadership to transition the journal to an electronic medium. That transition will be our focus as we begin our term as editors. Articles will be posted to the members-only section of the ICTM website as soon as they have completed the review process and been approved. Complete issues of the journal will likewise be distributed electronically.
Please take a look at the IMT’s updated website at http://www.ictm.org/imt.html and take note of the changes to the submission guidelines. Feel free to send questions to us at the journal’s new email address, [email protected].
Illinois Mathematics Teacher – Fall 2012 ...... 24
Confetti – It’s Not Just for a Party Anymore!
Rena Pate – Danville District 118, [email protected]
Many of our students think that math to tell what strategy they were using. You is some kind of a secret, that there is only might get the answer 5+3 when adding the one way to get the answer and you have to hats on the top row with hats on the bottom know the “trick” to get that one possible row. You might get 6+2 which represents answer. Unfortunately many math teachers the dotted hats plus the wavy hats. encourage these thoughts by providing a Hopefully you will also get the commutative limited number of kinds of problems and property problems for both. Children will possible solutions. Students aren’t want to know which is the “correct”answer challenged or encouraged to be creative and so you will have to remind them that since think outside the box to use any possible you didn’t establish strict parameters that strategy to solve a problem. We have to they are all correct. break that cycle by giving our students Now challenge students to come up problems that challenge their problem with other possible answers. You might get solving skills. 4+4= hats upside down + hats right side up. A simple overhead projector and a You might even get 8+0 =the hats plus the handful of confetti (you can find numerous other objects (there aren’t any other kinds at the dollar store) provide an endless objects)! Once students begin to realize that amount of mathematic possibilities. Have there are several possible solutions they will you ever noticed the “magic” that an challenge each other to come up with as overhead provides? How a tiny confetti many plausible answers as possible. bunny commands attention when enlarged The next example uses overhead by an overhead? Well, wait until you see coins. You might get the following answers: the special magic unique shaped confetti infuses into your classroom! Take the following party hat confetti for example:
3+4=7 Which could be any of: coins on the top row + coins on the bottom row brown coins + silver coins pennies + nickels heads coins + tails coins Ask students to provide you a 23 cents possible math problem that is represented by $.03 + $.20=$0.23 the confetti and remind them they will have
Illinois Mathematics Teacher – Fall 2012 ...... 25
Seasonal Confetti What is there an odd/even number of? Seasonal confetti or foamy shapes How many things might you eat? are also a lot of fun. Once your students are How many heads do you see? Arms? pretty good at identifying story problems Hats? you can switch and make your own story How many more snowmen than candy? problems from shapes. In the following Gingerbread than bows? example you might ask: Categorizing/Classifying Children get a lot of practice identifying attributes through working with pattern blocks and other shapes but it is important to encourage them to think about other ways to sort/order things. Every once in awhile it is fun to put random confetti and see what kinds of things kids will come up with. In the following example they might offer:
What is there the most of? Least? How many more sleighs are there than bells? (a trick) Bells than gingerbread? How many stars do you see? What is an easy way to count the stars? How many of these things could you eat? How many gingerbread eyes are there? If 2 gingerbread could fit in each sleigh, how many total gingerbread could the
sleighs hold? There are 5 different kinds of flowers. If each gingerbread was to get 3 bells, The star and moon can both be found in how many more bells would you need? the sky.
The rabbit and butterfly are both animals There are two flowers that are congruent. There are 15 things all together. All shapes but 2 have a line of symmetry. 7 shapes have a triangle somewhere within. Encourage and accept all reasonable explanations.
How many items all together? Which
has the most/least?
Which items have the same amount?
Illinois Mathematics Teacher – Fall 2012 ...... 26
Other Confetti Uses A Helpful Hint Once you get a good collection of Purchase a plastic bait box with confetti you will find many other uses dividers to keep the confetti separated and including: organized.
Ordinal Positions
AB, ABC and Repeating Patterns
Where Do I Go From Here? Once your students get creative and have experience looking outside the box they will learn to identify the necessary parameters needed to solve a problem. They
will also become adept at using multiple Subtraction approaches to problem solve. You can now expand that realization into general problem solving. When giving the problem “I need to read seven books this semester. I’ve already read four. How many more do I need to read?” encourage students to share all the possible ways they could get the answer. Would you use a part-part whole box? Draw a picture? Create an algebraic sentence? Teaching your students to challenge themselves and look at mathematics differently is a difficult but valuable skill. Expanding Patterns While children are having fun creating stories and “playing” with confetti, they are finding new ways to apply their mathematical knowledge. When your students are no longer limited by traditional problem solving parameters they will begin to apply a variety of strategies to the increasingly difficult problem solving scenarios they will encounter.
Illinois Mathematics Teacher – Fall 2012 ...... 27
Problem Solving in the Primary Classroom
Rena Pate – Danville District 118, [email protected]
When I was in second grade, the became my inspiration to write a book that words math and computation were would meet this need. synonymous. Each day, we completed a Take the following story problem worksheet of 30-50 computation problems examples: with one story problem at the bottom of the page. We called it the “dreaded story Alexandra has 4 cats. Her aunt gives her problem.” Even our teacher called it that! 3 more cats. How many does she have On most days her instructions were to just now? cross it off because it was too hard or complicated for us to do. On occasion our Rich has 5 cookies. He eats 2 of the teacher would be in “a mood” and would cookies. How many does he have left? require us to complete the problem. So, we would do what any second grader would do. These are typical story problems Half of us would add the two numbers found in most series today. While we are together and the other half would subtract jumping up and down wondering why our the two numbers. Of course, half of us kids aren’t excited about problem solving, would get the problem right and the other these are the questions our students are half would get it wrong. When I began thinking: teaching math, I realized that the only problem solving we were doing were “Who in their right mind has 4 cats and percentages and fractions because half if you did why would you get 3 more?” would get it right and 50% would get it wrong! That was 1968. Unfortunately, not “What’s wrong with Rich? Any normal a lot has changed since then. The only kid would have eaten all 5 cookies!” difference is that we have an exciting name like, “Problem of the Day” for the story These kinds of story problems aren’t problem. relevant to children today. They are also too Most programs do not offer daily simplistic and can usually be solved by multiple opportunities to problem solve, and adding or subtracting the two numbers. the problems are usually one step and Instead, we should be teaching our students simplistic. My Action Research Thesis was multiple ways to solve complex and multi- written about the importance of teaching step problems. As adults we realize that problem solving in the primary classroom. there are multitudes of ways to solve real Through my research I tried to locate and life problems, and if we want our students to identify some of the highest quality be future problem solvers we need them to materials available. Unfortunately, after realize this as well. conducting an extensive search of private I spend 15-20 minutes each day and textbook companies and scouring problem solving with my kids. My children material at NCTM conferences, the only quickly learn that it is just as important to be resources I was able to find were simple able to tell me how they arrived at the “Problem of the Day” manuals. This answer as it is to get the correct answer.
Illinois Mathematics Teacher – Fall 2012 ...... 28
Therefore, every student must be able to At the end of this article there is a explain his/her reasoning. After a student sample from the book When Do Dandelions models his/her strategy I challenge the rest Become Weeds? A Guide to Teaching of the class to come up with a different way Problem Solving in the Primary Classroom. to solve the problem. Modeling multiple It is a scripted lesson whereby the teacher methods of problem solving provides reads the script and draws (after student struggling students with clear examples they responses) what is written in bold – just like might use in the future. It also pushes more the sample on the chalkboard. advanced students to look farther outside the I use the scripted lessons with one box. response for each problem for the first Take the following example: There month or so of school so my students can be are 15 items in my school box. If 4 are exposed to a variety of problem solving pencils, 3 are erasers, and the rest are strategies. After that time, you are ready to crayons, how many are crayons? accept multiple methods to solve each The first student explains that he problem. used a part, part, part-whole box. He put 15 Once your students are experienced in the whole and 4 and 3 in the parts. He at identifying multiple ways to problem then knew it was a subtraction problem and solve, it is time to move to more complex once he subtracted 4 and 3 from 15 he and multi-step problems. Look at the would solve for the number of crayons. following samples: The second student explains that she counted up. “I knew I had to get to 15 items Different Kinds of Homes so I added the 4 and 3 to get 7. I then used 1. _____people live in an igloo. _____live the number line to see how much I needed to in a cabin. How many people are living add to 7 until I had 15.” there? Are there more igloo or cabin The third student shares that she people? How many more? drew a picture. “I drew the 4 pencils and 3 erasers and then kept drawing crayons until I 2. My apartment building has _____stories. had 15 items. Then I could count how many If _____people live on each floor, how crayons I drew.” many people live in my apartment Clearly each of these 3 students is building? working at a different capability of problem solving. All three are going to get the 3. It takes _____stilts to hold up one beach correct answer if given a question like this house. How many stilts would it take to on a test. Your goal is to assist the student hold up _____beach houses? who had to work much too hard drawing all of those pictures by showing her that there 4. A ski lodge has _____floors. Each floor are increasingly easier ways to get the has one less room than the floor below. correct answer. If the bottom floor has _____rooms, how We typically solve 4 different types many are on the top floor? How many of problems each day. While one Problem rooms altogether? of the Day isn’t nearly enough, time 3 limitations and attention spans rarely allow you to work on more than four problems a day.
Illinois Mathematics Teacher – Fall 2012 ...... 29
The Little Red Hen Notice that the number amounts are 1. One bundle of wheat could be ground purposefully left blank for two reasons. You into _____cups of flour. How many know your students best and can choose cups could she get out of _____bundles numbers that would be appropriate for the of wheat? level of problem solving they are ready to complete. The other reason is that it gives 2. Little Red can bake _____loaves each you an opportunity to teach your students day. How many would that make in a how basic problem solving skills will help week? them as their problems become more difficult and the numbers get larger. Take 3. Little Red baked ______white, Little Red Question 2 as an example. If I ______wheat, and ______cinnamon taught this lesson early in the week, I might loaves. How many loaves is that? Put put a 2 in the blank. As we problem solved, them in order from most to least? How my students would discover that using our many more does most have than least? doubles or counting by twos would be an easy way to get the answer. If I re-taught 4. Little Red changes her mind and decides this lesson at the end of the week I might put to share her bread with duck, dog and the number 8 in the blank. We could cat. If she cuts a loaf into _____pieces, initially use our doubles to get started and how many will each get? then use a hundred chart to add the doubles together. Another fun thing to do is let My problem solving topics now students choose the numbers the second time center around: current holidays and sports; you problem solve. When someone chooses characters and events in stories we are 99 (thinking it will be funny) you can reading in the classroom; parts of our school actually challenge students to find an easy day, etc. It is important to keep the topics of way that the problem could still be solved. high interest to students as the application Problem solving is a lot of work for for our problem solving goals the teacher and for your students. You will “Different Kinds of Homes” fits with begin this journey with the goal of raising a basal story and social studies unit where student achievement in problem solving. we study different kinds of homes around While you will easily accomplish this goal, the world. Problem 1 requires work with you will also discover that you and your number sense. Problem 2 teaches students students are now experienced, successful multiplication through repeated addition. AND eager problem solvers. Problem 4 is a much more complex problem where students might have to draw a diagram or create a formula to solve. About the Author “The Little Red Hen” takes a Rena Pate is a National Board childhood classic and requires students to Certified Teacher. She has a BS degree work with number sense, multiple from the University of Illinois and a patterning, and fractions. Questions 3 and 4 Master’s in Teaching and Learning even provide discussion opportunities on Mathematics. She has taught first grade for what kind of bread do students think she 23 years in Danville, Illinois. really did bake and who thinks that when She has presented at several ICTM she shared it was the right thing to do? workshops, regional NCTM in New Orleans and St. Louis and the Annual NCTM in
Illinois Mathematics Teacher – Fall 2012 ...... 30
Indianapolis. You can see her presentation November 29th. For more information, visit at the regional NCTM in Chicago, Illinois, her website at primarymathrules.com.
Below is a sample from the book When Do Dandelions Become Weeds? A Guide to Teaching Problem Solving in the Primary Classroom by Rena Pate.
Illinois Mathematics Teacher – Fall 2012 ...... 31 101 102 uses for a paper clip
Charlotte Schulze-Hewett – Harper Rainey College, [email protected] Hana Sallouh – Best Brains Learning Center, [email protected]
Paperclips have many uses besides paper. One pencil is held still, as a pivot, and just clipping papers together. This simple the oother is used to draw the arcs, as shown piece of twisted metal can be used in over in Figure 1. 100 ways, for example: a picture hanger, an emergency fish hook, a book mark, an ejector for a powered-off CD-ROM, or a zipper pull replacement. Even though there are plenty of unique ways to use a paperclip, I have managed to find one more—compass and straightedge constructions. From my experience, students Fig. 1 learning compass and straight edge constructions can become overwhelmed The markings on the paper will look when trying to remember the various steps just like the arcs and lines made in they must follow. For many students the tradiitional constructions. I find that if I most frustrating part of the process is demonstrate just one construction with a wrestling with a compass that slips, pokes paper clip, students grasp the concept and holes in their paper, and is a general are able to complete the activity with nuisance. I have developed an activity that minimal assistance. allows students to perform most compass The simplest constructions to do are and straightedge constructions without a the following: compass. the perpendicular bisector of a I typically use this activity as a (short) segment review of constructions, prior to an an anglee bisector of a angle assessment. I find that in this way, even an equilateral triangle students who have mastered using a a line perpendicular to a given line compass benefit from the activity. In this through a point on the line article, the constructions that are emphasized a line perpendicular to a given line are the bisection of a short line segment, through a point not on the line (but bisection of an angle, construction of a close to tthe line) perpendicular line through a point on a a line parallel to a given line through given line, and the bisection of a very long a point not on the line (but close to line segment. the line) In this activity, the compass is the circumcenter and incenter of a replaced by a paper clip and two writing (small) triangle utensils. A jumbo size paper clip is the altitudes and medians of a recommended, although a standard size (small) triangle paper clip will suffice. The paper clip lies I will describe a few of the constructions flat on the paper with a pencil inside the here, but anticipate that the rest will be self- paper clip at both ends, perpendicular to the evident. I wiill begin by constructing a
Illinois Mathematics Teacher – Fall 2012 ...... 32 perpendicular bisector of a short line verteex of the angle and make an arc that segment. For the purpose of the example, intersects the rays of the angle. consider line segment AB , which is 6 cm in length (Fig. 2).
Fig. 5 Fig. 2
Place the paper clip at each intersection Place one end of the paper clip at point A point, make arcs, and connect the vertex and make an arc (Fig. 3). with the intersection point of the arcs. We
recommend that the first arc be produced
using the inner loop (Figs. 6 and 7), or it
may be difficult to find the intersection point
of the last two arcs (Fig. 8).
Fig. 3
Now, move the paper clip to point B and make an arc that intersects the first arc. Use Fig. 6 a straight-edge to connect the intersection points of the arcs (Fig. 4).
Fig. 7
Fig. 4
This should seem like a fairly straight forward compass and straight-edge construction. The next construction is slightly more complex. Consider the bisection of an Fig. 8 angle, such as angle A (Fig. 5) which, for illustrative purposes, I have made an angle In FFigure 9, I show the result of the of 160º. The basic idea of the construction construction using the inner loop. While it is remains the same as in the traditional possible, albeit messy, to bisect an angle construction: place the paper clip at the
Illinois Mathematics Teacher – Fall 2012 ...... 33 using only the outer loop, for the next construction the inner loop must be used.
Fig. 9 Fig. 12 The third construction is that of a line perpendicular to a given line, through a point on the line (Fig. 10).
Fig. 10
The first step in this construction is to make Fig. 13 an arc centered at point P that intersects the sides at points A and B. Next, make arcs The preevious constructions were centered at points A and B, which should simple; now we will look at a final, more intersect each other above and below the complex, construction. A construction, line. Note that if only the outer loop is used which at firsst glance, appears to be for all three arcs, the last two arcs will impoossible to do with a paper clip, turns out intersect each other at point P, rather than to be merely challenging. Consider bisecting above and below the line (Fig. 11). a line segment which is 12.5 cm long. In the tradiitional compass and straight-edge construction, one opens the compass to slightly more than half the length of the line segment. However, using a jumbo size paper clip, the longest arc that can be made is slightly less than 4.5 cm. Thus, one might jump to the conclusion that it is not possible to bisect this line segment using the paper clip method. However, it can be done. Fig. 11 Let AB be a line segment 12.5 cm in length. Use the paper clip to make arcs The inner loop is used for the first arc (Fig. centered at points A and B; let C and D be 12), producing points A and B, and the outer loop is used for the last two arcs. This will the points where these arcs intersect AB result in an image similar to Figure 13. (Fig. 14).
Illinois Mathematics Teacher – Fall 2012 ...... 34 construct the perpendicular bisector for segments of any length.
Conclusion Students are often mystified by compass and straightedge constructions. The simplicity of a paper clip, in comparison to a compass, seems to make some students more comfortable with geometric constructions. In fact, I will occasionally Fig. 14 observe a student opt for a paper clip over a compass, when given the choice. Now, construct the perpendicular bisector of Another alternative to a compass would be to use a string. You can tie a knot, CD , which is also the perpendicular bisector creating a loop, similar to thhe paperclip. We of AB . Note: this idea can be used to invite you to find other creative ways to do constructions.
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Illinois Mathematics Teacher – Fall 2012 ...... 35 Quadrilaterals: Transformations for Developing Student Thinking
Colleen Eddy – University of North Texas, [email protected] Kevin Hughes – University of North Texas, [email protected] Vincent Kieftenbeld – Southern Illinois University Edwardsville, [email protected] Carole Hayata – University of North Texas, [email protected]
Students begin learning about Chief State School Officers, 2010, transformations in the early elementary Geometry: Introduction section, para. 4). grades. They might create simple Using transformations to strengthen tessellations by cutting and pasting your students’ understanding of geometry cardboard pieces, or investigate symmetry sounds like a good idea, but how can you by folding figures on patty paper (Common actually do this in the classroom? This Core State Standards [CCSS] 4.G.3). Their article describes geometry activities that knowledge and ease with transformations incorporate transformations to discover and continues to grow through middle school, justify definitions and properties of where the everyday language of turn, flip, quadrilaterals, and the relationships between slide, bigger and smaller is connected with the different quadrilaterals. We provide the corresponding mathematical terminology examples how you can help students use of rotation, reflection, translation, and transformations in their reasoning. The dilation (National Council of Teachers of lesson described in this article incorporates Mathematics [NCTM] 2000). By the end of CCSS geometry content from across the middle school, students should be able to grades. Specifically, it addresses the describe the effect of transformations on following standards in the Congruence two-dimensional figures (CCSS 8.G.3). domain (G-CO): In high school, however, students often study transformations in isolation, Experiment with transformations establishing few connections with other in the plane parts of geometry. This disconnects the G-CO.3 Given a rectangle, parallelogram, informal thinking of elementary and middle trapezoid, or regular polygon, school, and the deductive reasoning required describe the rotations and in a high school geometry course. Coxford reflections that carry it onto itself. and Usiskin (1971, 1975) originally G-CO.5 Given a geometric figure and a proposed the use of transformations to build rotation, reflection, or translation, conceptual understanding based on prior draw the transformed figure experiences before deriving proofs. using, e.g., graph paper, tracing Transformations also form the basis of paper, or geometry software. geometric understanding in the vision of the Specify a sequence of Common Core. For instance, for high school transformations that will carry a geometry “[t]he concepts of congruence, given figure onto another. similarity, and symmetry can be understood from the perspective of geometric transformation” (National Governors Association for Best Practices & Council of
Illinois Mathematics Teacher – Fall 2012 ...... 36
Prove geometric theorems using transformations of triangles and G-CO.11 Prove theorems about discuss how to derive the properties of the parallelograms. Theorems resulting quadrilateral. With teacher include: opposite sides are guidance, students synthesize and congruent, opposite angles are summarize their findings and investigate congruent, the diagonals of a how these properties relate the different parallelogram bisect each other, quadrilaterals to each other. Students then and conversely, rectangles are develop definitions to classify each parallelograms with congruent quadrilateral shape accordingly (CCSS MP diagonals. 3). The outline for this activity is as follows:
Completing the activities presented in this 1. Form heterogeneous groups of 3 - 4 article will help your students develop the students. following mathematical practices [CCSS 2. Assign each group one or two MP]: quadrilaterals to investigate parallel- ogram, rhombus, rectangle, square, 1. Make sense of problems and persevere in trapezoid, isosceles trapezoid, or kite. solving them; a. Give each student in the group an 3. Construct viable arguments and critique investigation sheet to record their reasoning of others; findings. Blackline masters for these 5. Use appropriate tools strategically. six investigations can be found at the end of the article. Blackline masters for all activity pages are b. In addition, give each group a sheet included at the end of this article. Examples of graph paper, a ruler, and a sheet of of student work and suggestions on how to patty paper, which students can use deliver the material can be found throughout to complete the prescribed rotations the text. in some of the investigations (CCSS MP 5). Discovering and developing an 3. During the investigation phase of the understanding of quadrilateral properties lesson, each group creates a quadrilateral by completing a set of transformations As mathematics teachers, we know on a given triangle. In the process, the that simply providing students with a list of students discuss the various properties of definitions and properties of quadrilaterals the quadrilateral shape they create with to memorize is an ineffective practice. This respect to the sides, the vertex angles, approach leads to minimal understanding of the diagonals, and the symmetry of the the quadrilateral properties and how they are quadrilateral. derived. In addition, little or no connections 4. Students are asked to go back and are made between the properties and how discuss how they could justify the the quadrilaterals relate to each other. various properties using their prior Transformations give students the knowledge of transformations (CCSS opportunity to develop the definitions and MP 3). In particular, students should properties of quadrilaterals through have developed the concept that discovery and to build connections among rotations, reflections, and translations the family of quadrilaterals. In the activity preserve congruency of segment lengths we describe, students create quadrilaterals
Illinois Mathematics Teacher – Fall 2012 ...... 37
as well as angle measures in an earlier When asked what they noticed about the lesson. verteex angles of the parallelogram they 5. The students determine how the created, the students responded that the quadrilaterals relate to each other by opposite angles were congruent. The teacher creating a family tree for the prompted the students to provide a quadrilaterals (see the last blackline justification for their reasoning (CCSS MP mastere ). In the process of creating a 3). In response, the students simply stated family tree, students formulate a that they measured the angles of the definition of each shape, making sure parallelogram and found that one pair of each definition specifies minimum opposite angles measures 40 degrees while properties while at the same time the other pair of opposite angles measured excluding any unwanted cases. 140 degrees (see Figure 1). The teacher asked the studeents if they would be able to An alternative to number 5 above is draw the same conclusion if the original the use of expert and jigsaw groups. Using triangle they started with had different angle this method, students are first grouped into measures. expert groups where they create a single At this point, students were quadrilateral and record its properties. Then struggling to move beyond the specific new jigsaw groups are formed which contain parallelogram they had created and think in at least one expert for each quadrilateral. terms of any arbitrary parallelogram. The Every member shares with the jigsaw group teaccher reminded the students that simply the properties of their particular shape. Then measuring the angles did not constitute a each jigsaw group creates a quadrilateral justification that the property would hold family tree. true for all parallelograms. The students were instructed to think back too the Justifying definitions and properties properties of transformations they had using transformations disccussed in an earlier lesson. One of the students in the group now stated (CCSS MP Discussion provides an opportunity 1) that when they started with the original for the teacher to guide students in the use of triangle, the vvertex angle (marked 40 deductive reasoning to prove quadrilateral degrees on their paper) would always be properties. We describe two instances of congruent to the opposite vertex angle student reasoning with transformations as an because they rotated the original triangle to example of how to facilitate this type of form the parallelogram. Since rotations discussion. The first group constructed a always preserve congruency, the opposite parallelogram (see Figure 1). verteex angles would be congruent. Using single and double notation marks for the two Figure 1. Student drawing from group angles, which formed the other vertex angle discussion of parallelograms of the parallelogram, the students were able to justify that the other pair of opposite verteex angles also had to be congruent by the fact that rotations preserved congruency. The group summarized their knowledge of the properties of parallelograms (see Figure 2).
Illinois Mathematics Teacher – Fall 2012 ...... 38
Figure 2. Student work from group in this group was whether would always discussion of parallelograms be congruent to . The students felt that this would always be true, but were unsure why. When prompted by the teacher to think back to the relationship between the sides and angles of triangles, the students realized that if they could show that the base angles of ∆ were ccongruent, and then the sides and opposite those angles would be congruent. The question thus became how to show that ∠ ′ ≅∠ . First, students noticced that ∠ is complementary to ∠ , because these are the base angles of origginal right ∆ . Realizing that ∠ corresponds to ∠ ′ undder the rotation, they concluded that the angles are
congruent. The students concluded that The second group in another class ∠ ′ is a right angle. Then they argued started their construction of a rectangle with that ≅ ′ and ≅ ′ because the right ACB. They found the midpoint M length of a segment is preserved under on , drew the median , and then rotation. This allowed the students to say rotated the triangle 180 degrees around that ∆ ≅ ∆ ′ . Hence, they knew that midpoint M (see Figure 3). ∠ ′ ≅ ∠ because the two angles are corresponding parts of congruent triangles, Figure 3. Student drawing of rectangle which proved tthat ≅ . To deepen ′ students’ understanding, the teacher prodded the sstudents to make explicit connections to the side-angle-side triangle congruence postulate. Using their prior knowledge of transsformations and applying this knowledge to a series of investigations, students gained a deeper understanding of the properties of the quadrilaterals and were able to justify the properties. Although the students used transformation in their reasoning, their final justiffiication is no less rigorous than a traditional two-column proof. This activity was highly engaging and had the benefit of students asking questions about the properties conjectuured. In one particular classroom, stuudents Measuring the sides and the angles, the demonstrated a deeper understanding of the students marked congruent sides and angles quadrilateral properties by creating a in the figure. One of the questions that arose quadrilateral family tree. Although more
Illinois Mathematics Teacher – Fall 2012 ...... 39
time was spent discovering and deriving lesson, students investigate the properties of properties than is customary, students a paarticular quadrilateral in groups. In a successfully demonstrated their largeer class, you may need to assign the understanding by constructing the family samme quadrilateral to different groups; in a tree for the quadrilateral shapes. The smaaller class, you could assign each group variations in student representations were more than one quadrilateral, or you could evident in the family tree constructions. assign only a subset. In the first stage of the Some groups organized their investigation, students mosst likely focus on quadrilateral shapes in a bottom-up fashion disccovering the properties. In the second starting with the most specific shape, the stagge, direct the attention of the students to square, and working back up to the more explaining and justifying these properties. general quadrilateral. Other groups After students have completed their organized their family tree using a top-down investigation, you may want to bring the method with the more general parallelogram class together to discuss the similarities and figure at the top and working down to the differences in the properties discovered. more specific figures (see Figure 4). This also provides students an opportunity to use mathematical arguments to justify their Figure 4. A top-down quadrilateral family conclusions, and to judge the validity of tree arguments made by their peers. Finally, students make connections between the different quadrilaterals by constructing a family tree. When using this lesson in your classsroom, you may want to keep the following suggestions in mind:
If a group of students initially fails to come up with any properties, you may need to use a question to point them in the right direction. For instance, “What do you notice when you measure the vertex angles?” Once a group of students has discovered a property, guide the studentts in constructing a justification using Most groups had an easier time transformations. placing the kite and the trapezoid in the If a group of students finds it difficult to family tree diagram than the rest of the generalize from their drawing of a figures. At the center of the discussions was specific parallelogram (or another the relationship between the rectangle, quadrilateral) to the general case, you rhombus, and square. After the teacher may want to suggest drawing another directed students’ attention to the properties parallelogram. that were similar across the figures, the students correctly concluded that every square has the properties of both a rhombus and a rectangle. In summary, the lesson can be delivered as follows. At the beginning of the
Illinois Mathematics Teacher – Fall 2012 ...... 40
Advancing geometric thought through and the properties of triangles. With the the use of transformations discussions, the teacher guided students to the informal deductive level by having them The van Hiele levels of geometric discover the properties of quadrilaterals and thought (1986) provide a sound justification create informal arguments deriving the of why transformations are a good bridge quadrilateral properties. Finally, students between initial student thinking and the unify their understanding of quadrilaterals abstract nature of high school geometry. The by creating the family tree. model includes five levels of geometric reasoning: visual, descriptive, informal Conclusion deductive, (formal) deductive, and rigor. Although a student progresses through the Incorporating transformations to levels linearly, there is no strict dependence develop understanding in high school on age. For example, a fourth grader and a geometry helps students move from the high school geometry student could be at the visual and descriptive van Hiele levels to the same level. The levels overlap as a student informal deductive and formal deductive transitions from one level to the next. Most levels. All students, regardless of their van students entering a high school geometry Hiele level, stand to benefit from an course operate at or below the van Hiele integration of transformation geometry in a visual level of understanding (Shaughnessy high school geometry course (Usiskin 1972; and Burger 1985). These students are able to Okolica and Macrina 1992). In the identify different shapes, but they may find quadrilateral activities described above, the it hard to recognize and reason with specific teacher used this approach to guide the characteristics of shapes. For example, students in building their understanding of doing transformations with everyday quadrilateral properties. When students can language as described in the introduction is connect geometric concepts to the prior at the visual level. Students at this level are knowledge and experiences that they bring often not yet ready for the abstract reasoning to class, they can formulate deductive required in high school geometry. Students arguments. The study of transformations in using the mathematical terminology for the early grades and the continuing transformations are at the overlap of the development of these concepts in middle visual and descriptive levels. Students and high school can provide the bridge discovering and deriving quadrilateral geometry students need to reach the properties are at the informal deductive informal and formal deductive levels of level. geometric reasoning. Transformations provide a bridge between the initial visual intuition of the References students and the more formal reasoning of the higher van Hiele levels. The Coxford, Arthur F., and Zalman P. Usiskin. quadrilateral activities provided entry to Geometry: A Transformation students at the descriptive level. Students Approach. River Forest, IL: Laidlaw, drew conclusions about the sides and angles 1971, 1975. of their original triangle and then rotated the National Council of Teachers of triangles to create the specified quadrilateral. Mathematics (NCTM). Principles and This provided students the opportunity to Standards for School Mathematics. apply prior experiences of transformations Reston, VA: NCTM, 2000.
Illinois Mathematics Teacher – Fall 2012 ...... 41
National Governors Association for Best Usiskin, Zalman P., and Arthur F. Coxford. Practices & Council of Chief State “A Transformation Approach to School Officers (2010) Common Core Tenth-Grade Geometry.” Mathematics State Standards for Mathematics. Teacher 65, no. 1 (January 1972): 21– http://www.corestandards.org/the- 30. standards Usiskin, Zalman P. “The Effects of Okolica, Steve, and Georgette Macrina. Teaching Euclidean Geometry via “Integrating Transformational Transformations on Student Geometry Into Traditional High Achievement and Attitudes in Tenth- School Geometry.” Mathematics Grade Geometry.” Journal of Teacher 85, no. 9 (December 1992): Research in Mathematics Education 3, 716–19. no. 4 (November 1972): 249–59. Shaughnessy, J. Michael, and William F. Van Hiele, Pierre M. Structure and Insight: Burger. “Spadework Prior to A Theory of Mathematics Education. Deduction in Geometry.” Mathematics Orlando, FL: Academic Press, 1986. Teacher 78, no. 6 (September 1985): 419–28.
Blackline masters for the activity pages are included below and on subsequent pages.
Building Quadrilaterals: Parallelogram
1. On a sheet of graph paper, draw obtuse ∆ . Draw one of the sides of ∆ along one of the grid lines. Be sure all vertices are placed at the intersection of grid lines.
2. Locate the midpoint, , of .
3. Draw the median to side .
4. Label the figure you have drawn by indicating congruent sides, angles, and measures using appropriate markings.
5. Rotate ∆ by 180° around point . Label the new image with the correct markings to indicate congruent sides, angles, and measures.
6. Discuss with your group the properties of the parallelogram that you created using the transformations above. Pay attention to the properties of the sides, the vertex angles, the diagonals, and the symmetry of the figure. Summarize your findings under the headings on the chart.
Illinois Mathematics Teacher – Fall 2012 ...... 42
Building Quadrilaterals: Rectangle
1. On a sheet of graph paper, draw scalene right ∆ . Draw both legs of ∆ along grid lines. Draw the right angle at vertex . Be sure all vertices are placed at the intersection of grid lines.
2. Locate the midpoint, , of .
3. Draw the median to hypotenuse .
4. Label the figure you have drawn by indicating congruent sides, angles, and measures using appropriate markings.
5. Rotate ∆ by 180° around point . Label the new image with the correct markings to indicate congruent sides, angles, and measures.
6. Discuss with your group the properties of the parallelogram that you created using the transformations above. Pay attention to the properties of the sides, the vertex angles, the diagonals, and the symmetry of the figure. Summarize your findings under the headings on the chart.
Building Quadrilaterals: Rhombus
1. On a sheet of graph paper, draw scalene right ∆ . Draw both legs of ∆ along grid lines. Draw the right angle at vertex . Be sure all vertices are placed at the intersection of grid lines.
2. Label the figure you have drawn by indicating congruent sides, angles, and measures using appropriate markings.
3. Reflect ∆ across the line containing . Label the new image with the correct markings to indicate congruent sides, angles, and measures. Use prime marks for the image vertices. What kind of figure do you have now? Justify your answer.
4. Reflect ∆ ′ across the line containing ′ . Be sure to also reflect . Label the new image with the correct markings to indicate congruent sides, angles, and measures. Use prime marks for the image vertices.
5. Discuss with your group the properties of the parallelogram ′ ′ that you created using the transformations above. Pay attention to the properties of the sides, the vertex angles, the diagonals, and the symmetry of the figure. Summarize your findings under the headings on the chart.
Illinois Mathematics Teacher – Fall 2012 ...... 43
Building Quadrilaterals: Square
1. On a sheet of graph paper, draw isosceles right ∆ . Draw both legs of ∆ along grid lines. Draw the right angle at vertex . Be sure all vertices are placed at the intersection of grid lines.
2. Label the figure you have drawn by indicating congruent sides, angles, and measures using appropriate markings.
3. Reflect ∆ across the line containing . Label the new image with the correct markings to indicate congruent sides, angles, and measures. Use prime marks for the image vertices. What kind of figure do you have now? Justify your answer.
4. Reflect ∆ ′ across the line containing ′ . Be sure to also reflect . Label the new image with the correct markings to indicate congruent sides, angles, and measures. Use prime marks for the image vertices.
5. Discuss with your group the properties of the square ′ ′ that you created using the transformations above. Pay attention to the properties of the sides, the vertex angles, the diagonals, and the symmetry of the figure. Summarize your findings under the headings on the chart.
Building Quadrilaterals: Kite
1. On a sheet of graph paper, draw scalene acute ∆ . Draw side of ∆ along a grid line. Be sure all vertices are placed at the intersection of grid lines.
2. Draw an altitude, , of ∆ from point to .
3. Label the figure you have drawn by indicating congruent sides, angles, and measures.
4. Reflect ∆ across the line containing . Label the new image with the correct markings to indicate congruent sides, angles, and measures. Use prime marks for the image vertices.
5. On a separate sheet of graph paper, repeat steps 1-4 with a scalene obtuse ∆ .
6. Discuss with your group the properties of the kites ′ that you created using the transformations above. Summarize your findings under the headings on the chart.
Illinois Mathematics Teacher – Fall 2012 ...... 44
Building Quadrilaterals: Trapezoid 1
1. On a sheet of graph paper, draw a large scalene ∆ .
2. Label the figure you have drawn by indicating congruent sides, angles, and measures using appropriate markings.
3. Locate a point anywhere on . Construct a line parallel to through point . Construct the intersection of this line with at point .
4. Discuss with your group the properties of the trapezoid that you created using the transformations above. Summarize your findings under the headings on the chart. Hint: What connections can be made between the measures of the angles in this figure and your prior experiences with parallel lines cut by a transversal?
Building Quadrilaterals: Trapezoid 2
1. Now draw a large isosceles ∆ . Draw the vertex angle at .
2. Locate a point anywhere on . Construct a line parallel to through point . Construct the intersection of this line with at point .
3. Discuss within your group the properties of the trapezoid that you created using the transformations above. Summarize your findings under the headings on the chart. Hint: What connections can be made between the measures of the angles in this figure and your prior experiences with parallel lines cut by a transversal?
Illinois Mathematics Teacher – Fall 2012 ...... 45
Names ______
Properties of ______
SIDES: VERTEX ANGLES:
DIAGONALS: SYMMETRY:
Illinois Mathematics Teacher – Fall 2012 ...... 46
Quadrilateral Family Tree
Your group will be building a family tree for all of the quadrilateral shapes that you have explored so far in this lesson.
Your family tree will be a diagram that connects all the different quadrilateral shapes together based on their shared properties. When designing your family tree it would be best to start with the more generic figures at the top of the diagram. These would be the figures whose properties were a part of all the other shapes. You will use the shapes that were cut out for the “Where Do I Belong Activity.”
As you come to more shapes with more specific and unique propertiees, you will need to place these underneath the first set of figures. Be sure to show connections between figures using line segments that branch between different quadrilaterals.
Your family trees should branch off from the most generic name for a 4-sided figure: quadrilateral. Once you have decided for sure what your family tree should look like, glue or tape the shapes in their appropriate place.
Quadrilaterals Family Scrapbook
Using paper, pencil, and pictures or photos, your group needs to design a scrapbook for the family of quadrilateral shapes. Your entire group will be responsible for creating one scrapbook. Each person in the group must create at least one page of the scrapbook.
Each shape must have at least one page in the scrapbookk. Each page should contain at least the following: A picture of the quadrilateral shape seen in the real world A list of the characteristic properties of that particular shape
In addition, the scrapbook needs to have a copy of the quadrilateral family tree that was created in class.
If your group chooses, you may also complete the scrapbook in PowerPoint on the computer.
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