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Assessment Strategies Using Calculator, Computer, and Internet Environments. Dr. David Ashley, dia059@smsu.edu math.smsu.edu/faculty/ashley.html Dr. Lynda Plymate, lsm953f@smsu.edu math.smsu.edu/~lynda. Department of Mathematics Southwest Missouri State University Springfield, MO 65804.
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Assessment Strategies Using Calculator, Computer, and Internet Environments Dr. David Ashley, dia059@smsu.edu math.smsu.edu/faculty/ashley.html Dr. Lynda Plymate, lsm953f@smsu.edu math.smsu.edu/~lynda Department of Mathematics Southwest Missouri State University Springfield, MO 65804
NCTM’s Assessment Principle “Assessment should support the learning of important mathematics and furnish useful information to both teachers and students.” Assessment should enhance students' learning. • Research indicates that making assessment an integral part of classroom practice is associated with improved student learning. • Activities that are consistent with (and sometimes the same as) the activities used in instruction should be included. • Assessment is a valuable tool for making instructional decisions. • To ensure deep, high-quality learning for all students, assessment and instruction must be integrated so that assessment becomes a routine part of the ongoing classroom activity rather than an interruption.
New Assessments to Incorporate/Accommodate Current Technology • New Types of Test Questions • New Assessment Formats • Testing Online • Changes to Curriculum and Instruction (e.g. CAS in Algebra and Calculus)
Martha made the pattern shown below on her TI-89 calculator. One line in this pattern has equation y = x - 1. Determine the equations of the other 7 lines in this pattern. Use the numerical limits on the x-axis and y-axis as references for your lines.
The following three graphs describe two cars, A and B. Complete each of the following tasks. • A. Decide whether each statement below is true or false. • i. The newer car is more expensive • ii. The larger car is newer • iii. The less expensive car carries more passengers. • On each of the graphs below, label two points that would represent cars • A and B.
The box plots shown below represent the ratings given to the 257 episodes, in the seven seasons, of Star Trek: The Next Generation (top plot is season 1 and bottom plot is season 7). These ratings, with 1 as the best and 257 as the worst, were determined by Entertainment Weekly magazine personnel. Use the center and spread in these plots to defend your choice of the “best season” for this program.
Name:_________________________ WORKING WITH EXPONENTSDr. Lynda Plymate • __________________________________________ ____________________________ • __________________________________________ ____________________________ • __________________________________________ ____________________________ • __________________________________________ ____________________________ • __________________________________________ ____________________________ • __________________________________________ ____________________________ • __________________________________________ ____________________________ • __________________________________________ ____________________________
You are to design your own quilt pattern or use one of the quilt patterns that I have provided for you. • You are to construct at least one quilt pattern using Geometer’s Sketchpad demonstrating the geometry required to produce the pattern and provide written documentation (30%)showing the step-by-step procedures involved in the construction. Be specific. You may use a printed script of your tool if you created one. However, even this must be annotated showing what the tool is doing) • Your team will assemble a 9-patch gird pattern quilt (15%) out of construction paper (or some material of your choosing) showing the various shapes and colors you would have in your cloth • Provide written documentation (Lesson Plans) (45%)(5 pages or more) of how you would incorporate this project into a thematic interdisciplinary unit involving the subjects of English, History, Science and of course Mathematics and the areas or key learning points that you would cover. These lesson plans must be sufficiently developed that another teacher such as a substitute could step in and teach for you.
Step-by-Step Procedure Involved in Construction • ProcedureConstructed line segment AB to measure 15 cm. • Rotated segment AB 270 degrees on point B, then made point C on that line and slid it to the end opposite point B, creating line segment BC, which is also 15 cm. • Created a line parallel to AB through point C, using the construct a parallel line tool. • Created a line perpendicular to AB through A, using the construct a perpendicular line tool. • Created the intersection of the parallel line in #3 and the perpendicular line in #4 to create point D. • Created line segments AD and DC. • Hid the perpendicular and parallel lines. • Constructed a smaller square to measure 5 cm exactly the same way we constructed the large 15 cm square ABCD. • Moved the 5 cm square so that the top and left side lay on top of lines AD and AB and point B. • Created line segments AF, FI, and HI. • Mirrored square AFIH along line FI. • Created line segments FJ, JK, and KI. • Mirrored square FJKI along line JK. • Created line segment JB, LB and KL. • Mirrored square AFIH along HI. • Created line segment HM, MN and IN. • Mirrored square HINM along IN. • Created line segments KO and NO. • Mirrored square IKON along KO. • Created line segments LP and OP.
Mirrored square HINM along MN. • Created line segment MD, DQ and NQ. • Mirrored square MNQD along NQ. • Created line segment OR and QR. • Mirrored square NORQ along OR. • Created line segment RC and PC. • Created line segment IJ. • Created line segment MI. • Created line segment OL. • Created line segment QO. • In square JBLK created midpoints of the sides, which were named points S, T, U, & V. • Created line segments TV and SU. • Created a point of intersection of lines TV & SU named point G. • In square IKON created midpoints of the sides, which were named points W, X, Y, & Z. • Created line segments XZ and WY. • Created a point of intersection of lines XZ & WY named point F. • In square MNQD created midpoints of the sides, which were named points A1, B1, D1, & C1. • Created line segments A1B1 and C1D1. • Created a point of intersection of lines A1B1 and C1D1 named point E. • Highlighted points A, F, I, & H and constructed a quadrilateral interior with the color blue. • Highlighted points J, F, & I and constructed a triangle interior w/ color red. • Highlighted points J, T, G1, & S and constructed a quadrilateral interior w/ color red. • Highlighted points T, U, B, & G1 and constructed a quadrilateral interior in blue. • Highlighted points G1, U, V, & L and constructed a quadrilateral interior in red.
Highlighted points H, I, & M and constructed a triangle interior in red. • Highlighted points I, X, F1, & W and constructed quadrilateral interior in red. • Highlighted points F1, Y, O, & Z and constructed quadrilateral interior in red. • Highlighted points O, L, & P and constructed a triangle interior in red. • Highlighted points M, A1, E1, & C1 and constructed a quadrilateral interior in red. • Highlighted points C1, E1, B1, & D and constructed a quadrilateral interior in blue. • Highlighted points E1, D1, Q & B1 and constructed a quadrilateral interior in red. • Highlighted points Q, O, & R and constructed a triangle interior in red. • Highlighted points O, P, C, & R and constructed a quadrilateral interior in blue.
History of Mount Rushmore The men from the White House Grade level: 4th grade Subjects: Social Studies/ Language Arts/ Math Materials needed: Internet articles from www.mtrushmore.nethttp://library.thinkquest.org/T0211461/history Paper Pencils Computer lab Objectives: The students will understand the importance of Mt. Rushmore. The students will understand why each of the four Presidents was chosen to be on Mt. Rushmore. The students will practice using ratios. The students will practice and improve their writing styles. Procedure: Ask the students if anyone knows what Mt. Rushmore is. Wait for their response. Ask the students if anyone knows who is on Mt. Rushmore. Wait for their response. Have students speculate why each of those Presidents was chosen. Write their answers on the board. Explain that we are going to learn about the sculptor, how big they are, how long it took, and why those Presidents were chosen. Pass out copies of the article from www.mtrushmore.net Read the article Mt. Rushmore Trivia to the class. Have discussion on what we just read. Pass out copies of the article from http://library.thinkquest.org/T0211461/history Have students who are comfortable and want to read aloud, read the article Mt. Rushmore: Presidents on the Rocks. Stop at the end of each section and have discussion about what was just read.
After the section on “Interesting Facts” talk about the size of the faces: • head = 60 feet • nose = 20 feet • mouth = 18 feet • eyes = 11 feet each • Inform the students that if the figures were sculpted from head to toe, they would be 465 feet. • Pose this question to the students, “How big would a 6 foot tall humans eyes be compares to the sculptures?” • Show the students how to figure this using the head as an example: • 465/6 = 77.5 times • head 60 ft/77.5 = .774…..ft, approximately 9 inches • nose 20 ft/77.5 = .258…..ft, approximately 3 inches • mouth 18 ft/77.5 = .23225….ft, approximately 2.75 inches • Have students try the problem for the eye on their own. • After approximately 5 minutes ask the class for their answers. • Give them the correct response and praise for those who got the correct answer. Then show all students how to get the answer. • eyes 11 ft/77.5 = .1419… ft, approximately 1.75 inches
Once they have read the whole article, have them write a 1 page paper on who might be the 5th President to be carved on Mt. Rushmore, if carving was to be continued and why. Tell them they will need facts and thorough reasoning. They must use complete sentences, correct spelling, and punctuation. • After they have completed the rough draft, have them come up to the desk (one at a time) and proof their paper. • Have the students make the corrections and have them proofread the revision. • During computer lab time have the students type up the paper and print it. • These will go in their published works journal. • Assessment: • We will have author’s chair where everyone has a chance to share their published work. • We will have a small quiz with several questions • Name three of the four Presidents on Mount Rushmore. • Where is Mount Rushmore located? • Who was the sculptor of Mount Rushmore? • How long is the nose of each of the Presidents? • What is Mount Rushmore made out of?
I found 'Geometry with Quilting' to be a fun project that effectively utilizes geometry skills in making something practical. This project allows a lot of creativity and is a great way to learn and apply geometric knowledge for visual/spatial learners such as myself. Before working on this project, I had no idea how much geometry was used in quilting. The various quilt patterns out there are amazing and really show ingenious uses of various shapes. I feel that this project would be a great way to show children in the classroom a practical way to use geometry outside of the school setting. Not only is the project fun, but it also can be used as a creative lesson for children who don't necessarily master objectives easily from a textbook format. It can be used to teach objectives about reflective symmetry, measuring and finding midpoints, the different types of triangles, angles, and other important geometry skills. Overall, this project is a great way to enhance the teaching of geometry. While this project definitely has many advantages in the classroom, there are also some drawbacks to teaching it. Some children, most likely boys, may not appreciate the project or lessons because quilting seems to be such a feminine hobby. Second, this project and a thematic unit may be too time consuming with all of the other objectives required in today's classroom. To understand the unit and adequately cover the geometry in this project, more time is required than would be if simply teaching terms and information out of a textbook. When state standards require many objectives to be mastered in a short school year, projects like this can put teachers behind in their teaching schedule. This seems to be a problem in every subject, though, and I feel this lesson is worth the time and should be incorporated when teaching geometry at the elementary level.
I have thoroughly enjoyed our Math 360 class this semester. The quilt project is a useful endeavor because it is a "real world" teaching project. It demonstrates how difficult that teaching effectively can be. With the tremendous pressure placed on teachers today to fit more and more material in less time, we need more practice doing interdisciplinary projects such as this one. I have enjoyed the teamwork aspects of this class, although, frankly, I did not carry my own weight in this project. Sally deserves most of my points for this project as she did most of the work. I offered help in the design and some basic suggestions, but she was the star of our show. The cons to the project were that quilts and learning about them would turn some children off. I envision very few ten year old boys waking early with Christmas-like excitement to hurry to get to school to learn about quilts. However, it is a perfect example of how we must teach everything required, whether we like the material or not. So, if we have to do it, why not make it fun?! Not all material will appeal to all children or teachers. Another con with a positive spin is the potential of one student of a group doing more work than the other. While it is "not fair ...boohoo " ...it is a fact of life that we must step up to the plate sometimes and carry someone else. In our professional lives (and our personal lives as well), there are times we will get the short end of a teamwork arrangement. We have two choices. ..pout or try harder. In our group, Sally tried harder. It was not that I didn't want to work hard or was bored with the class. I have three kids (8, 2, 3 months) and a wife to support, and I work fun time.
Although this project took more work than I was anticipating, I think we came up with a final product that can be useful in a future classroom. I thought the amount of work was reasonable and it was easy to split between two teammates. I thought the lesson plans were more effective as far as a future teaching tool, because we came up with lessons that I would actually use in my classroom. That was the major pro of this project for me: that I was putting the work into a product that will be useful later on. That also leads me to the major con of the project, which is that I felt the quilt making itself was very time consuming and I don't think that quilt will be something we will use again. Also, I think for this level of learning, we as college students did not necessarily need to be convinced of the geometric properties of quilts, which seemed to be the point of making the quilt ourselves. I think that simply designing a quilt block by using Sketchpad would have been sufficient to prove that we understood how a quilt fits in with math. However, we as teammates managed the workload so that the quilt making itself was not overwhelming, and we ended up with a neat design that has a lot of geometric properties as well as arithmetic properties, so it could be used to demonstrate how the Fibonacci sequence works at a higher level of education, perhaps. I thought that this was a unique and interesting project that made me think about quilts in a new way. I think that it could be modified to be more useful by requiring more lesson plans, to create a more comprehensive unit, and a less emphasis on the construction of the quilt itself.
Year Age Interest Tim Deposit Tim's Balance Tom Deposit Tom's Balance 1 21 9.00% $2000 $2,000.00 $0 $0.00 2 22 9.00% $2000 $4,180.00 $0 $0.00 3 23 9.00% $2000 $6,556.20 $0 $0.00 THE STORY OF TIM and TOM AT 9% Tim and Tom are interesting characters. Some people think there is an important moral to their story. Tim and Tom were twins. They both went to work at age 20 with identical jobs, identical salaries, and at the end of each year, they received identical bonuses of $2000. However, they were not identical in all respects. Early in life, Tim was conservative and was concerned about his future. Each year he invested his $2000 bonus in a savings program earning 9% interest compounded annually. Tim decided at age 30 to have some fun in life and he began spending his $2000 bonuses on vacations in the Bahamas. This continued until he retired at 65 years old. Tom, on the other hand, believed in his youth that life was too short to be concerned about saving for the future. For ten years, he spent his $2000 bonuses on vacations in the Bahamas. At age 30, he began to realize that some day he might not be able to work and then would need funds to provide for his support. So he began investing his $2000 bonuses in a savings program earning 9% compounded annually. This continued until he was 65 years old. Although separated for a few years, they were joyfully reunited at age 65 at a family reunion and exchanged many stories of the events in their lives. Eventually the conversation got around to retirement plans and savings programs. Each brother was proud of his savings and showed the other a spreadsheet describing his savings activities and accumulations. But they were amazed! Tom had made many more $2000 deposits than Tim. Yet,Tim had accumulated almost $200,000 more than Tom. How could that be? Using a spreadsheet fill in the chart and answer the questions.
Requirements: You and your partner are required to design and build a birdhouse from a eight foot by 10 inch by 1 inch piece of exterior grade lumber. You are to supply a set of design prints showing a top view, front view, back view, and side view of your design and how the cuts can be made from the piece of wood. You are to compute the surface area and volume for your birdhouse. You are to construct a scale model (out of a material of your choosing and the scale cannot be 1:1) of your group’s design. Your group will make a presentation (no longer than 3 minutes in length) to the class showing/describing your design, the amount of surface area and the structure’s volume, how you would mount the bird house, how you protect the box from predators and any other neat features of your project.
Your project is to visit three sites on the list and evaluate how well they measure up to the NCTM standards. A grading rubric is provide to assist you that was developed in MTH 479 by Emily McDaris. Some of these sites require you to explore further for a good evaluation. You should check out the site and visit as many links listed as you deem appropriate to get a good feel for what the site has to offer. You are to type a one page doubled spaced 12 pitch font with one inch margins summary for each of the three sites that you choose to visit. Your evaluation should state a score based on the grading rubric along with an explanation for the grade. Each site should be classified as to who you believe could use it (teachers, students or both) and what grade level it is appropriate for. Also state what you believe the general purpose is of the web site (lesson plan research, entertainment, student explorations, demonstrates the usefulness of mathematics), and what discipline does it emphasize. i.e. This site shows a high school student how you can use mathematics in art and architecture.
Once you have evaluated three of the sites listed you are to search the web and find an additional two web sites that you believe would be great sites under the NCTM standards and you would be willing to recommend to your peers. I want you to submit sites you believe are great sites to send a student to when they ask, “When am I ever going to use this stuff?” You are to write a one page summary for each site which addresses why you think this is a worthwhile web site and what grade level and discipline is united with mathematics.
Development: You are develop a 20 to 25 minute lesson involving using the internet and technology to reinforce concepts that have already been learned by Shae Johnson’s 4th grade class. The class will be divided in to two groups of teams, A and B. If you are identified as an A team you will choose a one of the concepts from the A list and you and your partner will develop your lesson from that concept. If you are identified as a B team you will choose one of the concepts from the B list and you and your partner will develop your lesson from that concept. Your lesson should be developed around the technology and should be student centered not teacher centered. It should last about 20 minutes and be detailed enough such that I could follow it and teach your lesson for you. Actual Teaching: You will have the opportunity to present your lesson to two 4th graders during the 20 – 25 minutes. You are to actively involve them and have your lesson be student centered not teacher centered. Keep you students involved and on task. I know that you will not have much time but try and find out what they understand and do not understand about your concept. At the end of 20 to 25 minutes you will get a new group of two students and you will repeat your lesson.
ExploreMath on the Web Interactive JAVA explorations and lesson plans • Quadratics: Vertex form y = a(x-h)2+k (http://www.exploremath.com/activities/Activity_page.cfm?ActivityID=14) • Exponential Functions y = Makx (http://www.exploremath.com/activities/Activity_page.cfm?ActivityID=4) • Logarithms y = logax (http://www.exploremath.com/activities/Activity_page.cfm?ActivityID=7)
Power Versus Exponential FunctionsLynda Plymate f(x) := x a g(x) := a x f(x) = x 2g(x) = 2 x
First Problem Prior to Exam Requires time to reflect and reason Non-routine problem solving (time to build a working strategy) Requires work with technology
Match the following scenarios with the graphs. Name & label (with appropriate numbers) the selected axes. • A 24 inch string when tied together can be made to form a infinite number of rectangles, with the area of the rectangle changing as the width of the same rectangle is made to get larger and larger. • The temperature of the filling in a frozen cherry pie increased dramatically when it is placed inside a preheated oven, then tapered off to a relatively steady hot temperature. • The population of frogs decreased as the pond became more polluted. • The diameter of the cocoon increased rapidly at first, then increased more slowly as the caterpillar prepared to change into a butterfly. • The temperature inside an oven increases when it is turned on, and then fluctuates a bit as the oven turns off and on briefly, trying to maintain a preset temperature. 6. The length of time it takes to paint the gymnasium changed as the number of people painting increased.
The Boston Marathon is one of the world's best-known foot races. Winning time in the Boston Marathon has decreased as runners get faster. The table shows time of the winning man, in minutes, for years 1959-1980. (Unlike women's winning time, men's winning time has not improved much since 1980, so that data is not given.) YearTime (min) i) Create a scatterplot to demonstrate winning time for the 1959 143 given years 1960 141 1961 144 ii) Find r and r2. Then discuss what each number tells you about 1962 144 the relationship between men's winning time and year. 1963 139 1964 140 1965 137 iii) Find the least-squares regression line; and then carefully plot 1966 137 it on your scatterplot. Be sure to include and name at least 2 1967 136 specific points on your line. 1968 142 1969 134 iv) By how much on the average did the winning time improve 1970 131 per year during this period? 1971 139 1972 136 v) Use your regression line to predict the winning time in 1990, 1973 136 a decade later. Is this prediction trustworthy? Explain. 1974 134 1975 130 vi) Complete a residual plot for your data. 1976 140 1977 135 1978 130 1979 129 1980 132
Problem Solving (College Algebra)Exponential Function Models A house has been invaded by 100 termites. The population of termites triples every two days. If the population reaches 800,000, the building will be in danger of severe structural damage. The house has also become infested with 2,000 cockroaches. This cockroach population doubles every five days. If the population of cockroaches exceeds 32,000, the house will be condemned. If the exterminator can only address one problem at a time, which is more pressing, the termites or the cockroaches? In your process of answering this question, you must also demonstrate (using pencil and paper manipulations rather than the calculator "solve" feature) the following 4 items: i) the specific function which models the number of termites present after t days, call it f(t) ; ii) the specific function which models the number of cockroaches present after t days, call it g(t); iii) the number of days it will take for the termite population to reach 800,000; iv) the number of days it will take for the cockroach pop. to reach 32,000.
Extra-Sensory Perception (ESP Simulation) A common test extra-sensory perception (ESP) asks subjects to identify which of four shapes (star, circle, wave, or square) appears on a card unseen by the subject. Consider a test of n=10 cards. If a person does not have ESP and is just guessing, he/she should therefore get 25% right in the long run. So, the proportion of correct responses that the guessing subject would make in the long run would be p = 0.25. Complete 100 simulated tests in which a subject guesses randomly on each of the 10 cards. In your simulation be sure to decide whether each card was guesses correctly; find the percent of cards guesses correctly by each subject; sort the 100 percents in ascending order; find the mean and standard deviation for your distribution of 100 tests; prepare a table showing the number of tests resulting in sample proportions falling in increments of 10% relevant to your data (like 0-10%, 10-20%, 20-30%, 30-40%, … ); and finally prepare a column graph to display those counts. i. Turn in enough printed pages to convince me you have successfully generated random numbers, identified whether the subject's responses are correct, sorted sample proportions, found the requested statistics, table and graph. ii. Referring to your simulated results, what percent of your subjects guessed correctly on 3 or more of their questions (which is better than random guessing)? iii. How many of the 10 cards would a subject have to identify correctly before you would say that less than 5% of all guessing subjects would do that well or better?
Examples of Quiz Problems: This is a problem for you and your lab partner to work on together. You are to use Sketchpad to solve this problem. You are given a line segment AB that is one of the diagonals of a square. Your job is to construct the square from the line segment you have been given and explain your reasoning. What you must do is open sketchpad and construct a line segments AB and then copy it to use as your basis for the construction. Then with that information construct the square, provide an explanation of what you did, and print your results.
You are to use Sketchpad to solve this problem. You are given a line segment AB that is the diagonal of a rectangle. You are also given line segment DE that is one of the sides of your rectangle. Your job is to construct the rectangle from the two line segments you have been given and explain your reasoning. What you must do is open Sketchpad and construct a line segments AB and then DE. Copy these segments and use your copies as the working segments. Then with that information construct the rectangle and provide an explanation of what you did. Print your results.
You and your partner are to use SketchPad and construct a Rhombus using three different techniques. Using a text box next to your rhombus describe how it was constructed. Make measurements to prove it is a Rhombus. Using your sheet of definitions, investigate whether the rhombus is a square, parallelogram, rectangle, trapezoid, trapezium, or a kite include measurements that support your statements. Construct the diagonals of the rhombus. What you can say about the diagonals that you can back up with measurements (Are the diagonals congruent? Do the diagonals bisect each other? Are the diagonals perpendicular? Does each diagonal bisect two angles of the quadrilateral?). What is true concerning the four triangles formed from the construction of the diagonals (are the triangles congruent, similar, equal in area)? Construct the midpoints of all four sides. Construct a quadrilateral formed by the mid points. What type is this quadrilateral? Find the midpoints of each side of this quadrilateral and connect them to form a quadrilateral. Continue this process until you discover a pattern, or the sides of the quadrilaterals become to small to measure. Back up your conjectures with measurements. How does the area of each quadrilateral relate to the original figure you started with? What can you determine concerning adjacent angles and corresponding angles for the original Rhombus. How may lines of reflective symmetry exists. How many degrees of rotational symmetry? What is the sum of the measures of the interior angles? Explore: Tell me all you can about Rhombuses!!!!
You and your partner must complete 2 problems from 1 through 3 (15 points each) using Sketchpad. • Construct a regular trapezoid that is not isosceles. What can you say about successive mid point quadrilaterals that are formed that you can back up with measurements? What can you say about the four triangles formed by the diagonals? Are they congruent, equal in area, or similar? You must have measurements to support your findings. What is the relationship concerning the intersection of the diagonals (Do the diagonals divide each other into a specific ratio?) What is sum of the interior angles of the trapezoid? Print your results. • Construct any triangle ABC. Construct D and E, the mid points of side BC and AB. Construct segment DF and segment EF, where F is any point on segment AC. Construct the polygon interiors of triangles AEF and FDC and quadrilateral BEFD. How are the areas of the triangles related to the area of the quadrilateral? Move point F back and forth along segment AC does it change anything? What happens when f is the midpoint of line segment AC? State any conjectures. Print your results • Space scientists have discovered a capsule from an extraterrestrial civilization. Within they found mysterious writings which noted intergalactic linguists translated into the area formulas below. Use SketchPad to determine if all four formulas always work(i.e. calculate the area as noted by the extraterrestrial civilization and then verify using what you know to be true). Make sure you manipulate your figures to confirm that the formulas will always work. Print your results for each figure showing your readings and conclusions. • The find the area of a triangle, use A = mh where m is the length of the mid-segment of the triangle and h is the height of the triangle. • To find the area of a trapezoid, use A = mh where m is the length of the mid-segment of the trapezoid and h is the height of the trapezoid. • To find the area of a rhombus, use A = rs where r and s are the lengths of the diagonals. • To find the area of a kite, use A =rs where r and s are the lengths of the diagonals.
