I found the following puzzle in Jacob's Geometry: Seeing, Doing, Understanding (3rd ed.). Given a square sheet cake, 9" on a side, divide it into 5 pieces that have the same amount of cake and the same amount of icing.
To be more mathematically precise, the problem is: Given a square, a units on a side. find a dissection of the square into 5 polygonal pieces, each with area a2/5 and each containing the same length of boundary of the original square, namely 4a/5. I think this is not too hard, but I will post my answer if anyone asks.
Extra credit: To generalize and sharpen this result, show that we can replace 5 by n, where n is any integer greater than or equal to 3. Also show that all polygons can be triangles or convex quadrilaterals, and that even so for any fixed n there are an infinite number of essentially different such dissections. (Two dissections are essentially different if one contains a polygon that is not congruent to any polygon in the other dissection.)
The Birthday Problem
A well-known problem asks for the smallest number of people (N) who must be in a room before it is more likely than not that two share the same birthday. The answer, surprising to most people who have not heard the problem before, is N = 23.
I thought it would be interesting to modify the problem where we ask for people who share that same day of the month for their birthday. While the answer is not as surprising as the original problem, the computation is much easier. Direct computation for the first problem using factorials will result in overflow on scientific calculators such as the TI-83. Also, the answer to the day-of-month problem (N = 7) is more suitable for empirical testing in small classes. Simply ask each student for their birth day (1 - 31) and record on a large month calendar. For N = 11 the probability of a match increases to almost 88%.
The formula for the probability of one or more matches amongst a group of N people is
Prob = 1 - (31)(30)...(32 - N)/31N
= 1 - 31! /[(31 - N)! * 31N]
I thought it would be interesting to modify the problem where we ask for people who share that same day of the month for their birthday. While the answer is not as surprising as the original problem, the computation is much easier. Direct computation for the first problem using factorials will result in overflow on scientific calculators such as the TI-83. Also, the answer to the day-of-month problem (N = 7) is more suitable for empirical testing in small classes. Simply ask each student for their birth day (1 - 31) and record on a large month calendar. For N = 11 the probability of a match increases to almost 88%.
The formula for the probability of one or more matches amongst a group of N people is
Prob = 1 - (31)(30)...(32 - N)/31N
= 1 - 31! /[(31 - N)! * 31N]
Mathematics and Humor
"Time flies like an arrow. Fruit flies like a banana." -- The Flying Karamazov Brothers.
Have you ever told a joke to someone who "doesn't get it"? If you patiently explain the referents you may get them to "understand" the joke, but they will probably respond something like "So, why is that funny?"
In the simple example above, you probably found this funny if (1) you are familiar with the maxim "Time flies like an arrow", (2) your knowledge of the English language allows you to understand that "flies" can be a verb meaning "passes swiftly" or a plural noun referring to a type of insect and that "like" can mean both "as" and "enjoy" (3) your knowledge of writing style leads you to expect that when the same word appears in two successive short sentences, it will usually have the same meaning in both sentences.
I think we face the same problem when we try to teach mathematical understanding. A proof is most memorable to us when, like in getting a pun, we make a connection between two or more apparently unconnected thoughts, what is often called an "Aha!" moment. Without previous deep knowledge of the constituent thoughts, the student may be able to follow the step-by-step logic, and may be able to remember the proof for tomorrow's test, but the proof will not be memorable, and both the theorem and the proof will soon be forgotten. One implication for pedagogy is that the curriculum must be carefully planned so that, when a mathematical topic is introduced, the students will understand the constituent parts and be able to appreciate their connection. Otherwise, we are mostly wasting our time.
I recently came across a proof of the Pythagorean Theorem that was new to me that gave me an aha! moment. This was given in Sanjay Gulati's excellent "Mathematics Academy" blog as a Geogebra demonstration. He does not indicate the original source of the proof. The aha! moment comes for the connection between the Pythagorean Theorem and an apparently unrelated theorem that I always teach in my elementary geometry class, the "crossed chords" theorem. The aha! moment occurs from looking at the following picture.
Then the crossed-chords theorem tells us that (c + a)(c - a) = b2, or c2 - a2 = b2.
Have you ever told a joke to someone who "doesn't get it"? If you patiently explain the referents you may get them to "understand" the joke, but they will probably respond something like "So, why is that funny?"
In the simple example above, you probably found this funny if (1) you are familiar with the maxim "Time flies like an arrow", (2) your knowledge of the English language allows you to understand that "flies" can be a verb meaning "passes swiftly" or a plural noun referring to a type of insect and that "like" can mean both "as" and "enjoy" (3) your knowledge of writing style leads you to expect that when the same word appears in two successive short sentences, it will usually have the same meaning in both sentences.
I think we face the same problem when we try to teach mathematical understanding. A proof is most memorable to us when, like in getting a pun, we make a connection between two or more apparently unconnected thoughts, what is often called an "Aha!" moment. Without previous deep knowledge of the constituent thoughts, the student may be able to follow the step-by-step logic, and may be able to remember the proof for tomorrow's test, but the proof will not be memorable, and both the theorem and the proof will soon be forgotten. One implication for pedagogy is that the curriculum must be carefully planned so that, when a mathematical topic is introduced, the students will understand the constituent parts and be able to appreciate their connection. Otherwise, we are mostly wasting our time.
I recently came across a proof of the Pythagorean Theorem that was new to me that gave me an aha! moment. This was given in Sanjay Gulati's excellent "Mathematics Academy" blog as a Geogebra demonstration. He does not indicate the original source of the proof. The aha! moment comes for the connection between the Pythagorean Theorem and an apparently unrelated theorem that I always teach in my elementary geometry class, the "crossed chords" theorem. The aha! moment occurs from looking at the following picture.
Harold Jacobs' Geometry
I've been considering a new text for a course in Euclidean Geometry that I teach for middle school teachers. I've been using Essentials of Geometry for College Students by Lial et al. The students seem OK with it, but I find it very boring. I supplement it with lots of my own exercises using Geometer's Sketchpad, paper folding, MIRA(tm), etc. to keep things interesting.
In looking for a replacement, the best book I have found so far is Geometry: Seeing, Doing, Understanding by Harold R. Jacobs. The latest (3rd) edition was published in 2003. Although I will probably use this book, I will transform many of the problems I assign from pencil, paper, ruler, and protractor to Geometer's Sketchpad. I would love it if the publisher W. H. Freeman would commission an update.
This is a high school text, but it is more challenging than Lial. The applications to "real life" are the most realistic and compelling that I have seen anywhere. I keep finding things that I didn't know, and ways of looking at geometry problems that I hadn't considered.
In one example on page 503 Jacobs shows a closed smooth curve bounding a convex region and consisting of circular arcs. One student said that the sum of the arc measurements must be 360 degrees, and the other doubts it because the curve is not a circle. From the nature of Jacobs' construction, it is easy to show that the sum of the arc measures is indeed 360 degrees. A good teacher could connect this with the fact that the sum of the exterior angles of a convex polygon is 360 degrees.
In another example, Jacobs gives an "Area Puzzle" where he guides students to prove a curious fact about triangle areas. If each vertex of a triangle (ABC in the figure below) is connected to a point 1/3 of the way from the next vertex (in CCW order, say) to the following vertex, and the intersections of these 3 segments (Cevians) are connected, an inner triangle (DEF) is formed. The area of DEF turns out to be 1/7 of the area of ABC. I have known this for some years, and even published a paper (with my brother Marshall and my nephew Michael) generalizing it to quadrilaterals and to ratios other than 1/3. The proof I used involved using analytic geometry to establish the result for a right triangle with vertices (0, 0) (1, 0), and (0, 1) and then arguing that the area ratio is preserved by affine transformations, so the result holds for all triangles.
Jacobs presents a neat synthetic proof that clearly shows where the strange ratio 1:7 comes from. He constructs 6 more triangles, each a translate of the central triangle, and then guides the student to show that the triangles can be dissected and reassembled to fill the original triangle. See the diagram below.
In looking for a replacement, the best book I have found so far is Geometry: Seeing, Doing, Understanding by Harold R. Jacobs. The latest (3rd) edition was published in 2003. Although I will probably use this book, I will transform many of the problems I assign from pencil, paper, ruler, and protractor to Geometer's Sketchpad. I would love it if the publisher W. H. Freeman would commission an update.
This is a high school text, but it is more challenging than Lial. The applications to "real life" are the most realistic and compelling that I have seen anywhere. I keep finding things that I didn't know, and ways of looking at geometry problems that I hadn't considered.
In one example on page 503 Jacobs shows a closed smooth curve bounding a convex region and consisting of circular arcs. One student said that the sum of the arc measurements must be 360 degrees, and the other doubts it because the curve is not a circle. From the nature of Jacobs' construction, it is easy to show that the sum of the arc measures is indeed 360 degrees. A good teacher could connect this with the fact that the sum of the exterior angles of a convex polygon is 360 degrees.
In another example, Jacobs gives an "Area Puzzle" where he guides students to prove a curious fact about triangle areas. If each vertex of a triangle (ABC in the figure below) is connected to a point 1/3 of the way from the next vertex (in CCW order, say) to the following vertex, and the intersections of these 3 segments (Cevians) are connected, an inner triangle (DEF) is formed. The area of DEF turns out to be 1/7 of the area of ABC. I have known this for some years, and even published a paper (with my brother Marshall and my nephew Michael) generalizing it to quadrilaterals and to ratios other than 1/3. The proof I used involved using analytic geometry to establish the result for a right triangle with vertices (0, 0) (1, 0), and (0, 1) and then arguing that the area ratio is preserved by affine transformations, so the result holds for all triangles.
Jacobs presents a neat synthetic proof that clearly shows where the strange ratio 1:7 comes from. He constructs 6 more triangles, each a translate of the central triangle, and then guides the student to show that the triangles can be dissected and reassembled to fill the original triangle. See the diagram below.
An Application of Triangle Geometry
I am always on the lookout for attractive, simple, and real applications of synthetic Euclidean geometry to share with my students. That's why I enjoyed an article in the March 2011 issue of The College Mathematics Journal by Robert K. Smither, "The Symmedian Point: Constructed and Applied". The symmedian point of a triangle is the intersection of the three symmedians, where a symmedian is the reflection of a median of the triangle in the angle bisector (at the vertex through which the median passes). See the picture below, where the dashed lines are the medians, the green lines are the angle bisectors, the red lines are the symmedians, and P is the symmedian point of triangle ABC.
Smither worked for the Navy in the post-WWII era, well before computers were widely available. His job was to design a system for locating mines that might be dropped by aircraft on a harbor. As a mine hit the water, it would be observed from three stations. On a map, rays would be drawn from each station location in the direction in which the mine had been sighted. In theory, the three rays would be concurrent at the location of the mine. In practice, due to measurement error, the three rays would not be concurrent, but would form a triangle ABC. The most likely location of the mine was assumed to be the point for which the sum of the squares of the distances to the sides AB, BC, AC was minimum.
The hand calculations required to locate the point in question using analytic geometry were horrendous and error-prone. By examining the results, Smither was led to rediscover the symmedian point, which turns out to be the point that minimizes the sum of the squares of the distances to the sides of the triangle. He also discovered a neat method of constructing this point, which is easier than using the definition and appears to be original.
Smither worked for the Navy in the post-WWII era, well before computers were widely available. His job was to design a system for locating mines that might be dropped by aircraft on a harbor. As a mine hit the water, it would be observed from three stations. On a map, rays would be drawn from each station location in the direction in which the mine had been sighted. In theory, the three rays would be concurrent at the location of the mine. In practice, due to measurement error, the three rays would not be concurrent, but would form a triangle ABC. The most likely location of the mine was assumed to be the point for which the sum of the squares of the distances to the sides AB, BC, AC was minimum.
The hand calculations required to locate the point in question using analytic geometry were horrendous and error-prone. By examining the results, Smither was led to rediscover the symmedian point, which turns out to be the point that minimizes the sum of the squares of the distances to the sides of the triangle. He also discovered a neat method of constructing this point, which is easier than using the definition and appears to be original.
E18. Alphabetical ordering of whole numbers
Suppose the integers from 1 to 10,000,000,000 are ordered alphabetically as spelled out in American English, according to the rules below. Find the first odd number in this ordering.
(1) Numbers are spelled out formally, using neither "and" nor common shortcuts. For example, 2400 is two thousand four hundred, not twenty-four hundred.
(2) To alphabetize, ignore any spaces or hyphens. For example, sixteen comes before six thousand.
This problem is one of the easier ones in Peter Winkler's excellent collection, Mathematical Puzzles: A Connoisseur's Collection. It is an excellent problem to give to students working together in small groups. Usually students will have to make many attempts along the way to finding the solution, giving everyone a chance to participate. Note the specification of American English. Many people don't realize that there is a difference between American English and British English in some number names. For example, "billion" has different meanings in the two forms of English.
It might make an interesting exercise to write an algorithm for converting an integer (say in the range specified in this problem) to the spelled out form.
(1) Numbers are spelled out formally, using neither "and" nor common shortcuts. For example, 2400 is two thousand four hundred, not twenty-four hundred.
(2) To alphabetize, ignore any spaces or hyphens. For example, sixteen comes before six thousand.
This problem is one of the easier ones in Peter Winkler's excellent collection, Mathematical Puzzles: A Connoisseur's Collection. It is an excellent problem to give to students working together in small groups. Usually students will have to make many attempts along the way to finding the solution, giving everyone a chance to participate. Note the specification of American English. Many people don't realize that there is a difference between American English and British English in some number names. For example, "billion" has different meanings in the two forms of English.
It might make an interesting exercise to write an algorithm for converting an integer (say in the range specified in this problem) to the spelled out form.
Curriculum for Overcoming Math Anxiety Course
(c) Peter. F. Ash, Ph.D. 2011
The following is the Curriculum for my Overcoming Math Anxiety course offered at Cambridge Center for Adult Education February 23 – March 9, 2011 over three two-hour classes:
1.Let's get personal
What brings you here? Why do you need to overcome math anxiety? When did your dislike or fear of mathematics first develop? Start keeping a "math journal".
2.Everyone can learn math
Is there something in your brain that means you can't learn math? What is dyscalculia? Overcoming handicaps.
3.Math phobia?
A serious fear of math may be a phobia, and may require treatment. A treatment you can do yourself, called TAT (Tapas Acupressure Technique) can help you. Our special guest lecturer shows you how.
4.One size doesn't fit all (even if your teacher thought it did)
Different people have different learning styles. If you know your preferred learning style you can learn math better. Are you a quantitative or a qualitative learner? Learning through different modalities: visual, aural, or tactile/kinesthetic.
5.Math myths
If you were taught with traditional methods, you probably learned that being good at math required prodigious memory and the ability to regurgitate what the teacher told you. You may believe that there is one way to solve a math problem, and that math must be done while sitting still and keeping quiet. Wrong!
6.The new way to learn math
Modern reform mathematics suggests that math instruction be focused on solving interesting complex problems which can be solved in different ways, that students work in groups and communicate their ideas to one another, and that students learn to do mathematics with deep understanding, not by rote.
7.A sound mind in a sound body
Research shows that regular aerobic exercise helps you to beat stress, improve memory, and sharpen your thinking. Schedule your exercise before doing your math and watch what happens.
8.Learning is not all in your head
Learning cannot be separated from movement. The fact that proper movement leads to optimal learning underlies Brain Gym®, We'll practice basic Brain Gym exercises to help reduce stress and make learning easier.
9.The mind-body connection
Learn to reduce stress and improve focus with meditation-based techniques. Use Zen meditation, yoga, TM, the relaxation response, or simple diaphragmatic breathing to reduce stress and empty your mind of chatter so you can learn better.
10.Music hath charms…
Playing certain classical music in the background can help you energize and focus. I'll play the CD and you'll hear if it helps.
11.Manipulate and understand
Learn what a mathematical manipulative is and how it helps visual and tactile/kinesthetic learners understand math concepts. Experience the power of multiple representations in math.
12.So, can I really do math?
Sure you can! You'll investigate a few math problems working in a group. Try out your new-found math anxiety reduction skills and enjoy some interesting open-ended problems.
13.Help! I need to take a math class
How to tell if you have a good teacher. What to do if you don't. Important study skills
.
14."Teach your children well"
What you can do so your children learn to like math, not to fear it.
15.Where do you go from here?
I'm here to help. Send me an email if you'd like a bibliography on math anxiety and math learning. Contact me if you are interested in math tutoring or math classes.
The following is the Curriculum for my Overcoming Math Anxiety course offered at Cambridge Center for Adult Education February 23 – March 9, 2011 over three two-hour classes:
1.Let's get personal
What brings you here? Why do you need to overcome math anxiety? When did your dislike or fear of mathematics first develop? Start keeping a "math journal".
2.Everyone can learn math
Is there something in your brain that means you can't learn math? What is dyscalculia? Overcoming handicaps.
3.Math phobia?
A serious fear of math may be a phobia, and may require treatment. A treatment you can do yourself, called TAT (Tapas Acupressure Technique) can help you. Our special guest lecturer shows you how.
4.One size doesn't fit all (even if your teacher thought it did)
Different people have different learning styles. If you know your preferred learning style you can learn math better. Are you a quantitative or a qualitative learner? Learning through different modalities: visual, aural, or tactile/kinesthetic.
5.Math myths
If you were taught with traditional methods, you probably learned that being good at math required prodigious memory and the ability to regurgitate what the teacher told you. You may believe that there is one way to solve a math problem, and that math must be done while sitting still and keeping quiet. Wrong!
6.The new way to learn math
Modern reform mathematics suggests that math instruction be focused on solving interesting complex problems which can be solved in different ways, that students work in groups and communicate their ideas to one another, and that students learn to do mathematics with deep understanding, not by rote.
7.A sound mind in a sound body
Research shows that regular aerobic exercise helps you to beat stress, improve memory, and sharpen your thinking. Schedule your exercise before doing your math and watch what happens.
8.Learning is not all in your head
Learning cannot be separated from movement. The fact that proper movement leads to optimal learning underlies Brain Gym®, We'll practice basic Brain Gym exercises to help reduce stress and make learning easier.
9.The mind-body connection
Learn to reduce stress and improve focus with meditation-based techniques. Use Zen meditation, yoga, TM, the relaxation response, or simple diaphragmatic breathing to reduce stress and empty your mind of chatter so you can learn better.
10.Music hath charms…
Playing certain classical music in the background can help you energize and focus. I'll play the CD and you'll hear if it helps.
11.Manipulate and understand
Learn what a mathematical manipulative is and how it helps visual and tactile/kinesthetic learners understand math concepts. Experience the power of multiple representations in math.
12.So, can I really do math?
Sure you can! You'll investigate a few math problems working in a group. Try out your new-found math anxiety reduction skills and enjoy some interesting open-ended problems.
13.Help! I need to take a math class
How to tell if you have a good teacher. What to do if you don't. Important study skills
.
14."Teach your children well"
What you can do so your children learn to like math, not to fear it.
15.Where do you go from here?
I'm here to help. Send me an email if you'd like a bibliography on math anxiety and math learning. Contact me if you are interested in math tutoring or math classes.
Car Talk Math Puzzler Solution Wrong!
On the NPR program Car Talk, hosts Tom and Ray presented a puzzler in which a truck has a broken fuel gauge. The trucker wishes to determine when his fuel tank is one-quarter full. The tank is in the shape of a circular cylinder on its side. The trucker can put a stick in the filling opening at the top of the tank, and use it as a dipstick to mark the height of the fuel. Obviously, the half-full mark is the radius of the circle from the end of the stick. Where is the quarter-full mark? A solution was requested that would use a minimum of advanced math. Solvers could use a pizza box, string, a pencil and a knife.
I loved the problem, and their method of solution which involved using the string and pencil to draw a circle the size of the tank cross section on the top of the pizza box, then using the bottom of the pizza box as a to draw a diameter of the box, and using the box as a square to draw a radius perpendicular to the diameter. Finally, they cut out the resulting semi-circular region, and balanced the semicircle on the pencilpoint to determine the center of mass of the region. They claimed that the quarter-full horizontal line would pass through the center of mass.
Not so! Look at the fuel tank "head on", so you see a circular region. Imagine drawing 3 horizontal lines on the region. The first line is the diameter of the circle, or the half-full line. The second line is the one that passes through the centroid of the lower semicircular region and is parallel to the first line. Let's call this the TARL (Tom and Ray line). If Tom and Ray are right, the TARL is the quarter-full line. However, they're wrong so there is a third line, the BL (bisector line) which is parallel to the previous two lines and divides the semicircular region into two pieces of equal area. If the circle involved has a radius of 10 inches, the TARL is approximately 4.24 inches from the half-full line, while the BL is approximately 4.04 inches from the half-full line.
The mathematical details are interesting, and can be shown without any explicit reference to integral calculus. See them at Scribd.
Why do clocks move clockwise?
Here's a question that can be answered with a little knowledge of mathematics and history, and some common sense.
Why do "normal" clocks always turn in the direction we are used to, and not in the opposite direction?
Why do "normal" clocks always turn in the direction we are used to, and not in the opposite direction?
Subscribe to:
Posts (Atom)
