MathNotations

Everything's a Square! Motivating Students to Use Deductive Reasoning

2008-07-23T15:56:00.005-04:00

Figures Not Drawn To Scale

Hi y'all! Enjoying a restful peaceful summer? Just a few thoughts...

1. Only one correct submission to this month's math anagram"

SIR "OY", E. NOET!

Please follow the rules here and email me by the end of the month.

2. Ok, so what's going on with the 3 "squares" in the above diagram? In a recent post, we looked at using "Figure Not Drawn To Scale" as an effective way to encourage student reasoning and to become more cautious about making assumptions. Making a variety of quadrilaterals all "appear" to be squares as in the above diagrams is consistent with this approach.

Even before teaching the formal theorems and definitions regarding quadrilaterals in "Chapter 5" of the text, why not begin with a preview activity? If you prefer to wait until students have the necessary definitions, theorems and postulates, then one can use this as an application. Your choice...

Suggested Questions:
Does the given information in each diagram guarantee that each is a square?
If you don't think so, your mission is to draw a quadrilateral with the given information but clearly does NOT look like a square.
Alright, think about the first one. After a minute share your thoughts, diagrams with your partner. Go!
Etc...

Comments
If using this to review the standard definitions and theorems on quadrilaterals, I would still encourage the drawing of diagrams to illustrate that the first two figures do not have to be squares.
Your thoughts...

Squeezing Circles Into the Corner: An Infinite Sequence Investigation in Geometry

2008-07-14T07:27:00.006-04:00

Another summer diversion from geometry...

The number of variations for tangent circles is endless and this is one of my all-time favorites. Math contests and SATs seem to have a preference for circles inscribed in squares or tangent circle problems and this one is along those lines. However, the real payoff comes from developing recursive thinking leading to an infinite geometric sequence and its sum! Students will be asked to intuitively "guess" the value of this infinite sum and to then verify their conjecture. Proving it requires nothing more than the classic formula for the sum of an infinite geometric series but, at the outset, this problem is eminently suitable for your geometry classes. Don't hesitate to use it in your "regular" classes. Questions that are deemed appropriate only for honors classes are often suitable for most students if the groundwork is laid (background, examples, etc.) and hints are given strategically.

PART I In the diagram above the larger circle has radius 1, the two circles are tangent to each other and to the two perpendicular segments (you can think of the larger circle being inscribed in a square if you wish).

(a) Make a conjecture from the diagram without computing: The ratio of the radius of the smaller circle to the larger is approximately
(A) 0.05 (B) 0.15 (C) 0.25 (D) 0.35 (E) 0.5
Note: This part may be omitted.

(b) Show that the radius of the smaller circle is exactly (√2 - 1)² = 3 - 2√2
How was your conjecture?
Note: Your decision about giving them the result like this. Obviously if they see part (b) on a worksheet, their estimate in part (a) will be pretty good! My intent was to focus on the method. Of course, feel free to rephrase this.

PART II
Of course we will not stop at 2 circles! Squeeze a third circle into the corner between the 2nd circle and the right angle. Determine its radius by using the result from part (a). [The key here is to think ratios!]

PART III
If we label the radius of the largest circle R₁, the radius of the 2nd circle R₂, the radius of the 3rd circle R₃, etc., we can now define an infinite sequence of these radii.
(a) Find a formula for the nth term of this sequence, n = 1,2,3,..

(b) What is the mathematical terminology for this type of sequence?

(c) Think intuitively here: From the diagram, what should be the "sum" of the original radius R₁ = 1 and the diameters of the remaining infinite collection of circles. [Another formulation: As n-->∞, this sum approaches what number?]

(d) Using the formula for the sum of an infinite geometric series, verify your conjecture in (c).

Comments:

As always, feel free to use this with your students and revise as you see fit. However, pls use the attribution in the Creative Commons License as indicated in the sidebar.
Finding the radius of the 2nd circle is a challenge by itself and the problem could stop there. The extensions can be assigned as a long-term project or for those wishing to do extra credit. I always liked having additional challenges for the students who were capable of going further, although relating this problem to geometric sequences or series is of importance. Of course, I am well aware of time constraints faced by the instructor.
Your thoughts...

"Any Way You Slice It" - A Classic Cube Dissection Problem to the Nth!

2008-07-09T18:58:00.006-04:00

The following series of questions was inspired by a recently released SAT question. The first two levels are appropriate for middle or secondary students. Level III requires more algebraic background or strong visualization skills. I could have attempted to include a graphic for some of this but I'll leave that to the experts out there!

LEVEL I
A cube is cut into 8 equal cubes by dividing each edge in half with three planes which are parallel to the faces of the original cube. Show that the total surface area of the 8 smaller cubes (when separated) is TWICE the surface area of the original cube.

Note: Most secondary students would attempt this algebraically or substitute particular values. To develop spatial sense, encourage them to find another solution, which is purely visual and elegant! Middle school students (or younger children) would greatly benefit from constructing a physical model of the cube from modeling clay (or something equivalent) and slicing it with appropriate tools. Better yet, one can avoid slicing by constructing the bigger cube from 8 smaller cubical blocks (There are many sets of plastic or wooden blocks available from catalogs).

LEVEL II
A cube is cut into 27 equal cubes by dividing each edge into 3 equal parts with planes parallel to the faces of the original cube. Show that the total surface area of the 27 cubes is THREE times the surface area of the original cube.

LEVEL III
Generalize the above relationships by dividing each edge of a cube into N equal parts with planes parallel to the faces of the original cube (N is an integer greater than 1). State a conclusion and explain! Again, try to find both an algebraic and a visual explanation.

Figure Not Drawn To Scale! An SAT-Type Geometry/Summer Diversion

2008-07-07T19:54:00.006-04:00

In the circle at the left, O is the center, A, B and C are on the circle and OABC is a parallelogram. If AB = 6, what is the length of segment AC (not drawn)?

(A) 3√2 (B) 3√3 (C) 6 (D) 6√2 (E) 6√3

POINTS TO PONDER

Is this an appropriate standardized test question?

Are you an opponent of multiple choice (aka, "multiple guess") questions. Why?

We can also say much about the issue of drawing figures that do not appear to be what they are? Is it just the testmaker's way of misleading or trapping students or is there a valid purpose to this?

Your thoughts about this problem...

SIR "OY", E. NOET! Our July Mathanagram!

2008-06-30T19:42:00.003-04:00

SIR "OY", E. NOET!

Laboring over the anagram of our Mathematician of the Month for hours, then having someone unscramble it in a few nanoseconds...
Oh well, I hope you enjoy this one. Remember there are rules which I will summarize below. To add to the usual requirements, you need to show how our Mathematician is somehow connected to our preceding star, thereby unlocking the hidden code in the puzzle. Otherwise all my efforts would be wasted!

DO NOT SUBMIT YOUR ANSWER AS A REPLY TO THIS POST

Remember to email me at dmarain at geeeemail dot com
Pls pls use Mystery Mathematician July 2008 in the subject line!

(PLEASE ENUMERATE!)

(1) The name of the mathematician
(2) Some interesting info/anecdotes re said person
(2.5) Explain the code embedded in the anagram!
(3) Your sources (links, etc.)
(4) Your full name and the name you want me to use when acknowledging your accomplishment
(5) If you're new, how you found MathNotations
(6) Your connection to mathematics

Probability and Counting Challenge -- You'll Need To Sit Down for This One

2008-06-29T06:28:00.003-04:00

Five people with first initials A, B, C, D and E were seated randomly in a row in a movie theater with no spaces between them. What is the probability that A, B and C were adjacent to each other in some order? (For example: "DBACE")

A potential SAT problem or is it a level above? A math contest problem or not difficult enough? More importantly, how much experience do most of our students have with these kinds of combinatorial problems? I know that some of our educators who visit here do these with their classes, but is it the norm? My feeling is that students need to see many of these developed over time in more than one course.

What method (s) do you consider the most effective for solving these kinds of problems? For teaching? Are these 2 questions really the same?

After this question is discussed with the class, how does the instructor assess the learning? Give them another one to try immediately or give a worksheet of these (if the text does not provide enough practice)? How could one raise the bar even higher?

Suggested Extension #1: This time we have 10 people seated randomly in a row (no spaces). What is the probability that 4 of them, say A. B, C and D, would be adjacent to each other?

Suggested Extension #2: N people seated randomly in a row (no spaces). What is the probability that a particular subset of M of these people would be adjacent to each other?

Your thoughts...

MEN-Y THEOREM? EMMY NOETHER REVEALED!

2008-06-22T11:11:00.003-04:00

I could say so much about Emmy, but our four winners this month have said it much better and have provided some excellent links. Some of my personal thoughts about Emmy are given at the bottom, in my reply to Alex.

If you like the idea of an occasional anagram, let me know. Creating meaningful anagrams out of mathematicians' names can be formidable (certainly time-consuming!).

Our winners are...

HYPATIA

Hi Dave,

I wasn't going to write, but I just couldn't pass this one up! I suppose a picture would give it away, so the anagram was a clever trick. I must say, it's about time -- a woman mathematician! And what a great one --Emmy Noether.

Now let's see - an anecdote. Well we could say that her obituary for the New York TImes was written by none other than Albert Einstein.

http://www-history.mcs.st-and.ac.uk/Obits2/Noether_Emmy_Einstein.html

For a short little bio, here's the following from http://faculty.evansville.edu/ck6/bstud/noether.html

Within the world mathematical community, Emmy Noether is widely regarded as the greatest of all woman mathematicians. She was born in the German university town of Erlangen, where her father, Max Noether, was a professor of mathematics. After receiving the Ph.D. degree from the University of Erlangen under Paul Gordan, Dr. Noether moved to the University of Göttingen, known in those days as the Mecca of Mathematics. There she developed as a world-class algebraist and taught a number of doctoral students who eventually became leading algebraists. Noether came to the United States in 1933, where she taught at Bryn Mawr College near Philadelphia and lectured at the Institute for Advanced Study in Princeton, New Jersey.

Emmy Noether's name is known to many physicists through Noether's Theorem, described by Peter G. Bergmann as a cornerstone of work in general relativity as well as in certain aspects of elementary particles physics. For details, see Brewer and Smith, page 16.

Her name is known to mathematicians largely in connection with the adjective noetherian, which applies in ring theory to properties associated with ascending chains of subrings. Specifics are given in Brewer and Smith, page 18.

It is probably true that most algebraists have never heard of Noether's Theorem in physics and that most physicists have never heard of noetherian rings.

But to do her justice, I'll also include the following links:

http://www.physics.ucla.edu/~cwp/Phase2/Noether,_Amalie_Emmy@861234567.html

http://www.sdsc.edu/ScienceWomen/noether.html

http://www.awm-math.org/noetherbrochure/AboutNoether.html

http://www-history.mcs.st-and.ac.uk/Biographies/Noether_Emmy.html

FInally, I love anagrams, actually puzzle of all kinds. So thanks for this one.

Hypatia

------------------------------------------------------

P. Miller

1. Emmy Noether

2. Worked for years w/o compensation

3. http://womenshistory.about.com/library/bio/blbio_emmy_noether.htm
------------------------------------------------------

TC--

Hi Dave,

I thought the name had to have a hyphen - after that wild goose chase,
I managed to find the name of Emmy Noether.
Thanks, but for you, I would have never learned about this abstract algebraist.

Interesting Anecdote; There is a lunar crater named for her (and for
many other mathematicians, I find!)
Also, her obituary in the NY Times was written by Einstein.
------------------------------------------------------

ALEX DAVIES

(1) The name of the mathematician

Emmy Noether

(2) Some interesting info/anecdotes re said person

Emmy was a brilliant mathematician who struggled for recognition from university authorities, because of her sex. Other mathematicians could see her talent, though - she was mentored by Hilbert, who said "she is superior to me in many respects."

Late in her life, she taught in a girls' school in Pennsylvania. "The New York Times printed a short obituary, as it always did when a Bryn Mawr teacher died, but shortly thereafter, they printed a long letter to the editor pointing out that Emmy Noether had not only been a teacher at a girls' college, but the greatest woman mathematician of all time," wrote Freund in a book. "The letter was signed: Albert Einstein."

(3) Your sources (links, etc.)

Acting on your hint, I googled "First female mathematicians," which took me to this top ten, on which Emmy was tenth.
Here is the quote from Hilbert.
Here is the end quote.
-------------------------------------------------------------------------------------------

By the way, here was my reply to Alex, which pretty much expresses my view of Emmy:

Thank you, Alex. It sure was nice of ol' Albert to acknowledge Emmy as the greatest 'woman' mathematician of all time. Imagine if some day, Emmy is recognized as the greatest mathematician/physicist of all time! Perhaps she would have acknowledged Albert as one of the best of his gender!!

SOMETHING NEW! Instructional Strategy Series: Teaching Average Rates

2008-06-17T07:28:00.009-04:00

The following is the first in a series of strategies for teaching concepts that often prove difficult for many students from middle school on. These are not based on carefully controlled research studies following clinical methodology for a dissertation. They are based on 30+ years of learning how to do it better!! I suspect that's why we refer to the practice of teaching. Our readers are encouraged to share their own favorite methods that have been helpful to their students or to themselves. These ideas are intended only as suggestions. Each teacher will, of course, bring her/his own ideas and style to bear on the lesson.

Most of you know the classic algebra word problem type that has appeared frequently on standardized tests and math contests:

THE BIG QUESTION
Jack averaged 40 mi/hr going to school and 60 mi/hr returning from school over the same route. What was his average speed in mi/hr for the round trip?

Since there has been a decrease over the past 25 years in the number of word problems to which our students are exposed, some youngsters may not get to see one of these until reviewing for SATs or in their physics class.

From watching how students approach this type of question, I'm getting a sense that we need to introduce the basic concepts earlier on in middle school, which I am sure already occurs in some programs. In planning to teach methods of solving these kinds of problems, I usually tried to return to basic principles of math pedagogy - keep it simple and start with concrete numerical exercises that built on prior knowledge. What does all this jargon mean?

Start with a review of averages, then move on to combined averages before attempting to explain the round-trip rate problem!

[Concerned that such development will take too much time? There won't be enough time to review homework and provide enough practice for the homework assignment? My supervisors never threatened to fire me if a lesson lasted for more than one day and if, heaven forbid, I did not assign homework that first evening! Some ideas just cannot be rushed.]

Suggested Question #1:
Jack had a 70 avg on some tests and a 90 average on some other tests. Can his overall average be determined?
More specifically: When do you think 80 will be the correct answer? When will it not?

Comment:
Question 1 is intended to provoke thought and encourage an intuitive response, not a calculated answer!

Suggested Question #2:
Jack had a 70 average on his first 4 tests and a 90 average on his next 6 tests. What was his overall average for the 10 tests?

Comments
Note that I am suggesting beginning with problems to which middle school students may better be able to relate than a rate-time-distance question. The first question above is fundamental in developing the concept of the original rate problem.

These questions should help many students focus on the essential idea that we need to know how many are in each sub-group!

Since most students connect average to dividing a TOTAL by some quantity, they should feel comfortable in solving the average grade question as follows:

(TOTAL PTS)/(TOTAL NUMBER OF TESTS) to arrive at an average of 82.

BUT DON'T STOP THERE! Stress the UNITS of this result to build the rate concept:

AVG PTS/TEST = (TOTAL PTS)/(TOTAL TESTS)

Since students generally do not attach units to the 82, stress that the combined average is 82 PTS PER TEST or 82 PTS/TEST! BTW, not a bad time to mention that PER MEANS DIVIDE!!

Suggested Question #3:
Jack averaged 40 mi/hr for 2 hours, then 60 mi/hr for the next 2 hours. What was his average speed (rate), in mi/hr, for the 4 hours?

[Note the incremental development (commonly termed scaffolding in today's world!). Rather than jump to the abstraction of the original problem, we move on to the next logical step - giving them both the rates and the times for each part of the trip. In this case, we use equal times to provoke their thinking about why the result is also the simple arithmetic mean of the two rates. Each of us needs to make decisions about how many of these examples are needed before moving on to the main question.

Depending on the background and ability level of the group, you may be able to skip one or more of these suggested questions. Further, you may already be thinking of placing these questions on a worksheet for students to try alone or in pairs, stopping and reviewing as needed.

Suggested Question #4:
Jack averaged 40 mi/hr for 4 hours, then 60 mi/hr for 2 hours. What was his average rate, in mi/hr, for the 6 hours?

Suggested Question #5:
Jack averaged 40 mi/hr for the first 120 miles of a trip, then 60 mi/hr for the remaining 120 miles. What was his average rate, in mi/hr, for the entire trip?
Key question: Why does it turn out that the answer is NOT 50 mi/hr?

Comments
Do you think your students would now be ready for the BIG QUESTION near the top of this post? OR do you think they would need at least one more interim problem? Again, could these questions have just as effectively been placed on a worksheet and given to students, working in pairs?

I'll leave the rest to our readers. Pls feel free to share your ideas, comments, thoughts and questions. There's no question in my mind that some of you would develop these ideas differently! Remember you can always email me personally at dmarain at geemail dot com (the last 4 words misspelled intentionally of course!). Unfortunately, I typically get little response from posts about instruction since most readers prefer to solve a challenging problem!

Final Comment: Note that I didn't once suggest that students use a short-cut for the original round-trip problem. Ok, so it is the harmonic mean of the two rates, and can be calculated
from the formula: 2R₁R₂/(R₁+R₂).
But who would want to use that (uh, SATs, GREs, GMATs,...)???

A Geometry Classic - Chord and Tangent Riddle

2008-06-16T06:24:00.004-04:00

Don't forget to submit your solution to this month's Mystery Mathematicianagram (ok, so I can't decide on a name yet!). We've received 3 correct solutions thus far and I will announce winners around the 20th.

As we wind down the school year, the problems below may come too late for students taking their final exams in geometry, but you may want to hold onto this classic puzzler for next year. I don't consider these overly challenging but I do feel they demonstrate some important mathematical ideas and problem-solving techniques. Further, encourage students to justify their reasoning since some may make assumptions from the diagram without verification. This will review some nice ideas from circles.

OVERVIEW OF PROBLEMS (see diagram)
For both questions, assume the circles are concentric, segment PQ is a chord in the larger circle and tangent to the smaller.

PART I
If PQ = 10, show the difference between the areas of the 2 circles is 25π.

PART II (the converse)
If the difference between the areas of the circles is 25π, show that the length of PQ must be 10.

Notes
(1) It is important for students to recognize that there are many possible pairs of concentric circles (varying radii) satisfying the hypotheses of these problems, yet the conclusions are unique! Some students will assume a 5-12-13 triangle is formed (not a bad problem-solving strategy), but stress that this is not the only possibility! Remember, we're not restricting the radii to integer values.

(2) There is a classic math contest strategy for these questions that mathematicians love to employ - the "limiting case." Can you guess what I mean by this phrase?

A Math Riddle that gets better with 'Age'!

2008-06-13T06:03:00.004-04:00

[Don't forget the Mystery Math Anagram for this month. Only two correct replies have been received thus far. I will announce the winners in a few days.]

Have you been wondering where the math challenges have gone on this blog? Here's one that I came across while reading David Baldacci's recent best seller, Simple Genius, just your usual tale of the dark world of mathematicians, codes and spies. Gee, math has become such an integral part of novels, TV shows and movies over the past few years, our students are going to think the life of a mathematician is really cool and exciting (which, as we all know, it is!).

Anyway, here is a paraphrasing of the problem (as long as I'm not copying the problem verbatim, the publisher granted me permission to discuss this):

Alex is as many months old as his grandpa is in years and about as many days old as his dad is in weeks. If the sum of their 3 ages is 140, how old is each?

Hint: This is a wonderful problem demonstrating the power of ratios. If you can solve it less than 20 seconds, then you're either an honorary member of Mensa or you could be the subject of Baldacci's next book!

Comments

(1) Like all riddles, the wording is somewhat convoluted and the mathematical assumptions are not explicitly stated. But that's part of the intrigue here. I will say that one needs to assume the ages are integers, but that's about it.

(2) In the novel, the problem is posed to a young mathematical prodigy named Viggie. While another mathematician in the room takes some time to solve it algebraically, Viggie comes up with the solution mentally in a few seconds. Can you!

(3) You may want to give this to middle school students, although the wording might frustrate them. You could demonstrate the idea with a concrete example or make it into a simpler problem:
Let's say that Alex is 96 months old, then his grandpa would be 96 years old. Now ask them to determine how old Alex's dad would be. This may be challenging enough...

(4) I'm naturally wondering what the source of this problem is. If anyone out there recognizes it, let us know its source!

MATH WARS - Advice for Parents

2008-06-09T20:27:00.003-04:00

A few weeks ago I received a request for an interview from Jan Wilson, education columnist for The Parent Paper, a local publication here in northern NJ:

Hello Mr. Marain --

I am the education columnist for the Parent Paper and I am writing an article about math instruction, specifically about moving beyond the rhetoric of the math wars to discover which ways of instruction work best for K-5 students. It's going to be a fairly general article, designed for the parent who hasn't thought a lot about math before but becomes concerned because her child hasn't master times tables in 3rd grade, or isn't doing long division in 4th, for example.

To see the full article, look here.

[Note: You will need to go to pp. 42 and 43 for the actual article entitled, "MATH WARS."]

The following is the reply I sent Jan from which she excerpted a few of my comments:

I am speaking both as a parent and a mathematics specialist with over 35 years in mathematics education at all levels. I have also been publishing a blog for math educators for the past year and a half. It was recognized by the Washington Post as one of the Top 10 Educational Blogs for 2007. Despite all the rhetoric, there are no bad programs out there in my opinion. The reform programs like Everyday Math and TERC do a fine job of developing children's number and spatial sense and address problem-solving as well. However, the district has to be committed to the expectation that children will practice their basic facts on a daily basis. Some children will master their times tables by the end of 3rd or 4th, some later on. However, they should all be expected to learn it by the end of 4th, even if some youngsters will take longer. Parents should not hesitate to ask the teacher and/or the principal if these expectations are in place. Further, they should ask if children are given some form of practice both in class and at home on a regular basis. Some children will learn from flash cards, others need to write each fact 5-10 times, others can benefit from games or software. Excellent online games like Timez Attack from Big Brainz can be played both in school and at home. The free version is quite good, but it will not work for every child. The only constant is that the expectation of learning these basics is stated in the currlculum and that this philosophy is actually implemented. If children's learning of basic facts is assessed regularly in class, then one can reasonably assume there is follow-through. Sometimes the only way to be sure of this is to talk to parents of children who have been through the program. Just remember: Playing games and problem-solving do not replace the need for children to memorize their facts. There is no way to get around this. If the district math program does not incorporate sufficient practice, then parents will need to supplement fact practice at home, even if it's only 10-15 minutes a day. Each child learns his/her own way, but each child must do something every day, using a reward system if needed to motivate them. In addition to skills practice supplementing existing programs as needed, parents should ask what kinds of problem-solving materials are used. Is the source of these problems restricted to what is provided by the publisher or are other resources utilized? For example, are teachers provided with the problem books from Singapore Math, which generally contains more difficult challenges than are normally found in our texts. Another question parents can ask if there is a new program is, "How have or will teachers be trained?" Is there a full-time Staff Developer working with teachers? Is there a math specialist for K-5 or 5-8? Short-term staff development is not nearly as effective as ongoing training, both for experienced and new teachers. It is important for parents to be aware that NJ Ask and other state testing will be undergoing significant changes in response to the recommendations of the National Math Panel, NCTM and organizations such as Achieve. All of these groups are calling for a reduction each year in the number of topics covered so that there will be more time for children to work toward mastery of important skills AND to develop greater depth of understanding. Teachers will be able to devote more time to provide both enrichment and reinforcement. This is an exciting opportunity. The Math Wars have been fueled by extremists on both sides. In the end, our children need a more balanced math education which will incorporate the best of the traditional and reformed.

Remember, I was writing this primarily for parents. These comments represent the same positions I have taken for the past 20 years and for which I have been challenged (a euphemism for attacked) by both "sides" for the same 20 years! Hey, I may be wrong but at least I am consistent!

Mystery Mathematician June '08 - An Anagram!

2008-06-03T07:49:00.003-04:00

And now for something completely different...

A Math Mystery Anagram!

No picture this month!
If you like this variation, let me know...

"MEN-Y" THEOREM

No, it's not Monty Python, although the anagram contains 'Monty'! There are many anagram generators/solvers online so you could try those, although I did not find them useful here - I had to do this scramble myself. Please allow poetic license on the spelling of 'many' - perhaps there is a hidden clue there!

This mathematician was unique in so many ways - it is about time we paid proper tribute...

Our wordplay enthusiasts out there (and most math people are!) will probably solve this quickly but DON'T FORGET THE RULES:

DO NOT SUBMIT YOUR ANSWER AS A REPLY TO THIS POST (read on)

Remember to email me at dmarain at geeeemail dot com with
(PLEASE ENUMERATE!)

(1) The name of the mathematician
(2) Some interesting info/anecdotes re said person
(3) Your sources (links, etc.)
(4) Your full name and the name you want me to use when acknowledging your accomplishment
(5) If you're new, how you found MathNotations
(6) Your connection to mathematics

Clocks & Modular Arithmetic - A Middle School Investigation

2008-05-31T15:48:00.005-04:00

[Did you think MathNotations was on hiatus? Actually, I've been working on a couple of investigations including an intro to the mathematics of circular billiard tables and the activity below -- hope you enjoy it...]

MathNotations has been invited to submit an article to Connect magazine. I'm considering something along the lines of the following investigation (the article would contain fuller explanations and additional teacher guidelines) and I would appreciate feedback particularly from middle school teachers. Feel free to suggest revisions, improvements, ...

If you have the time, as we approach the end of the school year, to implement some or all of the following, I would appreciate your observations. Also, what classroom organization (e.g., individual vs. small group) you used or what you would recommend. Thank you...

NOTE TO READERS OF MATH NOTATION: Your challenge is at the bottom!

CLOCK INVESTIGATION
Students are provided a handout with several clocks, numbered in the standard way from 1 through 12.

LEARNING OBJECTIVES/STANDARDS/TOPICS

Divisibility concepts (remainders, lcm, factors)
Repeating patterns (introduction to periodicity)
NOTE: Later on, when students study the unit circle in trigonometry, they will encounter similar periodic behavior.
Organizing data
Developing effective communication - writing in mathematics

Part I
Place a marker at 3:00. This will be your START position. For the first part of this activity, you will be moving your marker FOUR hour-spaces in a clockwise direction from your starting point. So after your first move, you will be on 7:00. With your partner, record the results of each move up to 15 moves. You could of course mark it directly on the clock or you could make a table such as:

Start....3:00
Number of Move (N).................Position
1......................................................7:00
2......................................................11:00
...
15

Note: It's good experience for students to see that we often start indexing variables from zero, so instead of Start...3:00, one could start the table
0.....................................................3:00

Question 1: Try to answer the following without actually listing all the moves: What will the position of your marker be after 25 moves? 50 moves? 75 moves? 100 moves? Explain your reasoning or show your method.

Part II
Same starting point at 3:00, but this time you will move your marker FIVE spaces clockwise each time. Again, record the results of each move up to 15 moves.

Question 2: You should now have discovered that after 12 of these moves, you have returned to your starting point. Explain why at least 12 moves were needed (stating that you tried every move up to 12 isn't quite what we're looking for!).

Possible explanation (they may do better than this!): Starting position is repeated when the total number of hour-spaces moved is a multiple of 12. Since the the number of hour-spaces advanced after each move is also a multiple of 5, the position will repeat after 12 such moves. Note that 12⋅5 = 60 is the LCM of 12 and 5.

Question 3: Again, try to answer the following without actually listing all the moves:
What will the position of your marker be after 25 moves? 50 moves? 75 moves? 100 moves?
Explain your reasoning or show your method.

Question 4: In part I, you discovered that positions repeat after 3 moves. therefore, not all positions from 1 through 12 are reached. In Part II, you probably noticed that every location is reached. Explain both of these results in terms of divisibility.

Question 5: In both parts you started at 3:00. What results would be the same if you started from the 12:00 position? What results would be different?

Question 6: Devise at least one variation of your own for these clock problems. Extra points for most creative!
Sample: In addition to the obvious (changing starting position or number of spaces moved, you may want them to consider moving counterclockwise or changing the clock itself to 13 hours or some other variation).

Note: Students do not often consider generalizations (see challenge below) using variables to represent starting positions or the number of spaces moved each time. Middle schoolers may benefit from an introduction to such generalizations. I recommend only varying one of the parameters (either starting position or spaces). This would be appropriate for the prealgebra or more advanced student.

CHALLENGE TO READERS OF MATH NOTATION
Try to develop a general formula for the position of the marker after N moves given an initial position (S), number of hours on the clock (H) and the number of spaces moved (M). Also, an expression for the least number of such moves required to return to one's start position.

Geometric "Connections" - How one problem leads to another...

2008-05-20T18:20:00.008-04:00

Answers to Problems from Previous Post:

(a) Substitute 0 for x, -1 for y in both equations.

(b) a > 1/2

(c) Two points above x-axis: a > 1; Below x-axis: 1 > a > 1/2; On x-axis: a = 1
Note: If 1/2 > a > 0, then the only point of intersection would be (0,-1).

(d) x = ±[√(2a-1)]/a; y = (a-1)/a

(e) Points: (±4/5 , 3/5); a = 5/2
Note: Pls check for accuracy!

Now for the connection...

In the previous post, we were given that the radius of the circle was 1 and the area of the triangle was 32/25. From this it can be shown that PQ = 8/5 and, in fact, the quadrilateral PQRS shown in the figure at the left is a square whose area is 64/25. The fact that this was a square intrigued me. I hypothesized that, up to similarity, these numbers were unique. This led me to the diagram at the left and the following converse of the previous problem.
Note: This problem is now unrelated to the parabola.

In the diagram above, points P and Q are on the circle, PQRS is a square and segment SR is tangent to the circle at T.
If the radius of the circle is r, show that the area of the square is (64/25)r², and, consequently, the area of ΔPQT = (32/25)r².

Comments:
(1) Students should not find this overly challenging using standard methods for solving circle problems (and the fact that it is closely related to the previous question).
(2) Of course, what really intrigued me is how, once again, the 3-4-5 triangle recurs! Ask your students to find a triangle in the diagram similar to 3-4-5. They need to draw something but this should occur naturally from the standard solution to the problem.

When Curves Collide Part II - Quadratic Systems Re-Explored!

2008-05-15T09:12:00.003-04:00

One of MathNotations more popular posts (hundreds of views) was published one year ago this week: When Curves Collide.

Here's a variation to review the essential ideas or to use as an assessment problem or just to challenge yourself. Parts (a) thru (d) require some theoretical analysis and algebraic skill. Part (e) is the main challenge...

An Investigation for Algebra 2/Precalculus
Consider the quadratic-quadratic system:

x² + y² = 1
y = ax² -1, a>0

(a) Show that (0,-1) is always a solution to this system.

(b) For what values of the parameter 'a' will there be 3 distinct solutions to the system?
Coordinate Interpretation: For what values of 'a' will the parabola and circle intersect in 3 distinct points?

(c) For what value(s) of the parameter 'a' will two of the points of intersection be above the x-axis? Below the x-axis (in addition to (0,-1))? On the x-axis?

(d) For the case that there are 3 distinct solutions, determine the two solutions, other than (0,-1), in terms of 'a'.

(e) Now for the main problem:

Assume the graph of our system has three points of intersection: P, Q and R(0,-1). If the area of ΔPQR is 32/25, determine the coordinates of P and Q and the value of 'a'.

(f) Can you think of an even more clever variation!

August Ferdinand Mobius - Mystery Math Man for May Revealed

2008-05-14T06:07:00.004-04:00

As promised, the contest for May ends around the 15th. Before sharing my own thoughts about Mr. Möbius, I will highlight our three winners for this month:

Susan Hoover

1. Our mystery mathematician (and astronomer!) is August Ferdinand Möbius.

2. He started out studying law because that's what his family wanted, but it was not to his liking, so he switched to mathematics, astronomy, and physics. He studied astronomy under Gauss and mathematics under Gauss's teacher Pfaff. Although his doctorate and his first academic posts were in the field of astronomy, his later, more famous, work is in mathematics, particularly topology and analytic geometry. His name is given to the single-faced, single-edged, two-dimensional surface known as the Möbius strip, although actual discovery and publication of that strip were by Listing.

3. Sources: http://www-history.mcs.st-andrews.ac.uk/Biographies/Mobius.html
http://www.britannica.com/eb/article-9053115/August-Ferdinand-Mobius
http://www.genealogy.math.ndsu.nodak.edu/id.php?id=18230

Erica Clay

1. August Mobius

2. "Before going out for a walk, he [Mobius] recited the German
formula "3S und Gut" composed of the initial letters of the objects
that he absolutely did not want to forget: Schlüssel (key), Schirm
(umbrella), Sacktuch (handkerchief), Geld (money), Uhr (watch),
Taschenbuch (notebook)."

3. http://scienceworld.wolfram.com/biography/Moebius.html

TC

The mystery mathematician is the most "one-sided" mathematician I have
heard of, i.e., Mobius.
Interestingly, when reading his biography on "Mathtutor: History of
Mathematics," I found that the Mobius Strip was actually invented by
Listing.

Congratulations to our three winners. Certainly, there are always some fascinating facts or anecdotes that surface when one delves into the backgrounds of these legends of math. I was particularly impressed by Möbius' initial interest in astronomy and that fact that his mentor was someone named Gauss! In addition to his contributions to topology, he also did research in number theory and his name is attached to some important concepts (Möbius Function and Möbius Inversion Formula).

A quote that revealed much to me about the nature of this extraordinary person came from his biographer:

The inspirations for his research he found mostly in the rich well of his own original mind. His intuition, the problems he set himself, and the solutions that he found, all exhibit something extraordinarily ingenious, something original in an uncontrived way. He worked without hurrying, quietly on his own. His work remained almost locked away until everything had been put into its proper place. Without rushing, without pomposity and without arrogance, he waited until the fruits of his mind matured. Only after such a wait did he publish his perfected works...

Components of the Effective (Math) Lesson Gr 5-12 - Part I

2008-05-12T14:10:00.006-04:00

One of the reasons I began this blog was to share the collective wisdom of experienced math teachers as a benefit to the novice. Well, here I am 18 months into MathNotations and I don't believe this has yet been specifically addressed. I expect the comments or follow-up posts to be even more beneficial than what I'm writing below.

Here's what I'm asking my readers --
In this post, I will begin enumerating one or two instructional components which I believe should be an integral part of most (math) lessons. Since I have strong antipathy towards jargon, I will try to avoid technical phrases like 'set', 'hook', although closure is ok.

Note that I put math in (..) to emphasize the point that I regard many of these suggestions as integral to effective lessons in general!

Note: These lesson components should be independent of teacher style, makeup of the class, content, etc.

Background
I do know that newbies often feel overwhelmed by all of the differing expectations coming from their immediate supervisor, colleagues, principal, other administrators, courses of study/syllabi, district technology initiatives, state standards, state standards, NCTM Standards/Curriculum Focal Points, standardized test specs -- just to name a few! I haven't even mentioned what they learned from their methods classes, the influence of their math teachers in their formative years, advice from just about everybody. When all is said and done, it seems that the number one concern on the part of most evaluators in the beginning is classroom management, effective delivery of content being number two. Of course, evidence of content knowledge becomes of greater importance if there is an immediate supervisor who has math certification.

How does one navigate through this morass without losing one's mind? Prioritize! Less really is more! Rather than attempt to build the perfect lesson to please the observer, be guided by what you know will lead to demonstrable evidence of learning. Yes, planning is critical. I will comment on that further.

Here then is just the beginning of what I expect to be an extended discussion and one which I am considering publishing as a pamphlet. Please adhere to the Creative Commons License in the sidebar if reproducing any of this.

DISCLAIMER
I am stating unequivocally that these are my own personal ideas of what makes an effective math lesson. I do not want anyone to say that I am telling anyone how to teach!

Each of you out there will have your own list, although I'd be surprised if there wasn't considerable overlap. The order of course will vary. These are the principles by which I was guided both as a classroom teacher and as a supervisor. At the beginning of the year, I would meet with teachers to discuss what I was looking for in the lesson. For clinical observations, I would also have a preconference to discuss specifics. This was particularly of critical importance before observing the non-tenured teacher.

THE BEGINNING
1) Class Opener - Critical first 5 minutes - Establishment of Routines

a) Allow students to socialize/decompress for a couple of minutes as they enter, but let them know what is expected of them; close door at late bell. Establish iron-clad routines for students to follow if they arrive after that - stick to it!

b) Math Warmup/Problem of the Day already on the board or projected on a screen using the overhead or PowerPoint (or Word) from the computer; the warmup can be used to review prerequisite skills for the upcoming lesson, SAT review, an opportunity for students to practice their communication (e.g., writing) skills in math, etc.

c) Answers to some or all of the homework exercises can be written on the board or projected on a screen from overhead or computer. Virtually every publisher of current texts provides ready-made transparencies both for WarmUps and answers to homework, not to mention PowerPoint presentations for every lesson! Some educators object to displaying answers like this as it invites students to quickly copy these on their paper. You may want to have selected answers displayed rather than all. There is no foolproof method here, so use your own judgment. The important thing is to busily engage students from the outset. While students are working on their warmup problem, the teacher is circulating, checking homework and engaging students. This personal interaction with students means so much (e.g., Lily, I saw you in the play on Thu night -awesome!).

Ok, folks, this is just a beginning...
Please contribute your suggestions!

A Very Simple Alphametic Message to Mom

2008-05-11T06:21:00.003-04:00

OX
XO
--------
MOM

Please make allowances for the spacing and my crude attempt to produce an alphametic for Mom's Day. The 2nd letter 'M' is supposed to be aligned under the 'X' and 'O', etc. For those unfamiliar with the definition and rules of alphametics, here is some information I copied from Mike Keith's wonderful site:

An alphametic is a peculiar type of mathematical puzzle, in which a set of words is written down in the form of an ordinary "long-hand" addition sum, and it is required that the letters of the alphabet be replaced with decimal digits so that the result is a valid arithmetic sum. For an example one can do no better than the first modern alphametic, published by the great puzzlist H.E. Dudeney in the July 1924 issue of Strand Magazine:



SEND
MORE
-----
MONEY

whose (unique) solution is:



9567
1085
-----
10652

There are two fairly obvious (but worth stating) rules which every alphametic obeys:

1. The mapping of letters to numbers is one-to-one. That is, the same letter always stands for the same digit, and the same digit is always represented by the same letter.

2. The digit zero is not allowed to appear as the left-most digit in any of the addends or the sum.

You may recall that on Pi Day, I linked my readers to Mike Keith's extraordinary opus, Poe, E.: Near A Raven. Mike is more than a Poe and Pi devotee, however, as the link above demonstrates. Another excellent site providing numerous examples of alphametics is Truman Collins' fascinating page.
I strongly urge my readers to visit both of these sites. There are enough puzzles there to keep you busy for decades!

Multiple Representations (Rule of 4) in Algebra 2 or Precalculus

2008-05-07T22:58:00.005-04:00

Did you overlook our Mystery Mathematician for May? I've received two correct responses thus far, but submissions can still be emailed until the 15th of the month. Don't forget to include the info requested in a previous post.

If (A+3) ÷ (B+5) ≥ 10 and B ≥ 7,
what is the least possible value of A?

DISCUSSION

The use of multiple representations of a concept or procedure in mathematics is highly recommended by NCTM and other math education experts. Also known as the Rule of Four, it suggests that instructors use some or all of the following, when introducing a new concept. This requires careful planning and considerable thought on the part of the teacher. Over time and with experience, it will flow. However, it does help to see many models of this heuristic for geometry, algebra, etc.

The Rule of Four suggests that a concept be presented
(a) Using natural language (words)
(b) Numerically (concrete examples, 'plugging in', use of data tables, etc.)
(c) Visually (e.g., using graphs, charts, concrete models)
(d) Symbolically (algebraical mode)

From my experience, many students will approach the problem at the top by ignoring the inequalities and simply plug in 7 for B. They've learned that this strategy usually works on standardized tests. It is our role as educators to challenge them to think more deeply. Create disequilibrium by provoking them with a question like,
"But to make a fraction small, don't you need to make the denominator as large as possible?" Of course this statement does not apply to this problem, but I'll wager that it would cause some to reconsider their initial answer!

Do you think that most students would quickly recognize that the relationship between A and B can be described by a linear inequality, which can be then be approached both algebraically and graphically? Do you think I need strong medication for asking you that question!

To deepen their understanding, one could ask:
How would you have to change the above problem so that one could ask for the greatest possible value of A?

I plan on posting further examples of the Rule of Four. I am aware that I have not fully demonstrated this technique for the problem above. I'm only hinting at it. More will likely come out in the comments...

A Geometry Tribute to Cinco de Mayo

2008-05-04T08:20:00.009-04:00

Correction: Jonathan pointed out that I did not specify the order of the vertices. Thanks, Jonathan! Here is the revised version in which A and C are opposite vertices as are B and D:

Consider parallelogram ABCD, three of whose vertices are A(0,0), B(2,3) and D(3,2).

Find the coordinates of C and the area.

Of course, we expect our Geometry students to celebrate even more by generalizing:

Note: This has been revised for the reasons stated in the correction at the top.

Three of the vertices of a parallelogram ABCD are A(0,0), B(a,b) and D(b,a), where b>a>0.

(a) Show that vertex C has coordinates (b+a,b+a).

(b) Prove that this figure is actually a rhombus.

(c) Show that its area is b² - a². Can you find FIVE ways? (ok, that's a stretch but anything is possible on May 5th!).

Coordinate Triangle Problem - Interface between Algebra and Geometry

2008-05-02T20:23:00.006-04:00

For Geometry or Algebra 2 students or anyone who wants a diversion...

The vertices of ΔABC are A(m,2k), B(k+11,k-2) and C(2k+6,k-2). The area of the triangle is 15.

(a) What restrictions need to be placed on k to insure there is a triangle.
(b) Given those restrictions, determine all possible values for k.

Comments:
(1) This is not intended to be a highly challenging problem. It can be used as review for a final exam, standardized tests, SATs, etc. Of course, on the SAT, the question would only ask students to grid-in one possible answer and would not generally ask about restrictions.
(2) You may want to ask your students why the value of m is irrelevant.
(3) There are two possible values for k in this problem. Challenge your students to write a revised version that would have more possibilities. Would the coordinates have to involve quadratic expressions in k?
(4) if anyone tries this in the classroom, please let us know how it went, specifically, student reaction. How was it implemented? As a warm-up, extra challenge at end of class, part of homework assignment, extra credit?

Mystery Mathematician #11 for May

2008-05-01T06:37:00.003-04:00

For now, we will be doing a monthly version of our International Math Idol (open to suggestions for naming the contest since I keep changing it!).

DO NOT SUBMIT YOUR ANSWER AS A REPLY TO THIS POST (read on)

Remember to email me at dmarain at gmail dot com with
(1) The name of the mathematician
(2) Some interesting info re said person
(3) Your source (links, etc.)
(4) Your full name and the name you want me to use when acknowledging your accomplishment
(5) If you're new, how you found MathNotations
(6) Your connection to mathematics

Our icon this month was multi-faceted but was best known for the opposite of that description!