Clocks & Modular Arithmetic - A Middle School Investigation

[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...

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!

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

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

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

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

If interested in purchasing my NEW 2012 Math Challenge Problem/Quiz book, click on BUY NOW at top of right sidebar.  175 problems divided into 35 quizzes with answers at back. Suitable for SAT/Math Contest/Math I/II Subject Tests and Daily/Weekly Problems of the Day. Includes both multiple choice and constructed response items.
Price is $9.95. Secured pdf will be emailed when purchase is verified. DON'T FORGET TO SEND ME AN EMAIL (dmarain "at gmail dot com") FIRST SO THAT I CAN SEND THE ATTACHMENT!

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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

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

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

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!