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Based on a reading of Google searches of visitors to this site, here are a few of the more common topics, rephrased as questions, that I have noticed. I also noted that, aside from pi day, which generated hundreds of visits from those looking for historical information about pi and the names of mathematicians, many are looking for sample MathCounts problems or references to books of these.


1. What is the largest 3-digit prime with 3 prime digits?
Ans: 773 if repeated digits are allowed; 523 if not. This question has appeared in one of my earlier posts. Must be a fairly common one that readers come across from math contests or from class. A quick list of primes up to 1000 can be found at VIAS Encyclopedia.

2. How are the following questions related:
How many different handshakes occur if each of 10 people shakes hands with each of the remaining people in a room? (can be expressed more clearly)
How many different segments can be formed by connecting the vertices of a decagon in all possible ways? (can also be expressed in terms of the number of chords formed by 10 points on a circle)
Ans: Both problems can be expressed as 9+8+7+6+...+2+1 or (9)(10)/2 or 10C2, the number of combinations of 10 objects chosen 2 at a time. In general, for n people or points on a circle: 1+2+3+4+...+(n-1) = (n-1)(n)/2 = nC2. The equivalence comes from the fact that each handshake or each chord is uniquely determined by selecting 2 people or 2 points. To avoid repetition we use combinations rather than permutations.

4. How many different seven-digit phone numbers (ignoring the area code) can be formed?
Ans: If any digit 0-9 were allowed in all positions then there would be 10⁷ possibilities since there are 10 choices for each digit (using the Multiplication Principle here). Subject to restrictions on zeros or other digits or other local considerations there would be fewer.
Here's a related question that I have not yet seen:
How many different IP addresses are possible worldwide if every computer, device etc., must have a unique one?
I believe that IP addresses are always of the form xxx.xxx.xxx where the first digit in each group is allowed to be zero, that is, one could use a 2-digit number in each group. For example, 66.19.35 is acceptable. I didn't research the rules so this may be incorrect.
Ans: If the rule of formation is accurate, there could be 10⁹ or one billion IP addresses. How long will it take for these to be used up? I know someone out there will tell us how this is or will be handled!

Update on IP Addresses: My belief about the form of IP addresses was dead wrong! The protocol should have been 4 groups of integers, each in the range from 0 through 255. There are now newer protocols to allow for the exponential growth of devices needing an address. See the comments for this post to learn more from those far more knowledgeable than myself!

Developing Algebraic Reasoning

The following sequence of problems deals with a fairly well-known pattern. Similar questions have appeared on SATs, on other standardized tests and in texts. The intent here is to provide an extended activity for students of diverse math backgrounds and abilities to develop a systematic approach to analyzing patterns. Students should also be encouraged to make a table of values in which the first column is the number of 'crosses' and remaining columns are reserved for other 'dependent' variables. This function-based approach is also an essential feature of this development.

Notes: There are many ways to approach these questions. Encourage students to share theirs! These questions involve pattern-based thinking, combinatorics, recursive sequences, arithmetic sequences and algebraic reasoning. Parts (d) and (e) are more challenging for some. Based on the pattern of the first 3 or 4 terms, some students will simply develop a linear formula of the form aN+b for the perimeter (which is somewhat harder than the area). It is important for our prealgebra and algebra students to recognize that any arithmetic sequence like 12,20,28,36,… can be described this way. My experience is that if a class has 20 students, there will be at least 5 different ‘counting’ methods discussed, Students often are very creative here and not all use ‘linear’ thinking!

The Genius of Archimedes: Parabolas, Tangents...

Pi day is over, but it seems fitting to continue exploring. Archimedes did more than develop an approximation procedure for pi! There are many excellent websites that explain the following in greater detail and discuss many more of Archimedes' theorems about parabolas and tangents. I attempted to draw a diagram using Draw in Word. It's crude but you'll get the idea. The object is to share this extraordinary piece of history of mathematics and have your students finish the proof that a light ray from the focus that strikes a parabolic surface is reflected in a ray that is parallel to the axis of the parabola. This is equally interesting in reverse: External light rays and other forms of electromagnetic radiation that are parallel to a parabola's axis are reflected to the focus, very useful for radar and other 'collection' devices.
Considering that Archimedes' proofs used only geometric properties makes his work even more astounding (now of course we can use coordinate geometry, calculus, etc.). This type of investigation is usually deferred to College Geometry courses, but I believe we can deliver it to motivated geometry, 2nd year algebra or precalculus students. If nothing else, it makes for a wonderful long-term project!

Ok, here goes...

In the diagram below, I've gone out of my way to make the reflecting ray NOT look parallel to the axis, even though we're trying to prove it is. This is to help students avoid assuming collinearity, when, in fact, that needs to be proved!

The two angles marked X are equal by a reflection principle (angle of incidence equals...). The two angles marked Y are equal because it can be proved that the tangent line at P is the perpendicular bisector of segment FP', where F is the focus and P' is the foot of the perpendicular from P to the directrix. I chose not to derive Archimedes' very subtle argument, but it is worth studying the proof. The proof starts by constructing the perpendicular bisector and showing that this line passes through P but no other point of the parabola, thus it is tangent. Alex Bogomolny's excellent and in-depth treatment (with java applets) of this topic (on cut-the-knot) is very worthwhile reading.

The student is being asked to prove that the reflecting ray is parallel to the axis. This is equivalent to showing that the line containing PP' and the reflecting ray are one and the same. The argument is straightforward, but students may want to continue learning more about the genius of Archimedes.

[Good luck copying this diagram (jpg). Some of you may find errors in my argument or an extremely simply argument for the parallelism, so pls share!!]

Advanced Algebra Challenge

Note: Many of the ideas below came from the excellent note in the Reader Reflections of the March 2007 Mathematics Teacher, contributed by Warren Groskreutz. One of his students derived and proved his own 'theorem' about certain radical expressions. I decided to develop these ideas into an activity. The last part is a bit different from 'Nathan's Theorem.' Mr. Groskreutz is to be commended for creating an environment in which such 'discoveries' can be made.

[Click on the small image below to enlarge.]

A Collection of Pi Poems...

Here are some Pi Poems from our students. They’re not ready for publication yet, but our teachers enjoyed these. I’ve removed their names of course. If you have student work you’d like to share, pls do so!

See Eric Jablow's comment on previous post for a very clever one!

How I love a sweet chocolate pi Sunday after noon =
3.141592654

It's a pain I wasn't competent at ratios, sines, and
stuff.

Now I want a puppy, dalmation, or poodle that's cute.

How I love a tasty delicious pi! Eating apple pie feels
dynamite, fantastic, perfect, excellent, and so fun! =
3.14159265358979323

Now I know a great geologist on planet Earth.

How I wish I could regularly be around great math!

This one unfortunately has an error in one of
the digits but I thought it was a really great effort
nonetheless...
Pie I like a peach blueberry or banana cream and lemon
meringue rasberry rhubarb mincemeat pie in sky apple ala
mode cherry or humble mud or pumpkin chocolate pecan oh
ruin your appetite

Wed 3-14 A Pi-Fect Day!

3.141592653589793238462643383279502884197169399375105820
97494459230781640...

Don't forget to celebrate Pi Day on 3-14-07 at 1:59 PM!! Considering how many out there are searching for information about Pi, this has become a huge event. Although we may not be quite ready for an official national holiday, many now refer to 3-14 as National Pi Day!

Here are three of the best Pi-Links I have found. You may want to visit them to learn more about one of mathematics most fascinating numbers - more than you may ever have wanted to know! The Wikipedia Pi article is also wonderful.

Exploratorium

A History of Pi

The Pi-Search Page

Enjoy the day but don't forget that in many other countries, Pi Day is celebrated on July 22nd in honor of the approximation 22/7! Personally, I've always been 'partial' to 355/113.

Parabolas, SATs, Quadratic Functions, Symmetry, Oh My!

The new SAT and other state math assessments are or will be including more Algebra 2 types of questions, particularly those involving quadratic functions. The following was inspired by a recent SAT math problem. As usual, my goal here is not to give conundrums and 'puzzlers'. I'll leave that to the expertise of Jonathan over at jd2718! My intent is to provide enrichment and extensions of questions that students are doing in class. More time is required for these than is normally given for an example presented by the teacher. Hopefully these can be used in the classroom.
The original question on the SAT gave a particular length for segment PQ (see below) and that may be a more reasonable start for most Algebra 2 students. The objective here is to have students apply and extend their knowledge of quadratic functions, graphs, coordinates, symmetry, etc. There are several approaches to this question. If instruction enables students to investigate this problem for 10-15 minutes, students may discover alternate methods that will deepen their understanding of the material. The teacher's role is to gauge the ability level and background of the group to determine how much structure/guidance is needed. This is not obvious at all and requires considerable pedagogical skill and experience.

Consider the graph of the quadratic function f(x) = x². Assume P, Q are points on the graph so that segment PQ is parallel to the x-axis and let the length of segment PQ be denoted by 2k.
If the graph of g(x) = b - x², intersects the graph of f(x) at P and Q, express the value of b in terms of k.

Notes:
Encourage several methods, i.e., pair students and require that they find at least two different methods. This is critical to develop that quick thinker who always has the answer before anyone else and does not want to deepen his/her insight. Many students will need to start with a numerical value for the length of segment PQ, say 4. Symmetry is a key idea in this problem, not only with respect to the y-axis, but also with respect to segment PQ! Some will see this quickly, others won't. It is our obligation to think this through in advance and be prepared to guide the investigation. Those who believe this kind of activity is a waste of precious time (so much more content could be covered) will never understand why I believe 'less is more' when it comes to learning math. Profound understanding can never be rushed. Short-cuts, IMO, are PART of a discussion, not the objective. Try it! Can you find at least THREE ways?

A Pi Day Scavenger Hunt

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 multiple choice, I/II/III case type 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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Note: This link will get you to all of the Pi Day posts/activities on MathNotations.


Here's an idea from one of the outstanding teachers in my department. We will be implementing it on Pi Day this year. Students have been notified in their math classes and on the PA of the following:
On 3-14, there will be questions about various historical decimal approximations to pi posted around the building. Students will have to find these questions, determine the answers and submit their results by the end of the day. In case of many ties, there will be a drawing to determine who wins the free Pizza Pi certificates! We will raise money for the prizes by baking pies and selling them on Mon and Tue in our cafeteria. We are using a wonderful book about Pi for our source of these approximations but I won't mention it yet in case some of our students are visiting this blog (I don't keep it a secret!). I might be slightly off in some of the details but you got the idea.

I've noted that many visitors to this blog are searching for engaging pi day activities. There are many out there already but educators are always looking for fresh ideas. It would be great if you could share these...

By the way, the Pi Activity worksheet I posted yesterday worked fairly well. Once the students got into it, they seemed to enjoy it. Read the comments regarding how they searched for the mathematicians whose birthdays were given. Interesting stuff regarding how some students know how to use 'refined search' techniques in Google but many don't.