Too Much 'Time on Your Hands?' Classic Clock Problems as an Application of RxT = D

Have you submitted your vote yet in the MathNotations poll in the sidebar?

Target audience for this investigation: Our readers and algebra students (advanced prealgebra students can sometimes find a clever way to solve these).

Let's resurrect for the moment those ever popular rate/time/distance classics. Hang in there -- there's a more interesting purpose here!

We'll start by using fictitious presidential candidates running in a 'race.' Any resemblance to actual candidates is purely coincidental.

R and J are running on a huge circular track. J can run a lap in one month whereas it takes R twelve months to run the same lap. To be nice, J gives R a 3-month head start. After how many months will J 'catch up' to (overtake) R?

Are those of us who were trained to solve these feeling a bit nostalgic? Do you believe that our current generation of students has had the same exposure to these kinds of 'motion' problems or have most of these been relegated to the scrap heap of non-real world problems that serve no useful purpose. Well, they still appear on the SATs, a weak excuse for teaching them, perhaps, but I can certainly see other benefits from solving these. Can you?

Ok, there are many approaches to the problem above. Scroll down a ways to see a couple of methods (don't look at these yet if you want to try it on your own):







Method I: Standard Approach (using chart)

..............RATE ...x.......TIME .....=.....DISTANCE
............(laps/mo).....(months)............(laps)

R.........1/12....................t........................t/12

J................1....................t...........................t

Equation Model (verbal): At the instant when J 'catches up' to R:
Distance (laps) covered by J = Head Start + Distance covered by R

Equation: t = 1/4 + t/12 [Note: The 1/4 comes from the fact that R covers 1/4 of a lap in 3 months]

Solving: 12t = 3 +t --> 11t = 3 --> t = 3/11 months.

Check:
In 3/11 months, J covers 3/11 of a lap.
In the same time, R covers (1/12) (3/11) = 1/44 lap. Adding the extra 1/4 lap, we have 1/4 + 1/44 = 12/44 = 3/11. Check!

[Of course, we all know these fractions would present as much difficulty for students as the setup of the problem, but we won't go there, will we!]

Method II: Relativity Approach
Ever notice when you're zipping along at 65 mph and the car in the next lane is going the same speed, it appears from your vehicle that the other car is not moving, that is, its speed relative to yours is zero! However, if you're traveling at 65 mph and the vehicle in front is going 75 mph, the distance between the 2 cars is ever increasing. In fact, the speedier vehicle will gain 10 miles each hour! This 75-65 calculation is really a vector calculation of course, but, in relativity terms, one can think of it this way:
From the point of view of a passenger in the the slower vehicle, that person is not moving (speed is zero) and the faster vehicle is going 10 miles per hour. We can say the relative speeds are 0 and 10 mph.

Ok, let's apply that to the 'race':

If R's relative speed is regarded as zero, then J's relative speed will be 1 - 1/12 = 11/12 laps/month.
Since R is not 'moving', J only needs to cover the head-start distance to catch up:
(11/12)t = 1/4 --> t = (1/4)(12/11) = 3/11 months. Check!
[Note: Like any higher-order abstract approach, some students will latch on to this immediately and others will have that glazed look in their eyes. It may take some time for the ideas to set in. This method is just an option...]

There are other methods one could devise, particularly if we change the units (e.g., working in degrees rather than laps). Have you figured out how all of this will be related to those famous clock problems? Helping students make connections is not an easy task. One has to plan for this as opposed to hoping it will happen fortuitously.

Here is the analogous problem for clocks:

At exactly what time between 3:00 and 4:00, will the hour and minute hands of a clock be together?

Notes:
(1) I will not post an answer or solution at this time. I'm sure the correct answers and alternate methods will soon appear in the comments.
(2) A single problem like this does not an investigation make. How might one extend or generalize this question? Again, these are well-known problems and I'm sure many of you have seen numerous variations on clock problems. Share your favorites!
(3) Isn't it nice that analog watches have come back into fashion so we can recycle these wonderful word problems!
(4) For many problem-solvers, part of the difficulty with clock problems is deciding what units to use for distance (rotations, minute-spaces, some measure of arc length, degrees, etc.). This is a critical issue and some time is needed to explore different choices here.

My Favorite Ole and Lena Math Joke

Have you submitted your vote yet in the MathNotations poll in the sidebar?

A few years ago we visited our niece and nephew in Minnesota and fell under the magical spell of the land of 1000 lakes. Minnesotan's self-deprecating humor is refreshing, innocent and, as corny as it may appear, I am proud to say I enjoy it!

There are even websites you can visit to read Ole and Lena jokes which have been around since the first immigrants arrived.

Here is one of my favorites from the book, Ole and Lena, Live via satellite, Adventure Publications (in Scandinavian vernacular):

Littler Ole: Our math teacher tought she could stump us with a hard question. She asked, "Dis may take you a vhile to figure, but how many seconds are dere in a year?"

Lena: Dat must haf taken a lot of figuring, Little Ole.

Littler Ole: No, I hardly had to tink. I told her twelve.

Lena: Twelve?

Littler Ole: January 2nd, February 2nd, March 2nd,...

Ok, stop groaning now and admit it made you smile!

Why a Poll? Why Now? A Significant Sample...

One of the primary reasons I started this poll was to provide K-12 educators an opportunity to express their opinion about a central issue in mathematics education: the teaching of algorithms, multiplication in particular, as well as the issue of the expectation of mastery.

Why now? The work of the National Mathematics Advisory Panel is essentially complete and their recommendations will soon be published. MathNotation's readers for the past year know that I sent numerous emails, repeatedly urging the directors of the panel to include several practicing K-12 classroom teachers on this panel, if only in an ex officio capacity. All such requests were respectfully denied. I published both my emails and the replies of the panel in full on this blog. Classroom teachers were not able to provide direct input to the Panel because they were not represented, their voices were not heard, other than the lone voice of a single 8th grade teacher who was invited to be on the panel. If interested, go to the Labels section in the sidebar and read the (5) posts tagged with National Math Panel.

Surely, the Panel, with the cooperation of NCTM, could have developed a survey of our nation's math teachers. A survey which would have allowed tens of thousands of educators to express their knowledgeable opinions of what they believe is best for the mathematics education of their students. But this did not happen...

This was an important part of why I created this apparently insignificant little poll, consisting of just one question among the many which need to be asked and answered. Just a drop in the bucket, but I believe it is a central question for our children. It addresses perhaps the heart of the Reform vs. Traditional debate in our country.

This poll is intended for all visitors, whether they be parents, students, scientists, research mathematicians, administrators, curriculum specialists, classroom educators or anyone else who cares about the future of our children. However, I particularly urge the classroom teacher of mathematics not only to use this forum to express their feelings but also to encourage their colleagues to vote. A statistically significant sample here can be used by anyone who may need data to inform curricular decisions. Perhaps others will heed the message too.

The poll ends on February 29th. Keep it alive - spread the word! We're receiving a steady trickle of votes but we need many more. I won't yet comment on the majority opinion, however, you can see the results yourself when you submit your vote.

Mystery Mathematician Week of 1-21-08 Revealed

DON'T FORGET TO VOTE IN THE POLL. THE RESULTS ARE BEGINNING TO MULTIPLY...

And, once again, the winner is
Lynx

Apparently, the clues I gave (RU, groups and simple) helped a bit. Being a Rutgers alumnus, I wanted to acknowledge one of its greatest research professors.

Here is her contribution...

Daniel Gorenstein taught himself calculus when he was 12 and taught math to army personnel during WWII. He successfully steered the worldwide project on classification of finite simple groups. He was a leader at Rutgers University, and they created a faculty award in his memory. The award is given to a faculty member who is outstanding in scholarly achievement as well as service to the university (Gorenstein was known for both).

Alas, I was unable to find any interesting anecdotes, like some of the other mathematicians have. He seems to have gotten all well with people.

--Lynx

Reminder: Be part of the NEW MathNotations Poll on Multiplication!

I'm posting this again (and I will probably be posting frequent updates over the next 30 days) to make sure everyone knows there is now a poll in the sidebar! The original post explaining this is here. Please choose the option that most closely matches your feelings about which multiplication algorithm(s) should be taught in Grades 3-5 as well as the issue of mastery.

I believe you're only allowed to vote once. After you submit the vote, the current tally is updated and the results appear in the sidebar in place of the survey options (only snippets may appear). Pls let me know if you're having difficulty reading the 4 options before voting.

This poll is an opportunity for your voice to be heard regarding a critical issue in mathematics education. I hope you will participate.

43x27? Which Algorithm should be taught? A MathNotations survey will end the debate!

Update:
A poll has now been created in the sidebar! Please express your preference. (Thank you, Jonathan, for making this suggestion).

As if the educational fate of our children could be determined by a poll of our readers...
Hey, you never know.

What's even more incredible is that individual teachers, schools, districts, and states still feel that this is a 'local' decision. After over two decades of this debate, children and their parents still have little to say about these kinds of curricular decisions that will impact on the mathematical futures of another generation. After all, the experts know what is best, right? Well, there are many many experts who all feel sure they know the answer to this question. Trouble is, they do not all draw similar conclusions and spend most of their time defending their choices or their particular agenda or favorite set of materials.

Finally, MathNotations will settle this debate once and for all. The results of this survey (assuming a minimum of 3 responses) will be used to influence national policy for years to come!

Here are your options regarding your preference for how multidigit multiplication should be taught in Grades 3-5 :

(A) Teach only the traditional algorithm and expect mastery
(B) Teach the 'partial products' method to develop understanding of place value and the traditional algorithm; teach the traditional algorithm as a more efficient method and require it; expect mastery
(C) Teach the 'partial products' method to develop understanding of the traditional algorithm; teach the traditional algorithm as a more efficient method; give students a choice of methods; expect mastery of at least one method
(D) Model other methods (e.g., 'lattice method') and encourage students to invent their own method; do not require any particular method or mastery

Now, don't miss this opportunity to be part of an historic decision. Your vote does count...

A 'Boring' Volume Problem or "If You Find Yourself in a Hole, Stop Digging!"

Important Note: It took forever but I finally posted the detailed video explanation of this problem here.


Please don't gag on my feeble attempt at humor in the title (my wife actually had bought a sign with that quote -- it's hanging on the dining room wall).

There are a couple of classic volume problems in calculus which have always been my favorites:

• The Volume of the Torus Problem (using 2 methods: cylindrical shells and by disks)

• The Hole in the Sphere Problem (also by 2 methods)

I always assigned one or both of these to my BC Calculus classes, most often as Extra Credit problems. Because of the extra points they could earn, most students tried these and submitted solutions. My feeling is that if a student could do both of these by both methods, they really understood disk and shell!

In this post we will focus on the 2nd problem as it always seems to generate curiosity and interest. I'm guessing that most of you know the puzzle version of this question that was answered by Marilyn vos Savant in her Ask Marilyn column over a decade ago. It's just possible that some calculus student in some second semester class is feeling some anxiety over this problem!

Here's one version of that famous conundrum. There are many approaches here, even the clever mathematical approach of assuming that the problem is well-defined and therefore independent of the radii involved (I expect at least one of our readers to do it that way!).

A hole is drilled (bored) completely through a solid sphere, symmetrically through its center. If the resulting hole is 6 inches in height (or depth), show that the remaining volume must be 36π inches cubed.

That's right, the answer is independent of the radius of the sphere and the diameter of the hole! The total volume of the sphere and the volume removed however do depend on the radii. Note that the volume removed is a cylinder with two spherical caps.


The original problem was worded ambiguously in Marilyn's column and then clarified somewhat. My version is not perfect but hopefully you'll get the 'picture', although a real picture would be far better. I will probably do a video presentation of the solution and a discussion of the problem because the diagram and the math expressions are cumbersome and it's not worth the time to play with Draw programs or LaTeX right now. I plan on presenting in detail the disk-washer and cylindrical shells method using a general depth of h inches for the hole.

For now, have fun playing with this. This is a well-known problem and therefore searchable on the web but try it yourself first. Try to use calculus to set up the integral and if you're brave you'll evaluate those integrals without Mathematica or the TI-89! Can you see why the answer for the volume remaining depends only on the depth of the hole?

Mystery Mathematician Week of 1-21-08

Do you see the big picture?
R U ready for a more contemporary mathematician? He belonged to a special group. I hope this is simple for you...
Remember to email me with your answer and include an interesting anecdote.
If you're reading this from a feed, don't forget to visit the site to see the image in the sidebar.

24x9 is one hundred more than 29x4. Why? Now generalize... An Investigation for Gr 4-9

Update: Read comments for an extension to 3-digit numbers. Have fun!


Continuing with our digit and number relationships, the question in the title is meant to provoke the reader/student to probe more deeply and try to understand the reasons behind number observations. Asking students to explain these kinds of results leads to fruitful dialog.

Target Audience:
Our readers of course
Upper elementary, middle school students through Algebra 1 (Grades 4-9)

Objectives:
(1) Develops understanding of and reinforces the distributive property (in both directions!) numerically and in algebraic form
(2) Develops understanding of and reinforces place value
(3) Develops meaning for multiplication, including multidigit operations
(4) Reinforces the meaning of multiplication as repeated addition
(5) Develops pattern recognition and generalization
(6) Motivates algebraic representation of number
(7) Practice with open-ended investigations

Correlation to Standards
Refer to the K-12 Benchmarks for Mathematics at the Achieve website.
Note: There are many links there. I will attempt to correlate to more specific expectations in the future.

ACTIVITY/INVESTIGATION FOR READER/STUDENT

Today we are going to investigate the effect of interchanging or switching the units' digits when multiplying a 2-digit number by a 1-digit number. There are many ways of thinking about this but our main focus will be on using the distributive property.

We will start by considering which is larger:
24x9 or 29x4?

Think about how you would explain your answer without obtaining the products. Anyone want to share how they thought about it?

So we've concluded that 24x9 is greater than 29x4. Makes sense doesn't it because 24x9 is the same as adding a set of NINE 24's and 29x4 is the same as adding only FOUR 29's. Obtaining the actual products, we see that
24x9 = 216
29x4 = 116
Thus 24x9 - 29x4 = 100
Is it possible to determine the difference of 100 without actually doing each separate product?
Today, you and your research team will devise such a method and explain why it works!!

Specific Instructions:
Your research project for today is to devise a method for finding the difference between two products in which we interchange or switch the units' digits. Make a table using at least 10 additional product pairs in addition to the ones we've given you as models. Make sure you include many different digits in the tens' places as well as the ones.

Product 1.......Result...........Product 2............Result............Difference
24x9...............216.................29x4....................116..................100
43x8..............344.................48x3....................144..................200

........


Conclusions: To find the difference between PQ x R and PR x Q, do the following:


Explanation for Method:



-----------------------------------------------------------------------------
Aside to the Instructor:
The following is a guide for this research.
Here is one approach to the question that uses the distributive property in both directions. The algebraic form is given in italics to the right of each step. For each line, think about which version of the distributive property we are using. [Answers given in brackets to the right].
24x9 = 24x4 + 24x5 [a(b+c) = ab+ac]
29x4 = 24x4 + 5x4 [Same]
Gain: 24x5
Loss: 5x4
Net Gain: 24x5 - 4x5 = 20x5 = 100 [ba-ca = (b-c)a]

AGAINST ALL ODDS -- THE GIANTS ARE ALIVE AND 'KICKING'...

My son-in-law is an even more rabid Giants fan than I am. As we watched the screen in disbelief, as the Giants let chance after chance slip off their fingers and toes, I wasn't thinking about math blogging. I wasn't thinking of all the negative numbers that defined this game for the ages. Looking back on it now, I cannot add any more to the incredible highs and lows and highs that we were feeling last night. It would also be insulting to the memory of MLK for me to relate this game to 'having a dream', so I'll simply end by saying...

GO GIANTS!

Digit Discoveries and Algebra for Middle School - An Investigation

The results below are well-known but, as usual, I am offering an investigation for the classroom that has many objectives:

(1) Digit properties of multiples of 9 (and their 'proofs')
(2) Review place-value and algebraic representation
(3) Investigate patterns based on data collection
(4) Develop inference and conjecture
(5) Introduce students to algebraic proof
(6) And, of course, practice for those open-ended questions we've all come to know and love...

Children are often fascinated by the discoveries they can make regarding 2- and 3-digit numbers. At some point in middle school all students should either discover on their own or be introduced to the remarkable properties of the number 9 in our base 10 number system. The investigation below will explore some of this.

Students are also fascinated by the results of taking a 2- or 3-digit number and reversing its digits. With or without calculators, students like to see how these numbers are related, particularly when they are added or subtracted. In this activity, they will have the opportunity to discover some of these properties and use basic algebra to explain why they work. Perhaps, this will also lead to questions about palindromes, but that's for another day...

The questions below are designed for middle schoolers through Algebra 1. The proofs require some basic algebra, so you can make those parts optional for the prealgebra group. For this group, having them state their conjectures and suggesting possible explanations are more than enough.

STUDENT/READER ACTIVITY/INVESTIGATION

(1) List all of the 2-digit multiples of 9. What do you notice about the sum of their digits?
(2) Using the fact that any 2-digit number can be represented algebraically as 10a+b, show/justify/explain/demonstrate/prove the following:

If a 2 -digit number is a multiple of 9, so is the sum of its digits AND
if the sum of the digits of a 2-digit number is divisible by 9, then the number is a multiple of 9.

(3) If you made sense of (2), why stop with 2-digit numbers! State and prove a similar result for 3- and 4-digit numbers!

Now for reversals:

(4) To be a mathematical researcher, one needs to do what the scientific researcher does. Collect lots of data first, then make conjectures and PROVE them! Choose at least 5 different 2-digit numbers, in addition to the examples below, and complete the table.

Number..........Reversal............Sum..........Difference (Larger-Smaller)
41.....................14.......................55...............27
33....................33......................66................0
72....................27......................99................45
Your turn - do this FIVE more times.

(5) Make conjectures about the how the sum and difference are related to the digits of the original number. Using the algebraic representation 10a+b for any 2-digit number, PROVE your conjectures (or disprove them!).

(6) 72 and 27 are not only reversals. They are are also both multiples of 9. Does this have to be true for any 2-digit multiple of 9? Explain! Further, is there a special property for the sum of the number and its reversal in this case. Make sure you verify conjectures for several cases before attempting to prove it.

(7) Make a similar table for 3-digit numbers. Is there an obvious relationship for the sum of the number and its reversal this time? The difference? Make conjectures and PROVE them!

If you feel this activity is useful, please comment, share it and rate it below. Enjoy!

New Feature -- Share and Rate Posts in MathNotations

Of course, everyone else in the blogosphere figured out how to do this months ago, but I'm a slow learner...

Please share your favorite posts from MathNotations (and hopefully give them a Thumbs Up!) by clicking on the links at the bottom of each post both on the site and from the RSS feed (if you use my feed). You have a choice of Technorati, Stumble, del.icio.us, or Digg.

Pls comment to let me know if you have any problems with this...

A Much Harder Mental Math 'Trick'?? - Algebra May Not Be Optional!

Update: For more background on the investigation below, I strongly encourage readers to visit Sol's wonderful blog wildaboutmath and, specifically, the post he wrote about the algebra behind multiplying 2- and 3-digit numbers. Further, he developed a video (mathcast) to demonstrate the method for 'mentally' mutiplying any two 3-digit numbers. Sol inspired me to develop this activity!


Target audience: Strong Algebra 1 or Algebra 2 students

Ok, I hope you enjoyed challenging yourself or your students to describe a step-by-step procedure for squaring three-digit numbers with a middle digit of zero. I believe many middle school students could devise a method using only operations on the digits, without the need to use algebraic representation or proof. I really believe that recording and organizing the necessary data and the search for patterns make that a valuable activity. Perhaps, even more importantly, the verbalization of the strategy or trick may be the most important benefit for our students! Finally, I strongly believe these kinds of investigations deepen student understanding of place-value and develop number sense.

But now we will move on to a much more significant challenge. I highly doubt that any of your students will be able to find the right combinations of digits to square any 3-digit number! One will still have to record the data and attempt to see a pattern, but, this time, a non-algebraic approach may be too formidable. I am fully aware that there are classical mental math methods for multiplying numbers (Vedic Maths and Trachtenberg to name a couple), but I didn't research those when writing this activity. I simply applied algebraic methods and looked for an algorithm that made sense to me. Thus my method may be similar to those others or not!

This challenge should be frustrating for some and many of you will question whether there is any benefit in looking for some strategy for squaring such numbers by some artificial-looking or convoluted combinations of digits. If the object here were to develop an easy mental math trick, you'd be right. BUT that is not my objective - the idea is to use algebra to motivate an alternate approach to multiplying 3-digit numbers. And, yes, there is some mental math involved, although, I suspect one will need to record intermediate results and do some paper and pencil arithmetic along the way (sorry, no calculator allowed other than to check your answer!). In the end, students should have a much deeper understanding of the meaning of digits, place value that is.

We begin with the algebra prerequisite for this problem, then move on to the real challenge!

READER/STUDENT CHALLENGE/ACTIVITY

(1) Show that
(p+q+r)² =
p² + q² + r² + 2pq + 2pr +2qr

Ok, here goes...
(2) Using the algebraic formula above, devise a method/strategy/algorithm/procedure/'trick' using combinations of digits to square ANY THREE-DIGIT NUMBER 'HTU'.
Note: Again, I reiterate, others have probably invented much cleverer methods for multiplication than this. The key here is to use the above algebraic representation!

In the end, you should be able to do an example 'mentally' like

463² = 214,369 (or should I write it as 21-4-3-6-9).
Lots of 'carrying' here or 'regrouping' as we say these days! If I give you the intermediate step, it may give it away. If I don't, you may give up! Let's see what happens...

508^2 = 258,064: Invent your own Mental Math 'Trick' and Prove It!

UPDATE: READ THE COMMENTS FOR A DISCUSSION OF NOT ONLY A POSSIBLE SOLUTION BUT ALSO THE PEDAGOGICAL IMPLICATIONS OF SUCH AN ACTIVITY.


What child (ok, adults too!) is not mesmerized by a magician. The Hindu-Arabic place value system allows for endless possibilities for mathemagic, aka, mental math strategies. Adults and children alike are fascinated by these as is apparent from the re-discovery of Vedic Maths. There are several excellent resources for this. You may want to check out The Vedic Maths Forum India Blog, which includes an interesting video showing youngsters demonstrating one of these methods in a dance routine! It's also no wonder that Sol's Impress your friends with mental Math tricks post over at wildaboutmath went viral! Sol discusses many of these wonderful mental math strategies and the response has been overwhelming.

I've personally always wanted to understand the basis for many of these mental math 'tricks' that have been around for centuries. Since a significant part of teaching is performing, math educators are often interested in any mathematical sleight of hand that provokes wonder in the child or older student. We would hope the student would want to know the WHY behind the trick but most are more interested in just performing the feat. What child doesn't come home and want to challenge their parents to square 45 in their heads or multiply 68 by 62 in 5 seconds or less! Hey, anything that turns kids on to mathematics is alright in my book. However, the purpose of this post is to have the reader discover a mental math method and use algebra to delve beneath the surface. Magic? Perhaps that is what teaching is all about...

STUDENT/READER ACTIVITY/CHALLENGE

Study the following:
101² = 10201
401² = 160801
508² = 258064
306² = 93636
909² = 826281


1. From studying the above examples, discover a mental math strategy which would enable you to square any 3-digit number whose middle digit is zero. Write the precise steps of the method. Be careful here - make sure your method works for all 5 of the above examples! Note: That last one is a bit harder to do mentally but give it your best.

2. Demonstrate (perform!) your method by having someone challenge you to square such a number! You need to be able to perform the trick in less than 5 seconds (ok, maybe 10 for old people like me!). Repeat your amazing performance at least 5 times!
Note: You may need to practice this for awhile before going on American MathIdol!

3. Ok, now PROVE IT ALGEBRAICALLY!
(Gee, that takes all the fun out of it.)

A couple of additional comments...

(i) Did you ever notice that all of these multiplication tricks actually require that the child knows the basic facts cold! Perhaps, these tricks could even serve as motivation to learn them!

(ii) When I discovered this trick, I was naturally excited and of course wanted to share it with my wife who has a droll way of putting my geekiness in perspective. After she picked a random 3-digit number and I did the math correctly, she seemed unimpressed and replied, "But can you cook a chicken?" I was tempted to suggest that I could square a chicken if it had a hole in the middle, but I knew she would not be amused....

Updates Week of 1-14-08

Lots of loose ends...

(0) CARNIVAL OF MATH XXIV AT ARS MATHEMATICA
Nice links including a couple from MathNotations.

(1) The math contest problem from the other day now has one possible solution posted - you need to scroll down quite a ways. There are some interesting comments to look at also, but my solution is not in the comments section. Check out tc's and Joshua's approaches.

(2) A SPECIAL THANK YOU TO SOL at wildabout math.
I am very grateful to Sol for highlighting MathNotations.
Sol has an excellent site that reveals his passion for mathematics and sharing it with students and adults alike. This is a must for anyone who loves math!

(3) A SPECIAL THANK YOU TO DENISE at Let's play math!
Denise selected MathNotations as one of her favorite blogs in her Math Bloggers Hall of Fame post from a few weeks ago. Denise knows she will always be one of my favorites as well!

(4) MATH CASTS
I'm awaiting the Mimio technology I mentioned a few weeks ago. When I receive it, I plan on learning how to use it to produce some rudimentary videocasts or 'math casts.' I've seen a few high quality ones online and I feel like this is a direction that makes sense for me. Surely this makes more sense than for me to stand in front of a digital camcorder and pretend to be teaching a lesson! This has recently been discussed over at Sol's excellent site, wildaboutmath. He has already developed some nice math videos I recommend you watch. This is a promising direction for the future that is currently in its infancy. Another excellent example of high quality math videos can be found at isallaboutmath. I've contacted the author of these outstanding videos that present math lessons in Flash, all with a touch of history.

(5) If you have a youngster who is struggling to learn the times tables or wants to get ahead, I strongly recommend you test drive Timez Attack, which I recently reviewed. I don't get anything from this other than the satisfaction of knowing I might be helping some child or frustrated parent out there. I believe that this is an exceptional product, particularly for the child with learning disabilities. Parents, math supervisors, educators, and directors of special services and other child study team members may want to check this out.

Leonardo Pisano -- The Greatest Mathematician of his time? No 'Fibbin'!

I didn't announce this week's Mystery Math Contest to see how that would play out. To be fair to those 200 or so subscribers to MathNotations who get a feed redirected through Feedburner, I guess I should mention it from now on, since the feed only shows new posts, not changes in the sidebar.

Anyway, we have two winners this week who correctly identified the one and only Fibonacci. BTW, this extraordinary gentleman had more names than George Foreman's sons! TC below mentioned one of his other names...

After visiting several websites, I found that he had devised some interesting word problems that seem to have survived to this day in one form or another. So now we know the rest of the story!

The following is from his most famous work, Liber abaci:

(1) A spider climbs so many feet up a wall each day and slips back a fixed number each night, how many days does it take him to climb the wall.
(2) A hound whose speed increases arithmetically chases a hare whose speed also increases arithmetically, how far do they travel before the hound catches the hare.
(3) Calculate the amount of money two people have after a certain amount changes hands and the proportional increase and decrease are given.


Here are our winners...

Mathmom!
Here's the info she supplied (the math symbols went a bit awry when I copied this from her email):

The mystery mathematician is Fibonacci. One interesting fact is that his real name was Leonardo Pisano.

Liber quadratorum, written in 1225, is Fibonacci's most impressive piece of work, although not the work for which he is most famous. The book's name means the "Book of Squares" and it is a number theory book which, among other things, examines methods to find Pythogorean triples. Fibonacci first notes that square numbers can be constructed as sums of odd numbers, essentially describing an inductive construction using the formula n² + (2n+1) = (n+1)². Fibonacci also proves many interesting number theory results such as: there is no x, y such that x² + y² and x² - y² are both squares, and x⁴ - y⁴ = z² has no non-trivial integral solutions.

He also defined the concept of a congruum, a number of the form ab(a + b)(a - b), if a + b is even, and 4 times this if a + b is odd. Fibonacci proved that a congruum must be divisible by 24 and he also showed that for x, c such that x² + c and x² - c are both squares, then c is a congruum. He also proved that a square cannot be a congruum. The Liber quadratorum alone ranks Fibonacci as the major contributor to number theory between Diophantus and the seventeenth century French mathematician Pierre de Fermat.

And our 2nd winner is
TC
who offered:

That's Leonardo Bigollo of Pisa.

This description of a famous sequence popularized by another famous Leonardo's code is interesting:

A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?

It is almost a rite of passage for most math students to discover the closed form expression to this (featuring the Golden ratio, of course).

Timez Attack - A Review and a View from an 11-year old

After doing my usual random search for web sites that might help my son learn his multiplication facts, I came across Timez Attack from Big Brainz. The company allows the user to download a free version which I did and I was immediately impressed. My 11-year old has struggled to retain his facts and learning software rarely engages him for more than a few minutes. However, after discovering how to navigate the game, he stayed with it for almost a half hour. Since attentional issues make it hard for him to sustain focus, I considered this promising. In addition to observing how he played the game, I tried it myself for awhile.

Having seen dozens of computer learning software products both online and on CD-ROM, I've reached the point where I can rapidly identify quality and Timez Attack is quality in every sense. I decided to email Ben Harrison, the president of the company, and share my thoughts about his product, the only item his company sells at this point. I explained that I would like to write an objective review of Timez Attack for MathNotations after I had an opportunity to try it myself and to observe a child using it. I asked him if he would be willing to give me access to the full version and he agreed. I haven't had time to thoroughly play around with this new version, but my son has been on it for about a week and he likes it. To derive the maximum benefit of this software, I feel I need to establish consistency for him to be on it for at least 15-30 minutes a day for about a month but that has not yet happened. I already see better retention but he is still only on a beginner level.

I strongly urge you to go to the Big Brainz website. There you will see an audio and video presentation of the software. From this home page you can download the free version or purchase the full version for $39.99.

PROS:

(1) Full video game format replete with creatures, a benevolent monster and exceptional graphics that maintain the child's interest. This is the only piece of educational software that I have seen that combines the professional graphics you expect from a video game with solid educational principles. Ben and his crew clearly put a great deal of thought and planning into this.

(2) The free version is outstanding. It is rare to see something of this quality given away. My son was engaged by the free version, although the full version has additional features that will sustain interest over a longer period of time. I may be incredibly naive, but I see this as more than a marketing ploy. To me it says something about the commitment of this company to actually help children learn. The software has already been well-reviewed, but it's possible you haven't heard about it yet. I strongly urge you to download the free version and just try it for yourself or allow your child to play around with it.

(3) The child has to demonstrate mastery of facts before moving on. Thus if working on the 2's, a mistake requires that the child respond correctly before the next fact is presented. Moreover, if, for example, the child corrected 3x2, the program might return again with the same problem or move onto 9x2, then return to 3x2 many times until the child demonstrates they can do the whole set without error and within a time limit. Remember how you used those flash cards - you made two piles and you'd keep going back to the troublesome pile until you got them all. Well, this simulates that and does it better! Throughout, the child gets hints but they must pay careful attention to see it. Further, each time the game is played, the program smartly retains those facts the child struggled with and reinforces frequently. Again, we're talking mastery here!

(4) The program saves the child's information, picking up exactly where the child left off from the previous play.

(5) The child is required to work through a consistent routine of capturing two cute little creatures, revealing the answer, then hurling an object at a wall revealing a picture of the multiplication fact (the child sees two rows of four objects when encountering 4x2). Only when both actions are taken does the friendly ogre allow the child to enter the answer. If incorrect, the ogre gently pushes the player and the child needs to try again (with hints). There is no violence here and the software is perfectly suitable for younger children who may be ready to jump ahead in their learning.

(6) I'm particularly impressed by an ancillary effect of this game. The child must concentrate in order to follow the correct sequence or to to see the answer when it's presented for an instant. I've already seen improvement in my son's ability to focus and attend as a result of this.

(7) Ben and his team seem committed to developing the best product possible, choosing to perfect this game rather than develop other games at this time. He has responded immediately to all of my questions in the several emails I sent. In fact, I shared my son's suggestions about the possibility of other creatures or aliens! He told me that some of this is already in the latest version of the full game and they are continuing to revise the game to make it even more engaging.

My instinct is that this software could be of benefit to all learners but particularly those children who have attentional or other learning issues. I asked Ben about the age range for which the software is intended and he replied:
"Timez Attack is fairly age-independent. It works extremely well for anyone who needs to learn multiplication. The value of the tool increases in cases where there the student is a teen, adult, or has ADHD, Autism, or other learning challenges. The value increases not so much because it works better but because it still works just as effectively where all other solutions have completely failed. It’s really fun when you see that happening."

It is important for me to say that this does not replace classroom learning -- it enhances it. The software could be used at home or in the classroom. Because it is not web-based, a school or classroom would have to purchase a site license, then have it networked. Several purchasing options are offered. However, the company allows the school to download the free version and network it. To quote Ben:
"As for schools, the free version is meant for them as well. They can install it on all their computers—and preferably on their server so that they can access all the built-in reporting/tracking tools."

You may also want to read the review on the website Super Smart Games.
In particular, you can read the interview they did with Ben Harrison - well worth reading. Also you can see videos of the game in action but nothing replaces downloading the game and playing it!

I can only add that I would love to see this same quality and commitment to excellence brought to many other educational software products. It's unfortunate but products such as Timez Attack are generally not seen as cost effective by most software publishers. I'm looking forward to Big Brainz developing other learning tools as well as continuing to perfect this one.

An Exceptional Math Contest Problem

Update: The solution I emailed to the director of the Math League is at the bottom after you scroll down a ways. Note how this ties into the difference of squares problem I posted the other day!

A very good friend of mine who runs one of the most successful math league contests in the US has given me permission to reprint the question below with the following attribution:
“Copyright Mathematics Leagues Inc 2007. May not be reproduced without permission of the copyright holder.”

This question was the last question on a recent contest. It certainly made me think! I felt the need to be rigorous about justifying one of the steps and that was the reason for the
M² - N² = 12 post from the other day. There are at least 3 methods that have been found thus far, according to my friend. Teachers often send in their own solutions and some even get included with the official solutions. My feeling is that questions like these are vital for our students intending to pursue higher mathematics, not just for those who happen to participate in contests. I also believe that variations on great questions like this will have an eternal life...

THE QUESTION

There's exactly one real number a for which ax² + (a+3)x + (a-3) = 0 has two positive integer solutions for x. What are these values of x?

Have fun with this!
For more information about the Math League, visit here.

Scroll down to see the solution I sent to the directors of the Math League...











First of all, I'll state the answer I obtained before giving the details:

A real number a which produces two positive integer solutions is a = -3/7 which does lead to the solutions x = 4 and x = 2. Proving that this is the only real value of a is far more challenging and I suspect that there's a much easier solution than the one I found. Anyway, here's my approach:

In order to simplify the original equation, divide both sides by a to make the leading coefficient equal to 1:

x^2 + (1+(3/a))x + (1 - (3/a)) = 0
To simplify further, let b = 3/a:
x^2 + (1+b)x + (1-b) = 0
From a well-known rule about the roots of a quadratic equation, the sum of the roots in this case equals -1-b and the product of the roots is 1-b. Since the roots must be positive integers, it follows that b is an integer less than -1. Remember for later:
b must be negative!

From the quadratic formula, we obtain the roots to be:
x = ((-1-b) +- SQRT(b^2+6b-3))/2.
In order for the roots to be integers, the expression b^2+6b-3 must be a perfect square.
I will rewrite this expression by completing the square:
(b+3)^2 - 12 and this must equal a perfect square , call it N^2.
Let b+3 = M, then we must have M^2 - 12 = N^2 or
M^2 - N^2 = 12. It's easy to guess a difference of perfect squares equal to 12, namely 16-4, but I will now prove this is the only solution:

Factoring, we have (M+N)(M-N) = 12 where M,N are integers.
The only pairs of factors to consider are: 12,1; 6,2; 4,3 as well as their negatives.
In order for M,N to be integers, the factors of 12 must BOTH be EVEN so the only possibilities are 6 and 2 or -6 and -2..
If M+N = 6, M-N = 2, then, by adding, M=4 and N=2. From b+3 = M, we would have b = 1, but this doesn't work since b must be negative (see above!)
Therefore M+N = -6 and M-N = -2. From this we have M=-4 and N=-2.
Then b = M-3 = -4 - 3 = -7.
b =-7 leads to the quadratic equation: x^2 + (1+(-7))x = (1-(-7)) = 0 or
x^2 - 6x + 8 = 0. Factoring we obtain the roots to be x = 4 and x = 2.

Since b = 3/a, it follows that -7 = 3/a or a = -3/7.

A Visualization for Factoring the Difference of Two Squares

M² - N² = (M+N)(M-N)

Given that my difference of squares problem from yesterday may have been overly ambitious for middle schoolers (and I will have more to say about that in the comments section from that post), I thought it was worth reviewing a diagram that many of you have probably seen before. There are many similar geometric representations of standard factoring and distributive formulas in algebra, but this one has always been one of my favorites. It would be more effective if I had been able to shade rectangles R and S using different colors, but I did the best I could on short notice.
It's often a good exercise for algebra students to invent similar diagrams for other formulas, although the use of manipulatives such as algebra tiles can be even more effective.

M^2 - N^2 = 12...Prove there is only one solution in positive integers and much more

Update:
(1) See the visualization for the difference of squares posted on 1-9-08.
(2) Read the comments in this post for considerable clarification and instructor guidelines and suggestions. Mathmom's and Eric's comments are particularly insightful.

This post can be developed into an activity for prealgebra through first-year algebra students (or even 2nd year algebra). The last part is more challenging.
The focus here is on developing a method/strategy that can be used to solve similar Diophantine equations. The other objective is to introduce the ideas and methods of proof. This problem may later be used to solve a recent math contest problem for which I obtained permission to discuss on this blog. I am fully aware that many students will 'solve' these equations by Guess-Test methods, but they need to go further.

STUDENT/READER PROBLEMS/ACTIVITIES

(a) Prove there is only one solution in positive integers for the equation:

M² - N² = 12

Note: If we omit the word positive, what would the solution(s) be?

(b) Determine all positive integer solutions:

M² - N² = 15

(c) Determine all positive integer solutions:

M² - N² = 36

(d) Let's investigate for what positive integer values of P, M² - N² = P has NO solutions in positive integers.

(i) Determine at least 5 positive integer values of P for which the above equation has no positive integer solutions.

(ii) (More challenging) Describe all values of P for which the above equation has no solutions. Justify your result.

Note: All students should have success with (i), although some may struggle to find 5 values. Part(ii) should challenge the student who has finished the other parts in rapid order and sits there complacently!

Additional Comment: If P is itself a perfect square, our equation is obviously related to the most famous equation in geometry. Thus, if P = 9 or P = 16, for example, students should recognize something! For this reason you may want to have students consider these values when doing this investigation. More to come...

Mystery Math Idol Week of 1-1-08 Revealed and a Proof!

Believe it or not, there were actually two winners this week. This probably translates to a total of two individuals in the math blogosphere who actually enjoy this feature! I will continue this feature until we run out of mathematical icons, however, by indirect proof, one can prove that the list is infinite:

Assume there are only finitely many mathematical geniuses to choose from:
M1, M2,M3,...Mk.
Since such mathematicians 'multiply' like the rest of our species, there will always be at least one other offspring from this list who is different from anyone on the list. This contradiction proves that the Name That Mathematician Feature will be around for awhile!

From looking at the picture in the sidebar, you now know that our mystery man is none other than (drum roll please...)

Ferdinand Eisenstein

Winner #1 (and still overall champion): LYNX

Here is her contribution:

Ferdinand Eisenstein was the only one of 6 children to survive. The rest died of meningitis. During school (aged 11-13) he solved 100 proofs during the time most students were expected to solve 11 or 12. He died of tuberculosis at the age of 29. (It seems many mathematicians have their lives cut short. What might have been discovered sooner if they had lived?) Gauss would later say the three top mathematicians were Archimedes, Newton, and Eisenstein. (Quite a compliment if I say so myself.) [http://fermatslasttheorem.blogspot.com/2005/07/ferdinand-gotthold-max-eisenstein.html]

In his autobiography, he writes, "As a boy of six I could understand the proof of a mathematical theorem more readily than that meat had to be cut with one's knife, not one's fork. "


Co-Winner: VLORBIK

that's eisenstein, of course

(the only one so far i knew

right away just by looking ...).

fun fact? well, according to

http://en.wikipedia.org/wiki/Ferdinand_Eisenstein

gauss rated him with newton & archimedes.

'Feed Me, See More' MathNotations in '08!

Puns are bad way to kick off the New Year, but, given the RSS problems I've been having with my posts feed I had to do something to turn it around! Well, my first New Year's resolution was to fix the problem and I believe I have. If you had not been receiving my feed for some time, it should be ok now. Of course, many of you have re-subscribed using the icons in the sidebar.

Some updates on 1-1-08:
(1) Please share your thoughts about your favorite math teacher. Perhaps the comments section of this post from the other day is not an appropriate forum but that's all I had available at this point. Someone probably has already created such a website!
(2) The Bingo post is getting some attention even during holiday time. I will be expanding on that but I await your thoughts.
(3) Did you notice the new Mystery Math Idol to kick off '08? This should prove difficult as he is not a household name for most, but his genius is still being recognized by algebraists and number theorists (hint, hint, wink, wink).
(4) The State of the Blog address -- I'd be interested in your thoughts about what you enjoy best about MathNotations. What changes would you like to see? What to keep? I've gone off in dozens of new directions but the meat and potatoes of this blog is still discussions of problems, concepts and instructional strategies.
(5) I'll be introducing Mimio technology shortly, perhaps some Mathematica Demonstrations, and reviewing an exceptional piece of software for helping students master their Times Tables. More videos too! Coming soon!

Have a healthy and happy!