Mathing...

Identifying Effective Representations

noreply@blogger.com (Mr. H) — Thu, 30 Jan 2014 20:08:00 +0000

At a recent PD, the Pre-Calc/Calc teachers at my site worked on the following question while we were on the topic of Identifying Effective Representations.

The verbal instruction given was: Solve the problem in as many ways as you can. The written instructions are as follows:

Solve the following problem, using alternative representations where appropriate.

A woman was 3/8 of the way across a bridge when she heard the Orient Express approaching the bridge behind her at 60 mph. Because she was a mathematics student, she quickly calculated that she could save herself by running to either end of the bridge at top speed. How fast could she run?

Go ahead. Try it out.

In the limited time given, we were able to find two representations that led to the solution. For me, it felt more like the same representation solved two different ways. The traditional way, if you could call it that, required more work as we solved a system of equations to get the solution. The other method we found required a little more thinking about the situation, but the path to the solution was much simpler and shorter. Can you find other effective representations?

The idea is that as we move to Common Core, it is not sufficient to just get an answer. Students are expected to solve a problem using multiple approaches and this was an example of that. How realistic is it that students will have the kind of insight that allows them to solve this kind of problem using multiple approaches and, more importantly, how do you go about teaching it in the limited time we have?

I would love to see your approach and your diagrams as you work on this problem. Link or comment below. I'll update this post with my work in a week or so.

Fun Little Area of Quadrilateral Question

noreply@blogger.com (Mr. H) — Thu, 18 Jul 2013 03:56:00 +0000

Fun little problem about the area of a quadrilateral tweeted by @daveinstpaul. Try it out.

If you're stuck, try the following applet and click on the checkboxes labeled "Show Hint ##"

mrhodotnet, 17 July 2013, Created with GeoGebra

Some questions that popped in my mind as I worked on this question:

Is the shaded quadrilateral unique? Is this the only quadrilateral that fits the criteria (given sides and angles)?
Is it possible to draw another diagram with the same given sides and angles but with different shape?
Isn't it the case that given only 4 sides of a quadrilateral, there isn't enough information to determine area? If so there may not be enough information to determine area, but if it were the case the question would be pointless.
Is it possible that the area of the shaded quadrilateral is constant regardless of the shape?
If the area is constant, let's choose the diagram that is the most calculation friendly and find the area.
I have the area now. How do I know it's constant? Can I show this? What principles are at work? What can I learn about the special cases (the diagrams that I drew) to show it for the general case?
Is it possible to show this using trigonometry? How should I break down the quadrilateral? A trapezoid and a triangle? Two triangles? Should I tackle the unshaded area first?
Let's label the unknown sides and use Pythagorean Theorem to find the relationship between the missing sides. Do the equations simplify?

It was some time later, while doing chores, when it hit me. There is a kind of joy and excitement when an idea strikes and it's immediately clear that it'll work. A simple auxiliary line segment is all it was, probably immediately obvious to Geometry teachers. It's interesting how this question was one auxiliary line segment away from being boring. Glad it was omitted from the original.

This is my first time embedding a tweet in a post and embedding a GeoGebra html5 applet instead of a java applet. People really make these things easy to do nowadays. Thanks Twitter and GeoGebra!

Update: Removed tweet embedding until I figure out how to embed without including media.

Monty Hall Problem through Philosophical Chairs

noreply@blogger.com (Mr. H) — Thu, 27 Jun 2013 23:22:00 +0000

This is an activity based on Philosophical Chairs that I first learned from AVID teachers at my school. I don't have my handout readily available but found one from San Diego COE on Philosophical Chairs. From their handout:

In theory, learning happens when students use critical thinking to resolve subsequent conflicts, which arise when presented with alternative perspectives, ideas or contradictions to what they have previously learned or believed. "Philosophical Chairs" is a technique to allow students to critically think, verbally ponder and logically write their beliefs.

Essentially, Philosophical Chairs is a way to organize discussion on debatable topics and then have the students reflect and justify their position in writing. These were yes/no questions and students had to choose between these two options and undecided. The desks are typically arranged in a horseshoe seating arrangement with yes and no on the sides, and undecided in the middle. 3 Signs are posted so that students can move to the corresponding area once they decide on a position with respect to the question/topic being discussed. They are called upon to justify their position in the "hot seat" in the middle of the room. Only one speaker is allowed at a time. Other students are invited or "volunteered" to join in with questions, comments, or present their arguments, however, they must summarize the previous speaker before they're allowed to do so. They will be given opportunity to move as their minds change.

The activity seemed tailored for ELA and Social Studies. If you do a quick search online you'll also find plenty of examples for those classes. Perhaps I haven't looked hard enough, but I haven't seen one for math yet. I didn't want discussion on whether reliance on calculators for computation has become a crutch for math students or whether all students should be required to take up to algebra 2 or any other math education topics. I wanted math topics. I was about to throw this activity into the cool-if-I-taught-other-subjects folder until I heard a presenter use the word paradoxical. I don't remember the context, but it gave me ideas.

The Monty Hall problem (wikipedia) isn't really a paradox, but it's infamous for being so counter-intuitive that even PhDs get it wrong. Here's a wording of the problem popularized by Marilyn vos Savant in her column "Ask Marilyn":

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

If this is your first encounter with this problem, stop reading now. Take some time and think about the problem.

The correct answer is you switch to door No. 2 for a better chance to win. It's not really paradoxical, it's really just counter-intuitive that when left with only 2 choices (door No. 1 and door No. 2), where the chance of winning seems to be 50%, it is more advantageous to switch. This is one of those problems where intuition works against you. Often, even after being presented with simulations and calculations, it's hard for people to wrap their minds around it.

I chose this activity so students would have an opportunity to get an understanding of the problem and be able to put into words their reasoning and also evaluate the reasoning of others. Instead of asking just one question, I would ask series of questions that led them to the answer. They also get an intuitive feel for why the initial conditions make a difference and opens the door for discussion about assumptions and about what probability really is. Plus, is there a probability or statistics class that doesn't cover or at least mention this problem?

I made the following modifications/additions:

Instead of presenting just one question, I would present several different scenarios to help students clarify their thinking
I told students that by the end of the day all students would get the correct answer so the goal is not to justify their position with a personal opinion, but to help each other understand the solution through discussion, questioning, and mathematical arguments. For this to happen, they must be open-minded.
Instead of yes/no/undecided, for this problem the students chose between yes/no/doesn't matter
To better maintain order with multiple questions and with the movement in the classroom, I organized each question into a round that consisted of: a) silent reading of question b) students move to the appropriate areas after deciding on their position c) call students to justify their position to start discussion d) ask for volunteers if necessary e) students ask questions f) students can switch positions before next round g) periodically I ask why students changed their minds

I had the following questions on PowerPoint and also on a handout with the wording of the problem.

(Original Problem) In the case above, Monty Hall (the host) asks you - do you want to stay with Door 1 or switch to Door 2? If you want a better chance of winning, should you switch from your original choice? Does it make a difference?
Imagine there are 1000 doors and only 1 prize. You pick a door. Now he eliminates 998 other doors that don't contain the prize so that 2 doors are left, one with the prize and one with no prize. Should you switch from your original choice? Does it matter?
Suppose there are 1000 doors. 500 (half) of which have the prize. You select a door, and all doors are eliminated so that 2 doors remain. The options are eliminated such that one has the prize and one doesn't. Should you switch from your original choice? Does it matter?
Suppose there are 1000 doors. 999 of which have the prize. You select a door, all doors are eliminated so that 2 doors remain. The options are eliminated such that one has the prize and one doesn't. Should you switch from your original choice? Does it matter?
Suppose there are 1000 doors. 1 of which has the prize. You select a door, the host opens a door without a prize and says you have the choice of 1 door or the 998 other doors. Should you switch from your original choice? Does it matter?
Suppose there are 3 doors and 2 prizes. You select a door and the host eliminates an option so that one has the prize and one doesn't. Should you switch from your original choice? Does it matter?
Suppose there are 3 doors and there’s only 1 prize. You select 1, and the host asks you if you want to keep the door you selected or the 2 that you didn’t select. Should you switch? Does it matter?
Suppose there are 3 doors and there’s 2 prizes. You select 1, and the host asks you if you want to keep the door you selected or take the other 2 that you didn’t select. Should you switch? Does it matter?
(Original Problem) Should you switch from your original choice? Does it matter?

I tried to emphasize mathematical justification. Some students will offer reasons like "I didn't switch because I trust my gut instincts" or "I don't like switching because I'm not a quitter." There will be times where students don't really have a position and can't provide an argument, just move on and return to the student later and ask him/her to summarize views presented by others. It takes a few rounds before students would automatically rephrase the previous speaker's argument before presenting their own. An interesting thing is most students will answer question 4 correctly but not 2. There might be some kind of bias at work here. Usually, by question 5 most students will choose correctly. At some point, someone realizes initial conditions make a difference in this problem and there's a discussion to be had about assumptions. To conclude this activity after discussion, I call on students for the correct answer to each question. Once the students are done with discussion, they write their reflection and justification in the back along with answering these activity related questions:

What strategy would give you an advantage: staying with the original choice or switching from the original choice? Explain.
What is the difference between the scenarios presented in questions 2, 3, and 4? Explain the strategy you would employ for each (switch/don't switch/doesn't matter).
What was the most frustrating part of today's discussion?
What was the most successful part?
What statements led you to change your seat or to stay sitting in your position today?
What is probability to you?
What would you change about your participation in today’s activity? Do you wish you had said something that you did not? Did you think about changing seats but didn’t? Explain.

After we conclude the Philosophical Chairs part, we would continue with simulations of the problem and perform an analysis using tree diagrams to calculate probability. If there's more time I would play the MythBusters clip on the Monty Hall problem.

This is not the standard Philosophical Chairs where you have a debatable topic and students can hold valid differing point of view based on their beliefs and values. But if you focus on the critical thinking, the formulation of problem, persuasive argumentation aspect of the activity, I think it can provide a framework for having good discussion about counter-intuitive topics in math. The main difference is that there is a correct answer in the end. The journey to get there is what's valuable.

Optimizing Cans

noreply@blogger.com (Mr. H) — Tue, 28 May 2013 06:36:00 +0000

This year, I gave my kids a project on optimizing cans. The goal is to minimize the amount of material/metal used for the can while holding the same content (minimize surface area while holding volume constant). This is the second time I gave this project. I like this project because it gives student the opportunity to make the connection between what they learn in class and the design of cans that they see and use daily.

As with any project, there are varying levels of achievement as you can see from the group picture below. For this project, the students had to create a poster with all the work (measurements, calculations, optimization, and conclusion) along with 2 cans (a replica of the original they'll use as reference and the optimized can). The cans have been paired for this picture with the replica of the original in the back row and the optimized can in the front row.

Group picture of original (back row) and optimized cans (front row)

The presentation of the project consisted of 4 parts:

Introduction (group members and purpose of project)
Presentation of the Math (measurement, calculations, optimization, and conclusion)
Show the volume of the original and optimized cans are the same (completely fill a can with rice then pour it into the other)
Q&A

Here are some questions I asked the students this year:

What is the purpose of this project in real world terms and in mathematical terms?
How did you find the radius of the can accurately? (I liked some of the different ways students had to determine the radius)
What do the parts of the formula for the surface area represent? (More students struggled with this question than I expected)
What was kept constant in this project and what was optimized?
How did you optimize?
What are critical numbers for and what do they mean in the context of the project?
What is special about the new radius that you found as opposed to any other radius from any of the possible combinations of radius and height?
What does your result mean for the company that created the product you used for this project? (Some groups had similar cans, while others had a more pronounced difference between the original and optimized cans. I loved some of the responses to this question)

After the all presentations were done I went back to their posters and checked their calculations and work. Below are pairs of cans from selected groups. A question for you is:

Just from looking at the cans, can you tell which pairs of cans were optimized correctly and which weren't?

In the pictures of pairs of cans below, the cans on the left are replicas of the original cans and the ones on the right are the optimized cans.

Pair #1

Pair #2

Pair #3

Pair #4

Pair #5

Pair #6

Pair #7

Pair #8

Pair #9

Pair #10

A good start can be with the question "which of the optimized cans is most unlike the others" and then proceed to "is there a pattern or similarities in the shape of the optimized cans" and "describe the relationship between the radius and the height mathematically?" I wish I had followed this line of thought during the year and asked better questions to summarize the entire project.

P.S. A few groups were cheering when they did the rice test and the rice filled their optimized cans without spilling or underfilling. I forgot how amazing it must feel for students to go through calculations that are challenging (to some) and have it work out perfectly when they build models using their final results. You can watch the anticipation as they slowly pour the last of the rice from one can to the other.

xkcd: Paths

noreply@blogger.com (Mr. H) — Sat, 11 May 2013 18:38:00 +0000

I guess I'm not the only one who worries about things like this.

Since we've been working on optimization problems in Calculus, it gave me an idea. The following is my extension to the above question.

How is this different from the problem in the previous comic? What are the assumptions, if any, that are made in the first comic? How would you solve this without Calculus?

Jeremy Lin vs The Ghost (Simulation)

noreply@blogger.com (Mr. H) — Tue, 25 Dec 2012 20:02:00 +0000

This is a follow up post to Jeremy Lin vs The Ghost (Finding Patterns).

So you've done the calculations and want to know if you're on the right track. Here's a simulator you can use to check your answer. Run the simulation a few times to get an idea of the range of values. If you increase the number of games simulated, you'll see the numbers stabilize. The simulator is as is with no warranties or guaranteed expressed or implied. :)

The default values verifies the answer in the previous post. Run it a few times. For a more precise answer, run it with a larger number of games simulated (1000, 10000, etc).

Probability of Success (%):		Presets: 1) Make: 4 Miss: 4 2) Make: 21 Miss: 7 Show Output
Number of shots made for win:
Number of shots missed for loss:
Number of games:

Jeremy Lin vs The Ghost
Jeremy Lin vs The Ghost (Finding Patterns)
Jeremy Lin vs The Ghost (Simulation)

Jeremy Lin vs The Ghost (Finding Patterns)

noreply@blogger.com (Mr. H) — Sat, 06 Oct 2012 21:52:00 +0000

This is a follow up post to Jeremy Lin vs The Ghost.

Note: For this post, I decided to stick to the basic rules of probability. There are formulas and models that could be used for this problem, but we'll focus on doing some exploration.

Before answering our question, let's tackle the same question with smaller numbers to see if we can find any pattern that we can use to help us arrive at the solution. Instead of finding the probability of making 21 shots and missing at most 6, for now, let's use making 4 shots and missing at most 3. We will still assume his probability of making a shot is 71%.

There are several arrangements possible for 4 shots made and up to 3 missed shots. Let's write out the arrangements and see if we can get some insight. To keep things simple, we will be using S for success (made shot) and F for failure (missed shot). Here are the possible arrangements for wins (4 made shots in 4 to 7 attempts):

Shots Attempted

Outcome Summary

4S & 0F

4S & 1F

4S & 2F

4S & 3F

Possible arrangements
(disregarding rules)

SSSS

SSSSF
SSSFS
SSFSS
SFSSS
FSSSS

SSSSFF
SSSFSF
SSFSSF
SFSSSF
FSSSSF

SSSFFS
SSFSFS
SFSSFS
FSSSFS
SSFFSS

SFSFSS
FSSFSS
SFFSSS
FSFSSS
FFSSSS

SSSSFFF
SSSFSFF
SSFSSFF
SFSSSFF
FSSSSFF

SSSFFSF
SSFSFSF
SFSSFSF
FSSSFSF
SSFFSSF

SFSFSSF
FSSFSSF
SFFSSSF
FSFSSSF
FFSSSSF

SSSFFFS
SSFSFFS
SFSSFFS
FSSSFFS
SSFFSFS

SFSFSFS
FSSFSFS
SFFSSFS
FSFSSFS
FFSSSFS

SSFFFSS
SFSFFSS
FSSFFSS
SFFSFSS
FSFSFSS

FFSSFSS
SFFFSSS
FSFFSSS
FFSFSSS
FFFSSSS

One thing to note is that certain arrangements are just not possible. For example:

SSSSF

is not a possible arrangement since 4 shots were already made (S), there wouldn't be a 5th attempt that results in a miss (F).

In fact, every arrangement that has 4 made shots (S) and where the last shot is a miss (F) is not possible, the game would end before the last shot was even attempted. In the above table, the arrangements which are not possible are bolded. A summary of the arrangements are in the table below.

Shots Attempted	4	5	6	7
Outcome Summary	4S & 0F	4S & 1F	4S & 2F	4S & 3F
Number of arrangements (disregarding rules)	1	5	15	35
Number of impossible arrangements (last attempt is F)	0	1	5	15
Number of arrangements (following rules)	1-0=1	5-1=4	15-5=10	35-15=20

The question is essentially "What is the probability that he makes 4 shots out of 4 attempts, 4 shots out of 5 attempts, out of 6 attempts, or out of 7." We will multiply the probability of those events occurring by the number of ways (different arrangements) it can occur and add them together.

We assumed earlier that the probability of making a shot is .71, so the probability of missing a shot (its complement) is 1-.71=.29.

For 4 successes in 4 attempts, we see there's only 1 way it can happen:
\[ \begin{align*}
& 1 * (.71)(.71)(.71)(.71) \\
& 1 * (.71)^4 \\
& 1 * .25411681 \\
& .25411681 \end{align*} \]For 4 successes in 5 attempts, we see there are 4 ways it can happen:
\[ \begin{align*}
& 4 * (.71)(.71)(.71)(.71)(.29) \\
& 4 * (.71)^4(.29) \\
& 4 * .25411681 * .29 \\
& 4 * .0736938749 \\
& .2947754996 \end{align*} \]For 4 successes in 6 attempts, we see there are 10 ways it can happen:
\[ \begin{align*}
& 10 * (.71)(.71)(.71)(.71)(.29)(.29) \\
& 10 * (.71)^4(.29)^2 \\
& 10 * .25411681 * .0841 \\
& 10 * .021371223721 \\
& .21371223721 \end{align*} \]For 4 successes in 7 attempts, we see there are 20 ways it can happen:
\[ \begin{align*}
& 20 * (.71)(.71)(.71)(.71)(.29)(.29)(.29) \\
& 20 * (.71)^4(.29)^3 \\
& 20 * .25411681 * .024389 \\
& 20 * 0.00619765487909 \\
& .1239530975818 \end{align*} \]Adding all together we get:
\[ \begin{align*}
&= .25411681 + .2947754996 + .21371223721 + .1239530975818 \\
&= .8865576443918 \end{align*} \]
So assuming Jeremy Lin's probability of making a 3-point shot is 71% and the shots are indepedent, then the probability of him making 4 shots before he misses 4 (up to 3 missed shots) is about 88.66% (assuming I made the right assumptions and did the calculations correctly).

Can we use this pattern to answer the question in our original post?

Question 1: Assuming each shot is independent of the others and his probability of making a shot is 71%, what is the probability that Jeremy Lin beats the ghost (essentially he makes 21 shots before he misses 7)?

I'll put up a simulator in the next post so we can check our answer when we arrive at it.

Jeremy Lin vs The Ghost
Jeremy Lin vs The Ghost (Finding Patterns)
Jeremy Lin vs The Ghost (Simulation)

Touch Mathematics Trigonometry in GeoGebra

noreply@blogger.com (Mr. H) — Sat, 01 Sep 2012 06:58:00 +0000

This is an applet I made last year while learning the various features of GeoGebra. I thought that I might learn a few things from trying to accomplish what Touch Mathematics's trigonometry flash applet (CC BY-NC-SA) does and I might use it myself in class with my students.

Dragging the large grey dot on the unit circle to see various angles in degrees and radian along with the value of the six trigonometric function at that angle. On the graph window, a trace of the values will slowly reveal the graph of the trigonometric functions.

To make the applet fit in this blog, I split the unit circle from the graph, but it can be easily recombined so that it looks just like Touch Mathematics version.

If you would like to use it in class, click File then Save As to save a copy to be used later. If close the unit circle window accidentally and would like it back, click View then Graphics 2 and it should reappear.

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Jeremy Lin vs The Ghost

noreply@blogger.com (Mr. H) — Thu, 26 Jul 2012 20:34:00 +0000

At the height of Linsanity, when I read everything I found about Jeremy Lin, I came across The Evolution of Jeremy Lin as a Point Guard at New York Times. As I read about a drill/game called "Beat the Ghost" that Jeremy Lin plays while training to improve his 3-pointers, I thought about questions I could ask my math students. Here's the text:

Lin’s perfectionist tendencies came out in a 3-point-shooting drill called “beat the ghost,” in which Lin earned 1 point for every shot he made at the arc and the “ghost” earned 3 points for every shot Lin missed.

On one occasion, Lin made 17 3-pointers but lost 21-17, then kicked the ball in anger, Scheppler recalled with a chuckle. He refused to stop until he beat the ghost. It took 14 games. When Scheppler tallied up all of the scores for the day, Lin had converted 71 percent of his shots from the arc. “That’s the beauty of Jeremy Lin,” Scheppler said. “It’s not about moral victories. It’s ‘I have to win.’ ”

Question 1: Assuming each shot is independent of the others and his probability of making a shot is 71%, what is the probability that Jeremy Lin beats the ghost (essentially he makes 21 shots before he misses 7)?

Question 2: According to the article, it took him 14 games to beat the ghost. Is this typical? On average, do we expect him to take this many games before he wins (he was unlucky with his shots) or do we expect it to take longer than 14 games before he wins (he was lucky he beat the ghost in only 14)?

Here's a video of Jeremy Lin's former teammate Steve Novak, the NBA leader in 3-pt field goal percentage last season (2011-2012), in his pre-game routine making it look easy.

It's not all that surprising when you read that one of his feats of marksmanship include making 96 out of 100 3-pointers during a pre-draft workout with the Spurs in 2006 (espn).

Question 3: Assuming each shot is independent of the others and his probability of making a shot is 96%, what is the probability that Steve Novak beats the ghost (essentially he makes 21 shots before he misses 7)?

You can probably guess where I'm going next.

Extension Question: Assuming each shot is independent of the others and a player's probability of making a shot is p, what is the probability that the player beats the ghost (essentially the player makes S shots before F misses)?

I haven't worked it all out yet. I'll get the calculations, diagrams, and simulations up when I get some time to finish them. In the mean time, I thought you'd enjoy doing some math as the school year approaches.

Jeremy Lin vs The Ghost
Jeremy Lin vs The Ghost (Finding Patterns)
Jeremy Lin vs The Ghost (Simulation)

Mr. Continuous Brow

noreply@blogger.com (Mr. H) — Fri, 29 Jun 2012 05:32:00 +0000

The top pick of the 2012 NBA draft has an unique feature which he trademarked recently. Best of all, it's awesome how his mom celebrated it at a game by wearing a mask. Here's my celebration inspired by a mathy conversation with students.

Is it differentiable?

Predictably Irrational (Experimental Design)

noreply@blogger.com (Mr. H) — Sat, 12 May 2012 14:51:00 +0000

Interesting TED Talk by Dan Ariely, a professor of Psychology and Behavioral Economics at Duke University, on irrationality in decision-making. I saw this a few years ago and found it excellent for starting conversations with Stats students about experimental design, bias, and (if later in the year) practice with hypothesis testing.

Here are some questions to start discussion:

Regarding the data summarized in a bar chart regarding % of drivers donating organs in the various European countries, was it collected through a study or an experiment? Explain.
According to the video, what was the difference between the countries with high percentage and the countries with low percentage of organ donors?
Sketch a possible experimental design diagram for the experiment involving doctors and pain medication hip replacement.
Sketch a possible experimental design diagram for the experiment involving subscription to the Economist. What was the population of interest? To what extent should the conclusion be drawn?
Sketch a possible experimental design diagram for the experiment regarding ugly Tom and ugly Jerry. Was there a control group in this experiment (based on just the information provided)? Explain.
What bias is being investigated in the video?

Here are some screenshots from the video to set context for question 4:
Treatment 1:

Treatment 2:

This is an AP Test style sample answer to question 4:

Lastly, check the conditions and perform an appropriate hypothesis test using the data from the video. Here's the screenshot with the results:

That's all I have come up with for now. I would appreciate suggestions for improvement.

This is one of the more memorable videos for kids, especially the ugly Tom and ugly Jerry's effect on Tom and Jerry respectively. Students got to see examples of experiments and how they were designed so that conclusions can be drawn. I think it would be even better if the students also got to read the original papers, especially if the papers use the statistical methods learned at the high school level.

Steve Novak vs Tyson Chandler

noreply@blogger.com (Mr. H) — Thu, 10 May 2012 15:59:00 +0000

The 2011-2012 season of the NBA has been one of the most memorable seasons, at least for this former and now renewed fan of the NBA. I stopped following the NBA after Michael Jordan's second retirement from the Chicago Bulls, partly because I didn't connect with any player or team. I'd catch a few games every now and then, but I haven't cared about any player or any team for quite some time... until Linsanity. The rise of Jeremy Lin has been a joy to witness. The impact of it all, especially on the Asian or just the Taiwanese community, will have to wait for another post. Today, we'll focus on a math question.

One thing about watching Jeremy Lin is that I also got to watch many of his teammates on the New York Knicks. Two players who had an impressive statistical season are Steve Novak and Tyson Chandler. One for having a league-leading 3-point field goal percentage and the other for league-leading field goal percentage. They were both very impressive, but who had the more impressive achievement (statistically speaking)?

To answer the question, we can choose to compare them with players from any time or just with players in this season. I've decided to compare them with the latter, it seems more reasonable to compare them to their peers. Rules of the game do change and player abilities also evolve over time.

A little digging and I find an great site for basketball statistics at basketball-reference.com. It took a little time to clean up the data and combine the stats for players who were traded mid-season. One other issue that popped up were outliers, players who had 0% because they attempted low number of shots and missed all of them and players who had a very high percentage because they made most or all of the few shots they had attempted. I decided (arbitrarily) to set the minimum number of attempts to 66 (the number of games in this shortened season) which is equivalent to about 1 shot attempt per game. Doing this eliminated these outliers and left a histogram that is approximately normal.

From the histograms above, Tyson Chandler seems to be further away from the pack than Steve Novak. In Statistics, there's a useful measure of the distance from the mean in terms of standard deviation that is useful here. It's the z-score. It gives us an idea about how typical or how extreme a value is relative to peers. Z-score is given by the quantity of the value (x) minus the mean (mu), divided by the standard deviation (sigma).
\[ \begin{align*}
z &= \frac{ x - \mu }{\sigma}
\end{align*} \]
Calculating the z-score for Steve Novak's 3-point field goal percentage, when compared to his peers, we get
\[ \begin{align*}
z_{novak} &= \frac{ 0.471631206 - 0.351442150 }{0.051557762}\\
&= 2.33115347
\end{align*} \]
Calculating the z-score for Tyson Chandler's field goal percentage, when compared to his peers, we get
\[ \begin{align*}
z_{chandler} &= \frac{ 0.678873239 - 0.441181706 }{0.060831197}\\
&= 3.9073953
\end{align*} \]
While both achievements are impressive, Chandler's field goal percentage is almost 4 standard deviations above the mean while Novak's performance is slightly higher than about 2 standard deviations. Chandler wins. It's not even close. One thing to keep in mind, even as we draw this conclusion, is that players who attempt a large number of 3-pointers are typically pretty good at it. Whereas, it's reasonable that even poor shooters will attempt 2-pointers so Tyson's "peers" may have a larger proportion of poor shooters thus lowering the average.

Now, if only I could get my hands on the data for the points scored in the first 5 career starts of a player. :)

PS: Apparently, Tyson Chandler's .679 field goal percentage is second (third?) only to Wilt Chamberlain's .7270 (1972-1973) and .6826 (1966-1967) on the all time list for highest field goal percentage in a single season. Unfortunately, Chandler did not meet the rate minimum requirements to show up on that list. For field goal percentage, the requirement is 300 FG. Chandler had 241 FG.

PPS: Chandler could also have made the true shooting percentage list. Once again, unfortunately, for true shooting percentage, the requirement is 700 PTS. Chandler had 699 (missed it by 1).

Graph of a Function and its Inverse (VIDEO)

noreply@blogger.com (Mr. H) — Sat, 05 May 2012 23:16:00 +0000

Using a technique from art class, this activity helps students visualize the reflection of the graph of a function about the line y=x when sketching a function and its inverse. Students who went on to calculus tell me this was one of the activities they remembered from my Pre-Calc class.

Be sure to turn on the annotations (if it is off) to see the directions and comments. This time, I decided to spare you from my voice and some random Creative Commons licensed background music I would usually add to videos.

I suppose you can also use this strategy in Geometry to help students visualize reflections. The novelty factor from doing this in a Pre-Calc class really helped students remember how a function and its inverse are connected graphically. This is especially useful when learning about logarithmic functions.

PS: This is my first time playing with annotations on youtube.

Mathy Comics - XKCD

noreply@blogger.com (Mr. H) — Fri, 04 May 2012 07:05:00 +0000

More than just mathy, it's also sciency and geeky. XKCD is a must read! When you visit the site, be sure to mouse over the picture and hold. The alt text will soon appear with a hidden punch line and commentary. Here are two of my favorites on:

Correlation and causation

Extrapolation

Andromeda Yelton (@ThatAndromeda) keeps a fairly complete directory of XKCD comics by topic called DCKX - Directory of Curricular Knowledge in xkcd. Handy for those who teach math and science.

Mathy Comics
Mathy Comics - SMBC (Saturday Morning Breakfast Cereal)
Mathy Comics - XKCD

AP Test Taking FAIL

noreply@blogger.com (Mr. H) — Fri, 04 May 2012 05:16:00 +0000

... or "this is what happens when you don't read or follow directions"

I've told it several ways. This one seemed to work well with kids:

Me: ...So a long long time ago...

Students: How long?

Me: Hrm. When you could still hear Boyz II Men on the radio.

Students: No one listens to radio anymore!

Me: Ugh. Anyways. There was a boy who came to the United States with his family hoping for better opportunities and a better education. After a recommendation from a counselor, he decided to take AP Computer Science in his sophomore year, which back then was in a programming language called Pascal and students worked on Apple IIes with green monochrome monitors and floppy disks.

He did well in the class, usually getting one of the higher scores on quizzes and tests and he was certain he would do well on the AP test. On the AP test, he breezed through the Multiple Choice section and felt confident about his performance, at least until the Free Response Questions section.

The first question in FRQ that he worked on is a function called square root. He thought long and hard about what the question was really asking and about the kind of answer the function was expecting. A satisfactory answer wasn't coming to him as quickly as it previously had in class and in practice tests. As time passed, he began to feel that a solution would elude him. Throughout his academic career he was successful because of his persistence, so he persisted.

About half-an-hour, maybe 45 minutes later, he decided that at the rate he was working, he'd never finish the test. He wasn't one to give up easily, but after some hesitation and indecision, the anxiety about not finishing the test got to him. He finally decided to move on and work on the next question and revisit the first question if he had time later. He turned over the page and looked at the next question and noticed that it said #1. Puzzled, he turned back to the first question he worked for close to an hour to see what was going on. He was shocked that it had no number. Was this a typo? Did College Board make a mistake? Upon further examination, the question that he worked on before wasn't even a question, it was just an example. His heart sank.

He doesn't remember if he completed the test, probably just blocked out the memory. He did get a score of 3 on that test, and the following year he felt it necessary to redeem himself by doing better on an even more challenging test (Computer Science AB), which he did. Eventually, the boy grew up to become a teacher and to tell his students a cautionary tale about the pitfalls of not reading directions carefully on the AP test.

THE END

This is a story I tell my students every year about the importance of reading and following directions. It took a little digging, but here it is. With a little annotation, it is pretty self-explanatory. Feel free to share it with your students.

Spiral Ink Reservoir

noreply@blogger.com (Mr. H) — Mon, 23 Apr 2012 02:11:00 +0000

What is the ink density (length-wise) along the spiral if the ink contained is exactly two times the straight ink reservoir?

What is the radius of the cylinder around which the spiral wraps around?

(via Wired)

Shikaku

noreply@blogger.com (Mr. H) — Sun, 18 Dec 2011 15:57:00 +0000

Shikaku (四角), which means four corners in Japanese, is a fun puzzle game where you fill a puzzle board with quadrilaterals (squares and rectangles) using the clues on the board. Each quadrilateral must contain only 1 number and have the area given by that number.

It's a fun little game to help students connect area with multiplication (a quadrilateral with area 6 can be 1x6, 2x3, 3x2, or 6x1), discover properties of prime and composite numbers (all prime numbers n can only be drawn with 1xn, or nx1 and no prime number can be drawn as a square), and reasoning (some quadrilaterals will have to be redrawn as you encounter limitations).

Here's a video of me playing this Shikaku puzzle.

You can find more puzzles on the Nikoli Shikaku page.

Note: Shikaku was created by the same folks popularized Sudoku.

(via Breedeen Murray)

UPDATE: Changed "came up with" to "popularized" and added link to wikipedia in comments.

Speeding Up Audio Podcasts with Winamp and PaceMaker

noreply@blogger.com (Mr. H) — Sun, 18 Dec 2011 00:02:00 +0000

Here's a time-saving tip for those of you Winamp users who follow more podcasts than you have time to digest. There's a plugin called PaceMaker that allows you to speed up and slow down MP3s without changing pitch. This is useful if you want to listen to podcasts without feeling like you're listening to Alvin and Chipmunks. I first learned of the PaceMaker plugin in college when I was learning to play the guitar. It was helpful to be able to slow down songs so I could get some practice improving finger coordination as I strummed new chords along with the song.

Here's what you need to use it:

If you don't have it, grab a copy of Winamp from the Winamp download page. I usually skip the MP3 bundle that comes with the download.
Install Winamp.
Once Winamp is installed, you'll need to visit the PaceMaker Winamp plugin page and grab a copy of the installer.
Install PaceMaker.
To enable the PaceMaker plugin, go to "Tools" select the "Preferences" menu item.

Scroll down on the left and click on "DSP/Effect" then select the "PaceMaker tempo controller" option in the right pane.

Adjust the tempo to speed up the audio podcast. Then just play your MP3.

I find that for most podcasts +40% is a comfortable pace. This is true if the podcast is on a familiar subject or if it's easy to follow. Some podcasts can be played at even higher tempos but at higher tempos I spend more effort listening carefully to words than paying attention to the discussion.

There's some simple yet useful math in here, but that'll be another post.

3D Video of Asteroid Vesta

noreply@blogger.com (Mr. H) — Mon, 05 Dec 2011 00:48:00 +0000

A stereoscopic anaglyph 3D video created by NASA's Dawn Spacecraft.

I think a commenter on Slashdot said it best:

Each time NASA releases images from some distant planet or asteroid, I'm floored. The number of things that have to go right, that have to not fail, millions of miles away, is immense. Kudos to the scientists and engineers who worked on imaging Vesta. Fantastic results!

(via Slashdot)

Mathy Conversations with Students

noreply@blogger.com (Mr. H) — Thu, 01 Dec 2011 22:48:00 +0000

I use many non-verbal cues in my classroom. Often, I'll use hand signals or facial expressions to convey my requests/commands to students quietly. The signals and expressions help me avoid interrupting the class just to address a single student when other students are already working.

Today, I gave a student a judgmental look to hint that he wasn't doing what he was supposed to be doing. Usually students just go back to work, but today he decided to point it out.

Jimmy: You always give me that look!
Me feigning innocence: What look?
Jimmy: The one where your eyebrows are like this (student signs the letter v with his hands).
Me acting clueless while giving a subtler version of the expression I gave him: Huh?
Jimmy: Yeah. Your eyebrows look like a V. It's like they're perpendicular.
Me giving my best perpendicular brows: That's impossible! I'll even take a picture and show you that they aren't perpendicular! Eyebrows can't be perpendicular.
Jill: Some girls will draw 90 degree eyebrows using a Sharpie!
Me signing a V on my forehead: Like a 90 degree unibrow?
Jill: No, they draw sharp corners for eyebrows like this (she signals a circumflex).
Me: Are you saying I have a continuous function for an eyebrow?
Jimmy: No, you don't have a unibrow, it's discontinuous.
Jill laughs: Yea. You have a removable discontinuity!
Me laughing at their insight: Because the limit exists! My eyebrows meet from the left and right and it's undefined in the middle.
Me sensing more students getting off-task: Alrighty, back to work!

A spontaneous opportunity like this doesn't come up often which is why I'm mad I missed an opportunity for review here. I should have asked "what would my eyebrows have to look like for there to be a non-removable discontinuity?" instead of hurrying the kids back to work. Maybe someone would have responded with "you just raise one eyebrow so they don't meet in the middle." I need to be more open to opportunities like this one. I love it when students can apply a concept learned in class in unexpected ways. I love the fact that I get to have these conversations at all. I love my job!

A Function and its Inverse (Office Supply Edition)

noreply@blogger.com (Mr. H) — Sun, 13 Nov 2011 01:42:00 +0000

A visual representation for the notation of a function and its inverse.

Doesn't work as well if you have the stapler on the outside and the staple remover on the inside. Would've been more useful if instead of x, there were a stack of papers. Maybe if I can find some time or if someone has the time...

(via MathFail)

PS: If you like this, also check out f(a bag of skittles) over at Sweeney Math.
PPS: Quick google search yields this thread over at reddit. Following comment is by trebor89.

The function f(y) = (staple remover (stapler (y)) is NOT the inverse; it's actually the function commonly known as Gödel's Umlautifier. Simply try it on some trivial input: f(a) = ä.

Kaprekar's Constant (6174) in GeoGebra

noreply@blogger.com (Mr. H) — Sat, 12 Nov 2011 22:40:00 +0000

While doing a little browsing, I came across Kaprekar's Constant. I remember reading about it a few years ago but I never did anything with it. I finally decided to play with it a little using GeoGebra. Here's a description from Wikipedia:

6174 is known as Kaprekar's constant after the Indian mathematician D. R. Kaprekar. This number is notable for the following property:
Take any four-digit number, using at least two different digits. (Leading zeros are allowed.)
Arrange the digits in ascending and then in descending order to get two four-digit numbers, adding leading zeros if necessary.
Subtract the smaller number from the bigger number.
Go back to step 2.
The above process, known as Kaprekar's routine, will always reach its fixed point, 6174, in at most 7 iterations. Once 6174 is reached, the process will continue yielding 7641 – 1467 = 6174. For example, choose 3524:
5432 – 2345 = 3087
8730 – 0378 = 8352
8532 – 2358 = 6174
The only four-digit numbers for which Kaprekar's routine does not reach 6174 are repdigits such as 1111, which give the result 0 after a single iteration. All other four-digit numbers eventually reach 6174 if leading zeros are used to keep the number of digits at 4:

Here's a screenshot of the GeoGebra applet:

If you'd like a copy just click on "File" then "Save As" and save a copy on your computer. Click on the "Play" button on the bottom left of the applet to begin the animation.

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

11/11/11 Creative Day

noreply@blogger.com (Mr. H) — Fri, 11 Nov 2011 19:11:00 +0000

If the number zero were 陰 (yin) and the number one were 陽 (yang), today (11/11/11) would make up the first hexagram of the 易經 (I-Ching/Yì Jīng) or the Book of Changes. The first of 64 hexagrams, 乾 (ch'ien/qián) is the only hexagram that matches our date with 6 yangs. It is sometimes translated as "the creative," "heaven," or "force."

The hexagram is depicted below on the left. In the middle is the Chinese character for ch'ien/qián in ancient script font and on the right is the same character is a modern script.

Drumroll...

I hereby proclaim 11:11:11 AM of 11/11/11 to be the Creative Moment of Creative Day... to be repeated every 100 years if we ignore the first two digits of the four-digit year. We sort of missed the first and only real 11/11/11.

Go forth and create something on this Creative Day! It's as good a reason as any.

PS: 3000-4000 years ago, I'm sure the authors of I-Ching had in mind this date and moment based on a calendar and time system not yet invented or adopted.

PPS: Somehow, the 40,271th cycle of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom on 315th rotation of the 2011th revolution of the 3rd wandering star big enough to be rounded by its own gravity around an yellow dwarf calculated using a system agreed upon on the 55th rotation of the 1582th revolution since some arbitrary time around a ball of fusing atoms is special.

Facts + Sensitivity => Impact

noreply@blogger.com (Mr. H) — Sun, 23 Oct 2011 05:57:00 +0000

Here's an interesting exchange between Neil deGrasse Tyson and Richard Dawkins. Be sure to watch the entire clip.

Two quotes to think about:

Tyson: Being an educator is not only getting the truth right but there's gotta be an act of persuasion in there as well. Persuasion isn't always here are the facts and you're either an idiot or you're not. It's here are the facts and here's a sensitivity to your state of mind and it's the facts plus the sensitivity when convolved together creates impact.

Dawkins: Just one anecdote to show that I'm not the worst in this thing. A former and highly successful editor of New Scientist magazine, who actually built up New Scientist to great new heights, was asked "what is your philosophy at New Scientist?" and he said "Our philosophy at New Scientist is this: science is interesting and if you don't agree you can f#@k off."

Mathy "60-second Adventures in Thought" Videos

noreply@blogger.com (Mr. H) — Fri, 21 Oct 2011 06:45:00 +0000

I came across a set of six "60-second Adventures in Thought" videos by the Open University on thought experiments in philosophy, two of which I think would work well in a math class.

This first mathy video is on Achilles and Tortoise, one of Zeno's Paradoxes. Great for Calculus.

This second video is on Hilbert's Infinite Hotel about infinite sets. At the high school level, it could be a good fit in a math club.

There are more videos linked at Brain Pickings on Grandfather Paradox about time travel, on John Searle's Chinese Room about artificial intelligence, on Twin Paradox about relativity, and on Schrödinger's Cat about quantum mechanics.
(via Vitor Pamplona)