In this Newsletter:
1. Solution for the range hood problem
2. Resource  Mathapedia
3. IntMath polls
4. Math puzzles
5. Friday math movie  8 math talks to blow your mind
6. Final thought  Despicable
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]]>In this Newsletter:
1. Solution for the range hood problem
2. Resource – Mathapedia
3. IntMath polls
4. Math puzzles
5. Friday math movie – 8 math talks to blow your mind
6. Final thought – Despicable
Last newsletter I posed a problem regarding a stove range hood with curved sides. I usually get about 10 responses to each puzzle, but this time there was just one reply, which said "It’s very hard!". Here’s my go at a solution. My first attempts were to make it much more complicated than it needs to be. 
Mathapedia is an interesting devlopment by Dan Lynch. It’s a system for developing online math documents, and it produces output for all sorts of devices, including tablets and phones. It easily produces interactive math documents, like this example: Interactive (run your mouse over any of the diagrams on that page). There’s also an interesting section on The Art of Digital Publishing. 
The recent IntMath poll asked readers,
What’s the main thing teachers should do so you learn better?
There is some good advice to follow here, especially for math teachers. Obviously, there are not many friendly math classes out there! Here are the results.
Create a warm and friendly classroom
26%
Be enthusiastic
24%
Be flexible
21%
Love learning
11%
Respect students more
11%
Set high expectations
8%
Total votes: 2000
Poll date: Feb  Apr 2014
Latest poll: The current poll asks what kind of math you will find most useful in the future. You can respond on any (inner) page of IntMath.
I already talked about last Newsletter’s puzzle above.
New math puzzle
For integers A, B, how many solutions (A, B) are there for
A^{2} + B^{2} ≤ 12?
Please leave your responses here.
There are some really good thought provoking videos here. I’ve featured some of them in the past. I hope you find something to blow your mind!
TED has an incredibly broad range of really good talks. This collection of provocative math talks has been around for a while now, but they’re still worth a look. 
Here’s a great quote from Rockefeller, 19th century billionaire businessman and philanthropist. He was also founder of both the University of Chicago and Rockefeller University and funded the establishment of Central Philippine University in the Philippines. [Source]
I know of nothing more despicable and pathetic than a man who devotes all the hours of the waking day to the making of money for money’s sake. [John D. Rockefeller]
Until next time, enjoy whatever you learn.
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]]>I am a consultant to cabinet manufacturers. I program their cabinet software to calculate and cutout their cabinet parts on CNC (computer numerical control) machines. A lot of cabinet companies create hoods that go over stoves that are curved on both sides and front. I need to calculate the joint of that curve as if it is laying flat. How do you calculate this?
John provided a picture of one of the typical hoods that he is talking about.
What he needs is a formula to plot the Y coordinate of the curve given the X or viceversa.
This was an interesting problem that involved some actual "real world" applications of math.
The requirement was for a "curved side" with no other specifications, so I proceeded as follows, based on the picture he supplied.
I happen to have a Chinese lamp shade which also has curved sides and it’s very similar to the image he sent, so I used that as a model to make sure what I had was OK. I like to work with concrete materials first to give me a better feel for what is going on.
I used a piece of paper to outline one side of the shade, then measured the side of the paper (laid flat) in 5 locations and using Cartesian coordinates, it looked like this:
For the diagram above, I used my favorite tool for this kind of job, GeoGebra (a free open source tool.)
In Geogebra, it’s simple to ask for a polynomial passing through the 5 points, using the following syntax:
Polynomial[A,B,C,D,E]
While it was potentially a quartic (because it passes through 5 points), Geogebra decided a parabola would do the job:
f(x) = 0.03x^2 – 0.64x + 14
NOTE 1: I moved the curve on its side because points AB are almost vertical otherwise, and that gives messy polynomials. It’s much smoother if the curve is almost horizontal.
If we had proceeded with the curve in upright mode, the points would have looked like this:
With the points almost vertical, the resulting polynomial is a quartic, and certainly not useful to us. (I have changed the vertical scale for this next graph).
We don’t get a smooth curve joining the 5 points.
NOTE 2: You can also use Microsoft Excel to find a polynomial for you as well. (See the bottom of the DJIA Graph page for the procedure).
So I printed out my curves on paper, and cut them out. The 4 shapes were like this:
And the final mockup (using paper):
Yes, it’s rough, but the joins appear to be all good (within the mm or so tolerance of my cutting and sticking…)
So one generic solution for the problem is a parabola:
f(x) = 0.03x^2 – 0.64x + 14
As a side note, this problem reminded me of the shapes of Singapore’s iconic Marina Bay Sands Hotel. (I live in Singapore).
[Image source]
John replied to my first solution giving more detailed specifications, as follows:
So we now have to figure out the equation of the joins for circular sides, when laid flat.
John also supplied some dimensions for a typical case. It will be 12 units high, and 9.37 units from the top join to the bottom join, in the horizontal direction, as follows:
The curve at O is vertical, and the arc is 12 units in the vertical direction from O. When we calculate the arc length p, we find it is:
This leads us to consider the following possible solutions which represent the curved side laid flat (and once again rotated by 90° for the reasons given above).
Of course, there are infinitely many possible curves passing through (0,0) and (16.4,9.37), but the curve needs to be horizontal at (0,0), so that when we turn it 90°, it will be vertical.
Which curve could it be? And, what is the equation of that curve? Remember, it has to fit exactly with the other curves that are at right angles to it, and give us a circular arc when joined.
After several failed experiments involving complicated attempts at projecting the curved surface onto a flat plane, I stepped back and observed the 3rd curve looks a lot like a parabola, and the other 2 looked like they weren’t going to give us a circular curved surface. The useful paper, Unwrapping Curves from Cylinders and Cones, was the deciding factor.
Assuming now it will be parabolic, and given that it needs to pass through (16.4, 9.37), we can simply use the general form of a parabola:
Then solve for a by substituting :
This gives us:
I’ve left it in fraction form so we can more easily see where the general formula comes from later.
I then graphed this from x = 0 to x = 16.4 (the arc length).
Based on the simple parabola, I cut out shapes as in the following.
Here is the resulting range hood model made from paper.
The height of the curved part is now 12 units (as required) and the distance from the bottom left point in the picture the the top of the curved join was just over 9 units, as required. (Of course, some tolerance is applied since we are working with paper.)
You can see when we view the model from the side that the curved edges are indeed circular arcs, and it’s vertical at the top of the arc.
John wanted a general solution for the curve.
If the radius of the circular arc is given by r, and point K is (m,n) then the arc length will be:
The required curve will be:
I’ll leave it as an exercise for the reader to prove we actually will get sides which are arcs of a circle when using a parabola. ^_^)
(1) Often the simplest answer is the best.
(2) The quadratic curve is very handy in math!
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]]>In this Newsletter:
1. New zTable interactive graph
2. Resource: CK12.org
3. Math puzzles
4. Friday math movie: Complex numbers in math class
5. Final thought: Giving
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]]>In this Newsletter:
1. New zTable interactive graph
2. Resource: CK12.org
3. Math puzzles
4. Friday math movie: Complex numbers in math class
5. Final thought: Giving
After I sent the last IntMath Newsletter, I made some updates to the script. When testing it, unfortunately some of my test mails (which contained text like "Lorem ipsum dolor sit amet") went out to a small number of you. I’m very sorry about that!
Some students get lost understanding what normal probability distributions are all about. You can investigate the meaning of the ztable using this new interactive graph. 
Here’s some background on the topic:
Normal Probability Distribution
And here’s some background on the new applet:
CK12.org is a large collection of free math and science resources. There’s lessons, selftesting quizzes, videos and interactive applets. As one of the students says:

It’s worth a look!
The puzzle in the last IntMath Newsletter asked about the percentage of pages in a book containing the number 5.
Correct answers with explanation were given by Michael, Nicos and Sai Krishna, and Sheldon had the right number of pages. It was interesting to see the different approaches taken by the solvers. Mathematics rarely has one "correct" way of going about things.
Also, some of you questioned whether "a 5" means "one 5 only" or "at least one 5". Interpreting math word questions is one of the biggest challenges people face – reading skills are important in math, too!
New math puzzle
I received a letter from John recently. His problem is a really interesting one because it involves many areas of mathematics. Here’s what he wrote:
I am a consultant to cabinet manufacturers. I program their cabinet software to calculate and cutout their cabinet parts on CNC (computer numerical control) machines. A lot of cabinet companies create hoods that go over stoves that are curved on both sides and front. I need to calculate the joint of that curve as if it is laying flat. How do you calculate this?
Here is a picture of the hood.
When viewed from the sides, the curves are arcs of circles, and the very top of the curve is vertical.
The width of the front face at the bottom is 60 cm, at the top is 30 cm wide and the (final) height of the curved portion is 50 cm.
Basically what I need is a formula to plot the Y coordinate of the curve given the X or viceversa when the piece of sheet metal is laid out flat.
Once you have a solution, you are encouraged to make a model of it so you can be sure it works as expected. Take a photo of your model and upload it to Flickr, Dropbox or wherever, and embed the photo in your response (using something like <img src="LinkToYourImage">), or if that’s too troublesome, email me your photo.
Leave your responses here.
This is a very creative way to present a lesson – funny, too. 
Everyone is searching for happiness, and too many people try to find it in gadgets, cars, or other physical objects. Winston Churchill said:
We make a living by what we get, but we make a life by what we give. [Winston Churchill]
Until next time, enjoy whatever you learn.
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]]>The post New zTable interactive graph appeared first on squareCircleZ.
]]>Normal Probability Distribution Graph Interactive
You can use this applet to investigate what the ztable values actually mean.
In statistics, when we plot the distribution of a measurement (say, the heights of people) we very often produce the familiar bellshaped curve.
For example, the mean (average) height for males in the US is 175.5 cm and the standard deviation is 7.4 cm. When we plot the heights of US males, we get a bellshaped (also known as Gaussian) curve. We know that around 68% of US males have heights between 171.1 cm (one standard deviation below the mean) and 182.9 cm (one standard deviation above the mean.
However, it is difficult to work out probabilities and proportions when the mean and standard deviations are different for each type of measurement.
The solution is to standardize the measurements which means we translate the mean to 0 and the standard deviation to 1. When we graph this resulting distribution, we get the standard normal curve. We can use a zTable to find probabilities of certain events occurring. The zTable tells us the area under the standard normal curve for particular values of interest, thus telling us the probability of an event.
The applet allows you to vary the mean and standard deviation, and upper and lower boundaries of the region of interest.
Here’s a screen shot:
You can also see the calculations for your particular values.
This math applet uses JSXGraph, jQuery and MathJax. It works on tablets, albeit a bit slow. One of the challenges when developing this was to make it as efficient as possible.
The link again: Normal Probability Distribution Graph Interactive
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]]>It was an April Fools day gag by Matthew Weathers. My favorite bit is where he slams Internet Explorer.
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]]>In this Newsletter:
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]]>In this Newsletter:
Happy New Year, everyone! I hope 2014 is kind to you, and that you learn a lot.
I usually don’t blog much during the end of year period, mostly because people are not very interested in math then! (How do I know? See Thanksgiving for less math where I show what happens to IntMath traffic each festive season.)
IntMath recently passed some milestones:
I love hearing from readers and was delighted to receive this from Rachel in Ireland:
“Dear Murray
I have never met you in person, but you need to know how much help your IntMath Newsletters were to me whilst I was studying. I returned to college as a mature student so needed every bit of assistance I could get, and found that through your newsletter and site. Thank you so much.”
You’re very welcome, Rachel!
On with the Newsletter.
Here’s a great alternative to expensive college texbooks: 
Most of the books are released under a Creative Commons license and are provided by a consortium of colleges and organizations in the US, including AcademicPub, Carnegie Mellon University, Community College Consortium for Open Educational Resources, Flat World Knowledge, Macmillan’s DynamicBooks, MathScore and MERLOT.
Apart from math, you’ll find open textbooks on a broad range of topics including art, science, economics, nursing, and sociology.
And you can’t beat the price!
However, note their disclaimer, which says:
Disclaimer: College Open Textbook has not yet vetted all of the open textbooks on this list for quality or accuracy of the content.
This page has always been one of the most popular on IntMath. I updated it so it also works on tablets and (large?) smart phones.
This interactive allows you to place a mask over the photos of "beautiful" people, and see how close their face is to the "ideal". See: Math of Beauty (you’ll find the mask interactive about 1/2 way down the page) 
Skin deep? On the page, I have a disclaimer that says "inner beauty is more important than external beauty.". As I said, this is not a serious piece of science!
Studies show one of the biggest impacts on student learning is the quality of the teaching. This poll is consistent with that finding.
The poll during Nov/Dec 2013 asked readers:
What would have the biggest positive impact on your enjoyment of school?
Better teachers
44%
Learn more about real life issues
34%
Reduce bullying
11%
More technology
11%
Total votes: 1200
Current IntMath Poll: Considering the above result, the current IntMath Poll asks what teachers should do to improve learning outcomes. You can answer on any (inner) page of IntMath.com.
The puzzle in the last IntMath Newsletter asked about the least 2 positive integers with certain remainders.
The question I asked is actually very old. It was posed by the Chinese military general, strategist and philosopher Sun Tzu in the 1st century CE. (Yes, the guy who wrote Art of War.)
Correct answers with explanation were given by Nicos and Hamid.
New math puzzle
A 2000page book has pages numbered consecutively from 1. What percentage of the pages contain a 5 in the page number?
Leave your responses here.
This reminds me of Spirograph images that brought art and geometry together, and always had an element of mystery to them. Watch: Four Cable Drawing Machine 
"Think of life as a school for your soul; you are here to learn in perfect well being. Here’s a tip for life’s pop quizzes: instead of asking why something happened, ask instead ‘what can I learn?’ For extra credit, ask ‘…and how may I serve?’" [Jackson Kiddard]
Until next time, enjoy whatever you learn.
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]]>This one’s a bit of fun and not overly scientific.
Why is it that we tend to regard certain people as beautiful regardless of their (or our) race or skin color? Some people feel this is because we are attracted to people whose faces are symmetrical and who follow certain geometric shapes, especially those based on Fibonacci ratios.
This page has always been one of the most popular on IntMath. I updated it so it also works on tablets and (large?) smart phones.
Go to:
Skin deep? On the page, I have a disclaimer that says "inner beauty is more important than external beauty."
This interactive used to be Flashbased, but of course Flash doesn’t work on mobile devices.
So it now uses jQuery (a javascript library) and jQueryUI (stands for "user interface") instead, with another small file so it works with touch gestures on tablets.
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]]>
[Image source]
I came across a similar concept recently with these sand art drawing machines. This one uses 4 cables to anchor the moving object. Quite mesmerizing – and an interesting idea for a combined mathart project in school.
[Hat tip to Don Cohen for alerting me to these.]
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]]>In this Newsletter:
1. When does log(x − 3) = log x − log 3?
2. Updated math interactives
3. IntMath poll
4. Math puzzles
5. Friday math movie  Smart failure
6. Final thought  Learning to learn
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]]>In this Newsletter:
1. When does log(x − 3) = log x − log 3?
2. Updated math interactives
3. IntMath poll
4. Math puzzles
5. Friday math movie – Smart failure
6. Final thought – Learning to learn
A recent comment from a Twitter colleague got me thinking about the value of graphical solutions of equations. When does log(x3) = log x – log 3? Do you agree with me? 
These are the latest conversions I’ve done from Flash to mobilefriendly javascript.
a. Mortgage calculator with graph
The most expensive thing most people buy is a house. It’s a good idea to know what will happen when it comes to the mortgage you’ll be paying back for much of your working life. 
b. Graphing calculator
The IntMath Poll during Sep/Oct asked readers:
Generally, how do you feel about school?
It’s sad that for many students, school is not a happy place. At least the majority of respondents said either they like it or are neutral.
I hate it and can’t wait to finish
28%
I really love school
26%
It’s OK
22%
I enjoy it
18%
I don’t like it
6%
Total votes: 3100
Current IntMath Poll: With so many students unhappy at school, how can we improve it? You can give your opinion on any (inner) page of IntMath.com.
The puzzle in the last IntMath Newsletter asked about the sum of an infinite series.
Correct answers with explanation were given by Tomas, Don and Mawanda but my vote for the clearest answer is Mawanda’s.
New math puzzle
Find the least 2 positive integers having the remainders 2, 3, 2 when divided by 3, 5, 7 respectively.
Leave your responses here.
We don’t deal with failure properly in most educational settings. This talk gives some options for improving that. 
Most teachers (and parents) tell students to "study hard". But even though some students put a lot of time into study, some produce better outcomes than others.
In a 2011 study, researchers found that if students knew more about their own learning approach, the more likely they were to do well. Also, those who used a variety of learning strategies did better than those who stuck to one or two.
How many of the following successful learning strategies can you say "yes" to?
For the ones you said "no" to, why not give them a try?
Until next time, enjoy whatever you learn.
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]]>I’ve never liked aspects of logarithm notation, and my tweet that day pointed out that despite having this convention in math:
7(x − 3) = 7x − 21,
it happens that
log(x − 3)
is mostly NOT equal to
log(x) − log(3).
Such inconsistencies in math notation cause unnecessary confusion for new learners. (See more on this issue at Towards more meaningful math notation.)
Here’s the Twitter exchange:
My reply contained a link to a Desmos graph showing where the graphs of y = log(x − 3) and y = log(x) − log(3) intersect.
You only really understand many math problems if you can do them graphically, algebraically, and numerically. But I get most understanding from a graph.
First let’s see a graphical solution for this problem.
One way of solving equations is to graph the left and right sides, and look for where the graphs intersect, since that gives us the solution.
The graph I pointed to in the link was something like the following, where the green curve is y = log(x − 3) and the blue curve is y = log(x) − log(3):
Zooming in on the intersection point, we can see the 2 curves cross when x = 4.5, and the yvalue is about 0.4 (actually, it’s log_{e}(1.5) = 0.405465…).
I answered Pat’s question with a link to a graph as I felt that trying to explain the solution algebraically within the limitations of a 140character tweet would be somewhat problematic.
I’ve been thinking about Pat’s comment that graphs "still leave a mystery here".
I spent most of my student days and early teaching life solving equations algebraically. That’s what most textbooks did, and that’s what the curriculum required, so we just went ahead and did it that way. We didn’t have access to computer graphing tools (or calculators early on), so there wasn’t much option – algebraic solutions were quicker than drawing graphs by hand or finding numerical solutions.
Most of the time I didn’t have much of a "feel" for what was going on when the equations became more complicated.
I certainly had a feel for the solution when solving linear, quadratic and other polynomial equations. If the number of terms in the polynomial was small, and it was a "nice" polynomial equation where the number of roots was equal to the degree of the polynomial, all was fine.
But then as things became more complicated, you needed a better understanding of what was happening so you wouldn’t miss some of the possible xvalues. For me, that understanding came from graphs.
For example, once we started doing trigonometry I didn’t understand why there would be an infinite number of solutions for this equation:
sin 3x = 0.47
However, once I saw the graph of y = sin 3x, and noted where it intersected with the graph of y = 0.47, it was immediately obvious there were many solutions.
Back to the log question. Let’s now see how to solve our equation algebraically.
log(x − 3) = log x − log 3
Using the 2nd log law on this page, we know we can write the difference between 2 logs as:
log a − log b = log a/b
So the right hand side of our equation can be written:
log x − log 3 = log (x/3)
Our equation has become:
log(x − 3) = log (x/3)
Since there are 2 log expressions with the same base that are equal to each other, we know the expressions in brackets must be equal, so we have:
x − 3 = x/3
Solving gives:
3x − 9 = x
2x = 9
x = 4.5
So assuming base e, the following is true:
log(4.5 − 3) = log 4.5 − log 3 = log 1.5 = 0.405465…
We set up a table of values as follows.
If we try any negative value of x, we get an error. Similarly, for log(x – 3), positive values of x give us trouble unless x is bigger than 3.
x  log (x − 3)  log x − log 3 

2.5  ??  −0.18232 
3.0  ??  0 
3.5  −0.69315  0.154151 
4  0  0.287682 
4.5  0.405465  0.405465 
5  0.693147  0.510826 
5.5  0.916291  0.606136 
6  1.098612  0.693147 
6.5  1.252763  0.77319 
7  1.386294  0.847298 
7.5  1.504077  0.916291 
It looks like we have found the only solution, x = 4.5.
The problem with numerical methods, like a lot of algebraic methods, is we are left wondering if there are any more solutions. You often get caught by not considering other values.
We’ve now solved the equation graphically, algebraically and numerically.
I find I get the best insights into the problem when I can see it, and so most of the time these days I choose to solve equations by drawing a graph (using computer graphing mostly.)
It’s quick, and it gives me the best feel for the meaning of what I’ve found.
(1) Different log bases
Does it matter which logarithm base we are using? Do we get the same xvalue?
Above I was using base e (the natural logarithms). Here is the graph if we are using base 10.
Base 10
We can see the intersection point is when x = 4.5, but the resulting yvalue is lower.
Base e and Base 10 solution comparison
In the following graph, we can also see where the base e graphs intersect (in lighter colors) and the log (base e) value is shown with a red dot.
(2) Alternative graphical approach
We could have also approached this problem by getting everything on the left side of the equation, as follows:
log(x − 3) = log x − log 3, when
log(x − 3) − (log x − log 3) = 0
That is:
log(x − 3) − log x + log 3 = 0
Here’s the graph of y = log(x − 3) − log x + log 3
We can see it cuts the xaxis at x = 4.5, and this is our solution.
But this doesn’t give us as much "feel" about what’s going on as the graphs given at the beginning of this article, where we graphed both sides of the equaiton separately and then looked for the intersection.
(3) ln x or log_{e} x?
I think “ln x” is terrible notation for natural logs. I have used log_{e} x throughout this article as it is much easier to identify that it is a logarithm expression, and so easier to understand. I ranted about this issue in Logarithms – a visual introduction.
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