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]]>I recently added a new Riemann Sums Applet to IntMath which demonstrates some of the ways it can be done, without calculus.
Here is some background to the concept.
For centuries, mathematicians tried to find efficient ways of calculating rates of change. They needed to solve a wide range of practical problems, including building faster ships, developing more accurate weapons and predicting the positions of the stars and planets for better navigation.
They were also interested in the reverse problem. That is, they may have known the rate of change, but they wanted to know the fundamental relationship between two changing quantities. For example, they may have known the velocity of an object at different times, and they wanted to know the position of the object in space during those times.
Since the age of the ancient Greeks, there were good approximation methods for such problems.
A Riemann Sum is an approximation of the area under a curve (that is, between a curve and an axis). It’s named after Bernhard Riemann, the German mathematician who worked on this problem in the mid19th century.
The principle behind Riemann Sums is simple, and was known since the time of Archimedes. (See Archimedes and the area of a parabolic segment.)
We just divide the area into smaller pieces using simple geometric shapes for which we can easily find the area, like rectangles.
If we find the area of each rectangle, and add those areas, we’ll get a good approximation for the area under the curve. If we take more rectangles, the error will reduce, and we’ll get a better approximation.
We can place the rectangles in different ways (just above the curve, just below, or midway like in the above image).
I recently added a new applet that allows you to explore this concept. You can change the placement of the rectangles, and you can even see how trapezoids and parabolas are used as very good approximations.
You’ll find the new applet here: Riemann Sums Applet
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]]>In this Newsletter:
1. Arc length of a spiral around a paraboloid
2. Padgett  from brain injury to math genius
3. Thank you, volunteers
4. Math puzzles
5. Final thought  Asking good questions
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]]>In this Newsletter:
1. Arc length of a spiral around a paraboloid
2. Padgett – from brain injury to math genius
3. Thank you, volunteers!
4. Math puzzles
5. Final thought – Asking good questions
What is the length of a spiral around a paraboloid? This is an interesting application of integral calculus, involving an environmentally friendly cooker design. 
This is a fascinating story. Jason Padgett was attacked by 2 men one night and savagely beaten. He ended up with severe concussion and an amazing new ability to "see the world through the lens of geometry". See the full story at: Brain Injury Turns Man Into Math Genius 
Don’t miss the gallery of his art while you’re there.
Last mail I asked for volunteers to help transcribing some calculus videos on IntMath.
Thank you very much to Abby, Rusty, Lisa, Kok Soon, and Suzanne for helping out. Good job and much appreciated.
The puzzle in the last IntMath Newsletter involved a bet between Brigette and Ric, concerning cars.
Correct answers with explanation were given by Tomas, Saikrishna and Michael, giving a good insight into the thought processes they used.
New math puzzle: A box contains 20 balls numbered 1 to 20. What is the probability that any two of these balls chosen from the box will have numbers that are relatively prime?
Leave your responses here.
A lot of people who write to me have trouble articulating where their confusion lies. In class, a lot of students shy away from asking questions because they don’t want to look stupid.
A common theme here is that students often don’t have the skills to ask questions.
Dan Rothstein and Luz Santana in the book Make Just One Change: Teach Students to Ask Their Own Questions make the following crucial point:
The skill of being able to generate a wide range of questions and strategize about how to use them effectively is rarely, if ever, deliberately taught.
Asking good questions is essential for effective learning. When was the last time you were in a math class that taught how to ask good questions?
Until next time, enjoy whatever you learn.
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]]>The post Arc length of a spiral around a paraboloid appeared first on squareCircleZ.
]]>A reader from Africa, Magnar, recently asked about running a string in a spiral around a parabolic dish, as illustrated in the following image:
Magnar wanted to know how to calculate the length of such a spiral.
There’s an interesting story behind Magnar’s question. He builds parabolic dish cookers in Africa. In this photo, he’s helping to install one of his dishes in a village. The dish concentrates the heat of the sun and the cook pot is placed at the focus of the parabola.
Image source (used with permission)
This is a great use of sustainable energy, it keeps the air cleaner (by avoiding open fires inside huts), and saves women from the backbreaking and timeconsuming task of gathering firewood every day.
Let’s look at some of the math behind this problem.
A paraboloid is the 3D surface resulting from the rotation of a parabola around an axis. The equation of a simple paraboloid is given by the folmula:
z = x^{2} + y^{2}
The surface generated by that equation looks like this, if we take values of both x and y from −5 to 5:
Some typical points on this curve are (0,0,0), (1,1,2), (2,3,13) and (3,4,25). That last point is on the top lip of the surface in the 3D graph given above, at the front, facing us. (In this graph, the x and yaxes axes are equally scaled, but the zaxis is not.)
We can also represent this paraboloid using parametric equations with variables t and r, as follows:
x = r cos(t)
y = r sin(t)
z = r^{2}
We could write this as an ordered triple, like this:
(r cos(t), r sin(t), r^{2})
We only need to let t take values from 0 to 2π (otherwise we just "paint" the surface more than necessary) and r takes values from 0 to 5 if our aim is to create the same curved surface as shown in the above graph.
Let’s get back to the original problem. We want to create a spiral around the surface of the paraboloid. We modify the earlier parametric equations to get a curve rather than a surface, like this.
x = t cos(t)
y = t sin(t)
z = t^{2}
Once again, we could write this as an ordered triple, as follows:
(t cos(t), t sin(t), t^{2})
We no longer create a surface with this expression. Instead, it will be a curve.
As the variable t takes various values starting from t = 0, it generates a spiral around the 3D surface.
Let’s see how, starting with a simple case.
Let’s consider the case where we fix the curve so it is in the xz plane only (it’s not actually spiralling in this case, to make things simple.) We write this as:
(t, 0, t^{2})
The graph of this is part of a parabola, starting at (0,0,0) and extending to (20,0,400), as shown.
What is the length of this simple curve?
We use a result from calculus, which is based on the Pythagorean Theorem.
The arc (or curved) length is given by:
(See Arc Length of a Curve and Arc Length of a Curve in Parametric or Polar Coordinates for background on these formulas.)
In this example where x = t and z = t^{2},
and
Also, in this example we have the end point t_{1} = 20.
Substituting gives:
(I used a computer algebra system to find this integral.)
This seems quite a low firgure for the length, since the zvalue at the end of the curve is 400, and going around the curve should add more than just over 1 unit to the length.
However, the above graph does not have equallyscaled axes, whereas the following one does.
We see that a parabola is "almost" straight, and a curve length of just over 400 for this curve is reasonable.
Let’s now see the effect of those trigonometric terms we saw before.
We now plot (t cos(t), t sin(t), t^{2}) for t = 0 to t = 20. (Of course, t is in radians.)
The start of the curve, when t = 0, is at the point (0,0,0). The point where the curve finishes, when t = 20, is:
(20 cos(20), 20 sin(20), 20^{2}) = (8.162, 18.259, 400).
For interest, looking from above (down along the zaxis), the above spiral looks like this:
The arms cut the xaxis evenly at 2π = 6.28, 4π = 12.57, and 6π = 18.85.
You can see the curve ends near (8, 18) as we claimed before.
We extend the 2dimensional case above now that we are working in 3 dimensions. In general, the length of an arc when using the parametric terms ((x(t), y(t), z(t)) is given by:
Substituting in our expressions for x, y and z, we have:
Now to examine the solar cooker problem.
Considering one of Magnar’s cooker designs, the parabola has a crosssection which is quite flat compared to the paraboloid we drew above. (Like most parabolic dishes, they are using flat panels to approximate the curved surface.)
In fact, the top lip passes through the point (25,45) in that design. The general form of a parabola can be written:
z = cx^{2}
Substituting (25,45) gives us .
We now plot the curve
from t = 0 to t = 45.
The curve starts at (0,0,0) and the top point is (23.6, 38.3, 45). We see there are about 7 spirals.
Here it is from the side:
And next the view from the top shows we finish near x = 24 and y = 38:
We have around 7 turns in the above spiral, but we may need more or less. If we want to achieve more spirals within the same x, y, and zconstraints, we need to multiply the variable within the trigonometric terms by a number.
Here is the case when we multiply the variables in the trig terms by 5. That is:
We get around 35 spirals this time:
Here it is from the side.
The length of this spiral is:
Similarly, if we multiply those variables by a number less than 1, we get less spirals. Here is the case when we multiply by 0.3:
This time we have just over 2 spirals (which is similar to the situation we had in Magnar’s first diagram at the top) and our length is:
This has been an interesting application of math where 3D geometry and integral calculus are being used to examine a good use of solar power. We need to make more use of the huge amount of energy available from the sun, and solar cook pots are a great way to do so.
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]]>The post From brain injury to math genius: Jason Padgett appeared first on squareCircleZ.
]]>Jason Padgett was attacked by 2 men one night and savagely beaten. He ended up with severe concussion and an amazing new ability to "see the world through the lens of geometry".
He began to create intricate geometric art, like this image:
See the full story at Discovery.com: Brain Injury Turns Man Into Math Genius. Don’t miss the gallery of his art while you’re there.
Padgett’s story reminded me of the work of Alan Snyder, the Sydneybased researcher who is able to temporarily "turn on" savantlike mathematical abilities in test subjects. (See Switching on your inner Rainman: Enhancing Creativity).
Snyder uses magnetic pulses through the brain to release what he believes are math and creative abilities inherent in all of us.
Padgett wouldn’t change his new abilities if he could.
"It’s so good, I can’t even describe it," he said.
It’s fascinating that Padgett has discovered the joys of math patterns, but it’s very sad it took a violent event to bring it about.
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]]>In this Newsletter:
1. Wu's squaring trick
2. Resource  PatrickJMT math videos
3. Google uses integration to speed up the Web
4. Volunteers wanted!
5. Math puzzles
6. Friday math movie: 5 Quick Math Tricks for Filmmakers
7. Final thought  walking and problem solving
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]]>In this Newsletter:
1. Wu’s squaring trick
2. Resource – PatrickJMT math videos
3. Google uses integration to speed up the Web
4. Volunteers wanted!
5. Math puzzles
6. Friday math movie: 5 Quick Math Tricks for Filmmakers
7. Final thought – walking and problem solving
Here’s a neat number trick, attributed to Scott Wu, that could be a good one to impress people at your next party (depending on your social group, of course…).
You can quickly square numbers over 25 in your head. It uses this identity:
n^{2} = (n − 25) × 100 + (n − 50)^{2}
For example,
62^{2} = 37 × 100 + 12^{2} = 3844
44^{2} = 19 × 100 + 6^{2} = 1936
Can you see how it works?
[Source: Wu's Squaring Trick, Quora]
I came across Patrick JMT math videos recently. His tagline is:

The extensive list of videos cover basic number and algebra, through logarithms, trigonometry and probability, to calculus.
These JMT videos are similar in style to the Khan Academy offerings, but I find them easier to follow because they’re easier to watch (no black background). They are "over the shoulder"type videos where you can see all the steps, and include plenty of examples.
The "JMT" stands for "just math tutorials".
The link again: Patrick JMT math videos
Google tries hard to provide superfast Web pages, and uses calculus in the process. They want everyone else to produce fast pages as well. See how integral calculus is involved. 
I have a small task that some of you may be interested in – and you could learn something from it, too.
I made 6 short videos about integration some time ago and I need some help to transcribe the soundtrack of the videos. It just means typing out the text of my commentary.
The sound quality is not very good on those videos (they were made using Flash and then converted for YouTube) and I hope to record the soundtrack again. In the meantime, the transcription will help users to follow what is going on.
If you are studying calculus now, or will be soon, (or are just interested), this is a good opportunity to learn some of basic calculus concepts while helping me out.
If you would like to be involved, please reply to this email, or reply in the comment box.
The puzzle in the last IntMath Newsletter asked you to find the intersection of a pulley belt with one of its wheels.
Correct answers with explanation were given by Colin, Nicos, Tomas and Don.
New math puzzle: Brigitte and Ric are sitting on their motorbikes at a rest stop beside a busy a interstate highway. They observe that the cars can be partitioned into three types, fast (120 km/h), medium (90 km/h) and slow (60 km/h). They also observe that one of each type passes every 10 seconds.
Brigitte remarks that since she always travels at 90 km per hour she can expect to overtake and be overtaken by equal numbers of cars. Ric says "nonsense", and they agree on the following bet.
Ric will pay Brigitte $1.00 for every car that overtakes her before the next rest stop if she pays him $1.00 for every car she overtakes. The next rest stop is 90 kms away.
Who wins, and by how much?
Leave your responses here.
This short video from Indy Mogul discusses some of the math involved in making videos. There are many connections between the arts and math! 
I’ve been doing a lot of programming lately, mostly rewriting the old Flashbased math applets on IntMath so they’re mobiledevice friendly.
Programming involves a lot of problem solving, where things often don’t work as expected, or not at all. It can involve staring at the screen for long stretches, trying to get to the bottom of the issue.
I find one of the best problemsolving methods is to go for a walk. It’s amazing how you can see things more clearly, and see the broader picture when away from the screen. Walking also increases your blood flow, and that’s essential for good brain functioning.
So next time you’re stuck on a math problem, go for a short walk. It will help your mental and physical health, and hopefully give you the insight you need.
Until next time, enjoy whatever you learn.
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]]>The post Google uses integration to speed up the Web appeared first on squareCircleZ.
]]>Google’s success as a company was built on optimization. They’ve put a lot of effort into making everything they do super fast.
Google wants all pages to be fast, so they provide a free Web page speed test facility, WebPageTest.org. You can test any page there and see why it is so slow (or fast). It’s great for those of us who are keen on speeding up our sites.
Of course, there are many things that will slow down a page, including slow Net connections, a slow server, size of the page (in kb), slow download speeds and slow browser processing speeds.
WebPageTest provides various test result measures, including connection time, download times and a metric called Speed Index.
Speed Index is an indication of how quickly the page elements appear on a page. In some cases, we see just a small part of the page and then see the browser "spinner" indicating more is to come (we are likely to leave in such cases). On other pages, we may see most of the page quickly, but the spinner indicates some more is coming.
Obviously the second cane is a better user experience, so it should get a better Speed Index score.
The Speed Index documentation explains how they calculate the score. It involves finding the area under a curve, which is a concept we learn when first meeting integral calculus.
They give a formula:
where
T is the "visually complete" time (total time to load and process the page); and
VC/100 is the "percent visually complete".
The areas are calculated at 0.1 s intervals (or similar) and added to produce the final index.
The area represented by the integral is the darker blue area in this chart, which represents a fast Web page (most of the content is visible quickly and the last 10% or so appears later).
[Image credit: WebPageTest]
So in fact, we are finding the area between 2 curves, the top one is the constant value 1 (or 100%), and the other is the VC curve.
Some closing ironies:
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]]>The post Friday math movie: 5 Quick Math Tricks for Filmmakers appeared first on squareCircleZ.
]]>Here are some mathrelated tips for better movies.
He discusses the fstop scale, what an appropriate shutter angle is, the 180degree Rule and Rule of Thirds, and how to read a histogram.
Actually, there are plenty of potential interesting math lessons arising from this. How about get students to make relevant videos about the math they are using, while making videos!
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]]>In this Newsletter:
1. Pulleys interactive applet: background
2. Teraplot LT
3. Math puzzles
4. Friday math movie: Falling in love with primes
5. Producer or consumer?
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]]>In this Newsletter:
1. Pulleys interactive applet: math background
2. Resource: Teraplot LT
3. Math puzzles
4. Friday math movie: Falling in love with primes
5. Final thought – producer or consumer?
I recently updated the pulley applet on IntMath. There was a lot of interesting math involved, and here’s a description of it: Pulleys interactive applet: math background A lot of the same math is used when writing computer games. 
Teraplot LT is a free 2D and 3D graph plotter for Windows 8 and Windowsbased tablets. It’s the scaled down version of the fully fledged Teraplot Graphing Software (which is a commercial product. They have a free 30 day trial). I enjoyed working with both products and found the output quality is quite good. 
The puzzle in the last IntMath Newsletter asked about solutions for an inequality.
Correct answers with explanation were given by Peter (who provided 2 different approaches), Emily, Tomas, Francis, Nicos, Cichy and John.
(Some people just sent a number only as their answer. There was no indication of the thinking behind their conclusion.)
New math puzzle
Let’s revisit the pulley applet. As explained in the article mentioned earlier, for the applet to look right, we need to know where the segments intersect with the circles.
The new puzzle is: Given 2 pulley wheels of different radius r_{1} and r_{2}, with centers that are distance d apart, find the point where the first circle intersects with the segment joining the 2 circles. (The segment is tangent to each circle, and the left hand circle is centered at (0,0).) That is, you need to find the point where the red dot is.
Leave your responses here.
[Hint: If you want to enter subscripts in your answer, do it like this: r<sub>1</sub>. This will come out like this: r_{1}. If that's too troublesome, just put r1. We'll understand!]
Comedian Adam Spencer gives us the lowdown on finding monster prime numbers. 
As mentioned earlier, reader Peter gave 2 responses to the math puzzle. In a separate comment, he said:
My first response was done on an iPad. I find it hard to be wordy on that medium. The second was done on my computer with a full keyboard. That would be an interesting survey question: Are you more thoughtful (and complete) when providing an answer on a smartphone, tablet, computer, or pen and paper?
I’ve often thought about this. I use my phone and tablet for consuming content 95% of the time. They are good for that, but certainly not good for creating serious content, especially if it involves images and several other technologies (which my writing nearly always does.) I’ll usually only respond to emails (or most other things) when I get home.
The other thing I’ve thought about is that as fewer people buy and use desktops and laptops, and move to mobile devices, it will mean less creators and more consumers. This is actually not good for our future.
What do you think? Please respond here.
Until next time, enjoy whatever you learn.
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]]>The post Friday math movie: Falling in love with primes appeared first on squareCircleZ.
]]>He mentions the largest current prime, which is around 12.5 million digits long.
Monster, indeed.
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]]>The post Pulleys interactive applet: math background appeared first on squareCircleZ.
]]>As I developed it, I used quite a bit of interesting math.
I get quite a few mails from people asking how I develop these applets, so here is some mathematical background.
The 2 circles are built using the basic formula for a circle:
The lefthand circle has center (0,0), (I could have placed it anywhere on the xy plane, but (0,0) seemed simplest), and the radius it has at the start is 5. So its equation is:
The right hand circle has center (15,0) and radius 2 at the start.
If the circle has center other than the origin, say (h,k), we use this formula:
So the equation of the right hand circle is:
[JSXGraph allows us to easily draw circles with different centers and different radii (without resorting to the formulas), but I still needed to make use of the formulas for later steps.]
I have hidden the axes for the applet to make it cleaner.
The red dot follows the belt around. It travels through 2 circular arcs, and 2 straight line segments.
I needed to find the intersection point of the arcs with the straight line segments. This could be done using either Nonlinear systems of equations or via other geometric constructions.
Of course these intersection points change when the radius of each wheel changes.
I assumed no gravity was operating on the belt, and that it is (very) tight.
The red dot follows 2 line segments. I needed to find the equation of those segments. So I needed to use the basic equation for a straight line with slope m and yintercept c:
To find the actual values of m and c, I needed to use the following formula, once I found the intersection points and :
The red dot travels around 2 circular arcs, and there are points (the circles) on the pulley wheels that rotate around a center as well.
I needed to use the following parametric equations, where r is the distance from the center to the rotating point, to achieve this:
As time goes on, the angle θ changes (it’s in radians, of course), and the point sweeps out a circle. The starting point is (r, 0) when θ = 0.
Of course, the whole point of the applet is to demonstrate the relationship between linear and angular velocity, one of the applications of radian measure. Our basic equation is:
v = rω,
where
v is velocity in meters/second (m/s)
r is radius in meters (m)
ω is angular velocity in radians per second (rad/s)
I find I understand new formulas more easily when I can see the units.
Pulleys are used to make our lives easier (e.g. gears on a bicycle, or for lifting weights) and they work when there is a difference in the radius between the 2 pulley wheels. So we may need to pedal faster on a bike when in a low gear, but the advantage is we don’t have to push so hard.
The applet shows the difference in angular velocity between the larger wheel (lower angular velocity) and the smaller wheel (higher angular velocity).
It turns out the ratio of the angular velocities is the inverse of the ratio of the radii of the 2 wheels.
In programming, we can come across 2 kinds of functions:
I also needed to use the concept of shifted trigonometric functions (ones where the curve does not start at 0).
This is actually a relatively simple applet. Programming such applets – and most games – involves a lot of interesting math!
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]]>