KaTeX can handle ASCIIMathML input, and fall back to MathJax when KaTeX gives up with this approach.

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]]>While KaTeX is a lot faster than MathJax, for me there are 2 downsides with KaTeX:

- KaTeX only accepts LaTeX input. However, I have used ASCIIMathML for all the equations and formulas on IntMath, because it is signiicantly simpler and easier to use than LaTeX. If I wanted to use KaTeX for my site, I’m stuck.
- KaTeX doesn’t recognize many math expressions yet, for example matrices, determinants, various symbols, "aligned" environments (where you can align everything on the equal sign), various accents, case definitions and so on.

So I set about writing a page that handles both of these issues.

I needed a system that recognised ASCIIMathML input, converted it to LaTeX, and then was processed by KaTeX so it looked like "real" math. In the cases where KaTeX cannot handle the math, it needs to fall back to MathJax so all the math is rendered properly.

Here is a demo of my solution:

You’ll see some equations appear quickly (those done by KaTeX), then the MathJax-rendered ones will appear. They have a green mark next to them so you can see what MathJax can do that KaTeX cannot.

This is deliberately a “heavy” page (with many equations) to push its performance.

Fortunately, ASCIIMathML comes with a script, ASCIIMathTeXImg.js, which converts the simple ASCIIMathML input to LaTeX, and then outputs each math expression as an image (using MimeTex or MathTex).

I modified the function AMTparseMath() in ASCIIMathTeXImg.js, so it no longer converts math to images, but outputs LaTeX.

My tweaks were:

- Create a span in the place of each piece of ASCIIMathML
- Give the span a unique id
- Try first to render it with KaTeX
- If that fails, put back the original ASCIIMathML and render it with MathJax

Here’s the relevant parts of the changed script.

var node = document.createElement("span"); thisId = "mathId"+counter; node.id = thisId; try { katex.render(texstring,node); } catch(err) { node.className = "mj"; node.innerHTML = "`"+str+"`"; MathJax.Hub.Queue(["Typeset",MathJax.Hub,thisId]); } counter++;

Of course, the main aim is speed, and if the user needs to download both KaTeX and MathJax, as well as ASCIIMathTeXImg.js, there is going to be a significant delay before any rendering occurs, especially on a mobile device.

On my phone, the first KaTeX equations on the demo page appeared in about 6 seconds, and the MathJax matrix came in around the 13 second mark.

But it is a feasible solution for mobile devices, especially if your more complicated equations are low down the page (so they will process in the background and be done before the user gets to them).

Another downside is that the user will need to download 2 sets of fonts – one for MathJax and one for KaTeX. One way out of that would be to specify one or the other.

You may also be interested in this demo, where you can see how much faster KaTeX does its thing compared to MathJax:

This page gives background on KaTeX:

This page is a sandbox where you can play with ASCIIMathML input:

This page uses the script mentioned above, ASCIIMathTeXImg.js, to send math in emails:

And finally, this page gives examples of how to enter math using ASCIIMathML:

Peter Jipsen and David Lippman for ASCIIMathML.

The MathJax team (especially Davide and Peter) for MathJax.

The Khan Academy team for KaTeX.

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]]>KaTeX is a new method for publishing LaTeX-based math on the Web. It's faster than MathJax, but not as robust (yet).

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]]>It’s a direct competitor to MathJax, which I’ve been using on IntMath for some time now.

You can see immediately that KaTeX produces math much faster than MathJax. This is because KaTex "renders its math synchronously and doesn’t need to reflow the page", according to their short blurb on Github.

There is obviously much less time required for processing. KaTeX doesn’t suffer from the page reflow jumps that MathJax has after each equation is created. (To be fair, there are ways around those instabilities in MathJax.)

I wrote a page so you can see the same input as processed by both methods. See

There’s a rough "time to process page" at the top. It’s not a fair comparison in the sense that KaTeX doesn’t handle several of the equations (yet), and MathJax handles them all.

The difference is stark on a mobile phone – around 1 second for KaTeX and around 30 seconds for MathJax.

It claims to "supports all major browsers, including Chrome, Safari, Firefox, Opera, and IE 8 – IE 11". So does MathJax, and it also copes with the vagaries of earlier IE versions.

So far the only differences I’ve noted are some font weight issues. In Chrome, the rendering seems "thin" to me. Sometimes it’s too much so, and the "equal" signs almost look like minuses, and some minus signs almost disappear.

But that is not such an issue in Firefox or IE. Here are some screen shot comparisons.

Using KaTeX in Chrome, where "=" and “−” are thin :

Using MathJax in Chrome is bolder, and easier to read:

Here’s how KaTeX looks in Firefox (somewhere between the first 2, and most pleasing to me):

KaTeX in Internet Explorer looks almost identical to the above FF rendering, but has some alignment issues. (Note where the *u* and *v* are in relation to the fraciton line.

It’s no surprise that IE falls over here and there. It almost always does. Here are 2 instances that I’ve spotted so far:

The right hand side should have been: (*a*^{2} + *b*^{2} + *c*^{2})^{3}

This surd also has obvious problems in IE:

As mentioned earlier, KaTeX doesn’t do everything yet (it chokes on anything starting with \begin, \align or \choose, and many symbols don’t work yet.

You can see what currently works on the Demo page.

This list on the GitHub KaTeX wiki shows what functions are currently available.

The setup for KaTeX is similar to what you have to do with MathJax, but there is no CDN version of KaTeX yet.

First, download the KaTeX zip.

Extract the zip amd upload it all to your server. (This is much smaller than MathJax, as there is no image fallback with KaTeX, so there aren’t thousands of images to upload.)

Next, in the head of your page, point towards the KaTeX javascript and css, something like this:

<link rel="stylesheet" type="text/css" href="/path/to/katex/katex.min.css">

<script type="text/javascript" src="/path/to/katex/katex.min.js"></script>

If you have only a few equations on your page, you can proceed as follows:

<p><span id="mykatex1">...</span></p>

<script> katex.render("f(a,b,c) = (a^2+b^2+c^2)^3", mykatex1); </script>

This will place the equation into your Web page, as properly rendered math.

A second equation would need a new id on the span, and in the script, like this:

<p><span id="mykatex2">...</span></p>

<script> katex.render("f(a,b,c) = (a^2+b^2+c^2)^3", mykatex2); </script>

If you want to present a lot of LaTeX on your page, it would be better to proceed as follows.

You would use the same class name for the DIVs (or spans) containing math. Your HTML would look something like:

<div class="math"> f(x) = \sqrt{1+x} \quad (x \ge -1) </div>

<p>Some other text here. </p>

<div class="math"> f(x) = \sqrt{1+x}, \quad x \ge -1 </div>

Next, we use javascript to iterate over all the DIVs with class "math" (they can be <p>, <span> or <td> too, as long as the class is "math" and it contains LaTeX only.

I’m using jQuery to do this, but there are pure javascript methods as well.

The following code just means:

- Iterate over all the objects with class name "math"
- Get the text from the object (this is the LaTeX that we want to convert)
- Get the type of element. If it’s a DIV, then add "\displaystyle" so it will be presented as math centered on the page.
- Try to convert it to math using katex.render.
- If it fails, output an error message.

(function(){ $(".math").each(function() { var texTxt = $(this).text(); el = $(this).get(0); if(el.tagName == "DIV"){ addDisp = "\\displaystyle"; } else { addDisp = ""; } try { katex.render(addDisp+texTxt, el); } catch(err) { $(this).html("<span class='err'>"+err); } }); })();

There’s also an option to generate HTML on the server, so "you can pre-render expressions using Node.js and send them as plain HTML."

To do this you use "katex.renderToString".

This is where the math will appear:

<p><span class="katex">...</span></p>

This is the script you use:

var html = katex.renderToString("c = \\pm\\sqrt{a^2 + b^2}");

Apparently Khan Academy will be using both KaTeX (for speed) and MathJax (for more complicated equations) in the near term. Hopefully they will continue to develop KaTeX, but there’s a long way to go to catch up with MathJax’s flexibility.

MathJax has promised :significant speed improvements” in their next version.

Interesting times.

The developers answer some issues here, including “Why is it so fast?” (because it uses CSS for positioning and does a lot fewer things than MathJax), and some of their plans for future development (including matrices – yay!)

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]]>In this Newsletter:

1. Carnival of Mathematics #114

2. Updated Graphs of tan, cot, sec and csc

3. Graphing letters

4. Math puzzles

5. Asking questions

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]]>In this Newsletter:

1. Carnival of Mathematics #114

2. Updated Graphs of tan, cot, sec and csc

3. Graphing letter distribution

4. Math puzzles

5. Final thought – The skill of asking questions

It was my turn to host the 114th Math blog Carnival. I hope you find something interesting in the 16 articles there!

The Carnival of Mathematics is a collection of recent math blog articles covering visual math, various rants and some math history, by various authors. See: |

The updated applets on the Graphs of tan, cot, sec and csc page now run on mobile devices, albeit rather slowly. This leads to one of the most popular pages on IntMath. |

We live in a great age where it’s possible to see a lot more in data that was ever possible before computers.

Prooffreader (by David Taylor) has some interesting data visualizations, including several displaying trends in baby names. The name "Shirley" sky-rocketed in the mid-1930s, and "Linda" even more so in the the late 1940s. "Brittany" and "Ashley" were hits in the late 1980s. There’s an interesting animation of the rapid rise in boy’s names ending in "n".

He also has some infographics on cancer and mortality rates.

One of the graphics that caught my eye was Graphing the Distribution of English, which demonstrates how letters are used in words in the English language.

For example, the image below for the first 2 letters indicates that "a" occurs mostly in the middle of a word, whereas "b" occurs mostly at the beginning.

Go check out Prooffreader – there are some interesting things there.

The puzzle in the last IntMath Newsletter asked about a strange pyramid of ancient Egypt. The correct answer with explanation was given by Tomas and Nicos. The difference in their approaches was quite interesting.

**New math puzzle**: Find the smallest rectangle (non-square and with integer sides) where the perimeter equals the area (ignoring units).

Leave your responses here.

This quote comes from Annie Murphy Paul. It’s a summary of the Rothstein and Santana book, *Make Just One Change: Teach Students To Ask Their Own Questions**:*

"This book makes two simple arguments: 1) All students should learn how to formulate their own questions. 2) All teachers can easily teach this skill as part of their regular practice. This inspiration for the first argument came from an unusual source. Parents in the low-income community of Lawrence, Massachusetts, with whom we were working twenty years ago told us that they did not participate in their children’s education nor go to their children’s schools because they ‘didn’t even know what to ask.’ It turns out that they were actually pointing to a glaring omission in most formal and informal educational enterprises. 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. In fact, it has too often been limited to students who have access to an elite education. Our goal is to democratize this teaching of an essential thinking and learning skill that is also an essential democratic skill."

—Dan Rothstein and Luz Santana,Make Just One Change: Teach Students To Ask Their Own Questions

Until next time, enjoy whatever you learn.

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]]>The Carnival of Mathematics is a collection of recent math blog articles covering visual math, various rants and some math history, by various authors.

The post Carnival of Mathematics 114 appeared first on squareCircleZ.

]]>Helix Bridge and Flyer, Singapore

114 is a **sphenic **number (the product of 3 distinct prime numbers):

114 = 2 × 3 × 19

It’s also a **repdigit**: 114 = 222_{7} (the digits repeat in the base 7 representation)

It’s the 19th number in the **Padovan** sequence, given by the recurrence relation *P*(*n*) = *P*(*n* − 2) + *P*(*n* − 3), and where the first 3 terms are 1.

114 is an **abundant number **(where the sum of its proper divisors is greater than itself):

114 < 1 + 2 + 3 + 6 + 19 + 38 + 57

That’s appropriate, as we have abundant posts for this, the 114th Carnival of Mathematics!

On with the show.

Now here are some animations that could be the basis for some interesting class discussion. See |

Andrea Hawksley gives us a nice roundup of the recent meeting on Origami, Science Math and Educaion. She features some of her own work, as well as other gems that were presented. See: |

6th meeting on Origami, Science, Math, and Education (6OSME)

While you’re there, mouse over Andrea’s name in the header of her blog. It’s cute.

That’s logical, as it’s the name of his blog! See

See:

BBC Sport’s Anti-Smartness Bias

In a similar vein, Gilead of “Tycho’s Nose” rants about inaccuracies (or part-truths) in background equations as seen in the new Doctor Who series. See: |

Here’s a rant of a different nature. Stephen Cavadino of "cavmaths" questions whether stem-and-leaf plots are a necessary part of the math curriculum. See: |

Mark Dominus provides us with a neat exploration of numbers that are permutations of each other. His first example is the decimal expansion of |

See:

When do n and 2n have the same digits?

Richard Elwes introduces us to an interesting knotty problem. It turns out this generally accepted picture is wrong… See how at: |

Jeremy Kun of "Math ∩ Programming" explores optimal greedy algorithms and matroids.

See:

When Greedy Algorithms are Perfect: the Matroid

This post, right here on "squareCircleZ", was inspired by a reader’s question. He makes solar cookers for use in Africa and wanted to know how to construct the spiral length around his cookers. It’s some "real-world" math that involves sustainable, cheap energy.

See:

Arc length of a spiral around a paraboloid

0 |
In a lot to do about nothing, Evelyn Lamb of the Scientific American blog, recounts her journey of discovery with her students, which involved decipering Plimpton, the 4000 year-old Babylonian tablet. |

So how did the Babylonians work in base 60, without 0 as a place holder?

See:

See:

How The Ancient Egyptians (Should Have) Built The Pyramids

α β γ δ ε ζ |
Here’s a summary of the Greek alphabet by SixWingedSeraph of Gyre&Gimble. We should spend more time on this topic in class, so students are more comfortable with this foreign script. See |

This post is a plug for an experimental MOOC (Massive Open Online Course). The blurb says:

"Citizen Maths is an online resource anyone can use — to discover how maths can be a powerful tool for solving those problems that come up at work and in your life."

See:

So, what is mathematics? Shecky Riemann of "Math-Frolic" has a quote from *How Mathematicians Think*, by William Byers, which ponders that math is a series of situations where we are led to observe,

I hope you’ve enjoyed Mathematics Carnival #114, coming to you from Singapore.

The next carnival will be at MathTuition88, slated for October 2014. See where and how to submit.

*Singapore Helix and Flyer* by ensogo, accessed from http://www.ensogo.com.ph/escapes/singapore-flyer-cruise-09092012.html

*BBC Sport logo*, by BBC, accessed from http://www.bbc.com/sport/0/

*Stem & leaf plot* by ck-12.org, accessed from http://www.ck12.org/book/Basic-Probability-and-Statistics-A-Full-Course/r4/section/7.2/

*Reflection*, by Marcia Birken, accessed from http://alumnae.mtholyoke.edu/blog/light-motifs-marcia-birkens-images-meld-math-and-art/

All other images are from the posts to which they link. If you have any objections to their use in this manner, let me know and I’ll remove them.

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]]>IntMath will be hosting the next Carnival of Mathematics, a collection of recent blog posts about math.

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]]>To submit an article, go to:

As it says on the submission form:

The Carnival of Mathematics accepts any mathematics-related blog posts: explanations of serious mathematics, puzzles, writing about mathematics education, mathematical anecdotes, refutations of bad mathematics, applications, reviews, etc. Sufficiently mathematized portions of other disciplines are also acceptable. We also accept YouTube videos, and non-blog based content, as long as it’s new and has been recently posted online.

The most recent carnival was hosted by Mike at Walking Randomly. It covered the full gamut from limits to Ninjas and 3D printed geometry to mathematically-related underwear. See

All the previous carnivals can be found on Aperiodical. See:

See you back here in mid-September for edition #114!

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]]>In this Newsletter:

1. Latest interactive math applets

2. Resource - Context Free Art

3. Color Blindness graph simulator

4. IntMath Poll results

5. Math puzzles

6. Final thought - What is the point of education?

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]]>In this Newsletter:

1. Latest interactive math applets

2. Resource – Context Free math-based art

3. Color Blindness graph simulator

4. IntMath Poll results

5. Math puzzles

6. Final thought – What is the point of education?

I recently added a new Riemann Sums applet to IntMath. It allows you to explore the concept of finding areas under curves. Go to: Riemann Sums Applet You can see how we find areas under curves via rectangles, using an interactive applet. |

Here’s some background on this important calculus topic:

This will bring back memories for older readers.

A subscriber asked if I knew of a log table with 9 decimal place accuracy. (Traditionally, log books came with 4 decimal place accuracy.) I didn’t, and so I created an interactive log calculator, which works in several different bases (2, See it at: Interactive Logarithm Table. |

There’s also an interactive log table which gives output up to 9 decimal places.

You may be wondering, why create electronic log tables when the real thing disappeared 30 years ago? Well, we can still learn from exploring such tables, in a way that an ordinary calculator cannot. And I don’t know too many hand-held calculators that can work in base 2, 20 and 100!

Here’s an idea for something interesting to do during the summer break. Create some math-based art, and learn some STEM skills in the process!

Context Free is software you can use to produce some beautiful math-based art. Build STEM skills while triggering your creative side. |

I wrote an article a few years ago about Color blindness and math.

My main point in that article was that math teachers should make sure their diagrams, charts and graphs can be understood by the 8% or so of male students who do not perceive the full color range.

The recent IntMath poll asked readers:

It looks like calculus is the most useful type of math for many of IntMath’s readers, just ahead of algebra. It’s unfortunate trigonometry is seen as least useful, since there are many Uses of Trigonometry.

**Poll results:**

Calculus | 27% |

Algebra | 22% |

Financial math | 21% |

Statistics | 12% |

Geometry | 11% |

Trigonometry | 7% |

Total votes: **2400**

**Current poll: **The latest poll asks which **financial math topic** will have the most impact on your life.

You can respond to this poll question on any (inner) page of IntMath, for example this one: World Population Live.

The puzzle in the last IntMath Newsletter asked about the probability of the numbers on balls in a box being relatively prime. There was only one correct answer with explanation, by Tomas. Tomas made some assumptions before tackling the problem. This is something we often need to do in “real world” math problems. Nicos’ answer was close!

**New math puzzle**: The Phollee pyramid of ancient Egypt had a square base, but the four triangular faces were all of different shapes and size. The lengths of three of the sloping edges were 150, 160 and 200 m. The 150 m and 160 m edges were not adjacent. How long was the fourth edge?

Leave your responses here.

For some of you, education is vital for your future. It is valued by your society and considered worthy of full effort. You feel as a result of your learning that you are growing as a person, your horizons are wider and you understand more about the world.

Sadly, for others, it is a "necessary evil", something that they need to survive when young, and get away from as soon as possible.

Arthur W. Forshay, director of the Bureau of Educational Research at Ohio State University, wrote:

"The one continuing purpose of education, since ancient times, has been to bring people to as full a realization as possible of what it is to be a human being.”

What does your education mean to you?

Until next time, enjoy whatever you learn.

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]]>The post Context Free math-based art appeared first on squareCircleZ.

]]>Context Free is software that you can use to generate fascinating art work. You give instructions using code called a "grammar". It’s quite a neat way to bring together math, art and programming, which are key STEM skills.

The Context Free Art site provides downloads of the (free) software (Mac, Windows and Linux), and also a gallery of around 1000 works of art generated using the program.

Everything is released using a Creative Commons non-commercial license, so you can use the images in your own work. You can modify other people’s code to create your own materpieces (but give credit to the original, of course).

Here are some of the output images from the gallery:

Drawing these images involves some interesting math topics, including:

- Trigonometry
- Logarithms
- Matrices
- Vectors
- Geometry
- Number theory
- Complex numbers

This one is called the Lorenz Attractor, by thijs:

To give you an idea what is involved, this is the code thijs wrote to produce the above shape:

startshape Lorenz_Attractor //1600*900 CF::Background = [b -0.9 sat 0.2] CF::Size=[s 60 33.75] CF::ColorDepth = 16 CF::MinimumSize= 0.3 CF::Impure = 1 A=10 B=8/3 C=28 d=0.002 vector3 D(vector3 V) =V[0]+d*( A*(V[1]-V[0]) ) ,V[1]+d*( V[0]*(C-V[2])-V[1] ) ,V[2]+d*( V[0]*V[1]-B*V[2] ) vector3 M(vector3 V1,vector3 V2) =(V1[0]+V2[0])/2 ,(V1[1]+V2[1])/2 ,(V1[2]+V2[2])/2 //------------------------------------ shape Lorenz_Attractor { V0=(-12.5, -17.4, 26) Next(0,V0)[r 100] } shape Next (i,vector3 Vi) { if (i<320000) { Vn=M(Vi,D(D(Vi))) Plot (i,Vi,Vn) [] Next (i+1 ,Vn) [] } } shape Plot(i,vector3 V0,vector3 V1) { P=V0[0,2] Q=V1[0,2] Z=V0[2] H=45+15*sin(i/3) Line(0.05,P,Q) [z Z b 1 h H sat .4] Line(0.15,P,Q) [z Z b -1 a -.950] Line(1.50,P,Q) [z Z b -1 a -.999] } path Line(S,vector2 P,vector2 Q) { MOVETO(P) LINETO(Q) STROKE(S,CF::RoundCap)[]

Here's a useful list of resources by Mikael Hvidtfeldt Christensen: Generative Art Links, containing links to some interesting software for creating several different types of math art, including 3D fractals, and there's also some interesting blogs.

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]]>The updated applets on the Graphs of tan, cot, sec and csc page now run on mobile devices, albeit rather slowly. One of the most popular pages on IntMath.

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]]>To make the page more mobile-device friendly, I updated it so now the interactive graphs use JSXGraph, rather than Flash.

The page explains how to sketch these trigonometric curves, mostly by considering the denominator (the bottom) of the function expression.

Each of the curves tan *θ*, cot *θ*, sec *θ* and csc *θ* has a discontinuity, or "gap" in the curve. For example, tan *θ* is defined by:

Since cos *θ* has value 0 at *θ* = … −3π/2 −π/2, π/2, 3π/2, 5π/2,…, it means tan *θ* will not exist for those values of *θ*. The result is there will be "gaps" in the curve at those points.

Either side of those gaps, the curve heads off to plus or minus infinity. Here is the graph of *y = *tan *θ*:

The page has animations that demonstrate where the values of the curves come from.

Here’s a screen shot of one of the animations:

On a tablet, the animations may run a little slow. After all, we’re asking the processor to perform a lot of calculations! Alo, the screen gets re-drawn multiple times during the animation, which also slows things down.

As a result, you can only animate one of the graphs at a time.

The link again:

Graphs of tan, cot, sec and csc

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]]>Here's some background on the new Riemann Sums applet I added to IntMath. You can see how we find areas under curves using rectangles.

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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 mid-19th 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 mid-way 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, Li-sa, 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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