Yousuf, one of my regular correspondents, got stuck on the following problem recently.

What is the area of the rectangle ADEB shown in the diagram?

The curve is the graph of y = 1/x² (for positive x), and r is some arbitrary value of x.

We’ll come back to this question a little later. I suspect his problem with this question was due to a rusty conceptual understanding of functions.

Functions Overview

A function is simply an expression involving variable(s).

We usually write a function of the variable x using the notation: f(x). A function has at most 1 value for each value of x.

For example, if f(x) = 5x² + 3, we can find the value of the function if we choose x = 0 as follows.

f(0) = 5(0)² + 3 = 5 × 0 + 3 = 3

Now, this is a good example of the notation problem I was talking about at the beginning. We write "f(0)" (f bracket 0 bracket) to mean "evaluate the function expression by substuting 0 every time we see an x" and we see this on the left hand side of this equation.

But on the right hand side, I have written "5(0)²" (5 bracket 0 bracket squared) and this means "5 × 0²". We need to be careful with this – writing 2 different concepts with what is essentially the same notation.

It is a shame that function notation is so clumsy and causes problems for newbies.

Let’s look at some more examples for our function f(x) = 5x² + 3.

f(2) = 5(2)² + 3 = 5 × 4 + 3 = 23.

f(10) = 5(10)² + 3 = 5 × 100 + 3 = 503.

If we were to substitute many more values of x and plot the dots on a graph, we would get the following:

Note: On the vertical axis, I put f(x), but I could have also put "y", since the convention in math is the vertical axis represents the function value. Often you’ll see it written y = f(x).

OK so far?

Now, let’s make things a bit more interesting. What is f(a)? We just substitute a everywhere there is an x in the original function, like we did before with the numbers:

f(a) = 5(a)² + 3 = 5a² + 3

Let’s do another. In this next case, f(a + 4), we are just replacing each x in the original function expression with a + 4.

f(a + 4) = 5(a + 4)² + 3 = 5(a² + 8a + 16) + 3 = 5a² + 40a + 83

Of course, we need to be careful to expand out the brackets properly!

A Different Function

Let’s change our function to .

This is the curve we met in the question at the the beginning of this article.

If x = 1, we replace every x in our expression with 1 and we have:

What f(1) means on a graph is the distance from the x-axis to the graph is 1 unit. The function value is the height of the graph for that x-value.

Now let’s do f(3a).

The value

represents the height of the graph when x = 3a. We need to be careful with the brackets.

Back to Our Problem

Here’s the graph again:

So how do we find the area of the rectangle BADE? The width of the rectangle is quite straightforward, as the distance from r to r − 1 is just 1 unit. But we need to find the height AD.

AD is just the function value f(r):

So the area of the rectangle is just

Area = width × height =

What if we needed the height BC?

We would just find the function value as follows.

BC =

Functions of 2 Variables

The functions above only have one variable and they describe a curve in 2-D space.

To describe a 3-D surface, we need to use 2 variables.

We write a function of 2 variables using this notation:

z = f(x,y)

The "z" indicates the height of the surface for particular values of x and y.

An example of a 3-dimensional surface is z = x² + 3 sin y.

More Information

See this chapter for a lot more examples of functions: Functions and Graphs. (2 dimensional)

This is an introduction to 3-dimensional Coordinate System.

See also Towards more meaningful math notation where I suggest an alternative to the current confusion.

Related posts:

1. Partial differentiation – what is it about?
2. 3D Grapher with contour plot
3. Towards more meaningful math notation

(Others include “pi” π = 3.141592653…, which is the circumference of any circle divided by its diameter; and “phi” φ = 1.6180339887…, which is the so-called “beauty ratio“). Both of these numbers are irrational (that is, their decimals go on forever and never repeat).

e is also an irrational number and it has value:

e = 2.718281828459…

The number e was “discovered” by several mathematicians (Oughtred, Huygens, Jacob Bernoulli, Mercator and Leibniz) but they didn’t quite know they had stumbled on it and didn’t know its significance.

There are some curious properties of e, one of which is that it’s the limiting value as n → ∞ of (1 + ¹/_n)ⁿ.

It can also be found by adding the infinite sum:

e = 1 + ¹/_1! + ¹/_2! + ¹/_3! + …

So what is e good for?

It is used extensively in logarithms (which was the only way to do difficult calculations for hundreds of years before calculators came along), exponential growth (of populations, money or drug concentrations over time) and complex numbers (which were used to design the computer or mobile device you are reading this on).

So happy “e” day (February 7th, or 2/7).

[For more information on e, see the MacTutor history.]

Related posts:

1. My infinity’s bigger than yours
2. Dinosaur Mathematics…
3. Nursing Entrance Test – for mathematicians or nurses?

Let’s have a look at some of the available math graphing tools.

Graphics Calculators

Here are the Texas Instruments TI-83 (left) and Hewlett-Packard HP 40gs (right) calculators.

Graphics calculators are handy, but the screen size is small and there is usually no scale on the axes. Cost is often prohibitive, at around US$100.

Online Graphers

Here’s a few places where you can graph your curve – for free!

The first one is on IntMath.com, and uses Scalable Vector Graphics (you need Firefox browser, or a plugin for IE).

• Plot your own SVG Math Graphs

The following are also free offerings. They are either Flash or Java applets.

• Wolfram|Alpha
• Flash Equation Grapher
• GraphSketch.com
• GraphApplet (warning – red text on a black background)
• GCalc

Computer Applications

Here are some of my favorites. In each case you need to download and install the software.

• GeoGebra (Free. I wrote a review of GeoGebra which shows how to get started with it.)
• Graph 4.3 (Free. Small download, from cnet.com)
• Scientific Notebook (US$90, but does much more than a graphics calculator)

Some problems with using graphics software

If you don’t have a good idea of what a function should look like before your use a graphics package, you can have all sorts of problems.

Here’s an example of a function which catches out the unaware.

Let’s graph it on Geogebra.

Hmmm – it appears to be empty. Is there an error? What’s going on?

We zoom out a few times and start to see 2 curves. Once again, is there a mistake? Why 2 curves?

I zoomed out because I had a good idea in my head what the graph should look like. Since I couldn’t see either of the 2 arms of this curve, and because of the 30 in the denominator, I knew I had to zoom out.

If I left my graph at that, I would still not have a good idea of what the function looks like. I have not chosen a view that shows the crucial features of this graph.

The default view in Geogebra was too close to the origin (0, 0) to see any of the curve.

Let’s have another go, this time using Scientific Notebook. This is what I get when I try to graph the curve.

I get a vertical line passing through 2 on the x-axis. But notice the scale on the y-axis. The number "4e+09" actually means 4 × 10⁹ or 4 billion. So Scientific Notebook recognizes that there are very large values of y involved in this function, and has shown us the limits of its internal coding.

This time if we zoom in, we can start to see the graph appearing as before. But I need to know to zoom in, otherwise I would miss it altogether.

What was that vertical line that appeared before? Was it a mistake? Why did it disappear when we zoomed in?

Next, we use my SVG Grapher. It’s similar to Geogebra in that its default view misses the curves, but is also similar to Scientific Notebook in that it shows a vertical line through 2 on the x-axis.

When we zoom out a bit, this is what we get.

We are still not showing all the vital features of the graph, and that vertical line is still there. What’s that about?

Next, I tried the Flash Math Grapher. Once again I needed to do some zooming. The x- and y- scales are strange (multiples of 13.01??, and is that the x-axis or the line y = -2.98?) , but at least I can get a pretty good view.

Next, let’s call in the big guns. Wolfram|Alpha gives us the following 2 graphs when we put our function in their search box (and they give us a lot more information about the function).

This time we get intelligent graphs that have appropriate x- and y- scales and actually show the curves. Note the first one does not have a vertical line through 2, but the second one does. Why?

That vertical line through x = 2

When the software draws the graph, it chooses regularly-spaced x-values and substitutes them into the function, plots the resulting dots and joins them.

In the above examples that have a vertical line, it means the software has chosen a value just slightly less than 2 (which will give a very small, negative value for y) and another one just slightly more than 2 (which gives a very large positive value for y).

Here’s an exaggerated version of what it’s doing (using only a very few data points joined by straight lines).

If the x-value chosen is exactly 2, the software will normally just skip that value (since it will return a "division by 0" error). The (almost) vertical line should not be included in the graph, since we cannot have x = 2 (this function is undefined for x = 2 since the bottom of the fraction would be zero).

This gap in the graph is called a discontinuity. It should be a gap – not joined by a line as above.

Some software handles this situation gracefully (like the Wolfram|Alpha example) while in others, you need to either understand why the vertical line is there, or in some cases, you can elect to include discontinuities or not.

Best answer

Here’s probably the best way to display the graph of this function.

Our graph has 2 asymptotes. When a curve gets closer and closer to a line but does not touch it, that line is called an asymptote

The first asymptote is the x-axis and the other is the vertical line, x = 2, which I drew using a dashed line of a different color (since it is not part of the graph).

I have shown the 2 asymptotes clearly and I have also labeled the x- and y-axes.

Here’s the function again for convenience.

I knew the graph was going to involve asymptotes since the x-variable is in the denominator (and we can’t have 0 in the denominator) and also considering when x gets really big, the value of the function will be really small.

If I didn’t know this (from graphing many of them on paper), I would have made a mess of graphing my function on a computer.

Graphing software and the future

Will graphing software change what we do in classrooms? Should it? Is it really necessary to sketch graphs on paper still?

If you can draw a quick sketch of a function on paper, it certainly helps your understanding for many types of math problems. The conclusion from above is that it is certainly worthwhile to have a good sense of what a graph should look like before graphing it using software, so we can manipulate the settings to show the graph properly.

Related posts:

1. Free math software downloads
2. GeoGebra math software – a review
3. GrafEq math graphing software

Preparing a syringe.

The process of pharmacokinetics has 5 steps:

• Liberation – the drug is released from the formulation.
• Absorption – the drug enters the body.
• Distribution – the drug disperses throughout the body
• Metabolism – the drug is broken down by the body.
• Excretion – the drug is eliminated from the body.

Of course, each drug needs to act on the body in a different way. Some drugs need to be absorbed quickly (like nitroglycerin if we are having a heart attack) and preferably eliminated quickly (otherwise toxins build up in the blood). For other drugs, we want slow absorption so we get maximum benefit and don’t lose a lot of it from excretion.

So when your doctor prescribes (say) "take 2 tablets every meal time", this is based on the desirable levels of drug concentration and known levels of distribution, metabolism and excretion in the body.

What’s the math?

When the nurse first administers the drug, the concentration of the drug in the blood stream is zero. As the drug moves around the body and is metabolized, the concentration of the drug increases.

There comes a point when the concentration no longer increases and begins to decline. This is the period when the drug is fully distributed and metabolism is taking place. As time goes on, the drug concentration gets less and less and falls below a certain effective amount. Time to take some more pills.

We can model such a situation mathematically with a differential equation. It has 2 parts – an absorption part and an elimination part. At first, absorption (increasing drug concentration) takes precedence and over time, elimination (decreasing concentration) is the most important element.

We have the following variables:

D = drug dose given

V = volume distributed in the body

C = concentration of the drug at time t

F = fraction of dose which has been absorbed (also called bioavailability)

A = absorption rate constant

E = elimination rate constant

t = time

Absorption part: This depends on the amount of the drug given, the fraction that has been absorbed and the absorption rate constant. It decreases as time goes on. The expression for absorption is given by:

A × F × D × e^-At

Elimination part: The elimination dynamic is affected by the elimination constant, the volume distributed in the body and the concentration left of the drug. The expression for this part is:

E × V × C

For our model, we need to subtract the elimination part from the absorption part (since the absorption part increases the concentration of drug and the elimination part decreases it). Our differential equation is as follows:

We now substitute some typical values for our variables (without units to keep things simple. Note C is a variable, the one for which we seek an expression in t.)

Solving this differential equation (using a computer algebra system), gives the concentration at time t as:

C(t) = 533.3(e^−0.4t − e^−0.5t)

We can see in the graph the portion where the concentration increases (up to around t = 3) and levels off. The concentration then decreases to almost zero at t = 24.

Pharmacokinetics is yet another interesting “real life” application of math.

[Based on: A First Course in Pharmacokinetics.
Photo credit: Syringe]

Related posts:

1. Intmath Newsletter – Graphs, pharmacokinetics, color blindness
2. The melting Arctic – a disturbing application of math
3. Today is “e” day

Color blindness is not evenly spread across different ethnic groups. Around 8% of Caucasian men, 5% of Asians, and 4% of African males have the most common type of color blindness — the inability to tell the difference between red and green.

As an example of the difficulties faced by those who cannot distinguish red and green, here is how a red apple is perceived by a deuteranope (someone with the most common red-green color blindness) .

Red apple as perceived by a normally sited person (left) and by a color-blind person (right).

Now we see how a green apple is perceived by a deuteranope.

Green apple (left) and as seen by a deuteranope (right).

As you can see, there is very little difference in the perceived color for the 2 apples and therefore it must be quite a challenge to go grocery shopping. We distinguish many vegetables (including the quality of those vegetables) by their color.

Note that color blind people see some color (it’s not blank or invisible), but they don’t distinguish green and red.

Check how color blind people see your images and text

I obtained the apple images on the right above using the excellent facility at VisCheck. You can upload any image and see how it will look to people with different kinds of color blindness.

Colors of the Rainbow

Here are the rainbow colors, with the way a color blind person would see them.

The Ishihara Color Test

Shinobu Ishihara published tests for color blindness in 1917. They consist of circles with numbers embedded using different colors, as follows.

Note: If you have trouble seeing numbers in the above plates, it does not necessarily mean you are color blind. Each computer monitor has different settings and capabilities. Color blindness tests should only be done using paper print-outs.

Color blindness and learning math

Let’s now look at the implications of color blindness for math students.

Here’s a pie chart that could be used as part of a statistics lesson. The colors are unsaturated, making it difficult to read.

And now for the version that a color blind person sees.

What if the teacher asked a question like "What percent is represented by the orange sector?" The brownish colors are almost indistinguishable and this would make interpretation of this chart quite difficult.

Notice that the order of the items in the legend does not follow the order of the sectors of the pie chart, so there is a double whammy for anyone trying to figure it out.

What colors should we use?

According to Okabe and Ito in the very useful article Color Universal Design, the following are good practices to follow when developing materials.

Principles of Color Universal Design

1. Choose color schemes that can be easily identified by people with all types of color vision.
2. Use a combination of different shapes, positions, line types and coloring patterns, to ensure that information is conveyed to all users including those who cannot distinguish differences in color.
3. Clearly state color names where users are expected to use those names in communication.

No matter what the background color of your presentation, Okabe and Ito advise to avoid red, orange, yellow, yellow-green and green, if such colors are important for distinguishing information. That last bit is important – you don’t need to avoid these colors if they enhance the visual appeal for normally sighted people.

A lot of presenters use color for emphasis, but it can be lost on some viewers.

For example, the following could possibly look all black and so the emphasis is lost.

Dark red characters amongst black text

The following at least will be clearer.

Dark blue characters amongst black text

But please don’t use blue too often on Web pages, since this is traditionally the color for links and should be avoided.

Back to the Pie Chart

Let’s go back to our pie chart and try to improve it. Here’s the chart again with better, more saturated colors. I have used magenta (equal amounts of blue and red, or #ff00ff in color hex) for one of the sectors. Note it looks light blue to a color blind person. Also observe how dark red (#c00000) and light green (#00ff00) appear.

Pie chart with saturated colors (left) and how it appears for a color blind person (right).

It’s better, but far from perfect.

The legend (the dots on the right of each image) adds extra cognitive load for everyone. If we label each sector as follows, it is easier to read and understand.

Line Graphs

Here’s a graph from a page I wrote illustrating Vector Addition. (We’re adding 2 vectors, A and B, and the resultant vector is R.)

I chose primary colors (blue and green for A and B), and "red" for the "resultant" (R), which is fairly commonplace. But look how disastrous it is for a color blind person:

In this case, the vectors are clearly labeled, but the colors do not help at all.

I changed the script to replace green with black. I’m still using red (in the original) for the resultant vector, but I think you’ll agree all colors are distinguishable now.

Redundant Coding

Here’s another kind of line graph that’s found in statistics. It shows a projection of the population by age for the USA.

Here’s how it looks to a color blind person.

As we’ve seen in the other examples, it’s better to label the lines in a chart (rather than use a legend), and to use different patterns (square, triangle, cross, etc) on each line. This is called redundant coding, since we are differentiating the information in more ways than one.

Here’s a better version of this graph, with redundant coding.

I didn’t change the colors (since that would have produced limited benefits).

It’s also best to use thick lines when drawing graphs, since thin ones are more difficult for color blind people to see.

Avoid Certain Color Combinations

Even normally sighted people have difficulty reading certain color combinations. Particularly on Powerpoint slides, it is irritating trying to read red or dark blue text on a dark background, something like this:

Bad enough for normally sighted people, but disastrous for color blind people, as follows.

The usual rules apply for Powerpoint slides – use highly contrasting colors – the best is black on white. Here’s the same text in a form which is easier for everyone to read (black on white):

Conclusion

We can’t assume everyone in an educational setting can see things the way we see them (in fact, that’s almost always the case on many levels, not just color perception!).

Proper labeling of graphs, charts, equations and avoiding indistinguishable colors will certainly help improve readability for all learners – not just the color blind ones.

I realize now I need to go back and fix a lot of my own graphics so they are more readable.

Don’t get me wrong – still use lots of color in your presentations, but keep in mind the needs of those 5% or so of readers who are color blind.

Please comment: Are you color blind? I’d love to hear your feedback or corrections on anything I have written here.

Image Credits

Red apple: CC BY 2.0

Green apple: CC BY-SA 2.0

Ishihara Plates

Population Projections

Related posts:

1. Intmath Newsletter – Graphs, pharmacokinetics, color blindness
2. Making math accessible for the blind
3. Friday Math Movie – Mixed Feelings (Vision through the tongue)

Historian and philosopher of science, George Dyson, talks about early computer concepts (including those by mathematician Leibniz), ENIAC (Electronic Numerical Integrator And Computer) in the 1940s, the use of computers to create atomic bombs, the huge frustrations involved in early computers, the mouse and punch cards.

It’s interesting when he talks about the computer “evolving” into a new kind of “symbio-organism”.

Related posts:

1. Friday Math Movie – Toys that make worlds
2. Friday Math Movie – George Bush fuzzy math
3. Friday Math Movie – the Web’s secret stories

I’ve got to make a window with a curved top. The width of the frame is the same all round, including the part around the curved portion.

What’s a formula for the length of the inner arc of the curved portion?

The required length is labeled HI in the diagram.

(This is not quite a “Norman window”, since it’s not a semi-circle on top.)

This is a typical “real life” question, in that we don’t have a lot of information to go on, so we’ll need to make some assumptions.

Solution – Example

Let’s consider a plausible example first. We assume the curves are arcs of circles. (They looked circular in the question. If they are not circles we could adjust later, but for the sake of the example, we’ll stick to this reasonable assumption.)

Let the total frame (length CD) be say 4 units wide and the edges of the frame be 0.3 units wide. I pick a point P in the center (horizontally) of the frame (I choose point P (2,1)), and draw 2 concentric circles, 0.3 units apart. The outer one is 5 units, the inner one 4.7 units. I could have chosen any radii for my concentric circles, of course, as long as the inner and outer radii differ by 0.3 units.

If I can find angle θ, it will be straightforward to find the arc length HI.

First, we find angle α by using the right triangle PMI.

cos α = 1.7 / 4.7 = 0.3617

Using the inverse ratio, we get that

α = arccos (0.3617) = 1.2007 (radians, of course. If we need degrees, it equals 68.79°)

[Why radians? They are more commonly used in science and engineering than degrees, and are best for this problem. For more, see Radians.]

Now we observe that 2α + θ = π (180°), since they lie on a straight line.

So angle θ = π – 2 × 1.2007 = 0.7402

To find the arc length HI, we just apply the arc length formula

s = r θ

(See arc length formula)

s = 4.7 × 0.7402

= 3.4789

So the required arc length is 3.48 units (correct to 2 decimal places).

General Solution

Let the total width (window and frame) = 2x, giving x for half the width.

Let the edges of the frame have width w.

The circles have radius R (outer) and r = R – w (inner).

Angle α = arccos ((x – w) / r)

Angle θ = π – 2 α (in radians)

Arclength HI is given by

s = r θ

= r × (π – 2 (arccos ((x – w) /r)))

This is the formula required by the glass manufacturer.

Related posts:

1. Intmath Newsletter – Length of an arc, what’s new, goals
2. Hubble math
3. Curved spacetime simulation

I’m always on the lookout for alternatives to images for Interactive Mathematics.

Scalable vector graphics offer many advantages over images. A scalable vector graphic is a small text file that your browser (if you are using a good browser) renders as an image. It solves each of the issues raised above. Readers can easily change the graph to whatever math function they want to see, and such graphics involve very small file sizes.

One easy way to implement of scalable vector graphics on a Web page is to use ASCIIsvg. Below is an example, showing the graph of y = sin e^x.

Note 1: You need to use Firefox browser (Chrome and Opera also work) to see the graph below. It will only work in Internet Explorer if you have Adobe’s SVG Viewer plugin.

Note 2: You need to view this page in the original blog post. It won’t work if you are viewing via an RSS feed in your aggregator.

You can mouse over the graph and it shows the coordinates of the cursor position. You can use this to determine particular points on your graph (for example, where the curve cuts the x-axis).

You can double click on the graph to see the code that was used to create it. You can also change parameters in this code then re-draw the graph. (Click the “Update” button after you’ve made your changes.)

If you can’t see a graph above and don’t know what I am talking about, this is a screen shot. If you view this article using Firefox (or Chrome or Opera) browser, you’ll see something like this:

Screen shot of SVG plot

How does it Work?

The graph uses the native SVG capability of (good) browsers (like Firefox, Chrome and Opera). SVG stands for scalable vector graphics, which as mentioned above, produces high quality images usually with very small file sizes. (You can learn more about vector-based art here: Vector Art).

For the graph above, the only code used on the Web page is this:

\begin {graph} width=400; height=300; xmin=-1.5; xmax=3.5;
ymin=-1.5; ymax=1.5; xscl=1; axes(); plot(sin(e^x));  \end{graph}

SVG is very powerful, but can be difficult to obtain the graphs you want. So Stephen Jipsen of Chapman University developed ASCIISVGsvg which is a javascript interface for SVG. That is, it makes our lives easier.

Earlier I wrote about the related solution ASCIIMathML (see Enter math in emails, forums and Web pages using ASCIIMathML). ASCIISVGsvg is included in some versions of ASCIIMathML.

How do I set up ASCIIsvg on my own Web page or blog?

You just need to download 2 files from ASCIIsvg main page:

1. d.svg (this is a very small file that the “embed” tag uses)
2. ASCIIsvg.js (this is the javascript that talks to the SVG rendering capability of your browser)

Then, add the code to embed an SVG graph, which will be something like this:

<embed width="117" height="117" src="d.svg"
script='initPicture(-2,2)
axes()
stroke = "blue"
plot(2x^2 - 3x)
'>

Finally, add a call to the javascript file (best at the bottom of your HTML page) something like this:

<script type="text/javascript"
src="http://mysite.com/ASCIIsvg_min.js"></script>

Advantages of Drawing Graphs using SVG

1. The old way: I drew nearly all of the graphs on Interactive Mathematics using mathematics software (either LiveMath, Scientific Notebook or similar) and then I took a screen shot using an image editing software. I then saved the image and uploaded it to the server. This is a many-step process that is fiddly and inconvenient, especially if you want to change anything on the graph.

   But SVG-based graphs are much easier to work with. You just change a bit of simple code and you get your new graph right there on the page.

2. Another key advantage of SVG graphs is the small file size, since it just uses a small bit of simple code. This is good for you (it downloads quickly) and me (it reduces bandwidth costs).
3. Readers can plot their own graphs on the Web without having to create images. (See how below.)
4. You can print high quality graphs from your browser

Disadvantages of Graphs using SVG

1. It is not (yet) cross-browser, since Internet Explorer does not render SVG. This is a big limitation, but since IE is now less than half of the browser market, maybe it won’t be a problem for much longer. (Well, we live in hope. )
2. You need the javascript to run on the Web page. So as I said above, you can’t see the graph if you are reading this via a feed.
3. Despite what I said above about file sizes, your browser will download the ASCIIMathML javascript, and this is not so small at 150kB. But if you use ASCIIsvg only and use file compression on your server, it comes down to 7 kB only, which is less than many images on the Web. Of course, your browser will cache it so it is a one-time download.

My Examples of Graphs using SVG

There are a few versions of ASCIIsvg floating around. I first started with the ASCIIsvg which is inbuilt into ASCIIMathML, but I found it was quite limited. (I spent too long trying to figure out how to change parameters and gave up.)

I settled on (pure) ASCIIsvg since it was much more transparent. A big downside is it uses the “<embed>” tag, which is not HTML-compliant.

Adding vectors: I produced a page in the Vectors chapter: Adding Vectors which produces a different example each time you re-load the page.

User-generated graphs: You can plot your own graphs on this page: Plot your own SVG Math Graphs

Resources for ASCIIsvg

You can obtain ASCIIMathML from here: ASCIIMathML.

If you only want graphs and not the whole math rendering engine, you can get ASCIIsvg here: ASCIIsvg main page

Get your questions answered here: ASCIIMath Google Group

More examples of graphs using SVG: SVG Gallery

Related posts:

1. Graphs using free math software
2. Enter math in emails, forums and Web pages using ASCIIMathML
3. Friday math movie – Trigonometric Graphs

This is Mistake #17:

And this is Mistake #01:

You can download (free) PDFs of these in color and black and white. There’s also an MP3 commentary on each one.

This page gives an overview of all resources available: Classic Mistakes.

Don’t miss the Freebies page, which has more posters.

Related posts:

1. IntMath Newsletter – resources, modeling and mistakes
2. Math and color blindness
3. The IntMath Newsletter – Oct 2007

Humans until quite recently walked great distances each day in search for food and to avoid being eaten themselves. We learned to learn in an environment where we were physically much fitter than today.

This video outlines the connection between keeping fit and the positive effects it has on learning. Note the suggestion to do something active before taking a tough class, and how balancing helps them to learn vocabulary. The “brain breaks” during math class are a great idea, too.

Researchers are finding that exercise can not only keep you fit, but make you smarter. A school in Illinois has developed a program that gets students moving.

[Sometimes this movie won't start properly. Wait till you see "Replay video" and click on that.]

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I found this video via Dave Sladkey’s Teaching High School Math blog. Dave is a math teacher at Naperville Central High School where they filmed this video.

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3. Friday math movie – Angle Dance