In this Newsletter<\/p>\n
1. LiveMath - now more stable
\n2. April Fool's Day math
\n3. Integration mini-lectures
\n4. Math tips - Radicals and Integration
\n5. From the math blog<\/p>\n
\nUpdate: <\/b> (April 2014) I no longer feature LiveMath on the IntMath site. LiveMath was good in its day (the late 1990s), but has not kept up with changes in technology.<\/p>\n<\/blockquote>\n
LiveMath has been a key feature of Interactive Mathematics<\/a> for over 10 years. With LiveMath software, you can draw graphs, find inverses of matrices, solve equations, and see animations of math concepts. And best of all, it doesn't cost a thing!<\/p>\n
You may have noticed links going to \"see LiveMath solution\" throughout the Interactive Mathematics site. This is where you can play with the concepts presented in that chapter.<\/p>\n
I recently changed the way that LiveMath is delivered and it is now more stable. Instead of viewing LiveMath in your Web browser, it now opens up in the LiveMath viewer.<\/p>\n
\n2. April Fool's Day Math<\/h2>\n
The origin of April Fool's Day is uncertain, but most historians agree that it started in the 1580s after Pope Gregory decreed that a new calendar should be used by everyone. <\/p>\n
Instead of the New Year starting in late March\/early April (after the Spring equinox), New Year was changed to 1st January. The people who hung on to the old Mar\/Apr date for New Year were ridiculed as \"April Fools\".<\/p>\n
Some April Fools gags that have involved mathematics (source<\/a>):<\/p>\n
\n
- The state of Alabama voted to replace the value of pi (3.14159...) with the value suggested by the Bible (3).<\/li>\n
- The authorities were going to replace our current base 60 time system (60 seconds in a minute, 60 minutes in an hour) with a base 10 system (100 minutes in an hour, 10 hours in a day).<\/li>\n
- The alignment of two planets would result in an \"upward gravitational pull making people lighter at precisely 9:47 a.m.\" on April 1st. The author of this prank \"invited his audience to jump in the air and experience 'a strange floating sensation.' Dozens of listeners phoned in to say the experiment had worked.\"<\/li>\n<\/ul>\n
See other fun April Fools hoaxes<\/a>.<\/p>\n
Here's a good quote about fools: <\/p>\n
\nIt's better to keep your mouth shut and be thought a fool than to open it and leave no doubt.<\/p>\n<\/blockquote>\n
--Mark Twain<\/p>\n
\n3. Integration Mini-lecture videos<\/h2>\n
I recently updated the Integration mini-lecture videos<\/a>.<\/p>\n
The movies now use Flash (just like YouTube) so they can be viewed on both Mac and Windows.<\/p>\n
\n4. Math tips<\/h2>\n
a. Simplifying radicals tip:<\/b> We may get an answer for an algebra problem that looks like this:<\/p>\n
√(18x<\/em>3<\/sup>)<\/span><\/dir><\/p>\n This can be simplified (we are not using calculator here).<\/p>\n
Taking the number part first:<\/p>\n
18 = 9 × 2<\/span><\/dir><\/p>\n So<\/p>\n
√18 = √(9 × 2) = √9 × √2 = 3 √ 2<\/span><\/dir><\/p>\n The trick here (and many students forget this) is that<\/p>\n
√32<\/sup> = 3<\/span><\/dir><\/p>\n Square root of a number<\/em> and square of a number<\/em> are just opposite processes so when finding the square root of a number squared, the answer is simply the number.<\/p>\n
Now for the variable part:<\/p>\n
x<\/em>3<\/sup> = (x<\/em>2<\/sup>)(x<\/em>)<\/span><\/dir><\/p>\n So <\/p>\n
√x<\/em>3<\/sup> = √[(x<\/em>2<\/sup>)(x<\/em>)] = √x<\/em>2<\/sup> √ x<\/em> = x<\/em> √x<\/em><\/span><\/dir><\/p>\n Putting it all together we have:<\/p>\n
√(18x<\/em>3<\/sup>) = 3x<\/em> √(2x<\/em>)<\/span><\/dir><\/p>\n Go here for more Exponents and Radicals<\/a> examples.<\/p>\n
<\/p>\n
b. Integration tip:<\/b> When you are learning integration<\/a>, one of the most important techniques to understand is substitution<\/strong>. <\/p>\n
We substitute so that we can make a difficult integration into a much easier one<\/strong>. You can see a basic example of this in General Power Form<\/a>.<\/p>\n
Whenever we need to integrate something that involves a function of a function, something like <\/p>\n
x<\/em> sin(x<\/em>2<\/sup> + 5),<\/span><\/dir><\/p>\n then we need to choose a u<\/em><\/span> so that we get an easier expression to integrate.<\/p>\n
Sometimes it is not clear what the substitution should be. The first thing to try is to put u<\/em><\/span> equal to whatever is inside the brackets<\/strong>. That usually works without having to try anything else. So in the above example, we would put<\/p>\n
u<\/em> = x<\/em>2<\/sup> + 5<\/span><\/dir><\/p>\n Then we find the differential<\/a> and get:<\/p>\n
du<\/em> = 2x dx<\/em><\/span><\/dir><\/p>\n From there we can transform our integral into a much simpler integral:<\/p>\n
(1\/2)<\/span> ∫<\/span> sin u du<\/em><\/span><\/dir><\/p>\n Do the integration in u<\/em><\/span>:<\/p>\n
(1\/2)<\/span> ∫<\/span> sin u du<\/em> = −(1\/2) cos u<\/i> + C<\/span><\/dir><\/p>\n Then substitute the function of x<\/em><\/span>:<\/p>\n
∫<\/span> x<\/em> sin(x<\/em>2<\/sup> + 5) = −(1\/2) cos (x<\/em>2<\/sup> + 5) + C<\/span><\/dir><\/p>\n
\n5. From the Math Blog<\/h2>\n
1) 21st century math skills<\/a>
\nA reader asks for advice on 21st century skills and how her students can learn them.<\/p>\n2) Friday math movie - Trigonometric Strange Attractor Evolution<\/a>
\nThis week's movie is a representation of a 5-dimensional space acted on by 5-D forces.<\/p>\n3) Pythagoras<\/a>
\nPythagoras' religious cult had a strict regime of silence, except when they were singing songs to Apollo. This strange (to our thinking) monastic existence gave us the Pythagoras Theorem (sort of), a better understanding of the physics of music and advances in astronomical thinking.<\/p>\n