May 2011 – Page 4 – Montessori Muddle

Solving Quadratics

Solving quadratic equations requires finding the factors, which is not nearly as easy as multiplying out the factors to get the unfactored equation.

Instead, you have to do a bit of trial and error, to figure out which pairs of numbers multiply to give the constant in the equation and then add together to give the coefficient of the x term.

factoring — Factoring quadratic equations.

It gets easier with practice. Or you could use the quadratic formula, where if the equation were:

$a x^2 + b x + c = 0$

The solutions would be found with:

$x=\frac{-b \pm \sqrt {b^2-4ac}}{2a}$

So in the equation used in the diagram:
$x^2 + 7 x + 10 = 0$

you get:

a = 1

b = 7

c = 10

Putting these values into the quadratic equation gives:

$x=\frac{-7 \pm \sqrt {7^2-4(1)(10)}}{2(1)}$

which simplifies to:

$x=\frac{-7 \pm \sqrt {49-40}}{2}$

$x=\frac{-7 \pm \sqrt {9}}{2}$

$x=\frac{-7 \pm 3}{2}$

With the whole plus-or-minus thing ( $\pm$ ), this last equation gives two solutions:

(1): $x=\frac{-7 + 3}{2} = -5$

and,

(2): $x=\frac{-7 - 3}{2} = -2$

Now, you may have noticed that the solutions are negative, but when the equation is factored in the illustration, the result is:

$(x + 5) (x + 2) = 0$

The difference is that, although we’ve factored the equation, we have not solved it. When I say solve the equation, I mean find the values of x that would result in the left hand side of the equation being equal to the right, which is zero. Since multiplying anything by zero will give you zero, and the two factors multiply each other, the left-hand-side of the equation will equal zero when either one of the two factors equals zero.

So:

(x + 5) = 0
x = – 5

and:

(x + 2) = 0
x = -2

Finally, we can plot the line:

$y = x^2 + 7 x + 10$

using the Graphing Calculator Pro app, or this somewhat crude Parabola-Line Excel Graphing Worksheet, to show that the line crosses the x axis at -5 and -2.:

$25 computer

Here’s a real computer, the Raspberry Pi, for only $25. It has only two ports, one for a monitor and another for a keyboard. I’d suggest it needs one more USB port so you could hook it up to external devices (like robots), if you can’t split the single USB.

Its intention is to bring computer hardware and programming into schools. I’d love to get hold of one.

(Articles from BBC and geek.com).

Illustrating the Multiplication of Quadratic Factors

all-terms-color — Each term needs to multiply the other two terms in the opposite parenthesis, so start with the reds, then do the two mixed colors, then the blues, and finally, combine the mixed colors.

I’ve been playing around with ways of showing how to multiply out quadratic factors like the one above. I’m still not perfectly happy with these animations but they’re the best I’ve come up with so far. A smoother, Flash or svg animation might work better though.

In this second version the terms being multiplied are highlighted. I like how the highlighting gives some more stability to the animation, but I’m always leery of putting too much color or bells and whistles because they tend to complicate the picture. In this case at least, I think all the colors have meaning and are useful.

all-terms-color-hi — Multiplication of quadratic factors.

Caminito del Rey

Impressive video of someone walking across the Caminito del Rey in Spain. The narrow walkway links two sides of an almost vertical slot canyon. It has fallen into disrepair, however, and parts of it are missing. In some places the concrete walkway is entirely gone and all that’s left is a rusty metal beam that traverses the absent span. Quite impressive.

Writing should first focus on the text

I have a great antipathy when my word processor tells me what to do, or, even worse, “corrects” my writing without my permission. So I avoid MS Word like the plague. OpenOffice is little better. Now I’ll admit that my writing is usually in great need of a good editor, but not looking over my shoulder, inserting little, irritating suggestions while I’m caught up in the turbulent rapids of self-expression. Getting into the flow of productive writing is difficult enough; I don’t need the extra distraction.

Instead I much prefer the plain text editors; Smultron has been a favorite of mine since I’ve been using Macs, and I spend a lot of time writing on the class Wiki (MediaWiki) and on this blog (WordPress), which both have very simple text-entry boxes.

WordPress and MediaWiki also process the text and make it presentable. Like most websites these days, this blog has a theme that tells it where to put the text, how to format it, what background to have, where to insert images, what to have in the header and footer, …. The theme I use was created by Karen Blundell and adapted to put in a couple of my own details, like the little citation thing, and the ability to name the reviewer and editors at the bottom of the post. I did spend a lot of time getting these things to work, but I did learn quite a bit about the inner workings of WordPress and CSS in doing them, and once they were done, I could forget about them entirely and just focus on the writing.

Similarly, with LaTeX, although it’s much more of a pain to figure out how to use. On the Muddle I use LaTeX to add mathematical equations, but it really is a fully-fledged typesetting program, designed for professionals.

cc-book-pages — Two pages from a booklet my class and I put together about a display of fossils. I used LaTeX to typeset.

My class recently created a little display of fossils collected from Coon Creek for a school fundraiser, and we put together a booklet for it. I had the students write their essays and put them up on our Wiki. Then I copied and pasted their text into a LaTeX document, added a couple chapters from some of my blog posts, and it did the rest to create a very nice looking book, complete with title page, table of contents and bibliography.

Setting up the LaTeX file was not trivial, since I’ve not used it in a number of years, and this was the first time I tried to format a book. But it creates beautifully looking documents, without all the mysterious formatting features that inevitably show up if you tried something this complex with Word.

There are, I’m sure, other software for publishing documents like this. However, LaTeX is free and so is the old version of Smultron. Smultron’s new version costs $4.99, but is probably worth it.