Montessori Muddle – Page 47 – Middle and High School … from a Montessori Point of View

Embeddable Graphs

Going beyond just polynomials, I’ve created a javascript graphing app that’s easily embeddable.

At the moment, it just does polynomials and points, but polynomials can be used to teach quadratic functions (parabolas) and straight lines to pre-algebra and algebra students. Which I’ve been doing.

Based on my students’ feedback, I’ve made it so that when you change the equation of the line the movement animates. This makes it much easier to see what happens when, for example, you change the slope of a line.

P.S. You can also turn off the interactivity if you just want to show a simple graph. y = x²-1 is shown below:

A Poem for Easter and Spring

Loveliest of trees, the cherry now
Is hung with bloom along the bough,
And stands about the woodland ride
Wearing white for Eastertide.

Now, of my threescore years and ten,
Twenty will not come again,
And take from seventy springs a score,
It only leaves me fifty more.

And since to look at things in bloom
Fifty springs are little room,
About the woodlands I will go
To see the cherry hung with snow.

–Alfred Housman (1896): from A Shropshire Lad.

Last Snow

winter-road_4892c — A winding road through a snow caressed landscape.

Spring is coming, and I really don’t mind. Winter’s etched landscapes, however, cannot be denied.

Mitosis and Meiosis Videos

More information on my other mitosis resources posts (including the mitosis dance).

Biology Videos

Sumanas, Inc. has an excellent collection of biology-related videos, including good coverage of mitosis and meiosis.

White Butterfly

Introducing Polynomials

If you recall, straight lines have a general equation that looks like this:

$y=mx+b$ (1)

This is called the slope-intercept form of the equation, because m gives the slope, and b tells where the line intercepts the y-axis. For example the line:

$y=2x-3$ (2)

looks like:

Now, in the slope-intercept form, m and b represent numbers. In our example, m = 2 and b = 3.

So what if, instead of calling them m and b we used the same letter (let’s use a) and just gave two different subscripts so:

$m = a_1$ and,
$b = a_0$

therefore equation (1):

$y=mx+b$

becomes:
$y = a_1 x + a_0$ (3)

Now, in case you’re wondering why we picked m = a₁ instead of m = a₀, it’s because of the exponents of x. You see, in the equation x has an exponent of 1, and the constant b could be thought to be multiplying x with an exponent of 0. Considering this, we could rewrite our equation of the line (2):

$y=2x^1-3x^0$ (4)

since:
$x^1 = x$ and,
$x^0 = 1$

we get:
$y=2x-3(1)$
$y=2x-3$

So in equation (3) the subscript refers to the exponent of x.

Now we can expand this a bit more. What if we had a term with x² in an equation:

$y=\frac{1}{2}x^2 + 2x - 3$ (5)

Now we have three coefficients:

$a_0 = -3$ ,
$a_1 = 2$ and,
$a_2 = \frac{1}{2}$ ,

And the graph would look like this.

Because of the x² term (specifically because it has the highest exponent in the equation), this is called a second-order polynomial — that’s why the graph above has a little input box where the order is 2. In fact, on the graph above, you can change the order to see how the equation changes. Indeed, you can also change the coefficients to see how the graph changes.

A second order polynomial is a parabola, while, as you’ve probably guessed, a first order polynomial is a straight line. What’s a zero’th order polynomial?

Finally, we can write a general equation for a polynomial — just like we have the slope-intercept form of a line — using the a coefficients like:

$y = a_n x^n + ... + a_2 x^2 + a_1 x + a_0$

You can use the graphs to tinker around and see what different order polynomials look like, and how changing the coefficients change the graphs. I sort-of like the one below:

References

WolframAlpha has more details on polynomials.

The embedded graphs come from my own Polynomial Grapher.