Using the Binomial Cube in Algebra

binomial-cube-0564 — Figuring out (a+b)³; with a binomial cube.

After working with the hundred-squares, ten-bars, and thousand-cubes to figure out how to add polynomials, we borrowed the binomial and trinomial cubes to practice multiplying out factors. It’s a physical way of showing factor multiplication.

Binomial Square

You can first look at the binomial cube in two-dimensions as a binomial square by just finding the area of the top layer of four blocks.

If you label the length of the side of the red block, a, and the length of the blue block, b, you can calculate the areas of the individual pieces simply by multiplying their lengths times their widths.

binomial-cube-algebra — Looking from the top down, the top layer of the binomial cube is a binomial square.

Adding up the individual areas you get the area of the entire square:

$A = a^2 + 2ab + b^2$

However, there is another way.

If you recognize that the length of each side of the entire square is equal to (a+b).

binomial-mult-factors — The length of each side of the cube is the sum of the lengths of the two squares.

Then the total area is going to be total length (a+b) times the total width (a+b):

$A = (a+b)^2 = (a+b)(a+b)$

We can multiply this out (using FOIL is easiest):

$(a+b)(a+b) = a^2 + ab + ab + b^2$

which simplifies to give the same result as adding up the individual areas:

$(a+b)^2 = a^2 + 2ab + b^2$

The Binomial Cube

We can do the same thing using the entire cube by recognizing that the volume of the cube is the length times width times the depth, and all of these dimensions are the same: (a+b).

binomial-cube-cubing — Using the full binomial cube.

Now the students can go through the same process of multiplying out the factors, and can check their work be seeing if they get the same number of pieces (and dimensions) as the physical cube.

Emission Spectra: How Atoms Emit and Absorb Light

Emission_spectrum-H-new — Emission and absorption spectrum of Hydrogen. Image adapted from the one by Wikimedia Commons User:Adrignola.

When a photon of light hits an atom three things can happen: it can bounce off; it can pass through as if nothing had happened; or it be absorbed. Which one happens depends on the energy of the light, and which atom it is hitting. Hydrogen will absorb different energies from helium.

The interesting thing is that each atom will only absorb photons with exactly the right energy. You see, when the light hits the atom, the atom will only absorb it if it can use it to bump an electron up an electron shell.

Oxygen-shells — An oxygen atom with two electrons in the innermost shell and six in the second shell.

An oxygen atom, for example, has eight electrons, but these fit into two different electron shells. The innermost shell can only hold two electrons, so the other six go into the second shell (which can take a maximum of 8). This is the “ground state” of the atom, because it takes the least amount of energy to keep the electrons in place.

However, if the atom gets hit by just enough energy, one of the electrons can be bumped up into a higher shell. The atom will be “excited” and “want” the electron to drop back down to the lower shell.

And when the electron drops back, it will release the exact same amount of energy that it took to move it up a shell in the first place.

energy-levels — A hydrogen atom's electron is bumped up an energy level/shell by ultraviolet light, but releases that light when the electron drops back down to its original shell.

For hydrogen, the energy to bump up an electron from its first to its second electron shell comes only from light with a wavelength of 1216 x 10^-10m (see here for how to calculate). This is in the ultraviolet range, which we can’t see with the naked eye.

However, the energy absorbed and released when the electron moves between the second and fourth shells is 6564 x 10^-10m, which is in the visible range. In fact, it’s the red line on the right side of the emission spectrum shown at the top of the page.

Emission_spectrum_H_annotated — The visible part of the emission spectrum of hydrogen and its corresponding electron shell jumps. Note that different colors of light have different energies, and that the blue light is more energetic (and has a shorter wavelength) than red light.

Since this emission signature is unique for each element, looking at the colors of other stars, or the tops of the atmospheres of other planets, is a good way of identifying the elements in them. You can also do flame tests to identify different elements.

The University of Oregon has an excellent little Flash application that shows the absorption spectrum of most of the elements in the periodic table (NOTE that the wavelengths they give are in Angstroms (Å) which are 1×10^-10m; or tenths of a nanometer (nm)).

spectrum-app — Interactive app that shows the absorption spectrum of the elements in the periodic table. From the University of Oregon. Wavelengths are in Angstroms (Å), and 1 Å = 0.1 nm.

Equations

The wavelength of light emitted for the movement of an electron between the electron shells of a hydrogen atom is given by the Rydberg formula:

$\frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)$

where:
$\lambda$ = wavelength of the light in a vacuum
$R$ = Rydberg constant (1.1×10⁷ m^-1)
$n_1$ and $n_2$ are the electron shell numbers (the innermost shell is 1, the next shell is 2 etc.)

— see an example calculation here.

The amount of energy that corresponds to a particular wavelength of light is given by:

$E = \frac{hc}{\lambda}$

where:
$E$ = Energy (J),
$\lambda$ = wavelength of the light in a vacuum (m),
$h$ = 6.6×10^-34 Js,
$c$ = speed of light in a vacuum (3×10⁸ m/s).

— via Wikipedia.

References

The National Institute of Standards and Technology (NIST) has more extensive emission spectra data.

Polynomials: Revisiting pre-Kindergarten

algebra-0489 — Working with the thousand cube, hundred square, ten bar and unit cells in algebra.

I sent a couple of my algebra students down to the pre-Kindergarten classroom to burrow one of their Montessori works. They were having a little trouble adding polynomials, and the use of manipulatives really helped.

The basic idea is that when you add something like:

$n^2 + 2n + 3n^2 + 4n + 5 + 2n^3 + 4 + 3n^3$

you can’t add a n³ term to a n² or a n term. You only combine the terms with the same degree (and same variables). So the equation above becomes:

$2n^3 + 3n^3 + n^2 + 3n^2 + 4n + 2n + 5 + 4$

which simplifies to:

$5n^3 + 4n^2 + 6n + 9$

The kids actually enjoyed the chance to run downstairs to burrow the materials from their old pre-K teacher (and weren’t they quite good about returning the materials when they were done with them).

And it clarifies a lot of misconceptions when you can clearly see that that you just can’t add a thousand cube to a ten bar — it just doesn’t work.

Parabolic Mirrors

Parabola_with_focus — Parabolic mirrors magnify by reflecting parallel rays of incoming light onto a single point. (Adapted from Wikimedia Commons User:Nargopolis).

We’re talking about light and sound waves in physics at the moment, and NPR’s Morning Edition just had a great article on how the enormous, ultra-precise, mirrors that are used in large telescopes are made.

Astronomical observatories tend to use mirrors instead of lenses in their telescopes, largely because if you make lenses too big they tend to sag in the middle, while you can support a mirror all across the back, and because you have to make a lens perfect all the way through for it to work correctly, but only have to make one perfect surface for a parabolic mirror.

ScienceClarified has a great summary of the history of the Hubble Space telescope, that includes all the trouble NASA went through trying to fix it when they realized it was not quite perfect.

LBT_3s — Large parabolic mirrors are used for magnification in telescopes. (Image via Wikipedia).

In addition, it’s interesting to note that you can also make a parabolic surface on a liquid by spinning it, resulting in liquid telescope mirrors .

How Long does it Take to Make a Vpython Program?

magnet-b-100 — Moving the magnet through a wire coil creates an electric current in the wire. Animation captured from the VPython program: Magnetic Induction - Coils.

My students asked me this question the other day, and while slapping together an animation of electromagnetic induction I gave it some thought.

This program itself is really simple. It took about 15 minutes.

But that’s not counting the half hour I spent searching the web for an image I could use to illustrate magnetic induction and not finding one I could use.

Nor does it count the four hours I spent after I got the animation working to get the program to take screen captures automatically. Of course, I must admit that figuring out the screen captures would have gone a lot quicker if I’d not had to rebuild all my permissions on my hard drive (I’d recently reformatted it), and reinstall ImageMagick and gifsicle to take the screen captures and make animations.