Jupiter and Venus in Conjuction

jupiter-venus-0847 — Venus (brighter) and Jupiter.

Jupiter and Venus have been sitting near the western horizon, shining so brightly that even I have noticed them. Phil Plait explains with some back-of-the-envelope math, why Venus is brighter even though it’s smaller than Jupiter. It’s a nice example of how a little math can do a great job explaining how the world (and others) works.

Everything You (N)ever Wanted to Know About Parabolas

So that my students could more easily check their answers graphically, I put together a page with a more complete analysis of parabolas (click this link for more details).

[inline]

Analyzing Parabolas

Standard Form	Vertex Form
`y = a x² + b x + c`	`y = a (x – h)² + k`
y = x² + x +	y = ( x – ) ² +

Intercepts:	Vertex:
Focus:	Directrix:	Axis:

Solution by Factoring:

y = x² x

[script type=”text/javascript”]
var width=500;
var height=500;
var xrange=10;
var yrange=10;

mx = width/(2.0*xrange);
bx = width/2.0;
my = -height/(2.0*yrange);
by = height/2.0;

function draw_9239(ctx, polys) {
t_9239=t_9239+dt_9239;
//ctx.fillText (“t=”+t, xp(5), yp(5));
ctx.clearRect(0,0,width,height);

polys[0].drawAxes(ctx);
ctx.lineWidth=2;
polys[0].draw(ctx);

polys[0].write_eqn2(ctx);

//polys[0].y_intercepts(ctx);
//write intercepts on graph

// SHOW VERTEX
if (show_vertex_ctrl_9239.checked==true) {
polys[0].draw_vertex(ctx);
document.getElementById(‘vertex_pos_9239’).innerHTML = “(“+polys_9239[0].vertex.x.toPrecision(2)+ ” , “+polys_9239[0].vertex.y.toPrecision(2)+ ” )”;
} else { document.getElementById(‘vertex_pos_9239’).innerHTML = “”;}

// SHOW FOCUS
if (show_focus_ctrl_9239.checked==true) {
polys[0].draw_focus(ctx);
document.getElementById(‘focus_pos_9239’).innerHTML = “(“+polys_9239[0].focus.x.toPrecision(2)+ ” , “+polys_9239[0].focus.y.toPrecision(2)+ ” )”;
} else { document.getElementById(‘focus_pos_9239’).innerHTML = “”;}

// SHOW DIRECTRIX
if (show_directrix_ctrl_9239.checked==true) {
polys[0].draw_directrix(ctx);
document.getElementById(‘directrix_pos_9239’).innerHTML = polys[0].directrix.get_eqn2(“y”,”x”,”html”);

polys[0].directrix.write_eqn2(ctx, polys[0].directrix.get_eqn2(“directrix: y”));
} else { document.getElementById(‘directrix_pos_9239’).innerHTML = “”;}

// SHOW AXIS
if (show_axis_ctrl_9239.checked==true) {
polys[0].draw_parabola_axis(ctx);
document.getElementById(‘axis_pos_9239’).innerHTML = “x = “+polys[0].vertex.x.toPrecision(2);
}

//SHOW INTERCEPTS
ctx.textAlign=”center”;
if (show_intercepts_ctrl_9239.checked==true) {
polys[0].x_intercepts(ctx);
ctx.fillText (‘x intercepts: (when y=0)’, xp(6), yp(8));
//ctx.fillText (‘intercepts=’+polys[0].x_intcpts.length, xp(5), yp(-5));
if (polys[0].order == 2) {
if (polys[0].x_intcpts.length > 0) {
line = “0 = “;
for (var i=0; i 0.0) { sign=”-“;} else {sign=”+”;}
line = line + “(x “+sign+” “+ Math.abs(polys[0].x_intcpts[i].toPrecision(2))+ “)”;
}
ctx.fillText (line, xp(6), yp(7));
for (var i=0; i 0) {
for (var i=0; i2 “+polys[0].bsign+” “+Math.abs(polys[0].b.toPrecision(2))+” x “+ polys[0].csign+” “+Math.abs(polys[0].c.toPrecision(2))+”

“;

solution = solution + ‘Factoring: ‘;

if (polys[0].x_intcpts.length > 0) {
solution = solution + ‘0 = ‘;
for (var i=0; i 0.0) { sign=”-“;} else {sign=”+”;}
solution = solution + “(x “+sign+” “+ Math.abs(polys[0].x_intcpts[i].toPrecision(2))+ “)”;
}
solution = solution + ‘

‘;
solution = solution + ‘Set each factor equal to zero:
‘;
for (var i=0; i 0.0) { sign=”-“;} else {sign=”+”;}
solution = solution + “x “+sign+” “+ Math.abs(polys[0].x_intcpts[i].toPrecision(2))+ ” = 0 “;
}
solution = solution + ‘

and solve for x:
‘;
for (var i=0; i‘;
}
document.getElementById(‘equation_9239’).innerHTML = solution;
}

else if (polys[0].order == 1) {
solution = solution + ‘
‘;
solution = solution + “y = “+” “+Math.abs(polys[0].b.toPrecision(2))+” x “+ polys[0].csign+” “+Math.abs(polys[0].c.toPrecision(2))+”

“;
solution = solution + ‘

Set y=0 and solve for x:
‘;
solution = solution + ” 0 = “+” “+Math.abs(polys[0].b.toPrecision(2))+” x “+ polys[0].csign+” “+Math.abs(polys[0].c.toPrecision(2))+”

“;
solution = solution + ‘ ‘;
solution = solution + (-1.0*polys[0].c).toPrecision(2) +” = “+” “+Math.abs(polys[0].b.toPrecision(2))+” x “+”

“;
solution = solution + ‘ ‘;
solution = solution + (-1.0*polys[0].c).toPrecision(2)+”/”+polys[0].b.toPrecision(2)+” = “+” x “+”

“;
solution = solution + ‘ ‘;
solution = solution + “x = “+ (-1.0*polys[0].c/polys[0].b).toPrecision(4)+”

“;

document.getElementById(‘equation_9239’).innerHTML = solution;
}

}

function update_form_9239 () {
a_coeff_9239.value = polys_9239[0].a+””;
b_coeff_9239.value = polys_9239[0].b+””;
c_coeff_9239.value = polys_9239[0].c+””;

av_coeff_9239.value = polys_9239[0].a+””;
hv_coeff_9239.value = polys_9239[0].h+””;
kv_coeff_9239.value = polys_9239[0].k+””;

}

//init_mouse();

var c_9239=document.getElementById(“myCanvas_9239”);
var ctx_9239=c_9239.getContext(“2d”);

var change = 0.0001;

function create_lines_9239 () {
//draw line
//document.write(“hello world! “);
var polys = [];
polys.push(addPoly(1,6, 5));

// polys.push(addPoly(0.25, 1, 0));
// polys[1].color = ‘#8C8’;

return polys;
}

var polys_9239 = create_lines_9239();

var x1=xp(-10);
var y1=yp(1);
var x2=xp(10);
var y2=yp(1);
var dc_9239=0.05;

var t_9239 = 0;
var dt_9239 = 100;
//end_ct = 0;
var st_pt_x_9239 = 2;
var st_pt_y_9239 = 1;
var show_vertex_9239 = 1; //1 to show vertex on startup
var show_focus_9239 = 1; // 1 to show the focus
var show_intercepts_9239 = 1; // 1 to show the intercepts
var show_directrix_9239 = 1; // 1 to show the directrix
var show_axis_9239 = 1; //1 to show the axis of the parabola

var move_dir_9239 = 1.0; // 1 for up

//document.getElementById(‘comment_spot’).innerHTML = polys_9239[0].a+” “+polys_9239[0].b+” “+polys_9239[0].c+” : “+polys_9239[0].h+” “+polys_9239[0].k+” “;

//standard form
var a_coeff_9239 = document.getElementById(“a_coeff_9239”);
var b_coeff_9239 = document.getElementById(“b_coeff_9239”);
var c_coeff_9239 = document.getElementById(“c_coeff_9239”);

//vertex form
var av_coeff_9239 = document.getElementById(“av_coeff_9239”);
var hv_coeff_9239 = document.getElementById(“hv_coeff_9239”);
var kv_coeff_9239 = document.getElementById(“kv_coeff_9239”);

//options
var show_vertex_ctrl_9239 = document.getElementById(“show_vertex_9239”);
if (show_vertex_9239 == 0) {show_vertex_ctrl_9239.checked=false;
} else {show_vertex_ctrl_9239.checked=true;}

var show_focus_ctrl_9239 = document.getElementById(“show_focus_9239”);
if (show_focus_9239 == 0) {show_focus_ctrl_9239.checked=false;
} else {show_focus_ctrl_9239.checked=true;}

var show_intercepts_ctrl_9239 = document.getElementById(“show_intercepts_9239”);
if (show_intercepts_9239 == 0) {show_intercepts_ctrl_9239.checked=false;
} else {show_intercepts_ctrl_9239.checked=true;}

var show_directrix_ctrl_9239 = document.getElementById(“show_directrix_9239”);
if (show_directrix_9239 == 0) {show_directrix_ctrl_9239.checked=false;
} else {show_directrix_ctrl_9239.checked=true;}

var show_axis_ctrl_9239 = document.getElementById(“show_axis_9239”);
if (show_axis_9239 == 0) {show_axis_ctrl_9239.checked=false;
} else {show_axis_ctrl_9239.checked=true;}

update_form_9239();

//document.write(“test= “+c_coeff_9239.value+” “+polys_9239[0].c);
setInterval(“draw_9239(ctx_9239, polys_9239)”, dt_9239);

a_coeff_9239.onchange = function() {
//polys_9239[0].a = parseFloat(this.value);
polys_9239[0].set_a(parseFloat(this.value));
polys_9239[0].update_vertex_form_parabola();
update_form_9239();
}
b_coeff_9239.onchange = function() {
//polys_9239[0].b = parseFloat(this.value);
polys_9239[0].set_b(parseFloat(this.value));
polys_9239[0].update_vertex_form_parabola();
update_form_9239();
}
c_coeff_9239.onchange = function() {
//polys_9239[0].c = parseFloat(this.value);
polys_9239[0].set_c(parseFloat(this.value));
polys_9239[0].update_vertex_form_parabola();
update_form_9239();
}

av_coeff_9239.onchange = function() {
//polys_9239[0].a = parseFloat(this.value);
polys_9239[0].set_a(parseFloat(this.value));
polys_9239[0].update_standard_form_parabola();
update_form_9239();
}

hv_coeff_9239.onchange = function() {
polys_9239[0].h = parseFloat(this.value);
polys_9239[0].update_standard_form_parabola();
polys_9239[0].set_order()
update_form_9239();
}

kv_coeff_9239.onchange = function() {
polys_9239[0].k = parseFloat(this.value);
polys_9239[0].update_standard_form_parabola();
polys_9239[0].set_order()
update_form_9239();
}

//draw_9239();
//document.write(“x”+x2+”x”);
//ctx_9239.fillText (“n=”, xp(5), yp(5));

[/script]

[/inline]

Converting the forms

The key relationships are the ones to convert from the standard form of the parabolic equation:

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

(1)

to the vertex form:

$y = a (x - h)^2 + k$

(2)

If you multiply out the vertex equation form you get:

y = a x² – 2ah x + ah² + k

(3)

When you compare this equation to the standard form of the equation (Equation 1), if you look at the coefficients and the constants, you can see that:

To convert from the vertex to the standard form use:

         $a = a$ (4)

         $b = -2ah$ (5)

         $c = ah^2 + k$ (6)

Going the other way,

To convert from the standard to the vertex form of parabolic equations use:

         $a = a$
(7)

         $h = \frac{-b}{2a}$
(8)

         $k = c - ah^2$
(9)

Although it is sometimes convenient to let k not depend on coefficients from its own equation:

$k = c - \frac{b^2}{4a}$ (10)

The Vertex and the Axis

The nice thing about the vertex form of the equation of the parabola is that if you want the find the coordinates of the vertex of the parabola, they’re right there in the equation.

Specifically, the vertex is located at the point:

$(x_v, y_v) = (h, k)$

(11)

The axis of the parabola is the vertical line going through the vertex, so:

The equation for the axis of a parabola is:

$x = h$

(12)

Focus and Directrix

Finally, it’s important to note that the distance (d) from the vertex of the parabola to its focus is given by:

$d = \frac{1}{4a}$

(13)

Which you can just add d on to the coordinates of the vertex (Equation 11) to get the location of the focus.

$(x_f, y_f) = (x_v, y_v + d)$

(14)

The directrix is just the opposite, vertical distance away, so the equation for the directrix is the equation of the horizontal line at:

$y = y_v + d$

(15)

References

There are already some excellent parabola references out there including:

Wonderful interactive graphs by Murray Bourne.
Nice simple explanations of the different forms of the equations.
More complex explanations from Wolfram Alpha.

Straight Lines

[inline]

[script type=”text/javascript”]
var width=500;
var height=500;
var xrange=10;
var yrange=10;

mx = width/(2.0*xrange);
bx = width/2.0;
my = -height/(2.0*yrange);
by = height/2.0;

function draw8587(ctx, polys) {
t=t+dt;
//ctx.fillText (“t=”+t, xp(5), yp(5));
ctx.clearRect(0,0,width,height);

polys[0].drawAxes(ctx);
//drawAxes(ctx);
ctx.lineWidth=3;
//ctx.beginPath();
//ctx.moveTo(x1,y1);
//ctx.lineTo(x2,y2);
//ctx.stroke();
//ctx.closePath();
//y2+=dyp(dy);

// add polynomial timing
// First move (from y=0 to y=1)
if (polys[0].c < 1.0) { polys[0].c = polys[0].c + 2.0*dt*change*move_dir; } // Second move (change from y=1 to y=0.5x+1) else if (polys[0].b < 0.5) { polys[0].b = polys[0].b + dt*change*move_dir; polys[0].order = 1; } // Third move (change from y=0.5x+1 to y=0.5x+4) else if (polys[0].c < 4.0) { polys[0].c = polys[0].c + 2.*dt*change*move_dir; //polys[0].order = 1; } else { //move_dir = move_dir * -1; end_ct = end_ct + dt; if (end_ct >= dt*40) {
polys[0].a = 0;
polys[0].b = 0;
polys[0].c = 0;
polys[0].order = 0;
}
}
ctx.textAlign=”center”;
ctx.textBaseline=”alphabetic”;
ctx.font = “14pt Arial”;
ctx.fillStyle = ‘#f00’;
ctx.fillText (“Equation of a Line”, xp(-5), yp(9));
ctx.fillText (“y = mx + b”, xp(-5), yp(5));
ctx.font = “12pt Arial”;
//ctx.fillText (“y = “+polys[0].b.toPrecision(1)+”x + “+ polys[0].c.toPrecision(2), xp(-5), yp(7));
ctx.fillText (“b = intercept = “+polys[0].c.toPrecision(2), xp(-7), yp(3));

//draw polynomials
ctx.lineWidth=3;
for (var i=0; i

Here’s a prototype for showing how the equation of a straight line changes as you change its slope (m) and intercept (b):
$! y = mx + b$

It starts with an horizontal line along the x-axis, where the slope is zero and the intercept is zero:
$! y = 0.0$

The line moves upward, but stays horizontal, until the intercept is 1.0:
$! y = 1.0$

The slope increases from horizontal (m = 0), gradually, until the slope is 0.5:
$! y = 0.5x + 1.0$

Finally, we move the line upwards again, without changing the slope (m = 0.5), until the intercept is equal to 4.0:
$! y = 0.5x + 4.0$

Now I just need to figure out how to make them interactive.

FOIL: Multiplying Factors

Multiplying out two factors can be a little tricky. The FOIL mnemonic is a quick method when you have two terms in each factor, such as in:

(a + b)(a + b)

FOIL stands for:

Firsts,
Outer,
Inner,
Lasts.

It applies to the multiplication of the binomial cube.

Another way of showing the process — step by step — would be like this:

FOIL-pt1 — Multiplying factors using FOIL.

After FOILing you combine the similar terms:

FOIL-pt2 — Combining like terms to get the final result.

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.

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 .

Gravity, the Electromagnetic Forces, and the Inverse Square Law

Calculating the forces between two charged particles (electric force), two magnets (the magnetic force), and two masses (the gravitational force) require remarkably similar equations. But, while electricity and magnetism are directly related (that’s why it’s called electromagnetism), gravity is its own fundamental force. Yet they all depend (inversely) on the square of the distance between the two objects creating the force, so they’re all said to obey some form of the inverse square law.

Gravitational Force (F_g)

The force exerted by two masses on one another is:

$F_g = G \frac{m_1 m_2}{d^2}$

where:

G is the gravitational constant (6.67300 × 10^-11 m³ kg^-1 s^-2
m₁ and m₂ are the masses of the two objects attracting one another.
d is the distance between the two objects.

Electrical Force (F_e)

The force exerted by two electrically charged objects on one another (like a proton and an electron), is:

$F_e = K \frac{q_1 q_2}{d^2}$

where:

K is the electrical constant, sometimes called Coulumn’s constant (8.9876 × 10⁹ N m² C^-2
q₁ and q₂ are the sizes of the charges (in Coulumbs) of the two objects attracting one another.
d is the distance between the two objects.

Magnetic Force (F_m)

The force exerted by two magnets on one another, is:

$F_m = \mu \frac{p_1 p_2}{d^2}$

where:

μ is a constant, (a little simplified)
p₁ and p₂ are strengths of the magnetic poles of the two objects attracting one another.
d is the distance between the two objects.

The magnetic force is a little more difficult to give a single equation for, because you need to factor in the shape of the magnets.

Inverse Square Laws

In addition to gravity, electric, and magnetic forces, light (which is electromagnetic radiation) and sound also obey inverse square laws.

Analyzing Parabolas

Solution by Factoring:

Converting the forms

The Vertex and the Axis

Focus and Directrix

References

Binomial Square

The Binomial Cube

Gravitational Force (Fg)

Electrical Force (Fe)

Magnetic Force (Fm)

Inverse Square Laws

Gravitational Force (F_g)

Electrical Force (F_e)

Magnetic Force (F_m)