Practicing Plotting Points on the Co-ordinate Plain

Pre-Algebra class starts next week, so in preparation for one of the early lessons on how to plot x,y co-ordinates, I put together an interactive plotter that lets students drag points onto the co-ordinate plain.

coord-plain — Students practice plotting points by dragging the red dot to the coordinates given.

Usage

The program generates random coordinate pairs within the area of the chart (or you can enter values of the coordinates yourself):

Clicking the “Show Point” button will place a yellow dot at the point.
When you’re confident you understand how the coordinate pairs work, you can practice by dragging the red dot to where you think the point is and the program will tell you if you’re right or not.

About the Program

This interactive application uses the jQuery and KineticJS javascript libraries. The latter library in particular is useful for making the HTML5 canvases interactive, so you can click on points on the graph and drag them.

When I have some time, after classes settle down, I’ll see if I can figure out how to embed this type of app into this (WordPress) blog. KineticJS is based off HTML5 canvases, which is what I use for the other interactive graphs I’ve posted, so it shouldn’t be terribly hard (at least in principle).

Finding Where Two Lines Intersect: Khan Academy Lessons

Khan Academy videos on how to solve systems of linear equations:

Use a graphing calculator to double check your lines, or use WolframAlpha, or use my simple line grapher.

by Graphing

Video: Solving Linear Systems by Graphing

Most algebra texts have good problem sets for practice (Khan Academy does not – as yet).

by Substitution

Moving from the more intuitive, visual, graphical method of solution to the more exact algebraic methods, we start with solving by substitution.

Video: Solving Linear Systems by Substitution

Practice Set: Practicing Systems of equations with substitution

by Elimination

Video: Solving Systems of Equations by Elimination

Additional Video: Solving systems by elimination 2 (on YouTube).

Additional Video: Solving systems by elimination 3 (on YouTube).

Practice Set: Practicing Systems of equations with simple elimination

Exponential Growth/Decay Models (Summary)

A quick summary (more details here):

The equation that describes exponential growth is:
Exponential Growth: $N = N_0 e^{rt}$

where:

N = number of cells (or concentration of biomass);
N₀ = the starting number of cells;
r = the rate constant, which determines how fast growth occurs; and
t = time.

You can set the r value, but that’s a bit abstract so often these models will use the doubling time – the time it takes for the population (the number of cells, or whatever, to double). The doubling time (t_d) can be calculated from the equation above by:

$t_d = \frac{\ln 2}{r}$

or if you know the doubling time you can find r using:

$r = \frac{\ln 2}{t_d}$

Finally, note that the only difference between a growth model and a decay model is the sign on the exponent:

Exponential Decay: $N = N_0 e^{-rt}$

Decay models have a half-life — the time it takes for half the population to die or change into something else.

Exponential Cell Growth

The video shows 300 seconds of purely exponential growth (uninhibited), captured from the exponential growth VAMP scenario. Like the exponential growth function itself, the video starts off slowly then gets a lot more exciting (for a given value of exciting).

The modeled growth is based on the exponential growth function:

$N = N_0 e^{rt}$ (1)

where:

N = number of cells (or concentration of biomass);
N₀ = the starting number of cells;
r = the rate constant, which determines how fast growth occurs; and
t = time.

Finding the Rate Constant/Doubling Time (r)

You can enter either the rate constant (r) or the doubling time of the particular organism into the model. Determining the doubling time from the exponential growth equation is a nice exercise for pre-calculus students.

Let’s call the doubling time, t_d. When the organism doubles from it’s initial concentration the growth equation becomes:

$2N_0 = N_0 e^{r t_d}$

divide through by N₀:

$2 = e^{r t_d}$

take the natural logs of both sides:

$\ln 2 = \ln (e^{r t_d})$

bring the exponent down (that’s one of the rules of logarithms);

$\ln 2 = r t_d \ln (e)$

remember that ln(e) = 1:

$\ln 2 = r t_d$

and solve for the doubling time:

$\frac{\ln 2}{r} = t_d$

Decay

A nice follow up would be to solve for the half life given the exponential decay function, which differs from the exponential growth function only by the negative in the exponent:

$N = N_0 e^{-rt}$

The UCSD math website has more details about Exponential Growth and Decay.

Finding the Growth Rate

A useful calculus assignment would be to determine the growth rate at any point in time, because that’s what the model actually uses to calculate the growth in cells from timestep to timestep.

The growth rate would be the change in the number of cells with time:

$\frac{dN}{dt}$

starting with the exponential growth equation:

$N = N_0 e^{rt}$

since we have a natural exponent term, we’ll use the rule for differentiating natural exponents:

$\frac{d}{dx}(e^u) = e^u \frac{du}{dx}$

So to make this work we’ll have to define:

$u = rt$

which can be differentiated to give:

$\frac{du}{dt} = r$

and since N₀ is a constant:

$N = N_0 e^{u}$

$\frac{dN}{dt} = N_0 e^{u} \frac{du}{dt}$

substituting in for u and du/dt gives:

$\frac{dN}{dt} = N_0 e^{rt} (r)$

rearranging (to make it look prettier (and clearer)):

$\frac{dN}{dt} = N_0 r e^{rt}$ (2)

Numerical Methods: Euler’s method

With this formula, the model could use linear approximations — like in Euler’s method — to simulate the growth of the biomass.

First we can discretize the differential so that the change in N and the change in time (t$) take on discrete values:
$\frac{dN}{dt} = \frac{\Delta N}{\Delta t}$

Now the change in N is the difference between the current value N^t and the new value N^t+1:

Now using this in our differentiated equation (Eq. 2) gives:

$\frac{N^{t+1}-N^t}{\Delta t} = N_0 r e^{r\Delta t}$

Which we can solve for the new biomass (N^^t+1):

$N^{t+1}-N^t = N_0 r e^{r\Delta t} \Delta t$

to get:
$N^{t+1} = N_0 r e^{r\Delta t} \Delta t + N^t$

This linear approximation, however, does introduce some error.

The approximated exponential growth curve (blue line) deviates from the analytical equation. The deviation compounds itself, getting worse exponentially, as time goes on.

Excel file for graphed data: exponential_growth.xls

VAMP

This is the first, basic but useful product of my summer work on the IMPS website, which is centered on the VAMP biochemical model. The VAMP model is, as of this moment, still in it’s alpha stage of development — it’s not terribly user-friendly and is fairly limited in scope — but is improving rapidly.

Plugging Latex Equations into Webpages

I’ve figured out how to put latex equations into this WordPress website, but have been struggling trying to get it on my other math based web pages, like the parabolas page.

Now, however, I’ve discovered CodeCogs, which provides an excellent Equation Editor that allows the inclusion of latex equations on any website (html page).

The Mathematical Logic Behind Billions, Trillions and Standard Form

A billion, in continental Europe is, a million squared ( 1,000,000² = 1,000,000,000,000), but in the English speaking world, a billion is only a thousand million (1,000,000,000). numberphile goes into the beautiful, mathematical logic of the longer form (i.e. the continental system).

Of course, the “simplest” system, which avoids all the potential for miscommunication, is the standard scientific notation, where 1,000,000,000 is written as 1×10⁹ (or just 10⁹).

↬ The Dish

Bending a Soccer Ball

Students from the University of Leicester have published a beautiful short research paper (pdf) on the physics of curving a soccer ball through the air.

It has been found that the amount a football bends depends linearly on the speed of the ball and the amount of spin.

— Sandhu et al., 2011: How to score a goal (pdf) in the University of Leicester’s Journal of Physics Special Topics

They derive the relationship from Bernoulli’s equation using some pretty straightforward algebra. The force (F) perpendicular to the ball’s motion that causes it to curl is:

$F = 2 \pi R^3 \rho \omega v$

and the distance the ball curls can be calculated from:

$D = \frac{\pi R^3 \rho \omega}{ v m } x^2$

where:

F = force perpendicular to the direction the ball is kicked
D = perpendicular distance the ball moves to the direction it is kicked (the amount of curl)
R = radius of the ball
ρ = density of the air
ω = angular velocity of the ball
v = velocity of the ball (in the direction it is kicked)
m = mass of the ball
x = distance traveled in the direction the ball is kicked

The paper itself is an excellent example of what a short, student research paper should look like. And there are number of neat followup projects that advanced, high-school, physics/calculus students could take on, such as: considering the vertical dimension — how much time it take for the ball to rise and fall over the wall; creating a model (VPython) of the motion of the ball; and adding in the slowing of the ball due to air friction.

↬ScienceDaily

A Review of Fractions: Based on Khan Academy Lessons

This is a basic review of working with fractions using lessons and practice sets from the Khan Academy.

1. Adding Fractions with a Common Denominator

The first topic — adding fractions –ought to be really easy for algebra students, but it allows them to become familiar with the Khan Academy website and doing the practice sets.

Now do the Practice Set.

OPTIONAL: Subtracting fractions with a common denominator works the same way. Students may do this practice set if they find it useful.

2. Adding Fractions with a Different Denominator

This is usually a helpful review.

The practice set.

3. Multiplying and Dividing Fractions

A good review that helps build up to working with radical numbers.

Multiplying fractions:

Do the multiplying fractions practice set.

Dividing Fractions:

Dividing fractions practice set.

4. Converting Fractions to Decimals

The last review is on how to convert fractions to decimals.

Now try the practice set for ordering numbers.

5. Next: Working with Square Roots