Tag: algebra

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.

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 (td) 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.

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

Starting Algebra too Early?

There’s been a push for students to take algebra earlier and earlier, yet there are some serious pedagogic arguments that early algebra might not be a great idea for many, if not most, students. A fascinating paper by Clotfelter et al., (2012) (pdf) showed pretty clearly that for a large number of students, taking algebra earlier actually resulted in worse performance in not just algebra, but the follow-up classes as well (geometry and pre-calculus for example), compared to students who waited to take the subject. Indeed the Charlotte-Mecklenburg School District (the district studied in the article) actually reversed their policy of having students take algebra in 8th grade.

Students affected by the acceleration initiative scored significantly lower on end-of-course tests in Algebra I, and were either no more likely or significantly less likely to pass standard follow-up courses, Geometry and Algebra II

— Clotfelter et al., (2012): The Aftermath of Accelerating Algebra: Evidence from a District Policy Initiative (pdf) via NY Fed.

The argument for early algebra comes from the correlation between early algebra and better performance on standardized tests, and more advanced math classes in high school. But the authors here indicate that forcing students to take algebra early does not result in the same outcomes.

The argument against early algebra is based on the research that shows formal thinking develops during adolescence, and the belief that to do well in algebra requires the abstract thinking skills that are seated in the maturing prefrontal cortex. Until students are ready for the abstract thinking required (which happens at different times for each student), they will struggle with algebra.

Algebra provides an essential foundation for further mathematics, which is why it is my strong preference that students progress by demonstrating mastery of the topics at their own pace rather than struggling through the class.

Reasons to Study Algebra: Economics


I hope you think that I am an acceptable writer, but when it comes to economics I speak English as a second language: I think in equations and diagrams, then translate.

— Krugman (1996): Economic Culture Wars in Slate

I sometimes get the question: Why do I have to learn algebra? Followed by the statement: I’m never going to have to use it again. My response is that it’s a bit like learning to read; you can survive in society being illiterate, but it’s not easy. The same goes for algebra, but it’s a little more complex.

Paul Krugman argues for the importance of algebra for anyone thinking about economics, the economy, and what to do about it. Even at the basic level, economists think in mathematical equations and algebraic models, then they have to translate their thoughts into English to communicate them. People who are not familiar with algebra are at a distinct advantage.

There are important ideas … that can be expressed in plain English, and there are plenty of fools doing fancy mathematical models. But there are also important ideas that are crystal clear if you can stand algebra, and very difficult to grasp if you can’t. [my emphasis] International trade in particular happens to be a subject in which a page or two of algebra and diagrams is worth 10 volumes of mere words. That is why it is the particular subfield of economics in which the views of those who understand the subject and those who do not diverge most sharply.

— Krugman (1996): Economic Culture Wars in Slate

P.S. In the article, he also points out the importance of algebra in the field of evolutionary biology.

Serious evolutionary theorists such as John Maynard Smith or William Hamilton, like serious economists, think largely in terms of mathematical models. Indeed, the introduction to Maynard Smith’s classic tract Evolutionary Genetics flatly declares, “If you can’t stand algebra, stay away from evolutionary biology.” There is a core set of crucial ideas in his subject that, because they involve the interaction of several different factors, can only be clearly understood by someone willing to sit still for a bit of math.

↬ Reddit

Momentum

A ball rolling down a ramp hits a car which moves off uphill. Can you come up with an experiment to predict how far the car will move if the ball is released from any height? What if different masses of balls are used?

Students try to figure out the relationship between the ball's release height and how far the car moves.

For my middle school class, who’ve been dealing with linear relationships all year, they could do this easily if the distance the car moves is directly proportional to height from which the ball was released?

The question ultimately comes down to momentum, but I really didn’t know if the experiment would work out to be a nice linear relationship. If you do the math, you’ll find that release height and the maximum distance the car moves are directly proportional if the momentum transferred to the car by the ball is also directly proportional to the velocity at impact. Given that wooden ball and hard plastic car would probably have a very elastic collision I figured there would be a good chance that this would be the case and the experiment would work.

It worked did well enough. Not perfectly, but well enough.