Demonstrating Taylor Series Approximations with Graphs

January 31, 2017

This little embeddable, interactive app uses nth order polynomials to approximate a few curves to demonstrate the Taylor Series.

Citing this post: Urbano, L., 2017. Demonstrating Taylor Series Approximations with Graphs, Retrieved December 11th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

Practicing Plotting Points on the Co-ordinate Plain

August 25, 2012

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).

Citing this post: Urbano, L., 2012. Practicing Plotting Points on the Co-ordinate Plain, Retrieved December 11th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

Solving Quadratic Equations

February 29, 2012

So here we have a little grapher that solves quadratic equations visually. Just enter the coefficients in the equation.

Quadratic Equation:       y = a x2 + b x + c

Enter:                               y = x2 + x +

Your browser does not support the canvas element.

Solution:

Analytical solution by factoring (with a little help from the quadratic equation if necessary).

Hopefully, this can help students learn about factoring and quadratics in a more graphical way.

Notes:

This is my first interactive post. I use Javascript in combination with HTML5. Now I need to figure out how to interact with the image itself, instead of the textboxes.

Citing this post: Urbano, L., 2012. Solving Quadratic Equations, Retrieved December 11th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

Straight Lines

February 26, 2012

Your browser does not support the canvas element.

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.

Citing this post: Urbano, L., 2012. Straight Lines, Retrieved December 11th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

Malthusian Growth

June 30, 2011

I think I may fairly make two postulata. First, That food is necessary to the existence of man. Secondly, That the passion between the sexes is necessary and will remain nearly in its present state. -- Malthus (1798): An Essay on the Principle of Population via The Concise Encyclopedia of Economics
"Malthusian" is often used as a derogatory term to refer to alarmist predictions that we're going to run out of some natural resource. I'm afraid I've used the term this way myself, however, according to Lauren Landsburg at the Concise Encyclopedia of Economics, Malthus is being unfairly maligned. He wasn't actually predicting catastrophe but wondering why the catastrophes don't usually happen. What Thomas Malthus did, in 1798, was point out that while populations grow at a geometric rate - the U.S. population, he noticed, doubled every 25 years - but resources, like food, only increase at an arithmetic rate. As a result, any naturally growing population will eventually run out of resources.

The red line shows geometric growth. No matter how much you start off with, the red line will always end up crossing the blue line.

The linear equation has the form: y = m t + b where y is the quantity produced, t is time (the independent variable), and m and b are constants. This should not look to unfamiliar to students who've had algebra. The geometric equation is a little more complicated: y = a^{gt} + c here a, g and c are constants. g is the most important, because it's the growth rate - the higher g is the faster the curve will rise. You can play around with the coefficients and graph in this Excel spreadsheet . At any rate, after the curves intersect, the needs of the population exceeds how much it can produce; this is the point of Malthusian catastrophe.

The intersection point is where the needs of the population exceeds the production.

The observation is, indeed, so stark that it is still easy to lose sight of Malthus’s actual conclusion: that because humans have not all starved, economic choices must be at work, and it is the job of an economist to study those choices. -- Landsburg (2008): Thomas Robert Malthus from The Concise Encyclopedia of Economics.

Citing this post: Urbano, L., 2011. Malthusian Growth, Retrieved December 11th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

What’s Needed for a Nation’s Peace

June 23, 2011

The Fund for Peace has been doing a lot of thinking about what it takes for a country to be considered peaceful, and what it takes for a state to fail. For the last seven years they've been putting together maps of the world with an index of how stable different countries are.

Finland - the most sustainable state, at least according to the Failed State Index.

While it's pretty in-depth and makes for rather sobering reading, it's worth taking a look at the criteria they've come up with to determine a country's stability. It may be useful to include some of this information in the cycle where we focus on peace. Their criteria for instability include:
  • Demographic pressure (such as having too many young adults, as we've seen in Egypt)
  • Amount of refugees and internally displaced peoples (refugees are people who've crossed international borders). Both leaving or entering refugees can undermine stability.
  • Historical Injustice - communities can have an understandably hard time forgetting the past, just look at Isreal/Palestine.
  • Brain Drain - when countries start to fail, the first to leave are the ones who can afford to. Yet these intellectuals and professionals, with their college degree are vital for creating a stable and prosperous country.
  • Inequality - especially when driven by active discrimination (wealth inequality is something to watch out for).
  • Economic decline - pushes trade into the black market and increases criminality and corruption.
  • Illegitimacy of the state - if people don't believe the people in government have everyone in the country's best interests at heart, and are only looking out for themselves and their friends, then there's probably going to be trouble.
  • Public Services go kaput - It's a really bad sign when the government can meet people's basic needs - like picking up the garbage.
  • The Rule of Law goes kaput - when you're ruled by the caprice of men, and your rights under the law are not respected, you may begin to consider and agitate for other options for government.
  • Personal Armies - forces that are tied to individual leaders, like private militias or super-secret police for example, are very damaging to a country's cohesion.
  • Fighting elites - healthy countries need robust arguments in their political class - think checks and balances - but it can go too far and lead to things like extreme nationalism and ethnic cleansing.
  • Invasion - both overt invasion and covert meddling in the affairs of a country are unhealthy for that state's stability.
It's also very nice that you can download their index data as a MS Excel spreadsheet, which you can let students analyze to answer their own research questions. For example, I was wondering what was the difference between the best, the worst and the USA, so I plotted this graph.

Comparing the best (Finland), worst (Somalia) and the USA using the Fund for Peace's Failed State Indicies.

The USA is much closer to Finland than Somalia, thank goodness, but should probably watch out for that Uneven Development (wealth inequality). I think something like this would make a good experiential exercise for the science of geography.

Citing this post: Urbano, L., 2011. What's Needed for a Nation's Peace, Retrieved December 11th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

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