A quick graphical calculator for straight lines -- based on This. It's still incomplete, but it's functional, and a useful complement to the equation of a straight line animation and the parabola graphing.

You can graph lines based on the equation of a line (in either the slope-intercept or point-slope forms), or from two points.

Straight Lines

Slope-Intercept Form	Two Points	Point-Slope Form
y = m x + b	(x1, y1) = ( , )	y - y1 = m ( x - x1)
y = x +	(x2, y2) = ( , )	y - = ( x - )

x intercept:	Slope:	Equation:
y intercept:	Points:

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Citing this post: Urbano, L., 2012. The Algebra of Straight Lines, Retrieved May 19th, 2012, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

Posted in MathematicsNo Comments » - Tags: algebra, graphical calculator, graphs, html5, math, straight line

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).

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:

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Solution by Factoring:

y = x² x

Converting the forms

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

(1)

to the vertex form:

(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:

		(4)
		(5)
		(6)

Going the other way,

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

		(7)
		(8)
		(9)

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

(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:

(11)

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

The equation for the axis of a parabola is:

(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:

(13)

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

(14)

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

(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.

Citing this post: Urbano, L., 2012. Everything You (N)ever Wanted to Know About Parabolas, Retrieved May 19th, 2012, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

Posted in Algebra, Mathematics, Technology, Useful websitesNo Comments » - Tags: algebra, graphs, html5, interactive posts, interactive websites, math

Flowing Data has a nice post that gets at how to make good images of data. It's called, "5 misconceptions about visualization". The misconceptions:

• Visualization is for making data flashy
• Software does everything
• The more information on a single graphic, the better
• Visualization is too biased to be useful
• [A Visualization] has to be exact

Citing this post: Urbano, L., 2011. How to Make Good/Useful Graphics, Retrieved May 19th, 2012, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

Posted in MathematicsNo Comments » - Tags: Art, data analysis, graphs

Flowing Data has an excellent set of interactive graphs showing how Americans spend their day. It's an interesting look into modern American culture.

Main takeaway: we spend most of our time sleeping, eating, working, and watching television. -- Yau, 2011: How do Americans spend their days?

I particularly like comparing the 15-19 adolescents to adults (more time in education and less time watching television; there's also a different sleeping pattern).

Citing this post: Urbano, L., 2011. How People Spend Their Day, Retrieved May 19th, 2012, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

Posted in MathematicsNo Comments » - Tags: culture, curious, data, graphs

The objective is to show the shape of the whole and to express the beauty and cosmic reach of the periodic system. --- Stewart (2006): The Chemical Galaxy

The traditional periodic table of the elements breaks the elements into rows as their chemical and physical characteristics repeat themselves. But since the sequence of elements is really a continuous series that gradually increases in mass, a better way of displaying them might be as the spiral, sort of like the galaxy.

When the chemical elements are arranged in order of their atomic number, they form a continuous sequence, in which certain chemical characteristics come back periodically in a regular way. This is usually shown by chopping the sequence up into sections and arranging them as a rectangular table. The alternative is to wind the sequence round in a spiral. Because the periodic repeats come at longer and longer intervals, increasing numbers of elements have to be fitted on to its coils. ... The resulting pattern resembles a galaxy, and the likeness is the basis of my design. It seems appropriate, as the chemical elements are what galaxies are made of. ... The ‘spokes’ of the ‘galaxy’ link together elements with similar chemical characteristics. They are curved in order to keep the inner elements reasonably close together while making room for the extra elements in the outer turns. --- Stewart (2006): The Chemical Galaxy

While the spiral version of the periodic table is not used a lot, it is scientifically valid. There are other ways of representing the spiral and the periodic table itself. It all depends on what you want to show. Indeed, Mendeleev's monument in Bratislava, Slovakia has the elements arranged as the spokes in a wheel.

Citing this post: Urbano, L., 2011. Periodic Table Spiral Galaxy, Retrieved May 19th, 2012, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

Posted in Art, Chemistry, Natural World3 Comments » - Tags: art and science, elements, graphs, periodic table

A couple days ago I had students present their physics lab reports to the class. They did a good job, but I think I need to emphasize the importance of including graphs in their results. It's much harder to look for trends and patterns in the data without charts, especially when presenting to an audience. An interesting political science study (via Yglesias) found that it's much easier to change people's minds when you show them graphs, even when people don't want to believe what you're telling them.

[P]eople cling to false beliefs in part because giving them up would threaten their sense of self. Graphical corrections are ... found to successfully reduce incorrect beliefs among potentially resistant subjects and to perform better than an equivalent textual correction. --Nyhan and Reifler (2011): Opening the Political Mind? The effects of self-affirmation and graphical information on factual misperceptions

Teachers know how hard it can be to correct misconceptions - people tend to stick with the first thing they learned - so it's good to see that graphical corrections can make a big difference. Fortunately, my physics students are changing over to math next week, so we'll be able to use their experimental data to draw lines, find gradients and do all sorts of interesting things.

Citing this post: Urbano, L., 2011. The Power of Graphs, Retrieved May 19th, 2012, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

Posted in Natural World, Social WorldNo Comments » - Tags: data analysis, graphs, physics, political science, science

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 linear equation has the form: 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: 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 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 May 19th, 2012, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

Posted in Mathematics, Social WorldNo Comments » - Tags: algebra, economics, graphing, graphs, math, math applications

If I present information to you orally, you’ll probably only remember about 10% 72 hours after exposure, but if I add a picture, recall soars to 65%. --Alex Lundry (2009): Chart Wars: The Political Power of Data Visualization

How you present visual information is important. And my students are discovering this as they work up their Independent Research Projects (IRP's) this week. In the spring they are fairly free to pick their topic and style of IRP. Some choose research projects, others term papers, and a few do things that strike their fancy, like writing fiction or programming games. In the end, they submit a written report and give a presentation. For research projects, I have one student who did a great job of coming up with a hypothesis and testing it. He even compiled a nice table of his data for his results section, but was reluctant to go through the effort of making a graph. After all, he claimed, anyone reading his report (or watching his PowerPoint presentation) could just look at the table and read the data off there themselves. My response was that people absorb the data much more effectively when it's presented graphically. Fortunately, Alex Lundry has a nice little presentation that reinforces this point. It also gives a few tips about what to look out for in graphics, because they can be used to mislead. The key quote (via The Dish) is this:

Vision is our most dominant sense. It takes up 50% of our brain’s resources. And despite the visual nature of text, pictures are actually a superior and more efficient delivery mechanism for information. In neurology, this is called the ‘pictorial superiority effect’ [...] If I present information to you orally, you’ll probably only remember about 10% 72 hours after exposure, but if I add a picture, recall soars to 65%. So we are hard-wired to find visualization more compelling than a spreadsheet, a speech of a memo. --Alex Lundry (2009): Chart Wars: The Political Power of Data Visualization

Here's Lundry's five minute presentation.

Citing this post: Urbano, L., 2011. Multi-modal IRP's, Retrieved May 19th, 2012, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

Posted in videoNo Comments » - Tags: graphic, graphs, interesting, IRP, presentations, video, visualization