jump to navigation

Are You Driving Less? August 30, 2013

Posted by Edward Deleon in : algebra, math, media , add a comment

Image

The University of Michigan’s Transportation Research Institute released a study on American driving habits. The link to the study is here.

The results? Americans are driving less. The report includes a rich data set to explore the results in more detail. Two graphs summarize the results.

This graph shows the decrease in the number of registered cars, trucks, and SUVs.

Image

Fewer cars on the road means less driving. Although the number increased in 2011, it’s still below the peak.

But even more compelling is the decrease in the total mileage.

Image

In our current issue of Math in the News we analyze the data in more detail to see why Americans are driving less. We reproduce the PowerPoint here as a Slideshare presentation.

The Martin Luther King, Jr. Memorial August 28, 2013

Posted by Edward Deleon in : algebra, geometry, math, media , add a comment

On the the 50th Anniversary of MLK’s great speech, we’re showcasing a previous issue of Math in the News, which focuses on the MLK Memorial. This presentation goes into the physics and math of the material used to construct the Memorial. This is a great case study in art, history, mathematics, and physics.

We reproduce it here as a Slideshare Presentation.

The Mathematics of Relativity August 25, 2013

Posted by Edward Deleon in : algebra, math, media , add a comment

The physics4me Blog has an interesting post about the physics of the element mercury. Here’s the link. In the video, the focus is on why mercury, which is a metal, is in liquid form, even at relatively high temperatures. This is so uncharacteristic of metals, which we often see as solids.

One of the reasons that mercury exists as a liquid has to do with Einstein’s relativity. In the Special Theory of Relativity, an unusual property of matter is that as it approaches the speed of light, mass increases. This radical expression shows how the mass changes.

Image

An object of initial mass M0 has speed v. For small values of v, the change in the mass is negligible. The c in the equation is the speed of light. As the speed of the object increases, suddenly its mass increases.

For simplicity, let’s make the initial mass and c equal to 1. And let’s use x for v. Here is a graph of the mass function.

CalcScreen1

As you can see the values of f(x) for small values of x show the mass is unaffected by relativistic effects. But as x approaches 1 (or as the speed of the object approaches the speed of light), the value of the mass increases dramatically.

CalcScreen2

The graph shows that x = 1 is an asymptote, since an object can approach the speed of light, but never reach it.

This is the atomic structure of mercury. Its atomic structure is such that all the electron orbitals are filled, making it more difficult for groups of mercury atoms to organize themselves into a solid structure.

Screen Shot 2013-08-24 at 10.18.51 PM

As noted in the video, because of the overall number of electrons and the resulting mass of the atom, the electrons near the nucleus must travel at very fast speeds, causing the relativistic effects. These electrons acquire more mass. The increased “heaviness” of the electrons, combined with the atomic structure of the atoms themselves, cause mercury’s melting point to decrease. And thus mercury remains a liquid, while most other metals are solids that require very high temperatures to melt them.

You can see how mercury’s melting point compares to that of other metals.

 

Melting Points

Metal

Fahrenheit (°F)

Celsius (°C)

Aluminum

1218

659

Brass

1700

927

Bronze

1675

913

Cast Iron

2200

1204

Copper

1981

1083

Gold

1945

1063

Lead

327

163

Magnesium

1204

651

Nickel

2646

1452

Silver

1761

951

Steel

2500

1371

Tungsten

6150

3399

Wrought Iron

2700

1482

Zinc

787

419

Mercury

-37.89

-38.83

Have You Used the TI-Nspire App? August 20, 2013

Posted by Edward Deleon in : algebra, geometry, graphing calculators, math, media , add a comment

Image

For those who use the TI-Nspire handheld, have you made the transition to the iPad app version? In this article we’ll look at some of the basics of using the App, using your knowledge of the handheld. (Media4Math+ has a library of video resources for the TI-Nspire, and we’ll be adding support for the Nspire App.)

Handheld App
When you create a new TNS document, you’ll see this screen.

Image

When you create a new TNS document, you’ll see this screen.

Image

Let’s create a Calculator Window and do some calculations. When you launch the Calculator Window, the iPad keyboard is active. (To deactivate it, press this button: Image.)

Image

Input the expression shown above. To activate the Menu, press this button: Image. Then convert the result to decimal form.

Image

Here’s how to create a Graph Window. Press this button: Image.

Image

Select Graphs.

By default the Keyboard is active and you can easily input your function.

Image

Press Enter to see your graph.

Image

Manipulating the graph is very easy and intuitive on the iPad:

Image

Summary

The iPad App of the TI-Nspire gives you all the functionality of the handheld and becomes a great presentation tool to use in a classroom setting. Sign up for Media4Math+‘s newsletter and follow this Blog for more information on the TI-Nspire.

Geometry Application: Funnel Cakes! August 18, 2013

Posted by Edward Deleon in : geometry, math, media , add a comment

Image

During state and county fair season, here’s a good opportunity to create a back-to-school activity that involves country fair fare: funnel cakes. But this activity is carb free.

In our current issue of Math in the News we look at the geometry of funnel cakes. This may sound a bit unusual, but the act of taking a funnel full of batter and turning it into a funnel cake is the same as taking a cone-shaped volume and transforming it into a cylinder-shaped one.

Image

We use the formulas for the volume of a cone and and the volume of a cylinder to answer this question: How long a cylinder of batter comes from the funnel? The results are surprising and reveal some aspects of the volume formulas, given that these formulas share similar variables.

We summarize this issue of Math in the News with this Slideshare presentation. Come see our huge library of content for Algebra and Geometry on Media4Math+.

Using Video Games to Explore Math Concepts August 14, 2013

Posted by Edward Deleon in : algebra, math, media , add a comment

Nearly all math students play, or have played, video games. While it takes a good number of math skills to program a game, the experience of the game doesn’t offer many opportunities for mathematical thinking. Here’s where you come in.

A very popular game these days is called Candy Crush. It is modeled on other games where lining up three same-colored objects (in this case candies) gains points, and the goal of a level involves overcoming obstacles by finding such combinations.

Image

Like many video games, Candy Crush is programmed to randomly generate a game board with an array of candies. For example, here is a level. Your goal is to find possible combinations of candies to line up three in a row. (You can also line up four and five in a row.)

CandyCrush1

Here’s we’ve highlighted one possibility. By moving the top red piece to the left, you align three reds. You can see other combinations.

CandyCrush2

But the randomness of video games can show certain tendencies. Certain levels are programmed to create favorable conditions for certain types of combinations. And here is where you can encourage students to explore the randomness of certain levels, and to think mathematically. Let’s look at an example.

Here is a higher, more challenging level from the game. The more advanced levels of the game require you to find four- and five-piece combinations. But the arrangement for this level offers few such options.

photo

But the game allows you to Reset the board, so that it randomly generates a different board. Simply exit the board and launch a Reset. Here is another version of the same level. This version offers you an opportunity for a four-piece combination, highlighted here.  There is no limit to how many Resets you can have.

photo-2

With this Reset, you can see a five-piece combination:

photo-3

The common conception of randomness is that a phenomenon is out of your control. But here is a situation where a random phenomenon can show certain tendencies. By looking at tendencies, you can begin to get an understanding of expected behavior.

For example, when you toss two dice, the numbers that land are random. But toss the dice 50 times and you begin to see a trend. You can see that some combinations of numbers are more likely than others.

Screen Shot 2013-08-14 at 7.41.46 AM

Here’s another example. This weather map is a historical map of hurricanes that have hit the Florida coast for the past century. You can see that some areas are more prone to hurricanes than others. Even though where a hurricane lands is random, the tendency of a hurricane has a pattern that can be shown as a probability.

Screen Shot 2013-08-14 at 7.27.41 AM

Encourage your students to think mathematically about random events. Yes, they’re unpredictable, but analyze iterations of such random events, and you can see a pattern. You can then begin to make predictions based on these observations.

Using Popular Culture for Applications of Math August 11, 2013

Posted by Edward Deleon in : algebra, math, media , add a comment

Movies provide a wealth of opportunities to explore data analysis, trends, and graphical analysis of real world phenomena. In our current issue of Math in the News we look at data for The Lone Ranger. Here is a movie that made nearly $100 million, yet it’s considered a box office flop. It’s been such a disaster that it has, for the near term, affected Disney’s stock price.

What went wrong?

Image

One of the first ideas isn’t to look at how much a film earns, but how much it makes relative to how much was spent to produce it. In the case of The Lone Ranger, the movie had an incredible $215 million production budget, which from the outset put great pressure on the film to be a blockbuster. Although the movie made over $40 million dollars in its first five days, it was a far cry from where it needed to be. Why?

A movie will never make more money than in its premier week. After that, revenue drops in a predictable pattern.

Image

We can model this behavior with a sequence, making predictions about audience loss and expected revenue. We go through such a model for The Lone Ranger and determine that the movie lost over half its audience from week to week. This is a huge loss in audience, and no movie, whatever the budget, can be considered successful with such a drop in attendance.

We look at the factors that influenced the lack of success for The Lone Ranger, as well as what factors make for a successful move.

Come see Media4Math+ for more math resources for Algebra and Geometry. Our current issue of Math in the News is also included here as a Slideshare presentation.

%d bloggers like this: