How good is your sense of time?

This GeoGebra creation is the result of a discussion involving the "hang time" of a kick in one of the NFL playoff games.  The thought smashed into my head later when I heard someone say something like "I don't know how long it was, but it felt like hours".

This is a simple creation (original available here). The duration ranges from 3 seconds to 15 seconds.

Triangle of Time

Sometimes GeoGebra spurs me on to a new way of looking at something old and familiar.

In this case, a clock. Yesterday I posted the clock I made in GeoGebra. Today I have tweaked it a bit to help pose a question.

The three hands are all pointing to 3 points on the circumference of the circle. These 3 points form a triangle. Can we work with the area of that circle?

1) The smallest area of the circle is zero. How often does that happen in a 12 hour time span?

2) What is the largest area? How often does that happen in 12 hours?

A stretch for trig or precalculus students might be generating a graph of the area as a function of time of day. A stretch for calculus students might involve determining exactly when the area is largest by maximizing that function.

A stretch for younger students might be generating this graph on their own. It is too bad that the politics of education make it virtually impossible for a math teacher to take time to work with students on questions like these. 

I suspect textbook publishers do not like questions such as this!

Can you build a clock?

Many of us take clocks for granted, but there is not a simple clock anywhere. behind every timepiece is a great deal of mathematics together with either metallurgy or engineering or chemistry or electronics or programming, or maybe all of those!

Here is a basic dial clock made totally within GeoGebra.

GeoGebra does have a feature that allows it to read the time off of your computer, but what happens then is whatever you do with it.

Focus, Directrix, and Conics

Back in the classroom I used to wish I had a better way to demonstrate the focus-directrix connections between the conics. The algebraic methods were time-consuming and, I know, contributed to "brain stoppage" by many students. Pencil and paper constructions, taken to a useful stage, would have taken days and days. If only there was a better way...

GeoGebra helped me begin to bridge that gap. My first go at it has produced the file here. Not perfect, but a lot better than what I had before. Check it out and EXPERIMENT!!!

Unfortunately, making this fit in the blog requires it to be tiny. Click here for full version

Electoral College: Perform Your Task Wisely

Many people seem confused by the process of selecting a President.

At no point has this country been other than a republic of states, and the US Constitution specifies the means for those states to elect a President. That process is the Electoral College.

No matter what is on your ballot when you vote, you are not voting for a President. You are voting for an elector, a member of the Electoral College.  ALL your voting takes place within your state. There is no vote for a national office.

Tomorrow the Electoral College meets to vote for the next President.

I pray and hope that it exercises this task with utmost care and diligence, and does not merely make its selection because of a misconstrued sense that the "public" voted for one person or another. The public did not vote for anything more than the electors.

Electoral College: this is your opportunity to confirm the wisdom of our Founding Fathers.

Let Math Test Questions Test MATH!

Here is a portion of the New York State Grade 5 Mathematics Reference Sheet:

Here is a question from New York State's 2016 Grade 5 Mathematics Test Released Questions:

Thank goodness that New York does not allow for calculators on this test, as the only mathematics being tested is the division of 240 by 4. Everything else is reading comprehension. Careful patient reading, as any student familiar with aquariums (aquaria, if you wish) would know that they size is typically communicated in gallons (at least in the US) and would then question the strangeness of this so called "pet store."

If scores on this test are used to measure (determine?) students' mathematics knowledge and skills, the test should be focused on those skills and that knowledge.

I entered the first sentence of the question into Microsoft Word and let it supply the readability statistics, and it measure the sentence at the Flesh-Kincaid Grade level of 7.6.  I then took the same sentence to https://readability-score.com  and it came back with these:

Please, then, at least accept the possibility that a student's reading abilities might, just might, have a bigger effect on their answering of this question than does their mathematical talents.

By the way, the results on readability for the previous sentence is here:

Does this question belong in a MATH test?

Math without GeoGebra is like a day without sunshine!

It might be the right time of year to push the use of GeoGebra as a classroom presentation tool. I created this interactive file back in March of this year, and pull it out now with a request for teachers to send in ideas for modifications of this file, or suggestions for new files (send to dave(at)davemath.com.  This file can be found here.

Any and all GeoGebra files can be downloaded and modified. 

GeoGebra is totally dynamic and its use is limited only by your imagination!