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!

  

Never be satisfied with guess and check!

(Please read my last blog entry either before or after this one. They go together.)
\[\begin{array}{c}\left( {nx + m} \right)\left( {px + s} \right)\\nx\left( {px + s} \right) + m\left( {px + s} \right)\\np{x^2} + nsx + mpx + ms\\np{x^2} + (ns + mp)x + ms\end{array}\]

The above should be recognized by all secondary teachers of mathematics as a generic example of using the distributive property (of multiplication over addition, as it is frequently phrased) to multiply a pair of binomials.

Simple and to the point. Anyone can do it.

What I wish to point out to those who do not notice is that \[(np)(ms) = (ns)(mp)\]

This might seem like a bit of obvious but but seemingly irrelevant trivia, except for the fact that this little fact is the key that unlocks what I believe to be the way that quadratic factoring should be done (and taught).

Take a close look at an actual example (using numbers).
\[\begin{array}{c}(2x + 3)(5x + 7)\\2x(5x + 7) + 3(5x + 7)\\10{x^2} + 14x + 15x + 21\\10{x^2} + 29x + 21\end{array}\]
This is how the multiplication of binomials should look. (Forget that mnemonic FOIL. Forget it now and forget it forever.)

Now we will see the same steps displayed in reverse order.
\[\begin{array}{c}10{x^2} + 29x + 21\\10{x^2} + 14x + 15x + 21\\2x(5x + 7) + 5(3x + 7)\\(2x + 5)(3x + 7)\end{array}\]

Take note the first step involves separating \(29x\) into \(14x + 15x\). Why choose 14 and 15? Because they add to 29 AND multiply to 210, the product of 10 and 21. The rest is just applying that same old distributive law.

Take a look at one from scratch, say \(4{x^2} + 43x + 63\)

Our first step will be to find two numbers that add to 43 and multiply to 252, which is the product of 4 and 63.

\(\begin{array}{c}(1)(252)\,\,\,add\,\,to\,\,253\\(2)(126)\,\,\,add\,\,to\,\,128\\(3)(84)\,\,\,add\,\,to\,\,87\\(4)(63)\,\,\,add\,\,to\,\,67\\(6)(42)\,\,\,add\,\,to\,\,48\\(7)(36)\,\,\,add\,\,to\,\,43\end{array}\)

Take note: the numbers on the left are no more than counting, 5 was skipped because it is not a divisor of 252, and we stop at 7 because we found the numbers we need. With these numbers we can continue: 

\(4{x^2} + 7x + 36x + 63\)

\(1x(4x + 7) + 9(4x + 7)\)

\((1x + 9)(4x + 7)\)

Please please recognize that this process is highly programmable. It is actually a system based on action, not a fallback "guess and check" that is pushed on kids far far too often. Also, take note that I do accept "1" as a meaningful numeral, and do not always jump at the chance to avoid writing it.

Here is a sample with negatives: \[6{x^2} - 7x - 5\]

We start by taking pairs of factors of \[ - 30\]. Since the two numbers must add to a negative, we will make the larger number in each pair negative and keep the smaller one positive.

\[\begin{array}{c}(1)( - 30)\,\,\,add\,\,\,to\,\, - 29\\(2)( - 15)\,\,\,add\,\,\,to\,\, - 13\\(3)( - 10)\,\,\,add\,\,\,to\,\, - 7\end{array}\]

We got our two numbers pretty quickly this time, and we can continue.

\[\begin{array}{c}6{x^2} - 7x - 5\\6{x^2} + 3x - 10x - 5\\3x(2x + 1) - 5(2x + 1)\\(3x - 5)(2x + 1)\end{array}\]

Please please remember that the special cases with leading coefficient 1 are special only to someone who knows the whole story. They are not special to a student seeing them for the first time. What may be seen by those in the know as a shortcut cannot be seen by beginners as a shortcut. A shortcut is never meaningful unless and until a longer route is known.

If you would like a seemingly endless list of practice problems (with check-ability), check this out.