What do you know about "cosine"?

Technology such as Geogebra can give tremendous opportunities to hook students into mathematics if it is introduced as a tool that uses basic mathematics. In the "olden days" one had to have a solid grounding in trig and coordinate geometry to even begin to deal with graphs such as those shown here.

Yes, this does use "cosine". I feel that it could be used to create a desire to find out what this thing called "cosine" really is. As a motivator.

A number of my blog entries have been geared towards using GeoGebra as a math motivator. I will be presenting how some of them were created at the annual conference of the Association of Mathematics Teachers of new York State this fall in Rochester. It'll be fun!!

What does "understanding" really mean? (or "the question not asked..")

This is from the June 2015 Algebra I (Common Core) Regents Exam in New York:

This jogger, based on evidence given, took two full minutes to achieve a pace of 3 mph.
The same jogger, 16 minutes later, maintained a 8 mph pace (7.5 min/mile) for a full minute.
This jogger maintained perfectly constant speeds interrupted only by intervals of constant acceleration and deceleration.
 Is this real?
I know that one of the big deals in math education is to make it "relevant" and "real".

The students confronted with this graph had to answer the following question:

Which statement best describes what the jogger was doing during the 9–12 minute interval of her jog?
(1) She was standing still.
(2) She was increasing her speed.
(3) She was decreasing her speed.
(4) She was jogging at a constant rate.

A truly mathematical question would have been to have given the graph (uncaptioned) to the students and then asked them the following question:

Which statement best describes the graph on the interval 9(1) f(x)=0
(2) f(x)) is increasing
(3) f(x) is decreasing
(4) f(x) is constant

Mathematically the questions are identical, but the second does not exhibit that debilitating push towards "relevance".

For those who prefer the question as stated in the exam, I would ask them to answer this question as a test to whether or not the truly understand the situation they have created:

At what time had the jogger traveled exactly half the total distance jogged?

Happy Birthday...

Today being my wife's birthday just gave me the idea to create another very short GeoGebra graphic.
I "borrowed" the graphic from the web (http://freepicsimages.com/happy-birthday-clip-art.html) and created this in 6 or so steps.
It involves a two spirals anchoring two corners of the picture. The mathematics might be more easily understood if it were to be presented using creations such as this.