Tuesday, June 30, 2015

Do not let the students do the LEAST they can...

At some point next year a slew of high school students will be learning about ellipses. I hope that they get to learn and explore using GeoGebra at a minimum. Whether it is used as a presentation device or a lab tool, it would be helpful.

Limiting student technology to a graphing calculator is, well, limiting.

the entire file for this can be found at http://tube.geogebra.org/material/show/id/1379965

Sunday, June 28, 2015


I call this my "Wonderbread" graph for, I hope, obvious reasons.

I had it posted online before done in Geometer's Sketchpad, but I had an error in the captions. Rather than fix that (since I no longer use costly Sketchpad) I have merely recreated it in Geogebra. The point on the smaller circle rotates three times as fast as the point on the larger circles.

It is just another example in my quest to convince educators and parents that guided explorations with software such as GeoGebra can be a prelude to, and a motivator for, standard mathematics education in school.

To deny young students such hands-on experimentation is to do a great disservice.

The complete file is available at http://tube.geogebra.org/material/show/id/1374049

Friday, June 26, 2015

Secants to tangents, what does that mean???

Mathematics can be learned just by having a questioning mind.
This display allows you to select a basic graph. It shows you two points on the graph, together with the line between them and a slope triangle for that line. When you drag the points so that they "crash", the line and the slope triangle may disappear. Begin to investigate that disappearance, and you are on your way to learning calculus.
Keep in mind: the points are actually taking small "steps" on your screen, and often avoid the "crash" by stepping over and around each other. Be patient!!  For quicker examples, pick the bottom one  on the list.

Sunday, June 21, 2015

Should this be familiar?

Water, water, everywhere.
What should happen when the water is turned off? That will follow...

This and others just trying to show how Geogebra can be used as a motivator and laboratory for school mathematics. A lot can be learned by getting involved and watching, seeing, experiencing, experimenting, and imagining with Geogebra.

Friday, June 12, 2015

It could be a raindrop...

When the graphic below comes up, just click on the start button. See what happens. Then try to imagine a logical way that the shape you see could be constructed (graphed, for the layperson.)
Only when you are truly stumped, should you click on "show all'"

Creations such as this one could be tremendous math teaching tools and motivators, if only the powers-that-be running our schools would give teachers the opportunity and technology to make it happen.

The above creation was made using FREE software (GeoGebra) readily available on almost all platforms. The complete file is available here.

Unfortunately, I cannot see such things being done in a New York State classroom until our dear governor gets off his high horse. Maybe he could challenge schools rather than try to control them.

Monday, June 1, 2015

This is scary...

Here is a question from the June 2014 New York State Algebra I Common Core Regents Exam as it was included in its page of annotated items, together with one of its sample responses.

I will claim that this answer is worth more than 2 credits.

This question was testing the knowledge that in any function, no domain element (first coordinate) maye be paired with more than one range element (2nd coordinate).

The student here demonstrated that (s)he knew this, and demonstrated it by visually connecting each domain element with its correct range element.  I recognize that the explanation was not given in sentence form, but the justification is there nonetheless. This answer should be worth full credit.

Here is a second sample response to the same question, followed by its sample score. 

In this second sample response, the student earned more credit, apparently due to the fact that (s)he used a complete sentence to demonstrate that (s)he did not know the fact that was being tested. 

The alleged rationale for this question is given here
This question asks the student to determine whether a function could be presented by four given ordered pairs given the domain and range of the function. “Domain” refers to the set of input values, while “range” refers to the set of corresponding output values. Additionally, the student must determine whether exactly one output is assigned to each input. As indicated in the rubric, a correct response will state “yes”, with a correct justification given supporting the student’s reasoning. The justification can be presented in either written form or mathematical form which could include creating a graph of the function. The determining factor in demonstrating a thorough understanding is using mathematically sound justifications for the response. 
This is quoted directly from the annotated file as referred to above. All these can be found here.

This absolutely befuddles me. New York State is apparently taking the stance that incorrect work with verbal support is better than correct work. I find it even more amazing that their score for the second response above presupposes that the student knew the definition but misapplied it. On what basis can they make that claim?  There is no information within the student's response that supports that claim. It is merely giving the student the proverbial "benefit of doubt".  I guess we better not give that benefit to someone who knows what they are doing!