Home, home on the SPREAD, where the deer and the antelope play....

I truly wish that New York had higher quality exams, as they are intended to be used to measure not only student performance but teacher performance as well.

The analysis continues with question 20 from the January 2017 regents exam in Algebra I (Common Core)

At issue here is choice (2), that refers to the spread of the data. Here, in Algebra I, the word spread should not be used.  Spread can be represented many different ways, from range to interquartile range to standard deviation. To the best of my knowledge, standard deviation is not part of the Algebra I knowledge base. Even so, range and interquartile range do not go hand-in-hand: it is possible for a set with a smaller range to have a larger interquartile range.

This choice should most probably have used the word "range" instead of spread.  For the record, the ranges are equal in these two sets, but the interquartile ranges are not (7 to 10, or 3 years for soccer players and 9 to 11, or 2 years for basketball players. The standard deviation for soccer is 2.05798 and for basketball is 1.81137. So using range, choice (2) is false, while using the other two measures, choice (2) is true.

Could it be the case that the word "spread' was used when the word "range" should have been used?

New York has to make better tests!

The above is the first section of a question from the January 2017 New York State regents exam in Algebra II (Common Core).

I believe the question needs the word "tsunami" rather than "tidal".  The presence of the word "tidal" in this context illustrates the need for improved "proofreading" in the creation of these exams.

Continuing on, here is question 24 from the New York State Geometry (Common Core) regents exam:

I suspect that the word "cone" here should read "inverted cone". When used by itself, the word "cone" refers to this:

Rarely have I seen a water cup used "point up". Let me correct myself: I have never seen a cup used that way.  Should a student solve the question as written, they could be perfectly correct and get an answer not listed. That situation should be avoided at all costs on a state exam.

Let's look at question 8 from the January 2017 NY regents exam in Algebra I (Common Core):

I found this question misleading, since the USPS charges 49 cents for up to 1 ounce and 21 cents for each extra ounce or fraction of an ounce. The best mathematical model would be 

\({\rm{Cost}} = 49 + 21(w - 1)\)

where w is the weight of the letter in ounces and the costs are measured in cents.

I suspect the question writer was trying to come up with a "real world' application of recursive functions. My advice would be to look again. Question 20 on this test would have made a much better model, as postage must take into account portions of an ounce but mp3 sales would not.

Now comes question 14 from the same Algebra I exam;

The mathematics in this question is basically asking "Which of the following is equal to 6(16)^t ?" The rest of the verbiage is due to the attempt to make the problem "real world".

Can't we just ask math questions to test math knowledge? 

Clarification needed!

Here is question 24 from the New York State Algebra II (Common Core) Regents exam from January 2017. Please look at it closely!

Now that you have read it carefully, take note that the domain of this question seems to run from -2 past 5. Also take note that if x is less than 1 or greater than 5, one of the sides must have a negative length. Since lengths cannot be negative, this graph can NOT be a model for the volume of a box, hence the question cannot be answered. 

A few thoughts to ponder....

Here is a question from the New York January 2017 Regents Exam in Algebra II (Common Core).

Please read it carefully, then answer some questions below.

1) How many rabbits were there four weeks ago?

2) What do t and P(t) have to do with it?

3) Suppose a student thought as displayed in this chart. Would they get it right?

now	5 rabbits
in 28 days	10 rabbits
in 56 days	20 rabbits
in 84 days	40 rabbits
in 112 days	80 rabbits
In 98 days	Between 40 and 80

On a different note, here is question 21. Read it carefully, then answer a couple questions below.

1) Can you tell me who gets away with no credit card payments for 73 months? Could I stretch it out another 300 months?

2) If this is supposed to be a "real world" question, can you tell me what world that is?

Technology gives us new ways to look at simple things..

Far too often I hear people talk about how hard math is, which kind of bugs me because math itself is not hard nor easy. It just is. When they say math is hard they are actually referring to their interaction with mathematics. With today's technology, a person's initial contacts with mathematics can and should be drastically different than was even possible even a just a few years ago. I know that many people have a bad taste for math because they spent years in school struggling to tread water while being placed in a depth that was just over their heads. The panacea effect of calculators (which never really helped) was really just a matter of putting flippers on a non-swimmer.

Every once in while I like to take new technology and use it to look for new ways to teach old concepts, hoping to enable beginners to gain a comfort with the shallow water. Even while doing this care must be taken, as one can drown in just an inch of water. Such an attempt is shown here.

This file is nothing more than part of an attempt to enable students to internalize the concept of slope. My goal is to expand this simple file over time so that a path can be blazed that will connect this simple concept with unique slope of a line, slope-intercept equations, parallel lines,  right-triangle trigonometry, and more.

Take note that this file requires the viewer to be able to count. Even such a simple item as the "slope formula" is not needed. 

This file can be found here.

Comparing GeoGebra and Desmos on a Graphing Task

This file I created just as a personal challenge. I make a point to try something new and different every day, and this file I took as a challenge merely because I came across a Desmos version (see it here). I prefer posting with GeoGebra as it seems to give me a bit more control. With GeoGebra I can label items on the graph and have decent control over what appears in different circumstances.

Desmos does a very good job filling the role of graphing calculator, and does it with a much better resolution than the typical handheld calculator.  But, I have to be honest, GeoGebra does a lot more. (I should say it can do a whole lot more: it does have a steep learning curve at the start. It is this learning curve that encourages me to push for GeoGebra's use at young ages. A steep slope becomes less steep if it is lengthened. Any ramp user knows that.)

For the moment: I have included in the GeoGebra graph the focus and directrix and a drag-able way of showing the relationship between them and the parabola itself. The equation is also present. I have not found a way of including those features (with labels) in Desmos' embeddable graph.  Help me if you can!!

Choosing from 4 wrong answers?

The question below is from the June 2016 Algebra II regents exam in New York. Some questions have been raised about it claiming it might confuse those who are unsure as to how to "count" multiple roots. (For clarity, a multiple root is a crossing root if the root occurs an odd number of times, and a tangent root if it occurs and even number of times.)

Actually, that issue is irrelevant in this question. Based on the accepted standard regarding the meaning of "arrows" at the end of the graph, choice 3 is the only choice satisfying the second and third bulleted items. The first bulleted item is irrelevant in this question.

Or is it? It is possible that choice 3 has at least one root of odd multiplicity of 3 or more, but without a scale on the graph it is impossible to tell.

Or is it? Something did not seem right about choice 3. (Continued below)

 

I had to do some investigating.

In GeoGebra I created a file including the graphic from choice 3 along with a cubic polynomial graph sharing the x-intercepts with those in choice 3.  Since there were no scales on the axes in any of the choices, the only information I could rely on was the relative positions of key points.

Here is the GeoGebra sketch (get it here if you wish):

The dotted blue line marks midway between the leftmost x-intercepts.

No matter what I do to change what is changeable (experiment with changing leading coefficient or absolute position of leftmost root) the polynomials maximum and choice 3's maximum lie on opposite sides of that line.

I will not claim that I have the definitive answer on this issue, but it appears to me that the more you know about cubic polynomial graphs the less likely you are to accept choice 3 as an answer to this question.  The problem could have been avoided if actual polynomial graphs had been given.

I believe I understand what the question writers intended with this question, but I must reject the question itself.

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