Limacon with inner loop

Continuing my push to use Geogebra to introduce younger students to "fancy math", here is a plot similar to my last two blog entries. Although the shape is determined by the relative speeds of Jack and Jill, just as in the previous two postings, this one was shifted a bit to the right just so that the origin is more centered.

The original file is here.

I will not be posting here for a couple of weeks as a different project will intervene. Merry Christmas!

5-Leafed Rose the simple way

Here is an extension of yesterday's post, where I have adjusted the graph to get 5 leafs instead of 3.
This file can be found here.
Below is a bit of mathematics that I would have discussed in a class. with the goal of graphing in parametric mode on a graphic calculator. For students not yet up to that, I would adapt this so that they can experiment with different rotational speeds, reverse directions, etc., and would not include the pre-plotted graph.

 \[\begin{array}{c} A = ( - .75,0)\\ Jack = ( - .75 + \cos ( - 2t),0 + \sin ( - 2t))\\ C = (.75,0)\\ Jill = (.75 + \cos (3t),0 + \sin (3t))\\ {\rm{red}}\,{\rm{point = }}\frac{{Jack + Jill}}{2}\\ {\rm{red}}\,{\rm{point = }}\frac{{\left( {\cos ( - 2t) + \cos (3t),\sin ( - 2t) + \sin (3t)} \right)}}{2} \end{array}\]

Going in Circles Can be Fun!!

A major push of mine is using dynamic geometry such as GeoGebra to introduce younger students to more interesting mathematical concepts.
Here, based on knowledge of circles, rotations, and midpoints, a student can learn something about 3-leafed roses. The file can be found here.

I would love to see how to do this in Desmos, or how to make it web-friendly in Nspire.

Matrix Transformations in GeoGebra

GeoGebra provides a great avenue for traveling through the world of matrix transformations of the plane. Here is just a little example. Imagine if our teachers could compile a folder full of such sample, ready to pull out and share at a moments notice.

This one can be obtained  here. I hope it helps somebody somewhere.

Know your box!

...(B)ut I find that a concentrated atmosphere helps a concentration of thought. I have not pushed it to the length of getting into a box to think,

Doyle, Arthur Conan; Books, Maplewood (2014-08-04). Sherlock Holmes: The Ultimate Collection (p. 20). Maplewood Books. Kindle Edition. 

For years I have heard the phrase "thinking outside of the box". Finally, I have found a reference to thinking "inside the box". The phrase above is actually said by Sherlock Holmes in The Hound of the Baskervilles, a novel by Arthur Conan Doyle. Thankfully, at last, I have a point to accompany the ubiquitous counterpoint.

A number of times I had the sense that the person using the phrase "thinking outside of the box" really and truly did not know nor understand the box itself.

Although I do believe that some people do develop an ability to do what is referred to as out-of-the-box thinking, I draw the line at pushing it as a primary goal of public education. In my mind, a primary goal of public education should be creating a firm box for thinking, and helping to enlarge that box over time.

Happy Thanksgiving

Special lines in a triangle

Here is a little tidbit to help in basic geometry. My original is here.
 

The 9 Point Circle

Back in my time in the classroom, discussion of the 9-point circle was very awkward, given that construction of even a single example using chalkboard tools was very time consuming and extremely imprecise.

This sketch, made in GeoGebra, is an example of how dynamic geometry opens up a whole new world for students and teachers of mathematics. The file itself can be found at http://tube.geogebra.org/m/2110147,

Competition is good

I am making a slow inroad into a comparison between GeoGebra and the TI Nspire . I have previously blogged about the capability to plot polar graphs using just basic geometry (see here for an example.)

It will be a while before I get up to speed, but in the meantime I will proceed just as I did at first using Geogebra: generating an animated gif. Any interactivity in my blog may come much later (that is, if TI has allowed for actively embedding documents.)

In the meantime, it's a start!  Here two points are rotating in opposite directions, one twice as fast as the other, and the midpoint between them is traced.

Your own roller coaster!!

I decided that I needed to explore a bit in the 3D mode of GeoGebra, so I did a bit of a model of a roller coaster. It is 3D, so it can be viewed from different perspectives
The file itself is is at http://ggbtu.be/m2000123.

Sun, Earth, and Moon

I liked this example of using GeoGebra as a modeling tool. I found it here, and just did a bit of tweaking for this blog entry.
Understanding this graphic can precede the understanding of any of the equations behind it. Although the creator of this used equations, I will post this week a version that does not use equations, but does use the rotation tool in GeoGebra

Throw a lot of darts......

Some math students have read about how a blind dart shooter can be used to approximate π.

The basic process is based on the fact that, when randomly thrown, darts hit portions of the target board at a rate corresponding to the area of that portion in relation to the whole board. here we have a circle (area πr²) and a square (with area 4r²). So the ratio of darts hitting the circle to the the darts thrown should be the ration of these areas, which is π/4.

Take note: to get a really good approximation, you have to throw a lot of darts, and awful lot of darts!

Disclaimer: a skilled dart shooter would mess up the pi!!

The original file is here.

Reducing Fractions (using GeoGebra)

I made this today just so i would have another example showing that GeoGebra can be used outside of geometry. My hope and goal is that teachers (and students) would create a digital notebook of helpful files, and use them. One does not need to create in GeoGebra to be able to use GeoGebra.
 
The original file is at here.

AMTNYS in Rochester

I will be giving a short presentation on "The Power of GeoGebra" at the AMTNYS conference in Rochester next month. I am gathering a selection of my creations to use as examples. You can see the collection as it grows by clicking here.

If the diagonals of a quadrilateral are...(under construction!)

This includes 3 diagonal properties, but the perpendicular setting seems to be in need of some TLC.
Bear with me.
 

If the diagonals of a quadrilateral are.. (part 3)

Last Friday I posted two GeoGebra files dealing with quadrilateral displays with congruent diagonals, and with diagonals that bisect each other.
In the attempt to make these one step closer to teacher-friendliness, I have combined them into one fairly self-explanatory file.

This should be added to in the near future.

The actual file can be found at http://tube.geogebra.org/m/1844613

Have fun.

If the diagonals of a quadrilateral...(Part 2)

In my classroom days, I spent a lot of time dealing with students on the diagonal properties of quadrilaterals and what they said about the quadrilateral. I wish I had dynamic geometry software back then.  

Far too often my students would see or hear the word "quadrilateral" and immediately picture a familiar 4-sided figure, such as a rectangle or square. Getting them to think outside of their comfort zone was not easy. Sketches such as these two would have greatly helped. (The 2nd one was posted here first on October 13)

How early can they do it?

While delving into Geogebra I feel somewhat obligated to give Desmos its fair chance. I decided today to just plot a triangle with movable vertices.

I did achieve that goal, although I doubt I could get a 3rd grader to follow along, I suspect the third grader could do it in GeoGebra.

I am looking for advice as to a user-friendly means of labeling the vertices. Such labeling occurs automatically in GeoGebra.

 

If the diagonals of a quadrilateral....

Question #1 from the August 2015 Geometry (Common Core) Regents Examination got me to thinking of some of the issues that arose in class back in my teaching days. This is step 1 in a creation that I hope will encompass all the key properties of a quadrilateral's diagonals, each phrased in the "If the diagonals of a quadrilateral are __________" style. Wish me luck!

The SSA problem: an aid for teachers

At some point tjis year every geometry teacher in the country will be confronted with the infamous SSA (side-side-angle) problem. I hope this helps.
Available online at http://tube.geogebra.org/m/1771489

GeoGebra or Desmos?

Here is my 3-leafed rose plot in Desmos. Below you will find the same plot in Geogebra. The GeoGebra version, to me, can be presented to students earlier in their math program.

GeoGebra and Desmos side-by-side

Yesterday I was experimenting and came up with a page that places Desmos and GeoGebra side  by side. My screen resolution is 1920 by 1080. I hope and plan to experiment with other sizes.

This experiment is just a stage in my quest of packaging lessons that involve both piece of software in an internal dynamic manner. I do not suppose I will ever get them communicating with each other, but at least I can set it up so that the teacher (or student) has a seamless way of using both in the same lesson.

Desmos vs GeoGebra?

Here is the best way I have discovered to plot a movable segment in Desmos. Take note that in order to create it I had to have prior knowledge of Cartesian coordinates and parameters. See below for more.

Here I have a version in GeoGebra where I have utilized the limitation of tools available to the student to just "plot a point" and "plot a segment". It is very simple. Make your own right here and now!


In my humble opinion, I could teach a lot of Geometry, starting with the simplest of notions, in GeoGebra, such that beginning students can "latch on" right away. This "latching on" would not be so quick in Desmos.

However, both pieces of software would be used in my classes!

Desmos calculator embedded in a blog!

Two posts in one day! What am I thinking!
Just read about something on the web anmd it tickled my brain regarding blogger and iframes,  so I said to myself "give it a try!".

Here is the page https://www.desmos.com/calculator embedded in the blog. It has shrunk, but all the parts are there.
A moral victory!!

Use the web wisely!

Readers of my blog know that I am a strong supporter of the use of dynamic GeoGebra in the classroom.  As we proceed through this new school year I will be continuing this campaign, but adding support for other online math options.  In order to gain my support, the material I propose must be:

1. accessible via the web without any special software installation.
2. capable of being embedded into blogs such as this one
3. interactive

My goal is to create libraries of "teachable items" for use by teachers, freeing them from the need to create their own items in order to focus on their students.

The graphic here appears inert, but clicking on it brings you to a page where you can experiment, modify, play around,.... learn.

Enjoy

You can't make this stuff up

Here is a photo I took 6 months ago in Florida. I will keep the town and store nameless.

Just looking at it should be enough. No comment should be necessary.

The last time I checked frozen water was very low-cal, and might be good for you, too!

I just hope Yogi Berra got to see it.

Create a Slide Show in GeoGebra!!

Here is just an example as to how Geogebra can be used to create slide shows involving dynamic geometry.
It is actually pretty easy?
I can explain it if you catch me at the AMTNYS conference in Rochester this fall.
Otherwise, download the file here, and pick it apart!!
 

Is this an ellipse?

GeoGebra allows us to do a lot of exploring. here is a construction that traces the result of 3 rotations. It looks like an ellipse. Is it?
For a larger display, visit the file at http://tube.geogebra.org/m/1634221
Carpe diem!
s

What is the shortest route?

Just imaging you have to visit 6 state capitals every trip, starting and ending every trip at the same state capital where you started.

GeoGebra has a fascinating capability, based on the famous "Traveling Salesman" problem, which is used here to find your route.

Drag the points around and see how quickly the new route is found!!

Pentagoning the Square???

This was created to help show that GeoGebra can be used to create presentations for the classroom. I certainly wish I had this when I was a classroom teacher!!

This particular one shows how a regular pentagon (equal sides, equal angles!) can be created from a square.
Have fun!!

I will be talking about this and a lot about GeoGebra in general at my presentation at AMTNYS in Rochester (currently I am scheduled at 3:45 on Sept. 12. See you there!!

Happy Labor Day!

Just a little message made in GeoGebra!

A limacon? ABOUT FACE! Call it a Rose!

This is something I discovered while just exploring with GeoGebra. It occurred as a result of my work exploring what I call "Floating Midpoints".
The complete file can be found here.

A Rose is a Rose is a Rose is a .....

Dynamic Geometry can be a great avenue for introducing seemingly complicated math topics to the younger students in lower grade levels.  It also can be used to deal with topics from different perspectives.
Here I am using GeoGebra to show how rose graphs, normally not seen in school until polar graphs are dealt with, could be introduced much much earlier.

In this graphic points M and n will be rotating around the circle in opposite directions. My next post will show what happens in some cases when the rotation of one of the points reverses direction.

gh

The SAS Problem is here!

Enter the lengths of two sides of a triangle and the measures of the angle those two angles form, and this will give you a sketch and the opportunity to show the missing measures.

It was designed with the goal of giving students/teachers and inexhaustible supply of practice. Students can create and solve their own problems, and be able to check their results. The twofold gain is allowing students to discover when they have errors all by themselves, and saving the teacher from the role of answer-checker.

File available at http://tube.geogebra.org/m/1517789

Motivate students and hold them accountable..

John Metallo has a short article in the Albany Times Union of Saturday. August 15, 2015. The entire article can be found here. In under 20 sentences, he pretty much nails the issue regarding student success in school.

The victory of Jason Day in the PGA Championship has reminded me that the road to success has three major parts: decent opportunities, good guidance (instruction), and the will to succeed. Far too often the latter is ignored when it comes to education "reform". I suspect that is because the powers-that-be feel they have no power to change the students' "will", but see an easy route via changing the "guidance" and "opportunities". You know the old theory of "change what you can."

Our New York governor seems to be all over "accountability" for teachers and schools, yet somehow absolves students of any responsibility in the matter.  I fully believe that students have the ultimate responsibility, but I also feel that the adults in their world can do much much more to help them get motivated.

The famous quote (prayer, call it what you wish) goes like this:

God, grant me the serenity to accept the things I cannot change,

The courage to change the things I can,

And the wisdom to know the difference.

This, as a code for life, might seem self-evident, but it is anything but. Taken literally, it can lead to a feeling of complacency. It presumes wisdom, but how do we measure our own wisdom? What if we sense, incorrectly, that there is something that we cannot change? Unless we acknowledge our lack of wisdom, we would be falsely serene.

I believe that, as a culture, we have underestimated our powers to improve student motivation. We have become complacent in that regard, passing the buck to causes we perceive as beyond our control.

Perhaps it is time to address the issue of focusing on student failure over and above school failure. Successfully eliminating student failure would, in essence, eliminate failing schools. Yet, doing such is impossible without addressing student motivation (the "will to succeed").

I hope that I am doing a small part with my work in GeoGebra and my inclusion of some of them in my blog postings. What I do know is that I would make sure that students spent whatever time they could working with GeoGebra, not because it is a panacea for mathematics instruction, but because it could be one of the best motivating factors we have.

ASA is Here!!

Last week I posted a GeoGebra file for the SSS problem in trigonometry. Here is a companion for the ASA setting. Another example of how dynamic geometry can be a powerful aid in school.
 
File available at http://tube.geogebra.org/m/1489891

Make sure you know all sides of it and see it from all angles!

This was created with the specific goal of supplying self-practice for students in which they can check their own results. It might also be adapted for classroom use.

It continues my quest to see dynamic geometry (such as GeoGebra used here) become an integral part of school mathematics.

Math is Art is Math is Art

Just another example of the interface between artistic design and mathematics.
Are schools connecting the two disciplines yet?
math is fun!!!
 

What Strength!!

Here is a picture of a pair of sneakers hanging from power (or telephone) lines on a road just outside of Valatie, NY. These sneakers have been hanging there for years. At least 5 years if not ten or more. 

During these years we have had storms blow through that have knocked down tree limbs and even entire trees. Ice storms that have made birch trees bend over to touch the ground. We have had snow falls of two feet or more, driving rains that have sent water around closed windows into rooms of houses. In addition, we have had summer heat waves well into the 90's, and thunderstorms that have sent lightning to the ground with a vengeance.

Despite all these weather extremes, these sneakers have stood their ground (or wire, if you wish).

All I want to know is: What kind of laces do they have? 

Birth of a Hexagon

The first image shown here was initially created in Geogebra, and then saved as an animated gif, and edited a bit for size on screen  (and to keep file size down.) The second is an embedding of the actual Geogebra file (available here)

My main point in creating it was twofold: 1) another example of how dynamic geometry can be used with students at younger and younger ages, and 2) a bit of practice for me in using a single slider as a time line for an animation.

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.

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

Wonderbread!

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

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.

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.

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.

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!

Who knows?

The truth is that most Louisiana teachers have not found the Common Core standards to be developmentally appropriate, nor have they liked the preponderance of informational texts at the expense of good literature, which historically has been the source of the development of abstract thinking.

The above statement is from an article by Glynis Johnston in the The Times of Shreveport, La., on May 24 of this year.

I do not question the opinions she expresses in her article. She is entitled to them.

The phrase "most Louisiana teachers" I find truly head-scratching. If she truly has access to real data that supports her claim, she should be writing for national media and addressing her state legislature if not Congress. It is hard to fathom how these "Louisiana teachers" could have judged these standards as not developmentally appropriate at a time when very few students have tried to develop under them. No state has come even close to having students progress through their k-12 years under the influence of Common Core.  

Ms. Johnston really amazes me when she claims to know what has been the historic source for abstract thinking.  Good literature? Define that phrase. As far as I know, what is good for one is lousy for another. What I can say, based on my 36 years as a high school teacher, is that students never got enough work on reading and writing for detail. The closest they cam to reading for detail was their textbooks in math, science, and history.  The top students, however, needed to do little reading because they "got it in class" while the weak students never even took their textbooks home.  It is hard to tell the impact of reading for detail is that environment.

Perhaps these rationales for dumping on the Common Core are well-intentioned. Who knows. 

On December 1 of last year, Amanda McElfresh, writing about Louisiana, states 

Teachers also said they've seen significant improvement in student performance since they started using Common Core in their classrooms. Are these two talking about the same state?

Speaking of Ms. Johnston, back on December 5 of last year she wrote an article entitled All La. teachers do not support Common Core. I guess it is an improvement scaling back from "all" to merely "most" in her statements.

Speaking of her article's title, I believe Ms. Johnston has fallen into an increasingly common error. She uses the phrase "All...are not...", which is logically equivalent to "None... are...". I believe she intended to acknowledge that some teacher's do favor Common Core, and the phrasing "Some... are..." is logically equivalent to "Not all... are...".  In her statement the "not" appears to be misplaced.

Then again, maybe back then she really was claiming to speak for all teachers in Louisiana. Who knows...

So what is that old subtraction anyway?

Most people old enough to be parents learned subtraction in the (what I think is poor) method with "borrowing" (or "renaming" as it became known, since it was never "given back"). Their work looked something like this:

Now imagine you asked an adult to explain what is going on. Better yet, ask 10 adults. I think that the phrase "you can't take 8 from 5" and "cross out the 4 and make it a 3", or similar phrases, will be heard. You will probably not hear "50 from 30" or "change 300 to 200". Why? Because people have been trained to think of subtractions such as this in terms of a gathering of individual digits and not 2- and 3-digit numbers.

For years the phrase "number sense" was bandied about, but not enough as far as I am concerned. Generations have been trained in "digit sense" while developing little in the way of number sense.  (I use the word "training" specifically because it connotes the sense of developing a skill or talent without the need to understand why you are doing it. When was the last time you heard of someone being trained to be a PhD?)

What is happening to the numbers in the above subtraction? Here it is written out in some proper mathematical notation. (Keep in mind that once you get (got?) past arithmetic, your math work was predominantly written line-by-line, one step at a time.)

\[\begin{array}{c}345 - 158 = (300 - 100) + (40 - 50) + (5 - 8)\\ = (300 - 100) + (30 - 50) + (15 - 8)\\ = (300 - 100) + (30 - 50) + 7\\ = (200 - 100) + (130 - 50) + 7\\ = (200 - 100) + 80 + 7\\ = 100 + 80 + 7\\ = 187\end{array}\]

If we can accept for the time being this line-by-line approach as having some legitimacy, consider this:

\[\begin{array}{c}345 - 158 = 345 - 100 - 50 - 8\\ = 245 - 50 - 8\\ = 245 - 45 - 5 - 8\\ = 200 - 5 - 8\\ = 195 - 8\\ = 187\end{array}\]

I include this because it is different, perfectly correct, and more easily followed mentally

My method for doing this subtraction is like this:

\[\begin{array}{c}345 - 158 = 245 - 58\\ = 195 - 8\\ = 187\end{array}\]

By having taught myself years ago to work with numbers left-to-right, the same way we say them, I can claim that I haven't "borrowed" (or "renamed") while doing a subtraction in decades.

If all you know is the handwritten version above, odds are you will find few shortcuts and will stumble doing subtraction mentally.  If you find yourself at a checkout counter with a clerk who has trouble making change, especially when you hand over a twenty, two quarters, and a dime for your bill of $19,58, consider that the clerk, who is quite possible younger than you, has only the "borrowing" method to use, which is very very awkward to do mentally.

An ellipse is born

I thought I'd just post a sampler of what is happening on my other digital stomping grounds. This starts with two circles that can rotate around two circles with the same center. They are rotating in different directions, but their orbits take the same amount of time. The end creation is an ellipse. Below is a derivation of a Cartesian equation as an #Algebra II or #PreCalc student might do it.

The important thing here is that dynamic geometry such as #GeoGebra can open doors to concepts not normally seen until later years. The basic construction here (minus the buttons and check boxes) can be handled by a middle school student. The entire GeoGebra file can be found here.

Time to Reverse Direction

I have read a lot about people blasting Common Core for de-emphasizing, if not leaving out, the old right-to-left "borrowing" (or "renaming") method of subtraction.

To all those people I should first point out that even on a calculator, numbers are entered left-to-right. Having students learn right-to-left for paper calculations, but left-to-right for electronic calculations, has helped create a generation that finds mental arithmetic almost totally impossible.  Conflicting methods all through school create havoc in the student's brain.

Imagine learning to write right-to-left. Quickly, spell your full name backwards.

The Common Core recognizes that teaching right-to-left subtraction does not promote understanding of what subtraction really is.

All over the web there are people blasting supposed new and different techniques for subtraction. Business Insider had one last October (see it here). In it the author, Andy Kiersz, writes

The "counting up" method (which is what's depicted in the textbook above) is not intended to replace the standard way. Instead, it captures some of the underlying aspects of subtraction and place value that allow borrowing and carrying to work.

His comment bothers me. It is based on the assumption that the old methods are fine, just in need of a facelift.

In fact, they were not fine. The whole mess of right-to-left arithmetic needs to be tossed. We say numbers left-to-right, we write numbers left-to-right, we enter them on calculators left-to-right, we even say, remember, and dial our phone numbers left-to-right. Why would anybody think that we should be calculating with them right-to-left?

Another article referring to the same example is by Ed Morrissey, also from last fall (see it here). In it he says "the “counting-up method” requires a paper calculation and more complicated cognitive judgments than simply subtracting and carrying over."  That analysis is highly debatable. Anyone grounded in left-to-right addition finds the 2+60+200+25 fairly easy.  The process does not require paper calculation at all. In fact the need to write down paper calculations will diminish as one gets more familiar with left-to-right arithmetic.

Thankfully, students are beginning to learn arithmetic in a manner that will help them internalize the concept making them stronger and better thinkers. The "New Math" back in the 60's tried to do that, but it never dealt with the "right-to-left" vs "left-to-right" issue. Back then it was apparently presumed that if people obtained a better number sense the old arithmetic methods would make sense. That was still climbing the uphill battle because right-to-left was against-the-grain.

The examples referred to in Kiersz' article begin to make full sense to someone who has begun to think and work with numbers left-to-right. In the "subtract 38 from 325" question, I would subtract 40, get 285, then add 2 back to get 287.  It is different from "counting up", but it's much easier to understand than "you gotta take 8 from 5 but you can't take 8 from 5 so you take the 2 in the tens place, make it 1, and and a 10 to the 5 in the ones place making 15,.....". (Apologies to Tom Lehrer). 

Common Arithmetic

If our Common Core testing fiasco yields nothing else, I hope it changes the way arithmetic is done in this country.

Scenario: you are told to add these 4 numbers. How do you do it?

If you proceed the traditional "paper and pencil" way, here are your thoughts:

7 + 8 = 15
15 + 6 = 21
21 + 3 = 24Write down 4 under units columnWrite 2 at top of tens column.2 + 4 = 66 + 7 = 1313 + 4 = 1717 + 1 = 18Write down 8 under tens columnWrite 1 at top of hundreds column1 + 3 = 44 + 2 = 6Write down 6 under hundreds column

This took 14 steps, and at no point did you say or even think any number even close to your answer.

Now we will do it left-to-right. Here are the thoughts:

300 + 200 = 500 500 + 40 = 540 540 + 70 = 610 610 + 40 = 650 650 + 10 = 660. 660 + 7 = 667 667 + 8 = 675 675 + 6 = 681 681 +3 = 684 Write down 684

Ten steps, and at each step you were actually thinking numbers that got closer and closer to the final answer.

Why is the first option the most common that people remember? Because single digit arithmetic is the first taught and learned, and it does not vary far from that.

If we treated transportation the way we have treated arithmetic, we would all still be walking.

Will it go round in circles?

The beginnings of a Ferris Wheel done in GeoGebra. Go to here. download it, and play with it. Finish it. Change it. Explore. Learn.

STEM is not synonymous with education

My local NBC channel, WNYT channel 13, has been bludgeoning its viewers lately with stories it relates to STEM, the Science-Technology-Engineering-Mathematics push, with its STEM13. One would think that I, as a career math teacher, would appreciate such coverage and clamor for more. But no.

There is a strong linkage between the four subject areas, but in promoting it uber alles (in its true translation as "more than anything else"), it has belittled each of the 4 subject areas, deeming them each subservient to some higher academic god.  That is to say nothing of what it does to history, language, art, music, physical education, and so on.

STEM has a place, once the basic footings of mathematics, science, history, literature, language, art, music, and physical education have been put in place.  (I leave out technology, for it, as Oxford dictionaries states, is  "the application of scientific knowledge for practical purposes, especially in industry". Seems to me, before it can be applied, science must be known.)

Sure, in the professional setting, all these areas are interwoven frequently. They naturally would, once the foundations have been laid.  But stressing them to the exclusion of anything else is putting the cart well in front of the horse. 

Gottfried Leibniz is quoted as saying "Music is the pleasure the human mind experiences from counting without being aware that it is counting." Just being able to appreciate that sentence requires one to know something about Leibniz, music, and math. The mathematics in music is astounding, almost magical.  If you have any doubts about how key mathematics is in music, take a look at the wikipedia page for Pythagorean tuning. That's right, Pythagoras. The same guy who worked with triangles.
To quote Wikipedia

According to legend, the way Pythagoras discovered that musical notes could be translated into mathematical equations was when he passed blacksmiths at work one day and thought that the sounds emanating from their anvils were beautiful and harmonious and decided that whatever scientific law caused this to happen must be mathematical and could be applied to music. He went to the blacksmiths to learn how the sounds were produced by looking at their tools. He discovered that it was because the hammers were "simple ratios of each other, one was half the size of the first, another was 2/3 the size, and so on".

Imagine if Pythagoras had no concept of ratios. Sure, there is a slight chance that such an event might spur someone to invent a concept of ratios. A very slight chance. 

While on the topic of music and math, stop and watch a short video from TED. It is part of a web page from the site ed.ted.com which can be found here. After watching that video, just ask yourself again as to whether or not music and math travel together.

It might be that our nation would be more prosperous with more "STEM" people. What is more important is that the foundation of mathematics be fully grounded, and the inclusion of music can only help make that happen. The tunnel vision of "STEM" seems to be pushing subjects like music to the sidelines. That has to stop.

The Family of Nobodies

Here is a question from last January's Algebra I (Common Core) regents Exam in New York state. Read it carefully.

This "predicting function" is based on household size as input and number of devices as output. Hence the household size falls into the domain.

Here I show a quote from www.cliffnotes.com (see it here)

When you first learned to count, you started with 1, 2, 3 and kept going until you couldn't remember what came next or grew tired of counting. These positive counting numbers (1, 2, 3, 4, ...) are called natural numbers. The ... means the number list continues on infinitely.

If you add the number 0 to the natural numbers, you get the whole numbers (0, 1, 2, 3, ...). You also get an example of how a number can be classified as more than one type. For example, the number 2 is both a natural number and a whole number. In fact, all natural numbers are whole numbers, but not all whole numbers are natural numbers. Why? The number 0 is a whole number but not a natural number.

I may be missing something here, but it seems to me that question 6 above is missing the correct answer. Unless you find it common to talk about the number of families with zero members, the answer to question 6 must be the natural numbers ("counting numbers" is an accepted synonym). The question asked for the most appropriate. As long as you are willing to include elements in the domain that are nonsensical, any of the answer choices are okay.

As for a benefit of including households with 0 members: Uninhabited homes can be considered as householder-occupied, vagrant iPhones can be given a place to live. Wolfram (seen here) even acknowledges lack of a standard, and recommends the following:

I do grant Wolfram a certain amount of respect in the world of mathematics.

While we are at it, a a standard convention in mathematics is, in the absence of a specified domain, to use the largest domain for which the function makes sense. In this case, that is the positive integers, or natural numbers, or counting numbers. However here, as in many elections, the correct answer is just not one of the choices.

Now we have question 13 from the same test.

Give me a break: 79 cents?  Not 75, not 80. Seventy-nine cents?

This question writer(s) seem to have concluded that 75 or 80 would have made the question too easy. For someone who can do arithmetic mentally, it would be easier. Or would it? This problem requires absolutely no calculations of any kind! That 79 just sends up a red flag saying "don't take this test seriously".

What's your plinth?

The following quote was allegedly used in the New York State 6th grade Common Core ELA test, according to Valerie Strauss of The Washington Post (see here)

As a result, the location of the cloud is an important aspect, as it is the setting for his creation and art of the artwork.  In his favorite piece, Nimbus D’Aspremont, the architecture of the D’Aspremont-Lynden Castle in Rekem, Belgium, plays a significant role in the feel of the picture. “The contrast between the original castle and its former use as a military hospital and mental institution is still visible,” he writes. “You could say the spaces function as a plinth for the work.”

I copied this text and pasted it into Microsoft Word  so I could check some readability measures. Irs Flesch-Kincaid readability level is grade 11.0.

I took it to and received the following:

I was surprised at the 10.9 to 11.0 discrepancy of the Flesch-Kincaid rating, but noted that they all came in quite higher than the 6th grade level of the test.

Let me add a disclaimer: until I read Ms. Strauss' article, I can honestly say that the word "plinth" was not part of my vocabulary. I know that in the absence of a dictionary, the essence of this paragraph would have been totally lost to me. Were students allowed a dictionary during this test? I would hope that the skills we are allegedly teaching our youth include the basic skill of looking up a word of unknown meaning. So do they get dictionaries? Do you know?

The New York State Testing Grade 3-8 Common Core English Language Arts and Mathematics Tests School Administrator's Manual (that's a title!) states the following:

Bilingual Dictionaries and Glossaries——English language learners may use bilingual dictionaries and glossaries when taking the 2013 Grades 3–8 Common Core English Language Arts and Mathematics Tests. These bilingual dictionaries and glossaries may provide only direct translations of words. Bilingual dictionaries or glossaries that provide definitions or explanations of words are not permitted. 

I am extremely curious as to whether or not the "other" language edition's of this test used a word as exotic as "plinth", and, if so, were they translated by the accepted dictionaries and glossaries as "plinth".

This is just indicative of a poorly run and disastrously implemented testing program.

What is beginning to really annoy me is that the political climate of our culture is making it very very difficult to be in favor of outside assessments while being dead-set against what is currently being done.  The testing system being implemented just has to get tossed. It may be making a few people richer in the short run, but in the long run it will help impoverish our society and culture.

If you have read this far, you might be able to tell me why these tests are being given in April, two months before the end of the school year? Who made that call? I'll bet it was for the convenience of some corporation. 

Cuomo wants to these tests as 50% of teacher evaluations, while testing only 80% of the school year. That is weird. But consider the source.

The first step in fixing this fiasco must be to get Cuomo out of office, and then get rid of any and all politicos who signed off on this debacle.

Last post continued: where did he go?

My last post adapted to GeoGebra. Experiment. Play around. Explain.

Which way did he go?

I have had this animated graphic for years. I do not remember where I first got it (I did not make it), but I do know that it has spurred a number of conversations where no one left feeling mentally competent.

The situation is simple: count the number of people both before and after the animation. Explain.

Mathematics and Creativity go hand-in-hand

Creations such as these can be introduced to beginning Geometry students, without even needing what is in red. Demonstrating the shapes, and how they are created using chords and midpoints, can be seen as a possible motivator for the later study of trigonometry and polar coordinates.  If we have a "core" be it "common" or not, that does not allow time for such investigations, we will lose the academic race to those cultures, countries, and schools that do allow for it.

This was made in GeoGebra in 10 minutes. If I had been able to use this in the classroom....

A Butterfly Effect...

Short extension of yesterday's post that merely allows you to change the direction of one of the moving points. A simple change creates a new shape. If I was still in the classroom, I would use the creative possibilities of GeoGebra as an aid in teaching the vocabulary and basics of Geometry in lower grades, and use it as a tool in posing questions and problems in the upper grades. All along one must keep in mind that GeoGebra is just a tool, not a teacher.
While I work on this I cannot but cringe at the lack of time teachers get during the school year to create, modify, improve, and implement any of their own works. Can't have creativity in a  "right-answer" subject such as math!

Math is simple...

I cannot forget that my primary goal is eliminating math phobia and generating "hooks" that can gab and keep interest. I use GeoGebra a lot with this in mind.
Here is a app showing how a limacon results when we trace the point exactly midway between two points rotating around a circle. One point is traveling twice as fast as the other.

What is best: weakly correct or strongly incorrect?

Here is a question from the New York Algebra 1 (Common Core) Regents Exam from January 2015. It was worth two points.

This solution is given as a model answer worth 2 points.

 Here is a model answer worth only 1 point.

 Here is a model answer worth 0 points.

Please take note that the sample answer that was incorrect is to be given more credit than an answer that was correct.

I have a serious problem with this as it seems to devalue the correctness of a response while overvaluing an explanation for an incorrect response. Euphemistically speaking, talking the talk seems to be more important that walking the walk.  Please take note that writing "fuction" where the word "function" belongs seems to be totally irrelevant.

This is not to belittle the value of being able to explain a process. Explanations are important. But the hierarchy should be:

1. Correct results well explained.
2. Correct results weakly explained.
3. Correct results not explained.
4. Incorrect results with support.
5. Incorrect results not supported.

Please take note that an incorrect result with support given is simpler to fix that an unsupported incorrect result. Just as a doctor will ask you "where do you hurt?", fixing a solution ought to begin with "where did it go wrong?" That is most easily answered when the result's alleged support is given. 

I refrain from using the concept of "explanation" in relation to an incorrect answer as such answers cannot be explained away. Hopefully they can be repaired.

Along with this dilemma is the misleading description of "function" as stated in the Common Core. Read this:

Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

The words "from" and "to" in the first sentence is misleading. The idea of "with" would be an improvement, as in "A function is a pairing of elements from one set (called the domain) with elements of a second set (called the range) such that each element of the domain is paired with exactly one element of the range."

The description used by the Common Core also gives the impression that a function must have a graph. Try to graph this function: y = the first letter in the word "x", where the domain can be any list of words.  It is a function, but can you graph it?

My favorite example of a function, that I used in class plenty, is the bar code scanner at the local supermarket. To get students to latch on to the notion, I asked them what would have to happen for it NOT to be a function? They quickly agreed that a single item should result in only one price.

Getting back to question 27, is it good to penalize a student for knowing a correct answer to a simple question? I have known many students who interpret "Explain your answer" as if it said "write your answer in sentence-paragraph form."  The answer is so obvious that they do what they think they should and move on.  And they get 0 points?

To put this question in perspective, evaluate all three responses as if they were given to a student by a tutor or teacher as answers when a student posed the question. Is the 1-point answer more valuable then? Naturally the 2-point response is best, but is a supported wrong answer more valuable than an unexplained correct answer?