The problem is posed with the following diagram:

What is the measure of ? That is, what is the value of ?

I’ve given a similar problem in my trigonometry class for the past few years, except that version of the problem has a side length included and the triangle is not isosceles.

A pdf of this problem is linked below:

Working from my experience with the other version of this problem, I began to write in values for the various unlabeled angles in the diagram – if we label the intersection of and as , then and are both , and are both , which makes and is .

I added in new variables and created a system of four equations with four unknowns, but it was a dependent system.

The solution for this problem that was devised by the student who gave it to me is after the jump…

His solution was primarily synthetic and not algebraic!

His first step was to draw line segment such that and are congruent, creating a small isosceles triangle in the interior of the larger one. Then and are both and is

Once this is done, we can add back in and look at triangle . Notice that and , so that is also , making isosceles with sides and congruent.

Now, if we draw , we know that , and are all congruent. With and congruent, then is isosceles. With vertex angle , then both and are as well, making equilateral with side congruent to , and .

Then, we add back into the diagram, is now as the original angle was and is .

So, with redrawn and , then in triangle , and , then must also be , making isosceles and sides and congruent.

But side is already congruent to in the equilateral triangle , so that means that and are congruent and is isosceles as well. Once we know this, the solution is trivial – with , then must be split equally between angles and , so and !

The reasoning here is somewhat tricky and convoluted, but I looked for a simple algebraic solution to this for hours and got nowhere.

In researching this problem I did run across a similar but different problem.

Chris Harrow, Mathematics Chair at the Hawken School in Cleveland, OH, has a math blog called CAS Musings. Here, he ponders a problem involving isosceles triangles with no angle measures given. This problem generates a system of 6 equation with 6 unknowns that he solves using Wolfram Alpha.

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Divide gently in a long division sauce pan:

Integrate briskly over a low flame:

Evaluate for and let stand at room temperature for 1000 terms for accuracy to three decimal places.

Add lemon zest to taste.

But this doesn’t look like a pi! or taste like a pi!

or a quarter pi!

Ah, but if we return to the original integral

and make the trigonometric substitution so that then:

and

which, if we return to the original substitution , we see that

So, , which means that:

and since , then

Enjoy your pi!

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I was really interested in this because I ran into this problem about four years ago when I locked my keys in my car (while it was running!) in Seaside. I went into the office of the hotel where I was parked and used their internet connection to search for a locksmith in Seaside. The one I called immediately started asking for personal information rather than the address where the car was.

I asked them where they were and it turned out they were in Salem. It wasn’t clear to me how they were going to help me from Salem, so I hung up and found a phone book. I called a local locksmith in Seaside listed in the (paper) yellow pages. They showed up in ten minutes and got in my car for $20. Problem solved (for me).

This is why I was so interested to see this article in the NY Times about locksmith internet scams. Apparently, the call centers that come up in a locksmith search farm the jobs out to independent contractors who often bait and switch by charging far more than the original quoted price. They also are typically short-term temps who don’t care about their reputation.

Some of these scammers go so far as to create fake digital storefronts that show up on Google maps as if they were an actual local business.

The moral of all this is:

“DON’T MISTAKE THE MAP FOR THE TERRITORY.”

In other words don’t think that, because something exists in a mediated form, that it will necessarily exist in the physical world. This problem has essentially no effect on Google’s revenue, so they have almost no interest in fixing or monitoring the problem. The internet is a wonderful tool, but be aware of the REAL physical local businesses in your area and support them in the real world.

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The original article by Mathieu Ossendrijver is posted at *Science* magazine.

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From “The Uses of Audacity”

Algebra is a world of principle, and a dramatic revelation of the power of principle. In fact, algebra, and even algebra alone, could provide a true and sufficient education out of which to understand the worth of living by principle…

…[Y]ou will have it in your mind that you can know something–truly know it, and not just believe it, or be informed of it–and maybe, since that is so, you can truly know something else. It’s interesting to wonder what such a something else might be.

…You will find that algebra shows you some truths. The first great truth is that there can be something real, and complete, and harmonious, and even, in some strange way, absolutely perfect right in your own mind, and made by you alone. You will see that you have a wonderful freedom not mentioned in the Bill of Rights, the freedom to decide what your mind will contain and how it will work. You don’t have to copy the rest of the world.

Algebra tells sad truths too. Where there is no balance, there is no truth. What is equal is equal, and between the equal and the unequal there is no conference table, no convenient compromise. In this terrible law there is a hinting question for all of life…

Algebra will show you the inexorable, the endless and permanent chain of consequence, the dark thread of necessity that brought you to a wrong answer because of a tiny little mistake back in the second line. I know how unfair that seems, and how scary that what seems unfair is nevertheless justice. Is life like that too, as all of nature seems to be? How then shall we live? What are the laws of the algebra of our living, and where do they exist, where created? Who can show us how to learn them?

It takes some serious living to see the truth hidden in algebra…

From “Wise Choices in Peoria”

[Some people]…assume that things like geometry and the multiplication table are taught in schools only out of tradition, and they are easily seduced into believing that such arts are useless to those who aren’t going to make some money from them.

But in fact the mathematical arts are the best studies in which to learn certain truths that are essential to the making of wise choices. It is in mathematics that we most readily see that the permanent relationship between principle and necessity is not subject to appeal, that every particular is a local manifestation of some universal, that there is a demonstrable difference between what we believe and what we know, and that experience can never do the work of logic. It is in mathematical studies that a child … can have his first inkling of Justice and Truth…

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Then, when I was teaching integral calculus last winter, we began a unit on separable differential equations. I was poking around looking for good application problems that would utilize separable ODEs and ran into the fundamental population differential relationship , followed by the relationship I had used for the logistic with defined as the “carrying capacity” or maximum growth for the population.

We went through the procedures for solving each of these ODEs (as well as the continuous mixing problems which follow a similar pattern) and then we moved on.

This year when I was teaching this topic again I was reminded of a paper my thesis advisor had given to me back in 2003 about the application of differential equations to modeling fish populations. I was intrigued by the profusion of models that could be generated by changing the constraints for a given relationship.

Since I didn’t teach integral calculus for another 10 years after I read that paper, I had essentially forgotten most of the equations, formulas and relationships that generated the graphs that had stuck with me. This year, while covering the separable ODEs with their applications, I began to look into the application of these relationships to fish populations and population in general.

I found two great resources that go through the set up of these relationships in a very clear manner, and each of them includes wonderful graphs showing the multiple solutions that result when the same differential relationship is solved with different initial conditions.

The opening section of Robert Borelli and Courtney Coleman’s *Differential Equations: A Modeling Approach* can be read here.

A student project from James Madison University written by Bailey Steinworth, Yuhui Wang and Xing Zhang can be read here.

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The paragraphs that are quoted on Diane Ravitch’s web page and which I think really get at the issue are as follows:

…Innovation and disruption are ideas that originated in the arena of business but which have since been applied to arenas whose values and goals are remote from the values and goals of business. People aren’t disk drives. Public schools, colleges and universities, churches, museums, and many hospitals, all of which have been subjected to disruptive innovation, have revenues and expenses and infrastructures, but they aren’t industries in the same way that manufacturers of hard-disk drives or truck engines or drygoods are industries. Journalism isn’t an industry in that sense, either.

Doctors have obligations to their patients, teachers to their students, pastors to their congregations, curators to the public, and journalists to their readers—obligations that lie outside the realm of earnings, and are fundamentally different from the obligations that a business executive has to employees, partners, and investors…

At the University of Amsterdam, staff and faculty have joined the students in protests against recent actions taken by the institution. Jerome Roos of the European University Institute connects the events in Amsterdam to a world wide crisis in higher education:

Structurally underfunded, severely over-financialized and profoundly undemocratic, universities everywhere are increasingly abandoning their most crucial social functions of yore — to produce high-quality research and educate the next generation of skilled, conscious citizens — and devolving ever more into quasi-private companies run by an utterly detached managerial elite.

To make matters worse, these managers…are actually being paid six-sum figures to push around insane amounts of pointless paperwork, forcing destructive workloads and unrealistic expectations onto increasingly precarious staff, treating students like simple-minded consumers and impersonal statistics, and putting immense pressure on highly talented researchers to spew out mind-numbing amounts of nonsensical garbage just to meet rigid quantitative publication quotas….

The protesters at UvA thus find themselves at the front-line of what is essentially a

globalfightback against the commodification of higher education and the steady reduction of knowledge and learning to an increasingly unaffordable consumer good. In many countries, this neoliberal logic has resulted in dramatic tuition hikes and budget cuts, combined with the metastization of a culture of top-down managerialism, creeping bureaucratization and the systematic precarization of academic labor — with all the attendant consequences of rising student indebtedness, the proliferation of work floor bullying, and deepening anxiety, depression and burnout among university staff.

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For a 2 out of 3 scenario there are six possibilities (AA, ABA, BAA, BB, BAB, ABB) and for the 3 out of 5 scenario there are 20 possibilities (AAA, BAAA, ABAA, AABA, BBAAA, BABAA, BAABA, ABBAA, ABABA, AABBA and ten more in which B wins).

The question then came up of how to generalize these results. The way I approached this was to consider how many ways one of the players could win and then double the answer to include the scenarios in which the other won.

One of the issues in this type of problem is that once the winner has won the required number of games the match ends, so that a best 2 out of 3 match could never end AAB, for instance. Another issue is that each of the possibilities of winning in a different number of sets should be considered separately. That is, winning in the minimum number of sets, then winning after having lost one set and so forth.

Taking all of these issues into account for a competition in which the winner must win out of leads to a general formula of .

Or – if the winner must win out of then the formula would be .

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The cross product for two vectors will find a third vector that is perpendicular to the original two vectors given.

The given vectors are assumed to be perpendicular (orthogonal) to the vector that will result from the cross product. This means that the **dot** product of each of the original vectors with the new vector will be zero.

So, given two vectors

and

we want to find a third vector

so that is perpendicular to both and

As I mentioned above this means that we want the dot product of with each of the two original vectors to be zero.

and

This gives us two equations to work with. Since we have three variables to solve for , we’ll need another equation to work with.

The website *Heaven’s in the backyard* introduces a third constraint that the modulus of the cross product vector be equal to 1.

This creates a third equation and allows us to solve for and in terms of and

As mentioned above the web page *Heaven’s in the backyard* does a nice job with the derivation of the values for and and ends up with the formula .

I don’t see this mentioned in the derivation, but it appears that the term that is factored out and defined to make the derivation work more smoothly is actually the modulus of the cross product vector .

Assuming that the cross product vector has a length of results in an answer that is multiplied by , because the actual perpendicular has a modulus equal to which is .

After working through some of these ideas, I became interested in where the cross-product came from.

At Wikipedia, they mention that Joseph-Louis Lagrange, the French/Italian mathematician of the late 18th century provided a formula for this in a paper from 1773 that was focused on the properties of a tetrahedron. The calculations related to the cross-product appear in the first few pages of the paper. Lagrange posits 9 “quantités” and then proceeds through a blizzard of calculations based on these quantities.

If each triplet of values , and is considered as the coordinates of a vector, then the first calculations are the cross-products of each vector with each of the others. In Lagrange’s notation, we could identify each of the cross products as

, ,

On the following page Lagrange identifies the square of the modulus for each of the cross-product vectors as , and

Several pages later Lagrange notes that the dot product of each original vector with the appropriate cross-product produces a zero result.

There are two things that fascinate me about this – (1) the depth of this seemingly simple question – how do you justify the cross-product formula? and (2) what was Lagrange up to in this paper? – what is the purpose of the multitude of calculations that he completes in the paper from 1773 (*Solutions analytiques de quelques problèmes sur les pyramides triangulaires*)?

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