Ubiquitous Numbers

One and Zero are always everywhere

The more you tutor math, the more skilled you become in finding good, clear ways to explain all types of math concepts to students. However, some concepts are more elusive than others. The difficulty of grasping a concept depends not only on the concept itself but also on the student who is assimilating it. One person, for example, can easily understand polynomial multiplication, and struggle with percentages, while someone else can find percentages very easy but have trouble with polynomials. There are also some concepts or topics that seem to be hard for a significant majority of students, like word problems for instance.
Substitution is a very powerful problem-solving technique, and it is widely used in a variety of situations. Students who find substitution easy have a clear advantage over students who have trouble understanding it. Substitution comes up in many different ways, some more complex than others. Some students understand the more basic forms of substitution but have problems applying the same techniques when dealing with more complex expressions. In fact, a very consistent general trend is the host of negative reactions students tend to show in varying degrees when facing bigger, longer, more complicated expressions. The more complex the expression, the more likely that students will get confused, or feel overwhelmed by it.
There is a particular way of using substitution to which most students react with a strong resistance: It involves transforming a given expression into another one that is equivalent in value but looks more complicated. This is done with the ultimate goal of simplifying the expression but it starts out by complicating it a little more. It is like climbing up a hill to find a way down the mountain.
The most common way of using this technique is by introducing a representation of the numbers Zero or One into the given expression.
Zero and One are very special numbers. They implicitly are everywhere in any given algebraic expression, even when we do not see them written out.
Zero is called the additive identity because zero plus any number is that same number ( x + 0 = x ).
One is called the multiplicative identity because one times any number equals that same number ( x∙1 = x ).
These two properties make Zero and One algebraically omnipresent in an implicit way.
Further, we have the following properties:
A number subtracted from itself equals zero ( x – x = 0 ).
A number different from zero, when divided by itself gives us one as the result ( x/x = 1 ).
These last two properties give Zero and One an infinite number of representations (“disguises” if you will) to show up in a formula. So, not only are Zero and One ubiquitous, they can come in a dizzying multiplicity of seemingly different forms.
The above properties of Zero and One, and their consequences, make them extremely useful in solving equations, and in manipulating algebraic expressions in general.
However, as I mentioned before, many students present a strong resistance to the idea of making an expression more complicated to be able to reduce it later. This is partly because they do not see the point of multiplying a number times one, or adding zero to it; partly because doing so seems to increase the problem in size and complexity; partly because they feel we are working backwards into some uncalled-for calculation, and finally because they believe they would not know what particular form of Zero or One they are supposed to introduce into the expression if they were doing the problem on their own.
Sometimes, when working with a student on a given problem, it is very easy for me to see a path to the solution using these types of techniques but I have learned to make sure the student does not feel like I was expecting him or her to be able to solve the problem in the same way. I only use these techniques when the student is completely stuck in the problem, not making any progress at all. When they see one possible solution, it gives them some perspective on the different factors playing a role in the problem. After showing them one possible solution method, if they do not feel comfortable that they would be able to successfully use the same method on their own, we focus on finding an alternate method that works better for them.

Tutoring Vector Calculus

Rich Problems

I really like tutoring Vector Calculus because the problems are very rich. Very often you get to do a lot of stuff in a single problem, like: partial derivatives; determinants; dot-product; graphing three-dimensional shapes, parametrizing curves and/or surfaces; substitution; double or triple integrals; polar, cylindrical, or spherical coordinates; trigonometric substitution and/or integration by parts. Plus usually there is some flexibility as to how to go about setting up the problems; with quite a few choices from ordering the variables to writing down the equations, and selecting the integration techniques. It is a lot of fun.

Math is not English

The order of operations messes with our reading habits.

The mathematical order of operations seems to be a source of confusion for some students, sometimes even frustration. For example, when presented with the expression
3 + 4(x-1)
some students ask: “Why can’t we just start by adding 3 + 4, and then multiplying 7 times (x-1)?”
My short answer is: “Because math is not English.” Then I ask: “Is there any parenthesis around the 3 + 4 sum?” When they say “no” I continue: “Then the parenthesis that is there right after the 4 claims that 4 for itself, for multiplication purposes. It will not let the 4 run away with the 3, oh no sir, no way! The multiplication operation has title to that 4, and to that (x-1) as well, and it does not care about the 3 the slightest bit. The addition operation holds a lesser priority than multiplication does, so it has to wait for its turn.”
I explain the PEMDAS rules using action verbs commonly applied to human situations, thus making the math symbols play the role of active, independent characters with human-like behaviors. This type of explanation makes my students understand the mathematical structure of the expression at hand but still some seem puzzled, or surprised, or even bothered by the fact that the applicable sequence of operation does not necessarily follow the simple left-to-right order. So in those cases I proceed with the following explanation:
There is a crucial difference between the way we read math, and the way we read English. This is very important. We always read English from left to right. Such a simple, linear, unidirectional way does not do it for math. It does not work. Reading math from left to right only, is insufficient, and inadequate. In math we have to read formulas and expressions not just from left to right but from right to left; from the top down; from the bottom up; from the inside out; from the outside in; and even around in circles. In short, every which way, else we run the risk of missing essential information about the structure of the thing. Reading math is not reading. Reading math is much more similar to what the eyes of a helicopter pilot do when they are flying over a mountain terrain, looking for a spot to safely land the helicopter. You look at everything, everywhere.
Written language mimics spoken language, going along with the flow of the story. English is perfect for telling stories. Math describes structures. It has an altogether different goal, so it cannot work the same way English does. Mathematical expressions do not resemble stories nearly enough the way they resemble gizmos, appliances, devices, or cars, objects made out of parts. The parts are connected to each other in a very specific way. Each part has its own function, and its own place within the whole thing. So, really, reading math from left to right only, makes as much sense as trying to “read a car” from left to right only.

The Square of a Sum

An arithmetic and geometric approach to Algebra

A very common mistake test takers make when they have been out of school for a while is that they automatically try to expand the square of a sum as if it was the same as the plain sum of the squares of the individual terms. For example, sometimes some students, when presented with say, (x + 5)², they wrongly expand it as x² + 25, leaving out the middle term 10x.
They forget they have to use the foil method to multiply the given sum times itself. Since typographically the sum of the squares looks like something that could equal the square of the sum, they guess they can expand the square that way, as if it was a real math rule, and then they get the problem wrong.
I try to preemptively address the possibility of this mistake, because both sums of squares and squares of sums show up a lot in the math section of standardized tests. For a long time I have used a drawing, with two little squares and two congruent rectangles filling up a larger square, to show the geometric interpretation (in terms of area) for the algebraic formula (a + b)² = a² + 2ab + b². I emphasize the middle term, 2ab, in this expansion, and I point to the corresponding two rectangles in the drawing.
The above approach is useful but many students forget the formula when it comes to actually applying it. Most do not realize it is a universal pattern where to plug any other expressions in.
So, lately I have been experimenting with a slightly different approach: I present two or three numerical examples first, before drawing the squares and rectangles. I ask the student to choose two numbers, and then I guide them through a sequence of calculations that allows them to actually compute and see the numerical difference between the square of the sum, and the sum of the squares. Then I ask them to multiply the original numbers, then to double this product. By asking all the questions in the right order, I have gotten comments from them like: “Wow!” or “How weird!” or “That is so funny!” or “This is very interesting!” or “Does it always work like that?” or “How can that be?” or “What is going on?” or “I’m sure there is a pattern here!” Then I draw the squares and rectangles with all the numbers in their proper places. So, creating the perception of a “mystery” with the numbers, and then explaining it away with the picture seems to work well. It makes sense for the students. They usually say: “Oh, right! Now I get it.”

Tutoring for the CSET

Matrix Multiplication, Fields, and Other Abstract Concepts

Lately I have been tutoring a few CSET takers. The CSET tests for math knowledge equivalent to what is expected from a math major. Consequently, the CSET covers a lot of topics, and some of them are quite abstract. One of the common CSET preparation guides out there starts with one of its most abstract questions right at the very beginning. The question asks to identify, among five possible options, one valid argument showing that the set of all invertible 3-by-3 matrices is not a field. So, this particular question checks whether the student knows what a field is (as an algebraic structure), and also that matrix multiplication is not commutative.
Many CSET takers who are not math majors have the goal of teaching middle school math. They are usually surprised by the amount of math in the CSET. At least two of them have made this remark to me: “Wow! Maybe I do not want to teach math. This is quite a lot, and very complicated.”
Individual reactions to that first question (on the set of invertible square matrices not being a field) vary widely from student to student.
One of them told me: “Let’s just skip this one. I do not want to waste time on this. If I see a question like this one in the test, I am just going to take a guess and that is it.”
However, some other students have the sort of driving curiosity that do not allow them to let go so easily, because they want to know what the question talks about. So, another student kept asking me question after question, during a couple sessions, until she understood the algebraic concept of “field”. As we went through several examples and counter-examples of number types, sets, functions, operations, and properties, along with the abstract names and notation, she kept saying: “Wow! This is mind-blowing! I never thought they would expect me to know so much stuff.” But she kept asking questions all the way until she made sure she understood that first problem in her practice test.
I very much enjoy tutoring for the CSET, precisely because the wide variety of abstract topics it covers. Just like Linear Algebra, and Vector Calculus, the CSET reminds me of my college years.

Look for Solutions with Less Math and More Logic.

One instance where often “less is more.”

The following question is an excellent guideline for solving math word problems:
“How can I solve this problem by doing the least possible amount of math?”

Oftentimes there are several pathways from the setting of a problem to its final solution. Some routes are safer, while some are riskier, more error-prone. Some routes are faster, while some are time-consuming. Some routes are clearer, while some may be confusing.
Usually, the routes with more elementary operations (especially long division), and bigger numbers, tend to be lengthier, longer, and riskier, because adding, multiplying, and dividing big numbers or expressions requires a laser-focus attention. There are plenty of opportunities for doing silly mistakes during these calculations. Besides, it is easy to lose sight of the big picture when worrying about the accuracy of the calculations.
Factoring whole numbers and algebraic expressions is a good habit because it allows you to simplify some expressions before diving into the calculations, so you can operate with smaller numbers, gaining time, and accuracy.
Using logic is a very good habit, too. Many problems lend themselves to solutions that involve more reasoning, and less calculation. This is usually a good thing because these solutions tend to be clearer, and shorter.
Organizing all the information about a problem in a way that makes sense to you, is an excellent habit because this way you keep track of where you are and what you are doing all the time through the problem, and having all these references available makes it easy to retrace your steps, and identify any possible mistakes.
Go visual at any opportunity. Pictures, drawings, charts, graphs, and tables often are a huge help in writing down the right equations, or even in avoiding equations altogether sometimes.
There are many problems you can solve with a drawing and a little logic. Just because the problem is a math problem, that does not mean you need to write down an equation to solve it.
Focus on your possibilities, on what you can do. Organize the information in a logical way, using a drawing, or a table. Above all, try to spend the least possible amount of time and energy doing long, detailed, time-consuming calculations. Instead, simplify the expressions, and ask yourself logical questions about the problem.

Missing Pieces of Information

Some search for doors, sometimes some do not want to see them

Last week I showed a student how to solve two linear equations in two unknowns. He knew perfectly well how to solve one equation with one variable but did not know how to combine two separate equations into one.
Also last week another student made the remark: “I do not know how to start solving this problem. What does ‘isoceles’ mean?” As soon as I gave him the definition of an isoceles triangle he successfully proceeded to solve the problem.
Earlier today another student asked me: “What is a frequency histogram?” When I explained the concept to him, he found it very clear. He said: “Just that? Documenting the numbers in a graph? That is pretty simple!”
Most times students take the initiative, and they spontaneously ask the meaning of terms they are not familiar with. Sometimes however, some students are near some sort of saturation point, and they do not want to even think about the remote possibility that maybe there is a concept they do not know, or a technique they have not seen, and they need this new information to solve the problem at hand. In these rare occasions they keep trying to solve the problem with only the insufficient tools they already have in their problem-solving toolkit.
Writer Kenneth Grahame said “The strongest human instinct is to impart information, the second strongest is to resist it.” So, I choose my words carefully when telling them there is something extra they absolutely need to know first before having any chance of solving the problem. Many times I let them finish their attempts, and check the solution in the back of the book so they realize their approach was wrong without me telling them so before hand, because that could increase their resistance.
There are several problem-solving techniques or approaches that seem indeed artificial, weird, or mystifying the first time around. Once you see how they work, and you use them a couple times, they become perfectly natural, and then you wonder why you never thought of that before.
A perfect example of this I saw also last week with another student.
It was a probability problem involving three coins. For me it is quite amazing to watch time and again how students keep trying to solve these problems by reasoning only about the three separate coins, as if the relevant probability space had only three points. The strong insistence in this naive approach is only matched in its consistency by the strong surprise students show the first time you show them the full eight-point probability space by branching out the development of the experiment at each successive flip, and recording the eight different combination triples. It is really interesting. Somehow these once missing pieces of information act like doors to a whole new realm of math knowledge when they are presented and opened. Many times the student’s reaction reminds me of that feeling of “Wow! I never thought that was a door!” I get when watching some sci-fi movies.

Writing upside-down, and sideways

An indirect measure of tutoring experience

Last week, a student made this comment to me:
“Wow! You write not only upside-down but also sideways!”

Then I realized I have gradually acquired this ability over the years as a direct result of my continued math tutoring practice.

As a tutor, many times you have to correct a result, or an equation the student has just written. You are sitting across the table from them, and the correction may be a minor one. Reaching for their notebook across the table, grabbing it, turning it, putting it in front of you, writing what you want to write, and giving the notebook back to the student may not be extremely time consuming; however, the couple seconds it takes to move the notebook back and forth across the table may sometimes add up to something of a hassle if you find yourself having to make a lot of corrections and/or suggestions to keep the session moving forward.
In such cases, especially when the corrections are minor, once your hand is on the notebook at the other end of the table, it is much easier just to write whatever you have to write, right then and there, without ferrying the notebook to your side of the table.
For me the process started inadvertently, just by changing minus signs to pluses. From there it went on to changing y’s into x’s, inserting parenthesis, adding missing zeroes at the end of a number, and things like that. Still easy stuff but increasing in difficulty a little bit at a time.
The easiest digits to write upside-down are 0, 1, and 8. Before long, you can write all the digits upside-down. One day, all of a sudden you find yourself writing whole formulas upside-down. By this time, most likely you have started writing a few things also sideways, because often the student sits next to you but at a 90 degree angle.
I never had any independent practice writing upside-down or sideways on purpose. The only times I write sideways or upside-down are during my tutoring sessions. So I can say this ability, in my case, is a direct result, and therefore an indirect measure, of my tutoring experience.

Finding Problems That Motivate Students

Time units may help with multiplication practice

According to my experience each child is a unique learner. Generally speaking, students are more motivated to solve a particular problem when the problem relates to something they find real, meaningful, or important. However, different children usually assign different degrees of importance to the same thing. So it is always helpful to find a topic that holds a student’s interest, and that can be easily connected to math.
For example, let’s say you want to help a student who is struggling with multiplication tables. You can start a session by asking how many hours are there in a day, how many minutes in an hour, days in a week, and so on and so forth. Most third graders will know the answers off the top of their head. Then you proceed to ask how many minutes are there in a day, how many hours in a week, and so on and so forth. Children will realize these are multiplication problems but each child will react with a different degree of enthusiasm –or apathy– to find the answer.
Students who somehow have a strong connection to time units will diligently work out these multiplication problems all the way down to the number of seconds in a week. With a little bit of help they will continue working until they get all the numbers right, even if they struggle with the multiplications tables all along. They do this because they want to know the answers. In their mind, these problems are real, not just an empty drill.
Some other kids do not care at all how many seconds are there in a day, so for them this particular set of problems will not be very motivating. You will have to find a different set of problems for them.
Even when students understand the concept of multiplication, and the basic rules to multiply numbers, they may feel that some questions do not justify doing all the work necessary to find the answer. It depends on how real or important the questions seem to them, because that is what makes them want to find the answers.

Calculus is harder than Algebra

A few sources of confusion to be aware of.

Sometimes I see students who used to get good grades in their Algebra classes but who are now struggling with Calculus.
They tell me they are confused by Calculus; they do not know when to apply the Chain Rule; they get uncomfortable around dx; they have no clue how to start the problems, and so on and so forth.
What most amazes them is they know they had no problems with Algebra, they understood Algebra well; so the question in their mind is: “What is it about Calculus that makes it so difficult?”
One problem is their implicit, non-accurate expectations. Sometimes they think they are doing something wrong because they have not gotten to the solution yet, after filling in one page with equations. However, they may very well be on the right track. They are not doing anything wrong. They just never expected for the solution to take so much work, and time. They were expecting to arrive to the solution after three or four steps, like they used to do in their Algebra work. Instead, with Calculus problems they may need to go through ten or twelve similar steps.
Sometimes they ask:
“Another integral? You mean, this is not the result yet?”
Apparently they cannot believe it. I tell them:
“Look, we have to keep going. We are not there yet but we are getting close. O.K.? Think of this as a little marathon. You were used to run the 400 meters, now we are going for three miles. We just have to keep going.”
Another feature that makes Calculus harder than Algebra is the huge number of options when it comes to selecting routes to the solution, many of which may turn to be dead-ends. Calculus incorporates all the operations of Algebra, including exponents, roots, and logarithms; plus all trigonometric functions; and it makes heavy use of function composition. All these ingredients can be thrown into a problem involving limits, or derivatives, or integrals. So Calculus really pushes you to get your Algebra up to speed. And because the expressions get more complex, there are more forks in the road at almost every step of the way. This is confusing for many students.
Now, in my opinion, the major source of discomfort when doing the transition from Algebra to Calculus, is that students who like Algebra like also order, and neatness. This proves to be a disadvantage when it comes to acquiring an intuition for Calculus.
They tell me:
“I do not know what to do, where to start, when to apply what rule, none of that! In Algebra everything was more clear, more precise.”
I ask them:
“And you liked that, didn’t you? Having a set procedure to follow in an orderly manner, right?”
They say:
“Yes!”
Then I say something like:
“Well, I hate to break the news to you but when it comes to Calculus, you are out there in the wild, and everything is moving, all the time, even if it does not seem that way. You have to get used to it.”
Many times they ask:
“But, why? Why is it that way?”
My explanation goes more or less like this:
“Well, Calculus was invented to deal with problems of a very physical nature: motion, volume, pressure, speed, weight, and the like, but not only that. The specific aim of Calculus was to provide answers as to how those physical magnitudes behave when they are changing, either growing or going down, and being related to each other at the same time. Originally, everything in Calculus happens in time, nothing stays the same, everything is changing, moving, decreasing, growing, building up, fading away, speeding up, slowing down, just like in the real world. So when you think of a variable x or y or v in Calculus, it is not just a number that is there, with some fixed value, but a number that wants to go somewhere else, a number that has already started to change, even if only a little tiny bit”
Somehow this bizarre wording paints a picture that makes sense for them, in that it puts their confusion into the proper perspective, eliminating a big part of it.
One of my students, after hearing this type of explanation, said:
“Oh! So, dx is the sneakiness of change! That’s why I don’t like it! But now I know I have to deal with it. I cannot just pretend it’s not there.”
And I said:
“That is exactly right. That is exactly what it is.”
Then she said:
“And, we always have to apply the Chain Rule, we cannot get rid of it, because the variables are connected to each other.”
And I said:
“Yes, the Chain Rule is always there.”
It is a big help to be able to tell students at least some of the reasons why they are confused, because when they do not even know the source of their confusion, everything gets exponentially more confusing for them.

Dealing with the difficulty of memorizing products of digits higher than five

Advantages of using Mayan and Roman numerals

When it comes to memorizing the multiplication tables, each child has his or her own pace. Some children find it easier than others to memorize the lists of numerical facts that make up the multiplication tables. Others need longer practice periods, maybe with the help of flash cards. Rote memorization is not everybody’s best act. Some children find these dry memorization exercises burdensome.
Here I offer a suggestion to help third and fourth graders who are not into rote memorization, to gain a better grasp on multiplicative manipulation of the higher digits.
The main idea is using the distributive property of multiplication over addition to calculate the result of a product of two “big” digits by breaking one of the “big” factors into smaller ones, doing two smaller multiplications, and adding up the partial results at the end.
The essential keys for this approach to be successful are:
First of all, do not even mention the phrase “distributive property.” That’s a big no-no. Do not do it. Show by example only. At this stage children do not need to know there is such a thing as a distributive property. It would be a waste of time. Avoid the confusion. Just do it. Show them how it works but do not try to explain why. The best way to understand why it works is for them to see how it works, period. Do not say aloud any abstract names like “property,” much less “distributive.” Stick to the numbers.
Second, use a standard “breaking scheme.” The number five is an excellent stepping-stone in this process. It is very natural for a variety of reasons. Mainly, because five is half of ten, the base of our numerical system, plus we have five fingers in each hand, so it is very easy to break any higher digit as the sum of five plus a lower digit. Children accept this fact very easily. Furthermore, the multiplication table of five is one of the easiest to remember.
Third, –and this is very important– spend enough time (at least half-an-hour) in a preparation period showing them, or reviewing with them, how to write numbers using Roman numerals and Mayan numerals. Do this before getting into any multiplication practice. This specific type of preparation has a dual purpose. On the one hand, it lowers their anxiety level. You have to understand they are under pressure. Their parents are at least concerned, maybe even worried. That is why they hired a private tutor in the first place. The child knows he or she is not doing great in the class when it comes to memorizing the multiplication tables. Some children may be even beginning to have some dents in their self-esteem, thinking that perhaps there is something wrong with them, or that they are not good at math, or whatever. The main idea in their head at this time is “multiplication is hard.” So when you –the expert– come along and start working with them doing lists of Roman and Mayan numerals, they go “Oh! This is really not that hard. This is easy.” Some children actually have fun with these numerals. Some prefer Roman numerals, while some prefer Mayan ones. The main point is now they are relaxed, at ease, and working with something they understand much better than the monolithic multiplication tables. That is the first goal of this preparation. The second purpose is for them to realize, or remember, or reinforce the idea of just how natural is the use of the number five as a breaking point, or a stepping-stone. Both Roman and Mayan numerals make heavy use of the number five, and of multiples of five, in this fashion. They consistently apply the principle of expressing higher digits as sums of five plus a lower digit. After doing this work with Roman and Mayan numerals, children are so much more receptive to the idea of using the number five as a standard "break point."
So this is the ideal moment for you to start practicing multiplication with the higher digits in this additive fashion. Here I give just one example to illustrate the main idea:
7*8 = 7 * (5 + 3) = (7*5) + (7*3) = 35 + 21 = 56
Do not expect them to know what to do. Guide them with questions. You are supposed to pause and ask as you write:
“Eight is five plus what? Three? Is that correct? O.K.”
“Now, how much is seven times five? Yes, thirty-five, perfect! Thank you.”
“And, how much is seven times three? Yes, twenty one, very good!”
“So, now we just add those two numbers. Can you please add 35 + 21 ? Thanks.”
Just by listening to you asking them these questions and watching you as you write down this multiplication process step by step, they get it, they understand it, and they end up empowered by knowing they can get the right result by themselves even if they do not have memorized the result, even if it takes them a little while doing it in steps like above. They are now much closer to self-sufficiency when multiplying higher digits.

Lowest Common Multiple

A Method and an Analogy to Clarify this Topic

During the last few weeks I came up with a way to explain how to calculate the lowest common multiple (LCM) of two or more integers or two or more polynomials.
Many students get confused by this LCM topic. One reason is the simplicity of the fact that, for any two expressions, their product is a common multiple, so, “Why look any further?” many students ask themselves.
They know their teacher told them in general the product is not the lowest common multiple of two expressions, so they know they are going to get marked down if they give that answer, but many do not know how to find the LCM.
Recently I improved my success rate at explaining how to find the LCM when I started using a table format, as follows.
In the head row I write the two or three polynomials or integers for which we are looking their LCM.
On the left margin I make a list (going down) of all prime factors of the expressions involved, without any repetition. Common factors get listed just once regardless of how many expressions they appear in.
Then, having one row per factor, and one column per expression, we fill in the table by writing the exponent each factor appears raised to in each expression, carefully including all exponents (even those with value zero or one).
After all exponents are listed in the table we make another column at the far right, under the heading “Maximum.” There we write the biggest number out of each row.
The next step is to form the LCM as the product of all individual factors listed in the table (in the leftmost column), each raised to its maximum exponent, as listed in the rightmost column (under “Maximum”). This last product is the LCM we were looking for.
Of course this process can be done without the table, but the table makes it explicit, and it helps as a visual aid for the student to see everything that is going on, all at once. It also helps in making very clear that we do not add the exponents, nor do any other operation with them, we only identify and select the biggest one for each factor.
Most students are happy with this process; the table is good enough for them. It gives them a clear method to follow, and it takes away the guessing and the mystery they formerly faced when trying to calculate the LCM. One of them even said: “You just saved my life with that table! Now I know how to do it!”
However, there are always a few students who also want to know why the procedure works, not only how to do it.
For those who ask “Why?” after seeing the table, I have this explanation ready:
“We have to imagine we are watching a movie about spies and intelligence agents, O.K.? Each expression is like a security checkpoint, where our agent has to show the proper clearances to get pass that point. The checkpoints have different sets of requirements. Each requires verification of a certain level of authority for each security category they are checking at that point. The factors of the expressions are the security categories, like “radioactive material,” “fire arms,” “chemical hazards,” and so on. The exponents are the different levels of clearance agents may have in each category. So when determining the LCM we are looking for the bare minimum possible set of clearance levels we need to give an agent for him or her to be able to make it through all the checkpoints, without any extra, unnecessary authority. They don’t lose their credentials when they go through a checkpoint. They only need to show their badges, they do not give them up. That is why we do not need to add exponents; we only need to select the highest from all the expressions for that particular factor.”
I have found this explanation works very well with all students with whom I have used it so far. One of them said: “Oh! I see. The x² from 3x²y is already included in the x³ from 5x³(x+1) because the exponent 3 is higher than 2. We do not need x⁵ or x⁶. Just x³ will be enough.” And I said: “That is exactly how it works!”

Quadratic equations in rotated form

Some long, time-consuming problems

The last three weeks have been very busy for me. I have been tutoring all math subjects, from fractions to Statistics and multivariate calculus.
Looking back over these past weeks it all seems kind of blurred but one topic stands out from the rest because, by coincidence, I had two sessions on the same topic with two different students, both during last week.
The topic in question is the rotation of quadratic equations in the two-dimensional coordinate (x,y)-plane. It had been a long time since I last taught this subject. It does not come up very often in my tutoring sessions, so I noticed the coincidence when I had two different students independently reviewing with me these geometrical transformations in the same week.
Also, each student separately made the same comment after we worked out problems of this type about quadratic equations: “Wow! This is a lot of work!”
They are right, it is a lot of work. The general problem starts with a quadratic equation like, for example, 5x²+2xy+10y²-12x-22y+17=0,
with a non-zero coefficient in the “xy” term.
The goal of the exercise is to find a specific angle, let’s call it θ, so that the transformed (rotated) equation in the alternate variables x’ and y’ lacks the x’y’ term.
The variables x and y are connected to x’ and y’ by means of these two equations:
x = x’ cos θ – y’ sin θ
y = x’ sin θ + y’ cos θ
Solving these problems requires several steps. I list them here, hopefully without going into too much detail:
First, finding the value of tan(2θ), the tangent of the angle double of θ.
Second, finding the measure of the angle θ itself.
Third, finding the values for cos θ, sin θ, and their squares.
Fourth, plugging those trigonometric values into the formulas below to find the new coefficients for the transformed quadratic equation:
A’ = A cos² θ + B sin θ cos θ + C sin² θ
B’ = 0
C’ = A sin² θ – B sin θ cos θ + C cos² θ
D’ = D cos θ + E sin θ
E’ = E cos θ – D sin θ
F’ = F
where A, B, C, D, E, and F are the coefficients of the original equation.
So you can see each one of these problems involves a lot of algebraic and trigonometric calculations. These problems are long, time-consuming, and you have to pay very close attention to all details to ensure an accurate result.
Anyway, in the video below you can see a room-size metallic structure (some kind of architectural sculpture) where Richard Serra, the artist, incorporated two congruent ellipses, one at the base of the room, and the other formed by the upper edge of the wall. The two ellipses are identical in shape but they are rotated with respect to each other. This is a real, tangible example of the rotation of a conic section. It is relevant to this post because quadratic equations represent conic sections, like the ellipses we see in the video. It is a very interesting structure. Take a look:

Making an Icosahedron

Geometry is fun!

Yesterday I helped one student with his Geometry project.
He had to build a 3-D model of a regular solid, so I showed him how to draw a net of equilateral triangles. We used the triangular net to cut out a template for icosahedrons.
The icosahedron is one of the five Platonic solids (the other four are the tetrahedron, the cube, the octahedron, and the dodecahedron). The icosahedron has twenty triangular faces, thirty edges, and twelve vertices.
Helping my student with this 3-D geometry project reminded me of a course I took in college, where we covered in detail the algebraic structure of the symmetry groups of the five Platonic solids. In that class each one of us built a few models of each Platonic solid, highlighting some of their features, like the cubes formed inside the dodecahedron by the diagonals of its pentagonal faces, for example. For me, that part of the course was a lot of fun.

In www.mathsisfun.com you can find ready-to-print templates for making paper models of the five Platonic solids.

In isotropic.org you can find similar templates, plus additional ones for the 13 Archimedean semi-regular polyhedra.

The following video shows how to make an icosahedron:



This other video shows a MatLab animation of an icosahedron turning itself inside out repeatedly displaying multiple icosahedral net configurations: