Linear Equations: Heads Up!

Section 8.2 of AoPS Introduction to Algebra introduces (i) the idea that a linear equation looks like   and (ii) the slope of the line.  With some students I get in a pinch because sometimes they are already somewhat familiar with graphing so in an effort to be efficient we rush into the definition of slope and then subsequently backtrack because of confusion.  

So this week I was prepared for my student.

We work the problem where we first calculate the slope of a line (Problem 8.7), using pairs of points to get the idea that the slope is the same everywhere on the line, no matter which points you choose.  But I was more careful.  Before getting into   I give him a heads up.  “Now I know this formula is going to look like it’s coming from outer space, but just bear with me for a second while we do these calculations.”  Then we calculate this unnamed ratio for 2 pairs of points and we find they have the same result, .  Then I ask him:

“What do you think?  Do you think this formula will be  no matter which points we choose?”

Thoughtful silence.  We had carefully traced the path of  Hopsalot the rabbit, so I reminded that between every consecutive point, we go right  units and up  unit.  Eyes light up.  He gets it, that the ratio is always going to be the same. 

Even better, we didn’t actually finish the problem, or name that this ratio is called slope.  I like leaving him with this epiphany for a while so he can internalize it a bit.  Then we can review this once more before proceeding.  

Since we use the slope to generate the point-slope form of the line, it’s imperative that students understand that the slope is the same everywhere on the line, no matter which 2 points we choose.  We use this property to generate the equation because it’s one of the few things that consistently describes every point on the line.    I’m going to give him this heads up and reiterate as much as possible that the slope is an inherent property of a line, no matter which 2 points you choose. 

Mistakes I Do and Don’t Correct

Math Mistakes I Will Not Tolerate:  Stream of Consciousness
Some students use the equals sign to mean “…and then…”  As in, I have   apples and then someone takes away and then I’m left with and then someone gives me more apples and then I have would look like this:

Now we have an illogical statement where on the left we have and on the right we see that it is equal to .  I think this is part of the reason so many students find math frustrating: they see illogical statements like this one and think math is a subject where rules can be made and broken at whim.  I am quick to correct errors like this.  

Math Mistakes I Allow:  Inefficient Exploration
My students who are new to algebra will not always choose the most efficient solution.  For example, when solving this equation for x:

an experienced student will first subtract from both sides and then divide both sides by .  Some of my less experienced students have more creative ways of solving the equation.  They may choose to first divide by sides by , yielding

Or they will begin by subtracting from both sides or any number of unusual attempts to isolate .  My mantra is “As long as it’s legal, we can try it.”  The most fundamental thing a student needs to learn about algebra is the idea that when you do the same thing to both sides of an equation, you still have equality of both sides.  This is my main focus, that if they do something to the left hand side of the equation, they must do the same thing to the right side.  If the cost is a little extra time spent in going down some inefficient rabbit holes, so be it.  

The other reason I tolerate Inefficient Exploration is that it won’t always be obvious how to solve a problem, until you try to solve it.  Many students when they encounter a problem they don’t know how to solve will freeze.  They may have some ideas, but since they don’t know if it will work, they don’t take the risk of trying.  I want my students to be like the one who decides to divide by first, just to see if it will work.  It might, it might not, but we’ll never know unless we try.  

AoPS Options: 4 Ways to Study!

Many parents contact me because they are confused by the options for studying AoPS curriculum. And it is confusing! Here is a summary of options with my list of advantages and disadvantages:

(1)  AoPS textbook only.   

Advantages: go at your own pace, more practice problems, if parent teaches it is least expensive option

Disadvantages: no group/community learning, may need to hire a tutor (becoming most expensive option)

(2)  AoPS online class.  

Advantages: student is on a set schedule, i.e., you know exactly when you will be finished with a course; student joins the AoPS online community of math loving students, opportunity to learn and practice Latex skills, writing problems evaluated by human teacher.

Disadvantages: students often do not take deadlines seriously and do not complete homework problems by deadline (homework extensions only compound this problem), little flexibility for family schedules, some students do not like text-based class, fewer homework problems.

(3) AoPS self-paced class.  (Prealgebra only)

Advantages:  very similar to online class without online classroom, but with great flexibility in timing

Disadvantages:  students historically tend to not complete the class, access limited to 9 months per half course.

(4)  AoPS Academy courses.

Advantages: In person classes with online video during pandemic

Disadvantages:  Limited campus sites

I support students who are using all these options with an online whiteboard and videoconferencing.  
For example with the textbook only option (1), I work through the problems with the students and then assign the Exercises, Review Problems and Challenge Problems as homework.  
I also support students who are taking online classes (options 2,3,4) with light homework help.  Sometimes I remind them of key points from lecture, a short prompt here and there to get them started.  I also model solid mathematical documentation on the whiteboard to help them avoid errors and improve their writing problem responses.  I also get them started with Latex (more exciting than it sounds!).

My 5 Types of Students

In the spirit of market segmentation, I’ve identified the 5 types of families who hire me:

1. Math-Maturity-Mismatch Students  These are the younger students who have high math ability, but they have the maturity of a typical kid, and they have trouble following through on their homework independently.  Students meet with me once or twice a week for “homework time” saving parents from nagging their kids so they can have a better relationship, and the students have more fun.
2. Socratic-Scaffolding Students  These students blank when they see a problem, and all they lack is the tiniest hint to get them started.  “Hm, today’s topic was triangle inequality.  Can you tell me the rule for triangle inequality?”  And suddenly, just like that, they are off to the races.  
3. “I Must Qualify for Nationals” Students  These are the students who are keen to level up in MathCounts, qualify for AIME, or just generally improve their math contest scores with an eye on college admissions.  
4. “You Look Like Me” Students  Yes, female students like that I’m a girl, and I really get them.  There’s something to be said for single-gender math education, especially in the critical middle school years.
5. Last Comic Standing Students  These students like me for my nerdy sense of humor.  And most give it as well as they get it.  If I use my standard “Problems involving FRUSTRums are very FRUSTRating” joke, they respond with their favorite one-liner:  “y=mx+b.”  

What Everyone Should Know About Math Contests

Math contests encourage a growth mindset.

Students who are accustomed to earning 95% or 100% on their worksheets and test papers understandably view that as a badge of honor, but the desire to maintain high grades can lead to wanting to “look smart” rather than “be smart.” This “tyranny of 100%” can encourage students to avoid challenge for fear of a lower grade. Because math contests are designed to identify only one winning student, even accomplished math students may earn far less than 90%. Regular participation in math contests that don’t impact GPA and where lower grades are common enables students to relax and learn for the sake of learning and not in order to “look smart.”

With academic support, math contests can replace a math curriculum.

I’ve had students who refuse to follow a traditional math curriculum, but were happy to learn their algebra, geometry, and discrete math in 3 years of preparing for MathCounts. With a strong coach who can explain and derive the concepts behind each problem, students can learn with greater retention. They don’t learn the Pythagorean Theorem once and then forget it. They are forced to deploy it in novel situations week after week at each MathCounts practice.

Math contests don’t always reward speed or math tricks.

Most modern math contests do not ask students to solve the identical multiplication problems in 10 minutes. Instead they reward thoughtful persistence and grit toward arriving at a solution. Each problem is different from the one previous and the one following. A geometry problem using similar triangles will be followed by a probability problem involving dice will be followed by a problem solved by modular arithmetic.

Math contests encourage academic bravery and risk-taking.

Often it isn’t obvious whether a particular strategy will result in a solved problem. It’s like navigating your way to the drugstore without a map by peering down a street and considering whether it might be there. A particular street may or may not have the drugstore, but you won’t know unless you take a risk and travel down the street. Likewise, in math, the most interesting problems don’t have a map showing you the way and often the student needs to take a risk and try something to know if it works.

Make math contests work for you

Many parents and students believe math contests aren’t for them because of the above reasons, but don’t let these misconceptions steer you away from this useful tool.

Why your gifted student should study AoPS

I’m your expert in all things related to Art of Problem Solving math curriculum and math contests generally.   I guide students who are taking the online AoPS courses and studying from the textbooks independently.  I help students prepare for AMCs and I coach a homeschooled MathCounts team every year.

You can also call me a Richard Rusczyk fan girl.  One of my favorite talks is one he delivered at Math Prize for Girls in 2009, when he showed this slide:

“I certainly wish your website and materials existed when I was in high school. I went through junior high and high school without ever missing a question on a math test, and then took [Math] 103 and 104 at Princeton, which was one of the most unpleasant and bewildering experiences of my life and poisoned me on math for years.”

–Princeton University alum

He continues:

“I want you to think for a minute what this student’s middle school and high school teachers thought when he went off to Princeton.  They thought, “We succeeded.  He went off to Princeton; we’re awesome.”  They never saw this.  I’m sure he didn’t go back to his middle school teachers and say, “Yeah what’s up?!?  You didn’t prepare me for this.”

“So they didn’t get this feedback, and this happens a lot.  Kids go through school, some very good schools, they get perfect scores on everything, and then they come to a place like MIT, a place like Princeton, they walk into that first year math class, and they see something they’ve never seen before: problems they don’t know how to solve.  And they completely freak out.  And that’s a bad time to have these first experiences.  Having to overcome initial failure.”

Don’t let this happen to your student.  Front load their math education by challenging them early in their academic career.  I’m talking elementary and middle school.

Answer the Question that was Asked

I like math contests for many reasons.  Here’s one more reason:

Even the earliest math contests train young students to answer the question that was asked, not the “answer” they came up with.  Here are some examples from Math Olympiad, designed for students in grades 4 – 6.

This question asks for the sum of the 5 prime numbers:

 

This question asks for the least of these numbers:

 

This question asks for the sum of two numbers:

 

When students learn this discipline early, they won’t miss key instructions like this one from a practice SAT:A student may get the correct estimate for height at 2 years old or 5 years old, but fail to answer the question that was asked.

 

Careful reading requires practice, and is not something a student sees when all they use are worksheets like this:

5 Ways to Show Your Work

I tutor students who are using the AoPS math curriculum and/or are preparing for math contests like MathCounts and AMC.

Today I’m sharing 5 Ways to Show Your Work in math that you can start doing today.

1. Line Up Equal Signs.  Nothing helps organize your thoughts like equal signs in a straight line.
2. Reorient Similar Triangles.  You found similar triangles.  Yay!  The last thing you want now is wrong ratios from wrong corresponding sides.  To avoid errors, redraw the triangles so that corresponding sides are oriented the same way.
3. Arrows, people!    Marie Kondo your cluttered diagrams with arrows.  Help your readers follow your reasoning.
4. Line Up Substitutions.  Everything that’s equal is in the same column.
5. Practice your 3D Drawings and Cross Sections.  Get used to slicing up solids and drawing parallel lines to vanishing points.  It’s art!  It’s math!

5 Ways to Help Your AoPS Student

I teach math to homeschooled students, and I limit my practice to Art of Problem Solving curricula and preparing for math contests like MOEMS, MathCounts, and AMC.  

Today I want to talk about a certain type of student that I help more than any other:  The Online AoPS Student.

AoPS offers terrific online classes for students who love math.  They are text-based, with no video or audio, just a live moderated discussion board where teachers ask questions and ideal student responses are posted.  They offer weekly homework, reading assignments from their textbooks, and more online practice with their Alcumus problems. My own kids and I have taken these classes and enjoyed them.  

So why are parents coming to me to help their online AoPS students?  I’ll answer this question with my list of 5 Ways to Help your AoPS Student

(1)  Read the textbook before class.

This is a tall order for any student, especially those in elementary and middle school.  Many don’t learn this style of independent study until college. Now is a good time to learn this education hack with a built in incentive:  if you arrive at your AoPS class more prepared, then your answers are more likely to be models and selected to posted for the class to see. As any AoPS student will tell you, they live for the fame and glory of being posted!  

(2) ACTIVELY read the AoPS textbook.

Reading a challenging textbook in any field does not look like curling up with a cup of tea under a warm blanket.  Reading a challenging textbook means sitting up in an uncomfortable chair at a desk or large table, with a pen and notebook ready to go.  Write out the problems and the solutions, copying them line by line as if you were solving them yourself.  Understand each step before you continue. Work through all the practice exercises.  Get your money’s worth!  

(3)  Learn in small chunks every day

It’s better to learn in smaller chunks every day than to cram a whole chapter’s worth of material the day the homework is due.  Sleeping helps with educational digestion.  

(4)  Start homework problems early

Read and attempt a few problems.  When you reach a wall, don’t bang your head against it.  Walk away from your homework, sleep on it and try again the next day.  Maybe take some time to solve a few alcumus problems. Schedule time every day to just take another look at your homework.  The truly difficult problems in life are not solved in an hour but often require months or years of study. Get used to it.  

(5)  Consider ditching the online class and going old school

The AoPS textbooks have plenty of practice problems and plenty of challenge problems.  Work through the textbooks at home, every day, for a less than an hour per day. Model clear mathematical technique by writing out the problems and solutions as you read.  Make slow but steady progress on your own schedule. We took over 1 calendar year (including summers!) each on prealgebra and algebra, but it was worth the extra time to ensure complete mastery.  In contrast the AoPS classes proceed at a blistering pace.  

Online learning is tough which is why I’m called to help.  I spend much of my time reteaching students the principles already described in the text.  I also assign practice problems from the textbook that align with the online homework problems.   I model clear problem solving documentation (AKA showing your work) so they can see for themselves how helpful it is.  

Old algorithms and new

I’m taking a number theory class this summer, so my mind is full of residuals and mods.  In that mindset it’s easy to forget all the other math I’ve learned in my life.  Here’s an example.  I was asked to find a divisor between and of this expression:

Because , , and all have prime factors in common, it’s easy to find some primes that evenly divide this expression.  But that alone isn’t enough to find a single number between and .

I considered all the divisibility formulas I had learned in the previous lecture:  Fermat’s Little Theorem, Euler’s Theorem, and Wilson’s Theorem, but nothing seemed to work.

It turns out, solving the problem is made easier by seeing that the first terms in the expression:  can be expressed as a difference of cubes.  Twice actually!

A difference of cubes can be factored as:

Substituting, we find:

But now we can do it again with the first factor:

Setting equal to all those yucky square and cubes to the right, we now we have

.

And so we have:

And voila, we have found another factor, , that divides this expression.

Good math problems force the student to use not only skills they have just learned, but also draw on older skills and older theorems that have been sitting around unused in your brain.

This is another reason I think math contests are so valuable to our students.  Students continue to use all the algorithms they learned earlier, keeping it fresh in their minds.   Is it unfair to test students on factoring polynomials in a number theory class?  Strong students welcome the opportunity to be reminded of old almost-forgotten formulas because hopefully they’ll recognize those differences of cubes again in the future.