AoPS Challenger problems and problem solving in general

[Kendall]

I’m wanting to do a better job with teaching problem solving, so I am having two of my daughters  do the Challenger problems from the Art of Problem Solving Introductory Algebra. MAth background-one in Alg 1 using Foerster and so far 7 chapters of AoPS Intro to Alg.  The other daughter is in Foerster Alg 2. I can't imagine doing these with a child just going through the chapter! 

 

Today they both worked on the 3^rd challenger problem in Chapter 2. There weren’t any in Chapter 1.

 

The problem was something like this:

 

5(x-4)- k(2x+1)

 

Find k so that the expression has the same value for any x value.

 

I put the girls in different rooms and went back and forth giving suggestions.  I wanted them to wrestle with it awhile, but because of frustration levels did suggest at first that they just try picking a value for k and find the value of the expression for different values of x.

I gave some other suggestions and commented on what they had tried when it was way off base.

 

I finally had them each distribute and then associate the x terms, factor the x out

X(5-2k) -20-k and then had to talk them through what value of k would make x have no effect on the outcome.

 

My questions are both general and specific.

 

How do you handle these problems when they either don’t seem to have any idea of where to start or get stuck? I think if they try awhile and get stuck I will have them try again another day, but when they don’t even know how to start and are frustrated do you give ideas or just work through it with them?'

 

What specifically would you have said to guide them to finding the answer to this one?

 

I gave a pep talk to each girl about the value of difficulty, yada-yada and told them that over time they would be more and more able to approach these problems with ideas. I encouraged them by telling them that I didn’t immediately see what to do when I first looked at the problem and that I had to really think about what it would mean. I have a degree in math (which I feel I should give back after the slowness with which I have figured out the answer to  some of the challenger problems!)

 

 

Thanks in advance,

Kendall 

 

 

 

 

 

[London]

Let me start by saying I heart AOPS because my kid hearts it. We have done AOPS since Pre-Algebra, and will be doing Calculus in the Fall. And we do not teach for AOPS either, so no conflicts here to report :)

 

It.is.HARD.  It is TIME CONSUMING. and It is NOT for everyone. But it is also amazing.

 

I love the problem solving skills it teaches and reinforces. Google Russian Math school's Ted video. I agree with the mentality that good math curriculum force you to learn how to approach an unknown instead of feeling like you are a failure and give in from the get go.

 

What worked for us is after a period of struggling with a problem (set  time limit, ours varied per subject), I or hubby get involved. We verbally and in writing walk him through our mental process for solving the problem. There are some questions that we can't answer honestly, and those we ask him to ask the teacher (AOPS online courses). But most of the time, one or both of us can walk him through it. As he has progressed through AOPS, his logic and critical skills have greatly improved. That has been the best benefit of AOPS for us.

 

Don't feel bad if you can't answer them, even if you are a math major. You could have a PhD and still struggle with some of the problems, IMHO.

[Monica_in_Switzerland]

What we are finding with AOPs is that often times the math is straight-forward, but only once the words are understood.

 

For the problem given, I probably would have asked leading questions to get them to use the word variable to describe x and constant to describe k. Then ask if they could solve for k as a first step, then go from there.

 

Identify what you know, clarify the words, identify what you don’t know, try something -anything- and see if it sparks an idea.

 

We’re only in the PA book, but I’ve already had several problems where I’ve had to tell DS, « I don’t know if this is the right way or not, but I’m just going to write some stuff down and see what happens... ». Being willing to go down the wrong path then come back with another approach is probably going to be the most important thing we learn with AOPS...

 

ETA. Oops, misread your original problem as containing an equal sign, so ignore anything I said specific to that particular problem. 🤣

Edited March 4, 2018 by Monica_in_Switzerland

[Julie of KY]

I have one child that "sees" math and can do all the challeging problems first time through the chapter. Others can "never" do most of them or it's not worth the struggle to get to that point. Many of them are very tough as you are   discovering. Also, while some of them are more of solving a problem, others are more theoretical problems that require understanding and writing out a solution that is a very different skill.

[SanDiegoMom]

I have a son who gets math, and these problems are perfect for him. He can usually do at least 3/4's of the challenge problems, and struggles through the rest sometimes getting them, sometimes not. 

I have a daughter who is decent at math, very smart over all, but doesn't see math like her brother.  We are doing some of the Aops Pre-A together, but none of the challenge problems. When I push her past her too far past comfort zone it usually is detrimental -- loss of confidence, frustration, and an aversion to that subject. The exercises in the book, even the ones that are not starred, often are still taken from math competitions, so I feel for this student those are enough challenge. 

 

When I do help her problem solve, I usually go over any of the tricks we have learned and ask her which one would be useful.  So for the exponent chapter of Pre-A I have a list of all the exponent rules next to her and any other problem solving technique that might be useful and ask her if any of these would work.  It helps her to see them listed out.  

 

We will probably only use AOPS Pre-A and then go to something else for Algebra.  She likes the videos and alcumus so that's why we are doing it now. 

[Gil]

How do you handle these problems when they either don’t seem to have any idea of where to start or get stuck?

I beat my kids over the head with the idea that the first step is to always, always, always take the time to gain an understanding of the problem in "Plain English".

 

For my kids, getting started can be the hardest part. So, for them this usually involves carefully reading and re-reading the problem. Dissecting the math-specific language and identifying what exactly you are looking for, then once you know what you are looking for go-back and carefully read and re-read the problem to ID what info/clues you are given to find it.

 

Consider that if this problem is at the end of Chapter 6, then chapters 1-6, not just chapter 6 have shown you EVERYTHING that you'd need to get this solution, so don't panic, think back about what you know so far.  In math we often have to use what we know, to figure out what we don't know.

 

--How does what you're given, relate to what you are looking for directly or indirectly?

--What do you know that could bridge between some piece of given information and some part that can be used to get the final answer?

In the problem you posed, I'd make them think carefully about what they're given. Which parts are variable? Which parts are constant, etc...

[Kendall]

Thank you so much everyone! You've given me great ideas and encouragement. I'm excited about using these problems to grow their math abilities and grow their ability to embrace difficulty. 

 

 

[daijobu]

 

 

How do you handle these problems when they either don’t seem to have any idea of where to start or get stuck? I think if they try awhile and get stuck I will have them try again another day, but when they don’t even know how to start and are frustrated do you give ideas or just work through it with them?'

 

What specifically would you have said to guide them to finding the answer to this one?

 

I gave a pep talk to each girl about the value of difficulty, yada-yada and told them that over time they would be more and more able to approach these problems with ideas. I encouraged them by telling them that I didn’t immediately see what to do when I first looked at the problem and that I had to really think about what it would mean. I have a degree in math (which I feel I should give back after the slowness with which I have figured out the answer to  some of the challenger problems!)

 

While it's nice for students to be able to think for days about how to solve a problem, I don't think it's appropriate for every student.   I think it's more efficient just to show them how if they give up, with a little Socratic questioning along the way.   Do this before frustration sets in.  

 

I do like how you model James Tanton's approach to solving a problem:  "Read the question, have an emotional reaction to it, take a deep breath, and reread the question. Have another emotional reaction."

 

My emotional reaction usually consists of "Oh my gosh.  I have NO IDEA how to solve this.  I don't even know what this means.  Okay, let's give this a try <or> Let's try to draw a picture <or> Let's reread the problem and try to figure this out."  

 

I wouldn't give them the "vitamins are good for you" lecture; I think they will only feel worse about themselves, IMO.  Just assign the problems and if they can't solve it, don't make a big deal.  Show them how it's done, and eventually they'll get the hang of it.  

 

I have a couple of freshman girls with no experience in competition math, so we started with the AMC 8.  I assign one exam a week for homework.  They can spend as much time as they like on the problems, whether it's 5 minutes and give up or days.  (I think it's closer to the former.)  Then we meet once a week, and I show them how to solve the problems they didn't get on their own.  No lectures about the "value of difficulty" just, "Here's how we solve it" with a bit of Socrates thrown in.  They've quickly gotten the hang of it, and I've leveled them up to AMC 10's now.  

[daijobu]

Sometimes it's helpful to state what you see in the problem that will trigger an approach at the solution.  

 

I remember Richard Rusczyk said in a video, almost as an aside, "The bisected angle here makes us think of the angle bisector theorem..."  Now whenever I see a bisected angle I try using that theorem, just to see where it gets me.  I try to articulate out loud those little clues.  My favorite is whenever you see right triangles inside of one another, check to make sure they aren't sharing an angle and are similar.  Or, if you are looking for integer solutions and you have the product of 2 variables, consider Simon's Favorite Factoring Trick.  Or, whenever I see a line tangent to a circle, I draw a radius to the point of tangency because it will form a right angle.  

 

It's like the metaphor of assembling your toolbox, and then deciding which tool to use in what situation.  "Whenever I see a nail sticking out, I reach for my hammer."  

[Kendall]

daijobu, Thank you so much for adding your thoughts. I'd forgotten about the AMC papers. This thread has given me more tools in my teaching/guiding toolbox! I'm excited! Thanks for mentioning explicitly telling them not only what you would try, but what the problem made you think of.