Saturday, February 24, 2018

Skills in Action

A reader challenged me to explain how to do competitive math.   I'm excited about these problems because they demonstrate a fundamental skill set that is developed during learning to read at age 4.  It's very similar to the skills that cognitive skill sets like the COGAT teaches.  It's 100% learnable.

What I like about this problem set is that a parent can work through these and have the exact same experience that your child goes through.  I'm hoping that parents who take the time to work through this material will have better training when your child shows you a figure matrix that is baffling.  It's also a good opportunity for the parent who discovers this website 6 or 7 years too late so that I can show that it's never too late.

I've gotten numerous questions about what follows Test Prep Math 3.   I like competitive math as a warm up for SAT test prep.  We dabble with pre-algebra, but usually only in a Algebra 1 setting.  Sounds hard, but this is the skill set.

Here we go.  Picture a competitive math worksheet with 40 problems on it, that has a 45 minute time limit.  I suppose if we were serious about competition, we'd train for learned strategies to address the time limit, but we're not serious about competition, just doing a bit of daily math.  I think 5 problems is asking a lot of an 11 year old.

Question 1:  F - T - L - T - ? - ?   Find the last 2 letters in this series.
  1. I have no clue how to do this.  Anyone you has seen this question type probably doesn't have any clue because it has unlimited subjects.  But I have the most important skill of all, which is the proper way to be Baffled, which is to not care that I'm clueless.
  2. I think for a minute about adult IQ tests.  Friday, Tuesday, Something That Begins with L, Thursday.  Fail.
  3. F = 6, T = 20, a difference of 14.   L = 12, T = 20, - 8 + 8?  Fail.  Skill 2 - don't care how many incorrect answers I get.
  4. I stop and think about the question a bit.  Kids only know arithmetic, language, geometry, and a tiny bit of algebra.  Pre-Algebra is fair game.  In the real world, I should have used Skill #3 which is to spend more time thinking about the question and less time getting incorrect answers, but in competitive math with no time limit, lots of learning can happen in dead ends.
  5. Going the geometry route, all the letters have a single vertical line.  F has 2 vertical lines and T has 1.  That's 3.  L has 1 vertical line and T has 1.  That's 2.  I forgot to look at the answer set.  My skills are rusty because the answer set is part of the question.
  6. The answer set is:
    1. L - T
    2. L - B ( I think this B has no vertical lines.)
    3. L - M
    4. T - P
  7. If B is the answer, using counting horizontal lines in the series, we get 2 - 1 - 1 - 1 - 0, but if we take pairs of vertical lines, we get 3 - 2 -1.   B is the answer, I accidentally stumbled on it, and I have no clue why I am correct.  But it was the best of a bunch of confusing bad answers.
This problem took me about 15 minutes.  It's very similar to the type of work I do on a daily basis.  I wonder if competitive math tests are structured so that super duper problem solving kids prioritize the questions and their time before answering and just skip this one.  Probably.  

When you work with your child and do a problem that is really hard for their age and skill set, just like the one above, here's the benefit you both gain:
  1. You get used to working with baffling things and don't get put off.
  2. You make a lot of mistakes and don't get put off.  In fact, in my failed attempts (attempts not include above), I learned a lot of interesting things and picked up a few mini-skills on the way to dead ends.
  3. The solver is forced to think creatively and view the problem from different angels.  It will take a lot more problems to learn creativity, but since I am making a habit of baffled and mistakes as skills by force feeding my child these problems over and over again, we'll get their eventually.
  4. I never looked at the clock or the solutions.  This problem is kind of tricky and fun.  The solution will end the learning process and reinforce the Rule #1 that it's not about learning or getting better at something, it's about being right or wrong.  Rule #1 will destroy your child's ability to learn.  Rule #1 is an anti-skill.
When I work with kids, a team will really help, and I'm the only one available for the other team member, so in practice I ask a lot of questions (as needed) and make suggestions for the next attempt (as needed).  I'm always baffled.  In practice, I'm suggesting skills and approaches from my toolset of exactly 5 approaches to any math problems. 

Why is it that when your child comes to you and asks what 'dispersed' means, you're more than happy to tell him, in fact you're so happy your bright little child has an interested in vocabulary and is not skipping over unknown words when reading, but when your child gets a math problem wrong, you're disappointed?  What a horrible destructive way to teach children to hate math.  Adding a time limit makes it even worse, because then a teacher can mark of a series of unanswered questions.  This is why schools can completely eliminate tests through Junior year in high school and produce kids who blow away college entrance exams.

OK, let's see what we get out of more baffling problems.

What is the remainder when the 15-digit number 444444444444444 is divided by 9?
  1. Are you kidding - this is too big to fit in the calculator.  Curse you competitive math test author.  The answer pick list is irrelevant.  Again, I have no clue.
  2. Too hard of a problem.  So I fall back to how we tackled any math - starting at age 4, when it's too hard.  We start with the easiest version of the problem and work our way back to the harder problem:
    1. 4/9 ~ r 4
    2. 44/9 ~ r 8
    3. 444/9 ~ r 3
    4. 4444/9 ~ r 7  this is good practice for division but a fail in solving the problem.
  3. Then I remembered that when I teach division, I always make the student turn 36 ÷ 9 into 3*3*2*2/3*3.  Now were trying to turn this problem into a more solvable, easier version of this problem.  Here's goes:
    1. 4*111111111111111/9 = ?  Still hard.  Fortunately, I can look back on the first fail and continue.
    2. 1/9 ~ r 1
    3. 11/9 ~ r 2
    4. 111/9 ~ r 3.  Get it?  Light bulb. 
    5. Continuing, I get to r 0 at 1111111111 which puts 111111111111111 (15 digits) at r 5.
  4. Unfortunately, I'm stuck having to multiply the whole thing by the remainder.  This stinks, I stink, and your child stinks, so we're going to have to take baby steps.
    1. Since 1/9 = 0 + 1/9, 4*(0 + 1/9) = (0 + 4/9) ~ r 4, which is what I got in the first fail.  Notice I'm checking the answer, which is skill #4 at the base of the cognitive skills pyramid.  I suppose this requires some pre-algebra.
    2. 11/9 = (1 + 2/9), so 4*(1 + 2/9) ~ r 8, again, just like above.
    3. 111/9 = (12 + 3/9) but 4(12 + 3/9) is going to give us 48 + 12/9, slightly confusing, and I have to go read the question yet again.  Oh yea, we're dividing by 9, and trying to find the remainder, so I can write 48 + 1 + 3/9 ~ r 3 just like expected.
    4. At some point, the lightbulb goes off, and I can just jump to 15 ones's/9 = (something big + 5/9), and I multiply by 4 and get 4*something big + 20/9 ~ r 2, which is not even on the answer list.  The choices are 4, 5, 6, and 7.  
    5. So starting over, which I'm totally used to because we do it all the time, I note that the 9 digit number 111,111,111/9 = 12,345,679 r 0, duh, should have thought this though.  This makes 111,111,111,111,111/9 = something big 6/9 (since 15 digits is 6 more than 9 digits), and 4*(something big + 6/9) = 4*big + 24/9 = 4*big + 2 + 6/9, giving me the correct answer of 6.
We've got 3 big solutions approaches that we start using when the child is about 3 years old.  

At some point, your child is looking at * * * * * * of something and you ask her to count.  She answers 12 or 5 or gives up, so you start small, like *, then * *, then * * *.  I teach addition, fractions, and multiplication this way.  It works in graduate school and it was by experimenting that I found it works really well at the youngest ages.  It works on pre-algebra.  It works on all forms of high school math.  It's required for competitive math.  Math books do this from chapter 1 through chapter 15, but we do it in 5 minute increments and don't really need a math book.

Next, when a problem is too hard, turn it into an easier problem.  This is the foundation of algebra.  You might as well start now.

Finally, notice that there are 3 steps to this problem.  If you've seen TPM, you know why I think 3 is so important.  It builds working memory.  For the age group for the problem above, we're probably beyond working memory, and if not, doing these problems will bring it back.  But the working part in 3 steps is where the little brain turns itself into a big brain by defining relationships and patterns and working abstractions into algorithms from one part of the problem to the second to the third.  You see all three in the solution above.  A genius can do it in one step only under one condition:  the genius worked through enough of these problems to get really good at devising and applying algorithms. Don't be fooled into thinking it's genetic.  The rest of us are happy doing the 3 steps one step at a time.  One step at a time is good for 99%.

Moving on, how about this problem.  What is the value of 1 - 2 + 3 - 4 + 5 - 6 + ... + 81 - 82?

This problem not only demonstrates the value of spending way more time exploring the question than trying to answer the question, it also demonstrates the value of what I call "Seeing".  I learned it from the COGAT.  It involves looking at the problem from different perspectives.  

I checked to see that there were an even number of elements to this equation, all equaling negative one when paired, and came up with -41.  Eight minutes of thinking about the equation and 4 seconds deriving the answer.  With 40 questions and a 45 minute time limit, I would have come in last on the competitive math exam.  Can you picture me sitting with a bunch of 6th and 7th graders? 

This next question is my favorite and a really great exercise on it's own to teach exponents.  I love this question.  This differs in an important way from the math I would give a younger child but is identical in nature to the non-verbal section in TPM.  It involves doing a lot of work, organizing and thinking about it, and then answering.  

If a and b can take on the values in [0,9] (meaning that they can each be 0, 1, 2, ... 9), then the expression ab can take on how many different odd number values?
  1. To start, I just created a grid with 0-9 on the rows and 0-9 in the columns and started calculating the expression based on inputs.   In a competitive math situation, this is a waste of time and requires thinking, but with most kids (and 9 year olds), I make them use the brute force approach because they usually have never seen aoutside of 42.   I've got a whole exponent crash course (including negative and fraction exponents), but this seems to be a good starter exercise.  
  2. The rows are a and the columns are b.  I didn't calculate the *'s but I could have.  
    *0123456789
    0?000000000
    11111111111
    21248163264***
    313981******
    4141664******
    51525125******
    61636*******
    71749*******
    81864*******
    91981*******
  3. This seems to be a fail.  Too hard.  I did notice that only one zero in the top row and one from row 2 and column 2 are going to be included.  What is zero raised to zero?  It's either one, zero, or undefined, but if you read the question again (and you should because it's a skill), it doesn't matter to the answer.
  4. After rereading the question yet again, I noticed that I only have to deal with ODD numbers.  With the exception of '1', the rows with 5, 7, and 9 qualify, and since 3*3*3*3 = 9*9, the row of threes where the exponent is odd also qualifies but not when the exponent is even.  And we can add 1 only once and ignore zero.  And that gives the correct answer  of 27 (the whole row of 5,7,9) + 1 (from the one) + 5 (from 3 row where it doesn't repeat a value from the 9 row) =  33.
  5. It's possible to jump to step 4 as a competitive math coach, but not a regular bright kid doing competitive math coach.
I'm guessing the question needs about 5 readings before this work can begin.  I've watched little mathematicians create charts to answer questions and it's very gratifying.

Finally, the last question is this.  If x and y are integers and 360x = y3, what is the minimum possible value for x + y?  At this point, we left all kids under 4th grade behind and we're just looking at algebra.  Or are we?  Yes, I'm running out of steam and have already covered all the really great problem solving techniques. 

  1. After 30 minutes with the question, I decided that x is just a function of y, so forget about x.  Just find the smallest possible value of y.  Or do algebra.  It's late, I've exceeded the maximum good thinking time of a grade school child of 25 minutes, and the Olympics are on.
  2. But I don't like 360, so I wrote 2*2*3*3*2*5x = y3.   Then I rewrote it to be 2*2*2*3*3*5x = y3.  You can see that if y is an integer, x has to be 3*5*5, making y = 2*3*5.  So x = 75, y = 30, and the answer is 105.  
I got the entire solution correct by following these steps:

  1. I had no clue what to do.
  2. I went off in the wrong direction by trying to use algebra, which I can, but doesn't solve the problem for a kid who doesn't know algebra.  Fail.
  3. I tried again.
  4. I spent more time looking at the question and eventually started to rearrange it in the hopes of finding an easier problem.  (I.e., I used one of the big five 5 math problem solving techniques.)
  5. I looked at it, specifically looking at the root primes against the exponent on the other side of the equation.  I used my power of seeing things differently.
  6. The answer emerged with no effort.
This is why studying for the COGAT is so critically important.  It's the easiest way to get the skills.  If you missed this opportunity, there are other opportunities including competitive math.  It seems harder and more complicated because your child is older and the math more obscure, but it's about the same.   If you did this when your child was younger, you would have blocked out all of the tears and frustration by now and just remember how it all worked out.  Same with bed wetting in the middle of the night.  Remember that?  Of course not.

Is there anything different between a child who does this problem successfully and one who gives up?  Not mathematically.  It's all in these base skills which are 100% learnable and needed for high school math.   If you want a strong competitor in a math contest, you'll need interest and a lot more practice, but if you just want a five on the BC Calculus without having to nag your child or hire a tutor, do a few problems and focus on the skills.

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