Marsbarn Designs...

And then you realize you need more math

So there I am living my geeky life, 
thinking that my math world is fairly healthy
and alive.. and then.. then I meet a new muse,
and I realize my math life is sad, and I need more
math, and more creativity, and I am in fact stunted.

I need more math, all the time.
What could make me say this?

Vi could make me say this.
Do you know Vi?  You should.
She is a goddess of creative math, and I am unworthy.

You can find her here.
She even has a mathematical food section here. 

I feel a nearly uncontrollable urge to start crocheting mathematical sequences or doodling.  Doodling would be good.  

Box template method of Algebraic equations

In working with mathematics, I love finding
alternate ways of solving a problem.  If something
doesn't make sense, there should be another way to work it out.

What makes me happy?  Discovering another method.

The girls who I tutor are just entering that new
world of Algebra. It's a land of mystery that either 
inspires adventure through problem solving, or
tears through confusion.  

You have an equation like:

     
 2x  +  4 = 4

        6

Now, the way I solve it, is to work on isolating
my variable, X.  I do this by working my way 
backward through the order of operations;
ie, I multiply to get rid of the divisor 6.

So, multiplying each side by 6, gives me:

2x + 4 = 24  

Then, I get subtract 4 from each side.

2x = 20

Finally, I divide both sides by 2, which gives me:

x = 10

This is the method I was taught, and it works for me,
but there it can be a frustrating method to teach.   

This is the new-to-me method the girls I work with
were shown in school:

You take your equation: 

     
 2x  +  4 = 4
      6

and you use a template to create a sort of map that follows the journey of x.
On the left hand side, you have X.  X is multiplied by 2, then has 4 added to it, and then is divide by 6.

Then, we write the inverse function of each operation along the bottom of the template.

Finally, we work from right to left, following the operations on the bottom of the template, writing the answer of each step in the box.  We are effectively backtracking through the equation.

What I like about this method is that it breaks down
the algebraic process into steps that are easy to follow.
There is less mystery as to where to start.  You start
with your variable, and follow the mathematical actions
that are done to it. For the beginner who is intimidated
by the idea of variables, and who struggles with order of
operations,  it is a confidence builder.  

I am waiting to see if this method can be expanded to 
more complicated equations, but I love how quickly
my students absorbed it.   It is a much better beginning
than I had. 

Pi Day

Cherry pie for Pi Day, March 14th.  (You know, 3.14)

So, I thought I'd get a post up in time for Pi Day, but
I am late.  Of course, Pi goes on forever, is eternal,
and irrational, so I think being a week late is okay.

Pi.  Pi is magical.  Honestly, it is.  It translates between
dimensions; distance, area, volume.  Pi does it all.
And yet, when I ask the 6th graders I know "What is Pi?"
They tell me, "3.14".  When pushed, they say, "It goes on forever."
I get shrugs when I ask "Where does it come from, what 
does it mean?"  "It's a constant." Is the best answer I get.
I would be lying if I said this didn't hurt my heart just a bit. 

So here is what I want them to know.
Pi is a wonder of the universe.  What does constant mean?
It means that all circles follow a rule.  One rule to govern
them all, and in the darkness bind them. 

Pi is a ratio.  It is the number of times the distance across
a circle (the diameter) can wrap around the outside
of the circle (the circumference).  You could find this 
measurement on your own, with a piece of string.  It's
a physical thing, a real touchable thing.  It is simple,
but profound, because it is true for every circle.

Pi is history.  It is practical and an object of obession.
Ancient Babylonians and Egyptians knew of Pi
through physical experiments (hello string).  
Archimedes (he who could move worlds) worked out
the first mathematical approach to Pi, which began
an endless obsession by mathematicans with the
irrational number.  Is a million numbers in Pi enough?
No, they keep expanding the decimal. 

Pi can catch the imagination, and make mathematics
come alive.  It makes so much possible with
just one piece of information: 3 dimensions of a
circle, with just the radius.  It can make a person fall
in love with problem solving.  

Is Pi Day just a gimmick?  A chance to eat pie?
I think it is what we need to inspire our children.
It is a chance to talk math with your child,
and make them think beyond the numbers, beyond "3.14" .

The brownie pan method

This visual method of multiplying fractions
was something that was shown to me last week
by the girls I tutor.  Like most people I learned
to multiply fractions by multiplying numerator
by numerator, and denominator by denominator.
That's pretty much the gold standard of fraction
multiplication. 

This method gives a visual sense of what it means
to multiply fractions.  For our example we'll use:
 2/5 x 1/3=

Draw a square.  Divide in into three vertical portions
and color in one of the thirds. This represents 1/3.  
For a real world styled word problem, think of this as
a brownie pan with only one third of the brownies
left in it.

Next, we represent the 2/5.  Divide the square
horizontally into five portions.  Using a different color,
fill in two of the fifths.  The two colors will overlap,
this is a good thing.

For the word problem, think some comes and eats
2/5ths of the remaining brownies.  How much of
the pan of brownies did they eat?

For our answer, count the total number of rectangles.
This number is our denominator.  So for our problem,
this number is 15.  The numerator is the number
of rectangles where our colors overlap. Here that 
would be 2.  So our answer is 2/15, or 2/15 of the
entire pan of brownies.

So, 2/5 x 1/3 = 2/15 

 

Math Group: the toothpick problem

We had our last Math Group meeting of the year 
on Sunday.  It's been a fun couple of months.

This week's problem used four toothpicks and had
two parts.  The first was to find out how many
different shapes that could be made with the four
toothpicks.  The rules were that in all of the shapes
had to be continuous, the joins between picks
had to be either 90 degrees or 180 degrees, and that
only two picks to touch at any join.  Also, the shapes
couldn't repeat.  To test their uniqueness, each new
shape was drawn on a piece of paper and rotated to
make sure that it didn't repeat.

The second part of the problem, was to see if the 
shapes could be arrainged in a continuous loop,
moving from shape to shape, by only moving one pick
at a time. 

It was a very playful, low stress problem, and was a
nice way to round up the year. 

Math Group

A couple of months back, I started a math
group.  I asked a couple of Sweet G's classmates
if they'd like to get together every other Sunday
morning to play with math, and several of them
said, "Yes".

Suprisingly, the only "No"s were from children
who are busy Sunday mornings.  Perhaps it was
the promise of snacks and funny voices that
enticed them.

We do have a lot of fun.  (And no, that's not
just my opinion.)   In between the breakfast 
treats and props (animal crackers and legos),
we work on problem solving techniques.  

My biggest challenge in our group meetings
is trying not to help too much.  With every
problem there is a point where the path
is unclear, and frustrations can mount, and 
there is a temptation to step in and point the
way.  That's the point where I need to step back
and give them a chance to move forward on their
own.  Like I said, it's a good time all around. 

The locker problem, or more math thoughts

Yes, another 6th grade math problem:

The locker problem goes like this:

A school has 1,000 lockers and 1,000 students.
The first student runs through the school and
opens all of the locker doors.

The second student runs through the school and
opens every second locker.

The third student runs through the school and 
"changes the state" of every third locker. (This
means she opens closed lockers, and closes open
ones.)

The fourth student runs through the school and
changes the state of every fourth locker.

And so on, until the 1,000th student goes through
the school and changes the state of the 1,000th locker.

How many locker doors are open?  Why?

What do you think?  Scared? Intrigued? Curious?

The book recommended taking a small sample
and charting what happens as the students move
through the school. Using a 25 block grid, I went
a step further and wrote down the number of the 
student on each square as they touched a given locker. 

If you do the experiment, you find that the 
first locker is open (only the first student ever
touches it), the second and third lockers are closed,
with two students having touched each.  The fourth
locker is open.  The fifth through 8th lockers are
closed.  The 9th locker is open.  This continues.
Analyzing the grid, I find that only lockers 16 and 25
are open.  All of the others are closed.

So.  Lockers 1, 4, 9, 16, and 25 are open.  They are
all perfect squares.  What makes a square number
special in this case?  It's the number of factors.
A non-square number has an even number of factors.
(For example, 12 has the factors of 1, 2, 3, 4, 6, and
12.)  A square number has an odd number of factors.
(25 has factors of 1, 5, 25).  

Not only can you determine how many lockers are open,
you can figure out how many students touched each
locker.  If you choose a locker, say number 100, you
can determine how many people touched it by counting 
its factors.  100 has factors of 1, 2, 4, 5, 10, 20, 25,
50, and 100, so 9 students touched that locker, and
the locker is open at the end of the experiment.  

I would love to say that I figured this problem out
in no time flat, but I didn't. I got stuck.
My first idea was to use prime factors, and that seemed
to work, a bit.  Until I realized it didn't work at all.
This is where having a complete model, one with 
not just open and closed door symbols, but the student
number that touched the door written on the chart,
really helped.  Still, even with my misstep, I really
enjoyed the time I spent charting and figuring out
this problem. So. Much. Fun.  

a new math obsession

Okay. I wrote a post all about a math problem.
Really.  The whole thing.  
You've been warned.  If you don't want to read a lengthy post about 6th grade math, you should go look in the archives. 

I have a new obsession.

Perhaps I missed this trick in middle school,
or (as I suspect) I was never shown it,
but I've found this pair of techniques to be 
fascinating and (dare I say it) elegant.

In my continuing exploration of middle
school math both to be a better tutor and
because I find it addictive (everyone has hobbies),
I've been playing with prime factorization.
We called them Factor Trees in my day.  You know,
these things:

My own ancient memories of them were that they 
were pointless seeming exercises in breaking down
numbers into their bits and pieces.  But in this here
modern world of math, they do things with them.

This first trick, I vaguely remember.
Start by rewriting the factors like this:

If you take all the prime factors both numbers have
in common  (two 2's and a 3), and multiply them out,
you have 12, which is the largest number that can
divide into both numbers evenly, the Greatest 
Common Factor.

Okay, the second trick is for finding the Least Common
Multiple. It's the number both numbers (180 and 48)
can divide into. I remember writing long lists of 
multiples until I would stumble upon a number they
had in common, or just multiplying both numbers
together and dealing with HUGE numbers. 

But this is a slick, pretty, and (dare I say it) easy
solution. Look at the prime factors again,

This time you take all of the factors you 
need to make both numbers.  Think of this 
as a building a kit to build two different
lego shapes.  You can share blocks, but you 
have to have all the right shapes.

So, we need four 2's, two 3's, and a 5 to 
make both numbers.  If you multiply
these numbers, you get 720, 
which is the smallest number both 180 
and 48 divide into evenly. 

Why you ask?  (Okay, why I ask.)
The four 2's, two 3's, and a 5,  
can be combined like this:
(2 x 2 x 2 x 2 x 3) x 3 x 5 = 48 x 15

or like this:

(2 x 2 x 3 x 3 x 5) x 2 x 2= 180 x 4.

 I don't know why I find this so cool, but I do.  

calculating new forms of insomnia

You know those nights when you cannot fall asleep?  
Your brain won't shut off, it just hums along with
lists and To Do's and budgets and worries.

Well, last week, I had a night when my brain would
not quiet, but instead of Big E's work or the To Do
list keeping me up, I kept thinking about....
math.  Yes, math.  

I thought, "If you bought a dozen items that each
cost 99 cents, how many different ways could you
calculate the total cost?"

For about ninety minutes, everytime I would come
close to drifting off, I'd think of an alternate 
method.

First I thought of these two: (You can click on the calculations to make them bigger.)

Then, I moved on to these two:

Seriously. I'd almost drift off, and then this...

Now the idea how to carry a "ten" became particularly
fascinating as the evening moved into morning.*(If you
have any interest in the rambling math thoughts of the
sleep deprived, see the footnote.) 

But the brain wasn't done... no, five methods weren't 
enough.  I thought, if I were to do this in real life,
I'd do this:

So, sometime after 12:30 AM I drifted off past the
math delusions into a very short night of sleep.

Total crazy pants.  Absolute, complete craziness.

 

 

 *Do you carry "ten" into the "tens" column, or do you carry
"one" into the "hundreds" column?  Of course you have to
carry twice, when you count up the "90"'s, you again get
that choice to carry "ten" into the hundreds or "one" into
the  "thousands".  Which leads to the thought of combining
the two carries into "eleven" carried into the "hundred"'s. 
Do these all count as seperate methods?  Who freakin'
knows!   

Using that other bit of my brain

I've been moonlighting on the crafty-arty scene.

You see I've been battling a demon of my elementary
years. (No, not that horrible climbing rope in gym;
that is completely beyond me. The rope won.)
I've been tutoring a 5th grader in math. 
And, I've thoroughly enjoyed myself.   

Shocking, yes?
I must admit to a bit of shock myself.

Here's the thing, I struggled with math throughout 
grade school. I am dyslexic and spent so much of my
time trying to decode what was being asked
and trying to keep my numbers in the correct order,
that I was constantly behind.  Addition and multiplication
were okay (if I could transcribe my answer properly),
but subtraction and division?  I can remember spending
ages just trying to work out problems like this:

                                8 ÷ 4

I knew the answer would be either 2 or 1/2, but 
trying to figure out the direction the problem was
meant to go was beyond me.   

When I got to college, I started over.
Honestly, I started in the lowest level math class they
offered and worked my way up to upper division math.
Suddenly, things clicked.  Math became beautiful.
(Don't get me wrong, it was still hard, and it was
never my best subject, but I could understand.) 

 So, tutoring?  Math?  I know teachers of mine who 
would be shocked.  But, here's the thing, I know how
it feels to not understand.   I can't be shocked by sudden
mental lapses or getting numbers confused; I have done
it all.  I still have moments where I can't read a number
straight, but math concepts?  Those I can do.
And I love, LOVE, to find alternate ways to think.
Going at an idea sideways is something I enjoy in art,
and problem solving, and teaching.  Math is a perfect
foil for this. 

Will I do more tutoring?  I don't know.  It just feels
good to stretch that analytical part of my brain,
and poke a childhood phobia in the eye.