Download Chapter 1

Math for Elementary Teachers
Expectaions!
What should your portfolio look like?
How much should it include?
What will we be learning?
How will what we learn apply to your life as a
teacher?
Your portfolio
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Your portfolio should be finished work.
We will be learning skills, concepts, and
strategies, each of which will be demonstrated
in your portfolio.
There should be several entries per “Lesson” or
chapter.
Each entry should be legible and tell the story of
the problem.
What should an entry include?
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Who knows about WASL expectations in Math?
When I took math, math was just about the skill:
we learned rules, concepts, strategies, and so
on, BUT we demonstrated those by just doing
the skill on homework and exams.
What about now?
Expectations for student work
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Now in school we emphasize multiple
modalities: students are expected to
demonstrate the skill, to draw a picture
demonstrating what is going on, and then to tell
the “story” in words.
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Your portfolio entry should do the same.
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Your portfolio entry should be NEAT!
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Your portfolio entry should not assume the
reader knows anything!
Class time
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We will spend a significant part of classtime
actually doing problems.
Some of those problems will be from the
“assignment”.
Some of those problems will be from examples
in the chapter.
We will model what ta portfolio entry should
look like in class.
What will you learn?
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You be the judge: how does a portfolio entry
add to your understanding.
Which brings us to: what will you learn? What
will you understand?
Learning/Understanding
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To teach math you have to do math.
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To do math it helps if you like it.
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So, one of the first things you should learn is
how to like math. Why it's fun.
Learning/Understanding
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To teach math you have to do math.
To do math, you have to understand a certain
body of mathematical discoveries.
Math has a long history of accomplishments
that we have inherited. You don't need to know
that history, but you need to know the
accomplishments.
Such as ...
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Sets and how they are used and why they are
important
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Fractions
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Decimals, ratios, proportions, and percent
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Integers
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Rational numbers
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Measurement
What does that includes
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The ideas behind each concept
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How we use them
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What skills are typically associated with those
concepts
How do we explain them?
What we probably won't cover
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Ch 4 and 5: whole numbers and number theory
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Ch 10 and 11: statistics and probability.
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Ch 12, 14, 15, 16: geomety
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This is largely a time problem. If we turn out to
have enough time because we progress quickly,
we'll add (in order): ch 4, 5, 12,14.
How does this relate to teaching?
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This class does not teach you how to teach
math.
This class teaches you (ideally) to understand
the math you will be teaching and to like it!
You may be motivated by your desire to teach,
but I can't emphasize this enough: you have to
develop an appreciation of math in order to be
effective at teaching it.
Starting on Chapter 1
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Chapter 1 is not on my syllabus.
It's purpose is to introduce some problem
solving strategies and give you some exposure
to how we use them.
Basically … it's about solving puzzles.
Polya's four steps
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Understand the problem
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Devise a plan
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Carry out the plan
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Look back (reflect)
Understanding the problem
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Although you may see many problems written
as mathematical equations of one form or
another that you need to manipulate (!)
This is not how most problems really occur.
Most problems are presented in words (ie word
problems): understanding the problem means
forming a mental model that can be translated
into an equation of one form or another.
Making a mental model
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This is often the
hardest part.
So there are a
number of strategies
Some of the first we
will consider:
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Guess and test
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Draw a picture
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Use a variable
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Look for a pattern
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Make a list
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Solve a simpler
problem
Relation of understanding to plan
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The book puts understanding the problem
separate from the strategy.
Often understanding the problem and devising
a plan (ie using a strategy) are not separate.
Some things about understanding the problem
are about English (or whatever human
language you are using).
But we bring the strategy to our understanding.
Once you understand
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Once you understand the problem and have
settled on a strategy,
Try it: apply your understanding and strategy.
If it seems like you are getting no where, you
can try to back up and consider the problem
again.
But don't get frustrated!
Check your work!
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Look back.
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Figure out if your answer is correct.
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Does it answer the problem statement
(question)?
Could you have done something different?
Process:
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Understand the original problem
Using your strategy create a mathematical
version of the problem
Solve the mathematical version
Translate your results back into English to
answer the question
So … back to the portfolio
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Your portfolio entry for a problem:
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State the problem in English
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Write down your understanding of the problem
leaving out any non-essential pieces
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Write down your strategy and why you are
choosing it
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Apply your strategy
Your strategy
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If your strategy succeeds, express your solution
as the answer to the problem
If your strategy fails, look back at the problem
and start again.
Examples
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This is a kid's game that you may have seen
Start with a triangle with some number of holes
and a collection of numbers.
Your job is to place those numbers in the holes
in such a way that each side adds up to a
predefined sum.
(E-manipulatives!)
Let's go over the example in the
book
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A triangle with 6 holes
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The numbers: 1,2,3,4,5,6
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Place them so each side of the triangle adds up
to 12.
Guess and check
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Guess and check just requires you to be orderly
and keep track of your guesses
Works fine so long as the possible guesses are
relatively small!
In this case how many possible guesses do you
think there are? 6*5*4*3*2*1= 720 (How did I
know that?)
So ...
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Although 720 is not a lot for a computer, it's a
lot for a person, even if we are clever about
keeping track or being systematic.
So let's try an approach that includes something
about our understanding of numbers.
Let's start with the largest number: 6. To make
12 we need 2 more digits that sum to 6. That
would be 5 and 1 OR 4 and 2.
Continued...
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Suppose we chose 6 at a corner and then for
the two sides we'll try 4,2 and 1,5 (that's our
guess)
Now we'll check: the base has a 2 and a 5. We
only have a 3 left, so that won't work.
Let's rearrange the sides: 2,4 and 1,5. Now our
base has 4+5 = 9. And we have a 3 left. So
that's our solution.
Notice ...
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I was a little luck in my choice of my first guess,
but using the properties of the numbers to guide
my guesses is a good strategy.
Let's try another on (go to e-manipulatives)
Another g&c example:
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Send + more = money
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Sun + fun = swim (digits 0,1,2,3,6,7,9)
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We'll start by noting that the s must be a 1,
since it's a carry out. That means that f must be
8 or 9. Since 8 is not one of our choices, f must
be 9.
Now, n+n = m, possibly with a carry. It can't be
0 (why?) It can't be 1 or 9 (used already).
Continued ...
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That leaves 2,3,6,7 to consider. Now it can't be
2 (why?) or 7 (why?)
So now we're left with 3 or 6.
So let's try n=3 first. Then m=6. But u can't be 0
or 2 or 7. So the test fails.
Try n=6. Then m = 2. u+u+1 = I or u+u+1 =
10+i. Our choices for u are from 0,7,3. If u is 0,
then I is 1, which is already used.
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If u is 3, then i is 7. And that leaves 0 for w.
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Success!
Another number puzzle
Send + more = money
Let E be 5, and select your digits from
0,1,2,3,4,5,6,7,8,9
There will be 2 unused digits.
This is a 4 digit AND I haven’t limited the number
of digits to just what is needed, so this is a
harder problem than the sun+fun problem
Draw a picture
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Pizza pie!
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Draw a pie with 1 line: how many pieces?
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Draw a pie with 2 lines: how many pieces? Do
you have any other choices of line placement?
Keep going.
Now write the explanation
What did you conclude from the sequence of
pictures that you drew?
Note that you really used 2 strategies:
you made a list
each entry in the list was a picture
Use a variable
Problem p 11: what is the greatest number that
evenly divides the sum of any three consecutive
whole numbers?
The problem doesn’t specify any particular
number, so we’ll use a variable to represent a
number, and since we are speaking of three
consecutive numbers, they can be represented
as x, x+1, x+2.
Now use algebra (simple!)
The problem speaks of the sum: x + (x+1) + (x+2).
Now we can reorder: x + x + x + 1 + 2
And combine: 3x + 3
And factor: 3(x+1)
What is the greatest number that evenly divides?
How do you know this is the greatest?
Use a variable
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Find the sum of the first 10, 100 and 500
counting numbers
Again! Make a list
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1+2+3+4+5+6+7+8+9+10 = 55
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Let’s generalize:
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1 + 2 + … + (n-1) + n
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Use a trick! What if we add these twice:
Continued
1+ 2 + 3 + … + (n-1) + n +
n + (n-1) + (n-2) + … + 2 + 1
This can be written (reordered):
(1+n)+(2+(n-1))+(3 + (n-2))+ …+ ((n-1) + 2) + n+1
Each on of these terms is the same. What are they?
How many of these terms are there?
What’s left to do?
(divide!) Why?
So this strategy used a variable, a list, and some
algebraic reasoning. I called that reasoning a
trick because there’s no obvious reason that
says you should try it. But that kind of reordering
is a common way of exploiting a pattern.
Could we use it on other similar
problems?
Lets look at adding the odd numbers:
1+3+5+…
We’re going to do this three ways.
one by letting n be an odd number (add the odd
numbers up to n
one by letting n be the nth odd number (add the
first n odd numbers)
one by looking at the pattern directly.
Another picture with algebra
Suppose we have a square with 5 dots on a side.
How many dots are there total:
5 + 5 + 5 + 5 = 20
But we counted the corners twice.
So 5+5+5+5 -4 = 16
What about n dots on a side?
N+n+n+n – 4 = 4n – 4 = 4(n-1)
There are other patterns you might see:
2 dots on a side: 4
3 dots on a side: 8
4 dots on a side: 12
5 dots on a side?
Weird probolem!
Find the largest eight-digit number made up of the
digits 1,1,2,2,3,3,4,4
The 4 is separated by 4 spaces, the 3 by 3, the 2
by 2, and the 1’s by 1 space.
How would you start?
(problem on p 18), 1.1.1
Magic square problem
Describe how you would solve a 3X3 magic
square with the digits 3, 4,5,6,7,8,9,10,11
Do it!
Look for a pattern
Probably the most useful!
We already used it a number of times
The find the number of downward paths problem
on p 21. You start at A. There is 1 path from A to
A (since you are already there).
The next “row” has two spots. You can get to each
one of those spots in one way. SO each of those
spots only has one path.
Continued
The next row has 3 spots. Can The leftmost spot
can only be reached by 1 path, and the same for
the rightmost spot. But the middle spot can be
reached by either of the two spots on the
previous level. So it has two paths.
This pattern is called Pascal’s triangle! Comes up
in surprising places.
More patterns
Find the ones digit in 33^99
3^0 = 1
3^1 = 3
3^2 = 9
3^3 = 27
3^4 = 81
3^5 = 243 etc
Continued
So … now we have a pattern for the one’s position:
If the exponent is one of the sequence 0,4,8,12,16, …
then the one’s digit is a 1
If the exponent is one of the sequence 1,5,9,13, … then
the one’s digit is a 3
If the exponent is one of the sequence 2,6,10,14 then the
one’s digit is a 9
Last sequence for the exponent: 3,7,11,15: one’s digit is
7
How to use the pattern
Now we just have to figure out which sequence 99
is a member of?
The first sequence are all numbers which divide
evenly by 4
The second sequence the numbers have a
remainder of 1 when divided by 4
Third? A remainder of 2. Last? A remainder of 3
Where does 99 belong?
Make a list
Look back of magic square problem
Or the 1,1,2,2,3,3,4,4 problem
Or …. We’re always making lists, most of the time
to detect a pattern and sometimes to be orderly
in guess and check.
Making a list all by itself will not necessarily lead
you to an answer! But if you just need to list the
possibility, an orderly list helps.
Solve a simpler problem
Example problem: coin weighing.
Suppose we have a pile of coins with 1 counterfeit
that is heavier. How to find the counterfeit in the
least number of balancings?
If we have 2 coins: easy: weigh them and
whichever is heavier, that’s it.
Continued
What about three coins? Take two. Weigh them. If one is
heavier, it is the counterfeit, If they are the same, the
one coin you left out is the counterfeit. 1 weigh
What about 4 coins? Solve two two coin problems (that’s
two weighings max)
What about 5 coins? Do the 2 coin problem first. If they
are the same, then do the 3 coin. 2 weights (1+1) If
the 2 coin problem is not the same, it would only take 1
weighing. (Note there are several ways to achieve the
minimum!)
How about 9 coins?
Take 4 coins: if they are the same, take the next 4.
If they are the same, the heavier is the odd man
out (2 weighs)
If the second 4 are not the same, then the heavier
is in the heavier pan: need one more weigh. (3
weighs)
If the first 4 are not the same, then the heavier is
in the heavier pan: need one more weigh (2
weighs). So! This one worst case gave us 3.
We may be able to do better.
Take the 5 coins and do the 5 coin problem. If the
heavier is in the first 5 coins we collect, then it
takes 2 balancings.
If not, then the first weighing was the same. Take
the odd man out, and add it to the remaining 4
and do the 5 coin problem. HA! Again we got 3.
Let’s try a different approach
Start with our 9. Instead of using the 4 coin or the
5 coin problem, let’s try using the 3 coin problem
Pick two groups of 3 coins. (There will be 3
leftover). If the two groups are the same, then
the heavier is in the last group, so 2 weighings.
If the two groups are not the same, take the
heavier, and find the counterfeit in one more
weighing (2 weighings).
Here’s one that looks really
difficult!
Find the sum ½ + ½^2 + ½^3 + …
Let’s make a list of varying sums and see what
they look like
Make a portfolio entry
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Let's make a portfolio entry (or two) for these
examples.