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

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?

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

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.

Your portfolio entry should do the same. Your portfolio entry should be NEAT! Your portfolio entry should not assume the

reader knows anything!

Class time

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?

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

To teach math you have to do math. To do math it helps if you like it. So, one of the first things you should learn is

how to like math. Why it's fun.

Learning/Understanding

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 ...

Sets and how they are used and why they are important

Fractions Decimals, ratios, proportions, and percent Integers Rational numbers Measurement

What does that includes

The ideas behind each concept How we use them What skills are typically associated with those

concepts How do we explain them?

What we probably won't cover

Ch 4 and 5: whole numbers and number theory Ch 10 and 11: statistics and probability. Ch 12, 14, 15, 16: geomety 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?

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

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

Understand the problem Devise a plan Carry out the plan Look back (reflect)

Understanding the problem

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

This is often the hardest part.

So there are a number of strategies

Some of the first we will consider:

Guess and test Draw a picture Use a variable Look for a pattern Make a list Solve a simpler

problem

Relation of understanding to plan

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

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!

Look back. Figure out if your answer is correct. Does it answer the problem statement

(question)? Could you have done something different?

Process:

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

Your portfolio entry for a problem: State the problem in English Write down your understanding of the problem

leaving out any non-essential pieces Write down your strategy and why you are

choosing it Apply your strategy

Your strategy

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

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

A triangle with 6 holes The numbers: 1,2,3,4,5,6 Place them so each side of the triangle adds up

to 12.

Guess and check

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 ...

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...

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 ...

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:

Send + more = money Sun + fun = swim (digits 0,1,2,3,6,7,9) 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 ...

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.

If u is 3, then i is 7. And that leaves 0 for w. 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

Pizza pie! Draw a pie with 1 line: how many pieces? 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

Find the sum of the first 10, 100 and 500 counting numbers

Again! Make a list 1+2+3+4+5+6+7+8+9+10 = 55 Let’s generalize: 1 + 2 + … + (n-1) + n Use a trick! What if we add these twice:

Continued

1+ 2 + 3 + … + (n-1) + n + n + (n-1) + (n-2) + … + 2 + 1This 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

Let's make a portfolio entry (or two) for these examples.