I was going to write a small summary of every chapter, but then I realized that would be a lot of information since there were so many chapters in this book. There were some chapters that were really small and I really didn't learn that much in that chapter. Therefore, I decided that I would break this review into the different parts of the book.

Part 1 - Numbers (1-6)

Part 2- Relationships (7-11)

Part 3- Shapes (12-16)

Part 4- Change (17-21)

Part 5- Data (22-24)

Part 6- Frontiers (25-30)

__Part 1- Numbers __
In these beginning chapters they talked about negative and positive numbers and how difficult those numbers can be for students that are just learning about them. Textbooks, online sites, etc do so many thing possible to have the negative numbers standing out, printing the numbers in red and putting parenthesis around them so they stand out. Us, as mathematics have come up with all sorts of funny little mental strategies to side step the dreaded negative sign. We will try everything to get those negative numbers gone from our equation. The author states that maybe why we are so confused about negative numbers is because a negative times a negative equals a positive.

3*1= (1) (1) (1)

3*(-1)= (-1) (-1) (-1)

-1*3=-3

-1*2=-2

-1*1=-1

-1*0=0

-1*-1=?

We add 1 to the number before it, therefore the next number would be 1.

The Commutative Property- Is this law so obvious?

Look at the array of 3*7 and 7*3. The number of dots are always the same, just the array is turned.

The last chapter of this part talked about Roman Numerals.

MDCCCVII - MDCCCLXXIV

1807 - 1874

The Roman Numerals are hard to read and cumbersome.

What are some good ways to represent numbers?

* Tallies, Roman numerals (this was a challenge)

* Hindu-Arabic system - all numbers can be expressed with the same 10 digits, just placed in the right spot.

* Place Value- Addition and Subtraction

In these chapters I really did learn a lot. Looking at the different types of numbers that many people used before our time was interesting. Some of these chapters were things that I already learned in other elementary math classes, but it was good to refresh my memory .

**Part 2- Relationships**
These chapters are the ones that I seemed to learn the most and enjoy the most. There were simple concepts in this chapter, that were put into prescriptive. Some of the diagrams and maps really seemed to help.

On page 47, they have a square that is 50x by 50x. We are always obsessed with solving for x. We do whatever is possible to move x to the other side or get x by itself.

Taking the square root of a negative. This is something that many people get confused about and this book does a great job on explaining that and showing pictures.

In chapter 9 they take apart the 50x by 50x square to show you how to come up with the answer and how the quadratic formula comes about. (This is shown in detail in my notes)

__Part 3- Shapes__
In the first part they start talking about the Pythagorean Theorem. There are many different types of proofs in this chapter. One of the proofs that shows the squares on the side of the side lengths is one that I actually proved in MATH 341, so I was very familiar with that one.

"The square on the hypotenuse is the sum of the squares on the other two sides"

The graph of the time of the sunset and the time of the sunrise was a very interesting graph. The person said that it looked like "opposing waves". That is true because that graph takes into affect the days growing longer and shorter with the changing of the season.

__Part 4- Change __
The problem that interested me the most was the one about the hike through a snow covered field. This was interesting because it took into consideration the different paths that the hiker could take. The hiker didn't want to walk in the snow for that much time, so what would be the shortest way for him to walk without walking in the snow for that long? This is the question that the student would be trying to figure out. In chapter 17, the pictures really show a great representation of this. This could be a problem that the students could work out together and figure out which one would best suit the hiker.

Reading about vectors took me back to MATH 345 since we talked about vectors a ton in that class. Looking at the relationship between two different concepts or objects. Looking at how vectors can be added and subtracted was a different concept to me. This something that we have talked about, but it would be nice to look deeper into this concept. In chapter 21 it takes about the concept of Tennis. That is a good way to think about vectors. (Running velocity, ball sails wide, and aim down the line).

__Part 5- Data__
This is part of the book that I seemed to know a lot about already. I have taken many courses at Grand Valley that has talked about data, collecting data, analyzing data, and making conjectures about data. One problem that I found to be very interesting in these chapters was the question about breast cancer in chapter 23. This is a question that I read and was like, " How can we figure that out?" But after looking at the problem for a little bit I figured out what needed to be done. The author asked 24 doctors to answer this question to the best of their ability, and most of them didn't get this right. What is wrong? Why did so many well educated people get this wrong? This is something that is a hard concept for people to think about. Looking at the probability of different things and figuring out what part needs to go first and what part needs to come second, last, etc.

__Part 6- Frontiers__
I learned what a twin prime was in this section. A twin prime is a pair of prime numbers close to each other, almost neighbors, but between them is an even number, keeping them from touching. For example, 11-13, 17-19, 41-43, etc.

The graphs on page 205 are very interesting. The graphs show the number of primes on one graph and the number of odd numbers on the other graph. There is a lot to be said about these two graph and a lost of comparing that could be done.

I THINK THIS WAS MY FAVORITE PART OF THE BOOK. When they talked about mattress math in detail. When I was first reading this chapter I really didn't know where they were going to go with it. But after reading a little bit farther I was literally laughing out loud to myself. This is a great way for students to visually see the different flips and rotations. This is a visual activity that could be acted out and made into an art project. The students could physically make the beds and cut to show the vertical or horizontal flip or rotation.

Overall, this was a great book that had a lot of information in it. It was a little bit on the longer side so I have to read through it really fast to get finished with it. It would have been nice to take my time and go into detail about every chapter!