June 2013 Brain Teaser Solution

Q: Between 1000 and 2000 you can get each integer as the sum of nonnegative consecutive integers. For example,
147+148+149+150+151+152+153 = 1050

There is only one number that you cannot get.
What is this number?

A: 1024

You cannot get 1024 because it is a power of 2
it is 2 to the 10th power
It is the only power of 2 between 1000 and 2000
For example 15 = 1+2+3+4+5
14 = 2+3+4+5
17 = 8 + 9
but 16 cannot be the sum of consecutive integers
neither can 2, 4, 8, 32, 64, 128, 256, 512, 1024, 2048 etc

Drawings of Number of Degrees in a Polygon

There are two less triangles in each shape than the number of sides
A triangle, is well, a triangle so 180 degrees
A square has two triangles in it so 360
A pentagon has three triangles in it so 540
The hexagon above was cut into two trapezoids so 2 x 360 = 720
See  the other post "Number of Degrees in a Polygon"

Number of Degrees in a Polygon

More Factoring with Algebra Tiles x^2 + 4x + 4

These algebra tiles form a square..hence the algebra that the tiles represent is a perfect square.
Look at each side of the square and you will see a length of x and then two little yellow squares making each side x + 2.
(x + 2)(x + 2) when FOILed is x^2 + 2x + 2x + 4 which when combining like terms becomes x^2 + 4x + 4.  If you look at the algebra tiles, there is the large blue square that represents x^2, four long green rectangles that are each x making 4x and 4 small yellow squares that represent 4.
see the post: http://mathconfidence.blogspot.com/2013/06/learning-factoring-with-algebra-tiles.html for more info and links on factoring polynomials with algebra tiles!

Learning Factoring with Algebra Tiles x^2 + 4x + 3

Algebra Tiles are a way of learning and teaching factoring skills.
The big blue square represents x^2 (x squared) and each green rectangle represents an x while the little yellow squares represent numbers (constants).
So if you look at both of these squares they are both composed of x^2 + 4x + 3
Factoring x^2 + 4x + 3 would be (x + 3)(x + 1) which if we FOILed would become x^2 + x + 3x + 3 and if we combined like terms would become x^2 + 4x + 3.
The rectangle on the left has dimensions x + 3 on the bottom and x + 1 at the top.
The rectangle on the right has the same total dimensions of x + 3 on the bottom and x + 1 at the top but is configured differently.
This is where Math is evidently a creative endeavor as two different students made these representations.
Intro to Algebra Factoring from Regents Prep
Another Factoring Example from Regents Prep
Student Set of Algebra Tiles from Amazon

SAT Book Wear and Tear in a PreEbook World

Do you think you studied enough?  Sometimes the amount of studying is evident by looking at the book.
This is what a used book that has been used can look like.
So far, the Official SAT Study Guide is not available as an e-book.
I will miss the dogeared pages and the falling off cover that come with repeated use of old-fashioned books!