X-Puzzles  This is usually introduced in Pre-algebra (7th grade).  It’s a simple pattern that is discovered, but tends to not be taught.

X-Puzzles Using the pattern in puzzles A and B, complete puzzles C, D and E. Did you get? Pretty easy, right?

X-Puzzles Try these!

X-Puzzles Look at these puzzles when different parts are missing

X-Puzzles - continued See if you can solve these harder ones.

X puzzles – WHY WERE WE DOING THIS???  When we did the area models  They always were set up for us  We knew what the coefficients of x were  We DON’T know how to fill in the blanks if they are missing  The X-Puzzles  Give you the coefficients that go in the missing spaces.  Allow you to have a visual for finding unknown values.  Are pretty straight forward once you get the hang of them.

Here’s an example of using the X-Puzzle Given: x x x2x2 369x 4x x 9 x4 Answer: x x +36 = (x + 9)(x + 4) Same numbers that were in the x-puzzle!

You try one!  Given: x 2 + 5x x 3x x2x2 6 x 2 x3 Answer: x 2 + 5x + 6 = (x + 2)(x + 3)

One more…  Given: x x - 24 Answer: x x – 24=(x + 2)(x - 12) x 2xx2x2 -24 x -12 2x

It even works with the difference of 2 squares (“b” term is missing)!  Given: x x 3x x2x2 -9 x -3 x3 Answer: x = (x - 3)(x + 3)

It also works if the constant term is missing!  Given: 4x 2 -8x 0 -8x 0 0 4x 2 0 x -2 4x0 Answer: 4x(x - 2) Since there is a column of zeros, we can get rid of it

Now, it’s time to see you do it on your own  Set up the x puzzles and the area models to factor the following polynomials. 1) x 2 + 3x + 2 2) x 2 + 5x + 6 3) x 2 - 7x ) x 2 - 8x - 9

Now, there are times when you’re given a coefficient in front of the x 2.  Don’t panic, you still have all the tools necessary to solve these, we just need to modify our x-puzzles.  Example: 2x 2 + 3x + 1  HOWEVER, you’ll need to look at the coefficient on the 2x 2 2x x 2x 2*?? 3 Thus, 2x 2 +3x+1=(2x+1)(x+1) 2*11 2x x 1 1

Let’s look at a harder one.  3x x + 10  Still, you’ll need to look at the coefficient on the x 2 3x 2 10 x 3x 3*?? 11 Thus, 3x 2 +11x+10=(3x+5)(x+2) 3*25 6x 5x 2 5

Here is the modification that is much easier.  3x x + 10  ALL YOU NEED TO DO, IS MULTIPLY YOUR OUTSIDE NUMBERS FIRST. (3x10) This goes on your x-puzzle where the product normally goes. 3x x 3x ?? 11 Thus, 3x 2 +11x+10=(3x+5)(x+2) 56 6x 5x 2 5

One more, but the steps are broken down.  2x 2 + 9x x x x Step 1: Draw the area model and x-puzzle Step 2: Fill in what information you can in both the area model and x-puzzle Step 3: Multiply the outside numbers. In this case, 2 and 10. ?? Step 4: Solve the x-puzzle and put those values into your area model 4x 5x Step 5: Find the missing pieces and write your final answer. 2x 2 + 9x + 10 = (2x+5)(x+2) 2x10=20

Try these three on your own 1) 4x 2 + 4x -3 2) 2x 2 + 7x + 5 3) 8x 2 – 14x -9

And that’s the basics to factoring with two visual tools