Algebra and Let's Play Baseball with FACTORING

 I have done two previous blogs on factoring and foiling with trinomial equations.  Today we are going to go all the way around the "bases" and do an equation using both methods.  We are going to begin with multiplying two factors together.  
The example is:   (4x +3)(2x + 5).

I am going to remind you that visual learners will find this "baseball" lineup hard to work with.  It doesn't look like a regular multiplication equation with more than two digits.  So let's line it up so that it looks easier to work with.
                   

            (2x + 5)

                 

            (4x + 3)  This step is very important.  Having the kids mark it with a red X will help them visualize what needs to be cross multiplied.

Because we are going with the baseball theme, I am inserting the baseball diamond below, even though it looks like a square with an X in the middle.  Of course, if you were drawing it, making the X in the middle red would emphasize the factors above.

I put the X's on the left side and the numbers added to them on the right side.  It is important that the numbers in each factor (   )  are connected along the same line as the box.  See how the 2x is on the same top line as the 5, and the 4x is on the same bottom line as the 3.

Now, we go back to the "Yes, MAM, I can foil."  The MAM stands for Multiply Add Multiply.

(I used a girl pitcher to reinforce idea of "MAM." Or you could tell the kids the coach is a lady.)

                                            So, the result of this inning is  

                        8x² + 26x + 15

But, we're not done.  Now we are going to complete be re-factoring the equation using our Who Let the Dogs Out method. (slide and divide) 
   1.   Remove the 8 from in front of the  x^{2     
   2.   Multiply the last number in the equation by 8 to get 120.
    3.   Leave the middle variable ALONE.
    4.   The new equation is:} x² + 26x + 120
    5.   Put the dog dishes out and put a bone (x) in each one:  ( x +      ) ( x +      ).
    6.   Because the equation has two positive numbers (important) you can put a + sign in each             bowl (set of parenthesis). 
    7.   Make your 1,2, 3 factoring chart.   Factor 120.    
                           
                              
    8.   As you can see from the chart, the only combination that adds up to the middle variable 
           of 26 is the 6 and 20. (See previous blog of factoring if you don't understand this).

    9.  So, the new factors are  (x + 6) (x +20).  Normally you would put the 8 back into the 
         factors here by dividing 6 and 20 each by 8.  But you can see that neither number is 
        evenly divided by 8.  So, you must put the 8 back in front of each x.  This results in the 
        following factors:  (8x + 6) (8x + 20).
  
  10.  Finally, back to factors, but they're not the same ones we started with.  What to do?
         Can we reduce each factor separately by dividing by the same number?  YES!!!
         (8/2x + 6/2) = (4x + 3)
         (8/4x +20/4) =(2x + 5)   (Do you see that each factor was divided by a DIFFERENT 
         number.   

          WE JUST MADE IT ALL AROUND THE BASES 
                          FOR A HOME RUN!