Factoring in Algebra

FACTOR THE FOLLOWING:
               3X² –10X – 8


This is a common problem in in the second semester of Algebra I.   Every teacher
has their favorite way of teaching it, but for many students it brings tears to their
eyes and pains to their stomach.


O.M.G., 2 negatives and the coefficient of x² is not 1!!


What is that now,  two negatives make a positive?


4 x 2 = 8, but 4 + 2 = 6,  3 is ½ of 6, oh darn, what
do I do anyway!!


So what's the question, "factor"?


Are you asking these questions, because you are in a Algebra I class, or you have
been there and now your children want your help.  In my last blog I talked about
problems that could go through many grades, and as the student learned more they
would be able to answer the deeper question.  This is one of those examples.


Most, if not all of us would agree that without knowing our multiplication tables
Math can be a struggle.  The process for solving the problem above can begin
with a game in the 3rd grade.  By the second part of the year most of the students
will have learned their multiplication tables through 12.  As a warm–up teachers
could start playing a game I called "I'M LOOKING FOR TWO NUMBERS".


The teacher asks: 
"I'M LOOKING FOR TWO NUMBERS WHOSE 
PRODUCT IS 24 AND WHOSE SUM IS 10."  




What is the student learning?


What words do I need to know the definitions of?

The student is learning to  "factor", break down into its divisors.
They learn and remember the following terms:

• Sum: to Add
• Difference: to subtract
• Product: to multiply

They realize there is more then one way to make 24, or 36,

or 16, or many other numbers as a product of 2 numbers .  This requires
them to further analyze the question:

6 x 4 = 24

3 x 8 = 24

2 x 12 = 24

1 x 24 = 24

But only 6 + 4 = 10

∴ the 2 numbers are 6 and 4

Now ask the same question, and end it by saying; 

•  The sum is 11
•  The sum is 14
•  The sum is 25 

Three knew questions without much thinking on the teachers part,
but great analysis for the students.  The game does not have to change
in any way, but as your students get introduced to integers we can
add knew sums.  For the above problem:

• The sum is –10
• The difference is 3
• The difference is 10 

When our students get to Algebra I the problem at the top of the
blog should not scare anyone.  We are looking for 2 numbers
whose product is –24, and whose difference is –10.

        3x² – 10x – 8  factors to
     
              (3x + 2)(x – 4)