Factoring Quadratics

Generic Format of a Quadratic Equation:     y = ax² + bx + c

 

Example: x² + 5x + 4

 

1.  Find the third term:   c = 4

 

2.  List factor pairs for c:           4 = (4 * 1) or (2*2) 

 

3.  Which set, (4 * 1) or (2*2), has the sum of b, 5?                  (4*1)

 

4.  Find the first term.  Factor it:            x² = (x * x)

It only has one factor, x.  Therefore, both binomials will start with x.

 

5.  In one binomial, add 4 (from the factor pair in Step 3).

 

6.  In the other binomial, add 1 (the other factor in Step 3).

 

7.  Therefore, we will have (x + 4)(x + 1)

 

8.  Check your work using FOIL (First Outer Inner Last/aka the Distributive Property):

           

(x + 4)(x + 1)

            (x * x) + (x * 1) + (4 * x) + (4 * 1)

            x² + x + 4x + 4

            x² + 5x + 4

 

 

Paraphrasing did help me internalize the concepts.   I was able to better break down the process of creating the correct binomials by listing each factor pair before creating the binomials.  I think creating this step-by-step process will allow for each student to have framework in place that will yield more success.

 

I think paraphrasing processes would be a great activity to reinforce computation, especially for computation of fractions.  This is a weak area for most students; therefore, this would be an opportunity for them to internalize the rules themselves rather than just “following the rules of the teacher.”

6-D-2: Applets

Applet: Pan Balance – Numbers

 

I really enjoyed the Pan Balance – Numbers applet.  It was an activity in which students create balanced equations using numbers.  When numbers are entered on one side, they must be balanced out on the other.  When a balanced equation is achieved, the equation is listed for the student.  Students should have ample time to play with this using all operations.  This applet was actually located in the Grades 3-5 activities and is a great visual to reinforce the concept of equivalence and also operations. 

 

However, the concepts that I would like to explore with this applet are the Properties of Equality.  Once students have created balanced equations, they would be directed by the teacher to add (subtract, multiply, divide) one side by a number and then do the same to the other side.  Students could then record the results of those operations.  The result, obviously, would be another balanced equation.  They could visually see that when something is done to only one side, an unbalanced equation occurs.  (This is the common mistake when students are learning procedural rules for solving equations.)  

 

After trying this many times, students will be able to discover the properties of equality rather than being told the vocabulary term and then its application.  They would truly see the properties in action and how they work.  This reminds me of the Countryman article that we read regarding problem solving and vocabulary needing to be experienced before noting specific mathematical terminology.  

 

To access this applet: http://illuminations.nctm.org/ActivityDetail.aspx?ID=26

Pascal’s Triangle: Formal Language

Pascal’s Triangle is a very useful tool to have in your mathematical toolbox.  Image available at: http://www.math.umass.edu/~mconnors/fractal/generate/pascal2.gif

 

The very top triangle in Pascal’s Triangle is an equilateral triangle.  (For this assignment, I use equilateral triangle to mean that there are equal amounts of numbers on each side, i.e. four numbers on each side).  There are two more triangles below this that, coupled with the one above, create a larger equilateral triangle with eight numbers on each side.

 

Below this 8-8-8 triangle to the left and right are two more triangles congruent to this one.  Within each of these, there are smaller triangles congruent to the smaller triangles at the top.

 

If we look at Pascal’s Triangle as a whole, we see triangular numbers.  For example, look at the rows of numbers.  In Row 0 there is one number.  In Row 1 there are 2 numbers; add those to the one number above, and you have 3 total.  In Row 2, there are 3 numbers; now a total of 6.  In Row 3, there are 4 numbers; now a total of 10.  These are the first numbers in the triangular number sequence, 1, 3, 6, 10, 15, 21…  This pattern is neither geometric nor arithmetic because it follows the sequence, +2, +3, +4, …  These totals (1, 3, 6…) are also specifically listed for us diagonally top to bottom (third diagonal row from the top in each direction).

 

Also in looking at the horizontal rows, I see a probability application.  These rows are the solution to a problem that I use in class, “A pizza shop offers 5 different toppings.  If you order one pizza per week, how weeks could you order a different pizza before you’d have to order one that you’ve already had?  (Sauce and cheese is a standard plain pizza.)”  The solution to this is 32, the sum of the digits in Row 5 (1 + 5 + 10 + 10 + 5 + 1).  The results are summarized in the table below (with combination notation below):

 

# of Toppings	0	1	2	3	4	5
# of Pizzas	1	5	10	10	5	1
	5C0	5C1	5C2	5C3	5C4	5C5

 

The natural extension to this is to increase the amount of toppings, thus creating a use for Pascal’s Triangle; it works for any amount of toppings.  It’s a nice tool for the students to have to solve problems quickly, though they must discover this application of Pascal’s Triangle for it to make sense and not just be an unfounded shortcut.

Order of Operations

Most people use the mnemonic device, Please Excuse My Dear Aunt Sally and theorganizer, PEMDAS.  However, I do not because some students believe that multiplication comes before division because M comes before D in the organizer; the same with addition and subtraction.  Here is what I use to alleviate this problem: Gary Eats Mc Donalds After School.  (Thank you Lucas!)  Here is the visual organizer:

G – Grouping symbols: brackets, parentheses, absolute value bars, fraction bar

E – Exponents

MD – Multiplication or  Divsion from left to right; I put an arrow above these going from left to right.

AS – Addition or  Subtraction from left to right; also an arrow above these letters.

Order of Operations Practice: http://amby.com/educate/ord-op/