Showing posts with label factoring. Show all posts
Showing posts with label factoring. Show all posts
Thursday, December 24, 2009
How Much Factoring In 1st Year Algebra?
SEASON'S GREETINGS
Math Notations 3rd Birthday- Thank You!
The American Diploma Project is and will be impacting on what is being taught in both Algebra I and II in the 15 states who have joined the ADP Consortium. The classic flow from Standards to Assessments to Course Content is leading to the type of content standardization in our schools which I envisioned decades ago. A natural part of this process is deciding what topics in our traditional courses need to be deemphasized or eliminated to allow more time for the study of linear and non-linear function models, one of the central themes of the new Algebra standards.This leads to curriculum questions like...
How much time should be spent on factoring quadratic trinomials in Algebra I?
My assumption is that factoring ax2+bx+c where a ≠ 1 is still taught in Algebra I. Please challenge that assumption if wrong! If we also assume there is sufficient justification for teaching this, then we move on to the issue of how much time should be devoted to instruction. Two days? More? Time for assessment?
Here are some arguments pro and con...
PRO
(1) It is required by the ADP Standards (see below).
(2) Learning only simple trinomial factoring of the form x2+bx+c is not sufficient for solving more complex application problems.
(3) The various algorithms, such as the "ac-method", which have been developed for factoring quadratic trinomials, are of value in their own right; further, the "ac-method" introduces or reinforces the important idea of factoring by grouping.
(4) Students gain technical proficiency by tackling more complicated trinomials.
(5) Students should be given the option of more than one method, not just the quadratic formula.
CON
(1) The AP Calculus exam generally avoids messy quadratics in their problems. If such occur, students normally go directly to the Quadratic Formula.
(2) The SATs generally avoid asking students to factor such quadratics directly, particularly since it is easy to "beat the question" by working backwards from the choices. Instead, they ask the student to demonstrate an understanding of the process.
Here's a typical question they might ask:
If 6x2 + bx + 6 = (3x + m)(nx + 3) for all values of x, what is the value of b?
(3)The ADP standards for Algebra I do include this topic but it does not appear to be stressed. The following are taken from the ADP Algebra I standards and practice test:
(3) Do other nations teach our traditional methods of factoring or are students told to go directly to the quadratic formula?
(4) Current Alg I texts seem to have deemphasized factoring in general and some have moved this topic to later in the book.
So I am opening the floor for your input here!
(a) How much time is spent on factoring quadratic trinomials in Algebra I in your school?
(b) Do you teach the "ac-method"? If yes, do you motivate it or teach it mechanically?
(c) Do you believe factoring quadratic trinomials is essential or should it be deemphasized?
By the way, here is an example of the ac-method:
Factor completely over the integers: 6x2 + 13x + 6
Step 1: Find a pair of factors of ac = (6)(6) = 36 which sum to b = 13.
Hopefully, students think of 9 and 4 without a calculator!
Step 2: Rewrite the middle term 13x as 9x + 4x (works in either order)
Then 6x2 + 13x + 6 = 6x2 + 9x + 4x + 6
Step 3: Group in pairs and factor out greatest common monomial factor from each pair:
3x(2x + 3) + 2(2x + 3)
Step 4: Factor out the common binomial factor 2x + 3:
(2x + 3) (3x+ 2)
Step 5: Check carefully by distributing.
Here is a "proof" of this method (some details omitted like the meaning of h and k):
Posted by Dave Marain at 7:14 AM 5 comments
Labels: ac-method of factoring, ADP, Algebra 1 end of course test, Algebra I Standards, curriculum, factoring, quadratic trinomials
Wednesday, January 9, 2008
A Visualization for Factoring the Difference of Two Squares
M2 - N2 = (M+N)(M-N)
Given that my difference of squares problem from yesterday may have been overly ambitious for middle schoolers (and I will have more to say about that in the comments section from that post), I thought it was worth reviewing a diagram that many of you have probably seen before. There are many similar geometric representations of standard factoring and distributive formulas in algebra, but this one has always been one of my favorites. It would be more effective if I had been able to shade rectangles R and S using different colors, but I did the best I could on short notice.
It's often a good exercise for algebra students to invent similar diagrams for other formulas, although the use of manipulatives such as algebra tiles can be even more effective.
Posted by Dave Marain at 5:58 AM 2 comments
Thursday, August 30, 2007
9991 M&M's were eaten by a group of freshmen...
Here's a warm-up you can give to your Algebra 2 (and beyond) students to welcome them back to math class after a summer of brain drain. NO CALCULATOR ALLOWED! This oughta' set the tone...
A total of 9991 M&M's were eaten by a group of Freshmen. Here are the facts:
(1) Each freshman ate the same number of M&M's.
(2) There was more than one freshman.
(3) Each freshman ate more than one M&M.
(4) The number of freshman was less than the number of M&M's each freshman ate.
How many M&M's did each freshman eat?
Work in your groups of 3-4. You have 3 minutes. First team to arrive at the correct number AND explain their method, gets to eat ______________.
Let me know how many groups solved it or your thoughts about the appropriateness of this question.
Posted by Dave Marain at 6:51 PM 14 comments
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