Showing posts with label Algebra 1 end of course test. Show all posts
Showing posts with label Algebra 1 end of course test. Show all posts

Thursday, December 24, 2009

How Much Factoring In 1st Year Algebra?


SEASON'S GREETINGS
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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):


Wednesday, April 1, 2009

ADP/Achieve Algebra I Practice Test Now Online! Links, Discussion...


This is not an April Fool's joke! As anticipated and mentioned previously on this blog, a complete practice test is now available for you to download in pdf format from the Achieve web site. Click on the bottom link on the right sidebar.

Student expectations are shown above (you may need to click the image to see a larger version).



From the site:

The ADP Algebra I End-of-Course Exam consists of algebraic topics which will be taken by students across all participating states. These topics are typically taught in an Algebra I course, and fall into four strands: 1) Operations on Numbers and Expressions 2) Linear Relationships 3) Non-linear Relationships and 4) Data, Statistics and Probability.



Composition of Test
47 operational items

  • 40 multiple-choice (1 pt ea)
  • 5 short answer ( 2 pts ea)
  • 2 extended response (4 pts ea)
I'm not permitted to reproduce any of the questions here but I will enumerate some of the topics tested:
Identifying linear functions (y = 2x vs. y = 2x vs. y = 2x2)
Choosing effective data displays
Absolute value graphs
Rationalizing denominators in radical expressions (square roots only)
Exponential functions (simple)
Graphical interpretation of systems of equations

For further discussion, click Read more below.


and some more...
Solve quadratic equation by factoring
Graph of linear inequality
Probability of independent events
Determine vertex of graph of a quadratic function
Interpretation of slope from problem situation (rate of change)
Solving linear equations
Identifying irrational numbers (square root form)
Multiplication of radical expressions
Solving absolute value equations

Download the entire document to see examples of the multiple-choice, short answer and extended response type questions. This should prove very useful for teachers and students as they prepare for the first administration of this test. From the partial listing of topics above you can see that this is a test reflecting an ambitious first year algebra curriculum which will surely raise the bar for those states and districts participating. I would expect most students will struggle the first time around with the content, format and a level of difficulty they may not have yet experienced. Over time and with experience students should perform at higher and higher levels. This is a learning experience for all of us and it should be viewed as an opportunity to be part of an exciting change in American mathematics education...

Additional Comments
  • Even if your district is not participating, this test is an excellent way to measure your Algebra curriculum against a highly regarded world-class standard. I strongly encourage you to include these sample questions in warmups, for classroom discussion or review for other assessments.
  • As nontraditional or difficult as some of the earlier problems may appear to be, read through the entire sample test. You will definitely see many traditional algebra exercises with which your students should feel comfortable.
  • Problem 31 is an interesting application of the Pythagorean Theorem and quadratic equations. Some students have a knack for guessing 'special' right triangles for these however. After discussing the algebra, remind them to consider multiples of 3-4-5. Since the legs differ by 4, it's worth trying sides of 3x4 and 4x4, then checking if the hypotenuse works. It does!
  • The test reflects an excellent blend of tradfitional and reform. The extended response and short answer questions should have a definite impact on instruction and assessment in your classes whether or not your district is on board with this test. Ultimately, I predict EVERY state will be on board!

    Your comments...
...Read more

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