1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Will Mainland HS ever fix its algebra page?

  1. Nov 20, 2011 #1

    Stephen Tashi

    User Avatar
    Science Advisor

    An algebra tutorial on the Mainland High School Algebra Lab stie
    (http://www.algebralab.org/lessons/lesson.aspx?file=Algebra_AbsoluteValueEquations.xml)

    Solves (example ii):

    [itex] |2x -3] = x - 5 [/itex]

    and obtains solutions [itex] x = -2 [/itex] and [itex] x = 8/3 [/itex]

    I've emailed them about this twice and they haven't corrected it yet. Are there algebra texts that actually teach the fallacious method of solution used on that page?
     
  2. jcsd
  3. Nov 20, 2011 #2

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    The method is fine; it is definitely true that
    |2x-3| = x-5​
    implies
    x=-2 or x=8/3​
    They've just forgotten to finish the problem from there.
     
  4. Nov 20, 2011 #3

    D H

    User Avatar
    Staff Emeritus
    Science Advisor

    That finishing off bit is a bit of an important point with this problem, since this problem has no solutions.
     
  5. Nov 20, 2011 #4
    I agree with D H.

    If you notice, they have provided incomplete procedure for solving these problems:
    it's not just one problem they need to fix. They need complete procedure and complete all other problems also.
     
  6. Nov 21, 2011 #5

    Stephen Tashi

    User Avatar
    Science Advisor

    The appropriate way to solve the problem would be to use the definition of the absolute value function - namely:

    By definition [itex] |a| = a [/itex] if [itex] a \ge 0 [/itex] and [itex] |a| = -a [/itex] if [itex] a < 0 [/itex].

    So we have two cases.

    Case 1) Assume [itex] 2x - 3 \ge 0 [/itex] This implies [itex] x \ge 3/2 [/itex] and
    [itex] | 2x - 3 | = 2x -3 [/itex]. So we solve [itex] 2x -3 = x - 5 [/itex] obtaining [itex] x = -2 [/itex] However, we have assumed [itex] x > 3/2 [/itex], so [itex] x = -2 [/itex] is not a solution.

    Case 2) Assume [itex] 2x -3 < 0 [/itex]. This implies [itex] x < 3/2 [/itex] and [itex] |2x -3|= -(2x -3) = -2x + 3 [/itex]. So we solve [itex] -2x + 3 = x - 5 [/itex] obtaining [itex] x = 8/3 [/itex]. But we have assumed [itex] x < 3/2 [/itex], so this is not a solution.

    This method has the advantage of teaching the formal definition of absolute value. It trains students to realize that "minus a quantity" need not be a negative number and all the conditions that are required of the solution are deduced.

    The method on the page uses some mumbo-jumbo about "opposites" and could be kludged up by telling students that they must "always check your solutions". That's not bad advice with any method, but why use a method that avoids proper deductive reasoning. The correct method can be generalized to solve equations like [itex] |2x - 3| = |x - 5| [/itex] by cases. I don't know how you apply the method of "opposites" to such equations.

    In view of the thread https://www.physicsforums.com/showthread.php?t=377783, we might get Bruce Tonkin to weigh in on this matter.
     
  7. Nov 21, 2011 #6

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Not "The". "A".

    Where do you get the idea it's improper? :confused: It is not difficult to conclude "x=y or x=-y" using "proper deductive reasoning" from the equation "|x|=y".

    The approach is perfectly good, and more efficient to boot.

    Both methods are correct. And you apply the method of "opposites" in exactly the same way for such a problem.

    (in fact, the "method of opposites" is reversible in this case: |x|=|y| is true if and only if x=y or x=-y)

    It is possible there are good arguments that teaching to solve by cases is pedagogically better -- but the one you give is simply not one of them.
     
    Last edited: Nov 21, 2011
  8. Nov 21, 2011 #7

    D H

    User Avatar
    Staff Emeritus
    Science Advisor

    What's important is to teach that (a) sometimes what appears to be a solution is not a solution; one has to check that it is a solution, and (b) students should always check their work.

    In this case I would have used the very advice presented in the next problem: "Before jumping into solving this problem, think about it first." (Their problem iii should have been the very first problem.) "Thinking about it first" in the case of problem ii says that to ensure that the left hand side is not negative we must have x≥5. This in turn means that one need not check the alternative -|2x-3|=x-5 because 2x-3≥8 for all x≥5. So there is at most one solution here, found by solving 2x-3=x-5. The solution, x=-2, is not a solution to the original problem. There are no solutions to this problem!

    While a bit verbose for this problem, eliminating entire branches from consideration by "thinking about it first" can save a lot of time on more complex problems.
     
  9. Nov 21, 2011 #8

    Stephen Tashi

    User Avatar
    Science Advisor

    If it's not difficult to do then why not do it? And why is that any simpler than showing the cases are x = y or -x = y ?


    It's efficient in the sense that it's incomplete.


    The correct method, using standard mathematical definitions gives the same thing. There is no standard definition for "opposite" in algebra.

    It seems to me that when a problem can be solved by a deductive approach using standard definitions, that the burden of proof for doing otherwise is on person who proposes an alternate technique.

    The advantages of the alternate technique in this case are that it does not burden students with learning the definition of the absolute value function, it does not require precise deductive thinking, and it reinforces the advice that one should check answers.
     
  10. Nov 21, 2011 #9

    D H

    User Avatar
    Staff Emeritus
    Science Advisor

    Stephen: You are arguing with Hurkyl over the correct way to say "tomato." Both his approach and yours will yield the same answers to any and all questions of this form. It's a bit silly to argue over which approach is the "right" approach since the two are ultimately equivalent.

    As for why they won't fix it, perhaps you are being a bit too argumentative in your messages or too adamant in telling them that the very approach they are using is wrong. That's just a perhaps; I don't know the history of your communications with them. Do you really care whether they use your approach, or Hurkyl's, or someone else's, so long as the approach they do use does yield the correct answers?
     
  11. Nov 21, 2011 #10

    Stephen Tashi

    User Avatar
    Science Advisor

    If you are grading a student's paper and he gets the correct answer, but doesn't show work that derives it deductively, does he get the same mark as a student who shows a deductive process? I'm not saying the answer to this question is necessarily "no". But it points out how the two approaches are not equivalent.

    A similar debate could be had over whether to teach students to understand the distributive law or just teach them "the FOIL method". On the one hand, given that there are many achievement tests in secondary education nowadays and that teaching precise reasoning is a burdensome task for teachers, there are pragmatic reasons for teaching students how to get the right answers quick at the expense of how to get right answers in some deductive manner. The deductive manner isn't likely to be the subject of a multiple choice question.

    Suppose a person is writing a proof and must state the consequences of [itex] |3x - \delta| < |4 \epsilon + 2| [/itex]. What will the correct approach be? To use the "method of opposites" and then try to "check your answer"? Granted that not all students of secondary mathematics are going to proceed into such a situation. And perhaps as students proceed in mathematics they must periodically suffer disappointments when the way they learned some things in high school is not recognized as valid.


    The "method of opposites" can be fixed so it uses valid deductions. The procedure on the Mainland site could be solved by introducing a theorem: If |a] =b then a = b if and only if a >= 0 and a = -b if and only if a < 0. Then the cases to work the example would be

    Case 1: 2x - 3 = x - 5 and 2x - 3 >= 0
    Case 2: 2x -3 = -(x-5) and 2x -3 < 0

    I don't see that's any simpler than the cases:

    Case 1: 2x - 3 >= 0 and 2x-3 = x-5
    Case 2: 2x -3 < 0 and -(2x-3) = x -5

    I don't see that introducting the theorem and avoiding the use of the definition of the absolute value function has any pedagogical advantage. In subsequent math courses, the definition of the absolute value function will be the more important concept.



    I'm not involved in secondary education now, but when I had contact with it, it always amused me to hear mathematics instructors enjoin students to "think logically" and contrast this with the way that the material is presented. It isn't crucial to me whether or how Mainland HS fixes anything. I do find it an amusing task to see if I can provoke them to do something. It's somewhat like buying a cheap product, having it break and then, just for the "fun" of it, trying to jump through all the hoops to get the manufacturer to honor a warranty. I'm a retired guy. I have time for such inane adventures.

    It's silly to argue with a closed mind, but reasonable people can argue profitably.
     
    Last edited: Nov 21, 2011
  12. Feb 22, 2012 #11

    Borek

    User Avatar

    Staff: Mentor

  13. Feb 22, 2012 #12

    Stephen Tashi

    User Avatar
    Science Advisor

    Several months ago, I posted about the topic on a woodworking site. A member who happened to be a mathematics instructor took up the cause and was able to get an email response from Mainland, who said they would take the page down and revise the lesson.

    Now I know which web sites hold the real power in mathematical affairs!
     
  14. Feb 22, 2012 #13

    Astronuc

    User Avatar

    Staff: Mentor

  15. Feb 22, 2012 #14

    Stephen Tashi

    User Avatar
    Science Advisor

    Thanks for the link. The page has been revised.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Will Mainland HS ever fix its algebra page?
  1. Fixed and Variable Cost (Replies: 10)

  2. A fixed point theorem (Replies: 3)

Loading...