Graphing Quadratics...

Discussion in 'Secondary Education' started by TxMaThTeAcHeR, Jan 26, 2011.

  1. TxMaThTeAcHeR

    TxMaThTeAcHeR Rookie

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    Jan 26, 2011

    Hi guys...
    So looking for some opinions. I just started teaching quadratic functions to my class. I teach at an alternative high school, and the class I am teaching this in is a graphing class. I started teaching quadratics functions and graphing them - what is the vertex, finding the vertex using -b/2a and then graphing by making a table. Students are struggling to grasp the whole concept. Anyone have any tricks for teaching this? We haven't gotten into factoring, roots, anything like that yet because I want them to get the whole table thing first..
    Thanks!
     
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  3. mmswm

    mmswm Moderator

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    Jan 26, 2011

    Go get a few dozen Styrofoam cups, some black electrical tape, and a 2-3 foot section of 2X4. "Graph" the function f=x^2 using the cups, using the 2X4 as as base, and the cups to represent the units. Use the 2X4 as a base, and draw an x-axis on a table using the electrical tape (line up the board with the tape) Plot points. So, for x=1, -1, x^2=1, put a stack of 1 cup along the electrical tape axis, then for x=2, -2, x^2=4, so place you stacks of 4 cups right next to the one cup stack (turn the cups upside down). If you do this for a couple of points, you can watch the parabola form out of cups. Now, you've got the parabola, slide the board so that the vertex lines up with x=some other number. Let them see how shifting the vertex doesn't change the shape of the graph. You can play with the position of the board. If you're really creative, you can rig a dowel rod to represent the y-axis and move the board up and down as well.

    My students have always loved that little demonstration.
     
  4. Aliceacc

    Aliceacc Multitudinous

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    Jan 26, 2011

    Start small: with y = x squared.

    Draw a parabola, then talk about its axis of symmetery, and how it turns at that point, appropriately called the turning point. And how, conveniently enough, it's on the axis of symmetery. And how it's incredibly convenient to have 2 points on either side of the turning point so we can see the shape.

    Set up a table, using 5 x values, with the axis of symmetry always as the 3rd.

    Then graph y = 4- x squared (easier for them to see than just y = -x squared.) Show how a negative a value makes it turn upside down.

    One thing that helps my kids a lot with the table is giving 2 lines for each x value. On the top line, they substitute in. Directly beneath that, they figure out the values as they go.

    So, for example, when x= 3, the middle column might say " 3 squared -2(3) +5 " and right beneath it, they have "9 - 6 + 5" with the answer 8 in the right hand column.

    Also, consider an extra column labeled "(x,y)" so they can see how to get the points to graph.

    Once you've graphed a few, go back and talk about the roots.

    Or, better yet, make your "Do Now" 3 or 4 quadratics to solve by factoring-- those same ones you're going to graph in class. After you've graphed a few, draw their attention to the similarities in the problems and ask THEM what they think the roots mean.
     
  5. Aliceacc

    Aliceacc Multitudinous

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    Jan 26, 2011

    OK, so mm gets the award for the far more creative lesson!
     
  6. mmswm

    mmswm Moderator

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    Jan 26, 2011

    Well, I stole it from somebody else: a professor that taught only college prep courses at the CC where I taught. I'll have to pass the award along to her.
     
  7. TeacherGroupie

    TeacherGroupie Moderator

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    Jan 26, 2011

    Then mm gets the award for knowing a good thing to... um, liberate... when she sees it.
     
  8. Ruby2011MA

    Ruby2011MA Rookie

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    Jan 27, 2011

    I have always been taught that the best teachers "steal everything that works"... This is going into my "long-term borrow" folder.
     
  9. G00d d00bie

    G00d d00bie Rookie

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    Jan 29, 2011

    Textbooks give the ideas. The authors are very clever. Teachers get the ideas similar to subliminal msgs. But if you try to write a text book with all these ideas, you run into problems. Teachers are needed to give the info. a bit incorrectly the way the students need to receive it to move. We learn the most from our mistakes.
     
  10. G00d d00bie

    G00d d00bie Rookie

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    Feb 4, 2011

    What are focusing waves? Or what wave am I trying to describe - I forgot.

    In addition to everything else, it might be of interest to solve a quadratic by the alternative method of iteration. I used to have my classes do this quite well in the 80's when calculators got cheap. Now with Excel it is even better.
    I am trying to recall how to use it to make the special kind of a standing waves. What do you call them? They are not really standing waves - focusing waves or something?
    I tried unsuccessfully to find the info. on the internet.
    The idea is that each input pulse will never be perfect, but repeatedly entering the value (of energy) into a mathematically converging environment will focus the enegy. Every mistake or random uncontrollable input error will produce a new value that is closer to the solution of the quadratic.
    For example, try to solve x2 -x - 6 = 0 by solving for x as follows:
    x = (x + 6)/x. How can you find x when you need to know x to find x? Guess a value like -9 for x, sub.it and you get a value of x = .333... that is closer to the solution. Now sub..333... and you get 19. Sub.19 and get 1.3157894, etc. Continue the iterative process and finally arrive at one of the solutions of 3.
    With the wave in the science lab, the random errors continually miss 3, but using each continues to focus on 3 producing a wave at "3," so to speak. It is a really cool wave. It just piles water up right there at the location consistent with 3, however that is interpreted through the set up. The water might just be piled up say 7/11 of the distance from one end of the trough (or maybe 11/17 of the distance - whatever 3 translates into through the experient set-up).
     
    Last edited: Feb 4, 2011

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