Conference rusure::math

    how about a simple geomtry?
                                      
                             +--------+\ 
                             |        | \ x
                           y |        |  \
                             |        |  /
                             |        | /z
                             +--------+/ 
    
    put a pendicular (sp?)  triangle on the side of a squar.
    give the area of square say 5.
    but the area of square is y^2
    but again y^2 = x^2 + z^2
    
    i.e 5= x^2 + z^2 
    i.e x^2 + z^2 -5 = 0
    let z= SQRT(x)
    then
        x^2 + x - 5 = 0
    
    i dont know if this is practical though?
    
    /nasser
    
    
                             
                                

    I am looking for a word problem rather than an abstract geometric
    construct.  For example, if the students knew physics, I could discuss
    parabolic trajectories (i.e. throwing a basketball through the basket) 
    as a means to show how the quadratic equation may arise in daily 
    life (not that any basketball player has actually calculated this).
    
    I have a few simple problems that I'd rather not disclose here until
    I see some other thoughts - I don't want to stifle creativity.
    
    If necessary, I'll give an example to show the complexity level I'm
    driving for.
    
    Also, if your example is demonstrable (stack of cards, ball bouncing,
    etc.), it'll be even better!
    

The one that comes most readily to mind is figuring out the locus of a 
pencil constrained by a loop of string moving around two fixed points - the
ellipse. I think you can get this across without calculus, just using the 
Pythagorean theorem.

Joe Fadfollower has a chainlink fence around his yard. He has 18 feet of fence
left over and was wondering what to with it when he thought of making a
compost bin. He decided to put it in a clump of trees away from the fence.
He talked to various people and concluded he needed a rectangular compost bin 
of 20 square feet. Joe remembers how hard it was to cut the fence material,
so he wants to use all of the leftovers. Can he do it? What will the 
dimensions be? What if he decided on 14 square feet, or 8 square feet?
What about 21 square feet? How big a bin could he make?
(the last question depends on math trivia they might have heard by that age
but might not. but the group of questions might let them discover it.)

	One of my favorites is the classic problem of
determining the golden ratio: 

	Find the ratio, w/h, of rectangle of width, w,
and height, h, with w > h and with the property that when a square
of height h is removed from one side of it, the remaining
rectangle has the same ratio of width to height as the
original.


        <-------------  w ------------------->

	+--------------------+---------------+    ^
	|                    |               |    |
	|                    |               |    |
	|                    |               |
	|                    |               |    h
	|                    |               |
	|                    |               |    |
	|                    |               |    |
	|                    |               |    |
	+--------------------+---------------+    v

	<----- h ----------->|<---- (w-h) --->


	i.e, letting h=1, want to solve for w in

		w/1 = 1/(w-1), 

or equivalently solve the quadratic equation:

		w^2 - w + 1 = 0

-Franklin               
            

    re. -1
    
    oops! Should be
    
    	w^2 - w - 1 = 0
    
    Perhaps I should audit the 8th grade class for algebra review!! 
    
    
    -Franklin

	Consider the Fibonacci numbers, the first of which is 0, the second is
	1, then each is the sum of the previous 2, like this:

		0  1  1  2  3  5  8  13  21  34 ...

	What happens to the ratio of adjacent numbers.  We have:

		0/1   1/1  2/3  3/5  5/8  8/13  13/21 ...

	Does this ratio approach a particular value ?  Towards answering this
	question, consider an arbitrary place in the Fibonacci number sequence:, and call it f.

		... a    b   a+b   a+2b ...

	If the ratio of adjacent numbers calms down to a particular value,
	we expect to see

			a/b  =  b/(a+b)

	and

			b/(a+b) = (a+b)/(a+2b)

	Solving for a and b by solving the first equation for a in terms of
	b, then plugging in for a in the second equation, yields a quadratic
	equation, the solution of which provides the desired ratio.

/Eric

    re .7
    
    and .7 is the same as .5 (as I'm sure you all knew) but maybe the 8th
    graders would enjoy discovering it. BTW just how old are 8th graders?

    i remember i was only 17 years old in my 8th grade, and my family
    were so proud of me, so i'd say add 1 year to that to get the average
    age of 8th grader, say 18 years old ?

    I really appreciate the examples listed in these replies; however, as
    stated in .2, I'm looking for *real, practical* examples - things from
    everyday life that a young child can and will grasp.  Fibonacci,
    geometry etc. are too abstract for what I want.
    
    For example, if it takes you x hours to lay 100 bricks and it takes
    your friend twice as long plus 1 hour to lay the same number of bricks,
    a)  what expression represents the number of bricks per hour the two of
    you can lay if you work together?  b)  If you and your friend built a 
    brick wall of 1000 bricks in 2 hours, how long would it take for you
    to do the same work if working alone?
    
    What I like about this problem is that it is tangible and easy for 
    students to jump into.  However, it still is a little too contrived - 
    intuitively, students feel that the problem is backward, ie they would
    most likely know how fast they can lay bricks themselves, but not how
    fast the team could do it.  That's why I'm looking for a better problem
    or at least one that's more practical.
    
    

    re .9
    
    17 or 18 years of age?
    
    This doesn't seem to tally with .10 who talks about a 'young child'.

    An eighth-grader in the United States would normally be 13 or 14 years
    old.
    
    --  Barry

Hmn, its not easy, is it...

Your example used sums of reciprocals to reach a quadratic. Other basic
approaches (none of which I'm happy with, a kid's life is pretty linear):

1)	Compound interest for two periods (assuming they have learned about
	interest). Your friend invests $100 at 10% simple interest for 2
	years. You invest the same amount but compounded yearly. At the
	end you are $1 ahead. Hmn, you say, what would the interest rate
	have to be so that I could have taken out that dollar at the
	beginning, invested only $99, and wound up with the same amount
	as my friend? Its 11.1111...%, but you need to solve a quadratic
	to get it.

2)	Compound interest for three periods (probably need a calculator
	for this one if its "practical"). You start up a little banking
	business. Your parents agree to loan you $1000 at the current market
	interest rate, compounded annually. You lend this money to your friends
	at 5% higher than the rate you get, also compunded annually. All the
	loans are payable in three years. What would the market interest
	rate have to be for you to clear $200 after 3 years? (the nice thing
	about this problem is the equations are cubic but they collapse to
	a quadratic, the bad thing is the arithmetic is rather ugly)

3)	Area. The fence problem was given in a previous note. The Golden
	Rectangle isn't "practical" but it is elegant and there's a story
	behind it, ancient Greeks and all that.

4)	Pythagorean Theorem - I don't know if they've had geometry yet, tho.
	Part I: You and your friend are practicing a killer football play. You
	start out side by side 10 yards apart, your friend on the left.
	(I.e. you're the quarterback and your friend's a halfback, right?)
	At the hike your friend heads straight downfield, counting. At the
	count of three - three seconds exactly from the hike - your friend
	reaches up without turning around or breaking stride and presto, the
	ball is there. The defense doesn't have a clue. An unstoppable play!

	Your friend runs at 10 yards/second. Your trademark bullet pass travels
	at 30 yards/second. When do you release the ball?
	(This is a degenerate quadratic because it shows up in the form
	 (3-t)^2 = c, of course many of them will expand and solve anyway...)

	Part II - in a real game, the other side would charge you, so you'd
	have to run straight towards the right sideline at 5 yards/second while
	maintaining your perfect concentration on that spot 30 yards downfield.
	When do you release the	ball now? When would you release it if you ran
	towards the left sideline instead?

    Do any of the kids know ohms law or a little electronics?

    I remember figuring out the resistance of a ladder of resistors
    when I was a kid:

	o---[R1]---+---[R1]---+---
		   |          |
     R ->	  [R2]       [R2]  ...
		   |          |
        o----------+----------+---

    Seemed like an exciting result at the time :-)

    When I learned the quadratic equation I had trouble with the part
    about "completing the square" - I found it a really tricky operation
    because I just couldn't "see" the idea.  My grade school teacher told
    me "you don't have to know that until such-and-such a grade" rather
    than explain it.

    - Jim

    Some things come to mind:
    
    1.
    Your going to build a doghouse.
    
    
    
                /\
              /    \
            /        \      
          /\           \ 
         /| \            \
        / |   \          /    given a an b find h.  
       /  |b  h\       /  
      /   |     \     /
     /    |      \   /
    ---------------
           a
    Solving for h would be fun and simple enough.  Have them figure it out
    first without the theorm to challenge them.  Knowing h also helps
    figure out how many shingles will be needed.
    
    2.
    
    You live in city A, you're going to City B (due east, known distance) and
    meeting a friend and then you and the friend are going to City C, which is
    due north (known distance) of where you live in City A.  How far is C 
    from B?
    
                C
                |\
                |  \ 
                |    \
                |      \
                A-------- B
                    
    
    3.  Using the ABC drawing above,  C is a radar station.  Target A is a
    known distance away, and A -> B is known.  If the radar signal travels
    at the speed of light (300*10**6 meters per sec), how long will it take
    from C to B and BACK ((round trip?)
    
    4.  Using a baseball field compute various distances to positions based
    on the knowns.
    
    
    I think with kids that age the problems should be something readily
    visualized and simple, at least at first.  Certainly with experience
    the complexity can increase.  I recall when I first learned about
    ratios as a child I went right home and measured every tree, pole and
    building around my house using my shadow and the objects shadows.  It
    was real fun at the time.  I used to enjoy figuring out how long a
    ladder had to be to reach a known height etc. with Pythagorean's
    theorem.  
    
    Dennis