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Q&A #19545


Graphing lines (slope/intercept)

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From: Claire (for Teacher2Teacher Service)
Date: May 17, 2008 at 19:49:25
Subject: Re: Graphing lines (slope/intercept)

Hi, Sandy --

Thanks for writing to T2T. To introduce linear equations to anyone, I'd first
start with a concrete situation that could be modeled that way. The most
important thing is to understand the concept of what a linear function is,
that something changes by a constant amount at each stage. That gives meaning
to the equations and graphs.

My favorite way is to use pattern blocks, one shape at a time, and connect
them together in a train, keeping track of the perimeter of the resulting
rectangles at each stage. With students I use the context of banquet tables,
asking them to find how many people can sit at tables formed by pushing, say,
square tables together.

No. of squares   Perimeter
1                   4
2                   6 You lose two units where they connect.
3                   8
4                  10
5                  12

Students readily observe that each new square added to the train contributes
two additional units of perimeter. To find the perimeter of the next larger
rectangle, just add 2. This is a recursive description of the pattern.

If we want to know the perimeter of a train of 100 squares, however, we don't
want to generate all the perimeters in between. We can simply use the number
of squares, double it, and add 2 (the ends of the rectangle). That is the
explicit/closed form, the linear equation:
P = 2n + 2 where n is the number of squares and P is the perimeter.
Students will come up with other equivalent ways to express the perimeter
based on n. The important concept is that each term in their expressions has
meaning in the physical situation.

I have students graph that and then find similar data when making trains of
the triangles, rhombuses, or hexagons. The trapezoids work, too, keeping in
mind that one of the parallel sides is twice the length of the shorter one.
By comparing the tables of data and the graphs for each different shape, they
develop the idea of the slope (the increase in perimeter/seats as each new
piece is added). The y-intercept (the constant 2) represents the two ends
that exist no matter how long the train. Incorporate the formal
vocabulary and notation at every step to bridge the connection from the
concrete to the abstract.

If you don't have pattern blocks handy, but have a computer with Internet
access, check out this pattern block applet:
http://nlvm.usu.edu/en/nav/frames_asid_170_g_2_t_2.html

I hope this is helpful. Please write again if you have more questions.

Good luck!

 -Claire, for the T2T service

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