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Q&A #19545 |
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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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