Introduction: This lesson can be adapted to suit the level of the student. Algebra students can work on skills such as collecting data, plotting points on a graph, and using formulas for volume. Higher algebra or calculus students can develop functional models and use more sophisticated techniques to interpret the data from the models.
Less Advanced Students will:
More Advanced Students will:
Problem: "Find the dimensions of the largest (in area) rectangle with perimeter 600 cm."
I have given this lesson to students ranging in ability from high school pre-algebra to college calculus. I can honestly say that I have never been disappointed that I did! Students of all levels seem to like the constructions, and then like to see the display of varying sizes of boxes at the front of the class. I often let them "vote" to see which they think is the biggest by placing scraps of paper in their chosen box. I am always surprised that the biggest box is never the unanimous choice! I guess it is a fine balance between being a deep box, and being a wide box!
The strength of this lesson lies in the multiple representations of the data. Namely we have the data from our physical model (corner size and volume ) being collected and recorded as:
Students seem to somehow gain an understanding of the many approaches to solving this simple problem. Those with higher levels of abilities can finally see in a very concrete way what it is that they have been studying about - yes, derivatives can be useful!
I always have a great time with this lesson, and really would hate to have to let a substitute teacher use it on a rainy day for a time killer. It deserves much more attention than that!
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