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Chapter 4: Why SlopesSlopes are defined and employed in mathematics to describe how the graph of one quantity versus another rises and falls. Slopes measure the rate of change of one quantity with respect to another: how fast a first quantity rises, falls or moves as the other quantity in the graph changes. A road which rises (or falls) three feet for each one hundred feet travelled horizontally, or three meters for each 100 meters travelled horizontally, is said to have a grade or slope of three percent. This slope gives the rate of change of height with respect to the distance travelled horizontally along the road. A graph of height above sea level versus distance along a road with this grade would be a line segment with a slope of three percent. Different sections of a road could have different slopes. That is, on different portions, height could change at different rates. Here the slope of the curve can be estimated by the slope of a straight edge or piece of wood that lies on it. How is a technical detail presented today in middle high school algebra courses. Why slopes are studied can be shown by a minimal-notation tour of easily-visualized, geometric interpretations of slopes and slope-related calculations in the context of skiing[3]. Briefly, the ski-based tour is as follows. A skier can tell from the slope of the ski (that is the slope sign or value), whether his or her travel is uphill, downhill or horizontal. In going over a hill top, the ski slope changes from positive to negative as the skier goes up and then down. Increases and decreases in the slope of the ski further show the skier how the steepness of the hill changes. All these observations have applications elsewhere, but the picture of a skier travelling over a hill y = f(x) gives a first image of them. Slopes to graphs and the areas under a graph have many interpretations in mathematics, physics and business. The interpretation depends on the quantities graphed. Speed, acceleration, constants of proportionality are all examples of slopes. Slope or rate of change calculations, and methods for the reversal of these calculations, all appear in the algebraic reasoning of business, science, engineering and technology. In calculus, the mathematical subject mainly concerned with slope-related calculation, slopes are called derivatives. Slopes may be computed or measured from formulas or graphs. Rules for obtaining or deriving slopes from formulas are explained in calculus courses and are called rules for differentiation. Using the rules in reverse, an ad hoc process in calculus with two names, anti-differentiation or integration, leads to well-known and not so well-known formulas for areas, volumes, perimeters or distances, weights and masses, etc. The etc. includes many quantities in physics and engineering, and sometimes business. The rules for obtaining slopes or derivatives for graphs given by formulas, and the rules for obtaining formulas from slope information, all depend on the algebraic way of writing and thinking. In the companion book Why Slopes and More Math, the geometric and physical interpretations show why slope-related calculations are of interest without saying how to calculate. The latter is a detail left to calculus courses. The why slopes chapters in this companion book provide a ski-based tour, a preview or review of calculus [3] for students of algebra, trig or calculus, itself. The appendices in this companion book give the proofs normally omitted from first courses in calculus. Chapter Sections: Next: Chapter 5, References - Past Efforts in Curriculum Design |
Foreword + Chapters 1 to 10 + 12
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