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Grade 9 Academic Guide to Slope

Mr. M. Couturier MPM1D. Grade 9 Academic Guide to Slope. Slope. Fundamentally, slope is a rate of change. You can identify a rate of change in the wording of a sentence, because you will most often hear one of the following: per, each, every; {sometimes a or an}. Slope.

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Grade 9 Academic Guide to Slope

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  1. Mr. M. Couturier MPM1D Grade 9 AcademicGuide to Slope

  2. Slope Fundamentally, slope is a rate of change. You can identify a rate of change in the wording of a sentence, because you will most often hear one of the following: per, each, every; {sometimes a or an}.

  3. Slope Here are a few examples of rates of change: Kilometers per hour (km/hr)‏ Miles per hour (mph)‏ Dollars an hour ($/hr)‏ Cents a minute (¢/min)‏ Words per minute (wpm)‏ Dollars each ($/unit)‏

  4. Slope Now lets put numbers: A person driving 75 km/hr is a rate of change between distance (km) over time (hr); (mph)‏ A person is paid $15/hr is a rate of change between dollars ($) over time (hr); A person can type 70 wpm is a rate of change between words typed (w) over time (min). Blood oranges are $2/each.

  5. Slope When given a standard linear equation, the slope of a line is represented by the letter m, in: y = mx + b

  6. Slope Since slope is a rate of change, it means that it is a comparison of one thing over another. In math, we represent this as a change in y (Δy) over a change in x (Δx). More specifically, we say: m = Δy Δx

  7. Slope The m is derived from the fact that it was René DesCartes, a French mathematician that formulated the idea. He was thinking of mountains and relating a “slope” or “monter” even though in French, this kind of slope is called: “une pente”.

  8. René DesCartes

  9. Slope So keeping this concept of the “m” in link with the mountain, we can imagine the four kinds of slopes that a skier may have to encounter. Let imagine a distance time graph, where the x (axis) is the time that a skier skis and the y (axis) is the height that the skier travels.

  10. POSITIVE SLOPE If a skier wants to “go up” a mountain, the skier, will have to “increase” his “altitude” as time also “increases”. (Note that time can only increase). Since we have: m = Δy = increase = Positive # Δx increase We therefore have, an increasing or positive slope.

  11. POSITIVE SLOPE

  12. NEGATIVE SLOPE If a skier wants to “go down” a mountain, the skier, will have to “decrease” his “altitude” as time “increases”. Since we have: m = Δy = decrease = Negative # Δx increase We therefore have, a decreasing or negative slope.

  13. NEGATIVE SLOPE

  14. ZERO SLOPE If a skier wants to “go cross-country”, meaning he can neither go up or down, then the skier’s altitude does not change as time “increases”. Since we have: m = Δy = No change = 0 = 0 Δx increase increase We therefore have, a flat-line or zero slope.

  15. ZERO SLOPE

  16. INFINITE SLOPE Technically impossible, I call it the “Star Trek: Beam me up Scotty slope”. Some call it undefined, but saying infinite is more meaningful. If a skier wants to “increase” their altitude “without any change in time”, then we have: m = Δy = increase = increase = ∞ Δx No change 0 We therefore have, an infinite slope.

  17. INFINITE SLOPE

  18. INFINITE SLOPE

  19. INFINITE SLOPE On a technical note, even the Beam me up Scotty example isn’t perfect because although they beam-up from one place to another, the amount time that it takes to do it is not equal to zero.

  20. FINDING THE SLOPE Now let’s do some actual calculations: Given two points on a line means that you are given two sets of (x,y) coordinates. We will always label one as (x1,y1) and the other as (x2,y2). The choice of which is which is yours. Since slope is rate of change and since we have to points to compare, we can find the slope.

  21. FINDING THE SLOPE Recall that: m = Δy = rise Δx run m = y2-y1 x2-x1

  22. FINDING THE SLOPE • Find the slope of the following graph.

  23. FINDING THE SLOPE • We are given many points but two are marked with red dots. Let us define point 1 as (x1,y1) = (-1,2) and point 2 as (x2,y2)=(1,4).

  24. FINDING THE SLOPE Using: m = y2-y1 x2-x1 we get: m = 4 – 2 1-(-1)‏ to yield: m = 2 2 m = 1

  25. FINDING THE SLOPE • So in conclusion, our slope, m=1 makes sense, because if the skier is rising (in altitude) as time increases and m=1 is positive.

  26. FINDING THE SLOPE • Find the slope of the following graph.

  27. FINDING THE SLOPE • We are given many points but two are marked with red dots. Let us define point 1 as (x1,y1) = (-3,-3) and point 2 as (x2,y2)=(3,-3).

  28. FINDING THE SLOPE Using: m = -3 - (-3)‏ 3 – (-3)‏ we get: m = 0 6 to yield: m = 0

  29. FINDING THE SLOPE • So in conclusion, our slope, m=0 makes sense, because if the skier is “cross-country” skiing; neither descending nor ascending as time increases.

  30. FINDING THE SLOPE • Find the slope of the following graph.

  31. FINDING THE SLOPE • We are given many points but two are marked with red dots. Let us define point 1 as (x1,y1) = (2,-1) and point 2 as (x2,y2)=(5,-2).

  32. FINDING THE SLOPE Using: m = -2 - (-1)‏ 5–2 we get: m = -1 3

  33. FINDING THE SLOPE • So in conclusion, our slope, m = -1/3 makes sense, because if the skier is “descending” as time increases.

  34. INFINITE SLOPE http://www.algebrahelp.com/worksheets/view/graphing/slope.quiz http://www.wtamu.edu/academic/anns/mps/math/mathlab/beg_algebra/beg_alg_tut23_slope.htm

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