Grade 9 Academic Guide to Slope Mr. M. Couturier MPM1D

Grade 9 AcademicGrade 9 AcademicGuide to SlopeGuide to Slope

Mr. M. CouturierMr. M. Couturier

MPM1DMPM1D

SlopeSlope

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

SlopeSlope

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

SlopeSlope

• Now lets put numbers:Now lets put numbers:– A person driving 75 km/hr is a rate of change A person driving 75 km/hr is a rate of change

between distance (km) over time (hr); (mph)between distance (km) over time (hr); (mph)– A person is paid $15/hr is a rate of change A person is paid $15/hr is a rate of change

between dollars ($) over time (hr); between dollars ($) over time (hr); – A person can type 70 wpm is a rate of A person can type 70 wpm is a rate of

change between words typed (w) over time change between words typed (w) over time (min).(min).

– Blood oranges are $2/each. Blood oranges are $2/each.

SlopeSlope

• When given a standard linear When given a standard linear equation, the slope of a line is equation, the slope of a line is represented by the letter m, in:represented by the letter m, in:

y = mx + by = mx + b

SlopeSlope

• Since slope is a rate of change, it Since slope is a rate of change, it means that it is a comparison of one means that it is a comparison of one thing over another. In math, we thing over another. In math, we represent this as a change in y (represent this as a change in y (ΔΔy) y) over a change in x (over a change in x (ΔΔx)x). More . More specifically, we say: specifically, we say:

m = m = ΔΔyy

ΔΔxx

SlopeSlope

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

René DesCartesRené DesCartes

SlopeSlope

• So keeping this concept of the “m” in So keeping this concept of the “m” in link with the mountain, we can link with the mountain, we can imagine the four kinds of slopes that imagine the four kinds of slopes that a skier may have to encounter.a skier may have to encounter.

• Let imagine a distance time graph, Let imagine a distance time graph, where the x (axis) is the time that a where the x (axis) is the time that a skier skis and the y (axis) is the skier skis and the y (axis) is the height that the skier travels.height that the skier travels.

POSITIVE SLOPEPOSITIVE SLOPE

• If a skier wants to “go up” a If a skier wants to “go up” a mountain, the skier, will have to mountain, the skier, will have to “increase” his “altitude” as time also “increase” his “altitude” as time also “increases”. (Note that time can “increases”. (Note that time can only increase). Since we have: m only increase). Since we have: m = = ΔΔyy = = increaseincrease = = Positive #Positive #

ΔΔx increasex increase• We therefore have, an increasing or We therefore have, an increasing or

positive slope.positive slope.

POSITIVE SLOPEPOSITIVE SLOPE

NEGATIVE SLOPENEGATIVE SLOPE

• If a skier wants to “go down” a If a skier wants to “go down” a mountain, the skier, will have to mountain, the skier, will have to “decrease” his “altitude” as time “decrease” his “altitude” as time “increases”. Since we have: m = “increases”. Since we have: m = ΔΔyy = = decreasedecrease = = Negative #Negative #

ΔΔx increasex increase

• We therefore have, a decreasing or We therefore have, a decreasing or negative slope.negative slope.

NEGATIVE SLOPENEGATIVE SLOPE

ZERO SLOPEZERO SLOPE

• If a skier wants to “go cross-country”, If a skier wants to “go cross-country”, meaning he can neither go up or meaning he can neither go up or down, then the skier’s altitude down, then the skier’s altitude does does not changenot change as time “increases”. Since as time “increases”. Since we have: we have:

m = m = ΔΔyy = = No changeNo change = = 0 0 = 0 = 0 ΔΔx increase increasex increase increase• We therefore have, a flat-line or zero We therefore have, a flat-line or zero

slope.slope.

ZERO SLOPEZERO SLOPE

INFINITE SLOPEINFINITE SLOPE

• Technically impossible, I call it the “Star Technically impossible, I call it the “Star Trek: Beam me up Scotty slope”. Some Trek: Beam me up Scotty slope”. Some call it undefined, but saying infinite is call it undefined, but saying infinite is more meaningful. If a skier wants to more meaningful. If a skier wants to “increase” their altitude ““increase” their altitude “without any without any change in timechange in time”, then we have: ”, then we have:

m = m = ΔΔyy = = increaseincrease = = increaseincrease = = ∞∞ ΔΔx No change 0 x No change 0 • We therefore have, an infinite slope.We therefore have, an infinite slope.




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

FINDING THE SLOPEFINDING THE SLOPE

• Now let’s do some actual calculations:Now let’s do some actual calculations:– Given two points on a line means that you Given two points on a line means that you

are given two sets of (x,y) coordinates. are given two sets of (x,y) coordinates. We will always label one as (xWe will always label one as (x11,y,y11) and the ) and the other as (xother as (x22,y,y22). The choice of which is ). The choice of which is which is yours. Since slope is rate of which is yours. Since slope is rate of change and since we have to points to change and since we have to points to compare, we can find the slope.compare, we can find the slope.


• Recall that:Recall that:

m = m = ΔΔyy = = riserise

ΔΔx runx run

m = m = yy22-y-y11

xx22--xx11


• Find the slope of Find the slope of the following the following graph.graph.


• We are given many We are given many points but two are points but two are marked with red marked with red dots. Let us define dots. Let us define point 1 as (xpoint 1 as (x11,y,y11) = ) = (-1,2) and point 2 (-1,2) and point 2 as (xas (x22,y,y22)=(1,4). )=(1,4).


Using: m = Using: m = yy22-y-y11

xx22--xx11

we get: m = we get: m = 4 – 2 4 – 2

1-(-1)1-(-1)

to yield: to yield: m = m = 2 2

22

m = 1m = 1


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




• We are given many We are given many points but two are points but two are marked with red marked with red dots. Let us define dots. Let us define point 1 as (xpoint 1 as (x11,y,y11) = ) = (-3,-3) and point 2 (-3,-3) and point 2 as (xas (x22,y,y22)=(3,-3). )=(3,-3).


Using: m = Using: m = -3 - (-3)-3 - (-3)

3 – (-3)3 – (-3)

we get: m = we get: m = 0 0

66

to yield: to yield: m = 0m = 0


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




• We are given many We are given many points but two are points but two are marked with red marked with red dots. Let us define dots. Let us define point 1 as (xpoint 1 as (x11,y,y11) = ) = (2,-1) and point 2 (2,-1) and point 2 as (xas (x22,y,y22)=(5,-2). )=(5,-2).


Using: m = Using: m = -2 - (-1)-2 - (-1)

5–25–2

we get: m = we get: m = -1 -1

33


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


• http://www.algebrahelp.com/workshehttp://www.algebrahelp.com/worksheets/view/graphing/slope.quizets/view/graphing/slope.quiz

• http://www.wtamu.edu/academic/annhttp://www.wtamu.edu/academic/anns/mps/math/mathlab/beg_algebra/bes/mps/math/mathlab/beg_algebra/beg_alg_tut23_slope.htmg_alg_tut23_slope.htm

http://www.algebrahelp.com/worksheets/view/graphing/slope.quiz

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



Documents

Grade 9 Academic Guide to Slope Mr. M. Couturier MPM1D