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A Slope of One

    Listen to a slope of one. Then, listen again.

    We'll use this as the unit of measure of slope. We'll judge other slopes in comparison to this.

    Listen again to a slope of one then describe in detail to yourself what you've heard.


Slope -- A Measure of Change

    If your description involved two ideas, a sequence of notes (increasing in frequency) and a speed at which the notes are sounded, it may be used to relates the two ideas. Slope does that. Slope relates or compares two ideas and how each changes relative to the other.

    Slope may be used to compare changes in distance to that of time, or changes in speed compared to time, or changes in size compared to time, or vertical distance compared to horizontal distance, or function value compared to inpute value.

    First, skip reading an accurate technical definition of slope and consider its purpose and the numerical scale used to measure it.

    Slope measures change and this page presents that change in three ways:
a number,
the speed at which a scale is played,
the tilt of a line displayed on a coordinate plane.


The Scale Called Slope

    You've listened to a slope of one but you've not listened to a slope of zero. Using two measurements, a scale or guage is possible. Two measurements are required -- the zero and the one. With these established, other measurements follow.

    Listen to a slope of zero and to a slope of one.

  • Do the sounds make sense to you?
  • What is the same in 0 and 1.
  • What is different in 0 and 1.
  • What should -1 sound like?
  • What should 2 sound like?

    Before you scroll down on this page, press RELOAD or REFRESH on your browser so the next graphic has animation, then, just sit there, examine the graphic, and think about slope.


The Slopemeter
Slope is 0
--- No vertical change, a constant horizontal change.
0/1 or 0/4 or 0/6 or 0
--- constant, no change
Slope is 1
--- Equal vertical and horizontal change.
1/1 or 4/4 or 6/6 or -3/-3 or 1
--- the unit of measure of change
Slope is -1
--- Equal but opposite vertical and horizontal change.
-1/1 or 1/-1 or 4/-4 or -4/4 or -1
--- the opposite of a change of 1
Slope is undefined
--- No slope
0/1 or 4/0 or 6/0 or undefined
--- Slope of a vertical line
--- No horizontal change but an infinitly large vertical change.

Slope is
--- vertical change compared to horizontal change
--- change in y compared to change in x
--- "rise over run"


Slope in Sounds



    Predict the sound of each slope before clicking on the line with the desired slope.


Slope As Numbers

    Experiment. Predict the sound of each slope using the slope of 1 as your standard or unit.




    Reread the definition of slope. Slope is crutial to the study of velocity, acceleration, and dilation.

    If you wish to know more about how the sounds on this page were created, read Slope on the HP. To learn how the last graphic was produced, read Image Map.

    If you wish to learn more about dilation of functions, read Dilation by a Constant.

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