PROPORTIONALITY STATEMENTS

The expression of how one quantity varies in relation to another is called a proportionality statement. In science and many other areas of thought people are looking for patterns that exist between different types of quantities. If there is a pattern to the data it is important to be able to deduce this relationshipLet’s start by graphing some given data. Graph this data on one quarter of a standard piece of graph paper. Place distance on the y-axis and time on the x-axis. Label your graph properly.

time (s) 1 2 3 4 5

distance (m) 0.2 1.6 5.4 12.8 25.0

The goal of this exercise is to find the pattern that exists (if one exists at all) between distance and time. The general proportionality statement is, y xn, while for this set of data it is d tn. In the first graph d vs. t1 was plotted. How does one decide if distance is proportional to time? Two quantities are proportional to each other if they produce a linear relationship when graphed. A linear relationship means that the ratio of any two pairs of data are equal or proportional. After graphing this data you should see that distance is not proportional to time.Plot d vs.t2 on another quarter of your graph paper. Is distance proportional to the square of time? Plot d vs. t3 on another quarter of your graph paper. Is distance proportional to the cube of time?

Since this graph is linear the proportionality statement is d t3. A couple of items to note, firstly we will only be studying data that includes the origin as a data point even though it will not be listed. Secondly, if we had graphed time on the y-axis and distance on the x-axis we would have obtained t d1/3 , after a graphical analysis.A proportionality statement can be analyzed more completely by finding the constant of proportionality, k.

y xn y = kxn (proportionality equation)

Therefore for our example d = kt3

We can determine the value of k by calculating the slope of the line in our final graph. This means the proportionality equation for the data given is d = 0.2t3 .

It is important to see the visual relationship in each of the graphs but there is a faster way to determine the

proportionality equation without graphing the data.

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xn

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Any two sets of data points could be used.

n = 3

Express all exponents in integer or rational form.

Use any set of data to calculate k.

k = 0.232.0 td

This equation can be rearranged to another form of the equation.

dt

td

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53

3

Write n values as fractions or whole numbers at all times in this unit. Write k values as decimals at all times.

Up to this point we have dealt with artificial data where the proportionality expression is perfectly upheld by all of the data points. In experiments that will be performed later the data obtained will not behave the same way and we will return to our graphing techniques to discover the proportionality expression. Graphing will determine the exponent while calculation of the slope of the best-fit line will provide the proportionality constant.

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710.1 dt

Of course there are many mathematical relationships between variables that have already been discovered such as the following:circumference = 2r surface area of sphere = 4r2 F = ma F = Gm1m2/r2

If one was to plot circumference vs. radius one would expect to find the slope to be 2Plotting SA vs. r2 should produce a straight line with slope 4Plotting F vs. a should produce a straight line with a slope of whatever the constant mass is. Plotting F vs. 1/r2 should produce a straight line with a slope of Gm1m2 (these are held constant).Homework: Booklet Proportion 1-4

don’t do graphing for any