Each of the relationships have their own patterns. Many of the patterns are similar and easy to recall.

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1 Proportional Reasoning Much of science and mathematics have a great deal to do with finding patterns. Many scientists spend a lot of time trying to determine how various variables are related mathematically for a given physical phenomena. If a relationship does exist then one can make predictions for various situations. Ones does not then need todo all the possible combinations that may not easily be done in the laboratory. This unit is designed to help one understand some of the simpler mathematical relationships that exist in nature. The unit will also point out some of the simple patterns associated with the various relationships. The following relationships will be developed: A) Direct B) Inverse C) Square D) Square Root E) Inverse Square F) No relationship Patterns that will be investigated for each are: 1) Factor Effect 2) Characteristic Plot 3) Constant Ratio 4) Straight Line Plot 5) Same Factor Effect Each of the relationships have their own patterns. Many of the patterns are similar and easy to recall. To better understand relationships, only two variables will be considered at one time. a) the independent variable - the one that you can often decide upon actual values before actually collecting data. b) the dependent variable - the one that changes as the independent variable is varied. The result of this variable depends on what the independent variable is. Tables: the first or left hand column of all tables will always be the independent variable, the second column will be the dependent. Always provide the unit of measurement under the variable. Mathematical operations (multiplying or raising to a power) on variables will always take place on the independent variable. Ratios: will be set up so the dependent variable is divided by the independent variable ( y/ x). Graphs: the independent variable will be plotted on the X (horizontal) axis and the dependent variable on the Y (vertical) axis. Always provide the unit of measurement under the variable.

2 The Factor Effect The following data pairs represent the mass and volume quantities for some material. In this case the volume is the independent variable while the mass is the dependent variable. A) Direct Relationship - this section illustrates patterns that show how the dependent variable changes as the independent variable is changed. SUMMARY: The factor effect pattern associated with the Direct Relationship is that whatever factor change occurs for the independent variable, the same factor change occurs for the dependent variable

3 B) Inverse Relationship - the following data pairs represent the Period and Frequency of a rotating object. The frequency has been selected as the independent variable. SUMMARY: The factor effect pattern associated with the Inverse Relationship is that whatever factor change occurs for the independent variable, the inverse of that factor is the factor change for the dependent variable.

4 Self Quiz: Assume that there is a direct relationship between the position a car has and the time it is on the road. Fill in the blanks of the following tables: Assume that there is an inverse relationship that exists between the following two variables, x and y, which have no units. Fill in each of the blank boxes in the following problems.

5 C) Square Relationship - the following data pairs represent the Area and Length of various rectangles that have the same width. The Length has been selected as the independent variable. SUMMARY: The factor effect pattern associated with Square Relationship is that whatever factor change that takes place for the independent variable will cause the dependent to change by the Square of that factor change. Self Quiz: Assume a Square Relationship exists for each of the following tables.

6 D) Square Root Relationship - the following data pairs represent the Period and its corresponding Radius of an object that is attached to the end of a string and rotating about a circular path. The radius is the independent variable. SUMMARY: - the factor effect pattern associated with the Square Root Relationship is such that whatever factor change occurs for the independent variable the dependent variable will change by a factor which is the Square Root of that factor change.

7 Assume a Square Root Relationship exists for each of the following tables of data.

8 The Characteristic Plot A) Direct Relationship - the following Mass-Volume data is from the Factor Effect section and will be used here to determine the characteristic plot for a Direct Relationship. SUMMARY: the characteristic plot for an Inverse Relationship is a curve of the type illustrated above (Hyperbola). As one of the variables increases the other decreases

9 C) Square Relationship SUMMARY: the characteristic plot for a Square Relationship is a curve of the type illustrated (parabola). The Dependent variable increases more rapidly than the independent variable does. SUMMARY: the characteristic plot for a Square Root Relationship is a curve of the type illustrated above. The independent variable increases more rapidly than the Dependent variable.

10 Direct Relationship -----Review The Constant Ratio As can be noted from the above table, the factor change for both variables is the same and the graph has a constant slope. Is there anything else that is constant? For this example a test of the product and ratio of the two variables will be illustrated. As indicated in the above table the ratio is a constant value. SUMMARY: the pattern found when there is a Direct Relationship is that the ratio of the dependent variable over the independent variable is a constant value. In this case, the Mass/Volume is constant. Which of the following sets of data represent an example of a Direct Relationship?

11 Inverse Relationship -----Review This time the factor change is not the same and the graph is not a straight line. Recall that the factor change of the dependent is the inverse of the factor change of the independent variable. The following illustrates a ratio of the two variables and a ratio of dependent over the inverse of the independent variable. The inverse Relationship also demonstrates a Constant Ratio. In this case the ratio of the period over the inverse of the frequency is a constant. SUMMARY: The pattern of a constant ratio continues. For the inverse relationship, the ratio of the dependent variable over the inverse of the independent variable is a constant. Which of the following sets of data represent an example of an Inverse Relationship?

12 Square Relationship -----Review The square relationship does not show a straight line plot nor does it show a factor change that is the same for both variables. Recall that the factor change of the dependent variable is the square of the factor change of the independent variable. The following two ratios are tested. The square Relationship also demonstrates a Constant Ratio. In this case the ratio of the Area over the square of the Length is a constant. SUMMARY: The pattern of a constant ratio continues. For the Square Relationship, the ratio of the dependent variable over the square of the independent variable is a constant. Which of the following sets of data represent an example of a Square Relationship?

13 Square Root Relationship -----Review The Square Root relationship does not show a straight line plot nor does it show a factor change that is the same for both variables. Recall that the factor change of the dependent variable is the square root of the factor change of the independent variable. The following two ratios are tested. The Square Root Relationship also demonstrates a Constant Ratio. In this case the ratio of the Period over the square root of the Radius is a constant. SUMMARY: The pattern of a constant ratio continues. For the Square Root Relationship, the ratio of the dependent variable over the square root of the independent variable is a constant.

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15 Straight Line Graphs Recall how the Direct relationship data had a constant ratio as well as a straight line plot. The constant ratio has also been found for each of the relationships. Since constant ratios can be determined for each relationship, is it possible to generate a straight line plot using the dependent and independent variable? The answer to this question can be found in a pattern already shown in the direct relationship results. Those results illustrated a constant of the dependent variable over the independent variable and also a straight line of the dependent variable versus the independent variable. The following demonstrates the plots of the dependent variable versus the independent variable raised to the power used to generate the constant ratio. Inverse Relationship Square Relationship Square Root Relationship

16 Indeed straight line plots do exist for each type of relationship. Note the pattern: The Same Factor Change One more pattern needs to be discussed. Each relationship had a constant ratio and a straight line plot both of which were first demonstrated in the direct relationship. In addition, whenever the independent variable doubled the dependent variable also doubled. Is it possible to find a quantity which as it is doubled the dependent variable also doubles? The pattern found in the last two sections will be used to develop the idea of a constant factor change. Inverse Relationship Square Relationship Square Root Relationship

17 The same factor increase can be determined for each relationship. Note the pattern: Which relationship does each of the following sets of data represent?

18 Proportions When each of the following can be found for a set of data pairs 1) same factor change 2) constant ratio 3) straight line plot scientists summarize those phenomenon by saying that a direct proportion exists between the two variables. The words is proportional to will be used several times in the following paragraphs and will be symbolized with the symbol (Dependent Independent) Direct Relationship The results of the direct relationship are summed up by saying that the mass is proportional to the volume or the mass the volume. The statement is summing up each of the following: I. When the volume doubles so does the mass II. There is a constant ratio of mass/volume III. There is a straight line plot of mass versus volume. Also the constant ratio expression (k = (mass/volume)) can also be written Mass = k * Volume Inverse Relationship The results of the inverse relationship are summed up by saying that the period is proportional to the inverse of the frequency or Period 1/Frequency The statement is summing up each of the following: I. When the inverse of the frequency doubles so does the mass II. There is a constant ratio of the period over the inverse of the frequency III. There is a straight line plot of period versus 1/frequency. k = (Period/(1/frequency))or Period = k * (1/frequency) Square Relationship The results of the square relationship are summed up by saying that the Area is proportional to the square of the Length or Area Length 2 I. When the square of the length doubles so does the area II. There is a constant ratio of the Area over the square of the Length III. There is a straight line plot of the Area versus Length squared k = (Area/Length 2 ) or Area = k *Length 2 Square Root Relationship The results of the square root relationship are summed up by saying that the Period is proportional to the square root of the radius or Period Radius I. When the square root of the radius doubles so does the period II. There is a constant ratio of the Period over the square root of the radius III. There is a straight line plot of the Period versus the square root of the radius k = (Period/ Radius ) or Period = k * Radius

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