Relationships Using First Differences to Determine if a Relationship is Linear

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1 MPM 1D Name: Relationships Date: Using to Determine if a Relationship is Linear Big Ideas: If the x-values in a table of values go up by a constant amount and the y-values also go up by a constant amount, then, the relationship between the two variables is linear. If the x-values in a table of values go up by a constant amount then we use first differences to determine if consecutive y-values are also going up by a constant amount. To accomplish this, always subtract the y-value immediately higher in the table of values from the one immediately lower. Example: These values of x are changing by one unit each time so we are allowed to use first differences in the y-values to see if the relationship is linear. x y (Always subtract first y-value from the second y-value. ) = = = 2 Conclusion: Because the first differences in the y values are equal while the x-values are going up by a constant value, as well, the relationship between x and y is linear. Check: Plot the data to see if your prediction from first differences is correct. Make your graph using all the techniques learned in class Remember: - A title that makes sense - Axes labelled with words and units - Axes scales that make sense - Use breaks in the axes if needed so that the points fill up the grid Practice Do the attached handouts.

2 Part A: Jody Jody is paid $8.50/hour to calculate perimeters of square tiles in a factory. i) Determine Jody s pay and record your answers in the Pay column of the table. Number of Hours 1 Pay ($) ii) Describe what happens to Jody s pay when the number of hours she works increases by one hour. iii) Construct a graph of Jody s pay vs. the number of hours she works. Include labels and titles. a) Which variable is the independent variable? Justify your choice. c) Describe the trend and relationship between the two variables by completing the statements below. As the number of hours increases, Jody s pay (increases; decreases; stays the same; moves randomly) so we see that the trend in the data is (down to the right; up to the right; there is no trend). In terms of relationship, the data show a (strong; weak; no) correlation. The relationship is (non-existent; positive; negative) and is (linear; non-linear). iv) Determine the first differences in the column of the table. What do you notice about the first differences?

3 v) Summarize your observations about the data for Jody by circling and completing the correct phrase within the brackets for each statement below. a) As the number of hours worked increases by one hour, the pay Jody earns (increases by an amount of ; decreases by an amount of ; stays the same). b) For Jody s data, the first differences are (not always the same; always the same). A linear relationship between two variables occurs when the first differences are (all the same; not all the same) and in this case, the plotted points show a (non-linear; linear) relationship. So, the relationship we see on the graph is (different; the same) as the relationship we predicted with first differences. Part B: Raj Raj, another employee at the factory, also works with the tiles. He helps to determine the shipping costs by calculating the area of each square tile. i) Determine the area for the missing square tiles and record your answers in the Area column of the table. Length of sides (cm) Area (cm 2 ) ii) Describe what happens to the area of each tile when the side length of a tile increases by one centimetre. iii) Which variable is the dependent variable in the graph for Raj s data? Justify your choice.

4 iv) Construct a graph of the area vs. the length of the sides of the tiles. Include labels and titles. vi) Use the graph to describe the relationship between the area and the side length of the tile. (HINT: Use Jody s information from question iii c) as a model to write your answer.) vii) Determine the first differences in the column of the Raj s table. What do you notice about the first differences? v) Summarize your observations. a) The trend shown by the plotted points is best described this way: As the length of the sides increases by one centimeter, the area covered by the tile (increases; decreases; stays the same) so the data move (up to the right; down to the right; randomly). b) For Raj s data, the first differences are (not always the same; always the same). A linear relationship between two variables occurs when the first differences are (all the same; not all the same) and in this case, the plotted points show a (non-linear; linear) relationship. So, the relationship we see on the graph is (different; the same) as the relationship we predicted with first differences.

5 Part C: Another Kind of First Difference Problem Laura is swimming lengths of the pool to prepare for a triathlon. The table below shows the number of lengths and the distances she swims. a) Determine the values of just the missing distances in the table below. b) For Raj and Jody, the independent variable changed by one unit for each successive data point. What is happening to the independent variable for Laura s data? Number of Lengths swum Distance (m) c) Now, complete the first differences in the table above. d) Based on your answers to b) and c) above, do you expect the relationship between number of lengths swum and distance to be linear or non-linear? (Choose ONE!) Justify your choice. d) Graph the data on the grid below. e) What is the length of the pool Laura is swimming in? Explain how you know. f) Complete this statement: As long as the independent variable changes by (the same; different) amounts for each successive data point AND the first differences for the dependent variable change by (the same; different) amounts for each successive point, then the relationship between the variables will be (linear; non-linear).