The table below shows the number of jumping jacks completed after a given period of time in minutes.
Time (minutes) Jumping Jacks
1 50
2 100
3 150
4 200
Considering the jumping jacks: 50, 100, 150, 200, what is the common difference?
Now, think of this table as a set of ordered pairs. This means that the first row can be placed in an ordered pair as (1, 50). The second row can be written as (2, 100). Using this, what is the slope of the line that connects the first two points?
What is the slope of the line that connects the 3rd and 4th point?
What is the slope of the line that connects the 1st and the 4th point?
Is the common difference (aka slope aka rate of change) constant?
Why is it or is it not constant?
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The common difference can be determined by finding the difference between consecutive terms in the sequence.
For the jumping jacks sequence (50, 100, 150, 200), the common difference is 50.
To find the slope of the line that connects the first two points (1, 50) and (2, 100), we can use the formula: slope = (change in y)/(change in x).
The change in y is 100 - 50 = 50.
The change in x is 2 - 1 = 1.
Therefore, the slope of the line connecting the first two points is 50/1 = 50.
To find the slope of the line that connects the third and fourth point (3, 150) and (4, 200), the change in y is 200 - 150 = 50, and the change in x is 4 - 3 = 1. Therefore, the slope is 50/1 = 50.
To find the slope of the line that connects the first and fourth points (1, 50) and (4, 200), the change in y is 200 - 50 = 150, and the change in x is 4 - 1 = 3. Therefore, the slope is 150/3 = 50.
The common difference, which is also the slope, is constant at 50. This is because for each additional minute, 50 more jumping jacks are completed, leading to a consistent rate of increase.