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Patterns, Functions, and Algebra
Session 5 Part A Part B Part C Part D Part E Homework
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Session 5 Materials:



Solutions for Session 5, Part B

See solutions for Problems: B1 | B2 | B3 | B4 | B5 | B6| B7 | B8 | B9 | B10
B11 | B12 | B13

Problem B1

On a basic level, slope measures vertical change over some horizontal distance -- a road can "slope up" or "slope down." The slope of a line describes how much vertical change (change in y) there is per horizontal change (change in x).

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Problem B2

On a line, the ratio of rise to run is always constant; on a curved object, this value is constantly changing. So, the slope of a curved object changes depending on the points selected, while the slope of a line is always constant.

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Problem B3

From P to Q, the ratio is roughly 0.6.

From P to R, the ratio is roughly 1.3.

From Q to R, the ratio is roughly 4.

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Problem B4

The ratios for the graph of a line would be constant throughout. For example, a line connecting P to R would have a rise/run ratio of 1.3, regardless of where a new point was located on it. This is different from the graph used in Problem B3, which has a slope that varies depending on which two points are used.

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Problem B5

The coordinates are R(1, 6) and S(4, 12).

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Problem B6

The slope is (change in y) / (change in x). The change in y is 12 - 6 = 6, and the change in x is 4 - 1 = 3. Therefore, the slope is 6/3 = 2. Note that the line may not appear to have slope 2, since the vertical axis is labeled by twos, while the horizontal axis is labeled by ones.

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Problem B7

Graph A has a larger slope, even though graph B appears steeper. This appearance is caused by the different scale used in the two graphs. If it were placed on the other scale, the line in graph B would appear nearly flat!

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Problem B8

All the lines have the same slope at all points. None of the lines intersect, and they are always the same distance from each other -- they are all parallel lines.

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Problem B9

A line with negative slope travels down and to the right ("decreasing"). Lines with negative slope may still intersect one another. Any line with negative slope will travel through the top left and bottom right quadrants of the graph paper.

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Problem B10

The total rise / total run is 10/14, which is roughly 0.71.

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Problem B11

To calculate step rise and run, divide both rise and run by 18. The step rise is 10/18 feet (6 2/3 inches) and the step run is 14/18 feet (9 1/3 inches). The ratio of step rise / step run is (10/18) / (14/18), which again equals 5/7 or 0.71 -- the step rise and step run are in proportion to the total rise and total run.

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Problem B12

The total rise and run can be estimated accurately by multiplying the rise and run of a single step by the total number of stairs, because each stair has roughly the same rise and run.

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Problem B13

There are a lot of answers here!

Number of steps: between 21 and 28
Height of each step: between 6 and 8 inches
Run of each step: between 9 and 12 inches
Total run: between 189 and 336 inches
Ratio of step rise to step run: between 6/12 and 8/9
Ratio of total rise to total run: between 6/12 and 8/9
Regardless of what you did, the ratios of step rise to step run and total rise to total run should be identical.

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