2.2.1: Preparation M.2

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In Collaboration M.2, we will be using linear models, which are models that can be used when the data lies along (or approximately along) a line. The following definitions are important for linear models.

Definition: The slope of a line is the ratio of the change in the vertical value (sometimes called output, y-value, dependent variable, or response variable) to the change in the horizontal value (sometimes called input, x-value, independent variable, or explanatory variable) as we move between any two points on the line. The slope of a line is therefore a rate of change. Often it is referred to as the ratio of the rise of the graph to the run of the graph, or “rise over run.” The slope can also be described as the rate at which the output values are changing (increasing or decreasing) as the input value increases by 1. It can be found by using any two points on the line. (Note: By convention, the input is on the horizontal axis and the output on the vertical axis.)

Definition: The vertical intercept of a line is the point at which the line intersects the vertical axis, where the input value is zero. This is also called the initial value if the input starts out at a value of zero. A vertical intercept is always written as a point in the form (0,#).

Definition: The horizontal intercept of a line is the point at which the line intersects the horizontal axis, where the output is zero. A horizontal intercept is always written as a point in the form (#,0).

Definition: A linear model has a constant rate of change (slope). (This is not true of all models and their corresponding equations.)

Two examples of graphical linear models are shown below:⁹

Figure 1

Figure 2

Graph showing an example of the Linear Model.

Y-axis ranges from negative 2 to 10.
X-axis ranges from negative 7 to 7.

Slope (rate of change) = 0.5
Horizontal intercept is shown at (negative 4,0)
Vertical intercept is shown at (0,2)

Notice that in Figure 1, the slope is 0.5. This is because every time we move 2 units to the right on the line, the line rises 1 unit, so the rate of change is the ratio 1/2 = 0.5. Alternatively, we could say that every time we move 1 unit to the right, the line rises ½ unit, so the rate of change is again the ratio (½)/1 = 0.5. Notice that we get the same slope either way, since all linear models have a constant rate of change.

In Figure 2, the slope is −1 because every time we move one unit to the right on the line, the line falls 1 unit (a change of −1), so the rate of change is the ratio (−1)/1 = −1.

(1) The following picture shows five skiers riding up a mountain on a ski lift.

$Image showing five skiers riding up a mountain on a ski lift, as described in the previous paragraphs.$

(a) Which of the skiers is on the steepest portion of the lift?

(b) If you wanted to be certain your answer to Question 1(a) is correct, what would you want to know about each of the line segments representing the lift in the picture? Consider the definitions for linear models that were just presented.

(2) Many old comic books contained ads offering money-making schemes to children reading the comics.¹⁰ As an example, suppose a company offered to send 30 greeting cards to a child to sell to relatives, friends, and neighbors. The child was to sell them for 50 cents each, and had to send $10 back to the company. Here is a graph of a linear model representing the situation. The input (horizontal value) represents the number of cards sold, and the output (vertical value) represents the child’s profit in dollars.

Graph showing an example of the Linear Model.

Y-axis ranges from negative 10 to 15.
X-axis ranges from 0 to 40.

0 = negative 10
20 = 0
40 = 10

(a) What is the vertical intercept? What does it represent in the context of this problem?

(b) What is the horizontal intercept? What does it represent in the context of this problem?

(c) What is the slope? (Hint: Since this line goes through both the horizontal and vertical axes, it would be easy to use both intercepts to find the slope.)

(d) How much money will the child earn if all 30 cards are sold and the agreed upon $10 is sent back to the company?

(3) The City of Minneapolis Construction Code contains the following requirements:

ROOF SLOPE:

The code requires a roof slope (rise divided by run) of 2:12 or greater for asphalt shingles. However, asphalt roofs sloped between 2:12 and 4:12 will require a double layer of felt paper that goes under the shingles to prevent leakage.

The slope of a roof is usually written in ratio notation. For example, a roof with a slope of 2/12 would be written as 2:12. Note that roofers write slope as 2:12 rather than reducing the answer to 1:6. Why do they leave it that way? It’s not because they are bad at math, rather it is because a denominator of 12 is preferred because there are 12 inches in a foot!

For roof slopes of 2:12 up to 4:12, two layers of approved felt are required underneath. For roof slopes of 4:12 or greater, one layer of approved felt is required underneath.¹¹

Suppose Joe lives in Minneapolis. He measures his roof, and from the gutter to the peak, it runs 15 feet horizontally and rises 4.5 feet vertically.

$Image of a roof, as described in the previous paragraph.$

(a) What is the slope of Joe’s roof?

(b) How many layers of felt must Joe put on his roof to meet code? Select one answer.

(i) He is not allowed to use asphalt shingles with felt.

(ii) He must use one layer of felt.

(iii) He must use two layers of felt.

(iv) His slope can’t be compared to the ones in the code, so it can’t be determined.

After Preparation M.2 (survey)

You should be able to do the following things for the next collaboration. Rate how confident you are on a scale of 1–5 (1 = not confident and 5 = very confident).

Before beginning Collaboration M.2, you should understand the concepts and demonstrate the skills listed below:

Skill or Concept: I can …	Rating from 1 to 5
read an input/output table.
plot points from an input/output table.
find slopes and intercepts of linear models.

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⁹ Graphs created at http://www.meta-calculator.com/online/

¹⁰ http://mentalfloss.com/article/57958/6-get-rich-quick-schemes-vintage-comic-books

¹¹ http://www.minneapolismn.gov/www/groups/public/@regservices/documents/webcontent/convert_272400.pdf

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