Interpreting Data as a Line

The Lesson Activities will help you meet these educational goals:

·  Mathematical Practices—You will make sense of problems and solve them, construct viable arguments and analyze the reasoning of others, and use appropriate tools strategically.

·  Inquiry—You will analyze results; communicate your results in tables, graphs, and written form; and draw conclusions.

·  STEM—You will apply mathematical and technology tools and knowledge to analyze real-world situations.

·  21st Century Skills—You will employ online tools for analysis and communicate effectively.

Directions

You will evaluate some of these activities yourself, and your teacher may evaluate others. Please save this document before beginning the lesson and keep the document open for reference during the lesson. Type your answers directly in this document for all activities.

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Self-Checked Activities

Read the instructions for the following activities and type in your responses. At the end of the lesson, click the link to open the Student Answer Sheet. Use the answers or sample responses to evaluate your own work.

1.  Linear Relationships

At your new job, the majority of your earnings come from commission. For every dollar’s worth of shoes you sell, you receive a 10% commission. In addition, you get daily pay of $9.50 for doing chores such as dusting the display fixtures, straightening stock, and other tasks that keep the working environment pleasant and organized. These tasks are shared by all the sales staff and take a minimal amount of time.

Naturally, you want to know how much you will earn at your new job. For this you need to understand the relationship between three things:

·  amount of sales you make in a day

·  amount of commission you receive

·  amount of earnings you receive

a.  The table contains the data for your first week’s sales. Complete the table by calculating your commission and earnings for each day of the week.


Sample answer:

Day / Sales / Commission
(10% of sales) / Non-Sales Pay / Earnings
(commission + non-sales pay)
Mon. / $2,200 / $220.00 / $9.50 / $229.50
Tues. / $2,000 / $200.00 / $9.50 / $209.50
Thurs. / $3,134 / $313.40 / $9.50 / $322.90
Fri. / $2,417 / $241.70 / $9.50 / $251.20
Sat. / $3,579 / $357.90 / $9.50 / $367.40

b.  Graph the values for daily sales and daily commission on the scatter plot tool. (If you need help, use the tool instructions.) Remember that sales is the independent variable to be graphed on the x-axis, and commission is the dependent variable to be graphed on the y-axis.

Make sure to check Line of Best Fit so the graph will show a line to fit your data values. Scale the graph to include the point (0, 0). Finally, be sure to title your graph and label the axes.

Export your graph and paste it in the space below. What form does your graph take?

Sample answer:

The graph is a straight line (linear) with a positive slope (the line slopes upward to the right).

c.  Assume that you don’t know the commission rate, but you do know the actual amount of daily commission you earned and the total dollar value of the shoes you sold. Based on these values, you can derive a formula for the commission rate.

Using the slope-intercept form of a line, write an equation to calculate the commission, y, based on sales, x. Hint: The y-intercept will be zero since you get no commission if you have no sales.

Sample answer:

Equation for commission, y:

y = mx + b, where m = slope and b = y-intercept

Using the two most extreme data pairs to define the slope:

slope =

Next, calculate the y-intercept. Since the commission is based solely on sales, the commission for zero dollars of sales will be zero. So, the y-intercept will also be zero, which means the y-intercept will be at (0, 0).

Therefore, the equation for commission, y, is:

y = 0.10x + 0

y = 0.10x

d.  Using the equation you found, calculate the daily commission for each day's sales and complete the table. Do the values for commission in the new table match with the values in the table from part a?

Sample answer:

Day / Sales (x) / Commission (y)
Mon / $2,200 / $220.00
Tues / $2,000 / $200.00
Thurs / $3,134 / $313.40
Fri / $2,417 / $241.70
Sat / $3,579 / $357.90

The values in the new table match the values in the table from part a.

e.  In everyday English, interpret the financial meaning of the slope in the equation.

Sample answer:

The slope is m = 0.10, which indicates that for every dollar’s worth of shoes you sell, you earn 0.10 dollars commission. So, the slope is the commission rate.

f.  Now, use the scatter plot tool again to graph the data points for daily sales and daily commission. Scale and label this graph appropriately. Export your graph and paste it in the space below. How does the new graph differ from the previous graph for daily sales and daily commission?

Sample answer:

They’re identical. The new graph has the same slope, m = 0.10. Both graphs also have a y-intercept of 0.

g.  How would you modify the equation for daily commission based on daily sales to write an equation for daily earnings based on daily sales?

Sample answer:

The equation for daily commission (y) is y = 0.10x.

daily earnings = daily commission + daily pay for non-sale activities.

Therefore, the equation for daily earnings is y = 0.10x + 9.50.

h.  Use the equation for daily earnings to calculate your earnings on a day when you sold $1,750 worth of shoes.

Sample answer:

The equation for daily earnings is y = 0.10x + 9.50, where x is the amount sold.

y = 0.10x + 9.50

y = 0.10(1,750) + 9.50

y = 175 + 9.50

y = 184.50

So, when you sell $1,750 worth of shoes, you earn $184.50.

i.  In everyday English, interpret the financial meaning of the y-intercept in the equation
y = 0.10x + 9.50.

Sample answer:

The y-intercept is 9.50, which indicates that you will earn a fixed amount of $9.50 even when the sale of shoes is zero dollars.

2.  Determining Women’s Shoe Size

This data table gives women’s shoe sizes based on the length of the foot from heel to the longest toe as measured on the Brannock device.

Length of Foot (inches) / Women’s Shoe Size
8.00 / 3
8.33 / 4
8.67 / 5
9.00 / 6
9.33 / 7
9.67 / 8
10.00 / 9
10.33 / 10
10.67 / 11
11.00 / 12
11.33 / 13

a.  In this case, which is the independent variable and which is the dependent variable? Explain your answer.

Sample answer:

The length of the foot is the independent variable and shoe size is the dependent variable. The shoe size may change depending on the length of the foot, but the length of the foot does not change depending on the shoe size.

b.  Graph the points using the scatter plot tool, with the length of the foot being the independent variable (horizontal axis) and the shoe size being the dependent variable (vertical axis). Notice that all the points fit exactly on a single line that rises from left to right. This indicates a positive linear relationship between the length of the foot and the shoe size.

Export your graph and paste it in the space below. How much change in foot length causes the shoe size to increase by one unit? Is the change consistent throughout the table?

Sample answer:

As the foot length increases by one-third inch, the shoe size increases one unit. This change is consistent throughout the table.

c.  What is the slope of the line?

Sample answer:

slope =

In this example, slope is calculated using the points (8, 3) and (9, 6). But the slope will remain the same no matter which points are chosen.

d.  How can you interpret the meaning of slope in context of the situation?

Sample answer:

The slope is m = 3, which indicates that for every increase of an inch in foot length, shoe size increases by 3 units. Or, for every increase of one-third inch in foot length, shoe size increases by one unit.

e.  Find the equation for the graph of the line. Hint: Substitute any two values from the data table for x and y in the slope-intercept form to find the y-intercept value.

Sample answer:

slope-intercept form:

y = mx + b

3 = 3 × 8 + b

3 = 24 + b

b = 3 – 24

b = -21

Substituting in the y-intercept and the slope, the equation of the line is y = 3x – 21.

f.  Imagine that you graphed a new set of data points by interchanging the dependent and independent variables. What would the slope of that line be? How could you interpret that slope in context of this situation? Is that interpretation logical?

Sample answer:

The slope, m, would be 0.33 (reciprocal of 3).

This slope indicates that for every increase of one unit in shoe size, foot length increases by one-third inch. This interpretation is not logical because foot length will not change based on the size of the shoes purchased. Rather, it’s the other way around: shoe size depends on foot length.

3.  Shopping Online for Shoes

Data table for the European adult shoe sizes.

Foot Length (cm) / European Adult Shoe Size
22.00 / 36
22.67 / 37
23.33 / 38
24.00 / 39
24.67 / 40
25.33 / 41
26.00 / 42
26.67 / 43
27.33 / 44
28.00 / 45
28.67 / 46
29.33 / 47
30.00 / 48

a.  Is there a linear relationship between European shoe sizes and foot length? Support your answer without graphing the data points.

Sample answer:

As foot length increases by 0.67 cm, shoe size increases by one unit. This relationship is constant throughout the table. This shows that there is a linear relationship between European shoe sizes and foot length.

b.  If you graphed the points, what would the slope of the line be? What would the y-intercept of the line be?

Sample answer:

The length of the foot is the independent variable (horizontal x-axis) and the shoe size is the dependent variable (vertical y-axis). So, the slope can be calculated as

slope = or 1.5

To find the y-intercept, use the slope-intercept form, y = mx + b.

Substituting y = 36 and x = 22 in the slope-intercept form:

36 = × 22 + b

36 = 33 + b

b = 36 – 33

b = 3

So, the y-intercept is b = 3

c.  How can you interpret the slope, m = 1.5, in context of the current scenario?

Sample answer:

For every increase of 1 cm in foot length, shoe size increases by 1.5 units.

d.  Suppose that you are looking online for shoes for your brother and find a style that you want to buy. However, those shoes are not available in the store where you work. An online store sells them, though, and the store provides information on shoe sizes based on foot length. The table lists the information provided by the store about men’s shoe sizes.

Foot Length (inches) / Men’s Shoe Size
9.25 / 6
9.50 / 7
9.75 / 8
10.25 / 9
10.50 / 10
10.70 / 11
11.25 / 12
11.50 / 13
11.75 / 14
12.25 / 15

Find the slope using any two set of values from the table.

Sample answer:

Foot length is the independent variable (horizontal x-axis) and shoe size is the dependent variable (vertical y-axis). So, the slope can be calculated as

slope =

e.  Check the slope for other sets of values in the table. Is the slope (rate of change) constant for all of the values in the table? If not, list the different slopes you can calculate using the values from the table.

Sample answer:

Slope is not constant for all values in the table. The slope changes depending on the set of values analyzed:

slope 1 =

slope 2 =

slope 3 =

f.  Use the scatter plot tool to graph all the data points from the table. Export your graph and paste it in the space below. Can the relationship between the shoe sizes and the foot lengths given in the table be described by a linear equation? Explain why or why not.

Sample answer:

When I calculated the slope, I saw that the rate of change is not constant throughout the table. The data points in the graph cannot all be connected with a straight line (although the best-fitting line comes close to all the points). So, the relationship between the shoe sizes and the foot lengths cannot be described exactly by a linear equation. The relationship is very nearly linear, though.

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