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Section 2.2 Creating and Critiquing Graphs

Subsection 2.2.1 Overview

You have seen that graphs can be helpful in representing data. In Lesson 2.2, you will specifically explore the importance of scale when graphing equations.

Student Page 2.2.2 Location, Location, Location

Through this activity your group will create a graph to convey information about the number of people in a location over time. You will also interpret a graph prepared by another group and try to determine a location that fits the graph. To prepare, think about a place you go with some regularity. How many people are in that place at various times during the day?

To maximize the interest level, each group must choose very different types of locations. It helps if the numbers of people who go to the place each day are large. Secrecy of choice is critical! Make sure other groups cannot overhear your group's discussions!

Complete the following with your group:

1.

Choose a location and a day of the week to analyze. Get your teacher's approval before continuing (to assure that your group chose a unique location).

2.

Estimate how many people are likely to be in the location at the beginning of each hour of the day, starting at 6 AM and ending at 5 AM the following day. Record the data.

3.

Create a graph to accurately represent the data.

  • Use the horizontal axis to indicate the hour of the day.

  • Use the vertical axis to indicate the estimated number of people in the location at the beginning of each hour.

  • Do not include any information on your graph that identifies the location your team is analyzing.

When you have accomplished the above, your teacher will share your graph with another group and provide another group's graph for your group to analyze.

4.

For the new graph:

(a)

Decide what location the graph likely represents.

(b)

Explain to each other why the graph could represent the location you think it does.

(c)

Write questions you have about the graph.

5.

What features of graphs help you interpret them? What makes these features helpful?

6.

Share the graph with the class when asked to do so. Ask questions to help you and others analyze the graph.

Location, Location, Location is adapted from Annenberg Learner, Teaching Math: A Video Library, 5-8, The Location, online website‚ÄČ18‚ÄČ, retrieved April 16, 2019.

Student Page 2.2.3 It's A Matter of Scale

How important are the scales chosen for both the horizontal and vertical axes to the appearance of a graph or a group of graphs? Think about this question as you work together on the following tasks.

Among groups, compare graphs for Student Page Exercise 2.2.3.3 and Task 2.2.3.4.a. What do you notice? Think about these questions:

Discuss the importance of scale.

Think about Location, Location, Location How important was scale in helping you determine the location of each place? Does the \(x\)-scale matter? Does the \(y\)-scale matter?

Directions: Each group member receives a card with an equation on it and both \(x\)- and \(y\)-scales to use to graph the equation. Cards are found in It's a Matter of Scale Card Set.

1.

Starting with zero at the origin, the point where the axes intersect, each tick mark increases by the amount of the scale provided. Label both the \(x\)- and \(y\)-axes showing the scale on the card.

Figure 2.2.3.1. Blank Graph for It's A Matter of Scale

2.

Create a table that fits the equation and \(x\)-scale you were given. Plot the points in the table on the coordinate grid. Graph the equation by connecting the points you found. Write the equation of the line on the graph you drew.

3.

(a)

Compare the graphs drawn by members of your group and class. What do you notice?

(b)

Look carefully at the equations plotted and the scales used. Should the graphs appear as they do? Why or why not?

(c)

Explain why the graphs appear as they do.

4.

Work with other members of your group.

Figure 2.2.3.2. New Blank Graph for It's A Matter of Scale
(a)

Graph the equations each person was provided on the same graph using the largest \(x\)- and \(y\)-scales provided in the set of cards.

(b)

What do you notice?

5.

Write about the importance of scale in graphing.

Student Page 2.2.4 It's a Matter of Scale Card Set

Directions: Print or copy the following, on separate sheets of paper. Cut the cards apart. Each group member receives one card with all four cards in a row assigned to the same group.

Cards to be used with It's A Matter of Scale. Note: This learning activity may not be accessible.

Cards continued.

Cards to be used with It's A Matter of Scale. Note: This learning activity may not be accessible.

Student Page 2.2.5 What's Wrong with this Picture? Critiquing Graphs

News articles and ads are sometimes accompanied by graphs. Sometimes these graphs are deceiving. Complete What's Wrong with this Picture? Critiquing Graphs, in light of what you have learned about scale. In each graph check to see if the \(\)- and \(y-\)scales are correct. Are the distances between each of the \(x\)-values to scale? Are the distances between each of the \(y\)-values to scale? How can you tell?

Plot each graph using an electronic tool. Set the \(x\) and \(y\) scales to be the same as those used in each graph. Compare the appearance of the graph you created with the designed graph.

Figure 2.2.5.1. Graphs for What's Wrong with this Picture? Critiquing Graphs

1.

Study the graphs in Figure 2.2.5.1. Some are accurate, some are not. For each graph:

(a)

Determine if it is drawn accurately. Recreate the graph electronically to help you decide.

(b)

Which graphs are good representations of the data?

(c)

Which graphs still need work to make them more accurate? What parts of these graphs need work? Describe any changes needed to help the viewer accurately interpret the information provided.

2.

Consider Table 2.2.5.2.

Table 2.2.5.2. 60th District Court Traffic Fines and Costs Schedule
Speeding on Interstate FINE
71-75 mph 115.00
76-80 mph 125.00
81-85 mph 135.00
86-95 mph 145.00
96 mph or above 155.00
Figure 2.2.5.3. Blank Graph for Task 2.2.5.2.c
(a)

The independent variable is the variable that changes independent of any other variable. Either you must manipulate this variable yourself, or it changes on its own; it does not rely on the other variable. In the table, what is the independent variable? Why do you think so?

(b)

The dependent variable is the variable that depends on another variable. What is the dependent variable in the table? Why do you think so?

(c)

Use Figure 2.2.5.3 to display the data in Table 2.2.5.2. In mathematics, the horizontal axis always represents the independent variable. The vertical axis always represents the dependent variable.

3.

What are important features of an accurate, informative graph?

Data sources for graphs, retrieved April 20, 2020:

Subsection 2.2.6 Important Features of Graphs

Through What's Wrong with this Picture? Critiquing Graphs, you had the opportunity to determine important features of accurate, informative graphs. Your list should include:

The Origin.

Both the \(x\)- and \(y\)-axes must include zero. The intersection of the two axes is called the origin. It represents \((0, 0)\text{,}\) the place where both \(x\) and \(y\) are zero.

Scale.

The distance between consecutive tick marks must represent the same distance as the distance between 0 and the first tick mark. The \(x\)-scale can be different from the \(y\)-scale. Use as much of the graph as possible while still using a scale that is easy to interpret.

Plotting Points.

Once the scale is chosen, the data points must be plotted accurately. Each point has 2 coordinates, \((x, y)\text{,}\) where \(x\) represents a value of the independent variable and \(y\) represents the corresponding value of the dependent variable. If the scales for \(x\) and \(y\) are chosen so that they are easy to interpret, it is likely the points will be relatively easy to locate and plot.

Labels.

Label both axes with the quantities they represent. The \(x\)-axis (horizontal) represents the independent variable. The \(y\)-axis (vertical) represents the dependent variable.

Title.

In formal documents, a title is necessary. For our purposes, a title on the graph is required when the graph represents data given in context.

Homework 2.2.7 Homework

1.

Optional. Go back to How To Learn Math For Students Directions. Complete How to Learn Math for Students Exercise 1.1.1.12. Completing the course helps you develop and build a growth mindset in mathematics. Keep the ideas you have learned from How to Learn Math for Students in mind as you complete Beginning Algebra Made Useful and any future courses you take in mathematics. Hand in your earned certificate of completion in class to earn privileges as designated by your professor for completing the course.

2.

One mile equals 2000 average steps while walking. If you walk at a rate of 3 miles per hour, you will average 100 steps per minute. ‚ÄČ23‚ÄČ

(a)

Complete Table 2.2.7.1.

Table 2.2.7.1. Table for Task 2.2.7.2.a
Number of minutes Number of steps taken
1 100
10
20
30
40
(b)

Plot the number of minutes versus the number of steps taken. Indicate the scales and titles for each axis.

3.

As of January 1, 2020, the minimum wage for non-tipped employees in Michigan is $9.65 per hour. How much will you earn if you work \(h\) hours? ‚ÄČ25‚ÄČ

(a)

Complete Table 2.2.7.2.

Table 2.2.7.2. Table for Task 2.2.7.3.a
Hours worked Amount earned
1
10
20
30
40
(b)

Plot the number of hours worked versus the amount earned. Indicate the scales and titles for each axis.

4.

The graph in Figure¬†2.2.7.3 indicates the average fuel efficiency in miles per gallon for a new car in the year shown. (Resource: Bureau of Transportation Statistics‚ÄČ28‚ÄČ, retrieved 4/21/20.))

Figure 2.2.7.3. Average Fuel Efficiency of U.S. Passenger Cars
(a)

Complete Table 2.2.7.4 for the data in Figure 2.2.7.3.

Table 2.2.7.4. Data of Fuel Efficiency of U.S. Passenger Carse
Year Adjusted year, \(x\) Fuel efficiency in mpg
1995 0  
2000 5  
2005  
2010  
2015  
(b)

Plot the data, Adjusted Year versus Average Fuel Efficiency.

(c)

Is the graph accurate for the set of data you listed in Task 2.2.7.4.a? Why or why not? Discuss both \(x\) and \(y\)-scales and other pertinent aspects of accurate graphs.

5.

Study Figure 2.2.7.5. Life expectancy at birth indicates the number of years a newborn would live if prevailing patterns of mortality at the time of birth stay the same throughout life.

Figure 2.2.7.5. Years of Life Expectancy at Birth
(a)

Create a table for the data in the figure. Year is the independent variable.

(b)

Adjust the values for the independent variable so 1960 is year 0.

(c)

Plot the data, adjusted year vs. Years of life expectancy at birth.

(d)

Is the figure accurate for the set of data you listed in Task 2.2.7.5.a? Why or why not?

6.

Find more graphs to analyze in print or online news sources. Find at least one graph that is complete and accurate. Find at least one graph that is misleading. In both cases, state briefly how you know the graph is accurate or how you know the graph is misleading.

learner.org/channel/schedule/printmat.html?printmat_id=123
huffingtonpost.com/raviparikh/lie-with-data-visualization_b_5169715.html
in2013dollars.com/New-cars/price-inflation
statista.com/statistics/256040/mcdonalds-restaurants-in-north-america/
statista.com/statistics/273550/data-breaches-recorded-in-the-united-states-by-number-of-breaches-and-records-exposed/
Resource‚ÄČ24‚ÄČ, retrieved April 21, 2020
verywellfit.com/pedometer-step-equivalents-for-exercises-and-activities-3435742
Resource‚ÄČ26‚ÄČ, retrieved April 21, 2020
paycor.com/resource-center/minimum-wage-by-state
paycor.com/resource-center/minimum-wage-by-state
bts.gov/content/average-fuel-efficiency-us-passenger-cars-and-light-trucks