Seeing Is Believing: Graphing as a Visual Tool - Picturing and Measuring - Graphs, Measures, Stats, and Sets - Basic Math & Pre-Algebra For Dummies

Basic Math & Pre-Algebra For Dummies, 2nd Edition (2014)

Part IV. Picturing and Measuring - Graphs, Measures, Stats, and Sets

Chapter 17. Seeing Is Believing: Graphing as a Visual Tool

In This Chapter

arrow Making comparisons with a bar graph

arrow Dividing things up with a pie chart

arrow Charting change over time with a line graph

arrow Plotting points and lines on an xy-graph

A graph is a visual tool for organizing and presenting information about numbers. Most students find graphs relatively easy because they provide a picture to work with rather than just a bunch of numbers. Their simplicity makes graphs show up in newspapers, magazines, business reports, and anywhere clear visual communication is important.

In this chapter, I introduce you to four common styles of graphs: the bar graph, the pie chart, the line graph, and the xy-graph. I show you how to read each of these styles of graphs to obtain information. I also show you how to answer the types of questions people may ask when they want to check your understanding.

Looking at Three Important Graph Styles

In this section, I show you how to read and understand three styles of graphs:

· The bar graph is best for representing numbers that are independent of each other.

· The pie chart allows you to show how a whole is cut up into parts.

· The line graph gives you a sense of how numbers change over time.

Bar graph

A bar graph gives you an easy way to compare numbers or values. For example, Figure 17-1 shows a bar graph comparing the performance of five trainers at a fitness center.


Illustration by Wiley, Composition Services Graphics

Figure 17-1: The number of new clients recorded this quarter.

As you can see from the caption, the graph shows how many new clients each trainer has enrolled this quarter. The advantage of such a graph is that you can see at a glance, for example, that Edna has the most new clients and Iris has the fewest. The bar graph is a good way to represent numbers that are independent of each other. For example, if Iris gets another new client, it doesn't necessarily affect any other trainer's performance.

Reading a bar graph is easy when you get used to it. Here are a few types of questions someone could ask about the bar graph in Figure 17-1:

· Individual values: How many new clients does Jay have? Find the bar representing Jay's clients and notice that he has 23 new clients.

· Differences in value: How many more clients does Rita have than Dwayne? Notice that Rita has 20 new clients and Dwayne has 18, so she has 2 more than he does.

· Totals: Together, how many clients do the three women have? Notice that the three women — Edna, Iris, and Rita — have 25, 16, and 20 new clients, respectively, so they have 61 new clients altogether.

Pie chart

A pie chart, which looks like a divided circle, shows you how a whole object is cut up into parts. Pie charts are most often used to represent percentages. For example, Figure 17-2 is a pie chart representing Eileen's monthly expenses.


Illustration by Wiley, Composition Services Graphics

Figure 17-2: Eileen's monthly expenses.

You can tell at a glance that Eileen's largest expense is rent and that her second largest is her car. Unlike the bar graph, the pie chart shows numbers that are dependent upon each other. For example, if Eileen's rent increases to 30% of her monthly income, she'll have to decrease her spending in at least one other area.

Here are a few typical questions you may be asked about a pie chart:

· Individual percentages: What percentage of her monthly expenses does Eileen spend on food? Find the slice that represents what Eileen spends on food and notice that she spends 10% of her income there.

· Differences in percentages: What percentage more does she spend on her car than on entertainment? Eileen spends 20% on her car but only 5% on entertainment, so the difference between these percentages is 15%.

· How much a percent represents in terms of dollars: If Eileen brings home $2,000 per month, how much does she put away in savings each month? First notice that Eileen puts 15% every month into savings. So you need to figure out 15% of $2,000. Using your skills from Chapter 12, solve this problem by turning 15% into a decimal and multiplying:

· $2,000 × 0.15 = $300

So Eileen saves $300 every month.

Line graph

The most common use of a line graph is to plot how numbers change over time. For example, Figure 17-3 is a line graph showing last year's sales figures for Tami's Interiors.


Illustration by Wiley, Composition Services Graphics

Figure 17-3: Gross receipts for Tami's Interiors.

The line graph shows a progression in time. At a glance, you can tell that Tami's business tended to rise strongly at the beginning of the year, drop off during the summer, rise again in the fall, and then drop off again in December.

Here are a few typical questions you may be asked to show that you know how to read a line graph:

· High or low points and timing: In what month did Tami bring in the most revenue, and how much did she bring in? Notice that the highest point on the graph is in November, when Tami's revenue reached $40,000.

· Total over a period of time: How much did she bring in altogether the last quarter of the year? A quarter of a year is three months, so the last quarter is the last three months of the year. Tami brought in $35,000 in October, $40,000 in November, and $30,000 in December, so her total receipts for the last quarter add up to $105,000.

· Greatest change: In what month did the business show the greatest gain in revenue as compared with the previous month? You want to find the line segment on the graph that has the steepest upward slope. This change occurs between April and May, where Tami's revenue increased by $15,000, so her business showed the greatest gain in May.

Using the xy-Graph

When math folks talk about using a graph, they're usually referring to an xy-graph (also called the Cartesian coordinate system), shown in Figure 17-4. In Chapter 25, I tell you why I believe this graph is one of the ten most important mathematical inventions of all time. You see a lot of this graph when you study algebra, so getting familiar with it now is a good idea.


Illustration by Wiley, Composition Services Graphics

Figure 17-4: An xy-graph includes horizontal and vertical axes, which cross at the origin (0, 0).

remember_4c.eps A Cartesian graph is really just two number lines that cross at 0. These number lines are called the horizontal axis (also called the x-axis) and the vertical axis (also called the y-axis). The place where these two axes (plural of axis) cross is called the origin.

Plotting points on an xy-graph

Plotting a point (finding and marking its location) on a graph isn't much harder than finding a point on a number line — after all, a graph is just two number lines put together. (Flip to Chapter 1 for more on using the number line.)

remember_4c.eps Every point on an xy-graph is represented by two numbers in parentheses, separated by a comma, called a set of coordinates. To plot any point, start at the origin, where the two axes cross. The first number tells you how far to go to the right (if positive) or left (if negative) along the horizontal axis. The second number tells you how far to go up (if positive) or down (if negative) along the vertical axis.

For example, here are the coordinates of four points called A, B, C, and D:


Figure 17-5 depicts a graph with these four points plotted. Start at the origin, (0, 0). To plot point A, count 2 spaces to the right and 3 spaces up. To plot point B, count 4 spaces to the left (the negative direction) and then 1 space up. To plot point C, count 0 spaces left or right and then count 5 spaces down (the negative direction). And to plot point D, count 6 spaces to the right and then 0 spaces up or down.


Illustration by Wiley, Composition Services Graphics

Figure 17-5: Points A, B, C, and D plotted on an xy-graph.

Drawing lines on an xy-graph

When you understand how to plot points on a graph (see the preceding section), you can begin to plot lines and use them to show mathematical relationships.

The examples in this section focus on the number of dollars two people, Xenia and Yanni, are carrying. The horizontal axis represents Xenia's money, and the vertical axis represents Yanni's. For example, suppose you want to draw a line representing this statement:

Xenia has $1 more than Yanni.


Now you have five pairs of points that you can plot on your graph as (Xenia, Yanni): (1,0), (2,1), (3,2), (4,3), and (5,4). Next, draw a straight line through these points, as in Figure 17-6.


Illustration by Wiley, Composition Services Graphics

Figure 17-6: All possible values of Xenia's and Yanni's money if Xenia has $1 more than Yanni.

This line on the graph represents every possible pair of amounts for Xenia and Yanni. For example, notice how the point (6,5) is on the line. This point represents the possibility that Xenia has $6 and Yanni has $5.

Here's a slightly more complicated example:

· Yanni has $3 more than twice the amount that Xenia has.

Again, start by making the same type of chart as in the preceding example. But this time, if Xenia has $1, then twice that amount is $2, so Yanni has $3 more than that, or $5. Continue in that way to fill in the chart, as follows:


Now plot these five points on the graph and draw a line through them, as in Figure 17-7.


Illustration by Wiley, Composition Services Graphics

Figure 17-7: All possible values of Xenia's and Yanni's money if Yanni has $3 more than twice the amount Xenia has.

As in the other examples, this graph represents all possible values that Xenia and Yanni could have. For example, if Xenia has $7, Yanni has $17.