In this Cyberchase video segment, Harry wants to visit his grandmother. He decides that the cheapest way for him to get there is to travel by unicycle, but he wonders if he can get there before dark. Using a line graph, he tries to predict the amount of time it will take to travel the twenty miles, assuming he travels at a constant speed. Once he sets out on his unicycle, he charts his progress on a new line graph. After the first hour he appears to be ahead of schedule, but he is not able to keep up the pace and soon finds himself falling behind.
Once you have collected a set of data, it is often helpful to organize and display the data graphically to help you analyze it. You can list the data in a table, or represent it visually in a graph, such as a bar graph, line graph, pie chart, or a stem-and-leaf plot. Sometimes a graph is helpful because it will illustrate something that may not be immediately apparent when looking at a list of numbers alone. Line graphs, in particular, can be a useful tool in making predictions.
Line graphs make use of ordered pairs and allow you to see how two variables relate to each other. Each variable will be plotted along either the horizontal axis (x-axis) or the vertical axis (y-axis). Usually the horizontal axis shows numbers representing the independent variable (the variable that stands alone and isn't changed by the other variables you are measuring). The vertical axis contains values for the dependent variable (the variable determined by the independent variable).
Connecting the plotted points creates a number of line segments. The slope of each line segment represents the rate of change, (the ratio of the change in the x value with respect to the change in the y value) over that period of time. If the points happen to all lie on one line, then the rate of change is constant.
When extending a line graph in order to make a prediction, you are assuming that a constant rate will continue indefinitely. For example, at a constant rate of 50mph, a car will have traveled 50 miles in one hour, 100 in two hours, and so on. So you can predict that after 10 hours, the car will have traveled a total of 500 miles. In some cases, when the rate is not constant, simply extending the line graph will not provide an accurate prediction. This occurs when the rate changes over time. Take the car example again: Suppose the car stopped for a break for an hour, then had to slow down to 30 mph for a half-hour in a construction zone, and then was stuck in traffic for another hour. In this case the actual distance traveled at the end of 10 hours will be much less than your prediction based on a constant rate of 50 mph.
For a lesson plan involving line graphs, check out Line Graphs Showing Change Over Time.
HARRY: I hope you feel better soon, Grandma.
My grandma is such a great person. She’s the one who helped me start my collections of old stuff. I really would like to visit her, but I don’t have the cash to buy a bus ticket. Maybe I can get to her house some other way. It’s too far to walk. But maybe I can use my unicycle.
Grandma lives 20 miles away. If I hit the road tomorrow by noon, can I get there before dark?
Maybe a graph can help. Here’s the distance I need to travel. Here’s the number of hours I could be traveling. When I go to Wave Hill, which is five miles away, it takes me one hour. So that means I can go 10 miles in 2 hours, and 15 miles in 3 hours. So if I connect the points and extend the line, that means I should get to Grandma’s in four hours. Great, Grandma will be so happy to see me.
The alarm didn’t go off!
Twelve on the dot! Gotta get going!
It’s 1:00. I’ve been traveling for one hour, and I’ve gone 8 miles. Wow, I’m ahead of schedule. At this rate I’ll get there before 3:00.
Yes, we ate at this great place that’s called Danny’s.
Yikes, I’ve only gone 10 miles in 3 hours. I’m 5 miles behind where I want to be. It’s 3:45. I’ve got 7 miles to go. I’ll never make it to Grandma’s house before dark.
GIRL: Hi Harry.
HARRY: Hi, Raney.
GIRL: What are you doing here?
HARRY: I was going to my Grandma’s house and my tire went flat, so I had to put air in it.
GIRL: We live near your Grandma’s house; we’ll give you a ride.
WOMAN: Sure, come one.
HARRY: That rocks! All right, I will make it to Grandma’s house by 4:00.
GRANDMA: Harry, what a wonderful surprise.
HARRY: Hi, Grandma, I’m happy to see you’re up and about. I thought you were sick.
GRANDMA: No, I feel much better than I did last night. As a matter of fact, I’m going to take a spin on my unicycle before it gets dark, what to join me?
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