4.1.2 Reasoning Graphically about the Relationship between the Two Quantities
4.1.2 • Reasoning Graphically about the Relationship between the Two Quantities
Activity
Three days in a row, a dog owner tied his dog’s 5-foot-long leash to a post outside a store while he ran into the store to get a drink. Each time, the owner returned within minutes.
The dog’s movement each day is described here.
- Day 1: The dog walked around the entire time while waiting for its owner.
- Day 2: The dog walked around for the first minute and then laid down until its owner returned.
- Day 3: The dog tried to follow its owner into the store but was stopped by the leash. Then, it started walking around the post in one direction. It kept walking until its leash was completely wound up around the post. The dog stayed there until its owner returned.
- Each day, the dog was 1.5 feet away from the post when the owner left.
- Each day, 60 seconds after the owner left, the dog was 4 feet from the post.
Your teacher will assign one of the days for you to analyze. You will sketch a graph that could represent the dog’s distance from the post, in feet, as a function of time, in seconds, since the owner left.
Then, meet with someone in your group who had Day 2’s movement and someone who had Day 3’s movement. Work together to explain your graphs and make changes where needed before checking your work. Be sure to sketch each other’s graphs.
Day _____
Sketch the graph.
Compare your work: Your answer may vary, but here is a sample for each day.
Day 1
Day 2
Day 3
Are you ready for more?
Extending Your Thinking
From the graph, is it possible to tell how many times the dog changed directions while walking around? Be prepared to show your reasoning.
Compare your answer:
No. The graph only gives the distance from the post. The dog could change from moving clockwise around the post to moving counterclockwise around the post. These changes could not be seen on the graph.
Self Check
The graph below shows a person’s elevation for a period of 10 minutes. Which statement below is true based on the graph?
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The person never stopped climbing.
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During the first 3 minutes, the person was hiking up a hill.
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The person slowed down for the second half of the hike.
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The person sped up in the first 3 minutes.
Additional Resources
Representing Everyday Situations with Graphs
Graphing functions to represent everyday situations can be useful to provide a visual for what is happening in a situation.
Example 1
The graph shows the distance , in miles, for the time of each of Mia’s runs , in minutes.
Notice that at 0 minutes, Mia has run 0 miles. At 30 minutes, Mia has run 3 miles. At 180 minutes, Mia has run 18 miles.
Try it
Represent Everyday Situations with Graphs
Sky spends 5 hours at the library. She is on their internet for an hour. Then she takes an hour break to read a book. She then is back online for 2 hours before her friend comes to talk for an hour. She spends her last hour at the library on the internet.
Draw this situation where the time spent on the internet, in hours, is a function of time spent at the library, in hours.
Compare your answer:
Here is how to represent this situation in a graph: