Modeling with Tables, Graphs, and Equations

 

Objective: When done with this lesson, you will have demonstrated how to model simple problems using tables, graphs, and equations.

Approximate completion time: 3 hours

 

 

Would you like to live in a house like the one pictured above? Click on the image for more pictures of the same house.
Of course, that house is only a model. You couldn't live in that house--it's simply too small for humans to live in it.
Why would anyone take so much time to build a little house if no one will ever live in it? Think for a minute about different reasons someone would want a model of a house.

In this unit we will use models in much the same way an architect uses a model. Rather than constructing an entire scenario in real life size and time, we will use tables, graphs, and equations to capture the detail and ideas of a situation. With a good model we will be able to solve problems, make predictions, and better understand a situation without needing the whole scenario to unfold in actual size and time.

Your work with tables, graphs, and equations in our previous units will help tremendously as we focus now on modeling.

 

Example 1

"Peggy Plum bought $185 in supplies to finish the plumbing in a basement bathroom. She charges $32 an hour for her work. Peggy will charge only for the parts she bought and her labor on the bathroom."
Table Model
Graph Model
Equation Model
hours worked
total charge ($)
0
185
1
217
2
249
3
281
4
313
5
345

Let h = # hours Peggy works
Let T = total charge ($)

T = 185 + 32·h

THINK DEEPER: Please open up a new word document and answer these questions about this Peggy Plum scenario.
(Save the file as "assignment 5.01.yourname.doc" and keep it handy. Please copy and paste in the questions and remember to make your work stand out with something like blue font.)

  1. Explain how each model could be used to determine the total charge for 3 hours of work.
  2. Explain how each model could be used to determine the total charge for 10 hours of work.
  3. What advantages do you see in using the table as a model? The graph? The equation?
  4. No model is perfect. List some disadvantages you see with each model.
  5. If it were your bathroom that Peggy Plum was finishing, carefully explain which model would appeal to you the most.
  6. If you were Peggy's financial and marketing consultant, carefully explain which model would you recommend she use when speaking with the customer.
  7. If you were determining the final bill for Peggy, explain which model would you prefer.
  8. Review: What is the dependent variable in this situation?

 

Example 2

"Lonny wants to plant some grass. The recommended amount of seed is 4 pounds for every 1000 square feet of lawn."
Table Model
Graph Model
Equation Model
lawn area
(ft2)
seed needed (lbs)
0
0
1000
4
2000
8
3000
12
4000
16
5000
20

Let a = area of lawn (ft2)
Let S = seed needed (lbs)

S = a ÷ 250

 

THINK DEEPER: Please open up your "assignment 5.01.yourname.doc" and answer these questions about this Lonny's grass seed example.
Please copy and paste in the questions and remember to make your work stand out with something like blue font.)

  1. Which model you like the best? Explain why.
  2. Explain how each model could be used to determine the amount of grass seed needed for 2000 square feet of lawn.
  3. Explain how each model could be used to determine the amount of grass seed needed for 1600 square feet of lawn.
  4. In what circumstances would the equation be the preferred model?
  5. In what circumstances would the table be the preferred model?
  6. If you were designing the label for customers who would buy this grass seed, explain which model you would use on the label to help customers know how much they need to buy.

 

Every model has strengths and weaknesses. While tables tend to be easy to read for exact numbers, many people prefer the visual look of a graph. An equation looks so simple, yet it packs a punch for what it can accomplish. The goal for this lesson is that you would be able to use all three of these types of models.

 

As you work on making models, the following notes might come in handy.

 

 

Example 3. Along time ago we looked at the fun story of the dragon and doubling donuts. At that time we were trying to predict patterns and use variable expressions to help us. Let's take another look. Here's a screenshot after 8 doublings.

THINK DEEPER: Please open up your "assignment 5.01.yourname.doc" and answer these questions this doubling donuts example.
Please copy and paste in the questions and remember to make your work stand out with something like blue font.)

  1. Describe the rate at which the number of donuts increases. How is it like the increases in the tables in the other two examples? How is it different?
  2. Create your own table that shows the information in the screenshot and the next few doublings.
  3. Without graphing, what do you think the shape of the graph would look like in this situation?
  4. Create your own graph to model this situation. Be sure to label your axes, title, units, and scale. (Hint: Does the number of donuts grow discretely or continuously? Will you need dots or a curve for this model? )
  5. Try to write an equation that would model this situation. Be sure to label your variables.
  6. Demonstrate that your equation works by using 8 for the number of doubles to show there would be 256 donuts. Demonstrate that your equation works by using 0 for the number of doubles to show there would be 1 donut.
  7. Explain which model you believe best shows the rate of growth in the number of donuts.
  8. Explain which model you like best for this situation.

 

For practice:

In your "assignment 5.01.yourname.doc"

Remember to make your work stand out with something like blue font.
Remember to save your work as "assignment 5.01.yourname.doc".