Section 4.1 Another Type of Function?
Subsection 4.1.1 Overview
As with linear functions, nonlinear functions can arise in context. In this lesson, we will analyze the numbers of performers based on group number in two books by Kathi Appelt, Bats on Parade and Bat Jamboree. Though written for children, the underlying mathematics is much more advanced. The simplicity of the context helps us makes sense of the more complex mathematics.
As you complete the Batty Functions activity, look for several nonlinear patterns that arise in the stories. ^{ 42 }
Student Page 4.1.2 Batty Functions
1.
Listen to the story, Bats on Parade, by Kathi Appelt. (Find online a YouTube video of someone reading Bats on Parade.) Read the scenarios found in or adapted from the story. Scenario A describes the characters in the story as they appear in the book. In Scenario B, you will also count the flag bearer as shown in the story. In Scenario C, the story is adapted to include more flag bearers.
 Scenario A
The first group has 1 marcher in 1 row. The second group has 2 marchers in each of 2 rows. The third group has 3 marchers in each of 3 rows, and so on.
 Scenario B
Each group that joins the parade also has a single flag bearer. Include the flag bearer in the number of marchers in each group.
 Scenario C
Each group that joins the parade has as many flag bearers as columns in the group. For example, the third group of bats marches in 3 rows and 3 columns and has 3 flag bearers.
(a)
In Table 4.1.2.1, draw a tile pattern to match each Scenario A, B, and C.
(b)
Write the numbers of tiles for each of Scenarios A, B, and C in Table 4.1.2.2. The step number above becomes the group number in Table 4.1.2.2.
Scenario  Step 1  Step 2  Step 3  Step 4  Step 5 

A  




Number of Tiles  




B  




Number of Tiles  




C  




Number of Tiles  




D  




Number of Tiles  




Scenario A  Scenario B  Scenario C  Scenario D  Scenario E  

Group Number 
Number of Marchers in the Group 
Number of Marchers in the Group 
Number of Marchers in the Group 
Number of Marchers in the Group 
Number of Performers So Far 
1  
2  
3  
4  
5  
6  
7  
8  
9  
10  
\(x\) 
2.
For each of Scenarios A, B, and C:
(a)
What patterns do you see in the data?
(b)
Are any of the patterns linear? How do you know?
(c)
How are the columns of data related to each other?
3.
(a)
Graph each set of data with Group Number as the independent variable and the number of marchers for the scenario as the dependent variable.
(b)
How are the graphs related to each other?
4.
(a)
For each data set, find an equation giving the number of marchers in group \(x\text{.}\) Write the equations in the last row of Table 4.1.2.2.
(b)
Are any of the equations linear? Should they be? Explain.
5.
Listen to another story, Bat Jamboree, also by Kathi Appelt. (Google Bat Jamboree to find this story online in a YouTube video.) Complete Table 4.1.2.1 and Table 4.1.2.2 for Scenarios D and E.
 Scenario D
Each group has the same number of performers as the group number.
 Scenario E
The number of performers who have appeared so far is the number of new performers and all performers who appeared before the new group. For example, when Group 3 performs, the number of performers who have appeared so far plus the new group are \(1 + 2 + 3 = 6\) performers.
6.
(a)
Compare the data for Scenario E to the data for Scenarios A, B, and C. What do you notice?
(b)
Is the data for any of Scenarios A, B, or C related to Scenario E? How?
(c)
Graph the data with Group Number as the independent variable and Scenario E as the dependent variable.
(d)
How is the graph related to the graphs in Task 4.1.2.3.a?
(e)
Determine an equation to model the data for Scenario E as it relates to group number.
7.
Share your work on Batty Functions. Resolve differences and ask questions before continuing.
Activity 8. Solving Functions Graphically.
In Batty Functions, you found many equations, most of them nonlinear. Relationships between some of the data sets helped you determine other equations. What questions might we ask about the data sets that can be answered using the equations? Brainstorm with your group. Write at least 3 questions about Batty Functions that can be answered using one or more of the equations. Share your questions with the class.
Questions that are frequently asked about Batty Functions include:
How many bats are in group 7? Group \(n\text{?}\)
If a group has 37 bats, what is the group number? If a group has \(b\) bats, what is the group number?
What is a reasonable domain for each function?
What is a reasonable range for each function?
Did you ask questions like these? Look at the graphs you drew for Batty Functions How can a graph help you answer these questions? How can a table help? Choose some of the questions about Batty Functions that you and others wrote. Answer as many of them as you can in context. If you still have questions, write them down to ask once the Batty Functions questions have been answered.
Look at the graph in Figure 4.1.0.3. Suppose the graph represents the height, \(h\text{,}\) of a ball \(t\) seconds after it is thrown. Ask and answer similar questions as those above for this context and graph. Can you find the answer exactly in each case? Why or why not?
Homework 4.1.3 Homework
1.
Some of the patterns from Batty Functions might be familiar.
(a)
Which patterns from Batty Functions have you seen before?
(b)
Where have they come up in the past? In this text?
(c)
Can you now find equations for previous patterns from this text that you were wondering about? List the context, data, and equation.
2.
Look back through patterns you have explored in this text.
(a)
Find examples of nonlinear patterns. List where you found the patterns.
(b)
Identify which nonlinear patterns are similar to one or more of the Batty Functions.
3.
How many total bats appear in Bat Jamboree? How do you know?
4.
Notice that most of the Batty Functions are nonlinear. Look at the equations you found to represent each of the Batty Functions.
(a)
The Batty Function described for Column B arises from the story directly. How did you determine the equation?
(b)
The Batty Function described for Column C is related to the first one. How?
(c)
The Batty Function described for Column D is also related to the first one. How did you determine the equation?
(d)
How did you determine the equation that represents the Batty Function described for Column F?
(e)
Are any of the equations above related to linear functions? How?
5.
Terri wants to build a rectangular pen for her goats. A barn will form the wall on one side of the pen. A garage will form another wall next to the barn. No fencing is needed on these sides of the pen. She has 15 yards of fencing to use to make the pen. Help Terri determine the dimensions that will make the area of the pen as large as possible. (Recall: For a rectangle,Area = Length x Width.)
(a)
Explain how Figure 4.1.3.1 is related to this context.
(b)
Some possible lengths to use all 15 yards of fencing are shown in Table 4.1.3.2. Complete Table 4.1.3.2 for other dimensions.
\(x\)  1  2  3  4  5  
\(15  x\)  14  13  12  
Area of pen  14  26  36 
(c)
What patterns do you notice in the table?
(d)
Write an equation that can be used to find the area of the pen based on the side lengths of the pen.
(e)
Graph the equation. For what value of \(x\) is the area largest? How do you know?
(f)
Is the area function linear or nonlinear function? Why do you think so?
6.
Holly wants to build a rectangular pen for her horses. One side of the pen will share the wall of the stable. The other three sides of the pen will be fenced. Her stable is 100 feet on one side. She can afford 150 feet of fencing.
(a)
Draw a diagram to illustrate this situation.
(b)
Find at least 5 pairs of dimensions that will use all of the fencing Holly can afford. Build an organized table with the dimensions. Also find the area of the pen in each case.
(c)
Determine an equation that can be used to find the area of the pen.
(d)
Determine the dimensions that give the largest area for the pen.
(e)
What is the largest area?
(f)
Is the area function linear or nonlinear function? Why do you think so?
7.
(a)
Make an open top box by cutting out squares and folding up the corners of an 8.5 by 11 sheet of paper. See Figure 4.1.3.3.
(b)
Record the dimensions and the volume of the box in Table 4.1.3.4. (Recall: For a rectangular prism (a box), Volume = Height x Length x Width.)
Depth, \(x\)  Height  Width  Area of Base  Volume 

0  8.5  11  
1  
2  
3  
4  
\(x\) 
(c)
Find equations for the height of the box, the width of the box, the area of the base, and the volume of the box. Write them in the last row of Table 4.1.3.4. How do you know your equations are correct? Use the figure provided to show how you found the equations.
(d)
What is the largest side length of a square that can be cut from each corner of the paper and still be able to make a box? Explain.
(e)
What are the dimensions of the box with smallest volume? What is the smallest volume? Explain.
(f)
What are the dimensions of the box with largest volume? What is the largest volume of a box created as directed in this problem? How do you know?
(g)
Build the box with largest volume. Are you surprised by its appearance?
(h)
Is the volume function linear or nonlinear function? Why do you think so? Is it the same type of nonlinear function as the functions in Exercise 4.1.3.5 and Exercise 4.1.3.6? Explain.
8.
(a)
Complete Table 4.1.3.5 to show the sums of consecutive even numbers beginning with 2.
Number of consecutive even numbers 
1  2  3  4  5  6  7  8 
Sum of consecutive even numbers 
2  \(2 + 4 = 6\)  \(2 + 4 + 6 =\) 
(b)
Graph the data in the table.
(c)
Find an equation to fit the table.
(d)
Is the function linear or nonlinear function? Why do you think so?
(e)
What is the domain of the function given the context?
(f)
What is the range of the function given the context?
(g)
What is the sum of the first 100 even numbers?
(h)
Will the sum of consecutive even numbers ever be 100?
(i)
Repeat Exercise 4.1.3.8 for the sums of consecutive odd numbers beginning with 1.
(j)
Have you seen either of these patterns before? Where?