Decaying Dice
Task
â€œExponential decayâ€ describes a situation in whichÂ a quantity decreases following a certain pattern. Weâ€™re going to investigate a situation that decays exponentially. Read through steps 16 below and answer questions ac before carrying out the investigation.
1. Get your hands on 30 6sided dice and put them in aÂ container.
2. Find a safe place where you can roll 30 dice at once, like a tray (or the container itself, if it is large enough).
3. Roll the remaining dice (or shake the container thoroughly).
4. Remove any dice that display a "1" and set them aside.
5. Write down how many dice remain.
6. Go back to step 3 and repeat. Stop when you have rolled the dice ten times.
Preinvestigation questions:

When you roll 30 dice, how many 1â€™s do you predict you are likely to roll? Why?

After you remove the 1â€™s on the first roll, what fraction of the original number of dice do you predict will remain?
 How many times do you predict you might have to roll the dice and remove the 1â€™s before you get to a rollÂ where you donâ€™t roll any 1â€™s?
Conduct the Investigation:
Construct a table to record the number of 1â€™s and the number of dice remaining after each roll, as shown.
Roll 
0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
Number of 1â€™s 
X  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â 
Number of Dice Remaining 
30 
Â  Â  Â  Â  Â  Â  Â  Â  Â  Â 
Now for some mathematical modeling:

Use technology to create a scatterÂ plot with number of dice remaining on the vertical axis and roll number on the horizontal axis. Discuss what you see.

Based on what you know about exponential functions and the behavior of dice, explain in words why an exponential model would beÂ appropriate for this situation.

Based on what you know about exponential functions and 6sided dice, write a function of the form \(d(x)=ab^x\)Â to model the relationship between the roll number, x, and the number of dice remaining, d(x). Your job here is to decide what numbers to use in place of a and b. This should not be based on the data you collected  it is a mathematical model for how dice behave.

Graph your function on the same set of axes as your scatterÂ plot. Compare and contrast the graphed model d(x) to the outcome of the experiment shown in the scatter plot.
 Suppose someone repeated this activity, except they started with 100 8sided dice instead of 30 6sided dice. Write a new functionÂ that would model this situation.
Here is a tool that could be used in this task:Â
IM Commentary
Here is an editable version of theÂ student handout:Â
https://docs.google.com/document/d/1RQcyCS6eDI1NM0jPBarTAbWPXEqW7hX8tJFRm06bzZM/copy
This task provides concrete experience with exponential decay. It is intended for students who know what exponential functions are, but may not have much experience with them, perhaps in an Algebra 1 course.Â
On carrying out the investigation and collecting data:
In steps 13, directions are provided for rolling dice onto a tray OR shaking a container. Once a teacher decides on a mechanism the class will use, unnecessary parts of these instructions could be deleted.
It would be best if students carried out the investigation using physical dice. The basic experiment would haveÂ students work in pairsÂ with each pair of studentsÂ given 30 dice and time to collect their own data.
If a teacher desires, the class could gatherÂ collective data instead of each pair of students only working with its own data. One possible method might be for each group to recordÂ data to a shared spreadsheet. Then, the function that models the number of dice remaining could be overlayed on all the data or on a classÂ average for each round. Alternatively, if insufficient diceÂ areÂ available, each student could be responsible for rolling only one or twoÂ dice and the class could conduct the experiment at the same time, recordingÂ on a common data table on the board how many total 1's are rolled (and discarding dice that show a "1").Â An advantage toÂ everyone working with the same set of data is thatÂ wholeclass discussion becomes easier.
Thirty was chosen as the starting number of dice as a compromise between a realistic total number of dice to expect a teacher to provide forÂ a class and enough data to show a pattern. Also, a starting value of 30 facilitates thinking about the model in questions a. and b. because it is a multiple of 6.Â
If it is impossible to carry out the experiment using dice, studentsÂ could use a simulator. In a spreadsheet, use =randbetween(1,6) in 30 different cells to simulate rolling 30 dice. (You could even use more cells to simulate starting with more dice.) Count the number of ones by using the countif function. It would be best if students built this themselves, but here is an example of one:Â
https://docs.google.com/spreadsheets/d/1uDooM_eUj7uLCoq5JR3pqkEUFxOLfXv4myEG4bUNA/copy
The worst possible case is to use data that someone else collected, but sample data is provided in the solution, and a teacher could give sample data to studentsÂ as a last resort. But we strongly recommend taking the time to actually run the experiment so that students have the tactile, concrete experience on which to base their understanding of how the number of dice change and, ultimately, how exponential functions behave.
Some comments on specific components of the task:
(c) Be on the lookout for students who reason that there will only be six rounds by reasoning that five dice in each round will roll a "1." Those students may misunderstandÂ how the number of dice rolled in each round decreasesÂ as 1's are discarded.Â
(d) TeachersÂ should interpret the word "discuss" in a way that makes sense for their class, with discussion taking placeÂ eitherÂ verballyÂ or in writing. If it is the case where many groups of students collected different sets of data, it is to be expected that you might get a data set that does not appear to behave exponentially (for example, it might appear to beÂ quite linear.) "Real" data can behaveÂ strangely, and teachers may want to have a few sample data sets that are wellmodeled by an exponential function on hand for students to work with instead of the data they collect, in the case that their dice don't behave.
(d) (A separate note on question d.) You could have students run an exponential regression here and compare the regression equation with their modelÂ if it makes sense for your class, but this is not necessary.
(e) This is a sophisticated question but important to address at least briefly, and students may use very informal language, for example "On every roll, you expect to remove the same fraction of the dice." It explains why we can jump right to writing an exponential model without considering whether the change in dice might be characterized better by, for example,Â a linear or quadratic relationship.
(f) Students may get stuck here and be unsure how to proceed. One suggestion is to encourage students to try some numbers in place of a and b, and see what comes out when x isÂ 0, 1, and 2. They should be given time to fiddle around and learn some stuff about how \(a(b)^x\)Â works for different values of a and b. For even more scaffolding, teachers could disclose that a should be 30 so that students only have to worry about figuring out b.
(g)Â It is suggested that teachers view this question as an opportunity for students to check and refine their model. For example, if their model uses b = 1/6 in the function that defines d(x)Â instead of b = 5/6, it should be obvious that something is wrong when they graph it along with the scatter plot.
(h)Â This question is intended to be used as a check for understanding.
Solution
Â

When you roll 30 dice, you are most likely to roll 5 1's. That is because for any given die, there is a 1/6 chance of rolling a 1, and 1/6 of 30 is 5.

After you remove 1/6 of the dice, 5/6 of the original amount would remain.
 Answers will vary.Â
Sample data:
Roll 
0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
Number of 1â€™s 
Â  4  6  3  2  1  2  2  3  0 
Number of Dice Remaining 
30 
26  20  17  15  14  12  10  7  7 
Now for some mathematical modeling:

A scatter plot based on the sample data.Â Â Students might notice that the number of dice remaining is decreasing, the yintercept is 30, the relationship does not appear linear, or the number of dice appears to be "leveling off."

The fundamental property of an exponential function ist hat it changes by equal factors over equal intervals. That makes sense here, because how many dice come up 1's is expected to be 1/6 of the dice rolled. So the dice remaining should be a constant factor of 5/6 of the dice rolled.Â

\({d(x)=30(}{5\over6})^x\)Â or possiblyÂ \(d(x)=30(0.83)^x\)

Â Students should notice that the function is a pretty close approximation of the data in the scatter plot.
 \({100(}{7\over8})^x\)or possiblyÂ \(100(0.875)^x\)
Decaying Dice
â€œExponential decayâ€ describes a situation in whichÂ a quantity decreases following a certain pattern. Weâ€™re going to investigate a situation that decays exponentially. Read through steps 16 below and answer questions ac before carrying out the investigation.
1. Get your hands on 30 6sided dice and put them in aÂ container.
2. Find a safe place where you can roll 30 dice at once, like a tray (or the container itself, if it is large enough).
3. Roll the remaining dice (or shake the container thoroughly).
4. Remove any dice that display a "1" and set them aside.
5. Write down how many dice remain.
6. Go back to step 3 and repeat. Stop when you have rolled the dice ten times.
Preinvestigation questions:

When you roll 30 dice, how many 1â€™s do you predict you are likely to roll? Why?

After you remove the 1â€™s on the first roll, what fraction of the original number of dice do you predict will remain?
 How many times do you predict you might have to roll the dice and remove the 1â€™s before you get to a rollÂ where you donâ€™t roll any 1â€™s?
Conduct the Investigation:
Construct a table to record the number of 1â€™s and the number of dice remaining after each roll, as shown.
Roll 
0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
Number of 1â€™s 
X  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â 
Number of Dice Remaining 
30 
Â  Â  Â  Â  Â  Â  Â  Â  Â  Â 
Now for some mathematical modeling:

Use technology to create a scatterÂ plot with number of dice remaining on the vertical axis and roll number on the horizontal axis. Discuss what you see.

Based on what you know about exponential functions and the behavior of dice, explain in words why an exponential model would beÂ appropriate for this situation.

Based on what you know about exponential functions and 6sided dice, write a function of the form \(d(x)=ab^x\)Â to model the relationship between the roll number, x, and the number of dice remaining, d(x). Your job here is to decide what numbers to use in place of a and b. This should not be based on the data you collected  it is a mathematical model for how dice behave.

Graph your function on the same set of axes as your scatterÂ plot. Compare and contrast the graphed model d(x) to the outcome of the experiment shown in the scatter plot.
 Suppose someone repeated this activity, except they started with 100 8sided dice instead of 30 6sided dice. Write a new functionÂ that would model this situation.
Here is a tool that could be used in this task:Â