few_shots: Example:
Problem/activity:
Task
Cruz and Erica were both getting ready for soccer.
Cruz ran 1 lap around the school.
Erica ran 3 laps around the playground.
Erica said,
I ran more laps, so I ran farther.
Cruz said,
4 laps around the school is 1 mile, but it takes 12 laps around the playground to go 1 mile. My laps are much longer, so I ran farther.
Who is right? Draw a picture to help you explain your answer.


Mathematical concept/skill options: 
A. Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
B. Extend understanding of fraction equivalence and ordering.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
C. Use equivalent fractions as a strategy to add and subtract fractions.
D. Develop Understanding of fractions as numbers.
Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.
E. Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
F. Understand decimal notation for fractions, and compare decimal fractions.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.


Your response:
{
  "explanation": "In this problem, students learn to recognize equivalent fractions. That is, three laps around the playground is 3/12 mile, which is equivalent to 1/4, which is one lap around the school.",
  "answer": "B"
}

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: im_task_000212

4
options: Use functions to model relationships between quantities.

Extend the domain of trigonometric functions using the unit circle.

Prove and apply trigonometric identities.

Build a function that models a relationship between two quantities.

Define, evaluate, and compare functions.

Interpret functions that arise in applications in terms of the context.

Build new functions from existing functions.

Construct and compare linear and exponential models and solve problems.

Understand the concept of a function and use function notation.

Model periodic phenomena with trigonometric functions.

Analyze functions using different representations.

Interpret expressions for functions in terms of the situation they model.
problem_activity: Task
“Exponential decay” describes a situation in which a quantity decreases following a certain pattern. We’re going to investigate a situation that decays exponentially. Read through steps 1-6 below and answer questions a-c before carrying out the investigation.
1. Get your hands on 30 6-sided dice and put them in a 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.
Pre-investigation questions:
When you roll 30 dice, how many 1’s do you predict you are likely to roll? Why?
After you remove the 1’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 1’s before you get to a roll where you don’t roll any 1’s?
Conduct the Investigation:
Construct a table to record the number of 1’s and the number of dice remaining after each roll, as shown.
{"table": [{"tbody": [{"tr": [{"td": [{"p": [{"_value": "Roll"}]}, {"p": [{"_value": "0"}]}, {"p": [{"_value": "1"}]}, {"p": [{"_value": "2"}]}, {"p": [{"_value": "3"}]}, {"p": [{"_value": "4"}]}, {"p": [{"_value": "5"}]}, {"p": [{"_value": "6"}]}, {"p": [{"_value": "7"}]}, {"p": [{"_value": "8"}]}, {"p": [{"_value": "9"}]}, {"p": [{"_value": "10"}]}]}, {"td": [{"p": [{"_value": "Number of 1\u2019s"}]}, {"_value": "X"}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}]}, {"td": [{"p": [{"_value": "Number of Dice Remaining"}]}, {"p": [{"_value": "30"}]}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}]}]}]}]}
Now for some mathematical modeling:
Use technology to create a scatter 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 be appropriate for this situation.
Based on what you know about exponential functions and 6-sided dice, write a function of the form \(d(x)=ab^x\) 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 scatter 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 8-sided dice instead of 30 6-sided dice. Write a new function that would model this situation.
Here is a tool that could be used in this task:


correct_option_index: 7