[br][table][br][tr][br][td][b]Factual Questions:[/b][/td][br][td][b]Conceptual Questions:[/b][/td][br][td][b]Debatable Questions:[/b][/td][br][/tr][br][tr][br][td]What method is used to estimate the value of pi in this activity?[/td][br][td]Why is a quarter circle used in the estimation of pi in this simulation?[/td][br][td]How would changing the simulation conditions, like target size, affect the ethical implications of using this estimation method?[/td][br][/tr][br][tr][br][td]What is the estimated value of pi after 1000 arrows were shot?[/td][br][td]How does the law of large numbers apply to the estimation of pi in this simulation?[/td][br][td]Could the archery simulation method for estimating pi be considered reliable compared to traditional geometric methods?[/td][br][/tr][br][tr][br][td]How does the ratio of arrows hitting the target to those missing help in calculating pi?[/td][br][td]What statistical concepts underlie the Monte Carlo method used in this simulation?[/td][br][td]What implications might the accuracy of this pi estimation method have for mathematical education and research?[/td][br][/tr][br][/table][br][br]

Scenario: The Archery Analytics Adventure[br][br]Background:[br]The kingdom of Numeria is hosting its annual Archery Analytics Adventure, a contest where archers and mathematicians team up. The goal is to estimate the value of pi using a method based on shooting arrows at a target.[br][br]Objective:[br]As a mathemagician and aspiring archer, your task is to use the cognitive activator applet to simulate shooting arrows at a target and use the results to estimate pi.[br][br]Investigation Steps:[br][br]1. Simulating the Shots:[br] - Use the applet to simulate shooting 100 and then 1000 arrows at a target.[br] - Record the number of arrows that hit the target and those that miss.[br][br]2. Understanding the Math:[br] - Recognize that the target represents a quarter circle within a square.[br] - Relate the proportion of arrows on target to the area of the quarter circle and the total area of the square.[br][br]3. Estimating Pi:[br] - Use the ratio of arrows on target to the total number of arrows to estimate the value of pi.[br] - Discuss how increasing the number of arrows affects the accuracy of the estimation.[br][br]4. Sharing Your Findings:[br] - Present your method and findings to the other contestants.[br] - Explain the statistical concept of Monte Carlo simulation as it applies to this activity.[br][br]Questions for Investigation:[br][br]1. Discovery Question:[br] - How does the law of large numbers apply to this archery-based estimation of pi?[br][br]2. Experiment Variation:[br] - What would happen if the target's size was doubled or the distance from which you shoot was increased?[br][br]3. Analyzing Patterns:[br] - Can you identify any patterns in the distribution of arrows that might affect the accuracy of your estimation?[br][br]4. Reflection:[br] - Why is it important to have a good estimation of pi, and how does this[br]