Hands-on Activity A Chance at Monte Carlo

Learning Objectives

After this activity, students should be able to:

• (advanced) describe the link between frequentist probability (or, frequentism) and geometry
• quantify how the proportion of areas is reflected in outcome of the experiment
• distinguish between the geometric definition of π is and the numerical representation of π
• assess the quality of an estimate using percent error (or standard error for more advanced classes)
• confidently design a simulated experiment

Educational Standards

Each TeachEngineering lesson or activity is correlated to one or more K-12 science, technology, engineering or math (STEM) educational standards.

All 100,000+ K-12 STEM standards covered in TeachEngineering are collected, maintained and packaged by the Achievement Standards Network (ASN), a project of D2L (www.achievementstandards.org).

In the ASN, standards are hierarchically structured: first by source; e.g., by state; within source by type; e.g., science or mathematics; within type by subtype, then by grade, etc.

NGSS: Next Generation Science Standards - Science

Common Core State Standards - Math

International Technology and Engineering Educators Association - Technology

State Standards

Materials List

Each group needs:

• LEGO MINDSTORMS EV3 robot, such as EV3 Core Set (5003400) at https://education.lego.com/en-us/products/lego-mindstorms-education-EV3-core-set-/5003400
• LEGO MINDSTORMS Education EV3 Software 1.2.1, free online, you have to register a LEGO account first; at https://www.lego.com/en-us/themes/mindstorms/downloads
• computer, loaded with EV3 1.2.1 software
• notepad for logging results
• A Chance at Monte Carlo Pre-Quiz, one per student
• Monte Carlo Simulation Worksheet, one per student
• Standard Error Flowchart, one per student
• A Chance at Monte Carlo Post-Quiz, one per student

Note: This activity can also be conducted with the older (and no longer sold) LEGO MINDSTORMS NXT set instead of EV3; see below for those supplies:

• LEGO MINDSTORMS NXT robot, such as the NXT Base Set 
• LEGO MINDSTORMS Education NXT Software 2.1
• computer, loaded with NXT 2.1 software

Worksheets and Attachments

Pre-Req Knowledge

Students should be comfortable with the following:

• computing the area of a square and a circle
• applying data analysis concepts: mean, percent error (advanced: standard error)
• (to be reinforced) the equation of a circle: x² + y² = r²
• (optional) uniform probability distributions

Introduction/Motivation

(Note: Show students the Monte Carlo Presentation to illustrate the concepts developed in this section.)

In the classroom, we always encounter problems that can be conveniently solved using the equations we learn. In particular, when we encounter a shape, we learn to break it down into triangles, rectangles and ellipses. That way, we can use the component geometries to define the composite geometry. With today's activity, we are going to explore ways that scientists and engineers attempt to make calculations when an equation is hard to approximate or even hard to define.

Let's propose a method for computing area that is based on chance. Imagine that we have drawn a leaf on a large sketch pad and we want to determine the size of the leaf in area. First, we bound the leaf with a known shape (for example, a square). Then we scatter ink drops over the entire sketch pad in a uniformly random pattern, meaning with equal probability of occurring anywhere. Knowing this, we can safely say that the number of ink drops landing inside the leaf, divided by the total number of ink drops inside the square, approximates the proportion of the area of the leaf within the square. Since the area of the square is known, the unknown area of the leaf is approximated. The method we have just proposed is an example of Monte Carlo sampling, named after the famous casino in Monaco, France. (Show students Figure 1 or slides 10-12 of the Monte Carlo Presentation, which illustrate this idea.)

(Show students Figure 2, also shown on slide 6 of the Monte Carlo Presentation). This demonstrates a coarse approximation of an arbitrary shape (the green area) in more detail. In this drawing, 24 red points are arbitrarily placed inside the unit square. The calculations shown approximate the green area and, logically, the result seems about right. We can visually confirm that the area is less than half, because if we draw a horizontal line midway through the box, we see slightly less green area above the line than white area below the line. Would it be a good idea to repeat the process with more than 24 points?

Monte Carlo methods are typically used for integrating complicated functions, such as counting the irregular area of a leaf, but today we challenge the application of Monte Carlo sampling by estimating the area of a simple shape whose geometry depends on the constant π. To do so, we will not conduct a physical experiment with paper and ink. Instead, we will use the LEGO MINDSTORMS EV3 Intelligent Brick to simulate the placement of points and then to count those points and report a ratio.

After taking a few trial estimates, we explore the quality of the mean estimate by considering either A) the percent error with respect to a reference value (3.14159...), or B) the standard error, or within sample variation. (Note: Option A is recommended for less-advanced students, while option B is recommended for more-advanced students.)

Procedure

Before the Activity

• If it is preferred to not develop a program in class, pre-program the EV3 bricks with the Monte Carlo Program.
• Make copies of the A Chance at Monte Carlo Pre-Quiz, Monte Carlo Simulation Worksheet, Standard Error Flowchart. and A Chance at Monte Carlo Post-Quiz.

With the Students

Developing the Experiment

1. Develop the geometrical setting of the classroom experiment:

• Provide each student with a worksheet.
• Ask students to focus their attention on the first two worksheet pages, beginning with the circle inscribed within a square with side length 2. Remind students that the ratio of the circle to square areas is π/4. They can verify this using their knowledge of the equations for finding areas of squares and circles.
• The ratio remains identical for the positive quadrant of the square. This quadrant is to simplify computations on the second page of the worksheet.
• Point out that we only know the "solution" for this geometrical problem if we know the value of π.
• Have students complete the first two worksheet pages.

2. (optional, time permitting) Explain how a computer can generate uniformly random outcomes:

• A computer can generate coordinates (x, y), each between 0 and 100. (Note: Since the EV3 only provides integer arithmetic, make all calculations in 100x numbers. The maximum integer seen on the EV3 brick is 215-1 = 32767, therefore 1002 + 1002 = 20000 is an acceptable value. If leading the class in designing the program, this detail may require some explanation.)
• Calculating whether x² + y² < 1002 for a given point (x,y) determines whether the point is inside or outside the circle. If x² + y² < 1002, the point is within the circle, and if x² + y² > 1002, the point is outside the circle.

3. Therefore, using a Monte Carlo sampling within this unit-area frame, the relative frequency of points inside the circle should be approximately π/4.
4. The computer can repeat this experiment several hundreds or thousands of times in order to improve the accuracy of the estimate of π. Additionally, we can change the number of simulated points during each experiment. How does the accuracy of the estimate change with number of points? (Answer: The accuracy of the estimate increases as the number of points increases.)

Collecting Data

(optional: For this step, students should be comfortable writing programs with the EV3 software. Alternatively, use the Monte Carlo Program.)

1. Students plan and implement a program for the EV3 brick using the LEGO software to simulate 100 random points. A portion of the Monte Carlo Program is shown in Figure 3.

   

2. Using a student-developed program or the Monte Carlo program, run five experiment simulations, each using 100 points.
3. Repeat step 1 of this section, changing the number of simulated points to 500, 1000, and 4000. Note that the EV3 brick needs to be reprogrammed for each number of points. Instructions on how to vary the number of points is included in the appropriate position on the Monte Carlo program.
4. For each set of five samples, estimate π by computing the mean of the set. For example, the mean of five estimates from the 1000-point simulation is:

(3.126 + 3.176 + 3.080 + 3.110 + 3.178)/5 = 3.1340

5. Assess the quality of each estimate by percent error, and/or the standard error. For the samples above, the percent error is calculated against a reference value for π

100 x | 3.1340 - 3.14159 | / 3.14159 = 0.24%

To calculate the standard error, students can use the flowchart, as shown in Figure 4.

6. Students complete the remainder of the worksheet to turn in.

   

7. As a class, discuss the results and conclusions from this experiment. Ask students:

• How is the quality changing with the number of simulation points? (Answer: The estimates for π are improving; both the percent error and the standard error are converging to zero.)
• What is the effect of averaging the five outcomes for each experiment setting? (Answer: Since each estimate is part truth and part error, averaging five outcomes reinforces the true component (π) and reduces the erroneous part.)

Vocabulary/Definitions

geometric constant: A number that arises in describing shapes at all scales (for example, pi or the golden ratio phi).

outcome: An event whose chance of occurring is governed by the probability distribution.

random: Suggesting variation from trial to trial; not deterministic.

simulation: A numerical/computational model of a physical scenario.

Assessment

Pre-Activity Assessment

Pre-Quiz: Have students complete the A Chance at Monte Carlo Pre-Quiz, which asks them questions on the following topics:

• understanding/familiarity with the use of computers and programming
• distinction between the value of a geometrical constant and its meaning
• insight into uses of random outcomes

Collect and grade the quiz to gauge students' prior knowledge on these topics.

Activity Embedded Assessment

Worksheet: During the activity, have students complete the Monte Carlo Simulation Worksheet. Through a series of familiar geometry problems, students are guided through the model scenario for the Monte Carlo simulation to compute the value of pi. The last two pages help students organize and interpret their results. Review their answers to assess their depth of comprehension.

Post-Activity Assessment

Post-Quiz: Administer the A Chance at Monte Carlo Post-Quiz to assess students' comprehension of the same topics as the pre-quiz, although the post-quiz asks deeper and more specific questions. Review students' answers to gauge their learning during the activity.

Troubleshooting Tips

Encourage students to collect all of their data (five samples each for a varying number of simulation points) up front, before moving through the statistical analysis.

Copyright

Contributors

Supporting Program

Acknowledgements

This activity was developed by the Applying Mechatronics to Promote Science (AMPS) Program funded by National Science Foundation GK-12 grant no. 0741714. However, these contents do not necessarily represent the policies of the NSF, and you should not assume endorsement by the federal government.

Additional support was provided by the Central Brooklyn STEM Initiative (CBSI), funded by six philanthropic organizations.