Subject: AP Statistics
Topic: Randomness Part 2
Teacher: Bill Arnot
Lesson objectives:
- Students will be able to recognize random outcomes in a real-world situation.
- Students will be able to recognize when a simulation might usefully model random behavior in the real world.
- Students will know how to preform a simulation by generating random numbers on a calculator.
- Students will be able to describe the simulation so that others can repeat it.
- Students will be able to discuss the results of a simulation study and draw conclusions about the question being investigated.
Assessment Strategy:
Students will be given problems to simulate and analyze. Their results will be reviewed before moving onto the next problem.
Materials:
Coins
Dice
TI nSpire CX calculator
Procedures:
- Tell story: Two brothers were having an argument over who would take out the trash. The older brother suggested they flip a coin to see who goes. The younger brother agrees, but only if it is the best 2 out of 3. The older brother calls heads and flips the coin twice. Each time the coin landed heads. The younger brother argued that the coin wasn’t fair. How could you test the younger brother’s claim?
- Discussion: Students will discuss in groups how to best proceed then share with the class. Students will quickly decide that taking the coin a flipping it a lot of times would be the best bet.
- Teacher: Hand out coins to each student. Have students flip their coin 10 times and have them report back their results (results in a lot of cases will not be 50/50). Ask class how many times they should flip their coin (20, 40, 100… 1000 times??). What would convince them that the coin was fair (or unfair)? Does it have to be exactly 50/50? How can we speed trials up?
- Demonstration: How to simulate a coin toss with a TI nSpire calculator.
Method 1
randInt(0,1, # of trials)->coinflips
This stores the coin flips into the set coinflips. Zeros can represent tails and ones heads.
Sum(coinflips)
This finds the sum of all the heads, which we can then divide by the number of trials.
Method 2
Create a set {0,1} and use the function randSamp(set, # of trials, 0) [zero represents sampling with replacement]
Store results as shown in previous method.
Have students complete univariate data analysis on their results. Should be uninteresting, as the data will fall pretty close to 50/50.
- Activity: Assign problems to students with increasing complexity. Have students use their building a simulation handout from the last lesson to help guide their work. Students may opt to use manipulatives or their nSpire calculator, but must complete at least 30 trials for each problem. Students who opt for the manipulatives will quickly tire and switch to their calculators.
Additional Information
- Content:
What is the content you are teaching and what are the big ideas?
Statistical Randomness.
What are the challenging concepts that students struggle with or are difficult to teach?
Creating a simulation to model a real-world situation by using random-digit outcomes to mimic the uncertainty of a variable.
Consider your state standards (GLCEs or HSCEs) as you develop the essential questions you are trying to address.
AP Stats Standards: II. Sampling and Experimentation
- Pedagogy:
What pedagogical strategies are you using and why?
Most of the lesson instruction will be student-centered. After learning the basic functions of randInt() and randSamp() learners will be given problems in which they will have to create the simulations.
What theories of learning inform your strategies?
Constructivism, learners build their knowledge through experimentation.
- Content & Pedagogy:
How do these particular strategies help you teach the content mentioned above?
The TI nSpire acts as a scaffold to help students complete thousands of trials for their simulations.
Why choose these strategies over other approaches?
This saves time over traditional methods of using random number tables or using manipulatives (dice, coins, urns,… ).
Are there any technical or physical constraints that figured significantly into your choices?
None, every student has been assigned a calculator.
- Technology:
What technology will you be using and why?
The TI nSpire CX calculator will be used to aid students in simulating problems involving random chance.
Is the use of this technology absolutely necessary to achieve your objective?
It is possible to teach this lesson without the calculator. However, we would not have enough time to engage with a number of different problems without it. Additionally, on the AP exam students will not be allowed the use of manipulatives. The use of a calculator will be needed.
- Technology & Pedagogy:
How does the technology you have chosen fit with your pedagogical strategies and theories about learning?
It gives the student the power to run as many trials for their simulations as possible as well as conduct analysis of the data on their calculators.
What types of learning strategies are employed by the technology?
Simulations and Problem Solving strategies are employed.
- Technology & Content:
How does your choice of technology help you teach the “big ideas” and address the essential questions underlying the concept your lesson addresses?
It allows students conduct thousands of trials with just a click of a button.
- Assessment:
What do you want your students to know, and how will you know when they know it?
- What is randomness?
- How do you create a simualtion that models a real-wolrd problem?
- How do I simulate a problem on my calculator?
- How do I assess my results?
How will you assess what students have learned?
I will evaluate each student’s results to the problems I assign.
What role does technology play in these assessments?
Students will create a display of their data on their calculators to assist in their analysis.
billarnot 