Battle of the Sums

Focus Questions:

What does it mean for a game to be “fair”?

How can we determine whether a game is in fact “fair”

Part 1: Battle of the Sums (30 min)

For this lesson we are going to be playing a couple of different games and then analyzing the games.  The first game that we are going to play is called “Battle of the Sums”

After the game is played, make a tally of how many times Player A won vs Player B. Do you think that this is a "Fair Game"?

Part 2: Rolling Dice

For this part of the lesson, we are going to explore the "relative frequency" for each sum when we roll two dice. You will need to make the following table in your squarebook.

	2	3	4	5	6	7	8	9	10	11	12
Tally
Total

Now, roll a pair of dice 36 times and keep track of how many times each sum is rolled. record this in the table.

When you are finished, add your data to this form

Check out the results here:

Part 3: Redesign the Game. (10 min)

Have students propose a more “fair” version of the game they just played.  There are a lot of ways to do it.  Some will try to say things like (roll only one die, ignore sums of 7 etc.)  Tell students that they challenge is to assign EVERY possible sum to one of the players.

Part 4: Sample Spaces. (10 min)
Watch the video below on "sample spaces". This will be a major focus of what we do next class as a way to "represent" probability.

Homework:

• Students may need to finish the game at home.
• Give students the “Sample Spaces” handout and tell them to watch the video at home.

Choosing Marbles

Focus Question: 

How do we determine the theoretical probability of some slightly more complex situations.

New Vocabulary:

• AND - All conditions described must be met.
• OR - Only one of the conditions described need to be met.
• NOT - The conditions listed must NOT occur.

Part 1: More Kalvin

For this part of the lesson you are going to do a quick formative assessment to see how well you understand the main focus questions from last week's lessons on probability.

You will do the assessment independently and can use calculators if you think that you need them.

Part 2: Simple Events

In this part of the lesson you will watch a video describing some of the basics around finding the theoretical probability of an outcome for a given event.  We have already talked about this a little bit, so this should be mostly a reinforcement of prior knowledge.

When you have finished watching the video, make sure that you complete the "On Your Own" questions.  Show your teacher when you have finished the questions.

Part 3: Choosing Marbles - Determining Theoretical Probabilities

The main part of this lesson is for you to now practice applying the concept of theoretical probability.  This is also in your textbook on page 30.

By the end of this problem you should be able to do the following:

• Determine the theoretical probability for a series of outcomes that contain the "OR", "AND", and "NOT" conditions.
• Develop a probability situation based on theoretical probabilities.

Homework:

1. ACE (p. 36) (6-7) (18-25)

Choosing Cereal

Focus Questions: 

• How does collecting more data helps you to predict the outcome of a probability situation?
• How can we use the results of a probability experiment to determine the "relative frequency" of an outcome?
• How does running a probability experiment help you determine how many possible outcomes there are as well as the likelihood of each outcome?

Kalvin would prefer to eat Cocoa Blast cereal for breakfast everyday.  His parents would prefer that he eats Health Nut Flakes.  In order to decide, his mother allows Kalvin to design a probability experiment.

Option 1: Flipping Coins

Kalvin will flip a coin.  If the results are heads, he will eat Cocoa Blast.  If the results are tails, he will eat Health Nut Flakes.

1. To run this experiment we will use our integer counters.  With your partner, designate one color to be "Heads" and the other color to be "Tails".  Make a note of your legend at the top of the table.
2. After you have completed your experiment, fill out this online form to contribute your data to our class data.
3. Here are the class results:
   
   
   RED = HEADS and BLUE = TAILS

Option 2: Tossing Cups

Kalvin will toss a cup.  If the cup lands on either end, he will eat Cocoa Blast.  If the cup lands on its side, he will eat Health Nut Flakes.

• Run your experiment 20 times.  Record the results in your table.  Please remember to treat the cups kindly as they need to be used in multiple classes.
• After you have completed your experiment, fill out this online form to contribute to the class data.  Here are the results of the class data:

Option 3: Matching Coins

Kalvin will toss two coins.  If the coins match, he will eat Cocoa Blast.  If the coins do NOT match he will eat Health Nut Flakes.

1. Run your experiment 30 times.  Record the results in your table.  Remember to use the "shake and drop" strategy.  
2. After you have completed your experiment, fill out this online form to contribute to the class data.  Here are the results of the class data:

At the end of this Problem you should be able to do the following:

• Describe how the accuracy of a probability experiment is influenced by the amount of data collected.
• Describe the difference between the probability of each outcome for an event and the relative frequency of each outcome for a probability experiment.
• Use the results of a probability experiment in order to predict the likelihood of each outcome.
• Extend the results of a probability experiment in order to predict the results of a similar probability experiment.

Here is what you MUST do before next class:

• Complete all of the questions on the handout.
• ACE (3, 6, 8, 10)
• Khan Academy: Simple Probability and Experimental Probability

Gee Whiz Everybody Wins - Blocks C and E

Focus Questions:

1. What does it mean if outcomes are equally likely?
2. How do I determine the theoretical and experimental probabilities for an event?

Part 1: Gee Whiz Everybody Wins

Let's play a game!  The rules for this game are very simple:

• Guess what color of block you are going to pick.
• If you are correct you win, if not you lose.
• Questions???

What do you think is in the bag?  How did you decide this?

Part 2: Equally Likely

Outcomes are said to be equally likely if the probability of each outcome is the same.  Take a look at the two spinners below and try to decide which one shows outcomes that are equally likely:

 

Now...take a look at the different situations below and try to decide whether the outcomes listed are equally likely.  Be prepared to defend your choices as this gets tricky:

Part 3: Theoretical and Experimental Probabilities

As you saw in the video, theoretical probability is a probability that is calculated by examining all of the possible outcomes for an event.  Experimental probability is calculated from the results of an experiment.

We typically write this as a ratio, a decimal, or a percentage.  The two things you need to pay close attention to are the total number of outcomes and then how many of those outcomes are "favorable".

Answer the following questions based on Gee Whiz Everybody Wins:

1. What was the experimental probability of picking each color?  Be sure to use "probability notation".
2. What was the theoretical probability of picking each color?  Be sure to use "probability notation".

Here is what you MUST do before next class:

• ACE (11-15) p. 19
• ACE (1-2) p. 36

Introduction to Probability - Block G

Our next unit is my personal favorite - Probability.  This is a unit where we get to explore "how likely" is something to happen.  This is a great skill to have in anybody's decision making skill set.

First of all, let's start by playing a game.  The rules are very simple:

• Predict what you will pick out of the bag.
• If you are correct, you win
• Questions???

Next, let's get a sense of what you will be doing in this unit.  Take a look at the "I Can..." statements.  On the scale above each statement, place a mark to indicate how well you think you can do each of the unit objectives.  Remember that this is the beginning of the unit so I am not expecting you to be an expert.

Now...let's get some basic probability language down.  Take a look at this video and complete the note-taking sheet that you have been provided with.

Here are some key words that you should be able to use next class:

• Event
• Outcome
• Certain
• Impossible
• Likely
• Unlikely
• Theoretical Probability
• Experimental Probability

What does it mean if outcomes are "Equally Likely"?

If two or more outcomes are "equally likely", it means that they have an equal chance of occurring.  Think about the game we just played.  Each block had an equally likely chance of being picked, but that didn't necessarily mean that each color had an "equally likely" chance of being picked.

Take a look at the spinners below and see if you can decide which one has "equally likely" outcomes.  Be prepared to justify your answer:

 

Now let's look at a variety of other events.  For each event, list all of the possible outcomes (if this hasn't already been done for you) and then decide whether the outcomes are all "equally likely" to occur.

As you can see, this can get a little bit tricky.

Now let's Practice

I have included a link to a Khan Academy activity that addresses the same things that are discussed in the video above.  You need to complete this before next class: Understanding Probability

Before next class you MUST have completed the following:

• Self assess your current level of understanding of each "I can..." statement
• Watch the "What is Probability" video and complete the note-taker page.
• Pass the first level of the "Understanding Probability" exercise on Khan Academy
• ACE (11-18) p. 19
• ACE (26-29) p. 23

Introduction to Probability - Block C and E

Our next unit is my personal favorite - Probability.  This is a unit where we get to explore "how likely" is something to happen.  This is a great skill to have in anybody's decision making skill set.

First of all, let's get a sense of what you will be doing in this unit.  Take a look at the "I Can..." statements.  On the scale above each statement, place a mark to indicate how well you think you can do each of the unit objectives.  Remember that this is the beginning of the unit so I am not expecting you to be an expert.

Now...let's get some basic probability language down.  Take a look at this video and complete the note-taking sheet that you have been provided with.

Here are some key words that you should be able to use next class:

• Event
• Outcome
• Certain
• Impossible
• Likely
• Unlikely
• Theoretical Probability
• Experimental Probability

Now let's Practice

I have included a link to a Khan Academy activity that addresses the same things that are discussed in the video above.  You need to complete this before next class: Understanding Probability

Before next class you MUST have completed the following:

• Self assess your current level of understanding of each "I can..." statement
• Watch the "What is Probability" video and complete the note-taker page.
• Pass the first level of the "Understanding Probability" video on Khan Academy

What Does it "Mean" to be Normal?

Focus Questions:

How do we determine what is "normal" for a given parameter?
What are some ways that we can compare the same data for two different samples?

Part 1: Interpreting Box Plots

The main focus for this part of the lesson is to look at how we can use a boxplot to determine what is "normal" for a set of data.

As a reminder, here is the general structure of a box plot:



Remember that there are two main things to keep in mind:

1. Each section of the box plot represents 25% of your sample (or in other words, if your sample was 100 people, each section represents the data for 25 people)
2. Each section represents the "range in data" for that 25% of the sample.  This is why the sections are different sizes, some sections of your sample have more "variability' (spread) than others.

Speaking of Variability....

There are a couple of ways that you can determine the variability of your data.  Variability refers to how similar your results are.  Generally speaking, the lower the variability, the more confident you can be that the results of your sample can be used to make predictions about your population.

So...strategies for determining variability:

1. Calculate the Range and Interquartile Range (IQR) - if you are comparing two sets of data from the same parameter, the data set that has the LOWER range and IQR is probably more reliable.
2. Look at the Box Plot - box plots that have less variability tend to have longer whiskers and shorter boxes.

Look at the box plots below.  Which set of data do you think has the least variability?

Hopefully, you selected "Seattle".  This is because if I look visually, all of the box plots are about the same as they all have shorter whiskers. But, if I calculate the range and IQR, I can see that the weather is Seattle is less variable.

Part 2: Determining what is "Normal"

Statistically speaking, there are three different ways we can determine what is "normal".  These are tools that you have used in the past:

1. Mean - add all of the numbers up and divide by the sample size (this is not shown on a box plot)
2. Median - put all the numbers in increasing order and find the number in the middle (this is shown on a box plot)
3. Mode - find the most common number (or categorical response)

Since there are a variety of ways to decide what is "normal", we need to decide when we should be using each one.  Here is a little guide for your data analysis pleasure:

• STEP 1: Decide what type of "parameter" you have:
• if it is categorical, use the MODE
• if it is quantitative, go to STEP 2
  
• STEP 2: Make a box plot of your data and analyze the general shape of the box plot:
• if it is generally symmetrical, use the MEAN
• if it is generally "skewed", use the MEDIAN

So...how do you know if your data is SKEWED?

Data is skewed (not symmetrical) when we have data that is highly variable.  The main reason for this high variance is the presence of "outliers" which are shown as a * on the box plot.  Outliers cause the overall box plot to change its shape.  So look for the following clues that your data is skewed:

1. You have some outliers
2. One whisker is much longer than the other
3. The box is shifted far to the left or right.

Look at the example below to see what a "skewed" box plot might look like.

Part 3: Talking the Talk

Next class you will be analyzing the entire set of male and female data from our 7th grade census.  As you work through the data, try to answer the following questions as you go:

• Are you analyzing the results of a census or a survey?
• If it is a census, what is the population?
• If it is a survey, what is the population? what is the sample? what sampling strategy did you use? how do you know that it is representative (unbiased)?
  
• What parameter are you currently analyzing?
  
• Is it a quantitative or categorical parameter?  How do you know?
  
• What did you decide was "normal" for this parameter?
• Is your data skewed or symmetrical?  How do you know?
• What measure of normal (mean, median, mode) should you use?
• Why did you choose this one?

What you MUST do before next class:

1. Read this blog post at least two times before next class.
2. Complete the practice page handed out in class.

Making Random Samples and Asking Statistical Questions

Focus Questions:

How do I make a random sample?
How do I ask a "statistical question"?
How big should my random sample be?

Part 1: Making Random Samples

In this part of the lesson, we are going to look at some different strategies for making random samples.

Consider this question:

Mr. Ray wants to give chocolate to three randomly selected students.  How could he do this in an "unbiased way"?

Our student data form has almost 300 entries, so some of the strategies that were suggested, might not work as efficiently.  So, we will use a random number generator to do the hard work for us.  In order to do this, you will need your own copy of the "Samples and Populations - Student Information Responses"

• Click on the following link: Samples and Populations - Study Copy
• Make a copy of the form and rename it so that is says "Samples and Populations - YOUR NAME Copy"
• File it in an appropriate place in your GoogleDrive.

We are now going to generate 6 different random samples as indicated on the spreadsheet.

Part 2:  Asking Statistical Questions

In this part of the lesson we are going to start asking statistical questions that we can use our data to help us answer.  When asking these questions, we need to keep in mind a couple of things:

• What is the population in which you are interested.
• Do you want to find out about one population or would you like to compare populations.

Here are some examples:

• Are 7th grade boys are statistically taller than 7th grade girls?
• Do girls on B-side have statistically more Facebook friends than girls on C-side?

Part 3: Comparing Sample Sizes

One question that has come up a lot in our class is how big should a sample be?  That is a great question.  Here is what we are going to do:

• As a class, we will choose one quantitative "parameter" to focus on.  Examples could be height, shoe size, number of siblings, age etc.
• Find the mean for each of your random samples of your given gender.
• Place a sticker on the number line to represent the mean for your sample size.

Here is what you MUST do before next class:

• Complete your random samples of both boys and girls.
• ACE (15, 16, 17, 19)
• We will do the survey as a "Do Now" next class.

Drawing Conclusions From Samples

Focus Questions:

How can we use the results of a survey to make inferences about a population?
What are some strategies that we can use to improve the accuracy of our inferences?

Part 1:

Part 2: Taking a Survey.  Please complete the following survey about honesty.