The Fundamental Counting Priciple is going to be used to count the odd​ non-repeating three-digit numbers in our number system. The digits come from the set​ {0,1,2,3,4,5,6,7,8,9}. The first digit​ (in the hundreds​ place) cannot be zero. Since the number is​ odd, the third digit​ (in the ones​ place) must come from the set​ {1,3,5,7,9}.
In order for the Fundamental Counting Principle to apply to this​ problem, this​ 3-part task must satisfy the uniformity​ criterion, meaning the number of choices for any particular part is the same no matter which choice were selected for the previous parts.  
The task of counting these numbers is going to be broken down into three parts. In which order must these be completed in order to satisfy the uniformity​ criterion?

Graded Discussion: The Monty Hall Problem
Requirements:
A minimum of two posts is required.
You must give a detailed explanation for each part in your first post to get full credit. Use complete sentences, correct punctuation, and correct grammar.
You will be able to see other students' posts once you post your own solution. Review their responses to determine if they agree with your own.
For your second post, reply to a classmate's post.
Complete required posts by the posted deadline.

Part 1: Prepare
Read this scenario:
You're on the game show "Let's Make A Deal" looking at 3 doors. Behind one is a brand-new car. Behind the others are goats! You get to choose one door: if it's the car, you win! You choose Door #1. But before he unveils its contents, Monty opens Door #2 and reveals a goat. You may have picked the right door! Monty asks if you'd like to make a deal. If you want, you can switch to Door #3. What would switching do: improve your odds, worsen your odds, or leave them the same? Answer this question on a piece of paper BEFORE viewing the following video.
Watch the video:
https://youtu.be/2Kgud9QcZs4
Read the explanation:

When you choose Door #1, your odds of winning are clearly 1/3. Therefore, the odds that the prize is behind Door #2 or Door #3 is 2/3. When Monty reveals that there is nothing behind Door #2, it doesn't change the original probabilities. There is still a 2/3 chance that the prize is behind door two or three. (Knowing the contents of door two doesn't change the odds once you've started playing.) Therefore, door three now has a 2/3 odds of winning.
Part 2: First Post

Title Your Post: Students name Monty Hall Problem Post
Answer each question:
1) Before you watched the video, did you think you should stay or switch. (4 points)
2) After you watched the video, did you change you mind? Explain your reasoning. (4 points)
Part 3: Second and Third Posts
Second and Third Posts (2 points): Read and reply substantively to two classmates' posts.

Graded Discussion: Counting Shapes
Requirements:

A minimum of two posts is required.
You must give a detailed solution for each part in your first post to get full credit.
It is not enough to just give the final answer.
You must show steps or explain your reasoning.
If it is easier, you can write your solution out on paper, take a picture, and upload it to the discussion as a pdf file.
The pdf file type is the only file type that will be accepted.
You will be able to see other students' posts once you post your own solution. Review their responses to determine if they agree with your own.
For your second post, reply to your own post.
Don't worry! You will not lose points for an incorrect answer in your first post as long as you post a reply with the correct answer in your second post.
Complete required posts by the posted deadline.

Overview
For this graded discussion, you will try to find your own systematic approach to solving two counting puzzles. Then you will watch a video solution. You will post your answers to the questions in STEP 4 below. You will create and solve your own puzzle. You must complete all the steps to get full credit.

STEP 1: Puzzle 1: How many squares are in the diagram?
Find and use a systematic approach to counting the number of squares of any size in the diagram.

STEP 3: Watch the video to see the solutions.

As you watch the video, make a note of the approaches used to solve each puzzle.
https://youtu.be/mqtosKgHwCo

STEP 4: Create a Post with your answers to the following questions.
[2 points] Describe your approaches to solving the puzzles. Did you use the same approach to both?
[2 points] How would you explain the approach used in the video? How was it systematic?
[2 points] Compare and contrast your approach to the approach used in the video. Did you get the same answers?
(Don't worry! You will not lose points if you didn't get the puzzle right the first time you tried it.)
[2 points] Create (or find) a counting puzzle for your classmates to solve.

STEP 5: Solve a classmate's puzzle.

Once you have posted, you will be able to view your classmates posts. Find one of the puzzles they posted, reply to their post with a detailed solution. [2 points

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Graded Discussion: Misleading Graphs
Requirements:
A minimum of one post is required.
You must give a detailed solution to each question in your post to get full credit.
It is not enough to just give the final answer.
You must show steps or explain your reasoning.
If it is easier, you can write your solution out on paper, take a picture, and upload it to the discussion as a pdf file. The pdf file type is the only file type that will be accepted.
You will be able to see other students' posts once you post your own solution.
Review their responses to determine if they agree with your own.

You are not required to make a second post, but you can reply to your own post with corrections.   Don't worry!  You will not lose points for an incorrect answer in your first post as long as you make a reasonable effort on the first post and post a reply with the correct answer in your second post.  
Complete required posts by the posted deadline.

Part 1: Prepare

Read this article about the way graphs can be misleading. https://www.statisticshowto.com/probability-and-statistics/descriptive-statistics/misleading-graphs/

Source of data: https://www.bls.gov/careeroutlook/2020/data-on-display/education-pays.htm

[2 pts] What is the ratio of the median weekly earnings of the holder of a high school diploma only to the median weekly earnings of the holder of a bachelor's degree?
[2 pts] What is the ratio of the area of the red rectangle to the area of the blue rectangle in graph A? (The scale on the horizontal axis is 1 unit per grid mark.)
[2 pts] What is the ratio of the area of the red square to the area of the blue square in graph B? (The scale on the horizontal axis is 1 unit per grid mark.)
[2 pts] What is the ratio of the volume of the red cube to the volume of the blue cube in graph C? (The scale on the horizontal axis is 1 unit per grid mark and the bases of the prisms are squares.)
[2 pts] Explain which graph is the most misleading and give a mathematical justification for your answer.

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A middle school​ counselor, attempting to correlate school performance with leisure​ interests, found that of a group of​ students, 32 had seen Movie​ A, 27
had seen Movie​ B, 24 had seen Movie​ C, 15 had seen Movies A and​ B, 11 had seen Movies A and​ C, 8 had seen Movies B and​ C, 3 had seen all three​ films, and 6
had seen none of the three films. Use a Venn diagram to complete parts​ (a) through​ (c) below.

Complete a Venn diagram considering the following classifications.
Complete the Venn diagram. Let A be the set of students that saw Movie​ A, B be the set of students that saw Movie​ B, and C be the set of students that saw Movie C.

Also, answer the following questions about the Venn Diagram
a) How many students had seen Movie C​ only?
b) How many students had seen exactly two of the​ films?
c) How many students were​ surveyed?
The following Venn Diagram is a graphical representation of the relationship between the conditions, and can help you in answering the questions.

Solved Venn Diagram for the Math Reasoning question