I can say that I learned a lot from the material in this module. I have shared my thought and ideas that I gained from the different activities presented. I will be sharing some optional blog discussion topics but that indicated the required information in bold red.
Annenberg Module--Probability
For this section of my blog, I will be sharing the my answers to the questions I worked through in the Annenberg module. For me, it helps to verbalize my thoughts.
Part A
Problem A1: Make a list of the topics and ideas that come to mind when you think of probability, including both everyday uses of probability and mathematical or school uses.
Now I will admit that of the different math concepts I have struggled with in the past, probability is one of them. It may seem simple, but I have learned that it can entail a lot of work. When I think about probability, I think about things like rolling dice, pulling colored pebbles out of a bag, and from a deck of cards. These are scenarios that I can remember from my elementary math classes. Essentially, I consider probability the method you use to describe the chance or likelihood of an even occurring. Probability is used everyday in a variety of ways. Meteorologists use probability to forecast the weather and tell viewers what chance is of it raining, snowing, etc.
Problem A2: What does probability have to do with statistics? Think
about ways that statistics might use probability, and vice versa.
I think probably and statistics are connected in many ways. We often use statistics to gather information about groups of people, so the results you end up with could help you make assumptions or predictions about how the data trend will continue. Essentially, the data we collect helps determine the probable outcomes of a situation that has been studied/surveyed. Probability influences statistics because we might make a prediction about something that interesting but we want to find out if it is really true. Therefore, you would conduct an study test your prediction or hypothesis.
Problem A3: What is a random event? Give an example of something that happens randomly and something that does not.I think probably and statistics are connected in many ways. We often use statistics to gather information about groups of people, so the results you end up with could help you make assumptions or predictions about how the data trend will continue. Essentially, the data we collect helps determine the probable outcomes of a situation that has been studied/surveyed. Probability influences statistics because we might make a prediction about something that interesting but we want to find out if it is really true. Therefore, you would conduct an study test your prediction or hypothesis.
I think random events are ones that can have a variety of outcomes. For example, rolling dice and picking cards from a deck would yield a variety of possible outcomes.
The next part of Section A involves playing a game called Push Penny. I didn't have any poster board on hand, so I used computer paper and tape. It took me a while but I finally created my lined game board. I was harder than I expected because you have to draw straight lines:
Problem A4: Suppose you wanted to find out whether you could develop skill at playing rounds of Push Penny. How might you design experiment to test this idea?
I'd say that you could keep playing the game for a consecutive number of days and keep a total of the score. With each game, you would compare your average scores with the previous day's scores.
Problem A5: Play 20 rounds of Push Penny (four pushes per round), and record your results from each round using the following format (five rounds are provided as an example):
Just by looking at my score chart, I wouldn't say that I have developed a skill in playing this game because my scores are kind of all over the place. Moreover, I never made 4 consecutive hits, so I think that indicates a lack of skill. If was really good at is game, I would think I'd have a higher number of hits per round.
b) If you don't think you've developed much skill in playing the game, do you think it is still possible to develop this skill?
To be honest, I don't really know. Ultimately, I think this is a "game of chance" where the quarter randomly lands on a spot. However, I do think that if someone really wants to get perfect scores, they can manipulate the way the slide the paper to move the quarter. You could practice pushing the quarter in different ways to see how hard or how far you need to push it to hit a line.
c) Give an example of a game where it is not possible to increase your game-playing skills.
I think any game where you have no control of the outcome or the outcome is completely random, would make it impossible to increase your playing skills. For example, randomly guessing a card from a full deck would yield a lot of possibilities--it would be hard to gain skills I guessing a particular card. I think the same applies to guessing a number that you might roll on a pair of dice. There are many possibilities.
Part B
Problem B1: This spinner uses the numbers one through five, and all five regions are the same size. Create the probability table for this spinner:
Problem B2: Suppose you toss a fair coin three times, and the coin comes up as heads all three times. What is the probability that the fourth toss will be tails?
The probability would still be 1/2 or 50% because the coin still lands on either heads or tails.
Problem B2 illustrates what is sometimes known as the "gambler's fallacy."
Using probability tables, we can predict the outcomes of a toss of one coin or one die. But what if there is more than one coin, or a pair of dice?
Even if you use more than one coin or die, the probability would stay the same. You are not changing the number of faces on the coin/die, so the possible number of outcomes would not change.
Problem B3: Toss a coin twice, and record the two outcomes in order (for example, "HT" would mean that the first coin came up heads, and the second coin came up tails).
My outcome: TH
| a) List all the possible outcomes for tossing a coin twice. How many are there? What is the probability that each occurs? HH, TT, TH, HT. There are 4 possible outcomes. Each outcome would be a 1/4 or 25%. b) List all the possible outcomes for tossing a coin three times. How many are there? What is the probability that each occurs? |
Problem B4:
a) Complete this table of possible outcomes. (If you're doing it on paper, you do not have to use blue and red pencils, but be aware of the difference between such outcomes as 2 + 4 and 4 + 2.)
The table has 36 different outcomes.
Problem B5:
a) For how many of the 36 outcomes will Player A win?
12/36 = 33%
b) For how many of the 36 outcomes will Player B win?
24/36 = 67%
c) Who is more likely to win this game?
Player B is more likely to win. He/she has more numbers to possibly roll.
To find the solution to these questions, I used the totals from the table and put the designated total next to either player A or player B. After looking at the solution, I realize that there are probably simpler ways of finding the answer. Here is my work:
This was a tricky question. It makes sense to pick different numbers the players must have to win. the tricky part is figuring out how many numbers they to win.
Problem B7: Complete the probability table:
Problem B8: Use the probability table you completed in Problem B8 to determine the probability that Player A will win the game. Recall that Player A wins if the sum is 2, 3, 4, 10, 11, or 12.
Player A has a 33% (12/36) chance of winning. I just added the total probabilities. Here is my work:
Problem B9: If you know the probability that Player A wins, how could you use it to determine the probability that Player B wins without adding the remaining values in the table?
Since you already have the probability that player A will win (12/36), you just subtract it from 1 (36/36).
(36/36) - (12/36) =24/36 or 67%.
Part C
This part brought up the idea of using tree diagrams to determine probability. I can remember working with these when I was in middle school.
Problem C1: Use this tree diagram to explain why the likelihood of getting exactly one head in two coin tosses is not the same as the likelihood of getting zero heads in two coin tosses.
Because there two combinations of heads/tails to toss (HT or TH). The tree indicates that there can only be one occurrence of flipping two tails (TT).
Problem C2: On a piece of paper, draw a tree diagram for three tosses of a fair coin. Label and tally all the possible outcomes as in the previous examples.
I will admit that since I haven't worked with tree diagrams in a long time, I kind of forgot how to create one. It was confusing at first but I'm glad I had the solution to guide me. Here is my diagram:
Problem C3: Complete the probability table for three tosses of a fair coin:
Problem C4: Complete the probability table for four tosses of a fair coin:
Problem C5: Do you think a tree diagram would help you create a similar probability table for 10 tosses of a fair coin? Why or why not?
It's possible but I would not want to draw out the diagram of tosses. It would take forever! I think doing this would b confusing, m easy to lose your place or repeat a set of possible outcomes.
Problem C6: Use the properties of Pascal's Triangle to generate the fifth and sixth rows.
Problem C7: Use Pascal's Triangle to determine the frequencies and probabilities for five and six tosses of a fair coin. If you wish, confirm your results with a tree diagram.
To me, this question was a little confusing. I had to look at the solution to understand exactly what they were asking here. Here are my diagrams:
For 5 tosses: For 6 tosses:
Problem C8: Write a formula for determining the number of possible outcomes of n tosses of a fair coin. I thought this was a tough one to figure out so I looked at the solution. After looking over the answer, it makes a little sense to me. The solution use P=n^2 where P=the possible number of outcomes and n=the number tosses. I was a little confused on why you would square the value but I thought about how this might be because coins have 2 sides. I'm still not sure. How did find this answer?
Problem C9: Extend Pascal's Triangle to the 10th row. Using the 10th row, determine the probability of tossing exactly five heads out of 10 coin tosses.
This question took me while to answer, mainly because I kept checking my answers to make sure I was on the right track. I first extended the triangle to the 10th row and was a little lost on what my next step was. I used the P=n2 to determine how many total combinations there are in flipping a coin 10 times. Next, I used row #5 of the triangle (252) and divided it by the total to get the probability of flipping 5 heads. Here is my work:
Problem C10: Use the binomial probability model to determine the following:
a) What is the most probable score you'll get?
b) What are the least probable scores you'll get?
c) What is the probability of getting at least two answers correct?
d) What is the probability of getting at least three answers correct?
Problem C11: Find the probability of getting at least two questions right on a 10-question True-False test (where you must guess on each question).
I will say that these sets of questions have left me stumped! After looking at the solutions, I am still confused. I e-mailed Dr. Hargrove about these, so hopefully she can help me understand the process a little better. Can anyone else explain how you got the answers?
Part D
Problem D1: What do you think the probability is that a random push will hit a line? Remember that each line is exactly two coin diameters apart. It may be helpful to experiment with a quarter on your Push Penny board and to examine this illustration. What percentage of the total area of the board is shaded?
I was confused at why the board was shown in shaded strips but I can see that since the strips on the board are equal measurements, the shaded areas make up half of the strips, making them 50%.
Problem D2: Here is the summary of scores of 100 rounds of another player's attempt to master Push Penny. Do these scores suggest that this player has developed some serious Push Penny skill?
Yes. The scores indicate that the player has made improvements. He seems to have successfully beaten the scores of the average player in specific numbers of hits. Here is my work to show this:
Problem D3: Use your data from Problem A5 to determine whether you were developing any skill for Push Penny. Then play another 20 times to see if your skill has improved over the course of this session. When comparing my first game scores and probability against my newest game results, I can't really notice much improvement. I stayed the same in my O number of hits and went down in my frequency of 1 number of hit. However, I made up for it for scoring higher for hits and the one 4 hits I made. So really, I maintained my game playing "skill". |
NCTM Article--Probability: A Whale of a Tale
 
I enjoyed reading this article because it presented some really great ideas on how teachers can introduce the concept of probability and games of chance to their students. I love the idea of using the Dear Mr. Blueberry book to
initiate a discussion about the occurrence of unlikely events such as having a
whale in your backyard pond! I have never read this book before but I can see
how children would be interested in reading it. I think that using quality
literature across the curriculum is a great way to engage students into
learning about other subject content. I like how the teacher was able to make
the transition from the whale scenario to other events that students could
relate to and present them with the words we use to describe the likelihood of
such events happening. I also like how students were tasked with drawing
pictures to represent four different probability statements they created. I
think this is a great lesson that helps introduce students to the idea of
probability and teaches them important vocabulary words associated with the
concept. They will use it as they progress in the complexity of the math!
The second activity described in the article was also very
interesting. I like how the lesson helped students practice the use of
probability in a hands on activity. Students
seemed to learn a lot by playing the game with dice because they discovered the
struggle of rolling a certain sum of numbers. This helped them see that there
are different possible outcomes associated with playing this particular game.
This was a great activity to introduce students to the idea of using
theoretical and experimental probability. The table (Table 1) that the class
created was a great visual representation of all the possibilities of sums the
dice would produce. It was even helpful for me! It’s good to know that the
students learned so much from this lesson.
The module checklist suggested that we create our own
probability line charts that display events that are impossible, certain,
likely, or unlikely to happen. Here are pictures of my chart:
Close ups:
 
Dice Toss
 
1) Ms. Kincaid wanted the students to make predictions about their experiment on the basis of mathematical probability. Discuss preconceptions that students exhibited about tossing dice even after discussing the mathematical probability. Discuss the instructional implications of dealing with these preconceptions.
The students made a few statements about the mathematical probability
associated with this activity. They seemed to understand that mathematical
probability refers to the expectation that something will happen while
experimental probability relates to probability that occurs when you actually
test or experiment with the event. Before the groups rolled their dice, they
predicted (using mathematical probability) that they would roll a sum of 7 more
than any other number because it has the highest number of possible ways to
reach that sum. They also believed that they would roll less sums of 12 than other numbers. I thought that maybe
the students would stick to these preconceptions, especially after completing
the activity. However, when Ms. Kincaid visited with groups after they completed
the activity, it was interesting to see some students discuss their results of
rolling other numbers using experimental probability. One student named Monique
later talked about how she thought a 2 would show up more than a 6 or a 12
because that’s what happened in her other times of rolling dice before. She
relied on her past experiences to make predictions about what she might roll—not
mathematical probability.
I think the preconceptions children have about these things can interfere with the way they understand math instruction. I can tell how important it is to help students see that there is a difference between what math tells versus what our own experiences tell us about the likelihood some something happening. |
I also think young students do need early exposure to probability. It doesn’t even have to involve numbers—they can make statements of what they expect will happen in a specific situation. With that being said, I think probability should be introduced to probability in Kindergarten or 1st grade. Perhaps this can prepare them for more complex objectives later on in their education.
3) The teacher asked the students, “What can you say about the data we collected as a group?” and “What can you say mathematically?” How did the phrasing of these two questions affect the students’ reasoning? The first question led students to consider the results the entire class gathered as opposed to what individual groups found. I saw how the students referred to the chart to draw conclusions about the outcomes. The second question led students to use mathematical reasoning to give answers of why a certain number possibly showed up more times than others. The child that compared the data to a rocket ship explained that the 7s were more likely to appear because there are more combinations to get a sum of 7 than the other numbers. Essentially, it seems like the students revisited their predictions (mathematical probability) and compared them to the results they gathered as a whole group after completing the activity (experimental probability).
I think the ultimate reason in Ms. Kincaid having students roll 36 times is because each die has 6 sides (6x6=36). 36 is the total number of combinations you can get when rolling both dice. This also gives students a higher chance of rolling more combinations. Moreover, this can create a larger sample size to collect from the whole class. Having students roll a lower number would not really reveal a variety of combinations.
In addition to having more combinations, having a high number like 36 rolls was an advantage because each student had equal opportunities to roll the dice. In each 4-person group, individual students had 9 turns (36/4 = 9). Rolling a total of 36 times also has is disadvantages because students can easily lose track of the how many rolls they have done, something that was made evident in the video. One group of students rolled too many times, so they had to figure out what numbers they rolled in recent turns. This is a drawback because this potentially changes the data.
5) Comment on the collaboration among the students as they conducted the experiment. Give evidence that students either worked together as a group or worked as individuals.
I thought it was great to see so many students working together as they completed the activity. I saw groups of students work to develop a plan of how they were going to record the number of rolls each person completed. One group (around the 6:36 time mark) talked about organizing the data using columns for each person in the group and recording each throw. The group that went over 36 rolls worked together to figure out how to correct the problem. One girl in the group was seen helping the reporter understand what numbers to erase.
6) Why do you think Ms. Kincaid assigned roles to each group member? What effect did this practice have on the students? How does assigning roles facilitate collaboration among the group members?
7) Describe the types of questions that Ms. Kincaid asked the students in the individual groups. How did this questioning further student understanding and learning?
I noticed that Ms. Kincaid asked students a lot of questions that led student to analyze their group’s data. Questions like “What did you find?” and “What can you say about the data we collected as a group?” helped students look at their results and make mathematical conclusions. Ms. Kincaid also asked questions that helped students analyze the data and compare it to their initial predictions. She did this by asking students things like “Is that what you expected?” and “Were you surprised?”. These kinds of questions helped students make connections between what they expected and what they actually experienced. I like how she encouraged the students to analyze the data for themselves rather than telling them what their results revealed.
8) Why did Ms. Kincaid let each group decide how to record the data rather than giving groups a recording sheet that was already organized? When would it be appropriate to give students an organized recording sheet? Discuss the advantages and disadvantages of allowing students to create their own recording plans.
I did notice that since the students were completing this activity in groups, they were able to help one another work through potential recording plans. If the experiment was conducted in a whole group type setting (where they don’t work with others), I would consider giving students pre-organized recording sheets. This would help ensure that students learn one accurate way to record data, also making it less likely for them to have to “fix” problems later on.
For further consideration….
Knowing what you now know about probability concepts in the elementary school, how will you ensure that your students have the background to be successful with these concepts in middle school?I think the best way to ensure that elementary students are successful with probability concepts in middle school is to present them with several opportunities to use the skills they have learned. To me, this means incorporating fun, hands-on activities to help engage students in the learning process. Ultimately, I think students are “more likely” (pun intended) to enjoy learning about probability when they experience or see the situations occur for themselves.
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