Personal Reflection:
I saw this on 9gag.com and being a numbers nerd (proudly!), I saved it. I wasn't immediately sure how I would use it. It's a great way to show the concept of an irrational number never ending. (By the way, could we give a few other numbers credit the same way we give π credit?) It's also a great way to show the concept of infinity. We could use this in a geometry unit anywhere from 4th to 10th grades. And yet, when I thought about it, I decided I'd most love to use this in a probability exploration. Thus, the activities below are targeted at 7th grade (6th and 8th are statistics years, FYI) and at the high school level. I will try to put appropriate 7th grade explorations in RED print and high school explorations will be in BLUE print.
Grade Level: 7th & HS
Course: Pre-Algebra, Algebra, Geometry, Prob/Stat
Standards: 7.SP.5, 7.SP.6, 7.SP.7, S-CP.2, S-CP.5, S-CP.6, S-CP.7, S-CP.8, S-CP.9, S-MD.6, S-MD.7
SMP: MP2, MP3, MP4, MP5, MP6, MP7, MP8
Skills: Compound Probability, basic probability, rounding, research, science, decision making, conditional probability, independent probability, frequency table, sample space, probability modeling, data collection
How to use this as a mad minute:
You have 60 seconds. Look at this image. What are your immediate reactions, thoughts, concerns, celebrations, etc?
How to use this as a warm up:
You could ask the students to consider one or more of the following:
1. List everything you know about π.
2. What is the difference between an irrational number and a repeating decimal?
3. What is the probability of randomly selecting the digit 3 from a set of digits 0-9? What is the probability of selecting the digit 1 from a set of digits 0-9? Are the probabilities equal? Why?
4. What is the probability of randomly selecting the digit 3, replacing it, and then selecting the digit 1 from a set of digits 0-9? Is this more or less likely than selecting each number independently? Why?
5. What is the probability of randomly selecting the digit 3, replacing it, and then selecting the digit 1, replacing it, and then selecting the digit 4 from a set of digits 0-9? Is this more or less likely than selecting each digit independently? Why?
How to use this as a mini-lesson:
As a 7th grade exploration.
Assuming you have 20 minutes with 7th graders who have NO background in compound probabilities, here we go!
0:00--Today we are going to talk about probability. We only have a short time, so let's agree that we are going to talk about "FAIR" probabilities, no trick coins, spinners, dice, etc. Given that, what's the probability that when you roll a die, you get a 3? (Allow for discussion, calling out, etc.)
1:00--Ok, so I heard answers as __________ (fractions? decimals? Percents? All three? This is where your fantastic impromptu skills come in. I would guess that students will use a fraction more than any other, mostly because the decimal and percent equivalents are not obvious or easy for them to manage mentally.) The probability of an event occurring is
(from mathisfun.com)So fractions are pretty easy to use when you know the number of 3s on a die and the total possibilities. Do you all know that you can also write probabilities as decimals and percents? (Get feedback, determine if you need to use another minute to discuss converting to decimals and percents.)
2:00--Awesome! Let's do a few more probabilities. This time I want you to try answering as fractions, decimals AND percents. Switch it up!
What's the probability of flipping a coin and getting heads?
What's the probability of being born a girl?
What's the probability of grabbing a marker out of my hand and getting red? (Hold out 4 markers of different colors.)
(Listen to responses and address accurate and inaccurate answers as well as any concerns or creative answers you need to discuss! Add your own if you need to, based on what you have around your room.)
3:00--What's the probability of selecting a letter from my last name and it being a ____? (Using SPRIGG as an example, I'd ask first about the S, the P, the R, the I, and then, the G. Most answers will be in fractions, depending on how many letters your last name has, which is fine!)
4:00--Let's talk about the possibilities of COMPOUND probabilities. This is when I ask you something more complicated, like, "Whats the chance I roll a 3 AND flip a coin and get heads?" Take a moment and write down your guess. What do you think the chances are that when I flip this coin and roll a die, I get a 3 and Heads?
5:00--How are we going to figure this out? We already know the chance of getting a 3 is 1/6 and the chance of getting a heads is 1/2. So what next? (Allow for discussion. Some students may already have learned this, which is a great stepping stone. List their ideas. Refer them back to the probability definition above.)
7:00--If we want to know the probability, we need to know the number of outcomes. Does anyone have a suggestion of how we can figure out the number of outcomes?
8:00--Ok, grab a partner and you have 2 minutes to try to list all of the outcomes that can happen if I flip a coin and roll a die. (While kids work, write the combinations down. You can put it on a projector, your dry erase board, on a sheet of paper for a doc cam, etc. Basically, you don't want to waste time doing it in front of them. However, you might want to show different strategies, such as a tree diagram, organized lists, etc.)
10:00--How many different outcomes did you find? (Get feedback. If answers are all over the place, you are going to need an additional one or two minutes for this lesson. If the answers are very divergent, ask groups to partner with a group whose answer is FAR from theirs and discuss and compare. Hopefully they will help each other see the best answer (12)! If they are close to 12, but maybe missed one or two combinations, continue.) How did you come up with those?
11:00--I did the same thing! (Show your work.) I got 12 combinations. What do you think? How many of those are combinations that show a 3 and a heads?
12:00--So, 1/12. We already knew 1/6 and 1/2 for each. What happened? Do you think this will happen every time? Should we try a different experiment?
13:00--What's the chance that I reach into a bag that has every letter in it, and I draw out an "H"? What about if I replace it and try again. What's the chance of drawing out an "I"? Do you want to make these lists? Can we figure out the probability without making the list?
15:00--Yes, it's 1/676. I have a 1/676 chance (.15%) chance of randomly making the word "HI" when I draw the letters.
17:00--Look at this graphic. read it carefully. Think about it for a minute on your own. (Show the π image.)
18:00--Talk about it with your friend. Based on what you just observed about probability, do you think this graphic is true? Why or why not?
19:00--It's a fascinating proposal! If you are interested, this will be on my website so you can investigate further. I also recommend researching the "Infinite Monkey Theorem" to help you understand what this image is suggesting as well as the probability of it happening. (Note: Teachers, the first GOOGLE link to infinite monkey theorem is to the winery. You may want to offer your own links on your website to avoid any issues.)
As a 10th grade exploration.
Assuming you have 20 minutes with 10th graders (Any level of high school, really) who have background in compound probabilities, and need a refresher before you jump in, here we go!
0:00--Today we are going to talk about probability. We only have a short time, so let's agree that we are going to talk about "FAIR" probabilities, no trick coins, spinners, dice, etc. Given that, what's the probability that when you roll a die, you get a 3? (Allow for discussion, calling out, etc.)
1:00--Ok, so I heard 1/6 (I hope!) The probability of an event occurring is
(from mathisfun.com)These are independent probabilities. If you look at the probability of a single even occurring, you can find it by looking at the number of positive outcomes divided by the number of total outcomes.
2:00--What's the probability of selecting a letter from my last name and it being a ____? (Using SPRIGG as an example, I'd ask first about the S, the P, the R, the I, and then, the G. Most answers will be in fractions, depending on how many letters your last name has, which is fine!)
3:00--Let's talk about the possibilities of COMPOUND probabilities. This is when I ask you something more complicated, like, "Whats the chance I roll a 3 AND flip a coin and get heads?" How many of you have done this in the past? Do you remember how to do it? (Discuss and do a quick review.)
5:00--Usually, in lower grades, you list the total outcomes. You might make a tree diagram, a table, a list, etc. But you already learned a trick to bypass this step. It's important, though, to make sure that you are finding all possible combinations. In high school you are asked to show probability distributions. This proves that you've accounted for all possible outcomes. Let's start with something simple. List the outcomes I can get if I flip a coin twice.
7:00--Great, you should have gotten HH, HT, TH, TT. Quick question. Are HT and TH different results? Should we count them as two different outcomes? Why?
8:00--Ok, great. 4 outcomes. Now you can make a probability distribution. You start with a question. "If I flip a coin twice, what is the chance I get heads?" You want to start with a table that shows the possible outcomes and probabilities.
9:00--Next, you translate that into a graph. Often the graphs are "curves" but can also be bar graphs.
10:00--What questions do you have?
11:00--Let's look at another one. This is based off of 10 coin flips.
12:00--Let's talk about this. Why do you think the graph looks like this? Is this common? Can you compute, by hand, how many times you'd get only 1 head? We'll come back to probability distributions later.
13:00--Ok, let's change directions. What's the chance that I reach into a bag that has every letter in it, and I draw out an "H"? What about if I replace it and try again. What's the chance of drawing out an "I"? Do you want to make these lists? Can we figure out the probability without making the list?
14:00--Yes, it's 1/676. I have a 1/676 chance (.15%) chance of randomly making the word "HI" when I draw the letters.
15:00--Look at this graphic. Read it carefully. Think about it for a minute on your own. (Show the π image.)
16:00--Talk about it with your friend. Based on what you just observed about probability, do you think this graphic is true? Why or why not?
18:00--It's a fascinating proposal! If you are interested, this will be on my website so you can investigate further. I also recommend researching the "Infinite Monkey Theorem" to help you understand what this image is suggesting as well as the probability of it happening. (Note: Teachers, the first GOOGLE link to infinite monkey theorem is to the winery. You may want to offer your own links on your website to avoid any issues. This is wikipedia, but has a good, basic intro to The Infinite Monkey Theorem.)
How to use this as a full lesson?
I'd start a full lesson exactly as I did above. I wouldn't feel the need to rush the kids and their answers or their work, though. So this may take more than 20 minutes.
As a 7th grade exploration.
Starting where we left off, with the "proposal" that all of "everything" can be found in π.
I'd print a page of the first ____ digits of π for the students. I'd give each student a sheet and ask them to get a marker or highlighter. I'd ask them to see if they could find the following sequences in the first ____ digits of π.
- The two digit month they were born.
- The two digit day they were born.
- The two digit ending of the year they were born.
- The six digit birthdate made up of the month, day and year they were born. Ex: July 4, 1996 would be: 070496.
- Their age.
- Their height in inches.
- Their height in centimeters.
Ask the kids to trade markers. (A new color would be helpful.) Then explain what ASCII is. ASCII is an acronym for the American Standard Code for Information Interchange. It is a set of digital codes widely used as a standard format in the transfer of text. (google def.) In other words, computers only talk in numbers, not in letters. So ASCII is a programming language that translates letters and symbols to numeric codes. Here's a conversion table:
Sprigg:
S = 83
p = 112
r = 114
i = 105
g = 103
g = 103
So Sprigg could be translated to 83112114105103103 (Note: This is not entirely accurate, as coding systems need a way to know the difference between 83 and 831.)
However, I can now search π to see if this string of digits appear. (It didn't appear in the first million digits.) I did this by going to http://www.piday.org/million/ and then opening the find tool bar, and typing in my string of digits.
I tried something easier, my nickname, Nat. 7897116 (Nope.)
I tried something easier, hi, as we explored before. 104105 (Nope.)
I was concerned that this wasn't working and tried pasting the digits in a word document to do the search. Nope. I checked by searching for a string I could see. It was found. So this method will work! It was on page 6 before it found my 4 digit code of my birthday month and day.
I really like this, even the failure part, because it illustrates to kids how difficult this probability will be. If the translation to ASCII of "hi" can't be found in the first 9 pages of digits of π, what's the chance it will find, "the name of every person you will love, the date, the time" etc.?
I'd return to discuss the probability of this happening.
Then I'd throw out the Infinite Monkey Theorem (which I'm 99% certain inspired this graphic). Ask the kids to discuss their thoughts and reactions to the theorem.
Then share with them this quote, "The relevance of the theory is questionable—the probability of a monkey exactly typing a complete work such as Shakespeare's Hamlet is so tiny that the chance of it occurring during a period of time even a hundred thousand orders of magnitude longer than the age of the universe is extremely low (but not zero)." (From wikipedia.)
If that doesn't give all of us some perspective on the concept of infinity, not much will!
As a 10th grade exploration.
I would do the lesson outlined above, with a few changes.
First, I'd still address the probability distribution in the mini lesson. I'd continue from the mini lesson. Rather than print pages of π, I'd get the kids to use technology. Just as I outlined above, if they go to http://www.piday.org/million/ and search for the following sequences.
I'd ask them to see if they could find the following sequences in the first ____ digits of π.
- The two digit month they were born.
- The two digit day they were born.
- The two digit ending of the year they were born.
- The six digit birthdate made up of the month, day and year they were born. Ex: July 4, 1996 would be: 070496.
- Their age.
- Their height in inches.
- Their height in centimeters.
Given the image, the kids need to know what ASCII is. Give them 3 or 4 minutes to use technology to search for what ASCII is and to find some examples. (Info is above if you need it.)
Definitely take a minute to review their findings and clarify with them!
I'd then ask the kids to use the table above, or any tool they found in their research to translate their name to ASCII characters. In this way, we can translate any word into strings of digits. For example:
Sprigg:
S = 83
p = 112
r = 114
i = 105
g = 103
g = 103
So Sprigg could be translated to 83112114105103103 (Note: This is not entirely accurate, as coding systems need a way to know the difference between 83 and 831.)
However, students can now search π to see if this string of digits appear. (It didn't appear in the first million digits.)
I tried something easier, my nickname, Nat. 7897116 (Nope.)
I tried something easier, hi, as we explored before. 104105 (Nope.)
I was concerned that this wasn't working and tried pasting the digits in a word document to do the search. Nope. I checked by searching for a string I could see. It was found. So this method will work! It was on page 6 before it found my 4 digit code of my birthday month and day.
I really like this, even the failure part, because it illustrates to kids how difficult this probability will be. If the translation to ASCII of "hi" can't be found in the first 9 pages of digits of π, what's the chance it will find, "the name of every person you will love, the date, the time" etc.?
I'd return to discuss the probability of this happening.
Then I'd throw out the Infinite Monkey Theorem (which I'm 99% certain inspired this graphic). Ask the kids to discuss their thoughts and reactions to the theorem.
Then share with them this quote, "The relevance of the theory is questionable—the probability of a monkey exactly typing a complete work such as Shakespeare's Hamlet is so tiny that the chance of it occurring during a period of time even a hundred thousand orders of magnitude longer than the age of the universe is extremely low (but not zero)." (From wikipedia.)
Ask your high school students to simply calculate the probability of the string of digits to form "hi" appearing in a row. (104105) That's 10^6 combinations! 1/1,000,000 chance! Of course it doesn't appear in the first 1 million digits! But what if those digits went on forever?
How to use this as an assessment?
I don't feel that this is an appropriate assessment tool. However, if you have just finished an in-depth exploration of probability at the high school level, you could definitely ask your students to respond to the graphic and back up their responses with mathematical thinking and research. That would, to me, be a big project, and a take-home assignment at the very least!
Please feel free to use any of these ideas and modify them to meet your needs. However, please acknowledge the original source of the items and my own lesson outlines. ©NatalieRSprigg 2013
No comments:
Post a Comment