Wonder of Pi

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.

                                                                                                      (Image obtained here)
10:00--What questions do you have?
11:00--Let's look at another one.  This is based off of 10 coin flips.

                                                                     (Image located here)

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.

This could take a VERY VERY long time.  Don't let it.  If they want to keep searching, let them do it at lunch, at home, etc.  Give them 10 minutes to explore.  Make a big deal out of it when kids find one!

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:

 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, 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.

Give them 10 minutes to explore.  Make a big deal out of it when kids find one!  Let them try anything they want.  They may not find anything.  I'd try ages, shoe sizes, a combination of siblings' ages, etc. 

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

What's in a Domain (name)?

Personal Reflection:

I have vivid memories in school of not remembering the difference between domain and range...that moment of fear on an exam--"Which is x and which is y?!?"  For some reason there was a mental block.

As a teacher of middle school for 9 years, I never had much of an occasion to talk about domain and range.  To me, it felt a bit over the heads, even to honors algebra students.

However, as more algebra has migrated to the lower grades, and student understanding in Algebra 1 has subsequently increased, I've realized that students are VERY capable of understanding the concept of a domain or range.  (Even if they do continue to struggle with identifying which is which!)

Grade Level: 8-10

Course: Algebra, Algebra II

Standards:  F-IF.1, F-IF.2, F-IF.3, F-IF.5
SMP: MP2, MP3, MP4, MP6, MP7, MP8
Skills: Algebra, Functions, Domain, Range, Graphs, Graphing, Relations


How to use this as a mad minute:
You have 60 seconds.  Explain this math pun using accurate mathematical vocabulary and concepts.

How to use this as a warm up:
You could ask the students to consider one of the following:
1. Is this response appropriate and mathematically correct?
2.  Give another example of a function (in function notation or a graph) that would have the same answer.
3. Give an example of a function (in function notation or a graph) that would NOT fulfill this answer.
4. Help, in words, pictures, notes, etc. a student understand the difference between a domain and range.

How to use this as a mini-lesson:
As an introduction to Domain and Range (Henceforth known as D&R in this post)

If I have 20 minutes, and the desire to introduce Domain and Range to students, I'm going to begin by making sure they understand what a function is, and some common variations might be.  Remember, these students are just in Algebra I or beginning Algebra II.  They probably know linear equations, but may not be familiar with Exponential, Quadratic, Absolute Value, Step, or many other types of functions.  To make this accessible, kids need scaffolding!

I'd start by giving them only the first quadrant of a graph, and asking them to graph something like f(x)=4x-9.  Hopefully this would kick start a conversation about accurate graphs, and appropriate graphing techniques, scales, etc.  Of COURSE kids can graph this line when only given the first quadrant, if they think about it carefully.  But, would it be more accurate to use all 4 quadrants?  Would it be easier?  Does it matter?

WHY do kids crave all 4 quadrants in this situation?

Because they know how to graph using the intercept and the slope!  If the intercept isn't available, they really have to think about the graph in more detail.  (Thinking is good, but not always easy, right?)  My remedial algebra kids struggle with turning an intercept and a slope derived from a point into a full line.  They like to connect two dots and stop, forgetting that this is only a line segment, and not a line that represents every solution to a linear equation.  This is important because students who truly understand domain and range will understand that linear equations never end!

This is where I'd steer the conversation.  Students intuitively want 4 quadrants for ease, but also because they want a "full" picture of the line, not a segment confined only to the first quadrant.  If they can identify that the line will continue forever in both the x and y directions, they are ready for the D&R vocabulary.  Don't make it difficult!  Just point out that since the line goes on forever in the x direction, the Domain is All Real Numbers.  Since it goes on forever in the y direction, the Range is All Real Numbers.

For novice students, this is a great introduction.  Visually they can see it easily!  Here's where I'd shift to using the visual approach for equations that they may not be familiar with, but can VISUALLY see the domain and range!

With only 20 minutes, the last 10 minutes will be a mad (and productive rush) to visually explore functions of all kinds.  I did a quick Google Search for Domain and Range of graphs and grabbed several interesting graphs.  Feel free to throw these into a ppt, or a worksheet, for kids to explore as a class, with a partner, in small groups, or even as a homework assignment. (I made a worksheet!)



Don't forget to share the image at the end and ask kids, now that they know what domains are, if the "pun" is accurate! 

As a review of Domain and Range (D&R)
20 minutes.  Ready?  Let's go!
0:00-Check in--Raise your hands if you remember what domain and range are.  Keep them raised if you know which one is which.  Keep them raised if you feel you can define or explain them to a friend.
1:00-Those with your hands raised, spread out.  Everyone else, FIND someone with their hand raised.  You have 1 minute to summarize in words, pictures, etc.  GO!
3:00-Ok, now MIX!  Make a new group of 4 or 5 with no more than 2 people from your previous group.  Discuss.  Did you hear the same things?  Do you "get it"?  Do you agree? You have 2 minutes to discuss and compare.
5:00--Head back to your seats.  Grab a dry erase board and explain D&R with words, tables, definitions, graphs, etc.  You have 1 min to get supplies and 1 min to write.  Go!
7:00--Hold them up!  (Here you will want to look for any that stand out.  I'll hope you have a few of these to discuss and point out.)  Here's one with great definitions!  (Hold it up, put it on your doc cam, put it on your chalk tray, etc.)  This explains.... (Review definitions.)
8:00--Here's one with great pictures!  (Repeat as above.)
9:00--Here's one with equations/function notation!  (Repeat)
10:00--Here's one with tables of values!  (Repeat)
11:00--Ok, let me tell you a story, and you determine the domain and range.  Imagine your parents help you open your first debit card account.  The account will open on January 1st with $200.  You can deposit money from the holidays, or from your job.  You can withdraw money to spend at the movies or on clothes, etc.  However, this is a very "safe" card and you can NEVER overdraft.  If you try to spend more money than you have available, the purchase will be declined.  Talk with a partner and then write what you believe the domain and range are on your boards.
13:00--Let's talk about it.  What did you say?  Why?
15:00--Great, can you think of a real life situation that would have a Domain of All Real Numbers, but a Range of y≤10?  (Answers will definitely vary.  It could be human height over time.  Kids may debate that some day our heights COULD be over 10ft.  It's just a thinking task!)  Does anyone want to share ideas? 
18:00--Ok, last but not least, check out this math pun I found.  Remember, I'm proud to be a numbers nerd!  (Show the image at the top.)  What do you think?

By the way, I'd probably give them the graphs above as a homework sheet for a review assignment, or use them as a warm up the next class period.


How to use this as a full lesson?
Once again, this is tough, because it depends on how much experience your students have with D&R.  Let's assume they have none.  Otherwise you probably wouldn't need a full lesson on D&R and the mini lesson above as a review would be enough.

If I were to introduce D&R for the first time, I'd begin with the mini lesson above.

However, after students explored the 15 images and discussed the domain and range, I would bring them back together to both review their thinking and to offer support in HOW TO USE NOTATION CORRECTLY!

I'd review each image one at a time.  I'd start by asking kids what they thought the domain and range were.  I'd talk about the proper notation to show that domain and range.  (Have they used inequalities recently?)  I'd also ask if they've seen a graph like that before.  If so, do they know what type of function it is?  I would definitely throw out the appropriate vocabulary, even if the kids aren't expected to know it.  It's a great prep for the future.  (This is an absolute value function.  This is an exponential decay function.)  If the kids ask more questions, you can decide if you have time to explore, or if you want to add it to your "Let's talk about this later" list!

I would NOT plod through all 15.  I'd go through the process with about 5 or 6 and then set the kids free to revise.  Then I'd ask them to partner up, compare, discuss and revise.  Then I'd ask them to partner with another pair, and repeat.  Then I'd ask them to partner with another group of 4 and repeat.  If the group WANTS to know what kinds of functions they are, they can come and ask you.

In my room, this would be a very very full 45 minute lesson, or a leisurely 60 minute lesson.  If you are on the block schedule, I'd continue by exploring domain and range given actual functions.  Can students identify the D&R of linear functions?  Are they ready for quadratics?

If you have technology, I'd use it!  Have them graph the functions and then determine the domain and range.  Definitely make lists and observations so they can start to develop intuition without having to graph the functions!


 How to use this as an assessment?

To me, this one is pretty obvious.  It's perfectly established to slap on a quiz or test and ask the students to reflect.  Of course, this will only work if you've established expectations for reflections!

You can ask a pointed question, "Is this accurate?"
You can ask an extension question, "Can you draw 2 graphs and write two functions that would also fulfill the answer All Real Numbers?"

You can spark creativity, "Add a third pane to this image.  In the third pane, have the first character DISPROVE the second character by showing an equation or graph that would NOT have an answer of All Real Numbers."

I hope these ideas help you assess your students' understanding of D&R.


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

Gratitude

New Shoes (Link to Life Article about this photo)

This isn't a typical post.  It's not about the standards or the lessons you can create from this photo.  It's a reflection about gratitude, and one I think we all need at this point in the year.  

I love this photo.  If you haven't seen it, the story is that this is a photo taken after a young boy was given a new pair of shoes by a Red Cross worker after WWII. 

The pure joy, appreciation and peace you see in his face brings warmth to my heart.  It's a reminder to me to be grateful for the little things, the things I have, and the things that haven't happened to me.  If only I could be this grateful for something like a pair of shoes.

As the year ends, we all long for vacation, a break, and time away from the classroom.  We can't wait for a break from students complaining, parents emailing, and administrative demands.  But what if we were grateful for the little things?

I know Oprah encourages everyone to find 5 things to be grateful for each day.  Here are mine:
1.  I'm grateful that a struggling student who ditched the final, missed about 5 weeks of class this semester, and failed spectacularly, still felt enough of a connection with me to stop by and apologize for missing the final, and to give me a small gift of 5 dry erase markers, an eraser and a bottle of Expo spray.  No matter how much we struggled to connect in mathematics, he must have felt a connection to me in some way.  I'm lucky that this young man reminded me never to give up on a student and his potential to be a wonderful person, despite his struggles in math.

2.  I'm grateful for the students who work hard and never give up.  For the young man who came in early today to ask for help creating his "cheat sheet" for the final because it was important to him to do well tomorrow.  He didn't wait until the last minute, he didn't complain, he was polite and kind and eager to learn.  I'm also grateful for the young ladies (TWO!) who put forth 100% effort on their finals today even though the grade on the final was unlikely to change their grade in the class.  They worked carefully, stayed late, and did an amazing job.

3.  I'm grateful for colleagues who strive to show others how much they care, strive to make others' jobs easier, and strive to support me as I work with kids.  Today I had three incredible opportunities to appreciate my colleagues.  First, our department chair organized a pot-luck lunch along with a "goodbye" celebration for a department member who is moving to another school.  He was touched by the card, the cake, and the camaraderie.   Two colleagues reached out to support me in teaching a single-gender math class next year offering support in any way that might help the class be successful.  Finally, I interacted with three different ISS team members as we administered finals to students on IEPs.  Each one communicated clearly, efficiently, and supportively in helping me to make difficult decisions regarding students and their final grades.  

4.  In the last several days, I have received numerous parent emails, even in difficult situations (students failing) who thanked me for my hard work and for doing everything I could to support kids.  When exhaustion sets in, and you are expecting a scathing email of frustration, a simple "Thank You" goes a long way.  This is a reminder, to me, to say THANK YOU to all of the people who are working hard to make my job easier every day.

5.  As I grade finals, it's easy to feel frustrated with the lower scores, the outrageous mistakes, or what appears to be kids who didn't even try.  But, taking a step back, I'm SO IMPRESSED by what the kids were able to achieve.  Start by taking out the dozen kids who skipped the final because they had worked so hard they already had over a 100%.  Those students should be enough to lift my spirits.  I am grateful to them for their hard work and for making me feel like what I offered met their needs and helped them reach success.  Then look at the students who said they were "terrible" at math and hated it.  The kids who haven't gotten anything over a C in math since elementary school who are celebrating a B, or even an A!  I'm so lucky to get to celebrate those achievements.  I'm so fortunate to get to help those students!  Even the students who only got Cs or Ds on the final left knowing SO MUCH MORE than they did when the year started.  Those students DID make progress and grow.  I'm lucky I have gotten to know each one of them.


So...what?  I hope you take a moment to find a few things to be thankful for as the year ends.  (I hope your gratitude is more substantial than "THANK GOODNESS IT IS THE END OF THE YEAR!")  :)  I also hope that if this struck you, you'll take a few valuable moments to share this with students so they, too, can find positive ways to celebrate what they have.

Penny Floor

Personal Reflection:

Finding photos like this on popular "student" sites like 9gag.com always makes me happy!  I know, right away, that this will get them to "wonder mathematically" and dive into exploring skills and concepts that they might not have ever wanted to consider in the past.  My "wondering" and wandering led me to questions like:  How much would it cost to tile that area?  Is using pennies the cheapest?  How does that compare to normal tile?  I hope that your students will wonder mathematically in their own ways!

Grade Level: 4-9

Course: Math, Pre-Algebra

Standards:  5.NBT.5, 5.NBT.7, 5.MD.1, 6.PR.3d, 6.NS.1, 6.NS.3, 6.EE.1, 6.EE.2, 6.EE.7, 6.G.1
SMP: MP1, MP2, MP3, MP4, MP5, MP6
Skills: Area, Unit conversions, exponential notation, estimation


How to use this as a mad minute:
You have 60 seconds.  How much do you think it would cost to cover the floor of our classroom in pennies?  Estimate accurately and be prepared to justify your estimate.


How to use this as a warm up:
You could ask the students to consider one of the following:
1. How much do you think it would cost to tile our floor in pennies?  Estimate accurately, show your work and steps and be prepared to explain.
2.  Do you think covering the floor with pennies would cost MORE or less than covering it with nickles?  dimes? quarters? dollar bills?  Why?
3. Do you think covering the floor with pennies would cost more or less than using a traditional floor covering such as tile or carpet?  Why?
4. Is there a currency in the world that would be better to use than pennies?  If you can't think of one, can you think of what properties it would need to have to be a better choice?
 

How to use this as a mini-lesson:
If I had 20 minutes to spend on this lesson, and the goal was for kids to estimate, use critical thinking skills, use their knowledge of area, and compute, this is what I'd do!

Show them the photos!  Let them talk with their friends about them for less than 60 seconds.
Give them a challenge.  Tell them that you will offer (____reward_____) for the student who is able to BEST estimate the cost of covering ONE DESK (or table) in your room in pennies.

"Best" estimate, as determined by me, would not only be accurate, but would be clearly justified and explained.  It would specifically address concepts such as measurement, how to compute area, etc.  I would NOT equally reward a group that "counted" how many pennies fit along each side and multiplied and a group that used measurements, staggering (to fit more in), repeated trials and samples, etc.  Although both may be right, one shows much more depth of analysis and thinking.  The first would be perfect for an answer in 5 minutes or less, and the second would be appropriate for a 20 minute exploration.

I'd offer them a selection of supplies:

1. Baggies of 10 or fewer pennies
2. Rulers or meter sticks
3. Tape
4. Paper

It would be up to the students what supplies they might use.


How to use this as a full lesson?

Use the EQUATE thinking routine.  Give them the photos with NO guidance.  Let them Explore, Wonder, and Question.

Record their questions and clarify!  Develop depth to the questions by extending them, drawing out detail and asking how they might go further.

For example, a student might ask, "How much would that cost?"  This is a great "wonder" but not enough to explore mathematically.  What are they going to investigate?  How can they extend this question?  As you draw out these ideas, make sure to ask them to list what information they need to answer their questions.  Feel free to share answers or to encourage them to FIND THEIR OWN answers.  This is a great time to use technology to your advantage.  Students can research on phones and iPods, or you can nominate a class researcher who will research the answers to these questions while you continue to work with the rest of the class.  Narrow down your questions to your top two or three questions.  Tell them they can choose from "These" deep understanding questions to answer in their next steps.  It could be something from the suggested warm ups with more depth, it could be calculating the cost of covering the floor of your room, their bedrooms, the hallway, etc.  For students who are advanced, ask them to compare the prices of using pennies to nickels and dimes with only a ruler and one of each coin. 

Establish expectations for products, time frames, behaviors, and jobs.

Set them free!  Help them question and explore and apply their mathematical knowledge to solve their problems and answer their questions.  

How to use this as an assessment?
You know your students best, and you know if you will have prepared them for this.  I know many teachers ask challenge questions such as, "How many boxes of Kleenex will it take to fill this classroom?" as assessment questions.  However, many other background experiences are necessary before students can attack these kinds of problems independently on an assessment.

If you feel you've provided similar learning experiences that would help a student successfully approach such a problem, go for it!

• Ask a specific and clear question.
• Be sure you know what kinds of answers you will accept.
• Determine how much written and verbal guidance you will provide during the assessment.
• Make sure you provide clear rubrics or standards for achievement.  Students need to know what will earn a passing, or excelling, grade!
• Make notes of what works and what doesn't so you can improve it for next time.

Every school grades differently, sometimes based on standards, some using IB or pre-IB rubrics, some on critical thinking and creativity, some on a strictly points-based system.  While I cannot help every one of you, in the future I will update this with my own question, instructions, and rubric!


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

But it's a dry heat...

Personal Reflection:

I love "math fails" but I think these are really just "editing" fails!  We all make typos, some are just MUCH BIGGER than others!  I found two temperature fails in a short period of time and decided to go with them!  I encourage my kids to "wonder mathematically" a lot, and my own mathematical wondering led to this exploration.

Grade Level: 4-9

Course: Math, Pre-Algebra, Algebra

Standards:  5.NF.3, 5.NF.4, 5.NF.6, 6.EE.2, 6.EE.3, 6.EE.9
SMP: MP2, MP3, MP4, MP5, MP7
Skills: Research, Conversion, Science, Temperature scales


How to use this as a mad minute:
You have 60 seconds.  Name 3 different temperature scales.  Would these temperatures be appropriate in any of those scales?


How to use this as a warm up:
You could ask the students to consider one of the following:
1.  Do you think the temperatures listed are in Farenheit or Celsius?  Why? 

2. Given the chart at the right, convert one of the "high" temperatures from Kelvin to Celsius and Farenheit.  (Assume the temperature is in Kelvin.)
3. Why do you think there are multiple systems of temperature measurement? 
4. Can you think of a place that might actually be 519 degrees?  Where?  Why?
5. The weatherman is pointing at a temperature of 519 degrees, but immediately below, it says the high temperature is 52 degrees.  Based on this, can you guess what error might have occurred?  Explain.  Do you think the 789 degree temperature was the same mistake?  Why or why not?

How to use this as a mini-lesson:
I would start with a conversation, asking students what they know about temperature, different temperature scales, etc.  You might want to ask what body temperature, room temperature, and freezing and boiling points are on different scales.   They really need a way to frame this number!
Feel free to use the Dan Meyer technique of saying, "Choose a number you are SURE is TOO BIG" for room temperature in Celsius," or a number that is TOO SMALL for body temperature in Farenheit.  This helps the students work on both estimation, number sense, and confidence in math.

Introduce the Kelvin scale and why it is used.  Offer major numbers such as freezing and boiling, room temperature or body temperature on the scale.  Talk about why we have a scale like this.

If kids have access to technology, set them free to answer these questions themselves and be prepared to come back and share their findings!  You could challenge them:  "Is there a place in the US that could reach these temperatures?"  "Is there a place on Earth that could reach these temperatures?"  "Is there a place in our Solar System?"  Ask them to use their knowledge to make informed decisions, not random searches.

Gentle guidance could help:

• What planets might be more likely to be warmer?  Why?  
• What kinds of environments are warmer? 

Depending on the level of your students, ask them to explore the temperatures of different planets.  What traits do they possess when they are hotter?  How do scientists record temperatures on other planets?  If it's a mini-lesson, it's probably not a major lesson or life goal, but a chance to spark interest, address a cool concept and move on.  Don't let the details stop you from letting the kids explore!


How to use this as a full lesson?

Use the EQUATE thinking routine.  Give them the photos with NO guidance.  Let them Explore, Wonder, and Question.

Record their questions and clarify!  Develop depth to the questions by extending them, drawing out detail and asking how they might go further.

For example, a student might ask, "Is that even possible?"  This is a great "wonder" but not enough to explore mathematically.  What are they going to investigate?  How can they extend this question?  As you draw out these ideas, make sure to ask them to list what information they need to answer their questions.  Feel free to share answers or to encourage them to FIND THEIR OWN answers.  This is a great time to use technology to your advantage.  Students can research on phones and iPods, or you can nominate a class researcher who will research the answers to these questions while you continue to work with the rest of the class.  Narrow down your questions to your top two or three questions.  Tell them they can choose from "These" deep understanding questions.

Establish expectations for products, time frames, behaviors, and jobs.

Set them free!  Help them question and explore and apply their mathematical knowledge to solve their problems and answer their questions. 

How to use this as an assessment?
You know your students best! I would not use this as a formal assessment.  You could, however, find a similar graphic and ask it as a constructed response item on an assessment of your own!  You could take any of the lesson options above and extend it with provided rubrics, more structured questions, etc.

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

EQUATE! Thinking Routines

The EQUATE thinking routine is a process I've used in my math classroom based on a constructivist approach.  This routine allows for students to explore a real life situation, wonder mathematically, ask questions, apply their mathematical knowledge, dive into the problem and solution opportunities and finally explain their results. 

This has worked as an assessment tool, a tool to develop interest in concepts, a tool for introduction to skills and formulas, and a great way to establish classroom routines and expectations.  You'll see me refer to EQUATE in other areas of my blog, and this is what I'm speaking of.  Please feel free to ask questions if any area is unclear!

(The above image is property of Andrew Sabatier, and is not my own.)

E—Explore—In this step the students discuss the situation and problem and come up with any questions they have for you. You may return to this piece over and over.

Qu—Question—In this step students ask you all of the questions that they have. You may

or may not have the answers.

(Now is a good time to have them go back and explore and see if they have more questions based on what they found out.)

A—Apply—This is when they will apply their knowledge and the information they gained from their questions to try to solve the problem. This may include investigations, trials, research, deriving formulas, doing calculations, etc.

T—Try Something—This is the “DO” piece. Even if they don’t know what they are doing, they need to take action. At any time they can come back to explore and question.

E—Explain—After they have arrived at an answer, they should explain it. How did they get there? Is it reasonable? How can they be sure they are right? What strategies did they use?


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

That's Some Inflation!

Personal Reflection:


I've heard recently about the collapse of Zimbabwe's currency and how they just keep printing larger and larger notes and that something like this probably wouldn't buy a cup of coffee.  (Apparently not true!) What causes inflation?  How does this connect to our own currency?  I know I wanted to investigate, so I hope students will too!

Grade Level: 4-9

Course: Math, Pre-Algebra, Algebra

Standards:  6.RP.1, 6.RP.2, 6.RP.3, 7.RP.1, 7.RP.2

SMP: MP1, MP2, MP3, MP4, MP5, MP6, MP7
Skills: Research, Conversion, Scientific Notation


How to use this as a mad minute:
You have 60 seconds.  What is this worth in US Dollars?


How to use this as a warm up:
You could ask the students to consider one of the following:
1.  Write one hundred trillion in scientific notation.
2.  As of May 21, 2013, 1 dollar in Zimbabwe is worth 0.00276 US Dollars.  How much is this note worth?
3.  What would 100 trillion US dollars be worth in Zimbabwe currency?
4.  If the US had 100 trillion dollars and divided it evenly among the citizens, how much would you get?
5.  Can you name the largest US bank note?  Why is that the largest one?
6.  Why can't poor countries just create these kinds of bills to pay off debts?


How to use this as a mini-lesson:
I would start with a conversation, asking students for their immediate thoughts and ideas.  PLEASE explore those and go with the flow!  They will ask amazing questions and take your class in directions you can't imagine!  If conversation stalls, try the warm up questions to get them talking and thinking.  They really need a way to frame this number!

Ask them to think about why the US doesn't create bills this large.  Is this even a real bill?  If so, could you cash it in at a bank?  Why or why not?

Why don't poor countries just make money like this and bring it to the US to cash it in? 

How to use this as a full lesson?
I would start with the warm up and mini lesson outlined above and then I would set the kids loose with a challenge.

CHOOSE ONE:
a.  Research inflation.  What causes it?  How is it controlled?  What happens when it isn't controlled?  How will inflation affect YOU in 20 years? 
OR
b.  Consider the IMP activity about the price of eggs.  (This is best for an Algebra 1 level class.)  It helps students to understand inflation and how prices grow and to predict the price of eggs in the future!  Link
OR
c.  Research the reasons for the collapse of the Euro and the financial crisis in Europe.  Create a 1 page poster that explains the BIG issues.  WHO?  WHAT? WHERE? WHEN?  WHY?
OR
d.  Choose a common item.  (Such as a pair of jeans.)  Find out the cost of that item in 5 different countries and convert it from the original currency to US Dollars.  (Use proportions to convert!)  Consider why prices vary so greatly in other countries.  Discuss why a pair of jeans would be more or less expensive elsewhere.

I'd ask students to be prepared to share their findings (perhaps in a jigsaw) with other students with about 15 minutes left in class.  This isn't intended to be a long term activity!

How to use this as an assessment?
You know your students best! I would not use this as a formal assessment.  You could, however, find a similar graphic and ask it as a constructed response item on an assessment of your own!  You could take any of the lesson options above and extend it with provided rubrics, more structured questions, etc.

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

Algebraic Notation in Real Life

Personal Reflection:


I ran across these two images and thought it was a perfectly fun way to talk about algebraic properties!  My students LOVE the Lady Gaga reference, and the new tweet to the right captures both the mathematical spirit AND a terrible pun!

Grade Level: 7-9

Course: Pre-Algebra, Algebra

Standards:  6.EE.2, 6.EE.3, 6.EE.4, 6.EE.6, 7.EE.1, 7.EE.2, 7.EE.3, 7.EE.4, 8.EE.7B
SMP: MP1, MP2, MP3, MP4, MP7, MP8
Skills: Algebraic notation, Mathematical Properties, Multiplying polynomials, "FOIL" method


How to use this as a mad minute:
You have 60 seconds.  Are these accurate?  Why or why not? 


How to use this as a warm up:
You could ask the students to consider one of the following:
1. What mathematical property is illustrated here?
2. Make up your own (appropriate) 4 letter word.  Multiply it as if it were a polynomial.  What happens?
3.  Can you multiply 4 letters to create a REAL 8 letter word?
4.  Rewrite MISSISSIPPI as a distributive property problem.
5.  Rewrite SASSAFRAS as a distributive property problem.
6.  Can your name be shortened using Algebraic notation?  Why or why not?

How to use this as a mini-lesson:
I would start by verifying that these two problems DO, in fact, work.  I would ask the students to justify the mathematical properties that are used in each step to expand the problems.  I'd challenge them to take a common chorus and rewrite it.  For example:

Come they told me, pa rum pum pum pum
A new born King to see, pa rum pum pum pum
Our finest gifts we bring, pa rum pum pum pum
To lay before the King, pa rum pum pum pum,
rum pum pum pum, rum pum pum pum,

How to use this as a full lesson?
I would start with the warm up and mini lesson outlined above and then I would set the kids loose with a challenge.

EITHER:
a.  Rewrite your favorite song using accurate mathematical notation.  Be sure to justify each line by noting the property you are using to shorten the song!
OR
b.  Find 10 long and repetitive words (such as MISSISSIPPI) and rewrite them using mathematical notation.  Bonus points if you are able to create an entire sentence of such words!

How to use this as an assessment?
You know your students best! I would not use this as a formal assessment.  You could, however, find a similar graphic and ask it as a constructed response item on an assessment of your own!


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

Swimming Pools of Saliva

Personal Reflection:

Let's start off with being honest.  I do my share of internet surfing.  Two sites I visit regularly for their awesome "kid" content are uberhumor.com and 9gag.com.  You'll see their watermarks at the bottom of most of my photos.

This one stood out to me right away because I thought, "NO WAY!"   (And I bet your students will think that too!)  So, I thought, "Let's find out."
  

Grade Level: 3-8

Course: Math, Pre-Algebra

Standards:  6.EE.2, 6.EE.3, 6.EE.9, 6.G.2, 7.RP.1, 7.RP.2, 7.G.1, 7.G.6

SMP: MP1, MP2, MP3, MP4, MP5, MP7
Skills: Estimation, Volume, Unit conversion, Scientific Notation


How to use this as a mad minute:
Get out those smart phones, those electronic devices, iPods, laptops, iPads, etc.  Is this true?  Can you find an answer in 60 seconds or less?  GO!


How to use this as a warm up:
You could ask the students to consider one of the following:
1. List the information would you need to gather to determine this is true.
2.  Estimate how much saliva you produce in 1 hour.  1 day.  1 week.  1 year.
3.  How big is a swimming pool?  How much water do you think it holds?
4.  Calculate the volume of a swimming pool that is 50m x 25m x 2m.  (Olympic average.)
5.  How much saliva do you think is in a single cubic meter of water?
6.  If the volume of an Olympic Pool is 2,500,000 L, how much saliva does one person produce each day?

How to use this as a mini-lesson?
You have to decide how much freedom to give your students and how structured you want this to be.  It could be a great inspiration for how to find the volume of a rectangular prism (true swimming pools that are more like trapezoidal prisms), or how to convert between metric and English measurements, or how to convert between large and small numbers, or even how to use scientific notation appropriately.


Mini Lesson 1:  Volume
Let's find the volume of different pools!  (Olympic, neighborhood, backyard pools)
Olympic:  50m x 25m x 2m  (Note: Olympic pools are not rectangular prisms, but this is an average.)
Neighborhood:  25m x 10m x 1.5m  (Kids could find out their own measurements!)
Backyard:  Radius=2m, Depth= 1.5m (Cylinder volume!)

Mini Lesson 2:  Unit Conversion
Saliva is measured in ounces.  How much is 2,500,000 Liters  in ounces?  How do you convert?  What proportions do you use?

Mini Lesson 3:  Scientific Notation
An Olympic Swimming pool has a volume of about 2,500,000 Liters.  Write this in scientific notation.  If the average person creates 1 liter of saliva every day for 79 years, how much saliva will they create in a lifetime?  Write your answer in scientific notation.  Which number is larger?

Mini Lesson 4:  Is this reasonable?
If you are trying to do this as a mini lesson, kids will need as MUCH information as possible.  You will need to tell them how much an average pool holds.  (Olympic=2.5 million liters)  You'll need to tell them how much saliva a person produces each day.  (Approximately 1 liter)  How long does the average person live?  (In the US it is approximately 79 years)  Can you use this information to determine if the average person would fill a swimming pool with saliva?


How to use this as a full lesson?
Expand any of the mini lessons above to include practice problems, a homework worksheet, or a continued exploration.  For example:  How long would it take to fill a bathtub with saliva?  How much saliva does the city of Denver create each day?  Each year?  Is the amount of saliva created each year by the population of China MORE or less than the amount of water in The Great Lakes?

How to use this as an assessment?
You know your students best!  If I were doing this with a GOOD group of middle school students who had mastered volume, this is the assessment I'd give them.

Look at this meme!  Is it true?  Use your technology to research both saliva production and the size of swimming pools.  Then use your knowledge of volume to answer whether or not this is reasonable.   Justify your answers with clear mathematical knowledge and computation.  Use the rubric to get full credit!

My rubric would require showing formulas for volume and how it was computed, how they got their numbers for the dimensions of the pool and the amount of saliva and the life span, including documenting sources.  I'd want them to explain answers in complete sentences with correct mathematical vocabulary.

If your kids aren't ready for this freedom, structure it for them!  (But, please, give them a chance and build the opportunities.  They'll get there, I promise!)


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

Introductions

Hi!

You may wonder, who's writing this blog, why should you care, and what can it do for you?

My name is Natalie Sprigg and I've just completed my twelfth year of teaching mathematics.  I LOVE working with kids, other educators, and the opportunity to create and inspire learning.  I've taught for 9 years at the middle level and 3 at the high school level.

I have always had the joy and opportunity to merge my own work and creations with the curriculum provided to me through my school district.  By nature, I'm a creative questioner and I like to find cool things in the world around me to use as inspiration for lessons.

As I begin, I am considering the following structure to each of my blog posts.  I will find inspiration around the web and in my daily life and then I will share it with you.  I will examine the item, share why it struck me, what grade level I would target, the type of math, standards and skills that would be appropriate, and then how to use the item as a Mad Minute, a warm up or introduction to a concept, a mini-lesson, a full lesson and an assessment.  Some of these will not be appropriate for every category.  Sometimes I'll outline an idea, and will come back to it later to add supporting documents.  Please share your feedback, your ideas, and your extensions!

I look forward to reflecting with you, to exploring tangents with you, and to creating really cool math lessons for kids!

Natalie Sprigg


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