Exponents...Rule?

Personal Reflection:

Oh my goodness, I love it.  I'm pretty sure I found this on http://math-fail.com/ (which is a pretty fun site if you have time to search for the good stuff).

I love that this shows someone, who looks like a teacher, making the same conceptual errors our students do!  What a perfect way to get kids engaged in discussing not just the "rules" but the "whys" and "hows" of exponential notation.

Grade Level: 6-9

Course: 6th Grade, Pre-Algebra, Algebra

Standards:  6.EE.1, 6.EE.2, 8.EE.1, N-RN.1, N-RN.2, A-SSE.3, 
Skills: Algebra, Exponents, Exponent Rules, Powers, Bases


How to use this as a mad minute:
You have 60 seconds.  Explain why this teacher's simplification is incorrect.

How to use this as a warm up:
You could ask the students to consider one of the following:
1. What is the meaning of an exponent?
2. What is the difference between 3-squared and 3x2?
3.  Where in real life do we use exponents?  Why?
4.  What is the difference between the original expression and what the teacher wrote?  (Note:  Only for students who more experience with exponents!)

How to use this as a mini-lesson:
Some might wonder why I listed this as a 6th through HS level standard or lesson.  Truly it is because of the depth of thinking and analysis you could ask each level to bring to the table.  Ideally, the skill of simplifying this expression is an eighth grade standard.  However, exponents and the use of them is introduced in sixth grade and is, of course, expanded through high school.

If I were teaching middle school, I'd begin by revisiting the meaning of an exponent and might even ask students to write examples and expanded forms.  I'd continue by asking them to replace g-squared with another substitute or variable.  If they realize that the replacement should expand to x*x*x*x*x*x*x and if they also can say that g-squared should expand to g*g, they can quickly arrive at the idea that this is really g*g*g*g*g*g*g*g*g*g*g*g*g*g.  What a great review!

I'd return by asking kids to create their own "mistake" problem and prove the right answer.


How to use this as a full lesson?
I wouldn't use this as a full lesson unless you were knee-deep in your exploration of exponents and their properties.  If that is the case, you are probably teaching an eighth grade math class!  And if that is the case, you probably have a district-mandated curriculum.

This is a great supplement to that!  If you have used your primary curriculum to build understanding of exponents and their properties, you could use this as an exit slip for your lesson and simply ask students to explain the mistake in the teacher's thinking.

If you'd like, use this to launch the lesson.  Your students should already understand the meaning of exponents, but have probably not experienced "nested" exponents.  You can simply ask students to make sense of the original problem, make sense of what was written, and compare their answers.  Kids would have to dig deeply, with scaffolded questions, to get there, but I'm confident they could, as long as they have a solid understanding of exponents and their meaning.  (See the mini lesson above for some scaffolded questions.)


 How to use this as an assessment?
If your students are ready for an assessment, I would definitely put this photo on an exit slip, quiz, or test with a simple, "Explain the error in thinking shown here."

*Remember to think about what a proficient answer would entail, and what might a student to go beyond your expectations!

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 2015

Workout

Personal Reflection:
I'm going to be honest right up front.  I'm not a runner.  So when I workout on a treadmill, it's a pretty slow jog.  The beauty of this post is that if you choose to use it with your students, they can laugh and criticize the person on the treadmill and you can promise left and right it isn't you, and you can agree, the person here is one step above a sloth!  :)

That being said, I can't be the only "numbers nerd" out there that is constantly doing mental math while working out.  Maybe you count your reps when lifting weights and calculate the total pounds you've lifted?  Maybe you set a swimming goal and are finding the fractional portion of your workout completed with each length or lap?  Maybe you figure out how many hours you'll have to stay on the treadmill to burn that milkshake you enjoyed last night?  I know I'm constantly looking at the numbers and doing a variety of calculations.  I couldn't help but take a few photos today thinking that the wealth of information included on the screen is invaluable.

Grade Level: 7-9

Course: Pre-Algebra, Algebra

Standards:  8.EE.5, 8.EE.6, 8.F.2, 8.F.3, 8.F.4, 8.F.5, 8.SP.1, 8.SP.2, 8.SP.3, N-Q.1, A-CED.2, A-CED.3, A-REI.10, F-IF.4, F-IF.5, F-IF.6, F-BF.1, F-LE.1, 
SMP: SMP.1, SMP.2, SMP.3, SMP.4, SMP.5, SMP.6, SMP.7, SMP.8

Skills: Algebra, Functions, Lines of best fit, slope, rate of change, real life application, computation, extrapolating data


How to use this as a mad minute:
You have 60 seconds. List all of the questions you could answer given the four photos provided.

How to use this as a warm up:
You could ask the students to consider one of the following:
1.  Write down three reactions you have looking at these photos.
2.  What information is provided in these photos?
3.  Is this person moving at a constant rate?  (Consider vertical movement as well as speed.)
4.  Is this person burning calories at a constant rate?
5.  Can you write an equation given the information above?  (Try it!) 

How to use this as a mini-lesson:
To use this as a mini-lesson, expand on the warm up questions above.  You can ask students to make data tables, graph the data provided, and to compare rates of change and determine if a line of best fit is appropriate.  Students can show the data in multiple ways and can then evaluate the graph, equations and tables to identify real-life meanings (What does the slope mean in the graph?  What does the intercept represent?)  A copy of all four photos on a single page is available here.

Note:  I didn't provide my 20 minute script because I feel this is best used as an EQUATE lesson, below.

How to use this as a full lesson?
I feel this situation is IDEAL for the EQUATE lesson routine.  As you may already know, this relies heavily on letting students explore what is most meaningful to them and then asking questions to guide their exploration and results.  I can only outline what "might" happen, but students have ways of amazing us!

EXPLORE--Provide a photo copy of all four photos on a single page.  Give them a few minutes to look at the photos and discuss them with their teammates.  You may structure this conversation and provide guidelines as your classroom expectations dictate.

QUESTION--Ask the students to brainstorm the types of questions they could explore given these photos.  What do they "wonder" about mathematically?  Is there enough information to answer those questions?  Remember to push your students to think mathematically and to ask questions appropriate for their grade level.  Asking if it is linear is a great start, but asking if they can use information to tell MORE than that is key.  Challenge them to ask and answer challenging questions.  Questions I might expect or encourage:

• Is the rate of linear (and/or) vertical movement constant? What about the rate of calories burned?
• How fast is this person moving?  
• Based on the information I can see, were they always moving at this rate?  
• How long will it take to "climb" a mile?
• How long will it take to "run" a marathon?  What about burning off my favorite meal?

 Note:  Some questions are much more basic than others.  The last few questions require analysis offered in the first two or three questions, but students could easily get "stuck" with having too much information or too many steps to explore.

Additionally, this is where students get to ask you for more information.  If you can provide it, great!  If not, push them to answer their question given the information provided or to do the research needed to answer the question.  For example, I would NOT tell them whether or not the person was moving at a constant rate.  I would, however, allow them to look up the number of feet in a mile or the length of a marathon.  (I wouldn't tell them, but I'd help them access appropriate resources!)

APPLY--Remember to encourage your students to apply their learning!  If students know how to write equations from a table of data, they should do so!  If students know how to make accurate graphs, they should do so!  This is a great time to reflect on the units you have worked on and what skills students have obtained.  This will encourage students to apply those skills to their problem solving process.

TRY SOMETHING--Encourage the kids to get working!  They may feel "stuck" or that they dont' know what to do.  Try anyway!  That's the goal.  Get going, try something and see what happens.  Of course, kids need guidance, but it is their job to take their exploration, their questions, their previous knowledge and apply it to work on finding solutions.

EXPLAIN--Students need to wrap up their exploration by explaining not only what they did, but why they did it, how their previous knowledge related to the problem, and what inferences they drew.  They need to be able to justify their answers and how they know that their solution is both reasonable and accurate.

How to use this as an assessment?
If your students have experience with looking a real life data to explore questions like the ones above, let them go!  Give them a challenge question and the data and make sure they don't just find a numeric answer but also that they explain their process and thinking!

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 

Giant's Causeway

Personal Reflection:

One of my most favorite places in the whole world (that I've never actually been to) is The Giant's Causeway in northern Ireland.

The summer after college I was a nanny for my cousins in a small town outside of Dublin called Dunboyne.  I was too young to get my Irish driver's license and ended up taking the kids to the city on the bus.  We did get to take weekend trips with the family to southern Ireland, but while I was there there was just too much unrest to visit up North.

Thus, this is the most amazing place in the world that I've always wanted to go to, but never have.  :)

The Giant's Causeway is a natural formation of rocks on the northern coast between Ireland and Scotland.  As you can see from the photos, these spires of rock form beautiful polygons, often hexagons, but reports are anything from quadrilaterals to nonagons. 

This, to me, is full of opportunities for great instruction.  I can see anything from estimation and basic polygon identification (3rd Grade) to tessellations and transformations. 

For this reason, I feel the EQUATE model is a perfect opportunity to explore these photos and this location.  Rather than focusing on a single grade, I encourage you to use the EQUATE thinking routine to apply appropriate standards at your grade level.

Grade Level: 3-HS

Course: Math, Pre-Alg, Algebra, Geometry

Standards:  3.MD.8, 3.G.1, 3.G.2, 4.MD.5, 4.G.1, 4.G.2, 4.G.3, 5.MD.5, 5.G.3, 5.G.4, 6.G.1, 6.G.2, 6.G.3, 6.G.4, 7.EE.3, 7.EE.4, 7.G.1, 7.G.6, 8.G.1, 8.G.2, 8.G.3, 8.G.4, G-CO.1, G-CO.2, G-CO.5, G-CO.6, G-CO.7, G-GPE.7, G-GMD.2, G-GMD.3, G-MD.1, G-MD.3
SMP: MP.1, MP.2, MP.3, MP.4, MP.5, MP.6, MP.7, MP.8
Skills: Estimation, Number sense, reasoning, modeling, geometry, geometric shapes, properties of shapes, area, perimeter, volume.


How to use this as a mad minute:
You have 60 seconds. Name all of the shapes you can see.

How to use this as a warm up:
You could ask the students to consider one of the following:
1.  Name the shapes you see.
2.  Does this fit the definition of a tessellation? Why or why not?
3.  Are these "regular" polygons?  Why or why not?

How to use this as a mini-lesson:
If I only had 20 minutes, I would use technology to explore this VERY COOL region.  This website has an awesome interactive map, some history, and the legend of the Giant's Causeway.

http://www.voicesfromthedawn.com/the-giants-causeway/

How to use this as a full lesson?
As I mentioned before, I feel that this is an ideal EQUATE lesson.  Although there is a ton of math that is obvious to an instructor, this captivates my interest because of the combination of legend, scientific history, and visual appeal.  I feel your students will also be drawn to these elements.  If you are comfortable, let the students dictate the direction of the lesson and exploration (within reason).

I would show these photos, let the students explore, discuss, etc.
Then I would list all of their questions, encouraging them to "wonder mathematically" about them.
Focused on grade-level appropriate standards, I would ask students to narrow down the questions to make sure they are relevant to things you have already explored or discussed in your class.
I would let the students ask YOU questions and you can provide the answers you feel are appropriate.  (How are they formed?  How big is the region?  How many are there?  You can provide as much or as little information as you wish.)
I would settle on a question (or two or three) for your students to apply their knowledge and continue to try to solve.  Encourage them to TRY something!  Draw on the photo, measure it, get online and do research, look up formulas that might be useful, gather information, start playing with the numbers, rules, formulas, photos, etc.
Finally, ask the students to Explain what they did, what they found, and how they approached the problem.

 How to use this as an assessment?
It is up to you if you think your students can use this as an assessment appropriately.

It could be something as simple as providing the first photo and asking students to outline as many different shapes as they can see and explain why they are different and what they are (Elementary School).

It could be more advanced, offering the size of the region, the size of an individual "step" and asking the students to estimate how many are in the entire region.  (Upper Elementary to Middle School.)

You could ask the students to find two similar "steps" and justify why they are similar (Middle/High).

You could ask the students to find the volume of two or three different "steps" and justify their solution methods.  (Middle/High).

You could ask the students to PROVE that two items are congruent or similar based on transformations such as rotations, reflections, etc.

Works Cited:
Photo 1
Description: Giant's Causeway and Causeway Coast
Copyright: © Philippe Croo
Author: Philippe Croo
Image Source: Philippe Croo  (Link)

Photo 2
http://farm3.staticflickr.com/2755/4427445338_7869405855_z.jpg?zz=1

Photo 3
https://garystravel.wordpress.com/page/107/


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 

Life's Complex Plane

Personal Reflection:
I love when someone, more creative than myself, is able to combine really awesome math with really deep thinking. I keep thinking that if we want to create critical thinkers, the people who create these kinds of images are the epitome of critical thinkers.  So...can we get kids to do the same?

Image Source

Grade Level: 6-9

Course: Pre-Algebra, Algebra

Standards:  6.NS.6.b, 6.NS.8, 7.RP.2.a 
SMP: MP2, MP3, MP4, MP7
Skills: Coordinate plane, critical thinking, graphing, analysis


How to use this as a mad minute:
You have 60 seconds. Explain your interpretation of this graph to a partner.

How to use this as a warm up:
You could ask the students to consider one of the following:
1.  If you had to give a title to each axis that would encompass the extremes, what would it be?
2.  Explain the relationships in each quadrant.
3.  Do sleepiness and joy have a direct or inverse relationship?
4.  Do you agree with the 4th quadrant?  Why or why not?
5.  Do you believe that dreams and reality are opposites?  Why or why not?  Use mathematics to back up your argument.

How to use this as a mini-lesson:
I'm going to assume (bad idea?) that you'll use this with kids who are familiar with the coordinate plane, the quadrants and how to read them.  This is not an introduction, but an elevation!  We've got 20 minutes?  Here we go!

0:00--Take 1 minute to read this, analyze it, and think about whether or not you agree with it.
1:00--Now, without talking, take the next minute to jot down your ideas, thoughts, etc.  You can draw, you can write, you can use notes, anything you want.
2:00--Now partner up and compare your thinking.  You have 60 seconds.
3:00--Ok, let's share out some ideas, thoughts and reflections.  What did you see?  Agree on?  Disagree on?
6:00--Ok, I'm interested in seeing if you can create your own "Complex Plane."  Let's start with 60 seconds of brainstorming opposites.
7:00--Let's list those where everyone can see them.
9:00--Ok, here's the challenge.  Take two pairs of opposites and put them on your axes.  (Dry erase boards, math journals, notebooks, etc.)  Now, try to imagine what each quadrant would represent.  (I'm going to include my own example, because it's not as easy as you might think!  First, the challenge is not to be swayed by the previous example.  I kept thinking of joy and sadness, or night and day, which both felt too close to the original.  I chose Hot & Cold and then Starving & Full.  I was thinking of temperatures of food, but without a title, that might not be clear.  Then I had to think, what food would be amazing hot and would make you full?  Not too hard for a high schooler to choose.  Pizza, Pasta, Cheeseburgers could all work.  Next, what would make you full when it is cold.  Ok.  Done.  But what food would you starve rather than eat?  Hot uncooked fish?  What food would you starve if you ate it cold?  That was the hardest.  Mine is not perfect, but it's my first effort.  I recommend you try this yourself several times before you ask the kids to do it!)

10:00, 11:00, 12:00--Check in on kids' progress.  Encourage them to keep going.  Ask them to create more than one if they struggled.
13:00--Let's partner up and share your results.  DO NOT EXPLAIN.  Ask your partner to study yours and then tell you what they think it shows.  Then flip.  You'll have 3 minutes total.
16:00-There is a fantastic website that creates graphical images like this almost every day.  thisisindexed.com.  Examine these two images.  Then try to create your own!

How to use this as a full lesson?
I probably wouldn't.  I think this is an engaging activity that sparks creativity, but is NOT destined for an entire class period.  However, if you wish, you can explore thisisindexed.com and select other images.  From there you can ask students to explain what is happening and to try to create their own.

Another option is for students to explore the site on their own (warning: a few images do pertain to more mature subject matter) and ask them to select their 4 favorites, analyze them and be prepared to explain them.  Browsing will take a lot of time!

 How to use this as an assessment?
I don't feel this is appropriate for a summative assessment.  Formative assessment will take place as you listen to student discussions and explanations.


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

Snow White Transformations Matrix

Personal Reflection:

Wow.  When I saw this, I was so excited!  Although it's been awhile since I worked with a matrix, or matrices, I thought I quickly understood what was happening.  As I explored further, I realized, I needed a refresher.  Plus, what an amazing way to illustrate the changes.  This "Snow White" has enough detail that we can clearly see the x and y transformations.  I love it and I've barely even begun to explore it.

I'm disappointed to report that the Algebra II courses in my school do not teach matrices.  Why is this a surprise to me?  I never saw a matrix until I set foot in College Algebra my first day of college.  Oops, that was a major mistake!  I know now that even the briefest exposure would have been invaluable.  That being said, as the standards make a transition to understanding properties of shapes and proofs of theorems through transformations, this type of analysis and thinking cannot be ignored.  I hope that by creating and applying this to mini lessons, lessons, and assessments, my colleagues will see the value and return to teaching matrices. 

This image was obtained from this site.

Grade Level: High School
Course: Algebra II (Most likely)
Standards:  N-VM.7, N-VM.10, N-VM.12, A-REI.9,
SMP: MP1, MP2, MP3, MP4, MP6, MP7, MP8
Skills: Properties of Matrices, Matrix Identities, Transformations of matrices, linear transformations 

How to use this as a mad minute:
You have 60 seconds.  Choose one transformation and explain why the image is accurate.  

How to use this as a warm up:
You could ask the students to consider one of the following:
1.  What does the identity matrix represent?
2.  What would the identity Snow White look like?  How do you know?
3.  Explain the Wicked Queen matrix.
4.   Personally, I think this would be more accurate, and more appropriate, if it were the Alice in Wonderland Matrix.  Why?
5.  Can you think of another transformation?  Draw the matrix and the resulting "Snow White" image.

How to use this as a mini-lesson:
20 minutes, assuming kids DO know about matrices and the basics of transformations.

0:00--This Snow White image caught my eye.  Take 2 minutes to read, study, explore, and think about it.  Make notes, list questions, draw pictures, etc.
2:00--Ok, anyone want to share immediate thoughts and reactions? (Hopefully the kids will share some of these thoughts, but if not, try to lead them in these general directions.  Remember that if they go in a direction that you don't expect, but is valuable, take that path and abandon mine!)
4:00--Let's talk about the identity.  Why do you think the artist never drew the identity?  Do you think you could?  Take a minute to do your best.  You may want to draw another one next to it to show the difference.  For example, draw Scaley and then draw the identity.
6:00--Can we talk for a minute about why the artist used the name Scaley?  Any guesses why?  (If kids don't see the connection between "y" and the "y" value and the "y" axis, don't push it.  You'll come back to it.  If they do, ask them to check and see if their theory works for all of the examples as you discuss and review.)
6:00--What does the identity represent in any matrix?  How does that relate to any geometric figure?
7:00--What's the difference between Scaley and Scalex?  Do you think someone unfamiliar with those could figure it out by looking at the two images?
9:00--What about the difference between Reflecty and Reflectx?
10:00--So, let's revisit the question about Scaley vs. Scalex and Reflecty vs. Reflectx.  Why those names?  Are they clear and obvious, now?  Did your hypothesis work out?
11:00--Let's look at the rotation example.  If the matrix is named "A", the element A_1,2 is -1.  Why did that cause the rotation shown?
12:00--What would happen if you made A_2,1 = -1?  What kind of rotation would you get?  Do you know?
14:00--What would happen if both elements were -1?  Can you predict?
15:00--What do you think the Wicked Queen matrix does?  Explain.
17:00--Take the next 3 minutes to draw a simple, yet distinct figure.  Notice that Snow White has a bow on the left side of her head, and a distinct top and bottom half.  Try to create at least 3 transformations by drawing the matrix first and then drawing the resulting figure.

How to use this as a full lesson?
Honestly, when I saw this, as much as I loved it, I really wondered why it wasn't Alice, from Alice in Wonderland, instead.  After all, who better to use for stretching and shrinking?

I would do the mini lesson above, but then show the two movie clips below.  One is 6 minutes long and one is 1 minute long.  (The first is from my personal favorite live action version from 1985!  Please let me know if these video clips expire or the links no longer work!)

I would challenge the students to create, while watching, a quick timeline of the transformations.  For example:
Normal size--drink potion--become small
Small--eat cake--become larger than normal
etc.

Then, have them get in groups and try to create matrices that would create the transformations.  They need to be able to back them up.  Give them 10 to 15 minutes.

After this, ask them to share their results.  (Poster?  Write it on the board?  Hold up their dry erase board?)  They have to convince you that their matrix is better than another group's.  (It might be more accurate, it might better account for her change in size in both directions, etc.)

Let there be a debate.  Let kids really argue for why theirs is the best.  You can decide if they should get time to revise their answers and try again, or if they have to stick with their first answers.  (Revisions would encourage kids to learn, assess, and fix!)

From there, I would choose a simple cartoon figure and ask students to perform the given transformations.  (Sample worksheet here.)  I selected Kirby because he doesn't require much artistic strength to create simple transformations.  Also note that the worksheet I made doesn't ask students to draw.  If I had better software to create matrices, I might have included problems like that, but I didn't.  Some of your more creative students may better show their understanding by drawing various Kirby images to match given matrix transformations.

How to use this as an assessment?
You could use the worksheet above as a quiz (though something requiring artistic ability is not very fair!)

You could also use the video activity as an assessment tool instead of a team project.  You would need to create a rubric and more specific directions!


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

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

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