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

Pancake Proportions

Personal Reflection:
I'm looking down the road a few weeks to when my 7th grade class begins a long haul with Ratios and Proportions.  (Stretching and Shrinking, Comparing and Scaling for those using the CMP books!)  It makes sense that we will spend a lot of time on this unit because it's what the CCSS emphasize as the fundamental skills in 7th grade, and what we should build all 7th grade learning around.  For that reason, I suspect many of my posts in the next few weeks will be around proportional thinking.  (And I'll be going back to some of my others, such as the FitBit post, the Treadmill post, etc.)

We participate in the BIC program, Breakfast in the Classroom, so I can't make this one a hands-on activity, but I would sure LOVE to.  I'm finding that BIC (while I support it in theory) is going to force me to change my style of "bribing" kids to get engaged because of all the food I like to incorporate.  :)  Anyway, here we go!

Grade Level: 6-8

Course: Math, Pre-Algebra

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

SMP: MP1, MP2, MP3, MP4, MP6
Skills:Writing ratios, analyzing ratios, analyzing proportional relationships, solving proportions, using proportions in the real world, solving for missing values using proportions


How to use this as a mad minute:
I've taken to noting that each of these not only depends on the amount of time you are willing to commit to a given activity, but also to note the proficiency level of your students.  I say this because I'm working with a population of students that is causing me to shift my thinking about what a "warm up" might look like, due to lower levels of proficiency, language challenges, etc.  For a quick check in, 60 seconds or so, I would ask:

• What is the unit rate for mix, milk and eggs for 1 pancake?

How to use this as a warm up:
The above question would also work well for a warm up, if that is the skill you've been working on.  However, I might ask the students to find the ingredients needed for a simple number of pancakes in order to highlight proportional reasoning and multiplicative relationships:

• How much mix would I need for 28 pancakes?
• How many pancakes would 6 eggs make?  What about 7 eggs?

How to use this as a mini-lesson?
As you may have discovered reading other posts, I usually find images that catch my eye because I am skeptical.  So my first thought was, does it make sense for this box of pancakes to make that many pancakes?  Is this truly in "scale" or proportion?  I would ask my students, is this in proportion? If so, how many cups does the entire box hold?

• Note, I don't think it is!  If we use the eggs as a guide, the recipe is scaled by a factor of 9, but 9 times 1 cup is 9 cups, which is not 3Q.  (A great way to work on unit conversions!  Have you seen the "big G" conversion chart?  I love it!)  (Here's one place I found the image.)
• If we also use the factor of 9, the box would contain 18 cups of mix, which I would assume is more of a "Costco" size box, not what we see here.
• Finally, a scale factor of 9 would make only 126 pancakes, not 155. 
• If we use the milk as our guide, the SF is 12.  That would mean we need 24 eggs and 24 cups of "mix".  That should also make 336 pancakes.    Hmmm.....

How to use this as a full lesson?
I don't think this could be used as a full lesson, but it depends on your students.  If you choose to use it, I would extend the warm up and mini-lesson into a full discussion AS WELL as setting aside time for students to present rebuttals and/or corrections to the "recipe."  A great interdisciplinary connection would be having the students write the company (can we tell which company this is based on the colors?  I think so.) with their discoveries.  I suspect that the company might respond with some coupons or other "swag"!!

How to use this as an assessment?
Any one of the questions listed above would be perfect to use as an exit slip, a mini-quiz or an assessment question!


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 2014

Frozen Equations

Personal Reflection:

If you are an educator, you've probably been using Pinterest for a few years.  In fact, you probably found this post through Pinterest!  I'm fairly certain I found this image on Pinterest, but possibly on one of my other regular "fun" sites such as 9gag.com.  In an attempt to track down the original, I found this site.  It is not where I got the image, but it is a nice connection to the world of CGI and more detail about the snow effects in Frozen!

The site, linked above, has this amazing introduction, 

"Snow is a challenging natural phenomenon to visually simulate. While the graphics community has previously considered accumulation and rendering of snow, animation of snow dynamics has not been fully addressed. Additionally, existing techniques for solids and fluids have difficulty producing convincing snow results. Specifically, wet or dense snow that has both solid- and fluid-like properties is difficult to handle. Consequently, this paper presents a novel snow simulation method utilizing a usercontrollable elasto-plastic constitutive model integrated with a hybrid Eulerian/Lagrangian Material Point Method. The method is continuum based and its hybrid nature allows us to use a regular Cartesian grid to automate treatment of self-collision and fracture. It also naturally allows us to derive a grid-based semi-implicit integration scheme that has conditioning independent of the number of Lagrangian particles. We demonstrate the power of our method with a variety of snow phenomena including complex character interactions."

Wow.  That's technical.  In my own words?  "Snow is hard to animate.  While past methods worked fairly well, wet and dense snow was challenging because it acts like both a solid and a fluid.  In order to address this, engineers created a model that uses two different geometrical methods to animate snow.  They are able to use a Cartesian coordinate grid, along with programming, to simulate how snow both gathers (forms snowballs) and breaks (falls, hits, etc).  This sounds SO advanced, but I believe the analysis is totally approachable by a typical middle school student.  This is why I snagged the gif above and saved it, knowing that it would give some concrete meaning to students who are interested in the use of variables AND scientific notation!

Grade Level: 6-8

Course: Math, Pre-Algebra, Algebra

Standards:   6.EE.6, 6.EE.9, 7.EE.4, 8.EE.4

SMP: MP1, MP2, MP3, MP4, MP6, MP7, MP8
Skills: Variables, Algebraic Reasoning, Scientific Notation

How to use this as a mad minute:
You'll definitely need to preview this and explain the basics to the students.  However, after a short intro, a 1 minute number sense and reasoning check in might look like:

Compare the top two expressions carefully.  When you identify the difference in the expressions, and the subsequent snow fall, describe how you think the change in the scientific notation changes the snowfall.

How to use this as a warm up:
Again, after an intro, so students understand what this gift shows and where it comes from, I would challenge them to work with a partner to try to define what each "variable" controls, and how the change controls it.  (Hint:  It might be useful to name the snow fall quadrants A, B, C, and D in order to know which is which.  I will use A in the upper left and rotate clockwise through.)

A sample answer might be, "The equations on the right have ThetaS of 7.5 x 10^-3 and the snow is noticeably more clumpy or less-liquid than those on the left." (There are several comparisons they could make, so don't stop! Also, I would recommend you DON'T provide this example to the students prior to them working and struggling, it will be more productive that way!

How to use this as a mini-lesson?
I would start the same way as above, but prepare extension questions for students.  The exploration I outlined above, might only take 5 minutes, but a 20 minute mini-lesson could easily flow.
First, I would encourage students to share out, and convince others of their observations.  This is a great way to work on descriptive language, choosing appropriate adjectives, and talking about scientific notation.

To push students further, I would ask them to try to create their own equation that merges some of the changes.  Then challenge them to describe the resulting snowfall, and possibly even illustrate the final image.

How to use this as a full lesson?
Continuing on the trend above, I would then push students to explore more independently.

If your classroom has technology, I would visit the site:  http://www.cgmeetup.net/home/making-of-disneys-frozen-snow-simulation/ and have the students watch the video.  Another option would be for them to find a clip of the movie online and try to match which of the four quadrants a particular scene might be using.

This is another video of how artists used mathematics in creating the imagery in the movie Brave.

Depending on your focus, you may push students to analyze the scientific notation.  How big are these numbers?  What does that tell you about the size of the changes?

A great reading extension is this article from The New York Times about the Columbia University mathematicians who are working with film studios to enhance their computer graphics.

How to use this as an assessment?
I would not use this as an assessment, as it's probably a student's first exposure to this type of analysis.  However, if you've provided similar learning experiences for students, the Warm Up lesson is probably an opportunity for authentic assessment and analysis.


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 2014

FitBit Math

Personal Reflection:

I have a FitBit.  This is a Fitbit Zip, the smallest and most basic of the systems.  It clips on to your waistband, pocket, bra or shirt and tracks your steps.  I assume, based on the info you provide when you set up your online account, it then translates that movement into a distance traveled and a number of calories burned.  (This post will give your students a chance to explore whether or not this is true!)

Anyway, we've all heard that a goal of 10,000 steps a day is a great way to increase movement and to stay healthy.  How far is that?  How many calories does that burn?  Is that consistent?

As you know, I do a lot of thinking about "how much I have left" when I'm working out.  This is similar to my treadmill post, but slightly different.


Grade Level: 6-7

Course: Pre-Algebra, 6th and 7th grade math

Standards:  6.RP.1, 6.RP.2, 6.RP.3, 7.RP.1, 7.RP.2, 7.RP.3
SMP:  SMP1, SMP2, SMP3, SMP4

Skills: Ratios, Proportions, Unit Rates, Problem solving, Real world problems

How to use this as a mad minute:
You have 60 seconds. Estimate the number of steps someone takes in one mile.

How to use this as a warm up:
You could ask the students to consider one of the following:
1.  What is the relationship between steps and distance?
2.  What is the relationship between steps and calories?
3.  What is the relationship between calories and distance?
4.  Estimate the number of steps you would take in 10 miles.
5.  Estimate how far you would have to walk (either in steps or in distance) to burn off a large McDonald's French Fries.  (500 calories.)

How to use this as a mini-lesson:

0:00--I'd like you to take 60 seconds to brainstorm everything you know about ratios or proportions.
1:00--Partner up and share your ideas with a partner.  Make sure to add anything you forgot to your list!
2:00--Can we make a list of properties of ratios and proportions and define them?
4:00--I'm going to show you three photos.  (Link here.) When you look at them, don't talk to your friends, but take a minute to write down your immediate "math wonders" about the photos.
5:00--Take a second to reflect.  Are your questions mathematical?  Are you focused on applications of math and not off-topic?  If so, please share them with your partner.  When both have shared, select two questions you feel are your "best" and write them on the board.
7:00--Here you will want to zoom in on the most relevant and appropriate questions.  I suspect that several will be able to be answered through solving proportions.  Feel free to "prime" groups as you observe to encourage them to think proportionally.  I will post "pretend" questions for the remainder of the lesson based on what I would expect kids to "wonder" about.  It looks like we have a lot of questions about the distance and the steps!  Do you know what a "unit rate" is?  Think about this, if you can buy 4 candy bars for $1, how much is each candy bar?  (Allow time.)  Finding the cost for one candy bar is the UNIT rate, how much for 1 of that thing.  I think we are wondering how many steps for 1 mile.  Can you set up a proportion that shows steps compared to distance? 
9:00--How could you change that to find out how many steps are in 1 mile?  (This assumes previous knowledge of solving proportions.  If I were introducing the skill for the first time, I would have spent at least a class period on setting up equivalent fractions and observing/discovering the property of cross products being equal.  Kids should know how to set up a proportion with a missing value.)  
10:00--Please solve your proportion and determine how many decimal places you need.  When are done, discuss with a partner and come to an agreement. 
12:00--(Discuss the answers as a class.  The exact answer, rounded to the nearest hundredth is 2148.22.  I rounded here because the next two places are zeros.  However, I would round to the nearest whole step, or even to the nearest ten.)  How did you solve?  Why?  How can you be sure that makes sense?  How did you round?  Why?  How can you be sure that is reasonable?  (Choose two students who rounded differently and ask them to explain.  I hope someone would note that 2150 is much easier to use in long term estimating than 2148 or 2149.22, I also hope someone would note that 2148 is easier than 2148.22 and there is no such thing as .22 of a step.  Finally, I hope someone would note that 2148.22 is the exact value and that the extra decimals are negligible.) 
14:00--Can you repeat this magic?  Can you tell me how many calories I'll burn in an hour?  Or how many steps it takes to burn 100 calories? 
16:00--Are both of the questions I asked Unit Rates?  Why or why not?  Be prepared to back up your answer!
17:00--Who thinks they are?  Who thinks they are not?  (Hold a mini debate, or allow students to change sides of the room.  Revisit the definitions you established if there is still a question at the end of your "debate".)
19:00--Can anyone, after looking at these examples, think of a time when they might solve a proportion to answer a real life question?

How to use this as a full lesson?
I always recommend extending the mini-lesson into a full lesson with further exploration.  First, I wondered if the FitBit readings and unit rates would differ for a different person.  I had to enter my height and weight in the online program when I registered, so I asked another user to share a few screen shots of HER FitBit.  Adding this second set of data opens another opportunity for exploration and extension.  For example, are the unit rates the same?  If not, what can you tell about each person's rates?  Can you graph the data?  Can you compare the two sets on a single graph?  Will one person "go farther" with the same number of steps or burn more calories with the same distance?  This is a great introduction to slope!  Slope is a rate of change, or a relationship between two numbers, just like a proportion!  Even if you don't calculate the unit rates for the second FitBit, you could definitely have students graph the data (assuming both started at 0,0,0) and talk about what the slope represents.  What does a steeper line mean in this real life situation?

I included both sets of FitBit data on FitBit Worksheet 2.

How to use this as an assessment?
If your students are proficient with unit rates, it would be perfectly reasonable to provide the FitBitMath1 or FitBitMath2 worksheets and ask them to calculate unit rates and explain the meaning of their answers.  Short, simple, effective.


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 

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 

Summer on Uranus

Personal Reflection:

This is another photo from uberhumor.com.  I find that this site, though not always student-appropriate, does have a mix of "items" that often lend themselves to discussion and exploration.

This is a pretty simple fact.  It is easy to research the accuracy of this fact and determine that if the orbit of Uranus around the sun is 84 years, then "summer" or 1/4 of that time, would be 21 years.  But to me, there's so much more to ask and so much more to explore.

This is also an ideal way to encourage students to use background knowledge to build a "case" for the accuracy of this fact and then confirm their answers electronically.

Note:  As a special addition below, I included some ideas for interdisciplinary connections!

Grade Level: Middle School 
Course: Pre-Algebra 
Standards:  Science:  5-8 Standard D, 9-12 Standard B  (Though math is involved, I don't think this relates to specific standards.) 
SMP: SMP.1, SMP.2, SMP.3, SMP.6, SMP.8

Skills: Critical thinking, research, questioning, Algebra, Computation


How to use this as a mad minute:
You have 60 seconds. Outline your immediate reaction to this and back it up with either scientific or mathematical knowledge.

How to use this as a warm up:
You could ask the students to consider one of the following:
1.  If "summer" on Uranus lasts 21 years, what do you know about it's period of rotation about the sun?
2.  Based on what you know about Uranus, what do you think "summer" looks and feels like?
3.  The Earth is tilted on its axis by 23º.  Uranus is tilted by 82º.  What does that tell you about seasons on the planet?
4.  The length of a day on Uranus is -0.718 Earth days.  What does the number tell you?  What does the negative mean?

How to use this as a mini-lesson:
Given 20 minutes, I would focus on discovery, exploration, and discussion.  For this mini lesson students will need access to the internet.
0:00--Let's look at this image!  (Show the graphic.)  Take a minute to think about it and discuss your immediate reaction with a friend.
1:00--What did you see or say to your partners?
2:00--Let's brainstorm.  What do you already know about seasons, Earth and Uranus?  Talk with a partner, write down everything you can think of, you have two minutes!
4:00--Partner up with another group and share your lists.  Add anything you don't have on your own.  Put a ? mark next to anything you are unsure about or disagree with.
5:00--Repeat combining two more groups.
6:00--Let's share out what you know!
8:00--Let's share items you were unsure of or to which you put a ? mark.  (Remember, you don't want them to ask, "Is this true?" just yet.  This is a valid question, but we are trying to build and confirm background knowledge.  Students WILL check out the validity of this number, but not quite yet.  Explain to them that they can answer this question shortly, but we are focusing more on things like, "Seasons are caused by the tilt of a planet on its axis." or "Uranus is tilted much more on its axis."  These are items that will help students answer the question on their own, eventually.)
9:00--You are going to have 3 minutes.  I want you to research anything you listed EXCEPT the length of summer on Uranus!  Make sure you find valid sites and make sure you document your sources!
12:00--Everyone had different questions.  Did everyone find their answers?  Were there any questions you were unable to answer?  Can anyone help them or tell them the answer?
14:00--Let's see if you can use your knowledge to answer the following questions.  I'm going to ask 2 questions and then give you and your team 3 minutes to answer them.   You should NOT use the Internet to answer.  1.  If summer is 21 years on Uranus, how long is a year?  2.  Does the length of summer (or a year) relate to how big Uranus is or how far away it is from the Sun? 
17:00--Confirm your answers using any resources you prefer!
19:00--So this simple graphic is TRUE!  Awesome!  What other questions would you like to explore now that you've seen this?

How to use this as a full lesson?
I really feel this is ideal for an EQUATE Lesson.  (Click link for explanation.)

I can see the students wanting to know if this is true, but that is far too simple and can be answered easily by Google.  It will require some fantastic questioning strategies from the teacher to guide students to more challenging or deep questions.  Off the top of my head, I would want to explore the following questions:

• Which planet has the longest and shortest "summers"?
• Which planet has the longest and shortest days?
• Does the size of the planet relate to the length of the "summer" or "day"?
• Do other planets have seasons like summer?  Why or why not?
• What does "summer" look like on other planets?  (For example, Earth is tilted on its axis as it rotates, creating seasons, but other planets are not tilted or are tilted nearly 90 degrees, this causes great variation.  Also, gas planets don't heave "seasons" in the same way as others, and in some planets, though the temperatures vary, there's not what we would consider a season!)

Please note:  Some of my questions seem basic, but, as any good teacher knows, the depth is in the WHY? So don't forget to ask!

I found this site to be useful.  (NASA--Planetary Seasons)

How to use this as an assessment?
I don't feel this is appropriate for an assessment.  (You may feel otherwise and of course, feel free to use it!)

Interdisciplinary Connections
Ray Bradbury's short story "All Summer In A Day" is a great connection.  However, it can be a bit disturbing and you should definitely preview it before using it with students.

There is also a short (30 min) movie version of "All Summer In A Day"

I would definitely work with your team to, perhaps, have students read the story, watch the movie, research seasons on planets and, it's up to you, then assess their understanding.  My first thought is to separate students and hold a debate about how accurate the story/movie are, and whether or not this is a fair representation.  Students would need to back up their arguments with scientific evidence about the seasons on various planets as well as other items of "accuracy" such as life on another planet.


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 

Nobel/Chocolate Correlation?

Personal Reflection:

Just based on the last two posts, you'd probably think I love chocolate.  I don't!  I just find things that spark my interest and I save them. 

This image caught my eye because, off the bat, there seems to be a pretty strong correlation between chocolate consumption and Nobel Laureates.  I thought it would be fun to investigate!

This image can be found here.

Grade Level: High School

Course: Algebra, Algebra II, Prob/Stat

Standards:  S-ID.5, S-ID.6, S-ID.7, S-ID.8, S-ID.9, S-IC.2, S-IC.3, S-IC.6
SMP:  SMP.1, SMP.2, SMP.3, SMP.4

Skills: Algebra, Line of Best Fit, Correlation, Causation, Statistics, Problem Solving, Reasoning, Critical Thinking


How to use this as a mad minute:
You have 60 seconds. Explain what this graphic implies in 1 clear and specific sentence.

How to use this as a warm up:
You could ask the students to consider one of the following:
1.  Does this image have all of the essential elements of a clear graph?
2.  Do you think the use of flags and country names enhances or detracts from the image?  Why?
3.  Do you see a possible correlation?  Why or why not?  If so, what kind?
4.  What does the "r" value tell you about this graph?
5.  Which country consumes the most chocolate?  The least?  Which country has the most Nobel Laureates?  The least?

How to use this as a mini-lesson:
I'm disappointed that the data isn't available. I would love to have kids use their graphing calculators and a data table to generate equations of best fit.  I guess we just have to trust the info that is provided.  I did find the original article.  Linked here.

0:00--Take a look at this and then take a minute to discuss it with a partner.
2:00--What did you notice?  What stood out to you?
3:00--Do you think it is fair to make the argument "The more chocolate you eat, the more likely you are to win a Nobel Prize?"  (Feel free to adjust the statement to better match what your students say!)
5:00--Do you see any data that might be considered an outlier?  (If your students know the mathematical formula for outliers, feel free to apply it!  I would just discuss "in general" rather than doing it in that much detail.)
6:00--Do you see any correlation?  Where?  (Hopefully they can tell you they see it visually in the data points, but also that they recognize the "r" value in the image.)  What does that mean? 
7:00--What questions do you have about this data, the study, or the relationship?  (Have them partner up, list their questions and then gather them back together to share out.  Write their questions down.)
10:00--If we draw a line of best fit, what would it tell us?
11:00--What would the slope tell us?
12:00--Work with a partner to write the line of best fit.  (Give an enlarged copy of the image.)
17:00--What is the difference between correlation and causation?
18:00--Can you think of other things that might have a correlation with no causation?  (Here's a site that has some great examples!)
20:00--Do you think this is an example of correlation without causation?  Why or why not?

How to use this as a full lesson?
I would definitely start with the mini lesson.  The ending question is a great point for the students to explore further.

Below I have 4 links to information about this "Nobel vs. Chocolate" image, research, etc.  I would ask the students to break up, study the information and be prepared to come back and share the information with others.  (I would do a jigsaw.) 

http://www.huffingtonpost.com/2012/10/10/chocolate-consumption-nobel-prize_n_1956163.html

http://www.bbc.co.uk/news/magazine-20356613

http://jn.nutrition.org/content/early/2013/04/24/jn.113.174813.abstract  (Full text is available in pdf link on the left.)

http://www.thescienceforum.com/news/31697-correlation-causation-chocolate-nobel-prize.html


A video that explains the image and research:

After their jigsaw, I would ask students to form an opinion about the graphic.  I'm thinking something in the range of:

• I think the research and data are accurate and logical.
• I think the reasearch is accurate but the causation link is missing.
• I think the research and data are inaccurate.

(Of course any other opinions are totally fine!)

I would then ask students to back up their answers with examples from the texts they read, their own background knowledge, correct mathematical vocabulary, etc.  I would ask them to do it in a 1 page poster.

 How to use this as an assessment?

See the lesson above, it includes an assessment tool.

Another option would be to simply give students the graphic on a test, as an activity, etc, and ask them to reflect on the image.  (I would provide a word bank or other guidance on the types of "reflection" you want them to do!  My word bank might include:  linear, correlation, causation, accuracy, misleading.)

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 

Infinite Chocolate

Personal Reflection:

This was one of the first images I've seen in a long time that inspired me to actually investigate and try to explain what was happening.  I didn't just try to use logic, reason, estimation, etc.  I actually got out graph paper, scissors, tape and MADE a paper chocolate bar.  If it can inspire me to build and play, I'm sure it will do the same for students.

I got this gif from this site.

Grade Level: 7-10

Course: Algebra

Standards: 7.G.1, 7.G.4, 7.G.6,G-CO.6, G-CO.7, G-CO.12, G-MG.1, G-MG.3
SMP:  SMP.1, SMP.2, SMP.3, SMP.4, SMP.5, SMP.6

Skills: Geometry, Area, Constructions, Modeling, Problem Solving, Ratio, Proportion, Scale

How to use this as a mad minute:
You have 60 seconds. Give me one reason you think this DOES or does NOT work.

How to use this as a warm up:
You could ask the students to consider one of the following:
1.  What is the area of the original chocolate bar in generic "units"?
2.  What is the area of the new chocolate bar in generic "units"?
3.  When the candy bar is broken up, there are 5 pieces.  Describe each piece using a correct geometric name and explain what properties each piece has to categorize them.
4.  Draw the 5 pieces on your paper (sketch!) and then label the angles.  (Acute, obtuse, right, straight.)
5.  What is the perimeter of the original candy bar?  Is the final perimeter the same or different?

How to use this as a mini-lesson:

Please note:  I included screen shots of the candy bar when it isn't moving for YOU to use, but largely, it would take away some mystery for the kids, so I probably would NOT show the photos to them.


0:00  Look at this awesome gif!  Have you seen this online?  If so, what did you think?  Watch and then turn and discuss with a partner.
1:00  If you haven't already discussed with a partner, focus on whether or not you think this works and why.
2:00  Ok, let's talk.  Who thought it worked?  Who didn't?  Why?
4:00  How could you prove your side?  What would you do?  (If they don't know, gently guide them toward making their own model)  What supplies would you need?  What information would you need?
6:00  Here's what I can give you:  Graph paper, rulers, scissors and markers.  You have 5 minutes to create your own ACCURATE candy bar.  (For your information, the side length ratio is 3.5:6, you can decide if this is helpful for your students, I think it would be, but could make construction challenging!)
11:00  Now that you have this candy bar, you want to "break" it accurately.  How could you "cut" this candy bar accurately?  (The bar, if students watch carefully, is cut on a diagonal from 1.5 "squares" up on the left through 1.5 "squares" down on the right.  But I would encourage kids to measure angles as well.) 
13:00  Next, we need to break the top piece into three smaller pieces.  How should we do that?  (This is a much easier "cut" since they are clear vertical and horizontal lines.)
15:00  Finally, we need to take out the extra square.  (I would have the kids label the pieces either by number, letter, or size.  I'll call them "single", "double", Small, medium and Large for my explanations.)
16:00  Now slide your medium piece up and your small piece over and down.  Fill the gap with your double piece.  Discuss what you see with your partner!
18:00  What did you see?  How do you explain the extra piece?  (Hopefully they see that the "squares" are not the right size or dimensions and to get them to "line up, the students need to shift the small and medium pieces "up" a bit leaving a long thin "gap" between the top and bottom.  Almost like the photo to the right.)
19:00 Does your extra piece "fill the gap"?  Is this real "infinite" chocolate? Why or why not?

How to use this as a full lesson?
I would definitely use the mini lesson above, allowing for more freedom if students are enjoying the exploration and discussion.  Depending on the grade level I would also ask appropriate questions and use appropriate vocabulary.

For example, if you repeated this experiment without the "squares" of chocolate and one large bar, could you prove congruence?  Why or why not?  If you can prove congruence, explain the criteria for congruence and back up your answer mathematically.  If not, what mathematical proof (not just modeling) could you use to justify why these are not congruent?

Why are the "small" and "Medium" pieces NOT similar?  Use definitions, properties, and measurements to back up your claim.

What is the ACTUAL area of the original?  What is the actual area of the final candy bar (minus the extra square)?  Does this prove that they are or or not congruent?

After these explorations, I would show the video linked here.  This is a similar optical illusion, trick, or manipulation.  Encourage your students to watch, either as a class, or on their individual devices.
I apologize in advance for the ad that precedes the video, but you can skip it after 5 seconds. 

I would ask students to watch out and consider these questions as they watch:
What are the original dimensions?
What is the original area?
What do you notice about the space in the "box" as he shakes it?
What do you notice about the cuts of the pieces?
What do you notice as he lays out the tiles the first time?
As he moves them, what do you see?
As he places them back the box what do you see?
As he "repeats" or "reverses" the trick, what do you see?

Can you explain his "trick" mathematically?

Use your number sense.  (I think you can see more "wiggle"room once the 3 squares are removed, and 3/63 is such a small percentage of change, it's not too obvious.  Add to that the fact that he has a very hard time at the end of the video making them all fit again!)

Ask the students to justify, model, draw, explain, etc.  They should use correct mathematical vocabulary, appropriate skill and relationships to their learning.  For example, can they discuss area and congruence?  Can they name shapes and angles?  Can they formally prove or disprove congruence?

How to use this as an assessment?
To use this as an assessment, I'd do the mini lesson at the beginning of a unit, refer to it throughout the unit as we are using vocabulary and talking about proof, and then I would show the video at the end.  I would assess the students on their explanation of the "trick" and how well they used what they had learned.

You would definitely want to create your own rubric before the assignment.  You would also want ample supplies for students, as well as multiple devices, as students will want to watch the video over and over as they work.  (Isn't it awesome that the video is nearly silent??)

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