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

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

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 

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 

Algebraic Notation in Real Life

Personal Reflection:


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

Grade Level: 7-9

Course: Pre-Algebra, Algebra

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


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


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

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

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

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

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

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


Please feel free to use any of these ideas and modify them to meet your needs.  However, please acknowledge the original source of the items and my own lesson outlines.  ©NatalieRSprigg 2013

Swimming Pools of Saliva

Personal Reflection:

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

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

Grade Level: 3-8

Course: Math, Pre-Algebra

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

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


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


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

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


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

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

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

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


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

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

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

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

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


Please feel free to use any of these ideas and modify them to meet your needs.  However, please acknowledge the original source of the items and my own lesson outlines.  ©NatalieRSprigg 2013