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 

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 

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

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