MakeOverMonday Week 8: Ferris Wheel

Dan Meyer says there are only 4 more makeover Mondays.  I have enjoyed participating and learning from others!  I report to school in the middle of this month ... I may not get to participate beyond this week.  I definitely want to take this idea of transforming textbook problems to my team.  I'll be working with different teachers this year - some I know from a distance last year, and some new faces.  I'm hoping we can put our heads together to create rich, worthwhile tasks for our students!  (I'm curious about the professional development they did as well ... it will be fun to put all of our new learning together!)

This week's problem is about a periodic graph.

So, #1 is all about proportional calculations.  #2 requires creating a graph; I can imagine students using the points from #1 as the basis of their sketch.  #3 is about the circumference of a circle.  So this problem is accessible to middle school students.  Students don't have to know anything about the sine curve or equation to approach this problem.  But if this problem is introduced after we have studied linear equations and transformations of those lines, I would start with enrichment stations ... from this desmos file.




The desmos file is an amazing collecting of applets and interactive spreadsheets.  Students manipulate a, b, and c (period, amplitude, and vertical shift) in the sine equation to match the movement of an engine, roller coaster, and scientific data (daylight hours).  The website says it is for upper math students and that students should be familiar with sinusoidal curves.  For this lesson, I don't think students need that understanding.  In our studying of linear functions, we would have noted that there are other functions that make other shapes or designs.  And that playing around with the parent equations of those functions you can transform them.

So ... first I would show a picture of a ferris wheel and/or a video of a ride on one.  I will invite students make notes about what they notice and what they wonder.  In groups, students would share, and then we would share out in class ... making a list of their thoughts.  I will invite students again to continue thinking about their ideas as we do some exploration activities.


Next, I would set up exploration stations (at least the engine and the ferris wheel).  I would invite students to explore, discuss, and develop ideas about what is happening.  I would have some guiding questions.

Then ... when we would go back to the textbook problem, just the picture and the seconds needed for a revolution.  "What are questions we might ask and answer from this picture?  Work with your group to develop 3 questions.  At least one must include a graph.  Answer your questions on a separate sheet.  Be sure to label you work carefully."


When groups are finished creating and answering their questions, I would collect those and sort them by types of questions.  I may create a few question cards myself depending on the groups' response to the task.  In our next class we would work through some of the student developed questions.  Last we would return to their noticing/wondering to discuss any ideas we had not yet explored.

http://www.mathdemos.org/mathdemos/sinusoidapp/sinusoidapp.html

MakeOverMonday Week 7: Postage Stamps

It is Week 7 of #MakeOverMonday problems!  Dan Meyer has issued the challenge to revise this postage problem and to be honest ... I'm struggling with this one as much or more than I struggled with the one last week.

In August, I know we will be introducing several parent functions to our Algebra 2 students.  Our Algebra 2 curriculum is built around the study of functions, consistently analyzing how the values for a, h, and k affect the graph, how the equations are used in context, and learning to manipulate and interpret the equations.

I'm thinking that both last week and this week's problems can be used as illustrations for basic functions.  (Although I'm still puzzled by last week's problem a bit.  I read someone's comment that tire marks should be a function of speed, and I agree.  It doesn't make as much sense to say speed depends on tire marks.  It makes much more sense to say tire marks depend on speed ... which would make the equation some sort of quadratic, instead of a square root function!)

OK ... so this week's is definitely a step function.  I know that my students will not have studied step functions in their Algebra 1 classes.  So I am curious how they will graph the data.   Here is the information presented from some textbook:

I would remove the term, "step graph," from the problem.  Instead, I want to put the data out there and talk about how it might be graphed.

Before we get started we might explore the history of the stamp briefly, and I would be curious if any of my students collect them.  (By the way, the American Philatelic Society has math activities on their website!)  I might also quiz students ... wonder if they know the current cost of a stamp ... I had to look it up!  Fortunately, Wikipedia has a record of stamp costs all the way back to 1863.

So ... after a brief discussion of the stamp, I would present data (I would add our current costs) and ask students to work in groups to graph the data.  I'll ask them to explain if the data is continuous or discrete.  I'll also ask them to identify the domain and range.

Each group will get an opportunity to share their graphs and explain why they graphed the data the way they did.  I am hoping that there will be some differences in the graphs so that we can discuss their reasoning.  I'd love to use "My Favorite No" strategy to discuss the positives and identify errors!

Depending on student work and our conversation, I'll end our session wrapping up the work involved in this function, explain how the greatest integer function works, and combine this work with an introduction to piecewise-defined functions.  I'll direct students to Cool Math's explanation of piecewise functions before giving students additional practice.


What would you do with this problem???

#MakeOverMonday ... Put on the Brakes!

Welcome to this week's MakeOverMonday challenge!  Our basic goal is to revise the posted textbook problem to make it better ... more worthwhile, more accessible, more engaging!  This week students are invited to take on the role of an accident investigator!

The problem we are revising this week is looks like this ...

I don't like this problem.  The formula doesn't match any that I could find online.  It does not take into account road conditions or adjusted drag factors.  In the problem there is a formula, a table, and a graph.  That leaves little for students to do or to interpret.

In my revision I would love to invite a local police investigator to share his work with our class.  Hopefully the officer would share video demonstrating a crash and how fault is determined.  If that isn't possible, there are some youtube videos that illustrate the main idea of this problem.

I chose to use just the graph and create a short activity for analyzing the graph. Analyzing graphs gets at a number of Common Core State Mathematical Practices such as making sense of problems, reasoning abstractly, constructing viable arguments, and modeling with mathematics. Analyzing graphs also addresses two Common Core State Standards.

• CCSS.Math.Content.HSF-IF.B.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. 
• CCSS.Math.Content.HSF-IF.B.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. 

Here is the activity ... I'd love to hear your suggestions!

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#MakeOverMonday ... Will the sauce spill over?

It's the fifth week of Dan Meyer's MakeOverMonday challenge!  Each week Meyer chooses a textbook problem from middle school math curriculum and invites anyone who would like to participate to suggest how to transform the problem.  He doesn't define the goal of the transformation other than to make it better!

This week is about volume.  The problem asks a great question ... "Will the sauce spill over when the chef adds the meatballs to the pot?"

The problem requires multiple calculating steps.

At first glance the problem reminds me of the book, Mr. Archimedes' Bath by Pamela Allen.  The book is quite primary and yet I find that sharing literature in math class helps to lighten the classroom and helps some students connect with the math.  I might share the story in preparation to work on this problem.

I would start to work on this problem by sharing the context and showing the picture of the sauce and meatballs.  Then I might set out a pot of water with two dozen small balls (or blocks).  "What do you think?" I'd ask.  Will this pot of water spill over if I add these balls to the pot?  I'd ask students to choose, "yes" or "no," will the water spill?  Then I would ask the students who say, "no, the balls don't fit" to estimate how many balls will fit.  And for the students who say, "yes, the balls will fit" I would ask them to estimate how many  more balls might fit.  I'll record their estimates on our chart paper.

Next I'll ask students to think for a minute about what information they have - both in my display of water and balls and in the sauce textbook problem, what information they wish they had, what step they think they would work to solve first.  I'll ask students to pair up and discuss their plans.

Then I'll challenge teams of students (2 - 3 students working together) to determine how to prove their estimations.  As they work I'll circulate to ask questions (in no particular order below ... )

• What is the volume of the pan?  How do you know?
• Why is knowing the sauce is 2 inches below the top of the pan significant?
• What is the volume of a meatball?
• How does the volume of one meatball affect the volume of the pot?
• What is the formula for the volume of a cylinder?
• What is the formula of a sphere?
• How are radius and diameter related?

Last ... we'll demonstrate ... we will toss in the balls and watch to see if the water spills.

What are you thinking?  What would you do with this problem?

#MakeOverMonday Week 4 Internet Cafes??

Dan Meyer chose a problem with this advertisement for our #MakeOverMonday challenge next week ...

And I could hear my students asking ... Whaaaat?  What's an Internet cafe?  Why not just go to the local coffee shop, bookstore, or almost any corner store where wi-fi is free?

So if I stuck with this problem I would show students the advertisement and ask ...
What questions come to mind?
What might you do with this information?
What other information might you need?
And at that point, I might ask, how much Internet time do you need per week?  Which of these cafes would be your "go-to" spot?

I also thought about changing it up a bit since Internet cafes' are not something my students would use.  In the March/April 2013 issue of The Saturday Evening Post there was this article:  "Howto Save a Bundle on Smartphone Service" by Jeff Bertolucci.  The article made me wonder why I pay so much for my Smartphone if I don't have to.  The article had a link to a chart outlining several prepaid plans and their basic costs.  I started to click on the links and ran across this ad for Republic Wireless.  I don't have teens at my house, but if I did, I might be interested as a parent in pursuing a plan something like this ... much less expensive than what I pay now!  (I might want to pursue this plan for myself!)

So my #MakeOver would go something like this ... Your parents are insisting that you take responsibility for some of your expenses as you head to high school.  They are going to get you a prepaid Smartphone with unlimited service through Republic Wireless - if you'll pay for half the cost!  They show you this ad and say:  Since you have to pay half the cost, which of these plans do you want to choose?  

At this point I'll show students the advertisement and say: Talk with your teammates to determine if one of these choices is better than the other.  How might you represent the information graphically?  Will it matter which of the 2 plans you choose?  If so, explain your choice and give a mathematically accurate reason.

Then for our emphasis on literacy/writing in math class, my prompt might be:  How might you represent your decision making process to your parents mathematically?  Write your parents a letter explaining the choice you have made.  Provide a clear and mathematically accurate explanation.

PS ... Are there Internet cafes' in your area?  Do you use the free wi-fi in your local coffee shops?  How many beverages do you purchase for that "free" benefit???

#MakeOverMonday Week 3 Bedroom Carpet ... Let's Save Money!

Dan Meyer has published his third #MakeOverMonday textbook revision challenge - Bedroom Carpet.  Check it out here!  (Hmmmm ... the problem says her sunroom, OK?)

There are two things I notice about the problem.

• First, the dimensions in the scenario are all in customary measurements and the picture is in meters.  My first revision would be to change the measurements on the diagram to customary measurements.  If I see that the prices are described in customary measurements, I am going to measure my room with feet/yards ... not meters!
• The second thing that catches my eye is this sentence, "Lesa wants to lay out the carpet so the nap is running in the same direction, with the minimum number of seams."  I am guessing that "nap" will need explanation.  And this creates the possibility of two solutions ... the nap could run from left to right in the diagram or it could run from top to bottom.  Thus my revision, is which way costs less?  Let's save money! 

Use the information given in the problem to determine the cost of the carpet if installed from left to right.  Also determine the cost of the carpet if installed from top to bottom (as in the drawing below).  Which installation will save Lesa money?  Would one way be more wasteful than the other?

A second revision might be to question Lesa's choice of indoor/outdoor carpet (ugh!) and suggest that students find an appropriate alternative and determine the cost.  The teacher could provide links to websites for flooring choices like this one. This revision could be combined that with their English class' assignment to create a persuasive argument for a specific flooring.   (If time is limited, and research not desirable, the teacher could provide costs for alternative flooring).

Additional revisions could include students measuring rooms in their own homes, bringing in the layout drawn to scale, and determining the cost of flooring.  Students could research flooring costs in their area, choose their own style, and determine the total cost.

Even though I don't expect to use a problem like this next year, I like the challenge of thinking about revision.  I even went on an Internet scavenger hunt this morning for a project in which students plan their first apartment.  I know such a project exists ... I had created one many years back but today I didn't find any links to share.

How are you revising this problem?  And if not this one, what problem are you revising?

Blogosphere ... Good Ideas Round 3

As I have been reading around the blogosphere, I am in awe of the many excellent ideas shared among educators!

Here are 3 ideas that caught my eye!

Our school is fortunate to have a 1:1 initiative for 9th and 10th grades.  (Last year only freshmen were issued laptops).  I am always on the lookout for meaningful ways to embed technology in my algebra classes.  I ran across Education Rethink: Fifteen Paperless Math Strategies.  I notice that several of the strategies involve a blog and/or a shared document.  I've already committed to learning how to use Google products more effectively - thinking the shared document will work.  One more note at this blog ... this team of two offer their visuals for free ... which may be helpful as I create materials for class!

Mary Dooms writes a poignant post about students and curiosity in response to Dan Meyer's post on The Unengageables. She says, "Students are curious. We just have to give ourselves permission to allow them to pose questions and wonder."  Mary goes on to talk about how we use the summer to recalibrate; this resonated with me!  Mary mentions the Annenberg series and Fostering Algebraic Thinking.  My stack of books is growing and the time I'm spending reading them is shrinking!  Her post inspires me to keep at the work of developing good problems - worthy of students' curiosity!  Check out her 7th grade textbook revision problem!

To follow up on developing good problems, a third site I found interesting is Inquiry maths! As we discuss textbook revisions at #MakeOverMonday and read professional books, developing rich worthwhile mathematical discourse is essential! I'm looking at the inquiries suggested for algebra and considering how I might use these to stimulate creative and analytical thinking. The author at Inquiry Maths says "Inquiry is built on inquisitiveness and curiosity. And for those to be articulated, students need to learn how to ask questions. When students inquire into their own questions, levels of motivation, engagement and confidence rise. Students become self-starters who take responsibility for their own learning. Importantly, they lose the fear of giving the wrong answer because they control the question under consideration."

So much to think about ... so little time ... even in the summertime!   The math blogosphere is an amazing source for professional development!

#MakeOverMonday Week 2 Checkerboard Borders

Dan Meyer published his second #MakeOverMonday textbook revision challenge tonight - Checkerboard Borders.  Check it out here!

The problem states:  In preparation for back to school, the school administration has planned to replace the tile in the cafeteria. They would like to have a checkerboard pattern of tiles two rows wide as a surround for the tables and serving carts.

I wonder ... when a contractor gets a tiling job, does he use math to determine the number of tiles needed?  What planning does he do before laying out a pattern?

I would give students just the context in italics above ... with some thinking time ... and then proceed:

1.  What questions come to mind?

2.  In teams, go to our 9th grade cafeteria and collect information that will be helpful to you.

3.  In teams, use the graph paper and colored pencils provided to create a diagram that fits the problem description.

4.  Determine how many colored tiles will be needed to create the pattern your team drew.

5.  Suppose the school administration changes its mind and now wants to tile the cafeteria in the 1100 building instead of the 9th grade cafeteria.  Create a generalization or rule that will work on any size cafeteria.  Does it matter if the cafeteria floor is square or rectangular?  Explain your thinking.

6.  The cafeteria manager looks at your diagram and comments that his serving equipment will need more than 2 rows of checkerboard pattern.  He requests 4 rows.  How does that change the rule you created for the contractor?

7.  Create a mini poster with your floor design, calculations, and generalizations.

I made two changes.  First, I asked students what they need to know from the problem situation to encourage their own thinking, curiosity, and problem solving ideas.  Second, I took away the square diagram, choosing instead to use real data.

How would you modify the problem?