Subway Scamwich? math task

If you haven’t heard Subway has three federal class-action lawsuits claiming their footlong comes up short.


With Stephen Colbert jumping all over the “Scamwich“, I thought it would make an interesting math task.

Act 1

Show the Colbert clip which features the journalistic reporting by the New York Post. Kids could do the math to discover how the Post calculated if customers “buy a $7 ‘footlong’ every other day for a year, an axed extra inch adds up to a loss of roughly $100.”


Act 2

Question: How did the New York Post calculate a loss of $100?

What information is useful?

  • one twelfth of $7.00 is 0.583
  • every other day for a year is 182 days.

Is this truly the loss? Have students read Chicago Tribune’s Eric Zorn article because he calculates it differently. He says the meat and cheese are pre-measured, so the same amount that goes into an 11-inch footlong goes into a 12-inch footlong. According to Zorn, customers are only being cheated by 8 percent of the bun, or 42 cents on a $5 sub.

Act 3

Perhaps Zorn and the NY Post are both wrong.  If Zorn’s assumption is correct, wouldn’t it be logical to also assume bread loaves are pre-measured as well? If so, the customer is still getting the same amount of bread. Therefore he is not being cheated.

What a great conversation to have with the kids.


Students completed the task this week and students gave me tremendous insight into their mathematical thinking. Some were spot on, others had serious mistakes.  The NY Post calculations were challenging for many students. I showed the Colbert clip, but stopped it at the point when the problem was posed. I posted the problem on the projector and the students worked in groups. A few argued the need to divide $7 by 11 because the customer only got 11 inches. MANY took 12 and divided it by 7 to get a per inch cost of $1.71. They didn’t check for reasonableness! A few thought to multiply $7 by 182 footlongs but that was their final answer. And a couple of students were very perplexed by the problem and had little understanding as to how to approach it.


Nearly everyone liked the “current events” and real world application. When I suggested that perhaps no customers are being cheated because the ingredients are pre-measured, one student wrote on the exit slip, “All that work and it really didn’t matter.” At the end of class I cleared up that misconception!


Exploring the beauty of numbers and ratios

I recently read A Mathematician’s Lament. The author would like to see math taught the same way as children learn to appreciate music and art. Explore the beauty by playing with it. Explore the beauty of numbers by manipulating them.

I think we need a better balance between pure and applied math. The Common Core stresses real-world application. Those scenarios answer the question, “Why do I need to know this?” but at times I think we’re missing the beauty of numbers. So when our math department met to continue our work on creating mathematical tasks I didn’t want to lose sight of that.

Our latest creation is a ratios and proportions task titled A Balanced Lunch. It’s influenced by Lure of the Labyrinth’s Managers’ Cafeteria Puzzle; however the students are required to delve deeper into equivalent ratios proportional reasoning. I’ll explain that in a bit. 

If you are not familiar with L of the L it is a fabulous puzzle-solving journey. Below is a screenshot from the Manager’s Cafeteria Puzzle. This puzzle’s goal is to determine the ratios between each of the four food items. The player has only enough “guesses” as there are number of missing food items. Twelve lights are lit to represent the twelve missing food items. I added the ratios to provide a clearer understanding of the puzzle. The puzzle has just enough information to solve.

Tray one establishes the "slop" to sushi ratio as 11:3. The middle tray shows a "slop" to turkey leg ratio as 11:7. The tray on the right shows the burger to turkey leg ratio as 9:7.
Tray one establishes the “slop” to sushi ratio as 11:3. The middle tray shows a “slop” to turkey leg ratio as 11:7. The tray on the right shows the burger to turkey leg ratio as 9:7.

The task goes beyond L of the L in a couple of ways. Besides wanting the students to use mathematical language when explaining their solution, they will explore graphing to discover that equivalent ratios form a line going through the origin. As an extension students will also create their own puzzle plus determine just the right number of values and combinations to be revealed at the start.

balanced lunch

When the students played the online game they were in pairs sharing a laptop. There was a LOT of math talk but much of it was imprecise language. For example a 1:1 ratio was stated as, “The numbers are the same”. I also heard, “This number is three times as big” when describing a 3:1 ratio. Sixth graders need to begin using more precise language so the task is designed with that in mind.

Here’s the task.

You can use any graph paper for graphing equivalent ratios but here’s one that has 10 lines per inch.

Let me know what you think.

Standards Based Grading–the best of 2012

I’ve previously written about SBG (Standards Based Grading in a Percentage Based WorldPowToon Presents Standards Based Grading, and Educating Parents about Standards Based Grading). It’s been my highlight of 2012 because I’m going solo with SBG in our building. In fact only one other teacher in our district is doing it. Our online grade book is point based and doesn’t support it. Parents and students are new to it. So why am I doing this?

I doing it because I truly want a reporting system that assesses specific standards based on a level of proficiency. That means not overly tainting the grade with behaviors such homework completion or extra credit.

Here’s my scale:

  • 4.0 exceeds the target. I know (can do) it well enough to make connections that weren’t taught.
  • 3.0 meets the target. I know (can do) everything that was taught without making mistakes.
  • 2.0 does not yet meet the target. I know (can do) all the easy parts, but I don’t know (can do) the harder parts.

Here’s my conversion when it comes time to report a grade:

  • 4.0 =100%
  • 3.5 = 95%
  • 3.0 = 90%
  • 2.5 = 80%
  • 2.0 = 70%
  • 1.5 = 60%
  • 1.0 = 50%

My grade book has three categories weighted as follows:

  • 95% Summative
  • 5% Practice (homework)
  • 0% Formative

The assessments are targeted by specific standards with problems grouped by level of difficulty. Here’s an assessment for 7G4 Circumference and Area where you can see how the problems are leveled.

Students had done a LOT of practice prior to this assessment. Even though “formative” is in the title, I ended up making it a summative because the vast majority of students met the target. A handful of kids need to arrange for an out of class assessment because they didn’t earn at least a three.

Here’s what it looks like in the grade book:


When I decide to report an official grade I create a summative assignment worth 10 points and convert the scale to a percent equivalent. Working with this point based grade book has been a bit of a challenge. The software thinks a score 3 out of 4 is 75%. It’s not; it’s a level of proficiency. It took parents and students several weeks to wrap their heads around that. Eventually they learned to ignore any grade (except the summative) and interpret numbers on a scale towards mastery rather than as a percentage.

I don’t want the above example to give you the impression that I only assess once. Here’s an example of where I assessed a standard three times over the course of a month.


Those of you familiar with E-school know it’s a student information management system with a mediocre grade book. I have to make due.

It has taken some time, but parents and students have learned to view the formatives not as a grade, but as information regarding how a student is progressing towards mastery.

On a related, but different note, next week our students will be taking winter MAPs. I’ll be interested in seeing how the kids do on the geometry portion of the assessment. This is our first year with the Common Core and my sixth grade standard and academic classes were living and breathing geometry for more than a quarter. When I get the results, I’ll post a summary and some thoughts.

Presenting mathematical thinking with Scratch

Dan Meyer asked us, “So what’s your idea and how will you turn it into a tool in 2013?”

Well, here’s mine: Use Scratch to present a mathematical problem and students post their solutions by remixing the project. Here’s an example using this question taken from the November, 2012 issue of Mathematics Teaching in the Middle School journal Palette of Problems.

Click on the link below to see what I’m talking about.

Scratch | Project | Train tunnel problem.

train problemwizard sketch

I love the idea of squeezing computer programming or computational thinking into my math class, but frankly it can take up a huge amount of class time. However I think this method won’t require a lot of time because, for beginners, all the student has to do is insert an image of their solution along with an audio file of their explanation. The stage and sprite are already set up! Students will get a taste of programming by opening up the Scratch project, plus if you or they choose to spend more time personalizing the project by all means go for it!

I could also see this as a quasi-Dan Meyer Three-Act Task. The level of abstraction in this problem may not be as deep as Dan’s, however it’s a different way of using media to present and explain mathematical thinking.

I would LOVE for others to create projects and add them to this list. Right now it is a whopping 1!

Please! If you have ideas for how to modify and grow this idea please share.