## [Makeover] Postage Rates

July 29th, 2013 by Dan Meyer

**The Task**

This is from Pearson’s Algebra 1 textbook for iPad.

**What I Did**

**Establish a need for the graph, in general**. Why are we drawing a graph? What’s the point? Does my ability to draw a graph serve any larger purpose than getting me points on an assignment? This task doesn’t have an answer to that question.**Establish a need for the**. Why are we drawing a*step*graph, in particular*step*graph? What’s the point. Does the step graph have any advantage over other graphs? This task doesn’t have an answer to that question.

Tom Ward has me covered on the first point. Nothing’s topping this aspirational save-the-date for his 2019 marriage to Ms. Stone. What will postage cost then?

Graphs and equations of data are useful when they let us predict something external to the data we know. We don’t know the price of postage stamps in 2019 so we can extend a linear model beyond the data and find out what it might be.

Mr. Ward will need to scrounge up 54 cents per invitation.

But he still hasn’t given us a reason to care about the *step* graph. For that we look at *internal* data. We tell the kid, “Hey, your graph is messed up. If you hand that graph to someone, it says the cost of postage in 2003 was 39 cents and the cost in 2005 was 41 cents. But the cost in both those years was only 37 cents.”

If you’re going to make a graph that tells the story of the data *accurately* you’re going to need a different model than a straight line. Enter the step.

**What You Did**

Aside from Tom Ward’s superb work, over on the blogs:

- Scott Hills seizes the opportunity to show students the benefits of a well-scaled axis.
- Beth Ferguson removes step graphs from the task, which is one way to handle the problem.
- Evan Weinberg goes digital, though I’m not sure what the digital medium adds here. He also just
*asserts*that the student’s graph “should be a step function,” which highlights the difficulty again of motivating a*need*for this function family.

Over on the Twitter:

@ddmeyer Get rid of text. Change table to pics of stamps with the years labeled. Ask to predict $ a letter costs this year and in 2020

— Mr Leibold (@leiboldmath) July 26, 2013

@ddmeyer investigate a longer time line and a line of best fit to determine future increases. Discuss "vertical distances" to each point.

— Jacob O'Brien (@jacobobrien82) July 25, 2013

@ddmeyer Ask students to guess the postage rate in 1995. Eliminate the table; look up data online. Compare with general price levels (CPI).

— Dave Radcliffe (@daveinstpaul) July 25, 2013

@ddmeyer switch to email? (Just kidding) :-)

— Jeff Cohen (@jeffcohen) July 25, 2013

**Featured Comment**

Instead of a wedding invitation, change it to a graduation invitation. Have the kids estimate how many invitations they would have to mail out. They could then calculate the cost of their invitations. You could also have them calculate the cost that their parents/grandparents paid for their graduation invitations.

on 29 Jul 2013 at 12:40 pm1 DonThe save the date card is cute but it seems like it skirts the edge of pseudocontext. Who cares what the postage will be in that case? You’re going to pay it. HS level students might not be aware of how insignificant a fraction postage is in a wedding cost but I bet they’ll intuit that it’s not the top ten.

I’d attach this to a more concrete item, like mailing concert tickets to purchasers. Budgeting isn’t sexy but you’ll have an easier time convincing me it’s something Ticketmaster would try to predict. Or maybe use a company that sells infomercial trinkets for the example.

If I make my “convenience fee” equal to 5x what I spend to complete a transaction and I set the price at the beginning of every year, how much do I need to increase my fees every year between now and 2020?

on 29 Jul 2013 at 1:04 pm2 Matt H.Or just buy a bunch of Forever stamps today and not bother with the model at all… =P

on 29 Jul 2013 at 1:33 pm3 MargretI had some teachers complete a nice problem to compare the increasing price of a stamp to the increasing price of a Hershey bar. (The data is available somewhere on the internet… can’t remember where off hand). This was a nice context for graphs and functions.. I also asked some version of “are forever stamps a good investment?” So, should Thomas buy forever stamps or put the money in the bank?

on 29 Jul 2013 at 1:47 pm4 Evan WeinbergMy whole reason for making over the problem as I did was that the step function was the most uninteresting part of the problem. A student would likely see this problem in the textbook, skip over the fact that these are postage rates, see ‘step function’, and immediately graph it using the definition without giving the rest of the problem much thought.

My redesign ends the same way (i.e. graph a step function) but gets students to think about the cost of sending things through the mail, make a guess, and get an idea of where their guess fits in the actual history of the cost of mail. The students could then see how the cost changes throughout the years, observe that the change isn’t linear, and then put together a table of values that shows that it is a step function.

I agree that the graph isn’t well motivated in the original task or mine, definitely a weakness. My goal was to at least get students playing a bit and getting a sense for the numbers beyond the given table before needing to do anything with it.

on 29 Jul 2013 at 2:27 pm5 Andy BrownMight we all stand to look back in time to the days of the Pony Express and other happenings in U.S. history? We might get involved with inflation-deflation too. Well it’s a different take on prove-you-can tasks.

on 29 Jul 2013 at 2:40 pm6 Mary DoomsIf a change of context is acceptable for motivation, why not frame the problem in terms of the federal minimum wage? It hasn’t been increased since 2009. Based on a linear model, what do you guess the president is proposing, what is Congress is proposing? What would you propose?

http://www.dol.gov/whd/minwage/chart.htm

Just a thought.

on 29 Jul 2013 at 5:03 pm7 TomMary, that’s a great context that is certainly not pseudo.

I went with a wedding because it was the first thing that came to my head and I knew I could easily make a few jokes and hook students in. Yeah, it’s a little forced but I think it gets the job done much more than the graph of data.

As for stamps being a small fraction of a wedding cost… Students likely have no idea about that. It’s a good discussion to have. If you have to drop $500 on postage for a wedding and then tell your students “that will probably be the cheapest part of the wedding” I think it will turn some heads. There’s also the “forever” stamp issue which is another important point to discuss as well.

Mathematically, my question remains: When in the world do we ever really need a step graph? Sure, it is more technically correct since we’re not talking a continuous function. But what’s so important about it?

on 29 Jul 2013 at 5:25 pm8 l hodgeI might go with something that is a little more personal that takes a little thinking to figure out the steps. Maybe the number of drivers in your family or the number of schools attended by members of your family as a function of year. Then ask how the graphs would change for families with twins, large numbers of children, large ranges of ages. Or go backwards by providing the step graph and asking questions about the family.

on 29 Jul 2013 at 6:06 pm9 Scott HillsI don’t do stamps when discussing step-functions in my own class. Knowing my audience will include 14-17 year olds I look at speeding ticket rates. We have a resource officer on campus and getting him to stop by sometime after the first 15 minutes generally isn’t a problem (at least not as the day is starting).

I get the driving students, who are generally the most jaded of the Algebra 1 kids, to answer if they’ve ever gotten a speeding ticket. (I’ve never had a class with driving age students who haven’t, btw.) then we discuss, with the officer’s assistance, the local speeding ticket rates (and if he isn’t available that hour, I have a copy of the chart)

I’ve had more than one kid remark that speeding just isn’t worth it (and for the doubters, a follow-up of similar rate increases to car insurance generally is convincing)

on 30 Jul 2013 at 2:13 pm10 pete capewellHow about world records for athletics or swimming in the context of setting a ‘Big Goal’ for an aspiring athlete that wants to be the world’s-best in 2020?

Field events give increasing step functions for record distances; races meanwhile give decreasing ones for record times. Nice to have the choice?

With the venue for the summer Olympics in the final phase of the bid process we could tell a story leading to Madrid, Istanbul or Tokyo…

(http://www.nytimes.com/2013/07/04/sports/olympics/madrid-shines-with-pitch-for-2020-olympics.html?_r=0)

on 31 Jul 2013 at 10:28 pm11 Shaun Errichiellothanks for the problem. I have been thinking about this all day! Here are my thoughts on the lesson. I haven’t finished, but I like the work around this problem.

http://putonyourmathgoggles.blogspot.com/2013/07/so-you-want-to-mail-letter-lets-do-math.html

on 05 Aug 2013 at 8:35 am12 dy/dan » Blog Archive » [Makeover] Ferris Wheel[…] a reason to give a damn. Here, as with our last Makeover Monday, you're asked to create a graph simply because we said so that's why. There isn't any sense that a […]

on 21 Sep 2013 at 3:25 pm13 Zach LairInstead of a wedding invitation, change it to a graduation invitation. Have the kids estimate how many invitations they would have to mail out. They could then calculate the cost of their invitations. You could also have them calculate the cost that their parents/grandparents paid for their graduation invitations.

on 22 Sep 2013 at 1:21 am14 Dan Meyer@

Zach, like it. Gives a reason to look backwards and forwards.