## The Frozen Code

Here are the locations of your chain of gelato shops:

Here is the temperature in the United States today (Fahrenheit):

So what do you do? You lower prices. An across the board cash discount? Maybe. But if you’re Gelato Fiasco, you institute The Frozen Code:

On each day that the temperature falls below freezing, we automatically use The Frozen Code to calculate a discount on your order of gelato dishes. [..] You save one percent for each degree below freezing outside at the time of purchase.

But how do you write this code using the language of variables that your pricing system understands? (Click through for Gelato Fiasco’s answer.)

How would you set this up as a mathematical learning experience for your students?

This is a series about “developing the question” in math class.

2014 Jan 8. Updated to add this important exchange with Gelato Fiasco on Twitter.

Featured Comment

John:

With the clear correlation between temperature and number of chain gelato shops, CLEARLY the temperature causes more gelato shops to be built…or does the number of gelato shops cause the temperature to decrease…am i right?

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.

1. #### Dan Meyer

January 8, 2015 - 7:15 am -

Gelato Fiasco discounts the price of its desserts by one percent every degree below freezing the temperature goes. Let y = the discounted price and x = the current price and t = the current temperature in Fahrehnheit. Write an equation using those variables that gives the discounted price.

2. #### Tom Hall

January 8, 2015 - 7:51 am -

Here’s my thoughts:

1. Show the picture of the temperatures.
2. Show the map of locations. Have students ponder what the locations could be. After some responses, share the answer of gelato shops.
3. Have students read the Frozen Code description. Ask them what the discount means (Where is below freezing? What would the percentage be for 30 degrees? 12 degrees?).
4. Take the map of locations and layer it over the temperatures using a transparency tool.
5. Ask students to consider where no discount will exist, some discount will exist, and where the greatest discount will exist. How do they know?
6. Discuss how prices are calculated at the gelato shops. Is it by size (small, medium, large) or by weight? Which pricing system is “better”? Why?
7. Project a receipt (real or fake) for a gelato at regular price.
8. Here comes the open-ended, choose your adventure, what do you notice part of the lesson. Have students choose different cities, estimate the discounted price, then calculate it.
9. Have students compare their work and see the different approaches used to find an answer. Picturing a seventh or eighth grade class, I could see some students using the regular decimal method to find the discount and subtract from regular price, using a proportion set up to find a discount, or multiplying the full price by 1-the discount as a decimal.

From here I would ask a series of questions to highlight similarities between methods and work towards a general rule (Tell in words how you find the discounted price). After writing down the statement, ask students what will always be the same and what could change based on location and temperature. Have students choose variables and connect them to what we have in specific cases. Finally, let students try to build the equation.

Key idea: Instead of making it some direct exercise in equation writing, make it a progression of activities from the concrete the the abstract. Find answers for specific cases before general cases. Create opportunities for students to respond and make sense of the data in the pictures.

3. #### Jeff Morrison

January 8, 2015 - 9:33 am -

I would even start by identifying the problem (Drop in Sales of Gelato when the temperature is colder).
Ask how to solve the problem – they may come up with creative ideas of their own.
Then offer Gelato Fiasco’s solution – discounts based on how cold it is. Each degree below freezing is a percent off (simple language, not too “mathy”)
Have students try to encode that mathematically, and derive an equation. Maybe by doing tables, etc.

Could also include a discussion about the difference between percent, and percentage point. A drop by 1 degree, then another degree sums, to add a second percentage point off. If the discount is 1%, then it drops another 1 degree, the phrase “another 1% off” is (technically speaking) inaccurate.
Maybe not the time for that though?

4. #### Nic Paolini

January 8, 2015 - 2:30 pm -

Dan Anderson’s comment inspired this thought in me for an extension: What if they used Celsius instead of Fahrenheit? How about Kelvins? Which is the better deal?

5. #### Dan Anderson

January 8, 2015 - 6:20 pm -

Extension 2: If they decided to institute demand pricing (ala uber surge pricing), how would you (the student) price the froyo based on temperatures greater than 72?

6. #### Pam Rawson

January 9, 2015 - 4:58 am -

With a lunchtime temp of 5 degrees here in Portland, Maine, yesterday was a perfect day to take advantage of the Frozen Code. Not sure too many of my students did, though. I can’t believe that I forgot about this marketing scheme. Thanks for the reminder, Dan. I’ll be using some form of this with my students next week. Perfect timing – seasonally and curriculum-wise.

7. #### John

January 9, 2015 - 5:50 am -

With the clear correlation between temperature and number of chain gelato shops, CLEARLY the temperature causes more gelato shops to be built…or does the number of gelato shops cause the temperature to decrease…am i right?

8. #### Bowen Kerins

January 9, 2015 - 7:07 pm -

Their formula is terrible! Price = Price – [Price * (…)]

Or is it…? Subtract Price from both sides…

0 = -Price * (something nonzero)

Therefore 0 = Price! It’s free, and there’s proof! QED

9. #### Mark P

January 10, 2015 - 12:53 pm -

I see some of the suggestions above as mathematical social studies. It is interesting to consider alternative strategies etc, but at the end of the period you’ve advanced their maths very little, if at all, by talking about things without doing the maths.

I also have a dislike of unchecked answers. They are the best way to embed incorrect techniques, because if students don’t know their working has flaws they will continue to make those mistakes, and indeed embed them deeper. (It’s why I distrust homework that is not marked as it is done.)

I would suggest

1) setting the scene pretty quickly

2) propose the “solution” come to by Gelato Fiasco

3) ask them to write that solution in words

4) ask them to convert words into variables (specifically not giving them letters to use as the variables, as they need the skill of picking their own).

5) ask for student answers as they come in, put on the board, in words and equations (allowing the slower ones to keep on going, but letting the fast having something to look at). We go through them seeing if they work or not. Explain that any that work are acceptable, no matter how ugly they look.

6) Correct any misconceptions, such as the price = price – discount. Quickly explain the mental process of writing equations. (It is important to discuss it as soon after they did it as you can — talking at the end of the period about what they did 20 minutes ago and since moved on from is not helpful.)

7) explain that the US uses Fahrenheit, not Celsius, so we’re going to have to change our zero point to be 32 and see if they can rewrite a correct equation in degrees F.

8) work through suggested solutions — inserting them into the previous correct ones on the board where possible, to mimic the process a skilled person uses.

I know this sounds very traditional, but it need not be if you pace it right, with engagement and discussion about the alternative correct answers. Some might allow students to come up and write their equations on the board, although I wouldn’t.

What I find most disconcerting about discovery and investigative learning is just how little gets done in a period. They may love it, but if you spend 30 minutes writing one equation then you’re selling them short.

10. #### Dan Anderson

January 12, 2015 - 8:04 am -

@Bowen
I’d bet that the person who wrote that formula has taken some programming classes. It’s the exact code that I’d write to update the price variable with the new price (the old price info would be overwritten). Ugly though.

11. #### Shernell Mabien

April 12, 2015 - 7:15 pm -

Hi, I am a student at the University of South Alabama and each week are comment on different blogs. I think it is interesting to do research like you have done. I think the Frozen code would be very helpful.