# Optimizing Box Volume with Marshmallows

The problem is a classic in algebra and calculus, and to be honest it really did not make sense to me until I built the boxes, when I was 35 years old.   Here is the problem:

• Given a standard 8 ½ inch x 11 inch piece of paper, determine a function which gives the volume of a box (without a lid) made by cutting squares from each of the corners and folding up the sides.   Let x be the length of a side of the square, and write the volume as a function of x. What are the units of x and v(x)?  What is the maximum volume of the box?

Granted this is an expanded version of the problem.  You couldn’t ask all of these questions on a standardized test.  But this problem is full of concepts like: domain, range, polynomials, graphing, roots, and volume just to name a few.  My classrooms have students at varying degrees of experiences in math.  For instance I will have students that are ready for calculus working alongside students who really don’t understand what volume means.  In fact if you ask some of them, they will reply, “L X W X H”.  If that is all students think about when they are asked to find volume, then they probably do not understand the concept of volume.

I have a bunch of 5X8 note cards, so I ask the kids to imagine cutting equal squares out of the corner of the note card and filling the resulting tray with marshmallows.  How many marshmallows could we contain in this resulting tray if we don’t pack them and we do not heap them above the box?  I let them fiddle around with the marshmallows.  I then give them 1 minute and tell them to write their name and their guess on a piece of small yellow post it note.  We then arrange them on the board in a makeshift line graph.  I do this so that students buy into the problem and practice estimation.  This guess cements their curiosity.   They start asking themselves.  “Am I right?” ” Is my answer the most?”  “Is my answer even possible?”  Some students will try and build the box so that it matches their answer.

As they are building these boxes some students will ask, “How big should my squares be?”  I respond, “How big can your squares be?”  I notice that the older I get the more I sound like a psychiatrist.  But here I am trying to lead students to the domain.  Eventually we will formalize the domain and the range for this problem.  But first we have to see how many marshmallows we can stuff in these boxes.

What we are finding so far is that unsurprisingly the number of marshmallows divided by 8 is just under the actual volume of each box in cubic inches.  Here we can talk about the gaps between mallows and ask questions like, does smashing them give us closer estimates?  How about piling them? (You are on your own here.  The kids ask this but I redirect them to the GeoGerba applet below:)

Finally, we talk about the function that represents the volume of the box for any square of length x cut from the corners of the 5 X 8 note card.  In this case V(x) = x(5 – 2x)(8-2x) (I will let you do the distribution to find the simplified version of that polynomial;).  And about the best place to explore the function is in Ted Coe’s GeoGebra worksheet  at http://www.tedcoe.com( His site is incredible!!!!! BTW). Here is the html applet.  I hope it works because it is awesome!!!
The power of this project comes in the form of having students tape their creations to a projected GeoGebra display of the function the class developed.  Every time I do this it brings a small tear to my eyes at its beauty. To me it shows the true power of technology in our classrooms.  The first time I saw this problem was Algebra II when I was 16 years old and our internet was floppy discs.  Along with the fact that we only used computers when instruction was complete.   I am sure my teacher at the time tried to show us the solution, but I did not register any understanding.  It was one of those problems that the teacher assigns and then when no one tries, he/she solves it for them on the board while all students nod their heads that they understand.  I saw it again in Calculus and I still don’t think I truly understood anything outside of (the algorithm) the fact that when I applied the derivative to the function and then set the resulting function to zero and solved I would get my maximum and minimum values.  The third time I saw this problem was in my masters program at the University of Nebraska-Lincoln.  Here is where I came up with the idea that maybe when we solve this problem we should actually construct the boxes. I am pretty sure that constructing the boxes actually makes this problem a Jo Boaler Low floor high ceiling  problem.  (YOUCUBED.ORG has changed my very perception of mathematics education).  I give this problem to all of my students 9 – 12 and they all love it.  Probably because they get to eat marshmallows, but I hope that at the very least some finally see the connections between stuffing marshmallows in a box and volume are the same thing.  If we get to the connections between algebra, calculus, and geometry…. that would be bonus for those that are ready to make those connections.  I am pretty sure I wish I lived right now.  Education is awesome!!!

# Who thought about that problem differently?

The question above should be asked by math teachers after almost every problem.   During my bell ringer today we had unexpected, crazy, rich discussion over two problems that  I never thought of as rich in mathematical ideas.  The problems were :

In a certain school, 45 percent of the students purchased a yearbook. If 540 students purchased yearbooks, how many students did not buy a yearbook?

and

If a machine produces 240 thingamabobs per hour, how many minutes are needed for the machine to produce 30 thingamabobs?

Solve these and see what you come up with.  Keep track of what you do.  In my classroom, I randomly choose students to come up and show their methods.  Sometimes they are wrong and we use incorrect answers as an opportunity to  show common mistakes and I  do everything in my power to show that these mistakes are helping everyone in class grow their brain (Thanks youcubed.org!!!) In this first problem, many students were blurting 243.  In this case students were multiplying 0.45 and 540 or switching up their proportional reasoning.  To handle this I told them to slow down and read the question again.  Does that answer make sense?  Also, please stop blurting answers! You are stealing someones opportunity to think.

Problem 1 method 1 The first student came up and told me that you first take 540 students divided by 45 percent to find the number of students per percent.  Then take that answer (12) times 55 percent for 660.  I asked them how they came up with 55 percent and they were able to tell me that 100 – 45 = 55 which is the percent of students who did not buy the yearbook. Then I asked, “Who thought of this differently?”  Surprisingly, many hands flew up.

Problem 1 Method 2 The next method was the straight up algebra.  If x is the total number of students, then 0.45x = 540.  Solve this and 540/.45 = 1200.  Since there are 1200 students we know that 1200-540 = 660 student that did not buy the yearbook.  Each time a new method popped up I would give it the name of the student who produced it.

Problem 1 Method 3 In the third method a student divided 45 and 540 by 9.  This student then proceeded to tell me that because of this every 5 percent is 60 students, so 11 times 60 is 660.  Here the 11 felt like witchcraft for some of the kids, so I had the student explain where the 11 came from and he was able to tell everyone that 55% has 11 groups of 5.  This method is similar to method 1 and this student may just have wanted to come to the board, but that is ok.  Kudos for thinking deeply.

Problem 1 Method 4 A student kept blurting out FISH.  At this students previous school they had been taught that proportional reasoning was a method called fish.  When you set up the proportion  45/540 = 55/x , where x is the total number of students.  We then proceed to cross multiply and divide. When you trace out the pattern of this the Christian fish symbol is produced. .

In all cases I was able to show students the connections between their solutions and use the language of mathematics.  In the word percent, per is divide and cent is 100.  Using labels helps you keep everything organized.  I had one student use 45/100 = x/540 to come up with 243.  Again, when I added the labels for each numbers 45( purchasers)/100 (entire class) = x (entire class)/540 (purchasers) several students in the class had an aha moment.

Problem 2 method 1 By the time we made it to the second problem, I had students practically running up to the front of the class to show what they did.  So I didn’t have to randomly pick them.  The first student told me it takes 8 boxes of 30 thingamabobs to package 60 minutes worth of product, so 60/8 means that it takes 7.5 minutes to produce one box.

Problem 2 method 2  On the second method a student told us that if 240 thingamabobs are produced in 60 minutes, then 120 are produced in 30 minutes.  Following this reasoning then 60 are produced in 15 minutes and 30 are produced 7.5 minutes.

Problem 2 method 3 It surprised me that the algorithm we teach is always the last method shown.  My personal opinion is that when we follow these “efficient” algorithms it slows down thinking .  In this method the student lets x = minutes for 30 thingamabobs.  Then 240(thingamabobs)/60(minutes) = 30(thingamabobs)/x(minutes).  Then students cross multiply and divide to get 7.5 minutes.  240x=1800 then 1800/240 = 7.5  I think “cross multiply and divide” is a “trick” that students don’t really understand.  I try and always show that it is really just common denominators and the multiplication of the entire equation by that denominator in order to simplify the fraction.

```Hans Rosling and his site gapminder.org are revolutionary and should be changing the way all of us look at local data!  I am surprised that his concept of the motion graph has not trickled down into more local applications.  Especially where these motion graphs can be made with Google spread sheets for free!!!!! (visualization may not work on phones or tablets) I have been part of negotiations since 2000.  Communicating to other teachers, community members, and school board members what is going on statistically in the negotiations process has always been our biggest problem. When I first saw Hans Rosling's 2006 TED talk, I knew that we needed this visualization in local education.```
```First off, play around with the graphic above. Notice how dynamically all the variables can be changed.  Check the box marked "Average".  This is the average for each year.  Compare it to other districts.  Are district above or below the average?  Change the variables.  This is data taken directly from district arrays.  What about teachers who are MA+18.  How do the districts compare when we change this variable?  Change the x-axis to Approximate base cost per contract day (12 X monthly insurance + base)/(contract days). Change the y-axis to ultimate max.  Now you are comparing what minimum costs districts pay per contract day and the most salary a district will pay to a new teacher.  Notice that the two large districts seem to spend less per contract day, but teachers can ultimately make higher salaries.  Does this suggest that larger districts are more efficient?  The data from Scottsbluff public schools and Gering public schools, skews the average.  But I threw those two schools in just to prove that we could create one of these visualizations that housed all Nebraska School districts.  This data is negotiations and compensation data from https://www.nsea.org/compensation.  I have spent well over 100 hours typing all of this data into a google spread sheet.  There are mistakes. Mainly because this is for instructional purposes only!!! PLEASE do not use this to make decisions if you are one of the schools in this visualization!! Although it is public data that can be accessed by anyone, I know that it is full of mistakes.  At the same time, I know that something like this would be useful in communicating data to local communities all across Nebraska. So I cannot keep it secret any longer!  It must be shared. Imagine what these visuals could do for local communities!!!! ```