Seriously, if you haven’t stumbled across Andrew’s Number Sense site, you need to check it out here. I recently has students work on Day 25, then hit them up with a follow-up. I showed them this image:
Then I asked them “Would it make a difference if we used one of these measuring cups instead of the one that Andrew used?” The response was overwhelmingly unanimous, yes.
This truly surprised me, I assumed that a majority of students would know the answer to this question. So I decided to run a lab the next day, where we could investigate which cup would actually hold more. I went home, scavenged my kitchen as well as my mom’s, and found a few different types of measuring cups to bring in. On the way to work I picked up candy corn, but I was hit with a math problem right away…
Which bag would you choose? Of course the first thing that caught my eye was 4/$5. As I was grabbing bags, I looked over on the next shelf and saw larger bags for $2.25. I did some quick math, and decided on which was the proper choice. The great thing about it was that I presented this problem to students before we actually started the candy corn activity. I was very proud of my students- even though many had the same initial reaction I did (4/$5), they quickly did some mental calculations and determined which was the best value. I did not have a single student who did not come up with the correct answer, which made Mr. Anderson a happy teacher.
So, back to the task on hand. I displayed the 1.4 cups to the students and asked them which would hold more candy corn. I allowed students to handle the items so they could examine them, notice differences between them and determine their answer. This is what one class looked like:
Why did so many students choose the glass measuring cup? It was not because of it’s overall size- I purposefully showed students the correct line for filling (and did this numerous times to make sure there were no misconceptions there). Students said that the size of the bottom of the cup and how the sides progressed from there was the reason they chose it. The plastic measuring cup came in a close second, and the reasoning was similar. Students picked it because of its drastic shape and size differences when compared to the others. Now it was time for the moment of truth: we filled one of the measuring cups with 19 candy corn….and were not full! This created a big problem for many of my students. They absolutely love Estimation 180, and Mr. Stadel is becoming an iconic star of the room. They could not initially believe it was wrong… until one student spoke up.
“Mr. Anderson, did you buy the same candy corn as Mr. Stadel?” I didn’t immediately answer this, but instead asked if there were any other reasons that could effect the number (I could tell that the student’s question got their minds rolling). Different dimensions of candy corn made by different companies, how the candy corn is “packed” into the measuring cup and whether whole candy corn was used are other questions that came out. I did admit that I did not buy the same brand, so students went with the reasoning of different dimensions to explain our situation (and that may be another follow-up investigation later this year). We finally agreed upon 25 candy corns for 1/4th of a cup.
I filled the tin measuring cup when we started this, and asked students to notice how full it appeared. They noticed things like packing, the space between the corn, and whether the candy corn was above the line of the sides or not. I then poured it into the steel cup, and students immediately said; “Ha! The tin cup is smaller!” Then they started examining things. It was good that I had them write down some of the observations from the first cup, they started comparing their notes and agreed that they actually were the same measure. There were similar reactions to the last two converting, and I had some staunch supporters of the glass liquid measuring cup being larger (it is harder to compare the side line of the corn on this one, and it visually looks larger). I asked them how we could compare these cups in a different way, and got a response of using water instead of corn. It would get rid of packing problems as well as being over the side line. I filled the liquid measuring cup, and proceeded to pour it into the small looking tin one. Students went ballistic, since I was demonstrating this over their table where all of their things were gathered. They were amazed that the little cup could hold it all. It as a struggle for me to show them the same for the last two cups (pouring with your off hand is a little shaky!), but in the end they all now understood that there was no difference in any of them. (Note: In the U.S., the quantity measured by dry and liquid measured less than a pint are the same. After that, there is a difference. A U.S. pint used for liquid measures is 473 milliliters, where as a dry measure pint is 551 milliliters, which means it is 16.5% larger. A U.S. dry measure quart is 16.4% bigger than its liquid counterpart, at 1101 milliliters vs. 946.)
I went on with class for the rest of the hour and asked students which measuring cup would hold the most as an exit ticket. Here were a sample of the responses:
I think that the experiment was a great experience for them, I’ll ask them a similar question in a couple of weeks to see if the concept sticks with them.