Which 1/4 Cup Holds the Most Candy Corn?

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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:Measuring Cups

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.

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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…

Candy Corn

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:

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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.

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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:

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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.

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Shoebacca -3Act Math

ACT 1:

Show them this clip and this picture:

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What questions do you have after seeing this Video and Picture?

Ask students to write down their questions, I normally ask students to find at least 3.  When I observe that most students have questions written, I ask them to share those questions with their neighbor.  I then throw up a Microsoft Word document and start typing down questions students supply.  Students from my classroom came up with all sorts of different questions, some we can easily answer and others that we can’t.  I am looking for a key question or questions to start this lesson.  If students do not ask one of these questions, I tell them that I hope I can answer most of the questions provided, but that I need them to consider one of these questions first.

  • What is the shoe size?  Is it like a 52?
  • How big of a foot would you need to wear that?
  • How big/tall would a person be to wear this shoe?

Any of these type of questions will lead students down the inquiry I hope to explore with them.

 

ACT 2:

This part can take a few different paths, depending on your focus for the day (you could use this idea with Proportions, Scatter Plots and Statistics).  I started this part by asking what type of mathematical operation or ideas they would need to find an answer.  The most common ideas students came up with are:

  • guess/ estimation
  • fractions
  • proportions
  • equations
  • graphs

These are all excellent ideas and I place them on the board as reminders to where we want to go throughout the investigation.  Depending on what focus I use this for, students normally fall into 3 strategies for finding our answer.

  1. Proportions: Most students want to make a proportion relating themselves to Shoebacca.  This is a great strategy to begin with, but since I teach 8th grade students, I want them to push themselves.  This strategy would work well with our 7th grade standards.
  2. Statistics: One thing that comes up when we start exploring Shoebacca is the fact that we can’t accurately predict how big someone’s foot depending on their height.  Students immediately think of mean (or average) and start taking down everyone’s foot size and height (this is also great review of measurement and conversion- students this past year put up their height in feet, inches and I asked for inches only, and for some classes, cm).  Then the next step students wanted to take was strategy 1, they made a ratio of their class average and made a proportion.  Once again, a great way to tackle this problem, but I am relentless- I want them to really explore different options of solving this problem and how those strategies relate and compare.
  3. Graphs: When I first designed this problem, this was naturally MY first thought.  One thing all great teachers do is anticipate how students will approach a problem and then tie it into the method they are presenting.  I anticipated the first two approaches and hope my students will make connections between them and this one.  One of the reasons I like looking at graphs is it allows students to use software to enter data, find trends, and display it graphically.  Students take measurements of their class, place it in a spreadsheet and create a graph.  They can then look at lines of best fit (before you automatically create it with the program) and make a conjecture based on their data.

Some other considerations:  The length of the trailer which will determine how big the shoe length is, can be varied.  I normally go with what the students discover- many will do searches on trailer sizes.  The most common lengths I get in class are 12′, 14′,16′, 18′ and 20′.  This variation of the trailer size could be a extension of the problem.

Here are the statistics for this year: 8th grade shoe 13-14

 

ACT 3:

These are the results my students got this year:

  1. Proportion: varies with individual   65″/10″ = ?/240″ so ? = 1560″ or 130′
  2. Statistics: 66.8″/9.7″ = ?/240″ so ? = 1652.8″ or 137′ 9″
  3. Graph: students drew a line of best fit and got 1080 or 90′

Some great discussions resulted in the difference in these estimates for height.

 

What is 3/5?

This past weekend, Tracy Zager tweeted a picture that 70% of 7th graders answered incorrectly:

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This led to a great conversation which also included Christopher Danielson, Michael Pershan and Curmedgeon.  It also sparked perplexity in me.  I found I could devise many different variations to this problem to explore different student thinking, but instead I took this week to determine if students understood fractions as equal parts or if they fell back on a quick rule to find an answer.  This was the results from day 1:

IMAG0396The results: 46.4% of my 8th grade students got it wrong, a much higher number than I anticipated.  Looking at the reasoning given by my students, it appeared that they were following an old rule they partially remembers of counting shaded parts.  I was not totally convinced this was the case or if it was because of the even partitioning that was done on the triangle.  I wanted to check their understanding of a fraction so I gave them this for day 2:

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When I looked at this, I expected small unit fractions and circle diagrams.  I was not sure what they would write explaining what the fraction was or an example of it outside of the classroom.  I was hoping they would talk about equal parts.  This is how they answered:

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One-half was the overwhelming fraction used, and they expressed that using a circle.  All diagrams had pictures where students created equal parts, many times painstakingly so (evidenced with many eraser marks).  Most students simply wrote it is one-half of the circle, none referenced equal parts.  For the real life example, a majority of my students expressed the fraction as pertaining to a circular food; such as pizza, pie, cake or even a cheeseburger!

This last part really told me about how students have learned fractions.  To gain student interest, teachers have fallen back on the standby of “the quickest way to student interest is food!”  That students chose pizza or pie was no surprise, but that students see fractional parts in terms of circle graphs was.  I need to make a point of expressing fractions with a different diagram- having students with such a narrow view of how fractions are used outside of the classroom limits them.  It is no surprise that when fractions are introduced as slope in 8th grade, students have a total disconnect.

 I still had a suspicion that students were only looking at the partitioning of the base instead of the area represented by them, I decided to use a diagram I thought of and that Curmudgeon reproduced and posted on his site:

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I was hopeful that students would not only compare the widths of the fractional parts, but also the length.  I was right and 77% of students answered the question correctly, here are the results.

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I was very happy and proud of the results, and then another thing happened that blew that away.  My students started questioning me as to why we were doing these warm-ups, what the correct answers were and were trying to anticipate what I was going to give them tomorrow!  I had not gone over anything in class and had not told them what the correct answers were.  They tried to pry information out of me, but I told them I had one more warm-up for them tomorrow and then we would discuss the week.  They started a student-lead classroom discussion about the problems, what was correct and were sharing ideas about why people answered a certain way.  I sat back and allowed them this time, it was great to see.

After class that day, I still had a nagging doubt that students only did so well because they were able to compare two diagrams, there was still an underlying issue of following this rule of counting three out of five pieces shaded.  So the next day when they entered the room, this is what they found:

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As I suspected, only 20% of them were able to answer it correctly.  Those student noticed that one piece was 1/4th the size of the circle, the rest came back with the old standby of three out of five pieces shaded.  I was crushed.  I asked in each class whether this represented 3/5ths and the first and immediate response was not from the people who said yes, but those that said no.  I said, “If this is true that the figure does not represent 3/5ths, why would that be?  Take a minute to think about your answer.”  I would call on someone who I knew answered incorrectly and when they contemplated the possibility of it being wrong, told me that one piece was bigger than the rest.  We then went back and revisited the problems over the week, and students realized that the triangle did not have equal parts either.  My students once again really dug into this discussion and other than a guiding question or two- took control and explored the problems and what the correct answers were.  I was very happy with the outcome of that day, and I hope the students made some connections that will follow them throughout their mathematical careers.  I will recheck their knowledge in two weeks and see if this experience will make a difference in their understanding of the concept.