As I read through the submissions to this problem, I had a few thoughts:

- I noticed that a lot of students used Venn diagrams
- I noticed that not everyone used a Venn diagram
- I wondered whether many students had to do some work to figure out that a lot of kids are counted two or even three times.

Let’s start at the bottom of my list. When I first solved the problem, it seemed like realizing that some students are counted two or three times would be a really important part of making sense of this problem. I know it was something that I had to think about when I realized that the 14 kids who ate carrots, the 13 who ate cucumbers, and the 16 who ate tomatoes added up to more kids than there are in the class! Yet only one submitter mentioned it. One! He wrote:

“The hardest part of this problem was figuring out how to make the total number of people be less than 27 because I had too many people in the diagram.”

I’m really curious whether other students ran into that same problem and didn’t mention it. It’s exactly the sort of thing we love to hear about. Did any of you have discussions about this in your classroom as part of understanding how the problem works?

The Venn diagram was definitely the most popular method of solving that we saw. It’s a very helpful way of organizing information. As one student explained:

“My first idea was to write a “c” for carrot, a “k” for cucumber, and a “t” for tomato. Then I wrote “k+t” for cucumber plus tomato, “t+c” for tomato plus carrot, “c+k” for carrot plus cucumber, and “c+k+t” for people who got all of the choices. I did different combinations, but I couldn’t keep up with my information because it wasn’t organized. Then, someone told me to use the Venn diagram. It was a lot easier.”

Easier, indeed! A number of students provided pretty good explanations about how they filled out their Venn diagram. In this excerpt, note the use of “because” and “For the same reason”. Those are good words and phrases to use to remind yourself to give reasons for the decisions you made.

First, I drew a Venn Diagram with three intersecting circles. In the middle section where all three circles intersected, I put in the number 5 for the number of people who took all of the toppings. In the section that overlapped the vegetables carrot and tomato, I put the number 4 because even though 9 people took both vegetables, the five people who took all three of the toppings count, so 9-5=4. For the same reason, in the section that overlapped the vegetables cucumber and tomato I put the number 3 instead of the number 8. I did the same in the section that intersected with the two vegetables cucumber and carrot, putting in the number 2 instead of 7…

Drawing a picture or diagram was also popular. Here are two descriptions of that process:

I drew 27 salads and put on the toppings. I started with the kids that had all the toppings and then I added the four more that just had tomatoes and carrots then I added the three that had just tomatoes and cucumbers than I added the four that just had tomatoes then added the two that had carrots and three had just cucumbers. Four had no toppings at all.

First I made a list of 27 students. Then I counted out how many students chose carrots and how many chose carrots and tomatoes. I wrote this down like

1 2 3 4 5 6 7 8 9 etc. ca ca ca ca ca ca ca ca ca t t t t t t t t t etc. I did the same with the cucumbers and the rest of my tomatoes to find my answer. There were three numbers left over that didn’t have any types of vegetables that went with them, so that told me that three kids did not get any vegetables with their lunch.

I wish these students had said more about how they decided where to put the toppings. When the first one wrote, “then I added the four more that just had tomatoes and carrots”, where does the four come from? I’m also wondering if they made a reasoning mistake somewhere that caused them to end up with an answer of 4, or whether that was just an arithmetic error.

Finally, a bunch of students used this method:

First i added the sum of everyone who had cumber, carrot, and tomato. Then i added up the sum of all the students that had cucumber and tomato, tomato and carrot, and carrot and cucumber. I subtracted the two numbers and got 19, then i added 5 and got 24. then the difference of 27 from 24 is 3. So 3 students took none of those vegetables.

When I first read the solutions that use this method, I wondered if it is mathematically sound, or if the students got lucky. Then I tried to figure out whether or not it works. What do you think? Did these students get lucky, or is there some sound reasoning behind the steps that they took? Let me know what you think!