Whether you like to solve problems yourself, or are looking for some tidbits for your children or students, I hope this post is informative.
I’ve been reading Jo Boaler‘s brilliant new book, Mathematical Mindsets. While there’s tons of great information and research there, I’ve been thinking lots lately about her charge to develop more “low floor, high ceiling” tasks into math lessons–problems that are “challenging, but accessible” to a much broader spectrum of students than typical exercises. In particular, Boaler encourages teachers to use problems that are easily understood, relatively simple to begin, and yet hold deep potential for advanced exploration. Boaler notes that these problems tend to be very difficult to find.
Here I offer an adaptation of an Ask Marilyn post toward this goal. While the problem was initially posed in terms of singles at a party; I rephrased it for younger students. Solving it helped me see variations that I hope address Boaler’s low floor, high ceiling call.
THE PROBLEM :
Paraphrasing the original:
Say 100 students stop by the lunchroom for a snack. Of these, 90 like apples, 80 like pears, 70 like bananas, and 60 like peaches. At the very least, how many students like all four fruits?
MY FIRST SOLUTION:
The Ask Marilyn post offered only the answer–zero–but not a solution. To prove that, I made picture. Since the question was the least number of commonly liked fruits, I needed to spread out the likes as much as possible. Ninety liked apples, so when I added the pears, I made sure to include the 10 non-apples among the 80 who did like pears giving
That made 30 (at the bottom) who liked only one of apples or pears, so when I added the 70 bananas, I first added them to those 30, leaving
That made 40 who liked all three, so the 60 peaches could match up to the other 60 who liked only two of the first three, confirming vos Savant’s claim that it was possible in this setup to have no one liking all four.
FIRST: As a minor extension, one of my students last year would have said the problem could be “complexified” slightly by changing the numbers to percentages. (I loved my conversations with that student about complexifying vs. simplifying problems to find deeper connections and extensions.) With enough number sense, students should eventually be able to work with absolute numbers and relative percentages with equal ease. Mathematically, it doesn’t change anything about the problem.
SECOND: The problem doesn’t have to be about a single minimum number of students to like all four fruits. While there is a unique minimum, there are many other non-optimal arrangements. I wonder how students with developing problem solving skills would approach this.
THIRD: In my initial attempts, I had used many different variations on my tabular solution above. Only in the writing of this post did I actually use the above arrangement, and that happened only because I was trying to come up with a visually simple representation. In doing so, I realized that the critical information here was not what was told, but what was not said. Where 90, 80, 70, & 60 liked the given fruits, that meant a respective 10, 20, 30, & 40 did not. And those added up to 100, so I knew that any variation of “not-likes” that also added to 100 could be distributed so that the minimum number who liked all four would also be zero. So there is an infinite number (if I use percentages) of variations of this problem that have the same answer. I also realized that any combination of 2 or more fruits whose “not likes” added to 100 could produce the same results. My ceiling just rose!
FOURTH: To make the problem more accessible, I could rephrase this in terms of setting out fruit and exploring many different possible arrangements. i could also encourage learners to support their developing problem solving by translating the problem into pictures.
I’m ready to pose a new variation. I’d love to hear your thoughts, insights, and variations for raising the ceiling in this problem.
LOW FLOOR, HIGH CEILING VARIATION?
Say 100 students stop by the lunchroom for a snack. Of these, 90 like apples, 80 like pears, 70 like bananas, and 60 like peaches. The lunchroom staff knows these numbers, but doesn’t know how much of each fruit to put out. But putting out too much fruit would be wasteful.
- What advice can you give them? Show how you know your solutions are correct.
- Draw some pictures of the possible numbers of students who like the different fruits.
- Is there more than one possible answer?
- It is possible that some students might not like any fruits offered. How many students might this describe?
- Some students might like all four fruits. For how many students might this be true? How many answers are there to this?
For students who manage all of these, you can challenge them to
- How can you change the initial numbers in this problem without changing most of the answers?
- Can you create the same scenarios with more or fewer types of fruit?
- If the numbers are too big for very young students, you could drop all of the initial numbers by a factor of 10. How many will see this scaling down simplification (or its scaling up complexification)?