South Hadley Middle School
South Hadley, MA
The following is adapted from a story by Marie Appleby, a mathematics teacher at South Hadley Middle School in South Hadley, Massachusetts. The original version appeared in ENC Focus: A Magazine for Classroom Innovators (vol. 6, no. 2, 1999), entitled "Inquiry and Problem Solving."
Graphics included with the online version are helpful in understanding the mathematical problem being described in the story. There are also additional materials, such as student journal entries and a description of a new mathematical puzzle. The full story can be found online at:
Petals Around the Rose: Building Positive Attitudes About Problem Solving
The sound of five rolling dice caught everyone's attention.
I continued rolling the dice until a small group of curious students gathered around me.
"What are you doing?" asked Sarah.
"What do you think she's doing? She's playing with dice!" croaked Danny disdainfully.
With a gleam in my eye but keeping a straight face, I started my routine: "The name of the game is Petals Around the Rose. The name is very important. For each roll of the dice, there is one answer, and I will tell you that answer."
I rolled the dice: five; one; two; two; four. "The answer is four," I said without changing expression.
I paused and then rolled the dice again: five; one; three; three; three. "The answer is ten."
"How did you get that?" asked Sarah suspiciously.
"You must be a mind reader or something," chimed in Jamie.
"Can I roll the dice next?" asked Danny, trying to take control of the situation. Danny rolled the dice three times, and I gave only answers.
By this time, the students were hooked. Here were materials they knew very well but presented in a way that was new to them.
Early in each school year, I want my students to examine the methods they use to approach problem solving. The Petals Around the Rose problem confronts students with a lot of data and an answer, but no formal question and no explicit conditions of the problem.
I am always intrigued by the observations students make and about the hypotheses they sometimes mutter out loud. I do want them to be observant -- to check for similarities and differences in the outcomes of the rolls of the dice -- and to generate many possible explanations or rules for the game. I also want them to feel the joy of working on a seemingly difficult problem in math and solving it. More importantly, I want them to begin making judgments about right and wrong solutions without depending on the teacher for verification.
Two days later, there was an opportunity to continue the game.
"Can we play Petals Around the Rose?" asked Jamie.
Her question was quickly followed by a chorus of "Can we?" from nearby students. I began: "The name of the game is Petals Around the Rose. The name is very important. For each roll of the dice, there is one answer, and I will tell you that answer."
I rolled the dice: five; two; four; one; five. "Eight."
"There's no eight there!" moaned Jamie.
"You didn't tell us the rules," wailed Danny.
Looking at my students' puzzled faces, I thought back to how I had learned the game at an educational conference. The head of our math committee had pulled five dice from her pocket and started rolling them. She repeated the same directions and kept a noncommittal face.
My colleagues included teachers from language arts, social studies, science, and industrial arts, and we were all trying to figure out how the answer fit the name. It was aggravating. I remember trying many different possibilities including combining dice in arithmetic ways, disregarding some dice, looking for patterns in the way dice landed, and on and on.
The language arts teacher, an admitted "math-phobe," was the first to find success. I admit I wondered what he had tried that worked so easily. The "ah-ha" feeling that hit me when I saw the Rose and the Petals around it was wonderful and gratifying. I decided that feeling was something every student should experience.
When guesses from my students started flying fast and furiously and seemingly without much thought, I halted the game. There would be no more playing of the game that day, but I asked, "What do we know about Petals Around the Rose?"
As the students generated a list, I wrote the items on the board, as I do with other investigations:
The next day Ted arrived first to class and said, "Test me! Test me about Petals Around the Rose. I think I've got it!"
Ted was well-liked and a hard worker but not the star of the math class. While his "test" was going on, the other students came into class and were both surprised and delighted that he had found a solution. I handed the dice to Ted, and he ran the game for the next ten minutes with small groups of students while other students and I were getting organized for the class.
As each day passed, I felt the excitement grow as I watched more and more students solving the problem. They could pick up a set of dice from the materials table and help small groups at the beginning or end of class or any transition time in between. The game was no longer mine; it belonged to the students.
There are other activities that carry students to an "ah-ha" discovery or experience. However, Petals Around the Rose offers a change of pace, a non-traditional problem that engages the students in light-hearted activity while requiring them to use several important strategies:
Petals Around the Rose has developed a feeling of success and power in my students and a positive view of problem solving.