Before teaching this lesson, play Petals Around The Rose for
yourself. You will only be able to teach this lesson effectively if you
have solved the problem on your own before giving it to students.
As students enter the classroom, stand at the front of the room
and roll dice. Continue rolling until a student finally asks, "What are
Say, "The name of the game is Petals Around the Rose. The name is important. I will roll five dice, and I will tell you how many petals appear."
Roll the dice so that all students can see the results. If
possible, use transparent dice on the overhead projector so that all
students can see the roll. After each roll, tell the students how many
petals are showing.
For example, if you roll the following, inform students that there are ten petals:
Continue for several rolls. As necessary, repeat the lines
above, especially when students ask for a hint: "The name of the game
is Petals Around the Rose. The name is important. I will roll five dice, and I will tell you how many petals appear."
As you roll the dice, encourage students to keep track of the
rolls and the number of petals. Explain that a table of results will
make it easier to identify any patterns and discover a rule.
Should a student suggest that she knows the rule for determining the number of petals, do not
ask her to share it. Instead, roll the dice and ask her to identify the
number of petals. If she answers correctly for several rolls in a row,
declare her to be a Potentate of the Rose and tell her, "Now
that you are a Potentate, you are sworn to secrecy about the rule. You
must never reveal the method for determining the number of petals. Only
those who solve it themselves should know the secret." (To help
maintain secrecy, you can allow students who figure out the rule to
roll the dice as a reward.)
You may want to ask students to generate a list of questions
for which they would like to know the answer. Questions could include:
- What happens when you only roll one die?
- What happens when the dice are rearranged from least to greatest? ...or vice versa?
- What happens when only red dice are rolled? ...green dice? ...blue?
- What is the least number of petals possible in a roll? ...the fewest?
- What happens if you change the value of just one die but leave the other four alone?
Depending on the questions on this list, you might want to
answer some or all of them. However, do not reveal too much. Arranging
the dice in order from least to greatest, for instance, does not affect
the number of petals, so you should feel free to do so. But rolling
just one die and reporting the number of petals would likely give away
the rule, so you probably should not do that.
It may be that no student will determine the rule within the
first ten rolls or so. To keep excitement high, you might want to stop
the game and say, "Okay, we are not playing the game any more today.
We’ll return to it tomorrow when we have more time. But let’s make a
list of what you’ve learned about the game so far." Have students
generate a list of observations, which may include:
- The name of the game is very important.
- The answer is always even.
- The color of the dice has no effect.
- Answers are generated by rolling the dice.
- The game uses five dice with pips (dots).
- The word "around" is important to the rule.
As is likely obvious, the important part of Petals Around the Rose
is not the problem, but rather the strategies that students employ.
Students might use any number of strategies to think about the puzzle,
- Keep an organized list.
- Guess and check.
- Consider a simpler problem.
- Generate a table of observations.
- Draw diagrams.
In addition to revealing various problem-solving strategies,
this problem also teaches students persistence, since they are not
allowed to know the rule unless they discover it on their own.
Have all students write a paragraph in their math journals about the
strategies they used to determine a rule. Even students who did not
discover the rule should write about what they tried.
- Students who determine the rule for the game may become bored as
other students are still trying to figure it out. You might want to
pose the following challenge questions to students who discover the
- What is maximum number of petals that could occur in one roll? What is the minimum number?
- In how many different ways could zero petals result?
- What is the average number of petals per roll? How do you know?
- Ask students to invent a similar game.
- Play Bears and Fishes, a game that uses similar rules. With each roll of the dice, repeat the following:
Polar bears come in pairs. They sit around a hole in the ice like petals on a flower.
A center pip represents a hole in the ice. The number of
pips around it is the number of polar bears who "sit around a hole in
the ice." (Note that if a 1 is rolled, there is a hole in the ice, but
no polar bears.) The number of fish refers to the number of pips on the
of the dice; consequently, a roll of 1 has 6 fish, a roll of 3 has
4 fish, and a roll of 5 has 2 fish. For the following roll of five
dice, there would be 4 bears and 14 fish. (Note that it is only
possible to see the fish on those dice with a "hole in the ice.")
Students will quickly realize that the rule for determining the
number of bears is the same as the rule for determining the number of
petals around the rose. Students will have a more difficult time
determining the rule for finding the number of fish.
Questions for Students
1. When five dice are rolled in Petals Around the Rose, what is the rule for determining the number of petals?
2. What strategies did you use to determine the rule?
[Students may respond by saying any of the following strategies: keep an organized list, guess and check, consider a simpler problem, generate a table of observations, draw diagrams, etc.]
- Determine a rule for Petals Around the Rose
- Consider various problem-solving strategies
- Use an organized list to identify patterns
Common Core State Standards – Mathematics
Grade 4, Algebraic Thinking
Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule ''Add 3'' and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.
Common Core State Standards – Practice
Make sense of problems and persevere in solving them.