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. Play Petals Around The Rose
As students enter the classroom, stand at the front of the room and roll dice. Continue rolling until a student finally asks, "What are you doing?"
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, including:
- 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.
- Determine a rule for Petals Around the Rose.
- Consider various problem-solving strategies.
- Use an organized list to identify patterns.
NCTM Standards and Expectations
- Describe, extend, and make generalizations about geometric and numeric patterns.
- Represent and analyze patterns and functions, using words, tables, and graphs.
- Model problem situations with objects and use representations such as graphs, tables, and equations to draw conclusions.
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.