## Petals Around the Rose

- Lesson

A puzzle involving five dice and a non-standard pattern is used to promote problem-solving skills.

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 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.

- Five standard dice
- Overhead dice (optional)

**Assessment Options**

- 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.
- Roll 5 (or less) dice, and ask students to find the number of petals around the rose. Also, have them write why they think that is the number of petals around the rose.

**Extensions**

- 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 rule quickly:
- What are the possible 5 dice rolls for a 6 petals?
- 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*bottom*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**

- When five dice are rolled in Petals Around the Rose, what is the rule for determining the number of petals?
- 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.]

**Teacher Reflection**

- Was this activity successful? What could have been done to make it more successful?
- Which students were the first to solve the puzzle? Were you surprised at how quickly some of your lower performing students were able to solve the puzzle?
- How did your lesson address auditory, tactile and visual learning styles?
- What strategies did students use to figure out the rule? How did you ensure that various strategies were considered?
- How did students illustrate that they were actively engaged?
- Did you solve the puzzle on your own before using this lesson with students? How did that affect the delivery?
- Was students’ level of enthusiasm high or low? Explain why.

### Learning Objectives

Students will:

- 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

- CCSS.Math.Content.4.OA.C.5

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

- CCSS.Math.Practice.MP1

Make sense of problems and persevere in solving them.