## Notes to a Mathematician

### Michaél Dunnivant

#### Description

This activity introduces how to express likelihood as a ratio in fraction form. After exploring the concept of likelihood, students write a -Note to a Mathematician- to analyze what they have observed about the likelihood of simple events.

#### Objectives

The student writes notes, comments, and observations that reflect comprehension of content and experiences from a variety of media.

The student solves problems by generating, collecting, organizing, displaying, and analyzing data using histograms, bar graphs, circle graphs, line graphs, pictographs, and charts.

The student uses models, such as tree diagrams, to display possible outcomes and to predict events.

The student predicts the likelihood of simple events occurring.

The student designs experiments to answer class or personal questions, collects information, and interprets the results using statistics (range, mean, median, and mode) and pictographs, charts, bar graphs, circle graphs, and line graphs.

#### Materials

-Graph Paper (see Associated File)
-Standard dice (one per student or pair of students)
-Non-standard dice
-Spinners (fair and unfair)
-Coins
-Small package of candy in colored pieces
-Lottery (strips of paper labeled each with one person's name in a bag)
-Data Diary (one per student) or journal paper
-Crayons or other marking pencils
-Product Rubric for assessment (see Associated File)
-Copies of Analyzing Data Chart and Record Sheet (see Associated File)

#### Preparations

The teacher needs to:
1. Make one Data Diary per student by simply stapling five sheets of folded 11 x 17 copy paper book-style with a construction paper cover. You may also decide to simply provide journal paper.
2. Preview the two online student lessons that reinforce the concepts introduced in this lesson as listed in the Weblinks section of this learning activity.
3. Download the Graph Paper for the class graph from the Associated File and make an overhead transparency. Make copies for students to use at stations.
4. Download the Analyzing Data Record Sheet from the Associated File and make one copy per student. You may also want to make a transparency of this to model the recording process.
5. Have a chart ready for students to record the results of their games. This will be used to organize the results from the previous class session and to create the class graph to be analyzed. A transparency is easier to store for use in future lessons. You may have students post their results on this chart before this learning activity.
6. Download the Product Rubric from the Associated File to use with the assessment.
7. Before introducing this lesson, it is important that your students have had ample experience with concrete lessons to explore probability. The learning activity called Drawing Bugs Game provides such an opportunity (see Weblinks.) In that lesson, students explored probability by playing a drawing game with a standard die, examining and predicting the likelihood of rolling particular numbers on the die, and beginning to analyze the results as graphed. This lesson takes that data and introduces how to express the probability of an event as a ratio in fraction form. Students should not be expected to grasp this abstract concept of numerical ratios without many concrete experiences with simple events.
8. Prepare the probability investigation stations with the appropriate materials as listed:
-2 standard die
-non-standard die
-spinners
-coins
-lottery
-bag of colored candy

#### Procedures

Session 1
1. Discuss the results of dice-rolling games and experiments conducted and graphed in previous sessions, such as the results of the Drawing Bugs Game, another Beacon Lesson Plan. (If students are to be successful with the concepts of this lesson, ample concrete lessons about probability are necessary. See Teacher Preparation.) Ask, -What is the likelihood of rolling any one number on a standard die? Have you solved this problem in an earlier lesson?- As the discussion proceeds, tell students that today's lesson will examine what mathematicians say about the likelihood of events when rolling a die and conducting other simple experiments.

2. Let's look at the results of other experiments we have conducted. Display the graphs that show results of previous experiments. Also display the graph for recording results of today's experiment on the overhead. (see Associated File) Tell students that one thing mathematicians do to figure out the likelihood is to conduct many, many experiments, just like we did with our Drawing Bugs games, spinner games, and coin experiments. (see Teacher Preparation) Ask, -What do we need to do to get the big picture of what happened with all of our games?- (collect the data, organize the information, graph it, analyze it)

3. Each student should report the results of one of their games as their graph displays. These graphs should be housed in their Data Diaries or journals and ready to be shared with the class. Tell the class that first, the results will by graphed on the class graph, then they can be analyzed to find out the likelihood for each event. Before students share their results by reporting the number of -wins- for each number on the standard die, ask students to predict the class total number of -wins- for each number. Record this to refer to later in the lesson. As students report the results of their experiments, chart the number of wins for each event in the appropriate columns on the -Analyze Data Chart and Record Sheet- (see Materials) so they can be calculated. (This step could be done before the class session by posting the chart where students could record their results as they completed their personal graphs.)

4. Ask students if they can tell which number had the most -wins- by looking at the chart. Ask, -What should be done so we can analyze the class results more easily?- Lead students to consider that the results need to be displayed so we can analyze them and this can be done with a graph. Before creating a graph, students can calculate the total number of wins for each number by totalling the results in the column for that number as charted on the overhead. Allow time for students to calculate. If students work in groups (of no more than four students in a group), have students check their work with other students before reporting the total number of wins for each event to the class.

5. As students finish their calculations and the class agrees on the answers, ask for volunteers to graph the data on the overhead class graph. This is a good time to talk about the increments used to display the results on the graph. How much does each increment need to equal so that all the results fit on the graph and make sense with the data? How should we scale the graph? How should we label it to communicate the results most effectively? Take advantage of this opportunity to refine students' knowledge about graphs by modeling.

6. Guide students while analyzing the data as a class. Discuss what the class data shows. Did each number on the die have the same number of wins? Why did a particular number have more or less wins?

7. Tell the students that now they are ready to analyze the results of this and other experiments. Distribute the Analyzing Data Record Sheet to each student. Direct students' attention to the -Event- column showing each particular side of a standard die. Look at the first event in the column, rolling a one. Ask, -What was the number of 'wins' for that event?- Allow time for students to work in groups to analyze the numerical data from the class graph. As they find that information, direct them to record the total -Number of 'wins' for that event- in the appropriate column as you model it on the overhead. Ask, -What was the total number of events? altogether- Model how to calculate the total number of events based on the class results and record that number in the column -Total number of events.-

8. Allow time for students to calculate and record the -Number of 'wins' for that event- and the -Total number of events- for each event of the die, rolling a two, three, four, five, and six, respectively. When the students have completed this task, then discuss why some numbers had more -wins- than others did. Explain that mathematicians have determined that even though some numbers in our experiment may have had more -wins- than other numbers, there is a certain likelihood of rolling each number based on lots and lots of experiments with rolling dice. Also, the more we roll the die, the closer we would get to this probability.

9. Tell students that they are ready to examine that probability. Ask, -How many different outcomes or possibilities are there when you roll a die?- (6) Show this as the denominator in fractional form in the appropriate column on the -Analyzing Data Chart and Record Sheet.- Ask, -How many different ways can you roll a one?- (1) Show this as the numerator in fractional form. (1/6) Record this ratio in the -Mathematical probability in fraction form- column on the overhead. State that this is the ratio or likelihood for rolling any one number on a standard die, 1/6 or one out of six probability.

10. Ask students, -If the likelihood of rolling a one on a standard die is 1/6 or one out of six probability, what would the likelihood be for rolling any other number on a standard die?- Allow groups to discuss what the likelihood of rolling each of the other numbers on the die would be and chart it on their Analyzing Data Record Sheet.

11. Talk about how to represent the likelihood for rolling numbers on a die with a model. A model in this case is a visual representation of the likelihood of each of the events. Ask students how we could show this fraction (1/6) in a model or picture? As an example, draw a picture or model, such as a circle graph divided into one-sixth sections, to also represent the ratio in fraction form. Color one of the sections. Read the ratio as, -The likelihood of rolling a one on a standard die is a one out of six probability or chance.- Write the ratio in fraction form in the -Mathematical probability in fraction form- column.

12. Ask, -If the likelihood of rolling any one number on a standard die is one out of six, what number do you predict you will roll the next time you roll a die?- State that knowing the likelihood of an event helps you to make better predictions about the outcome of that event. This also helps you make better decisions. Allow students time to record models and predictions based upon what they know the likelihood is for rolling each number of the die in the appropriate colomns on their Analyzing Data Record Sheet.

Session 2
1. Review the concepts from the previous lesson by asking questions about: likelihood, events, nuumber of wins for events, mathematical probability in fraction form, models, predictions.
2. Introduce the probability investigation stations for students to explore the concept of likelihood expressed as a ratio in fraction form with other events. They are as follows:
a) two standard dice,
b) nonstandard dice,
c) spinners and results from previous class experiments,
d) coins and results from previous class experiments,
e) lottery - strips of paper in a bag with a different person's name on each strip,
f) a bag of candy with different colored pieces.

Model as necessary until your students are ready to investigate the probability stations in small groups more independently. In their investigations, students should decide on the particular event for an experiment and record it in the first column. Then they should decide on a ratio in fraction form to express the mathematical probability, draw a model to display the possible outcomes, and make predictions about the event. Students could also refer back to previous experiments and analyze the data they have collected thus far from the coin toss and spinner experiments. Students record their findings on the Analyzing Data Record Sheet or in their Data Diaries on their self-made chart.

3. Continue the station work (if possible in another session ) until students explore each one.

#### Assessments

After exploring all of the stations, students compose a -Letter to a Mathematician.- The notes should include examples of their observations about likelihood based upon one or more events from the stations. The note should address the following:
-tell how they solved problems about likelihood with data that was graphed
-explain the model they used to display possible outcomes
-predict the likelihood of future events based upon the data
-justify their predictions based upon the data

Assess the content of the -Notes to the Mathematician- using the Product Rubric in the Associated File. (NOTE: This assessment is formative. The purpose of the note is to reflect comprehension of content and experiences like a professional might write to himself. The intent is not to assess the student's letter-writing format. If you decide you want students to write a more formal or friendly letter, you would need to teach that skill in this lesson.)

(This is a good opportunity to involve people in your community. Collaborate with a parent or community member that is a university professor, high school math teacher, or other professional from a lab or company in your community to come to your class and facilitate further investigations into probability and likelihood. This gives students many pictures of how math works in the working world. Furthermore, if the students have a real audience for their notes, they will be more motivated to improve the quality of their work.)

#### Extensions

This lesson is the second in a two-part series of lessons that examine likelihood. The first, Drawing Bugs Game (see Weblinks) provides ample concrete opportunities for students to collect data which is necessary before they express likelihood as a ratio in fraction form as presented in this lesson.

Web supplement for Notes to a Mathematician
Pin the Tail on the Tiger

Web supplement for Notes to a Mathematician
Lions and Tigers and Probability, Oh My!

Web supplement for Notes to a Mathematician
Chances Are…

Web supplement for Notes to a Mathematician
Drawing Bugs Game