D2. Probability
Describe the likelihood that events will happen, and use that information to make predictions.
Learning Situation 1: Planning Board or Spinner?
Total duration: approximately 3.5 hours
Summary
Students are asked to predict the number of times their favourite daily physical activity will be chosen over a 20-day period, after which they perform a simple probability experiment using a spinner and record the results in a table.
| Overall Expectations | Specific Expectations |
|---|---|
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D1. Data Literacy Manage, analyse, and use data to make convincing arguments and informed decisions, in various contexts drawn from real life. |
D1.2 Collect data through observation, experimentation, and interviews to answer questions of interest that focus on qualitative and quantitative data and organize the data using frequency tables. D1.3 Display sets of data, using many-to-one correspondence, in pictographs and bar graphs with proper sources, titles, and labels, and appropriate scales. D1.5 Analyze different sets of data presented in various ways, including in frequency tables and in graphs with different scales, by asking and answering questions about the data and drawing conclusions, then make convincing arguments and informed decisions. |
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D2. Probability Describe the probability of events occurring and use this information to make predictions. |
D2.1 Use mathematical language, including the terms “impossible”, “unlikely”, “equally likely”, “likely”, and “certain”, to describe the likelihood of events happening, and use that likelihood to make predictions and informed decisions. |
Learning Goals
The purpose of this learning situation is to enable students to:
- develop an intuitive understanding of the concepts of chance and probability;
- perform a simple probability experiment and interpret the results;
- use appropriate vocabulary to describe the probability of an outcome.
| Learning Context | Prerequisites |
|---|---|
|
In Grade 3, students can be supported to develop their probabilistic thinking through simple experiments. In this activity, students develop an understanding of the concept of chance by comparing a situation where the outcomes are known with a situation where the outcomes are random. They use their intuitive understanding of probability to predict the number of times their favourite physical activity will be chosen over a 20-day period, and compare their prediction to the results of a probability experiment. |
To be able to complete this learning situation, students must be able to:
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Mathematical Vocabulary
event, outcome, frequency table, prediction, impossible, unlikely, equally likely, very likely and certain
- empty calendar
- Appendix 3.2
- Appendices 3.3 and 3.4 (1 copy per team)
- paper clips (1 per team)
- thumbtacks or brads or pencils (1 per team)
- markers
- cards (1 per team)
Preparatory Activity
Duration: approximately 75 minutes
As a class, facilitate a discussion about the importance of daily physical activity by asking questions such as:
- Why is it important to be physically active every day? (To stay healthy.)
- What parts of the body benefit from these physical activities? (Heart, lungs, bones, muscles, etc.)
- Do you think physical activity helps us function better in the classroom? (Yes, because physical activity helps us feel good, stay awake and pay attention.)
If necessary, invite students to do research on the Internet to determine the benefits of daily physical activity.
Next, tell students that over the next five days they will have the opportunity to participate in five different physical activities (see Appendix 3.2) and then they will have to decide which one they like best. Introduce them to the first activity and give them the opportunity to practice it. Do the same with the other activities over the next four days. Once students have had a chance to participate in each activity, conduct a survey to determine which one they like best. Record the results of the survey in a frequency table.
Ask students to represent the data from the frequency table in a bar graph in a one-to-one or many-to-one correspondence. Then ask them to determine what decision the class should make to reflect the results of the survey (for example, We should do the Active Role Model activity more often, because it is the activity that the majority of students prefer. Also, if you do an activity that you like, you will do it more often). Display the frequency table and bar graph prominently.
Before Learning (Warm-Up)
Duration: approximately 30 minutes
Prepare an empty calendar representing four weeks of class. Glue the symbol of the students' favourite physical activity (for example, Active Role Models) in the four boxes corresponding to Wednesdays.
Display the empty calendar and introduce students to the following situation:
I would like to plan the daily physical activities that we will do for the next four weeks. Since the survey showed that Active Role Models is the daily physical activity that the majority of students prefer, I could schedule it every Wednesday. This would ensure that we do it once a week. On the other days, I would choose one of the other four activities. However, there may be other ways to go about planning daily physical activities over a four-week period. What do you think? Do you have suggestions?
If they suggest using a daily vote, tell them that they may always choose the activity preferred by the majority and that students who prefer another activity will always be disappointed.
Ask students to identify which of the suggested methods involve chance (for example, raffle, spinner) and point out that these methods also allow for less popular activities to be chosen.
Show students a spinner with all five activities represented on it (see Appendix 3.4). Ask them if the spinner is a fair way to determine which activity will be chosen. (Yes, because the needle can land on any of the five sectors of the wheel and all sectors are the same size.)
To encourage students to compare the use of the planning board and the use of the spinner as a means of choosing the daily physical activity they will do, ask the following questions:
- If I plan using the table, how many times will we practice the preferred physical activity Active Role Models in a 5-day period? in a 20-day period? (We will practice this activity 1 time per 5 days and 4 times per 20 days.)
- If I do the planning using the spinner, how many times will we do the preferred physical activity Active Role Models in a 5-day period? in a 20-day period? Explain your reasoning.
Ask students to discuss this last question with another student for a few minutes. Then lead a whole class discussion to emphasize the fact that when using a spinner, the outcome depends on chance and that the second question cannot be answered precisely as the first question was. Encourage students to predict the outcome after 5 days and then after 20 days (using their intuitive understanding of probability).
Write some students' predictions on the board and ask them to explain them. Examples of possible predictions include:
- One student predicts "3 out of 5 days" because she thinks the activity is "lucky"; she adds that it could not be "5 times out of 5 days" because there are other activities.
- One student predicts "1 in 5 days" and says that all 5 activities on the spinner have the same chance of being chosen because the sectors are of equal size.
- One student predicts "2 out of 5 days", noting that it is more likely that the needle will land on the other activities than on the preferred activity because there are 4 other sectors.
Active Learning (Exploration)
Duration: approximately 45 minutes
Group the students. Explain that they will be conducting a simple experiment to see how many times the spinner will land on Active Role Models for 5 trials, and again for 20 trials.
Distribute to each team a copy of Appendices 3.3 and 3.4, a piece of cardboard, a paper clip (for the needle) and a thumbtack or pencil. Read the instructions in Appendix 3.3 and ensure that students understand the task at hand. If necessary, explain how the spinner works. Ask students to complete the task, suggesting that they take turns spinning the needle.
Circulate among the teams, observe their work and ask questions such as:
- Why did you make this prediction?
- Would you say it is impossible, unlikely, equally likely, very likely, or certain that the needle will land on the sector corresponding to the preferred activity?
- Would you say it is impossible, unlikely, equally likely, very likely, or certain that the needle will land on a sector other than the preferred activity?
- Are the results similar to your predictions?
- Did the results after 5 trials influence your prediction of the results for the next 20 trials? Why or why not?
- Would you say it is impossible, unlikely, equally likely, very likely, or certain that the needle will land on a sector corresponding to the preferred activity more often during 20 trials than during 5 trials?
Consolidation of Learning
Duration: approximately 60 minutes
Prepare a table on a large sheet of paper or on an interactive board for teams to record the number of times they obtained the preferred physical activity for each of the four weeks. Ask each team to take turns recording their results.
Using the collective results on the board, encourage students to analyze the data by asking them questions that address each of the three levels of comprehension.
Reading Data (Level 1)
- What is the most number of times a team landed on the preferred physical activity? (10)
- What is the least number of times a team landed on the preferred physical activity? (1)
Reading Between the Data (Level 2)
- Why do you think the results differ so much from one team to another? (The results depend on chance.)
- How do the results compare to the predictions?
- How many teams obtained a result with the spinner that would allow us to do our favourite activity more often (less often) than we would if we used the planning board? (More often: 3 teams. Less often: 3 teams.)
Reading Beyond the Data (Level 3)
- Based on the results presented in this table, would you say that using the spinner, it is impossible, unlikely, equally likely, very likely, or certain that our preferred activity would occur more often than the other activities? Why? (Since 5 of the 6 teams got the preferred activity fewer times than the other activities, in other words, less than 10 times, we could say that it is unlikely to get the preferred activity more often than the other activities)
- Based on the results obtained, would it be better to use the planning board or the spinner to decide what physical activity to do each day? Why? Answers may vary, depending on the results obtained by the students and whether they prefer the certainty of using the planning board or the risk of using the spinner. With the planning board, students are certain to do their preferred physical activity exactly once a week. With the spinner, they have a 1 in 5 chance of doing it every day. In terms of probability, this corresponds to once a week, but in reality, it may be less than or more than once a week.
Differentiated Instruction
The activity can be modified to meet the needs of the students.
| To Facilitate the Task | To Enrich the Task |
|---|---|
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Follow-Up at Home
At home, students can build a spinner that would allow them to choose, for example:
- a daily chore around the home;
- a snack to be eaten after school;
- an activity to practice on the weekend.
They must then perform 20 trials with the spinner, record the results in a table and present the results to the class.
Source: translated from Guide d’enseignement efficace des mathématiques, de la maternelle à la 3e année, Traitement des données et probabilité, p. 263-270.
Learning Situation 2: We Need a Name for Our Mascot!
Total duration: approximately 1.5 hours
Summary
In this learning situation, students conduct a probability experiment using a spinner to choose, from among five selected names, the name of the stuffed animal that is or will become the class mascot (or reading corner, art corner, etcetera).
| Overall Expectations | Specific Expectations |
|---|---|
|
D1. Data Literacy Manage, analyse, and use data to make convincing arguments and informed decisions, in various contexts drawn from real life. |
D1.2 Collect data through observation, experimentation, and interviews to answer questions of interest that focus on qualitative and quantitative data and organize the data using frequency tables. |
|
D2. Probability Describe the likelihood that events will happen, and use that information to make predictions. |
D2.1 Use mathematical language, including the terms “impossible”, “unlikely”, “equally likely”, “likely”, and “certain”, to describe the likelihood of events happening, and use that likelihood to make predictions and informed decisions. |
Learning Goals
The purpose of this learning situation is to help students develop an intuitive understanding of the concept of probability:
- to say whether the probability of events occurring is "impossible", "unlikely", "equally likely", "very likely" or "certain";
- to predict the frequency of the different possible outcomes of the experiment;
- to compare the results of the experiment to their predictions.
| Learning Context | Prerequisites |
|---|---|
|
From Kindergarten to Grade 2, students have described the extent to which they believe an event can occur by using certainty words (impossible, possible, and certain). They have used visual models (frequency line) to place these events on a continuum, which helps them to understand that some events have a greater probability of occurring than others. In this learning situation, Grade 3 students will use their intuitive understanding of probability to predict different possible outcomes for a simple experiment. They then perform the experiment and compare the probability of each outcome to their prediction using terms such as impossible, unlikely, equally likely, very likely, or certain. |
To be able to complete this learning situation, students must:
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Mathematical Vocabulary
data collection, frequency table, tally table, impossible, unlikely, equally likely, very likely, certain, results, probability, probability line
Materials
- mascot
- Appendix 2.1 (1 copy per team)
- cardboard (1 per team)
- scissors (1 per team)
- glue sticks (1 per team)
- paper clips (1 per team)
- pencils or brads (1 per team)
- sheets of paper
- large sheets of paper (2)
- Appendix 2.2 (1 table per student)
Before Learning (Warm-Up)
Duration: approximately 30 minutes
Have students choose a mascot (for example, stuffed animal) for the classroom (or reading corner, art corner, etc.). Once the mascot has been chosen, tell them they need to give it a name. Write the list of names suggested by the students on the board. Select five of these names with them.
Write the five names on a large wheel divided into five equal sectors. Group the students in a circle, place the wheel in the center and present the following situation:
I thought of an experiment that would help us choose which of the five names we would give to our mascot. I placed the five names on this spinner. We could spin the spinner 20 times, record the results and give the mascot the name that comes up the most.
Before exploring, encourage students to think about the probability of certain outcomes by asking questions such as:
- In your opinion, is it impossible, unlikely, equally likely, very likely , or certain that the needle will land 20 times on the same name? Why? (In my opinion, it is impossible that the needle will land 20 times on the same name because there are five names that it can land on and you never know which one it will be.)
- Is it impossible, unlikely, equally likely, very likely, or certain that the needle will land at each of the names at least once? Why? (I think it's equally likely because all five sectors of the spinner are the same size, so the needle will eventually land at each one)
- Is it impossible, unlikely, equally likely, very likely, or certain that the needle will never land on the name Mrs. Tomato? Why? (It is impossible because the needls, after several turns, will eventually land on the name Mrs. Tomato)
Following each question, ask a student to indicate on a probability line the location that corresponds to their answer choice.
The justifications offered by students reflect the intuitive nature of their understanding of the concept of probability, and it is important at this stage of their learning to let them express themselves without placing too much emphasis on the mathematical rigor of the argument presented.
Active Learning (Exploration)
Duration: approximately 45 minutes
Have students try the experiment. Group them in pairs and give each team a copy of Appendix 2.1(spinner), a cardboard, a pair of scissors, a glue stick, a paper clip (the needle) and a pencil or brad clip. Have them cut out the spinner wheel, glue it to the cardboard, and write one of the five names in each sector of the wheel.
If necessary, explain to students how to install a spinner on the wheel, either by placing the tip of a pencil in the center of the wheel and inside the bend of the paper clip or by securing the paper clip with a brad clip. Allow them some time to try to spin the needle, then tell them that each team will experiment with the game and that the results will be compared when all teams have finished.
Tell students that before they begin, they should predict the number of times the needle will land on each sector and record these predictions on a sheet of paper. Remind them that the total number of results must equal 20. Circulate and observe the responses of the different teams to see how well students are using probabilistic thinking. If necessary, ask a team to explain their predictions, but do not encourage them to change them.
Some students may tend to predict a higher frequency for the name they prefer. Others may simply guess at random. Finally, some students may use their intuitive understanding of probability and predict a roughly equal distribution of frequencies among the five names.
When all teams have recorded their predictions, have them complete the 20 trials and record the results. Let them use whatever method they choose to record these results. Point out that if the needle lands on the line between two sectors, they should simply spin it again. Circulate, observe and intervene as needed.
| Possible Observations | Possible Interventions |
|---|---|
| A student tries to "control" the strength of his flick to get the name he prefers. | Remind the student that the purpose of the experiment is to let chance determine the name of the mascot. |
| A team does not know how to record its results. | Encourage these students to think about the different methods of recording data that have been used previously in data activities. |
| A team makes more than 20 attempts. | Remind them that no more than 20 trials should be done and ask them to remove the data for the extra trials. |
Then ask each team:
- to use the results of the experiment to indicate which name should be given to the mascot;
- to compare the results obtained with their predictions.
Consolidation of Learning
Duration: approximately 20 minutes
Ask a few teams to present their results. Facilitate the mathematical exchange by asking questions such as:
- What do you think the mascot should be called? Why? (We should name it Rikiki because that is the name that has come up the most.)
- If you repeated the experiment, do you think your results would be the same? Why? (Probably not, because the results depend on chance.)
- Are your results similar to your predictions? Explain your answer.
- What do you think the distribution of frequencies among the five names should look like? Why? (Because the five sectors of the spinner are the same size, the needle should land about the same number of times on each sector, so there should be a roughly equal distribution of frequencies among the five names.)
- Several teams did not have a very even distribution of frequencies among the five names. How can this be explained? (Since the results depend on chance, uneven distributions can also occur. Perhaps the number of trials should be increased).
Suggest that students combine the results of all the teams to see what the frequency distribution is among the five names. Take this opportunity to model the use of a table as a method of recording data. Prepare a large frequency table like the one in Appendix 2.2. Ask each team to tell you the frequency they obtained for each name and model how to record this data in the table using tallies. Ask students to record the data at the same time in the frequency table that you will have distributed (Appendix 2.2). Then determine the frequencies for each name and record these data in the table.
Review with students the conventions of the frequency table (title, column or row designation, categories, tally, and frequency within each category). If necessary, compare this table to some of the other recording methods used by the teams in the exploration. Then encourage students to analyze the data, asking questions such as:
- Is the frequency distribution among the five names fairly even?
- What do you think would happen if we performed the spinner experiment 10 times?
Note: The goal is to get students to recognize intuitively that in any chance-related situation, the more trials one has, the more one can expect to get an equal distribution of frequencies among the possible outcomes. Of course, this is true as long as those outcomes are equally likely. Students will need several such activities over the years to fully understand this idea.
Point out to students that according to the cumulative results of the experiment, such and such a name is the one with the highest frequency. Tell them that they now have to make a decision, in other words, they have to decide which name they want to give the mascot. Examples of answers students might give include:
- The spinner has decided. You have to choose the name that comes up most often.
- We can repeat the experiment to get more results and see if the name will be different.
- The name can be chosen in other ways; for example, you can indicate the name you prefer by a show of hands and then choose the name that corresponds to the majority's choice.
Decide with students what the mascot will be called and place it in an appropriate location in the classroom.
Differentiated Instruction
The activity can be modified to meet different student needs.
| To Facilitate the Task | To Enrich the Task |
|---|---|
|
At the beginning of the exploration, provide students with a copy of the frequency table in Appendix 2.2 (p. 230). |
Ask students to represent their data in a horizontal or vertical bar graph. |
Follow-Up at Home
At home, students propose four activities they would like to do with their parents this weekend. Since they must choose only one, they design a probability experiment using materials of their choice to randomly determine which one will be chosen. Specify that the experiment must include a minimum of 20 trials.
For example, a student may choose to use:
- a red cube to represent going to a restaurant;
- a blue cube to represent watching a movie;
- an orange cube to represent playing a computer game;
- a yellow cube to represent a visit to his grandfather.
The student then puts the cubes in a bag, picks one and records the colour in a table. They put the cube back in the bag and repeats the experiment at least 19 more times, then determines the frequency for each colour. As a class, students can explain what means they used to ensure that the choice of activity was random.
Source: translated from Guide d’enseignement efficace des mathématiques, de la maternelle à la 3e année, Traitement des données et probabilité, p. 217-225.