D2. Probability:
Describe the likelihood that events will happen, and use that information to make predictions.
Learning Situation: Impossible, Possible or Certain?
Total duration: about 3 hours
Summary
Students develop an understanding of the words impossible, possible, and certain by using them to describe the probability of events occurring.
| Overall Expectation | Specific Expectation |
|---|---|
| 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”, “possible”, 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:
- classify everyday activities according to the likelihood of events occurring (impossible, possible and certain);
- to understand that the classification of events can vary depending on the person who experiences them and the interpretation they give to the words.
| Learning Context | Prerequisites |
|---|---|
|
In kindergarten students learn to use the words never, sometimes and always (certain, possible and impossible) and likely/unlikely to informally describe the likelihood of any event occurring. In Grade 1, they begin to develop an intuitive understanding of the concept of probability by expressing the likelihood that an event may occur. Students learn to describe this likelihood using the words certain, possible and impossible. Using a probability line helps them to better understand the difference between these words. |
To be able to complete this learning situation, students must:
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Materials
- large sheets of paper (for example, chart paper)
- labels impossible, possible and certain
- blank sheets of paper (one quarter sheet per student)
- masking tape or rope
- stickers (two per student)
- large sheets (one per student)
- glue
Mathematical Vocabulary
certain, possible, impossible, probability, probability line, pictograph
Before Learning
Duration approximately 90 minutes
Ask students to describe what they have done since they woke up that morning (for example, brushing their teeth, eating, getting dressed, taking the bus). Record these activities on a large sheet of paper (for example, chart paper). Then ask them to list some of the activities they do during the day (for example, going to recess, doing math), and then some of the activities they do after school and in the evening (for example, going to daycare, playing sports). Record these activities on other pieces of chart paper or other large sheets of paper.
Prepare a label for the words certain, possible , and impossible. Select three students and give each student one of the labels. Invite them to take turns placing the label next to one of the sentences that describes an activity they do in a typical week according to the probability on the label. Then ask them to read the sentence aloud, adding the likelihood that the event will occur (for example, "It's possible that my parents will pick me up after school.") Encourage students to clarify the meaning of the term used by asking questions such as:
- You say that it is possible that your parents will pick you up after school. What does the word possible mean? (The student might say that his or her parents pick him or her up on occasions, so it is possible that they will pick him or her up today)
- You say it is certain that the sun will rise tomorrow. What does the word certain mean? (The student might say that the sun rises every day, so it is certain that it will rise.)
Then ask two students to read the same sentence but have each choose a different word to describe the likelihood of the activity occurring in a typical week. Have them take turns reading the sentence aloud with the chosen word and justifying their choice. Assist them, if necessary, by asking questions such as:
- Why do you say it’s possible for you to play at your friend's house tonight?
- Why do you say it's impossible for you to play at your friend's house tonight?
Distribute a quarter of a sheet of white paper to each student and ask them to illustrate an activity they do during the week and then describe it in one sentence. Be sure to have drawings of different activities.
Active Learning (Exploration)
Duration: approximately 60 minutes
Group Exploration
Use masking tape or string to represent a probability line. Place the impossible label at the left end of the line and the certain label at the right end. Then place the possible label in the middle. Ask students to sit in front of the probability line. Choose a student drawing that is likely to generate a variety of responses regarding the likelihood of this event occurring. Follow this procedure:
- Read the descriptive sentence (for example, "I'm going to gym class").
- Emphasizing probability terms, ask a student the following question, "Is it certain, possible , or impossible that you will go to gym class?"
- Ask the student to sit in front of the word on the probability line that corresponds to their answer and then explain the choice (for example, "I say it’s possible to go to gym class today because I go almost every day of the week with Mrs. Michelle").
- Ask other students to sit in front of the word on the probability line that corresponds to their answer and explain their choice.
| Possible observation | Possible intervention |
|
A student sits in front of the word impossible, but the teacher knows that students go to gym class a few days a week. |
Ask the student to explain why he or she says it is impossible for him or her to go to the gym. The student responds, "I don't have gym class." Then ask the student if he or she goes to the school gym a few times a week. Following this question, the student will most likely change his or her seat and justify the change by saying, "I was sitting in front of the word impossible because I forgot I had gym class a few times a week. I have to sit in front of the word possible instead." |
- Finally, to help students move to a more abstract representation, draw a probability line on the board. Give each student a sticker. Have them place it on this line at the location that represents their position along the probability line on the floor.
Note: This activity can also be done using the interactive whiteboard.
Facilitate a discussion by asking questions such as:
- Why do your answers to the same question differ?
- Why do some students say it is certain to happen as opposed to others who say it is possible or impossible for them to go to the gym today? (Some students are familiar with the week's schedule as opposed to other students.)
Then ask students to return to their seats in front of the probability line, select another drawing that might encourage different responses, and repeat the process above.
Encourage students to compare this probability line with the one corresponding to the previous drawing by asking questions such as:
- Why is the distribution of the stickers on the two probability lines different? (They correspond to different activities.)
- What sport do students in the class most often play in a typical week? (Many students play soccer since there are 17 stickers on the probability line for the possible and certain categories and only two stickers for the impossible category.)
- What is similar about the results on the two probability lines? (In both cases, four students indicated that it is possible that they would do the activity.)
Individual Exploration
Duration: approximately 20 minutes
Choose five drawings and make copies. Give each student the five drawings and a large piece of paper. Have them draw a probability line on the paper and write the words certain, possible and impossible in the appropriate places. Then have them glue each drawing above the word that best describes the likelihood of the activity occurring in a typical week. Circulate and ask questions such as:
- Why did you put this picture above the word certain? the word impossible?
- Does the word possible mean the same as the word certain? What is the difference?
Consolidation of Learning
Duration: approximately 20 minutes
Display the probability lines on the board. Encourage students to compare them by asking questions such as:
- Why is the same drawing not in the same place on all the probability lines? (Because the likelihood of an activity occurring is not the same for all students)
- Is there an activity that students are certain to do? (No, because several students did not place a drawing above the word certain)
- If you were to classify these activities again tomorrow, would your answers be the same? (Yes. I say it is certain to go to gym class because I go every day of the week and I would say the same thing tomorrow because the situation would be the same)
- If another group of students describe the likelihood of these activities occuring, would their answers be the same? (Maybe not because there may not be anyone in that group who goes to gym class every day of the week like I do)
Extension
Have students choose one of the five activities represented on the probability lines on the board and create a pictograph to represent the data for that activity for the entire class.
Differentiated Instruction
The activity can be modified to meet different student needs.
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To Facilitate the Task |
To Enrich the Task |
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Ask students to categorize the activities into two categories, such as:
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Follow-Up at Home
At home, students can create a probability line and place some of the family activities they have completed over the course of a week.
Source: translated from Guide d’enseignement efficace des mathématiques, de la maternelle à la 3e année, Traitement des données et probabilité, p. 181-189.