D2.1 Use mathematical language, including the terms “impossible”, “possible”, and “certain”, to describe the likelihood of complementary events happening, and use that likelihood to make predictions and informed decisions.
Skill: Using Mathematical Vocabulary to Express the Likelihood of an Event Happening
In the primary grades, probabilistic thinking is developed daily through the learning of certain words or expressions that describe the frequency or likelihood of certain events, for example:
- We always have Saturday and Sunday off from school.
- It's possible that I play ball at recess
- It is unlikely to rain today.
Source: translated from Guide d’enseignement efficace des mathématiques, de la 4e à la 6e année, Traitement des données et probabilité, p. 128.
In the primary grades, one of the first manifestations of the meaning of variability occurs when pupils are able to indicate whether it is certain, possible or impossible that a given event will occur (for example, It is impossible that cows begin to fly). Some young students find it very difficult to do so. For many, what is possible becomes certain. For example, if the teacher tells them that it is possible that they will bring their dog to class the next day, the students will be very disappointed if this does not happen, since in their minds, it is certain that the teacher will bring it. Students need to understand that if an event is possible, it is as likely to happen as not happen. Other students have difficulty distinguishing between what has never happened and what may never happen. For these students, if an event has never happened, it is impossible. For example, students may think it's impossible for them to go a whole night without sleep because they've never done it. However, such an event is not impossible.
Source: translated from Guide d’enseignement efficace des mathématiques, de la 4e à la 6e année, Traitement des données et probabilité, p. 132-133.
Skill: Making Predictions and Informed Decisions
According to the big ideas Collection and Organization of Data and Data Relationships, the recommended problem-solving approach uses the inquiry process. This approach fosters critical probabilistic thinking by encouraging students to formulate conclusions from data collected in probability experiments and to question their intuition about the probability of one of the outcomes. Such an approach helps to avoid the development of some of the misconceptions about probability that are too often found in both students and adults.
Source: translated from Guide d’enseignement efficace des mathématiques, de la 4e à la 6e année, Traitement des données et probabilité, p. 128.
Knowledge of concepts related to probability helps students better understand all kinds of everyday situations, such as understanding weather forecasts, the possible outcomes of an experiment, the probability of winning at the time of a drawing or game. The big idea Probability emphasizes the importance of probabilistic thinking in informing decision making in situations where the outcome is uncertain due to the fact that it is related to chance.
Source: translated from Guide d’enseignement efficace des mathématiques, de la 4e à la 6e année, Traitement des données et probabilité, p. 127.
Concept of Probability
Probability describes the degree of certainty with which a particular outcome or event can be predicted to occur in a situation of variability. Developing an understanding of the concept of probability is a long process that begins in the early grades. However, it is very important to build on an intuitive understanding of the concept first. In the primary grades, students are able to recognize intuitively that, in a given situation, certain events are possible and that the degree of certainty that they will occur lies on a continuum from impossible to certain. The probability line model is an effective visual means of describing this continuum.
Students place a point on this continuum to describe how certain they are that a particular event will occur. The further to the right of the line, the more certain they are. If students place the point in the middle, it means that they believe that the event has as much chance of occurring as not.
As students gain experience in solving situations of variability, some tend to develop misconceptions. One of the most common misconceptions relates to the imposition of implicit limits on variability. For example, in a game of heads or tails, students get tails five times in a row. If they are then asked to predict the result of the next toss, many will tend to indicate that it will certainly correspond to the heads side because their intuition leads them to believe that there is a limit to chance favouring the tails side. Teachers should help them understand that chance has no memory, that is, each result is independent of previous results and cannot be predicted with certainty. It is therefore just as likely to get tails as to get heads on the next roll. This kind of misunderstanding persists among many adults who believe, for example, that certain numbers have a better chance of being part of the winning lottery numbers simply because they have come up more often in the past. A good understanding of the concept of variability makes it possible to recognize that this is not the case. To help students fully grasp this concept, teachers should introduce them to many simple probability experiments and encourage them to question themselves and remain objective with regard to their intuition.
Source: translated from Guide d’enseignement efficace des mathématiques, de la 4e à la 6e année, Traitement des données et probabilité, p. 133-134.
Knowledge: Impossible, Possible and Certain
Image Vertical bar graph titled Favourite Activities of Grade 2 Students Before Bedtime. The vertical axis, Number of Students, is numbered from 0 to 30 in increments of five.
The horizontal axis, Favourite Activities Before Bedtime, is composed of six categories: Read a book, with 13 students, Watch television, with 28 students, Play video games, with 26 students, Play board games, with 11 students, Taking a relaxing bath, with 9 students, and Other, with 7 students.
- The words impossible and certain describe the two ends of the continuum representing the possibility of an event or outcome occurring. They are fixed points that serve as anchors.
- An impossible event is an event that can never happen. A certain event is an event that will always happen.
Note: The inverse of these two propositions are not necessarily true. Thus, in the context of an experiment, even if a result never occurred, it does not mean that it is impossible for it to occur. Similarly, the fact that one always obtains some result after a certain number of trials does not necessarily imply that one is certain to obtain it after all trials.
- The word possible describes all events or outcomes that are neither certain nor impossible.
- The location of an event or outcome on the probability line reflects the assessment of the likelihood that the event or outcome could occur.
- An event or outcome that is located in the center of the probability line is as likely to occur as not.
Source: translated from Guide d’enseignement efficace des mathématiques, de la maternelle à la 3e année, Traitement des données et probabilité, p. 25.
Knowledge: Complementary Events
Complementary events are events that cannot happen at the same time. For example, if the likelihood of selecting a red marble from of a bag is certain, then its complement of not selecting a red marble from of a bag is impossible.
Source: The Ontario Curriculum, Mathematics, Grades 1-8 Ontario Ministry of Education, 2020.