Mathematics · Grade 5 · Probability · Term 3

The Language of Likelihood

Let's learn the words we use to talk about chance, from events that are impossible to those that are certain to happen.

TL;DR:Let's become smart guessers! This topic explores how we can use math to predict the future, moving beyond simple luck to understand the language of likelihood.

• Ontario Curriculum Expectations
• Ontario Curriculum: Grade 5 Mathematics - Data - D2. Probability: Describe the likelihood of events happening using mathematical language.

About This Topic

This topic, 'The Language of Likelihood,' aligns with Canadian provincial curricula, such as the Ontario Mathematics curriculum's Data strand and British Columbia's focus on Financial Literacy and Chance. For Grade 5 students, this unit represents a significant step from using qualitative descriptors of chance (e.g., likely, impossible) to applying quantitative measures. The core of this topic is understanding and differentiating between theoretical and experimental probability. Students will learn to represent the probability of a single event as a fraction, connecting their knowledge of fractions to real-world scenarios involving uncertainty.

The instructional focus should be on hands-on experimentation. By conducting trials with spinners, dice, or coloured counters, students discover that experimental results can vary from theoretical predictions, especially with a small number of trials. This exploration builds a foundational understanding of randomness and the law of large numbers: the more trials you conduct, the closer your experimental results will get to the theoretical probability. This topic is crucial for developing critical thinking and data literacy skills, enabling students to make informed predictions and analyze situations involving chance in their everyday lives, from games to weather forecasts.

Key Questions

1. Identify an event that is impossible, unlikely, has an even chance, is likely, and is certain to happen today.
2. Explain why the chance of rolling a 7 on a standard six-sided die is impossible.
3. Compare the likelihood of picking a red marble from a bag with 5 red and 1 blue marble versus a bag with 1 red and 5 blue marbles.

Learning Objectives

• Represent the probability of a single event using a fraction.
• Differentiate between theoretical probability and experimental probability.
• Conduct simple probability experiments to gather data.
• Make and justify predictions about outcomes using both theoretical and experimental data.
• Describe the likelihood of events using language such as 'impossible,' 'unlikely,' 'equally likely,' 'likely,' and 'certain.'

Key Vocabulary

Probability	The measure of the likelihood that an event will occur, often expressed as a fraction or percentage.
Outcome	A possible result of a probability experiment.
Trial	A single performance of a probability experiment, like one flip of a coin or one roll of a die.
Theoretical Probability	The probability of an event calculated by comparing the number of favourable outcomes to the total number of possible outcomes.
Experimental Probability	The probability of an event determined by the results of an experiment, calculated as the number of times an event occurs divided by the total number of trials.

Watch Out for These Misconceptions

Common Misconception

The gambler's fallacy: If a coin lands on heads five times in a row, it is more likely to land on tails next.

What to Teach Instead

Each coin flip is an independent event. The coin has no memory of past results, so the probability of getting heads or tails is always 1/2 for a fair coin.

Common Misconception

All outcomes are equally likely, regardless of the setup. For example, in a bag with 7 green and 3 blue candies, the chance of picking either colour is the same.

What to Teach Instead

Probability depends on the number of favourable outcomes compared to the total number of outcomes. Since there are more green candies, the probability of picking a green one is higher (7/10) than picking a blue one (3/10).

Common Misconception

Experimental results must exactly match the theoretical probability.

What to Teach Instead

Theoretical probability describes what is expected to happen over a very large number of trials. In a small experiment, results can vary significantly due to chance.

Active Learning Ideas

Four Corners

Spinner Challenge

Students use paper clips and pencils to create spinners with unequal sections (e.g., 1/2 red, 1/4 blue, 1/4 yellow). They first calculate the theoretical probability of landing on each colour and then conduct 50 spins to find the experimental probability, comparing the two sets of results.

45 min·Pairs

Four Corners

Mystery Bag Predictions

Prepare several opaque bags with a known quantity of different coloured cubes or counters. Students calculate the theoretical probability of drawing each colour from a bag, then conduct 20 trials (replacing the cube each time) to test their prediction.

30 min·Small Groups

Four Corners

Dice Roll Tally

Students predict how many times each number will be rolled if they roll a standard six-sided die 60 times. They then perform the experiment, tally the results, and write a reflection comparing their prediction to the actual outcome.

40 min·Individual

Real-World Connections

• Weather forecasting, where meteorologists state the 'percent chance of precipitation.'
• Playing board games or card games that involve dice or drawing cards randomly.
• Sports statistics, such as a player's batting average in baseball or free-throw percentage in basketball.
• Quality control in manufacturing, where a sample of products is tested to predict the defect rate of the whole batch.
• Making everyday decisions, like choosing a checkout line at the grocery store based on a quick guess of which one will be fastest.

Assessment Ideas

Exit Ticket

Exit Ticket: Show students a picture of a spinner with unequal sections. Ask them to identify which colour is most likely to be spun and to write the probability of landing on that colour as a fraction.

Peer Assessment

Design a Game: Students create a simple game of chance using a die or a spinner. They must write out the rules, calculate the theoretical probability of winning, and then have classmates play the game to determine the experimental probability.

Quick Check

Confidence Checklist: Students use a checklist to rate their understanding of key vocabulary and their ability to explain the difference between theoretical and experimental probability to a friend.

Frequently Asked Questions

What is the difference between theoretical and experimental probability?

Theoretical probability is what we calculate should happen based on the possible outcomes (e.g., a 1 in 6 chance of rolling a 4 on a die). Experimental probability is what actually happens when you do the experiment and record the results.

Why do our experiment results not match the fraction exactly?

Chance and randomness mean that short-term results can be unpredictable. However, if you were to repeat the experiment thousands of times, your results would get much closer to the theoretical probability.

Can a probability be 0 or 1?

Yes. A probability of 0 means the event is impossible (e.g., rolling a 7 on a standard six-sided die). A probability of 1 means the event is certain (e.g., picking a red candy from a bag containing only red candies).