2. With or without replacement

Learning Goals:

At the end of this lesson, students will be able to:

List all outcomes for two-step or three step experiment
Describe what happens to the probability of an event with replacement
Compare results between experiments undertaken using ICT and concrete materials

Syllable Outcomes addressed:

List all outcomes for two-step chance experiments, with and without replacement, using tree diagrams or arrays; assign probabilities to outcomes and determine probabilities for events
sample, with and without replacement, in two-step chance experiments, eg draw two counters from a bag containing three blue, four red and one white counter
sample, with and without replacement, in three-step chance experiments, eg draw three counters from a bag containing three blue, four red and one white counter

Syllabus Working Mathematically Indicators:

compare results between an experiment undertaken with replacement and then without replacement (Reasoning)
explain the effect of knowing the result of the first step on the probability of events in two-step chance experiments, with and without replacement (Communicating, Reasoning)

Prior Learning:

How to calculate probability for a chance event
Convert between decimal, percentage, and fraction representations of a rational number
Provides reasoning to support conclusions that are appropriate to the context
Express favourable outcomes in the form of P(Events) = number of favourable outcomes/ total number of outcomes

Teaching Strategies

Technology use
Use of concrete materials
Class discussion and consolidation of knowledge
Group work
Predict, Observe and Explain

Lesson Sequence

Resource Material:

Bag
Marbles
Cups

Warm up:

From the previous lesson, students should now be familiarised on how to list and express outcomes for two chance experiments. For this warm up activity, students will be conducting a similar activity to activity 2 from Lesson 1, where instead of candies in two different cups, students are tasked in pick out two marbles from one single bag but with every marble they pick, they must put the marble back into the bag before they choose another. The teacher should draw to student's attention that this is called an event with replacement. Students are to work in groups of 3-4 and are tasked to form a hypothesis as to how replacement affects the probabilities of the second draw. Students can further extend this warm up activity by picking out 3 or 4 marbles at random and see how that further affects the probability.

Main Activity (ICT):

Once students have formed a hypothesis, the teacher will gather the classroom for a quick discussion to share what students have found and what kind of hypothesis they have come up with. Prompting students with questions like: What happens to the probability after you replace the marble? What happens to the sample size? Right after, the teacher will notify students that they will be using a computer to test probability but at a larger scale. Once all students have access to a computer, students will use an online marble bag simulator and recreate their trial similar to the real life marbles they had just worked with. Students are then tasked to validate their activity and compare their online results with their actual results from their hypothesis.

The teacher will then lead a quick discussion to see if students' online trials differ from their actual results. After this, students can turn on the 'multiple features' tab to test trials on a larger scale. Students are to form another hypothesis but with 3 marbles instead.

Conclusion:

Near the end of the lesson, students are to write in their own words how replacement changes the probability of drawing objects from a bag.

Activity sourced from: http://www.shodor.org/interactivate/lessons/ReplacementProb/

Lesson Evaluation Questions

Will all students be able to understand the notion of replacement after the one activity?
Does the ICT really help enhance students' understanding?
Does having students in groups and creating a hypothesis evaluate every students' understanding of replacement.

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