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Lesson 9: Using frequencies and probabilities

Introduction

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Aims of professional development

Teachers should:

be prepared to teach Lesson 9: Using frequencies and probabilities
understand how the lesson and resources have been designed to ensure that everyone feels that everyone can succeed
understand how the development and closure of the lesson should support students developing their confidence

In this professional development, you will first provide the teachers with an overview of the lesson, and then they will work in pairs to complete the main activity (as students), discussing the design of the student task as they do so. You will then lead a detailed discussion on the design and teaching of the lesson.

Explain that the big idea for this lesson is that mathematical situations may look very different but their mathematical structure may be very similar or the same. The aim of this lesson is to develop students’ algebraic thinking by using bar models to represent mathematical structure and the way the lesson is developed and brought to a close is designed to provide a ‘way in’ for students, so that all students can make progress and have some success (Key Principle 5).

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Reminder: Teaching for Mastery key principles

https://www.masteringmaths.org/mm-approach/key-principles

Visit the Mastering Maths website to read more about the Key Principles

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Lesson 9: Using frequencies and probabilities

Overview

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Objectives of Lesson 9

Interpret and construct frequency tree diagrams

Use approximate values to produce a probability model

Calculate probabilities using probability tree diagrams

Use representations to reveal mathematical structure

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Lesson 9: Outline plan

Develop a collaborative culture in which everyone believes everyone can succeed

Is 10 minutes enough for this introduction?

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Lesson 9: Research questions

Pedagogic focus

In which ways do the activities of the lesson help students to believe that everyone can succeed?

Maths focus

How do the collaborative activities in the lesson help to develop students’ confidence?

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Introduction to the lesson: A realistic context

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Introduction

How will you explain what false positive and false negative mean?

Explain to the teachers that the lesson begins with a context which students should be able to relate to.

First they should explain that Starla and Toby work in a medical science department developing tests to detect whether or not someone has a disease. They have a blood test for a specific disease, but they want to develop tests that can be self-administered. The tests are trialled to see how successful they are at correctly identifying whether or not someone has a disease. Starla and Toby have developed a test and trial it with the 20 people that work in their lab. Their colleagues have a blood test first, to confirm whether or not they have the disease. They may need to discuss the meanings of false negative and false positive but, when the lesson was trialled, most students appeared to understand these concepts. Treating someone for a disease they don’t have (false positive), may, in some cases, have no ill effects, but someone with a disease not receiving treatment could pose more of a problem (false negative).

Explain that, in the first student activity, the students will go through the process that Starla and Tob’s colleagues experienced. Each student is given one or more of the ‘Cards’, depending on how many students are in the class.

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Teacher activity: Lab trial results

Place your card(s) in the right place.

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Lab trial results

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Is this a good test?

Do YOU think this is a good test?

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An animated slide

How soon would you show the students the frequency tree diagram?

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Video: Frequency tree diagram

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Discussion: Frequency tree diagram video

What do the students do?

How does the teacher value the students’ prior knowledge?

What can you say about the classroom culture?

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Explore/discuss 1:

Creating probability models

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Official trial results

Would you ask the students to work out the missing values before holding a class discussion?

Teachers should now tell students that having trialled the test on their colleagues, Starla and Toby carry out an official trial involving 1001 people. 202 people have the disease and 151 of them get a positive test result. 398 of those that don’t have the disease get a positive test result. They ask students what the remaining values should be, encouraging them to use a calculator. This is intended as a whole-class activity but some teachers might like to print out the incomplete diagram and ask students to fill in the missing numbers themselves - or they could ask students to draw out the incomplete diagram on their mini whiteboards, if these are being used. The photograph on the right is taken from a lesson in which the teachers chose this approach (it was a previous version of the lesson, which used slightly different numbers).

The slide used in the lesson is animated, so that each answer can be discussed in turn and to emphasise that, for a frequency tree diagram, the numbers at the ends of arms from the same starting point must add up to give the starting point value, so the missing values can be found by subtracting.

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Introducing the probability tree diagram: Pair work

Work in pairs to make a list of the main teaching points included in this slide.

0.2 is an approximation of 202÷1001. Is this a good example?

Now the lesson introduces the idea of probability, and in particular how a probability model is developed from the actual frequencies recorded in a trial.

On the left of this slide is the previous frequency tree diagram. The teacher introduces the probability tree diagram, first discussing why 0.2 has been used, rather than 0.202, and explaining that the frequencies are real data and often result in numbers that are not nice to work with. 202 is approximately 200 and 1001 is approximately 1000 so 200 ÷ 1000 = 0.2 gives a good approximation.

Explain that we use approximate values to represent the trial data in a probability tree diagram and use these values to make reasonable predictions of what might happen if the tests were used beyond the trial.

Teachers should now ask students to work in pairs to complete the probability tree diagram for the official trial results, writing a decimal value on their mini whiteboards for each of the letters a to e.

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Introducing the probability tree diagram: Main teaching points

The teaching points in this slide include:

Frequencies tell us how often an event occurred in a specific context (like a trial). We can approximate them to provide us with a probability of an event occurring beyond that context, which can be used to make predictions.
The probability on each arm of a probability tree diagram is found by dividing the frequency on the end of the arm by the frequency at the start of the arm and then rounding.
The probability on each pair of branches of the probability tree diagram have to add up to 1.

Would it have been better to start with, for example, 4020, for the official trial?

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Examples: what have these students done?

Students A and B

Students C and D

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The probability model

Emphasises that the probabilities on each pair of arms add up to 1.

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A common misconception

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A valid response

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Explore/discuss 2

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Scaling up

Why have these students introduced percentages? What will you do if your students use percentages?

What mistake did the students make?

Teachers now explain that we can use probability models to scale up and predict what might happen if we extend to a bigger population, for example, the whole country. They ask students to use the probabilities to make some predictions about what would happen if 10 000 people used the tests.

They ask students how many people are likely to have the disease (0.2 × 10 000 = 2000) and check that they understand where 2000 comes from and then they ask students to work out what goes in the other ovals and write the numbers on their mini whiteboards.

Explain that the slide is animated, so they can go through the answers with the students (click for them to appear), allowing enough time for them to explain their methods.

They should emphasise that the numbers at the ends of arms from the same starting point should add up to the starting point value (so the 500 and 4000 values can also be found using subtraction).

Explain that, when the lesson was trialled, most students found this step relatively easy. In the example shown here, though, the students appeared to be confused by what percentages to use on the lower branches of the tree. (The use of percentages is not necessary and may be a distraction, but this was an idea that came from the class and the teacher went along with it).

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Teacher activity: take the role of students

What do students need to do?

Where might they struggle?

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Students working in pairs

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Students collaborating: Video

The image on this slide is a placeholder.

Tell the teachers that you are going to watch a video clip together (Slide 27). The video is 9541 Lesson 9 Pair-work ONLINE.mp4. Explain that the video shows students working on the scaling up activity the teachers have been working on.

Encourage the teachers to focus on the students and what they did, thinking about what this demonstrates about their understanding.

Encourage them to discuss how much they saw of the misconceptions they predicted, and what other misconceptions (or confusions) became evident. Ask them to think about how the collaborative activity helps to develop the students’ confidence.

Once you have watched the video return to the questions and hold a short discussion. Encourage everyone in the group to speak and try to make sure that everyone’s contribution is valued by everyone.

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Now watch the video

What do the students do?

How do they collaborate?

How does this collaborative activity in the lesson help to develop students’ confidence?

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Discussion of Row A

How much would you emphasise this misconception?

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Discussion of Row B

What would you do about checking that the students have filled in all of Row B correctly?

Moving onto discussing Row B, the teacher now tells the students that there are some mistakes in this probability tree diagram, asking them how we can tell that some of the values are not correct just by looking at the diagram (the probabilities on each pair of branches don’t add up to one).

The teacher asks the students to identify the incorrect values and discuss how they could have been obtained (by truncating decimal values obtained from dividing frequencies rather than rounding). Teachers might like to emphasise that when we know one of the values on a pair of branches we can subtract from 1 to find the other, rather than needing to go back to the data.

There is no slide to support checking the answers, but the teacher should also check that students can explain why 7500 people are likely to not have the disease and 3000 people are likely to get a false positive.

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Explore/Discuss 3

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The new scenario

How would you engage students in the class discussion?

This is the new scenario. The teacher discusses with students that concussion, where someone receives a blow or jolt to the head, is a common injury in playing hockey and repeated concussion injuries can be serious. The probability of having an injury from playing hockey that results in concussion in any one season is 0.4. Each year there are two hockey seasons, a winter season, which runs from September to April (with a winter break from December to February) and a summer season, which runs from May to August.

The teacher tells the students that the missing values for the probability tree diagram are shown on the sticky notes., and asks them to explain why the values are either 0.4 or 0.6 (as the probability of having an injury from playing hockey that results in concussion in any one season is 0.4 so is the same for both the winter and the summer seasons). In the discussion, the teacher should reinforce structure and how the probabilities on each pair of branches add up to one.

One way to involve students might be to ask different students to come to the board and drag a sticky note to the correct place in the tree diagram.

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Multiplying along the branches

How would you explain that you need to multiply along the branches?

Note that this slide is animated. The correct probability tree diagram and the question on the left are presented first. The teacher asks students to calculate the probability of a player having an injury resulting in concussion in the winter and then also in the summer season.

Discuss that they would want to check that students have identified the correct branches (this is animated) - students may think that they need to add the two 0.4s to get 0.8, so it is important to address this.

Ask the teachers how they would explain that students need to multiply 0.4 by 0.4 (rather than adding), suggesting that, if students are struggling to understand why we need to multiply, they might discuss an example involving 100 injured players. We would expect that 40 (0.4 × 100) of the 100 injuries in the winter season would result in concussion. We only work with these 40 injuries now, as we are interested in injuries that result in concussion in both seasons. We would expect (0.4 × 40 =) 16 of the 40 players who had a concussion injury in the winter season to have an injury resulting in concussion in the summer season so 16 out of 100, a probability of 0.16.

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Using probabilities in two scenarios

What is the purpose of this whole class activity?

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Explore 4

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Work through the Using Probabilities activity

Work in pairs

Take the role of students

Note any difficulties your students might encounter

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Instructions for students

Consider whether you might reduce anxiety by asking students to complete just one of the two questions. You could then discuss both questions.

In this last part of the lesson, before the exam questions, the teacher gives each pair of students a copy of the ‘Using probabilities’ handout, perhaps holding a brief discussion about the fractions in row D, depending on the students. For example, they may need reminding that ⅔ + ⅓ = 1, or that when fractions are multiplied, the numerators are multiplied and the denominators are multiplied. Some students may want to convert the fractions to decimals; consider whether you will discourage them from doing this.

The teacher asks students to work in pairs to complete the probability tree diagrams for rows C and D and then use the probability model to complete the final column. They should encourage students to explain their thinking to each other.

Once you think they have had enough time, hold a short discussion.

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Discussion of Row C

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Discussion of Row C: incorrect values

Will this slide be needed?

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Row D: Using fractions

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Five statements

What is the value of this activity?

To end the lesson the class reviews some true or false statements. This whole-class activity is included to bring students’ thinking together. The teacher asks them to consider the five statements, stating whether they are true or false.Students write ‘True’ and ‘False’ on their mini whiteboards and note down the statement number under the correct heading.

When discussing statement 1, teachers should emphasise the probability scale and what it means when the probability of an event is equal to one.

When discussing statement 2, they might ask students to explain the difference between frequency tree and probability tree diagrams.

With statement 3 they will want to emphasise both the links between fractions, decimals and percentages and how connections can be made between the concepts of probability and ways of expressing a part of a whole.

When discussing statement 4 they could highlight that a frequency tree diagram is a way of representing data that can be displayed using a two-way table.

Statement 5 introduces a theoretical probability, which, while not discussed in this lesson, should be familiar to students. Teachers should emphasise that probability is not definite; it provides us with a model.

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Practice questions and video

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Practice questions

Hand out to the teachers the practice questions (Slide 43). There are two practice questions: tell the teachers to give students a couple of minutes to work on the questions individually.

As students work on the first practice question, teachers might encourage them to note down any calculations they use. They will then discuss students’ thinking and check that the correct values (‘22 and 23’ and ‘4, 18, 7 and 16’) have been completed.

Tell the teachers that they should then discuss students’ thinking when completing the second practice question, checking students’ understanding that in probability tree diagrams the probabilities on each pair of branches should always add up to one. They should establish that the 0.25 should be 0.35 (as 1 – 0.65 = 0.35) and that the probabilities on the first set of branches for the second throw need to exchange places, as the probability that the dice will land on 4 is 0.65 (and that it will not land on 4 is 0.35).

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Practice questions: Some student responses

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Video: a classroom culture in which everyone believes everyone can succeed.

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Classroom culture: discussion

What does the student talk, in both the classroom mathematics activity and the interview reveal about the classroom culture?

What does the teacher say about developing the classroom culture?

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Lesson 9: Using frequencies and probabilities

Closing thoughts

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Lesson resources

What can you change in the lesson and what should NOT be changed?

What do you need to do to prepare for teaching the lesson?

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Lesson 9: Research questions

Pedagogic focus

In which ways do the activities of the lesson help students to believe that everyone can succeed?

Maths focus

How do the collaborative activities in the lesson help to develop students’ confidence?

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Aims of professional development

Teachers should:

be prepared to teach Lesson 9: Using frequencies and probabilities
understand how the lesson and resources have been designed to ensure that everyone feels that everyone can succeed
understand how the development and closure of the lesson should support students developing their confidence

For a reminder of what has been discussed today, please have a look at the self-study materials on Desmos

https://www.masteringmaths.org/mm-approach/self-study-materials

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Thank you