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Introduction to Lesson 6: �Frequency charts and averages

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Aims of professional development

Teachers should

be prepared to teach Lesson 6: Frequency charts and averages
understand how the lesson and resources have been designed to encourage understanding as well as fluency
understand how the development and closure of the lesson should support student’s understanding of summary statistics

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.

Developing both fluency and understanding is an important part of the mastery approach (Key Principle 4) and in this lesson, time is spent interpreting and comparing various data sets represented using frequency charts and summary statistics. By exploring these different representations, students are supported in developing a deeper understanding of the way in which the mode, median and mean represent the average of a set of data and the distinction between measures of average and range as a measure of spread.

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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 6: Frequency charts and averages

Overview

When working with frequency charts, students often struggle to see the relationship between the data presented as a list and represented on a chart. The use of sticky notes in this lesson, to both record and display data values provided by students, helps students to identify their own data values within a frequency chart and develop their understanding of the relationships between different representations of data. Using multiple representations provides insight into the mathematical structure of data sets and exposes the way in which the three averages provide a summary (Key Principle 1). Developing both fluency and understanding is an important part of the mastery approach (Key Principle 4) and in this lesson, time is spent interpreting and comparing various data sets represented using frequency charts and summary statistics. By exploring these different representations, students are supported in developing a deeper understanding of the way in which the mode, median and mean represent the average of a set of data and the distinction between measures of average and range as a measure of spread.

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Objectives of Lesson 6

Construct and interpret frequency charts

Find the averages and range for a set of data

Understand how the mean, median, mode and range summarise data

Use different representations of data to provide insight into mathematical structure

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Lesson 6 Outline plan

How does using students’ own data develop fluency?

Key Principle 4: Develop both understanding and fluency in mathematics

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Lesson 6: Resource questions

Pedagogic focus

In which ways does the lesson develop the students’ understanding and fluency related to averages?

Maths focus

How do students use different representations of data to develop their understanding?

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Establishing the context: �Keeping active

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Sticky notes

How well would your students relate to the context?

At the start of the lesson, students use sticky notes to construct a frequency chart showing how active the class has been in the last five days. All students are actively involved and can see where their own data value features in the chart. By asking all students to make a contribution and comment on the results, a collaborative culture is encouraged (Key Principle 5).

Teachers hold a short discussion about exercise and then give each student a sticky note, asking them to write down the number of days out of the last five days that they did something physically active for at least 10 minutes (a whole number between 0 and 5), and to put their initials in the bottom right corner. Then students stick their sticky notes on a class frequency chart. (Slide 9). Ask the teachers how well their students would relate to the context.

Tell the teachers that then they should hold a class discussing, encouraging all students to make a contribution to the discussion about something that they notice.

Students may not comment on any statistical measures, and it is not necessary to introduce these at this stage. Being able to identify their own sticky note and as a result recognise where their data value is represented in the chart can play an important part in developing their understanding of the relationships between different representations of data.

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Understanding the data: Watch the video

How does the teacher establish what the students already know?

What does the student’s calculation on the mini whiteboard reveal about his understanding?

Does anything surprise you?

In which ways does the activity develop the students’ understanding and fluency related to averages?

What questions would you ask to help you establish what they do and do not understand about the frequency chart?

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Video: discussion

What did you notice?

Did anything surprise you?

How does the teacher establish what the students already know?

What does the student’s calculation on the mini whiteboard reveal about his understanding?

In which ways does the activity develop the students’ understanding and fluency related to averages?

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Explore 1: Fitness class

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Fitness class exercise survey

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Putting the data into a chart

How will you check that students understand how the chart is constructed?

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Interpreting and summarising the data

What might students notice?

Why does Chloe suggest finding the average?

In the first slide here, Abi, Baz and Chloe start to discuss how to interpret the data and Abi asks what the chart shows; tell the teachers to ask a couple of students to comment on the chart. They may, for example, say that no-one in the class exercised on just one day or that most people exercised every day for the last five days.

In the second slide, Chloe suggests finding the average. Suggest to teachers that they ask their students what Chloe means by this and assess their understanding of how an average value provides a summary of a data set (perhaps this is why Chloe suggests finding the average). Students may refer to the mean, median and mode during the discussion. These average values will be explored further as part of the discussion of Abi’s, Baz’s and Chloe’s statements in the slides that follow.

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Teacher activity: Work in pairs

Work in pairs

What has each of Abi, Baz and Chloe done?

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How can they all be correct?

Would you provide calculators?

How would you help your students get started?

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Three slides to support discussion

What misconception does Yaima’s question reveal?

Discuss these slides with the teachers. They are intended to support class discussion of the three averages.

Students’ initial thoughts when thinking about averages may be to calculate the mean (as this is the most frequently used measure of average), which Abi has done. Establish that Baz has identified the mode, which is the most common response. 4 of the 11 people in the fitness class exercised for at least 30 minutes on all 5 days.

You might like to draw attention to Yaima’s question in the third slide, which focuses the class’ attention on a common misconception: students can sometimes mistakenly identify the mode as the frequency itself, rather than the value it represents. Chloe has found the median value. Some teachers are uncomfortable with making misconceptions explicit, but in the Mastering Maths approach, this technique is seen as valuable because, as Malcolm Swan explained in this article, ‘learning activities should expose current thinking, create ‘tensions’ by confronting learners with inconsistencies and surprises, and allow opportunities for resolution through discussion’.

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The three averages: Watch the video

What do you notice?

What can you say about the prior learning of the student who responds?

How would your students respond?

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Video: discussion

What did you notice? Anything surprising?

What can you say about the prior learning of the student who responds?

What would you do next?

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Make a note

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Explore 2: Fitness class

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An animated slide

What might students notice about the data?

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Abi’s statement

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Baz’s statement

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Chloe’s statement

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What can you read straight off the chart?

Which class is fitter?

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Modelling the matching activity

There is a mistake on this slide.

Explain to the teachers that when this lesson was trialled, this slide was not included, but many students appeared to need a bit more help in getting started with the main student pair-work task. So this slide models the process of matching charts and summary statistics tables.

Students are told that the two tables give a summary of the data represented in the two charts. Ask students which table goes with which chart and how they know. Emphasise that the mode is the easiest average to identify from the chart.

Once the tables/charts have been correctly matched, ask students to complete the missing values in the two tables. Encourage students to use the chart directly when calculating the mean. Emphasise how to identify the median when there are an even number of data values. Remind students of the importance of the total frequency when finding both the median and the mean.

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Explore 3: Working in pairs

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The main pair-work task

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Try the activity

Take the role of students

Work collaboratively

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Two students working on the task: watch the video

What do you notice?

In which ways does the lesson develop the students’ understanding and fluency related to averages?

How do students use different representations of data to develop their understanding?

How do the students approach the task?

How are the students collaborating?

What misconceptions are revealed? How would you mediate these in your own class?

How is this activity contributing to the students’ fluency?

Tell the teachers that you are going to watch a short video (just under three minutes) of two students working on the task. The video does not show their entire journey, but provides examples of their reasoning and collaboration.

Ask the teachers to bear in mind the research questions.

Pedagogic focus: In which ways does the lesson develop the students’ understanding and fluency related to averages?

Maths focus: How do students use different representations of data to develop their understanding?

Also ask them to make a note of anything they find interesting or surprising.

You might like to provide some further questions to structure the discussion, such as:

How do the students approach the task?

How are the students collaborating?

What misconceptions are revealed? How would you mediate these in your own class?

How is this activity contributing to the students’ fluency?

You might want to play the video right through and then hold a short discussion before playing it through again.

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The responses of one pair of students

Where, and why, did they change their minds?

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Review and exam questions

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Going beyond the answers

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Slides to support a detailed discussion

How would you handle the discussion if the students are tired?

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Is there a best average?

How were the numbers in these lists chosen?

To begin, ask the teachers to look at this slide and think about how the two data lists were put together.

Discuss bringing students’ thinking together, summarising the three different averages. Ask students to find the mean, median and mode for the two data lists:

Data List A: 1, 2, 3, 4, 5 �Data List B: 1, 2, 3, 4, 4, 4

After a couple of minutes, teachers will discuss the three values for the mean, median and mode for the two data lists and highlight the different ways in which they measure the average. They might like to discuss how the mean is calculated using all the data values, unlike the median and mode, and how there can sometimes be no mode (as in the case of the first data list) or more than one mode. Discuss identifying the median when there is an even number of data values (as in the second data list) compared with identifying the middle value as in list A.

Once the values for the three averages have been identified for the two lists, teachers should ask students which average they think best describes each of the data lists. They contrast data list A, where the mean and median have the same value, with data list B, where the three values are different. They discuss how the mode may be considered as the best average for list B since half of the data values are 4, or the median could be thought to be the best average as it takes into account the values that are less than 4. They point out that the mean may not be the best measure of average, especially when the data is skewed as it is here, and reinforce how taking each measure of average separately, without the context of the other two, can sometimes give a misleading summary of the data

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Why is the range important?

How were the numbers in these lists chosen?

Discuss this slide, beginning by asking the teachers to look at this slide and think about how the two data lists were put together

The range uses the extreme values to describe how spread out the data and the range is the same for both, but the lists are very different.

Teachers might like to ask students to determine the mode, median and mean for the two data lists and write them on their mini whiteboards, pointing out that the range sometimes provides an unreliable measure of how variable the data is, as demonstrated here by two sets of data with the same range but different average values.

It is important that students recognise the distinction between measures of average and range as a measure of spread, and how using the range and average values together can provide a more reliable summary of the data.

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Exam questions

When completing the first practice question, it is important that students explain what Bridgit has done wrong and not just state that she is incorrect because the median is actually 15.

When discussing the first question, teachers should establish the importance of ordering the data and how putting data into a frequency table helps with this. They may like to identify that there is no mode for the list of numbers.

When completing the second practice question, students may just total the frequency column to give an answer of 30 (the number of students). If they do, teachers might like to ask them the total number of points scored by the five students who each scored 5 points.

Teachers should discuss why working out the total number of points scored is a useful thing to do (because we can use it to calculate the mean).

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Lesson 6: Frequency charts and averages

Closing thoughts

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Lesson resources

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

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Lesson 6: Research questions

Maths focus

How do students use different representations of data to develop their understanding?

Pedagogic focus

In which ways does the lesson develop the students’ understanding and fluency related to averages?

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Aims of professional development

Do you feel prepared to teach Lesson 6: Frequency charts and averages?

Do you understand how the lesson and resources have been designed to encourage understanding as well as fluency?

Do you understand how the development and closure of the lesson should support students’ understanding of summary statistics?

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