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Introduction to Lesson 12: �Area and volume

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Aims of professional development

Teachers should

be prepared to teach Lesson 12: Area and volume
understand how the lesson and resources have been designed to value and build on students’ prior learning
understand how the development and closure of the lesson should support students developing their confidence.

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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 12: Area and volume

Overview

Students at this level usually know how to find the area and volume of 2-dimensional and 3-dimensional shapes, and can recall and apply the relevant formulae correctly.

Valuing and building on students’ prior learning is an important part of the mastery approach (Key Principle 2). In this lesson, time is spent discussing why their area and volume calculations work; this establishes what students already know and supports them in developing a deeper understanding.

This lesson’s focus on how scaling the dimensions of rectangles and cuboids affects the area and volume of these shapes provides an opportunity for students to see the links between mathematical concepts such as area and volume, proportionality, enlargement and similarity. Helping students to make connections across the curriculum is an important aspect of teaching for mastery (Key Principle 3).

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Objectives of Lesson 12: Area and volume

Understand the effects on area and volume of scaling one or both dimensions of a rectangle and one, two or all three dimensions of a cuboid

Understand and apply conservation of area and volume

Use relationships between similar figures to determine areas and volumes

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Lesson 12 Outline plan

20 minutes is a long time. How will you hold your students’ attention?

Key Principle 2: Value and build on students’ prior learning

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Lesson 12: Research questions

Pedagogic focus

How is the lesson developed and brought to a close in ways that values and builds upon what students already know?

Maths focus

What evidence do you observe of students’ prior learning about area and volume and how do they work with or modify this?

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Establishing the context: �Soft play blocks

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Checking vocabulary

How long would you spend on this?

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Valuing prior knowledge: watch the video

How does the teacher establish what the students already know?

How does the teacher value prior knowledge?

What do the students appear to know?

Now watch this video (just under seven minutes) showing how one teacher introduced the lesson and valued students’ prior knowledge. You might want to watch the video, hold a short discussion, and then watch it again.

Ask the teachers to consider the research questions as they watch the video:

How is the lesson developed and brought to a close in ways that values and builds upon what students already know?

What evidence do you observe of students’ prior learning about area and volume and how do they work with or modify this?

Also ask them to make a note of anything they find interesting, and in particular, anything they think exemplifies a mastery approach in FE.

You might like to have some questions ready to provoke discussion, for example:

How does the teacher question the students to elicit their prior knowledge?

How does the teacher value prior knowledge?

What do the students appear to know?

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Video: discussion

What did you notice?

Did anything surprise you?

How does the teacher establish what the students already know?

How does the teacher value prior knowledge?

What do the students appear to know?

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Surface area

This is the slide used in the video. Note that, following the trials of the lesson, the word ‘surface’ was added because teachers thought it would be less confusing for students.

Teachers should ask students to think about the 20 cm by 10 cm by 18 cm block in terms of area and write, on their mini whiteboards, everything that they know about surface area for this block. Give students a couple of minutes to think individually and then share their thinking with a partner.

As students work, the teacher circulates and notices what they are writing on their whiteboards. If it’s clear that many of them are struggling to get started, the teacher should spend some time revising and reviewing areas of surfaces of 3-dimensional shapes. Students should identify that the six faces of the block consist of three pairs of different-sized rectangles.

Point out to the teachers that students may calculate the volume of the cuboid and it is important to value this and build on students’ prior learning (Key Principle 2) when exploring the volume of blocks later (see Slide 11).

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An animated demonstration

13

18 cm

20 cm

10 cm

18 cm

20 cm

10 cm

Side

Top/Base

Area = 200 cm²

Area = 360 cm²

Area =

180 cm²

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Soft play blocks

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Surface area

Non-slip fabric

Non-slip fabric goes on the base of each block.

Side

Top

Side

Non-slip fabric on base

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The base area

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Introducing a misconception

How would students think about this?

Explain that that two employees, Charlie and Dee, are discussing the area of non-slip fabric needed for different sized blocks and suggesting what will happen to the area when you double one or both of the sides. The students are asked to discuss in pairs which employee, Charlie or Dee, is correct.

Consider this slide with the teachers, asking them what the point of the slide is. (It forces a discussion about what happens when sides are doubled and makes the misconception explicit).

Ask the teachers how they think their students might respond and how they themselves would respond.

The lesson plan suggests that the students discuss this in pairs and then the teacher moves on to the next slide. In some trials of the lesson, however, the teachers asked the students to respond to the question on mini-whiteboards which, although it took a little longer, had the advantage of helping the teacher understand her students’ thinking better. It also engaged all the students.

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What did they do?

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Comparing areas

10 cm

Area = 100 cm²

20 cm

10 cm

20 cm

Area = 200 cm²

Area = 400 cm²

10 cm

Show the teachers this animated slide, which takes a visual approach.

Discuss with them how they would use it.

The lesson plan suggests that teachers ask for a show of hands from students who think that the 20 cm by 20 cm base has double the area of the 10 cm by 10 cm base and those that think the 20 cm by 10 cm base has double the area, and to follow up by asking for students’ reasoning. It is important to explore different ways of thinking about the areas and acknowledge different methods of explaining the relationships between the three different base sizes to value students’ prior learning (Key Principle 2) and to encourage a collaborative culture where students share their understanding (Key Principle 5).

Once it has been established that the area of the 20 cm by 10 cm base is twice the area of the 10 cm by 10 cm base, establish how many times bigger the area of the 20 cm by 20 cm base is than the 10 cm by 10 cm base.

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Which ones are in proportion?

What would you discuss about the star logo?

This slide intends to focus the students’ attention on different shapes involved and, eventually, on the fact that doubling both sides results in a similar shape.

Ask the teachers how they would use the logo to emphasise that the similar shapes are in proportion.

Discuss with them that students may comment that the logo on the 20 cm by 10 cm base has been stretched and so is no longer in proportion, whereas the logo on the 20 cm by 20 cm base is still in proportion. Using the term proportion in their description suggests that they have made the connection between similarity and proportionality. Emphasising the connections between mathematical concepts is an important aspect of mastery teaching (Key Principle 3).

Emphasising similarity will be important as the lesson goes forward.

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Emphasising mathematical structure

21

10 cm

18 cm

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An animation to show doubling the volume

What happens to the volume if you double one of the 10 cm sides?

10 cm

18 cm

20 cm

10 cm

18 cm

Doubling one length

doubles the volume

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Another misconception?

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Bringing thinking together

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Six or eight times bigger? Watch the video

Now watch this video (B653 Lesson B Misconceptions ONLINE.mp4) showing how one class worked on this activity. Depending on time, you might like to show the video more than once.

Ask the teachers to consider the research questions as they watch the video:

How is the lesson developed and brought to a close in ways that values and builds upon what students already know?

What evidence do you observe of students’ prior learning about area and volume and how do they work with or modify this?

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

You might like to have some questions ready to provoke discussion, for example:

In which ways does this clip exemplify the mastery approach (collaborative culture, mathematical structure, slowing down, making misconceptions explicit)?

How did the students approach the question?

How did the class discussion a) value the students’ contributions and b) bring their thinking together?

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Video discussion

What did you notice?

Did anything surprise you?

In which ways does this clip exemplify the mastery approach?

How did the students approach the question?

How did the class discussion a) value the students’ contributions and b) bring their thinking together?

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Explore 1 and 2: Working in pairs

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Overview of the student pair-work

The students fill in missing areas and volumes
They place the cards
There is a class discussion
Students fill in some of the empty cells
There is another discussion

Discussion

Hand out the cards

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Teachers working in pairs

Follow the instructions for students

Fill in some of the empty cells

Think about:

the numbers that were used
the design of the grid

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Scaffolding: talking through how the grid is structured

Explain to students that they are going to be working with a grid containing rows and columns. Explain that the smallest block they have been discussing is in the top row/first column of the table because the area of the base is 100 cm² and the volume of the block is 1800 cm³. Draw students’ attention to the second, third and fourth rows where the volume of the blocks in these rows will be 3 times, 9 times and 27 times bigger than the 10 cm by 10 cm by 18 cm block. Ask students what is happening if the area of the base stays the same but the volume is 3 times bigger/9 times bigger/27 times bigger and establish that the height of the block is increasing by a factor of 3/9/27.

Check that students have understood how the table works. For example, any block in the column ‘Area of base is 3 times bigger’ will have a base with an area that is three times the area of the base of the 10 cm by 10 cm by 18 cm (smallest) block.

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Review/discuss 2

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Class review and discussion

How would you emphasise conservation of volume?

A focus on mathematical structure rather than answer-checking

Now teachers should bring together the students’ thinking (Slide 32) using two slides provided. Explain that all the cells in the table do not need to have been completed in order to be able to review students’ work and draw together their collective thinking. Rather than revealing all the blocks and then discussing the questions posed on the slide, teachers should pose the questions at appropriate points in the discussion based on students’ responses, focusing on mathematical structure rather than answer-checking.

The first part of the discussion (top slide) aims to check that the students have completed the calculations correctly and then goes on to ask some probing questions which are designed to go beyond answer-getting and develop mathematical understanding.

The second part confirms where the cards should be placed. The teacher will also discuss what goes in the column and row headings and why. The discussion continues to encourage students to think about conservation of volume.

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Completed grid

Discussion at the end of the pair work.

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Reviewing, generalising and exam question

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Beginning to generalise

These questions appear on clicking. What is their role?

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Generalising: 2D shapes

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A reminder and generalisation (volume)

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Putting the ideas into practice

What would you point out to students?

The next three slides can be used to consolidate student learning and emphasise how the relationships between similar figures enable us to find areas and volumes without the need to calculate. It is important that teachers emphasise that we can use similarity to determine area/volume when we don’t have enough information to calculate the area/volume using dimensions, provided we know the area/volume of one of the similar shapes.

In this first slide, two similar 2D pentagons are used (Slide 38). Pentagons are familiar mathematical shapes and students may think that the area of the shapes can be calculated; and, crucially, students will recognise/accept that they are similar. The teacher can use this example to emphasise that we are told that the shapes are similar and can ask the students how this helps them work out the area of the bigger shape.

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From area to volume

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Area, volume and similarity in unfamiliar shapes

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Exam question

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Lesson 12: Area and volume

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 12: Resource questions

Pedagogic focus

How is the lesson developed and brought to a close in ways that values and builds upon what students already know?

Maths focus

What evidence do you observe of students’ prior learning about area and volume and how do they work with or modify this?

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Aims of professional development

Teachers should

be prepared to teach Lesson 12: Area and volume
understand how the lesson and resources have been designed to value and build on students’ prior learning
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