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Introduction to Lesson 8: �Algebraic equations

This professional development resource is designed for use by a lead teacher or head of department working with a group of teachers to prepare them to teach CfEM Lesson 8: Algebraic equations

Before running the professional development, you should ensure that you have printed

CfE_L8_Lesson_plan: one each
CfE_L8 Lesson_overview: one each
CfE_L8_Completing_descriptions_handout: one between two
CfE_L8_Practice_questions_handout: one each

You might also like to create a handout with examples of students’ work (see slide 36).

The professional development includes three videos. Make sure you have speakers available so the audio can be heard. It is recommended that you download the videos onto the device you will be using to make the presentation. Watching the video should enrich the teachers’ understanding of how the lesson might play out.

A further two documents support the lesson study for this lesson.

Lesson 8 observation form (one for each observer)
Lesson 8 lesson study (this is for you).

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Aims of professional development

Teachers should

be prepared to teach Lesson 8
understand how the lesson and resources have been designed to emphasise mathematical structure
understand how the development and closure of the lesson should develop an understanding of mathematical structure

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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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Overview of Lesson 8: Algebraic equations

Explain the rationale for this lesson. Students are usually familiar with using letters to represent variables but can struggle to understand the mathematical structure of an equation.

In this lesson algebraic equations are used to model real-life situations and students explore the effects on the meaning of equations and expressions when the definitions of the variables change.

Developing an understanding of mathematical structure (Key Principle 1) through mathematical representations is a key part of the teaching for mastery approach and in this lesson the use of representations is encouraged when exploring relationships.

Possible incorrect equations are discussed, to expose common misconceptions and provide insight into students’ existing knowledge (Key Principle 2). Providing different possible solutions can give students a starting point, building confidence and encouraging them to share their understanding (Key Principle 5).

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Objectives of Lesson 8: Algebraic equations

Make connections between real-life situations and algebraic equations

Solve equations and interpret the solutions

Use representations to provide insight into mathematical structure

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Lesson 8 Outline plan

This is a long introduction. Is it too long?

Key Principle 1: Develop an understanding of mathematical structure

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Research questions

Pedagogic focus

In which ways does the teacher encourage students to think about mathematical structure?

Maths focus

How do students use representations when working with equations?

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Establishing the context:

Multipacks of face wipes

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Packs and multipacks

Will the students relate to this context? Why were facewipes chosen rather than, for example, crisps?

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Do this! Write an equation

Using

p = the number of packs and

m = the number of multipacks��Write an equation

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Write an equation

Are these equations what you would expect?

Note that in a previous version of the lesson, the variables were p, for the number of packs, and b for the bag of packs.

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Understanding equations: Watch the video

Now show the teachers the video 8500 Lesson 8 Number of packs ONLINE.mp4. (Note that in this version of the lesson:

there was a bag of packs, so the variables used were b and p)

Also note that in the updated version (which you are using)

- Wendy has been replaced by Sade

- Sade’s equation is not considered until later

- The PowerPoint explicitly states that Tia’s equation is wrong and Vinay’s is correct.

Ask teachers to make a note of anything they find interesting or surprising that they might like to discuss later.

Also ask them to bear in mind the research questions:

Pedagogic focus: In which ways does the teacher encourage students to think about mathematical structure?

Maths focus: How do students use representations when working with equations?

You might also raise a few other questions now, or you may prefer to keep these for a second viewing of the video. Some suggestions include:

In which ways does the teacher value prior learning?

How do students explain their incorrect equations?

How would you describe the students’ struggle?

When did the students’ thinking change, and how?

Finally, in this video (just over eight minutes minutes), we see the teacher introducing the lesson. Clearly some parts of the video have been cut out and in reality it took him much longer than nine minutes to get to this stage of the lesson.

The slides that follow the video are the same as the ones in the video (or nearly the same), and depending on time, you may decide to spend longer on the video and skip Slides 15 to 18, and restart at the next part of the video in Slide 19.

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Video: discussion

What did you notice?

Did anything surprise you?

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Video: discussion

In which ways does the teacher value prior learning?

How do students explain their incorrect equations?

How would you describe the students’ struggle?

When did the students’ thinking change, and how?

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Two different equations

Why do so many students make the same mistake as Tia?

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Unpicking the equations

Why does this slide explicitly state that Tia is wrong?

The video you have just watched includes a version of these slides. It might be worth discussing this one with the teachers because the slides now state that Tia’s equation is wrong and Vinay’s is correct. Ask the teachers why they think it was changed.

In the video, the teacher switched between slides a number of times so that the students could see the different equations. Suggest to teachers that they might like to add in a copy of Slide 15 here for the discussion.

The left hand slide suggests substituting some numbers for m and p to check whether Tia’s equation works. For example, If Tia’s Dad was sent 2 multipacks of wipes, he has a total of 12 packs of wipes. 2 x 6 = 12, so you need to multiply the number of multipacks by 6 to get the number of packs. Remind the students that the number of packs is always bigger than the number of multipacks.

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Equivalent equations

How much time would you spend on this slide? What would you discuss?

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Using representations: an animated slide

How does the pictorial representation contribute to the students’ understanding?

m = 1

p = 6

m = 2

p = 12

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A task for teachers

What is the relationship between x and y?

x = the cost of a pack

y = the cost of a multipack

y = 6x �or �x = 6y?

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Looking at cost

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Working out the cost: Watch the video

Now show the teachers the video 8520 Lesson 8 Working out the cost ONLINE.mp4 (about 5 minutes).

Ask teachers to make a note of anything they find interesting or surprising that they might like to discuss later.

Also ask them to bear in mind the research questions:

Pedagogic focus: In which ways does the teacher encourage students to think about mathematical structure?

Maths focus: How do students use representations when working with equations?

You might also raise a few other questions now, or you may prefer to keep these for a second viewing of the video. Some suggestions include:

How did students collaborate to come to a common understanding?

How did students use representations?

How does the activity emphasise mathematical structure?

How does the teacher encourage students to think about mathematical structure?

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Video: discussion

What did you notice?

Did anything surprise you?

How did students collaborate to come to a common understanding?

How did students use representations?

How does the activity emphasise mathematical structure?

How does the teacher encourage students to think about mathematical structure?

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Cost: three equations

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Explaining Tia’s equation

The emphasis is on why Tia’s equation is correct.

Why?

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Comparing scenarios

x = the cost of a pack

y = the cost of a multipack

p = the number of packs

m = the number of multipacks

p = 6m

y = 6x

How are these scenarios different?

Ask teachers to think about how this second scenario of the cost of wipes is different to the relationship between the number of multipacks of wipes and the total number of packs of wipes.

Comparing the two equations side by side may help students to understand the different mathematical structures. Establish that with the first scenario there are always six times as many packs as bags, so if we know the number of bags, we multiply by six to get the number of packs (and if we know the number of packs, we divide by 6 to get the number of bags) .

With the second scenario, rather than counting the number of packs, a pack of wipes now has a value, its cost, and so the cost of a multipack is six times the cost of a pack. If we know the cost of a pack, we multiply by 6 to get the cost of a multipack (and if we know the cost of a multipack, we divide by 6 to get the cost of a pack).

The total cost of multipacks is my and the total cost of the packs is px. As m = p/6 and y=6x, my = p/6*6x = px.

This slide was included in the version of the lesson that was trialled but most teachers and students found it confusing.

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Explore/Discuss 1: �What do the equations mean?

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Do this activity as a group

Take the role of students

d = number of ‘deluxe’ multipacks

ordered�

b = number of ‘basic’ multipacks

ordered�

w = cost (in £) of a multipack of ‘deluxe’ wipes�

z = cost (in £) of a multipack of ‘basic’ wipes

w + z = 15

(i)

d = 2b

(ii)

bz = 12

(iii)

Twice as many ‘deluxe’ multipacks as ‘basic’ multipacks were ordered

Together, a ‘deluxe’ multipack and a ‘basic’ multipack cost £15

The total cost of the ‘basic’ multipacks ordered is £12

What does the equation tell us?

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Deluxe and Value packs: Watch the video

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Video: discussion

Why, and how, did the teacher check the students’ understanding?

In which ways does this feel like a classroom in which everyone believes everyone can succeed?

What student misconceptions did you notice?

How did the teacher help them to overcome their misconceptions?

What did you notice?

Did anything surprise you?

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Explore 2: Writing and solving equations

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Focusing on the variables

Why should you discuss the four variables before asking the students complete the sheet?

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Teacher activity: Complete the sheet

Take the role of students

How would your students respond?

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What did they do?

Pair A

Pair B

Pair C

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Discuss 2:

Explaining thinking

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The answers

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Solving equations

How will you use these slides?

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What is the equation?

How does attention to the formal equations benefit students?

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Review:

Understanding mathematical structure

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Checking understanding

Ask students to write the equation before you show this slide.

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Checking understanding

Would you point out that these are expressions, not equations?

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Two exam questions

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Lesson 8: Algebraic equations

Closing thoughts

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Lesson plan and PowerPoint

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

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Research questions

Pedagogic focus

In which ways does the teacher encourage students to think about mathematical structure?

Maths focus

How do students use representations when working with equations?

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Aims of professional development

Teachers should

be prepared to teach Lesson 8
understand how the lesson and resources have been designed to emphasise mathematical structure
understand how the development and closure of the lesson should develop an understanding of mathematical structure

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: Any Questions?