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Introduction to Lesson 10: Geometric reasoning

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 10: Geometric reasoning

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

CfE_L10_Geometric_reasoning_handout (one between two)
CfE_L10_What_do_we_need_to_know_handout (one between two)
CfE_L10_Practice_questions_handout (one each)
CfE_L10_Lesson_Overview (one each)
CfE_L10_Lesson_Plan (one each)�

The professional development includes four short videos. We recommend that you download these onto the device you will be using in the cluster meeting. Make sure you have speakers available so the audio can be heard.

10501 Lesson 10 Prior Knowledge ONLINE
10521 Lesson 10 What do we need to know ONLINE
10541 Lesson 10 Is angle Z 30 degrees ONLINE
10560 Lesson 10 Is PQR right-angled ONLINE

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Aims of professional development

Teachers should:

be prepared to teach Lesson 10: Geometric reasoning
understand how the lesson and resources have been designed to build on what students already know
understand how the development and closure of the lesson should support students in demonstrating their understanding

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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 10: Geometric reasoning

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Objectives of Lesson 10

Understand what it means for lines to be parallel

Calculate missing angles

Use angles to determine whether lines are parallel

Develop fluency and understanding when reasoning with angles

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Lesson 10: Outline plan

Is 15 minutes about right for the initial poster activity?

Valuing and building on what students already know

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Lesson 10: Research questions

Pedagogic focus

In which ways does the teacher develop the lesson to value and build on what the students already know?

Maths focus

How do students demonstrate their knowledge and understanding?

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Lesson 10: Geometric reasoning

Introduction

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Prior knowledge

The lesson starts by telling students that they will begin the lesson by reviewing what they know about angles.

Teachers will distribute A3 paper and some coloured pens (with tips that are not too thick), and ask students to work in pairs to produce a poster to show what they know about angles.

When the lesson was trialled, some students appeared to struggle to get started, so the ideas on the slide were added and can be used as as a starting point. However, even though they struggled, and some needed a bit of prompting from teachers, all students managed to produce something.

Encourage the teachers to allow students to work on their posters and try not to intervene. The aim of this activity is to provide them with information about what students appear to know already and to get them thinking about geometry. By establishing what students already know, this existing knowledge can be built upon (Key Principle 2).

As students work, teachers should identify a couple of students to share their posters with the rest of the class.

Then the class discusses students’ posters. Students should be encouraged to add anything that is missing to their poster during the discussion.

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Student posters: examples

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Video: Exploring prior knowledge

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Exploring prior knowledge: group discussion

What did you notice and did anything surprise you?

How engaged were the students?

How did the students share their ideas and reasoning?

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Triangles

Would you discuss other kinds of triangles?

This slide is provided to prompt a discussion about angles in a triangle. Most students know that the angles of a triangle add up to 180˚. The teacher asks students when the three angles in a triangle would be equal and also discusses isosceles triangles.

Ask the teachers if they would want to ask students what other types of triangles they know (e.g. right angled and scalene triangles) and establish their properties.

When discussing the properties of angles in triangles, ask students when the three angles in a triangle would be equal. Establish that an equilateral triangle has three equal sides and three equal angles that are each 60°, while an isosceles triangle has two sides of equal length and two equal angles.

You may want to ask students what other types of triangles they know (e.g. right angled and scalene triangles) and establish their properties. Emphasise that regardless of the type of triangle, the angles always add up to 180°.

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Parallel lines: an animation

Will you discuss corresponding angles on non-parallel lines?

This slide is provided to check students’ understanding of what ‘parallel’ means.

This slide is animated. Play the animation, which is designed to emphasise the mathematical structure (Key Principle 1) of parallel lines on a transversal: essentially, the second line can be thought of as a copy of the first.

Discuss with the group what else they would want to highlight. For example will they check that students know that arrows are used to denote parallel lines?

Tell them that you would expect there to be some discussion of corresponding angles: they should establish that when the lines are parallel, corresponding angles are equal. They might like to emphasise that the angles are not equal if the lines are not parallel.

The text boxes are also animated.

Establish that when the lines are parallel, corresponding angles are equal.

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Angles on parallel lines

How do you know the lines are parallel?

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Angles on non-parallel lines

These angles are corresponding, but not equal.

These angles are alternate, but not equal.

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Perpendicular lines

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Do the students want to add anything?

Would you provide some diagrams of angles on parallel lines on the board?

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Explore and discuss 1:

What do we need to know?

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Under what circumstances are the alternate angles equal?

How would you emphasise that, although the lines look parallel, we don’t know if they are?

This what Petra needs to know.

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Under what circumstances are the lines parallel?

Do we know anything about the lines A1 and A2? ��Do we know anything about the corresponding or co-interior angles?

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Summing up

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Further examples

Would you print this out?

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Teachers working in pairs

Work in pairs

Write down what you would need to know to answer the question

Discuss why each of the four examples is included

How you would balance the tension between getting your students started and allowing them some time for productive struggle?

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Teachers feeding back

What are your general comments?

Why is each of the four examples included?

How would your students cope with this activity and how would you balance the tension between getting them started and allowing them some time for productive struggle?

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Video: What do we need to know?

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What do we need to know? group discussion

What did you notice and did anything surprise you?

How engaged were the students?

How did the students talk about mathematics and where did you see evidence of understanding?

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Handout Student responses 1

What have these students done? Where do you see evidence of an understanding of geometric reasoning? Where do you see evidence of misconceptions?

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Explore 2:�Reasoning questions

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A question to model what the students will do

Would you point out that the lines are parallel, and ask what this means for the angles?

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Mistaken thinking

What is this slide for?

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One way (of several ways) to approach the question

How would you use your own students’ responses to build up their self-belief?

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Video: Is angle z 30˚?

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Class discussion: is angle z 30˚?

What did you notice and did anything surprise you?

How engaged were the students?

How did the students respond to the teacher’s questions and to one another?

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Student pair-work

Why is it important to be explicit about how the students should work in their pairs?

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Teachers working in pairs

Take the role of a student

Work in pairs to answer the questions and give reasons

Discuss the examples chosen and the way they vary

Consider how your students will respond

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Teachers feeding back

What are your general comments?

Why were these examples chosen and how do they vary?

How might your students respond?

Is there too much variety in these examples?

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Video: Is triangle PQR right angled?

“We actually have to use our brains”. How do you think the student who said this was feeling?

The graphic on this screen is a placeholder. Here is the video: 10560 Lesson 10 Is PQR right-angled ONLINE. It is recommended that you download the video onto the device you will be using in the cluster meeting.

The main part of this video focuses on two students. They are given some help (by the video team) but essentially the geometric reasoning is all their own. The video provides some valuable insights into how students might develop their reasoning and what barriers they might encounter on the way.

Watch the video together with the teachers, and ask them to look out for anything they notice or anything that surprises them. In particular ask them how the students:

work together and help each other
use their poster
self correct?

Also look out together for the comment about having to use your brain towards the end.

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Class discussion: Is triangle PQR right angled?

What did you notice and did anything surprise you?

How does the students’ reasoning develop and what gets in their way?

How did the students reason and self-correct?

How did the students use their poster?

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Discuss 2: �Reasoning questions

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Whole class discussion A

Yes □�No □�Can’t tell □

Alternate angles on parallel lines.

50˚

Angles in a triangle add up to 180˚.

90˚

Is triangle PQR right-angled?

How does the animation contribute to the students’ understanding and fluency?

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Whole class discussion B

Yes □ �No □�Can’t tell □

95˚

Angles on a straight line add up to 180˚.

Alternate angles are NOT equal.

H₁ is NOT parallel to H₂.

What is another way to do this?

Is line H₁ parallel to line H₂?

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Whole class discussion C

J₁ is parallel to J₂.

Corresponding angles are equal.

65˚ + 115˚ = 180˚.

Co-interior angles add up to 180˚.

K₁ is parallel to K₂.

Yes □ �No □�Can’t tell □

65˚

What might confuse students?

Is line K₁ parallel to line K₂?

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Whole class discussion D

Is triangle TUV isosceles?

Yes □�No □�Can’t tell □

Angles on a straight line add up to 180˚.

Triangle TUV has two equal angles.

It is isosceles.

65˚

Corresponding angles on parallel lines are equal.

65˚

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Handout Student responses 2

What have these students done? Where do you see evidence of an understanding of geometric reasoning? Where do you see evidence of misconceptions?

This activity is optional.

The two student responses provided on Handout Student responses 2 provide an interesting contrast. These are taken from an earlier version of the lesson, which explains why the diagrams on the handout and hand drawn. Give this to the teachers now and ask them to discuss what the students have done. The one on the left appears at first to show little evidence of understanding, but the students have made an effort and done some working out. Ask the teachers how they would work with the pair of students who had completed these responses,

The response on the right is mostly well answered, and might surprise teachers. Assure them that this came from a pair of students in a class of GCSE resit students and the lesson took place just before Christmas in 2022, so it is ‘real’!

Ask them to look for examples of correct student reasoning and also of misconceptions.

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Review and exam questions

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Reviewing and summing up

What is the point of these two slides?

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Exam questions

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Lesson 10

These slides support a discussion of the answers to the practice questions

When discussing part (a), teachers should emphasise the importance of not assuming an angle is a right angle because it looks like one. Highlight that the question states that lines AB and BC are perpendicular, so we know that angle ABC is 90°. x is therefore equal to 40°, as 90° – 25° – 25° = 40°.

For question 1b, remind teachers to check that students have identified an angle that is 125° and that their reasoning is correct. They could have selected b or d.

When answering part (ii), students may use the fact that angles around a point add up to 360° (360° – 125° = 235°). Students should identify the values of angles a, b and c using the properties of angles on parallel lines as well, to show that a + b + c = 235°.

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Lesson 10

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Lesson 10: Geometric reasoning

Closing thoughts

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Lesson plan and PowerPoint

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 10: Research questions

Pedagogic focus

In which ways does the teacher develop the lesson to value and build on what the students already know?

Maths focus

How do students demonstrate their knowledge and understanding?

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Aims of professional development

Teachers should:

be prepared to teach Lesson 10: Geometric reasoning
understand how the lesson and resources have been designed to build on what students already know
understand how the development and closure of the lesson should support students in demonstrating their understanding

You can remind yourself of today’s discussion by working through the Desmos activity on the Mastering Maths website

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

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Thank you: Any Questions?