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Math lesson plan · 10th grade (Year 11 in England) · 45 minutes

Circle theorems

Geometry

CCSS HSG.C.A.2CCSS HSG.C.A.3England: GCSE Higher Geometry

Objective and success criteria

Students will: Circle theorems.

  • I can answer questions on today's skill on my own, without help.
  • I can explain each step and say why it works.
  • I can check my answer and spot a mistake.

Before and after

Needs first: Interior and exterior angles of polygons; Angles with parallel lines

Leads to: Circle theorems: angle at the centre and angle in a semicircle; Circle theorems: cyclic quadrilaterals and angles in the same segment; Circle theorems: tangents and radii

Materials: Rulers and protractors, Squared or dotted paper, Shape cut-outs, Mini whiteboards and pens, Printed student sheet (below).

Warm-up · 5 min

  • Quick retrieval from earlier learning (Interior and exterior angles of polygons; Angles with parallel lines). Students answer on mini whiteboards.
  • Show answers together and note who needs support in the main task.
  1. A line crosses two parallel lines. One angle is 55°. What is the alternate angle to it, in degrees? → 55Prompt: Alternate angles are equal (Z shape). Corresponding angles are equal (F shape).
  2. What is the sum of the interior angles of a polygon with 8 sides, in degrees? → 1080Prompt: Split the polygon into triangles from one vertex.
  3. A line crosses two parallel lines. One angle is 79°. What is the alternate angle to it, in degrees? → 79Prompt: Alternate angles are equal (Z shape). Corresponding angles are equal (F shape).

Hook · 5 min

  • Show a photo of a building, a tile pattern or a road sign and ask what shapes and angles students can spot.
  • Share the objective: Circle theorems. Read the success criteria together.

I do: model it · 10 min

  • Work through the example out loud, saying why you do each step, not just what you do.
  • Example: ABCD is a cyclic quadrilateral. Angle A = 110°. What is angle C?
  • Step 1: Opposite angles in a cyclic quadrilateral add up to 180°.
  • Step 2: 70°
  • Answer: 70
  • Pause after each step and ask a student to say what comes next.

We do: guided practice · 10 min

  • Do these together. Students try each step on whiteboards before you reveal it.
  • Use the hints as prompts rather than giving the answer.
  1. A triangle is drawn in a semicircle with one side the diameter. One of the other angles is 50°. What is the third angle (not the right angle)? → 40Prompt: The angle in a semicircle is 90°.
  2. Two angles at the circumference stand on the same arc. One is 65°. What is the other? → 65Prompt: Angles in the same segment are equal.

You do: independent practice · 10 min

  • Students work on the student sheet. Circulate and check the first two answers of every student early.
  • Students who finish: write a question of their own for a partner, with the answer.

The 8 questions are on the student sheet below, with answers in the key.

Exit ticket · 5 min

  • Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
  1. A triangle is drawn in a semicircle with one side the diameter. One of the other angles is 67°. What is the third angle (not the right angle)? → 23Prompt: The angle in a semicircle is 90°.
  2. Two angles at the circumference stand on the same arc. One is 27°. What is the other? → 27Prompt: Angles in the same segment are equal.

Watch out for

  • Thinking a square turned on its corner is no longer a square.
  • Reading the wrong scale on a protractor.

Support and stretch

Support

  • Review first: Interior and exterior angles of polygons.
  • Use concrete materials: rulers and protractors.
  • Give the first step of each question, or a partly worked example to finish.
  • Pair with a confident partner for the guided practice.

Stretch

  • Move on to: Circle theorems: angle at the centre and angle in a semicircle; Circle theorems: cyclic quadrilaterals and angles in the same segment; Circle theorems: tangents and radii.
  • Explain a common mistake in writing and how to fix it.
  • Write a word problem that uses today's skill, then solve a partner's.

Homework and more practice

Free online practice with hints and printable worksheets for this skill: eurekami.com/math/grade-10.

Free lesson plan from eurekami.com/lesson-plan-builder

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Circle theorems

Name

Date

Score/ 8

  1. Question 1

    A tangent meets a radius at the point of contact. A triangle formed with the centre has an angle of 59° at the centre. What is the angle between the tangent and the line to the centre (the third angle)?

    Answer

  2. Question 2

    Two angles at the circumference stand on the same arc. One is 34°. What is the other?

    Answer

  3. Question 3

    A triangle is drawn in a semicircle with one side the diameter. One of the other angles is 55°. What is the third angle (not the right angle)?

    Answer

  4. Question 4

    Two angles at the circumference stand on the same arc. One is 54°. What is the other?

    Answer

  5. Question 5

    ABCD is a cyclic quadrilateral. Angle A = 69°. What is angle C?

    Answer

  6. Question 6

    Two angles at the circumference stand on the same arc. One is 26°. What is the other?

    Answer

  7. Question 7

    A triangle is drawn in a semicircle with one side the diameter. One of the other angles is 65°. What is the third angle (not the right angle)?

    Answer

  8. Question 8

    Two angles at the circumference stand on the same arc. One is 39°. What is the other?

    Answer

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Answer key: Circle theorems

  1. 1. 31

  2. 2. 34

  3. 3. 35

  4. 4. 54

  5. 5. 111

  6. 6. 26

  7. 7. 25

  8. 8. 39