The Art of Angles

# 4. Untangle Polygons and Angles

Suggested Learning Intentions

• To develop logical reasoning to solve problems involving lines, angles, and polygons
• To articulate mathematical reasoning through providing justifications of steps in problem solving
• To use visual representations to support problem solving

Sample Success Criteria

• I can provide justifications for mathematical reasoning when problem solving
• I can use a range of manipulatives to explain and justify my thinking
• I can match justifications to a visual representation of a problem
• I can reference mathematical facts when proving an answer in problem solving
• Rectangle of paper for each student
• Hilma af Klint artwork: docx PDF
• Statement and reason chart: docx PDF
• Triangle problem: docx PDF
• Art of Angles Quiz - single question template: pptx

Facilitate a discussion with students about the mathematical term ‘proof’. Ask students to share their understanding of what proof means in everyday life. Explain that mathematical proofs allow others to understand the steps taken in solving a problem.

In this learning sequence students will learn to give justifications for their reasoning, rather than formal proofs.

Provide students with copies of af Klint’s artwork (available in the Materials and texts section above). Invite students to identify any angles they think appear in the art. Ask students to give a reason for their thinking, using question prompts such as:

• What makes you say that?
• Can you identify where that is shown in the artwork?
• What mathematical fact links to the artwork and makes you say that?
• What mathematical fact would someone need to know to understand why you think you are right?

Focus on student reasoning rather than whether responses are correct.

An example student response is that the object looks like a cube so the diagonal lines make 45-degree angles because 45 is half of 90. It is acceptable for students to make assumptions of 90 degree angles in the cube as long as they record that assumption.

Students record their thinking on a statement and reason chart (available in the Materials and texts section).

Further support for teaching justification of solutions can be found in the Literacy Teaching Toolkit.

1. Angles inside

This activity is inspired by Angles Inside at the NRICH Project.

Display the interactive Angles Inside problem to students. Move the point to show how the coloured angles change. Provide students with a rectangular piece of paper and invite them to draw any triangle and mark the red, green and blue angles as shown in the interactive tool.

The tool is presented with less visual distraction at GeoGebra.

Make this statement: ‘Ali made a conjecture: The green angle and blue angle added together give the red angle.’

Challenge students to prove whether Ali is correct. Guide students to record their thinking on a statement and reason chart (available in the Materials and texts section above).

Students might use the fact that the corner of a rectangle is a 90-degree angle. They could record that as:

 Statement Reason The rectangle has 90-degree corners. The definition of rectangle includes that is has four corners, each of 90 degrees.

Enable students who have difficultly recording their reasoning in writing by suggesting they write the steps they could follow as statements in order. Support students to verbally provide reasons why they made each statement. Record some reasons for them based on their verbal statements to model justification.

Enable students who have trouble visualising the steps to take by suggesting they cut the rectangle to move the angles and allow hands on comparison of the angles.

Extend students who are able to record their reasoning by supporting them to use more formal proof language. Students may find it beneficial to view this introduction to mathematical proofs

2. Triangle puzzle

Provide students with a copy of the triangle problem (available in the Materials and texts section above). Ask students to find angle a, and justify how they know they are correct. Guide students to record this on their Statement and reason chart (available in the Materials and texts section above). Encourage students to work through the problem to find some of the other unknown angles b-g.

Differentiate this task by setting different angles to solve for groups of students. The required solutions involve varied levels of logical and mathematical thinking. All students can learn valid reasoning skills through this task even without getting any of the marked answers. Consider having students work in teams to focus on reasoning.

Enable students who are overwhelmed by the lack of information on the diagram by encouraging them to draw the known information using a protractor to develop a scale diagram.

Enable students who have trouble visualising possible conclusions to make by providing multiple copies of their scaled drawing of the problem and encouraging them to cut out individual triangles. Demonstrate tearing corners off the triangle to show the internal angle sum of a triangle is 180 degrees as previously done in the Assemble your Angles stage.

Enable students who find the number of parts to the problem overwhelming by guiding them with a prompt to guide which angle they should try to find next. ‘If you know c then can you find d?’, ‘If you know b then can you find f?’, ‘If you know g then can you find e?’, ‘If you know g then can you find d?’.

Extend students by encouraging them to record their reasoning using some steps of more formal proof methods.

Areas for further exploration

To make more of a focus on problem solving in this learning sequence consider the following problems from NRICH.

Constructing triangles: Construct a triangle using randomly generated numbers as the side lengths.

Straw triangles: Use straws of different lengths to investigate the conditions that determine whether a triangle can be made from three lengths.

Angle hunt: A short problem to practise reasoning skills such as ‘if I know … then I also know …’.

Angle of overlap: A short problem to practise reasoning skills such as ‘if I know … then I also know …’.

Triangle in a corner: A short problem to practise reasoning skills by combining interior angle properties and straight-line properties.

Triangle mid points: Practise using visualisation to solve a problem. Staring with the midpoints of the sides of a triangle, draw the original triangle.

Take the right angle: Consider the angles made by the hands of a clock

Interactive quiz

Present students with the question, ‘Which of these shows co-interior angles?’

Ask students to record four possible options that could be the answer to the question, of which three are incorrect and one is correct. Facilitate a discussion about the types of correct and incorrect answers that might be displayed in a multiple choice quiz.

Invite students to share an incorrect answer they would provide as an option. Ask the students to identify why this option would confuse or trick someone completing their test.

The key idea here is to challenge student reasoning by having them know what the tricks will be and show others what not to do wrong. Teaching counter examples is explained in detail in the Literacy Teaching Toolkit.

Guide students to make a multiple choice quiz about the key terms covered in the learning sequence using PowerPoint. Provide students with a list of terms to guide their thinking. You may include the terms provided through the Reflect and Consolidate phases of the stages used in your class.

Instructions can be found at the Digital Technologies Hub.

1. Create a slide that presents a single question and four possible answers in separate text boxes.

2. Set the slide to not progress like an ordinary slideshow in the ‘transitions’ ribbon. Uncheck all ‘advance slide’ boxes (this step needs to be done to each new slide).

3. Create four new slides, one for each answer.

4. On each slide comment on whether the answer is correct or incorrect. Especially for incorrect answers use one or two sentences to explain why. Consider explaining how this is a trick answer and what to look out for next time. Each slide should have a text box to return to the question or move to the next question.

5. Create a slide to present the next question.

6. Insert the hyperlinks between pages.

• Select a text box.
• Open the ‘insert’ ribbon.
• Choose ‘place in this document’
• Select the slide that should open when the textbox is clicked.

Encourage students to use images in their quiz. Remind students of the online tools they have used during this sequence (e.g. Geogebra and Robocompass.) Each student can choose a method to create their own images, however using a drawing tool, such as Paint, would be less accurate than the mathematics tools.

Differentiate the task by considering the number of questions each student is expected to produce. Fewer questions with more reasoning about the incorrect answers should be encouraged.

Enable students by providing a set of questions that could be asked.

Enable students by providing them with the single question template to create their quiz.

Extend students by providing single incorrect answers and challenging them to think of a question it could be matched to.

To finish, ask students to expand their visual maths wall created during the Reflect and Consolidate phase of the Playing with polygons stage. Provide a list of key terms that students should include on their wall to guide their thinking towards the most important vocabulary words. These may include: justify, proof.

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Available at: https://www.geogebra.org/classic#geometry
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Srinivasan, P. K., 2011. NRICH: Squareo'scope Determines the Kind of Triangle. [Online]
Available at: https://nrich.maths.org/1389
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State Government of Victoria (Department of Education and Training), 2019. Justification of a solution. [Online]
Available at: https://www.education.vic.gov.au/school/teachers/teachingresources/discipline/english/literacy/Pages/lim_justificationofsolutions.aspx
[Accessed 15 March 2022].

TED-Ed, 2012. An introduction to mathematical theorems - Scott Kennedy. [Online]
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University of Cambridge, n.d. NRICH: Angle Hunt. [Online]
Available at: https://nrich.maths.org/7133
[Accessed 15 March 2022].

University of Cambridge, n.d. NRICH: Angle of Overlap. [Online]
Available at: https://nrich.maths.org/5669
[Accessed 15 March 2022].

University of Cambridge, n.d. NRICH: Angles Inside. [Online]
Available at: https://nrich.maths.org/13644
[Accessed 15 March 2022].

University of Cambridge, n.d. NRICH: Constructing Triangles. [Online]
Available at: https://nrich.maths.org/8098
[Accessed 15 March 2022].

University of Cambridge, n.d. NRICH: Take the Right Angle. [Online]
Available at: https://nrich.maths.org/1159
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University of Cambridge, n.d. NRICH: Triangle in a Corner. [Online]
Available at: https://nrich.maths.org/11627
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University of Cambridge, n.d. NRICH: Triangle Midpoints. [Online]
Available at: https://nrich.maths.org/483
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