Expected level of development
Australian Curriculum Mathematics V9: AC9M9SP01
Numeracy Progression: Understanding geometric properties: P7, Proportional thinking: P7
Previously, students will have been introduced to the concept of similarity and the conditions for similar triangles.
At this level, students will develop and use their understanding of the relationship between the corresponding sides of similar right-angled triangles. They will learn the terms 'base', 'altitude', 'hypotenuse', 'opposite' and 'adjacent' in relation to right-angled triangles. They will understand that the trigonometric ratios of right-angled triangles – sine (sin), cosine (cos) and tangent (tan) – in relation to the property of 'similarity' remains constant despite the size the triangle. Therefore, students will demonstrate their knowledge by applying the appropriate trigonometric ratio to solve problems involving these right-angled triangles, such as finding an unknown length or angle.
The concept of Pythagoras’ theorem will be relevant and useful in following measurement and trigonometry units of study.
Students will gain an understanding that the relationships between the sides of the right-angled triangle are the bases of the trigonometric ratios, and use this knowledge in future to explore the unit circle.
Teaching and learning summary:
- Revise similar figures and ratio.
- Revise application of Pythagoras’ theorem.
- Revise the types of triangles and vocabulary.
- Demonstrate the concept of using a common ratio to solve problems.
- Introduce the trigonometrical ratios and their meaning, recognise patterns and use reasoning.
- Use the properties of similarity to solve problems.
- Investigate real-life examples where trigonometry is used to give relevance and stimulate depth of thinking.
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Teaching strategies
A collection of evidence-based teaching strategies applicable to this topic. Note we have not included an exhaustive list and acknowledge that some strategies such as differentiation apply to all topics. The selected teaching strategies are suggested as particularly relevant, however you may decide to include other strategies as well.
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Concrete, Representational, Abstract (CRA)
The CRA model is a three-phased approach where students move from concrete or virtual manipulatives, to making visual representations and on to using symbolic notation.
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Explicit teaching
Explicit teaching is about making the learning intentions and success criteria clear, with the teacher using examples and working though problems, setting relevant learning tasks and checking student understanding and providing feedback.
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Mathematics investigation
By giving students meaningful problems to solve they are engaged and can apply their learning, thereby deepening their understanding.
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Worked examples
A worked example is not just a pre-worked question that is given to the students. There are several types of worked examples and ways of using them.
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Teaching resources
A range of resources to support you to build your student's understanding of these concepts, their skills and procedures. The resources incorporate a variety of teaching strategies.
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Introduction to trigonometry
This lesson plan guides teachers on how to introduce trigonometry to students through an investigation of similar triangles.
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Right-angle triangle trigonometry
In this unit students build an understanding of ratios in right triangles that leads to naming cosine, sine and tangent as trigonometric ratios.
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History of trigonometry
A three-part history of the origins of trigonometry.
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Pythagoras' theorem and Bhaskara
A visual demonstration of Bhaskara’s proof for Pythagoras’ theorem.
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Pythagoras' theorem and irrational numbers
A series of lessons and activities that includes an introduction and exploration of Pythagoras' theorem and its applications.
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Assessment
By the end of Year 9, students can apply Pythagoras’ theorem and use trigonometric ratios to solve problems involving right-angled triangles.
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Trigonometry
From the AMSI Supporting Australian Mathematics Project, this resource gives detailed explanations for teachers and students of the curriculum content with worked examples and assessment questions.
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Pythagoras’ theorem: phone finding
Students use Pythagoras’ theorem to explore practical technology that is reliant on mathematical concepts.
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