Mathematics / Year 9 / Measurement

Curriculum content descriptions

solve spatial problems, applying angle properties, scale, similarity, Pythagoras’ theorem and trigonometry in right-angled triangles (AC9M9M03)

Elaborations
  • investigating the applications of Pythagoras’ theorem in authentic problems, including applying Pythagoras’ theorem and trigonometry to problems in surveying and design
  • applying the formula for calculation of distances between points on the Cartesian plane from their coordinates, emphasising the connection to vertical and horizontal displacements between the points
  • understanding the relationship between the corresponding sides of similar right-angled triangles and establishing the relationship between areas of similar figures and the ratio of corresponding sides, the scale factor
  • using images of proportional relationships to estimate actual measurements; for example, taking a photograph of a person standing in front of a tree and using the image and scale to estimate the height of the tree, discussing the findings and ways to improve the estimates
  • investigating theorems and conjectures involving triangles; for example, the triangle inequality, and generalising links between the Pythagorean rule for right-angled triangles, and related inequalities for acute and obtuse triangles; determining the minimal sets of information for a triangle from which other measures can all be determined
  • using knowledge of similar triangles, Pythagoras’ theorem, rates and algebra to design and construct a Biltmore stick used to measure the diameter and height of a tree, and calculating the density and dry mass to predict how much paper could be manufactured from the tree
  • investigating how autonomous vehicles solve spatial problems using algorithms based on geometric properties relating to angles, distances and scale
General capabilities
  • Critical and creative thinking Critical and Creative Thinking
  • Numeracy Numeracy
ScOT terms

Similarity (Geometry),  Trigonometric ratios,  Distance,  Ratios,  Scale (Proportions),  Pythagoras theorem

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reSolve: Mathematical modelling: The steepest slopes on Baldwin St

In this sequence of three lessons, students measure and represent steepness in multiple ways, connect gradient to tangent, and consider how mathematical information is communicated to real audiences; using the context of Baldwin Street, the world’s steepest residential street. Key learning goals, pedagogical tools and design ...

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Introduction to trigonometry

These lesson plans guide the teacher on how to introduce trigonometry to students through an investigation of similar triangles.

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Mathematics of bushfire

In these classroom activities students will be introduced to some of the basic mathematical principles that underpin wildfire science, with an emphasis on how theoretical concepts are used to aid our understanding of the real world, and bushfire in particular. They will learn about the complexities of the fire management ...

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reSolve: Tree Biomass

In this sequence of two lessons, students investigate how many trees would be required to supply paper for their school for a year. Students use similar triangles, Pythagoras' Theorem and algebra to design and construct a Biltmore stick, used to measure the diameter and height of a tree. They measure trees, calculate their ...

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reSolve: Pythagoras' Theorem - Lunch Lap

In this sequence of two lessons, students apply Pythagoras' Theorem to explore a practical problem involving optimising paths to lunch carts. In the first lesson, students investigate the length of a path that touches three sides of a rectangle, starting and finishing at the same point. They model the problem, use Pythagoras' ...

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reSolve: Monte Carlo Simulations

This sequence of two lessons explores how statistical techniques that rely on randomly generated data can be used to solve problems. In the first lesson, students compare different methods for calculating the area of an irregular shape, using the context of oil spill maps. They are introduced to the Monte Carlo method for ...

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reSolve: Gradient and Tangent

This sequence of two lessons investigates gradient and angle by applying the tangent ratio to find the angles represented by a road sign or the angle of a street. In the first lesson, students research what a road grade is and determine the actual angle of a road given its grade. They then construct their own road sign ...

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reSolve: Pythagoras' Theorem - Phone finding

This lesson challenges students to apply Pythagoras' Theorem to explore a practical real-world problem. Students explore technology reliant on mathematical concepts. The lesson is outlined in detail including curriculum links, vocabulary, materials needed, sample answers, discussion points and student resources such as ...

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reSolve: Mechanical Linkages: Similar Triangles

This sequence of lessons explores the geometry of similar triangles using two real world objects: ironing boards and pantographs. In the first lesson, students investigate different ironing board leg lengths and pivot positions using similar and congruent triangles. In the second lesson, they use their knowledge of parallelogram ...

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MathXplosion, Ep 50: How to use a tetrahedron to solve the tree problem

How can you place four trees exactly the same distance apart from one other? By making a model! By using miniature trees to make a model of the problem, it becomes clear that a 2D solution is impossible. We learn how objects can help us visualise the problem situation, which in this case requires a 3D solution: a tetrahedron.

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Maths inside bees and beehives

Bees are necessary for assisting many plants to produce the food we eat, including meat and milk. Colony collapse disorder, which describes the disappearance of beehives, could have catastrophic effects on food production. Australian scientists are applying their maths and science knowledge to build up a picture of a healthy ...

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Mystery man Pythagoras meets his match

What do you know about Pythagoras? Join Vi Hart as she not only explains his theorem but raises some legends about his dark past! Follow Vi's timeline of famous mathematicians to find out in which century Pythagoras lived. See how Vi shows a proof of his theorem and raises what was a big dilemma for Pythagoras: the irrational ...

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Laptop wrap: Talking trigonometry

In this laptop-friendly resource, students consolidate their understanding of trigonometry by investigating practical applications of the ratios, highlighting the process they used to find a solution.

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Trigonometry

This is a website designed for both teachers and students, which addresses content on trigonometry from the Australian Curriculum for year 9 students. It contains material on working with the similarity of right-angled triangles, the three basic trigonometric ratios, and solving problems with trigonometry. There are pages ...

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Similarity

This is a website designed for both teachers and students, which addresses similarity from the Australian Curriculum for year 9 students. It contains material on enlargement transformation and similar triangles. There are pages for both teachers and students. The student pages contain interactive questions for students ...

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TIMES Module 22: Measurement and Geometry: scale drawings and similarity - teacher guide

This is a 41-page guide for teachers. It contains an introduction to scale drawings and similarity, and in particular the tests for triangles to be considered similar. Applications of similarity are included throughout the module.

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MathXplosion, Ep 10: What is the strongest shape?

Are triangles really the strongest shapes ever? If so, why? Learn how and why right-angled and equilateral triangles have been used in engineering, architecture and design through the ages.

Interactive

Renovate, Calculate!

A student resource that explores the use of mathematics in the trades. Highly interactive investigations into ratio, areas of special quadrilaterals and right-angled trigonometry.

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reSolve: Real World Algebra: Linear Functions

This sequence of two lessons explores real world, algebraic, numeric and graphical representations of linear functions, focussing on the gradient of linear functions. Students use images of everyday objects and inscribe straight lines, then use Geogebra to explore the algebraic representation of their lines. Each lesson ...

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reSolve: Rectilinear Polygons

In this sequence of three lessons, students use geometric reasoning to establish relationships between angles in polygons and go on to make generalisations using algebraic expressions. Students explore and enumerate right angles in a series of rectilinear polygons and generalise their findings. They then explore the number ...