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WS02 - Proof and conjecture

Mathematics, Year 10

By the end of Year 10, students recognise the effect of approximations of real  numbers in repeated calculations. They use mathematical modelling to solve problems involving growth and decay in financial and other applied situations, applying  linear,  quadratic and exponential functions as appropriate, and solve related equations, numerically and graphically. Students make and test conjectures involving functions and relations using digital tools. They solve problems involving simultaneous linear equations and linear inequalities in 2 variables graphically and justify solutions. Students interpret and use logarithmic scales representing small or large quantities or change in applied contexts.  

 

They solve measurement problems involving surface area and volume of composite objects. Students apply Pythagoras’ theorem and trigonometry to solve practical problems involving right-angled triangles. They identify the impact of measurement errors on the accuracy of results. Students use mathematical modelling to solve practical problems involving  proportion and scaling, evaluating and modifying models, and reporting assumptions, methods and findings. They use deductive reasoning, theorems and algorithms to solve  spatial problems. Students interpret networks used to represent practical situations and describe connectedness 

 

They plan and conduct statistical investigations involving bivariate data. Students represent the distribution of data involving 2 variables, using tables and scatter plots, and comment on possible association. They analyse inferences  and conclusions in the media, noting potential sources of bias. Students compare the distribution of continuous numerical  data  using various displays, and discuss distributions in terms of centre, spread, shape and outliers. They apply conditional probability  to solve problems involving compound events. Students design and conduct simulations involving conditional probability, using digital tools. 

Measurement

AC9M10M03

solve practical problems applying Pythagoras’ theorem and trigonometry of right-angled triangles, including problems involving direction and angles of elevation and depression

Space

AC9M10SP01

apply deductive reasoning to proofs involving shapes in the plane and use theorems to solve spatial problems

Space

AC9M10SP03

design, test and refine solutions to spatial problems using algorithms and digital tools; communicate and justify solutions

Annotations

 

1. Presents a portfolio of learning to demonstrate their understanding of triangles and ability to formulate proofs involving congruent triangles and angle properties. 

 

2. Identifies the necessary prior learning for formulating triangle proofs. 

 

3. Investigates right-angled triangles and identifies the difference between similar and congruent triangles, connecting the problem to known theorems. 

 

4. Generalises the least amount of information required to define pairs of congruent triangles. 

5. Generalises the properties of similar triangles. 

 

6. Lists the theorems and rules for congruency in triangles.

7. Uses deductive reasoning skills and demonstrates knowledge of the SAS rule, to prove triangles are congruent. 

 

8. Uses deductive reasoning skills to determine that there is not enough information given to prove triangles are congruent. 

 

9. Uses deductive reasoning skills and demonstrates knowledge of the SSS rule to prove triangles are congruent.

10. Applies knowledge of congruent triangles and angle properties to solve problems, and applies deductive reasoning to prove solutions.

11. Applies knowledge of similar triangles and deductive reasoning to find the height of the ladder. 

12. Applies knowledge of similar triangles and deductive reasoning to find the width of the void.