Exploring complex trig problems involving bearings and three dimensional applications Investigate the circle, tangent and chord theorems leading to the idea of geometric proof Use these skills, with technology, posing as a sports scientist, to improve the chances of converting a try in rugby
Conceptual Framework
Form measurement, generalization and justification in the context of Scientific and Technical Innovation
Statements of Inquiry
Understanding form and generalizing geometric relationships allows us to solve real-life problems
Factual
What are the relationship between sides and angles in triangles?
What is the relationship between angles inscribed in circles?
Conceptual
How can geometry help us find optimum solutions to a real life problem?
Debatable
To what extent does the use of mathematics help improve performance in sport?
Description
This unit will explore the relationships between the sides and angles of a triangle, namely trigonometry. They will use trigonometric ratios and the sine and cosine rules to find missing sides and angles in triangles. The unit will also address how to prove a variety of Circle, Chord and Tangent Theorems. The unit will culminate with an investigation into the best position to place a rugby ball in order to score a conversion after a try in Rugby. The students will be asked to be sports scientists, using Scientific and Technological Innovation in order to prepare a report for a Rugby coach. The students will generalise a rule connecting the position the try was placed and the position of the conversion kick in order to maximise the angle created between the player and the two posts. The rule will be justified by using formal geometric proofs and percentage errors in measurement.
Learning Outcomes
Review the parts of a right-angled triangle and use Pythagoras to find missing sides
Investigate and consolidate understanding of the three primary trigonometric ratios – sine, cosine and tangent
Find the sine, cosine and tangent for an angle of a right triangle – exact or approximate values using a GDC
Use SOHCAHTOA to find missing sides and angles in right angled triangles
Find the exact values of sine, cosine and tangent for angles of 30, 45, 60 using special triangles
Solve real life problems involving right angled triangles
Use the sine and cosine rules to solve problems involving non-right triangles
Use the sine and cosine rules to solve problems involving bearings and other real-life applications
State, prove and apply the following theorems about angles in a circle: angles inscribed in a semicircle, angle at the centre is twice the angle at the circumference, angles inscribed in the same segment, angles in a cyclic quadrilateral, alternate segment theorem
State, prove and apply the following theorems about chords in a circle: perpendicular bisector of a chord passes through the centre (and the converse), intersecting chords theorem
State, prove and apply the following theorems about tangents to a circle: radius drawn to a tangent at the point of tangency, tangents drawn from a point outside the circle