Mathematics proficiencies

Understanding / Fluency / Problem-Solving / Reasoning

Year 9

Number and algebra: Archers in the plane

Summary of task

Students were asked to find the coordinates of a point where an archery target could be placed such that it is equidistant from three archers.

Proficiencies

At this year level understanding includes describing the relationship between graphs and equations, simplifying a range of algebraic expressions and explaining the use of relative frequencies to estimate probabilities and of the trigonometric ratios for right-angle triangles.

At this year level fluency includes applying the index laws to expressions with integer indices, expressing numbers in scientific notation, listing outcomes for experiments, developing familiarity with calculations involving the Cartesian plane and calculating areas of shapes and surface areas of prisms.

At this year level problem-solving includes formulating and modelling practical situations involving surface areas and volumes of right prisms, applying ratio and scale factors to similar figures, solving problems involving right-angle trigonometry and collecting data from secondary sources to investigate an issue.

At this year level reasoning includes following mathematical arguments, evaluating media reports and using statistical knowledge to clarify situations, developing strategies in investigating similarity and sketching linear graphs.

1 Fluency
Calculates the midpoints and gradients of the intervals accurately and the gradients of the respective perpendicular bisectors of the intervals through the use of recalled formulas 2 Understanding
Connects the relevant plane geometry and coordinate geometry concepts by identifying the need to determine the respective midpoints and gradients of the intervals 3 Understanding
Identifies the need to determine the respective equations of the perpendicular bisectors of the intervals to determine the coordinates of the position in which the target should be placed 4 Understanding
Interprets the problem as a geometric problem involving the intersection of the perpendicular bisectors of the intervals joining points A and B and points A and C 5 Problem-Solving
Determines the equations of the perpendicular bisectors accurately, solves the equations simultaneously to determine their point of intersection, and communicates the solution effectively

  • Annotations

  • 1
    Fluency
    Calculates the midpoints and gradients of the intervals accurately and the gradients of the respective perpendicular bisectors of the intervals through the use of recalled formulas
  • 2
    Understanding
    Connects the relevant plane geometry and coordinate geometry concepts by identifying the need to determine the respective midpoints and gradients of the intervals
  • 3
    Understanding
    Identifies the need to determine the respective equations of the perpendicular bisectors of the intervals to determine the coordinates of the position in which the target should be placed
  • 4
    Understanding
    Interprets the problem as a geometric problem involving the intersection of the perpendicular bisectors of the intervals joining points A and B and points A and C
  • 5
    Problem-Solving
    Determines the equations of the perpendicular bisectors accurately, solves the equations simultaneously to determine their point of intersection, and communicates the solution effectively
1 Understanding
Connects congruency of triangles, Pythagoras’ Theorem and distance on the Cartesian plane to establish a proof that the target has been located to be equidistant from all three archers 2 Reasoning
Describes and justifies the strategy used to determine the point equidistant from all three archers

  • Annotations

  • 1
    Understanding
    Connects congruency of triangles, Pythagoras’ Theorem and distance on the Cartesian plane to establish a proof that the target has been located to be equidistant from all three archers
  • 2
    Reasoning
    Describes and justifies the strategy used to determine the point equidistant from all three archers