Elaborations on Level Six: Geometry and Measurement

Measurement

AO1: Measure at a level of precision appropriate to the task.

This means students will be able to identify and use a unit of measurement the meets the requirements of a given task, for example, the length of a fence paling in millimetres. This involves understanding of the accuracy required in a given context including appreciation of the practical consequences of both less accuracy and greater precision. This includes the effect of using different measures of length on the accuracy of the resulting area and volume measures. Students need to understand the necessary compromise at times between accuracy and adequacy, that is, something could be measured more precisely but there is greater effort required and no gain to meeting the demands of the task. This objective also includes making sensible estimates where appropriate, for example, estimating the number of rolls of wallpaper or litres of paint needed for a given space.

AO2: Apply the relationships between units in the metric system, including the units for measuring different attributes and derived measures.  

This means students will know the commonly used units including the role of prefixes as conversion factors of base units, e.g. kilo meaning one thousand, micro meaning one millionth. Below is a list of units for key attributes that should be expected.

Students are also expected to know derived measures that describe rates involving the units above and other common units. Attributes and the derived units used to measure them include speed (kilometres per hour km/h, metres per second m/s), fuel and energy consumption (litres per 100 kilometres L/100km, joules or calories per minute ), unit price (cents or dollars per gram), and density (kilograms per cubic metre kg/m³ , grams per cubic centimetre g/cm³). More complicated derived measures such as those for pressure, force, and power are not expected at Level Six.

Students should be able to connect the units for volume (capacity) and mass, e.g. Find the mass of 345mL of water, and convert between the simple derived units above, e.g. 140 km/h =  m/s.

AO3: Calculate volumes, including prisms, pyramids, cones, and spheres, using formulae.  

This means students will connect the formulae for the volume of prisms, including cylinders, as area of the cross-section or base multiplied by the third dimension, e.g. for a rectangular based prism v = l x w x h (l x w is the area of the cross section). This involves recognising how to apply the formula for prisms given any orientation of the solid that is presented. Similarly students should connect the formulae for the volume of pyramids, including cones, as the area of the base multiplied by one third of the height, e.g. for a cone v = πr²h (πr² is the area of the base). Students should know and apply the formula for volume of a sphere as v = 4/3 πr³. At Level Six students are expected to work with decimal measures as well as whole number measures.

Shape

AO1: Deduce and apply the angle properties related to circles.

This means students will know and apply the sum of interior angles of a triangle (180°) and the angle between a radius and tangent (90°) to deduce the angle properties related to circles.

The angle properties of circles expected are:

1. Angle at centre to any chord is twice the angle at circumference.
2. Angles at the circumference to any chord are equal. .
3. Angle between a chord and a tangent equals the angle in the opposite segment. .
4. For a triangle in a semi-circle the angle at circumference equals 90?..
5. Opposite angles in a cyclic quadrilateral add to 360?. .

Students are expected to connect at least two of these properties to find unknown angles in a given problem and to communicate their reasoning, citing the angle properties used.

AO2: Recognise when shapes are similar and use proportional reasoning to find an unknown length.  

This means students will know what properties of shapes are conserved as they are enlarged (or reduced) to scale. In particular this refers to angles and the ratios of side lengths within a figure and between a figure and its enlargement. They should also apply knowledge that area increases by the square of the scale factor and volume increases by the cube of the scale factor. Students should solve problems in which they find unknown lengths of shapes that are both regular and irregular. For example: Given that the ellipses are similar, find the unknown length of the major axis.

Given the rectangles are similar find the values of c and d.

AO3: Use trigonometric ratios and Pythagoras’ theorem in two and three dimensions.  

This means students will , given the required measurements, be able to connect trigonometric ratios (sine, cosine or tangent) in two dimensions with the demands of three dimensional problems, usually applying Pythagoras’ theorem in two or three dimensions in doing so. The problems should involve finding either an unknown length or angle. For example, find the angle <abc and the length of the line bc.

This objective also applies to problems where points are described using co-ordinates in two dimensions (four quadrants). For example, a triangle has corners at (2,3), (1,7), and (5,5). Find the lengths of its sides.

Position and orientation

AO1: Use a co-ordinate plane or map to show points in common and areas contained by two or more loci.  

This means the students will be able to use algebra and graphing to find a point in common with two intersecting lines when given their equations and connect this understanding with contexts that can be modelled with simultaneous linear equations in two variables. A loci is a set of points satisfying a given condition so common examples are lines, circles and ellipses, parabolas, and hyperbolas.

They should be able to sketch the locus for a given condition and recognise when that condition meets that of a conic section, e.g. perimeter of the flight area of a jet taking off and returning to a moving aircraft carrier (ellipse). Using graphing techniques students should be able to find the point or points in common between a line (given two points) and a conic (given the condition) and describe the area bounded by common lines and conics, for example, the grazing area of a tethered animal constrained by a wall.

Transformation

AO1: Compare and apply single and multiple transformations.  

This means students will be able to draw, with the assistance of technology where available, the results of transformations acting successively on a figure, for example, frieze patterns. The transformations involved are reflection, rotation, translation and enlargement. They should recognise when combinations of transformations give the same or a different result, for example, reflection then translation has the same result as translation then reflection (glide reflections), and acknowledge this in describing which transformations result in a figure being mapped onto a given image. Students should also connect the result of translations and reflections on lines and parabolas with the similarities and differences in their equations, for example, the image of y = x² + 3 reflected in the x-axis is y = - (x² + 3) or y = - x²- 3.

AO2: Analyse symmetrical patterns by the transformations used to create them.  

This means students will apply their knowledge of variant and invariant properties under these translations in explaining how they determined which translations were involved in a given mapping. This includes attendance to equality of lengths and angles, and order (direction). For example, students should describe how the following frieze pattern may have been created from the arrow element... (possibilities include)