Geometry and Measurement
Level 1
Measurement
AO1: Order and compare objects or events by length, area, volume and capacity, weight (mass), turn (angle), temperature, and time by direct comparison and/or counting whole numbers of units. This means students will classify or sort objects by their characteristics. These characteristics include shape, size, colour, texture, weight, and temperature. Students should be able to justify why they have sorted objects in the way they have and be encouraged to develop increasingly sophisticated classifications such as colour and size.
Click to download a PDF of second-tier material relating to Level 1 Measurement (115KB)
Shape
AO1: Sort objects by their appearance. This means students will classify or sort objects by their characteristics. These characteristics include shape, size, colour, texture, weight, and temperature. Students should be able to justify why they have sorted objects in the way they have and be encouraged to develop increasingly sophisticated classifications such as colour and size.
Click to download a PDF of second-tier material relating to Level 1 Shape (79KB)
Position and orientation
AO1: Give and follow instructions for movement that involve distances, directions, and half or quarter turns. This means students will be able to follow instructions given in distances, for example 14 steps; direction for example facing the library; and angle (turn) for example do a half turn clockwise (right). They should become proficient at following a series of instructions. Students should also be able to give instructions.
AO2: Describe their position relative to a person or object. This means students will describe their position using positional language such as next to, in front of, behind, between, to the right/left and simple diagrams and maps. Their descriptions should become increasingly precise in terms of distance from the landmark (in steps) and location of that landmark on simple schematic maps.
Click to download a PDF of second-tier material relating to Level 1 Position and orientation (106KB)
Transformation
AO1: Communicate and record the results of translations, refl ections, and rotations on plane shapes. This means students will physically carry out translations, reflections, and rotations on shapes and discuss what patterns they see. Translations are shifts of a shape along a line, for example repeating a potato print across the top border of a page. Reflections are images of a shape as though it is reflected in a mirror. Rotations are turns, so when an object is turned about a point, either inside or outside of itself, the image is a rotation of the original shape. At level one rotations can be described as fractions of a full turn, for example half and quarter turns.
Level 2
Measurement
AO1: Create and use appropriate units and devices to measure length, area, volume and capacity, weight (mass), turn (angle), temperature, and time. This means students will recognise that the attributes length, area, volume and capacity, and weight can be measured. At Level Two students are expected to recognise that measurement units are countable and therefore able to be partitioned and recombined in the same way as other units of one, e.g. If a 8 unit length is cut from a 14 unit long strip the remainder will measure 6 units. Units of measure have other characteristics including being a part of the attribute they measure and uniformity (same size). When measuring the units need to fill a length, space, time etc, with no gaps or overlaps (this is known as tiling). Students should create measurement devices, e.g. rulers, rod towers, scales, to quantify the attributes of objects in numbers of units. In doing so they should develop an understanding that the marks on a linear scale show the endpoint of units and that scales always have a baseline (zero). Less tangible attributes such as turn (angle), temperature and time should also be measured. While the focus at Level Two is on students’ understanding the role of units in measurement it is also expected that students will encounter simple standard measures such as metres, centimetres, kilometres, minutes, seconds, kilograms, litres, etc, through using everyday measurement instruments.
AO2: Partition and/or combine like measures and communicate them, using numbers and units. This means students will perform and communicate calculations involving like measures. Like measures involve the same units for the same attribute. This allows the result of joining or separating units to be anticipated using additive number strategies, e.g. A box has a volume of 36 cubes. If a 3 by 4 cube layer is put in the empty box then there will be space for 36 – 12 = 24 more cubes. At Level Two students should be able to use numbers and common symbols to communicate measurement results, e.g. my lunchbox holds 60 cubes. I took 13 minutes to walk home. My pencil is 14 cm long. I weigh 26 kg.
Click to download a PDF of second-tier material relating to Level 2 Measurement (119KB)
Shape
AO1: Sort objects by their spatial features, with justification. This means students will sort objects by selecting an attribute or attributes by which to classify items and allocating the items into groups by commonality of that attribute. At Level Two students should be able to find their own system to classify items, using attributes like shape, colour, size, texture, thickness, material, purpose, etc and justify their allocation of items into categories, e.g. "all of these shapes have three sides". In doing so students should develop geometric language for attributes such as "side", "corner", "centre", "face", "edge", "curved", "straight", "larger", "smaller", etc.
AO2: Identify and describe the plane shapes found in objects. This means students will be able to identify plane (flat) shapes in objects and structures around them and consider why the given shape is suitable for its purpose, e.g. wheels are circular so they roll freely, floors are usually rectangles because they are easier to build and things fit efficiently, etc. They should consider how three dimensional objects are built from flat shapes through pulling packets apart and constructing solids of their own, e.g. nets for cubes.
Click to download a PDF of second-tier material relating to Level 2 Shape (83KB)
Position and orientation
AO1: Create and use simple maps to show position and direction. At Level Two students should be able to use simple schematic maps, e.g. plans of their school, road maps of their local area. This involves finding their current position on a map by connecting landmarks they can see with locations on the map. Similarly it involves finding the place that matches a given point on the map and describing how they would move from one point to another. Descriptions of movement should include features such as main compass directions (N, S, E, W), half and quarter turns, and approximate distances in whole numbers of metres (e.g. about 12 metres) . Students should use simple co-ordinates (e.g. B5) to specify locations on schematic maps.
AO2: Describe different views and pathways from locations on a map. This objective requires students to see schematic maps as a two dimensional representation of the real world. By looking at a map students should be able to anticipate landmarks they will see from a given location and in which direction (N, S, E, W) those landmarks will be seen. From a map they should give a set of directions, using distances in whole numbers of metres and quarter/half turns, that will take a person from one position on the map to another, e.g. turn right and walk about 25 metres.
Click to download a PDF of second-tier material relating to Level 2 Position and orientation (126KB)
Transformation
AO1: Predict and communicate the results of translations, reflections, and
rotations on plane shapes.
This means students will experience physically moving shapes so that they can predict the
location and orientation of the shape after it has been translated, reflected or rotated, e.g.
draw/show what this shape will look like if I give it a half turn about its centre.
Students should be able to identify how many mirror lines a shape has that maps it onto itself,
e.g. a square has four mirror lines. Translations are images of a shape as it is shifted along a
line, e.g. 
... Reflections are
images of a shape as it is reflected in a mirror (sometimes called a flip), e.g. 
Note that the line may outside the object or within it. Rotations are images of a shape as it is turned about a point outside or within it,
e.g. 
.
Level 3
Measurement
AO1: Use linear scales and whole numbers of metric units for length, area, volume and capacity, weight (mass), angle, temperature, and time. This means students will recognise that length, area, volume and capacity, weight, angle, and temperature are the characteristics (attributes) of objects people most commonly measure in everyday life. Time is a special attribute since it is not tangibly attached to physical objects. Measurement involves quantifying an attribute using units. Units of measure have characteristics including being a part of the attribute they measure and uniformity (same size). When measuring the units need to fill a length, space, time etc., with no gaps or overlaps (this is known as tiling). At Level Three, students should be familiar with common units in the metric system for the attributes listed. These units include, metres, centimetres, millimetres, and kilometres for length/distance, square and cubic centimetres, and metres for area and volume, kilograms and grams for weight, quarter and half turns for angles, degrees Celsius for temperature, and seconds, minutes, hours, days, etc for time. It is not expected that students will know the size relationships between measures though they should have opportunities to explore these relationships, e.g. 15cm = 150mm. Using measurement instruments involves reading linear scales (an analogue clock face is three connected linear scales). Students should understand that the marks on a linear scale show the endpoint of units and that scales always have a baseline (zero). Part of measurement is selecting the scale, with the precision of unit, suitable for a task. Level Three students should apply their additive and multiplicative number strategies to measurement problems that involve whole numbers of units, e.g. how many cubic centimetres will fit in this packet?
AO2: Find areas of rectangles and volumes of cuboids by applying multiplication. This means students will begin by measuring the areas of rectangles and other shapes using square units. This is because square units of the same size tessellate, that is join together with no laps or overlaps. That means that the measurement is consistent whereas the use of a non-tessellating unit would give variable results due to gaps and overlaps. Similarly, volume is measured in cubes of the same size. At Level Three students should apply whole number multiplication to make the process of counting squares or cubes more efficient..
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4 x 6 = 24 square units |
3 x 4 x 6 = 72 cubic units |
Click to download a PDF of second-tier material relating to Level 3 Measurement (354KB)
Shape
AO1: Classify plane shapes and prisms by their spatial features. This means students will be able to classify is to define the characteristics of things and use these characteristics as a basis for sorting. Plane figures are those that lie flat, so have only two dimensions. So circles, triangles, and hexagons are all plane shapes. At Level Three students should be able to classify plane shapes by the following characteristics, number of sides and angles, e.g. all triangles have three sides, parallel or non-parallel sides, e.g. a trapezium has one pair of parallel sides, equal or unequal side length, angle size (less than, equal to, greater than a right angle), lines of mirror symmetry and order of rotational symmetry (e.g. A square maps onto itself four times in a full turn). Prisms are solid shapes that have a fixed cross-section. A loaf of bread can be seen as a rectangular prism since the slices are the same rectangle. So prisms are classified by their cross-section, e.g. a triangular prism has triangular cross-sections. In this way a cylinder can be seen as a type of prism though its cross-section is a circle. .
AO2: Represent objects with drawings and models. This means students will make drawings of objects can take the form of isometric projections, plan views or nets.
Note that a net is a flat shape that folds to form a solid. Many different nets form the same solid, e.g. there are eleven different nets that form a cube. At Level Three students need to create two-dimensional drawings of three-dimensional models, as above, and be able to recreate the model when given another person’s drawings of it. Models may be built with interlocking cubes, plasticine, connecting geometric shapes or other materials, e.g.toothpicks and plasticine.
Click to download a PDF of second-tier material relating to Level 3 Shape (95KB)
Position and orientation
AO1: Use a co-ordinate system or the language of direction and distance to specify locations and describe paths. This means students will use co-ordinate systems that are used on maps to specify location and direction (e.g.Greensborough Reserve is at D1, Ruakura Road runs West-East). The scale of a map indicates distance.
At Level Three students should be able to:
- Give the location of something using co-ordinate references, e.g. A3.
- Find the location of something given a co-ordinate reference, e.g. Find Daphne Street at E8.
- Use features of a map to describe movement that would get someone from one location to another, including distance and direction. This includes turns (right, left relative to orientation), main compass directions (N,W,S,E) and approximate distances in metres or kilometres.
- Follow a set of directions given in terms of turns and distances (as above) and show that path they walked on a map of the area.
Transformation
AO1: Describe the transformations (refl ection, rotation, translation, or enlargement) that have mapped one object onto another. This means students will explore describe transformations. “Transformation” is a generic term used to describe actions on shapes that result in some form of pattern, usually symmetric. A reflection is the image of a shape as seen through a mirror line either inside or outside the shape, sometimes called a “flip”. A rotation is the image of the shape turned about a point either inside or outside the shape. A translation is the image of a shift of the shape along a line, and an enlargement is the image of the shape made bigger or smaller by some scale factor. At Level Three students should be able to compare the image of a shape with the original and describe the transformation. This can include a sequence of two transformations. For example:
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A' is the image of A |
B' is the image of B |
C' is the image of C |
Level 4
Measurement
AO1: Use appropriate scales, devices, and metric units for length, area, volume and capacity, weight (mass), temperature, angle, and time. This means students will work with the commonly used units of the metric system and measurement devices for:
- Length- kilometre, metre, centimetre, millimetre, using rulers and tape measures
- Area- square kilometre, hectare, square metre, square centimetre
- Volume/Capacity- Cubic kilometre, cubic metre, litre (cubic decimetre), cubic centimetre using jugs, measurement cylinders
- Weight- tonne, kilogram, gram, using scales
- Temperature- degrees Celcius using thermometers
AO2: Convert between metric units, using whole numbers and commonly used decimals. This means students will apply their knowledge of decimal place value to convert between units for the same attribute, e.g. between units for weight. They should know the meaning of prefixes used in the metric system that act as “scalars” on base units, e.g. “kilo” means one thousand, “centi” means one hundredth. Conversions are restricted to whole number and simple decimal (tenth, hundredth, thousandth) scalars, e.g. 5kg = 5000g or 300ml = 0.3L, that act on whole number or decimal measures involving tenths, e.g. 0.6ha = 6000m2 or 675mm = 67.5cm.
AO3: Use side or edge lengths to fi nd the perimeters and areas of rectangles, parallelograms, and triangles and the volumes of cuboids. At Level Four students should apply their multiplicative strategies to find perimeters and areas of commonly used polygons and volumes of cuboids where the lengths of sides and edges are given as whole number measures. Calculations required are:.
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Rectangle: Area = base x height |
Parallelogram: Area = base x height |
Triangle: Area = 1/2 (base x height) |
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Cuboid: Volume = base x height x depth |
Students should express the areas and volumes using symbols, e.g. 48cm3 (48 cubic centimetres).
AO4: Interpret and use scales, timetables, and charts. This means students will be literate in getting required information from the following:
- Scales, such as thermometers, analogue (and digital) clocks, rulers, protractors, weight scales, capacity containers.
- Timetables, such as those used in transport (12 or 24-hour time), tides, broadcast programming, telephone books (international calls), sports events.
- Charts used to convey measurement information, such as weather reports, cooking recipes, Guiness Book of Records, statistics on living organisms.
Click to download a PDF of second-tier material relating to Level 4 Measurement (150KB)
Shape
AO1: Identify classes of two- and three-dimensional shapes by their geometric properties.
This means students will use geometric properties to identify classes of shapes. Classes are categories of two or three-dimensional shapes. Shapes are sorted into classes according to defined geometric properties, such as number and relationship of sides (e.g. equal and parallel); number and nature of angles (e.g. four right angles); symmetry, number, nature, and shape of faces and surfaces (for 3-dimensional shapes). Classes can be included within other classes, can intersect or be disjoint, e.g. all squares are rectangles or No triangles are pentagons. At Level Four students should be familiar with:
- Classes of polygons defined by the number of sides; triangles (3 sides), quadrilaterals (4 sides), pentagons (5 sides), hexagons (6 sides),...octagons (8 sides),...
- Classes of 3-dimensional shapes defined by the nature of faces and surfaces; prisms (constant cross-section) and cylinders, pyramids and cones, regular polyhedral (identical faces).
- Classes of 2-dimensional closed curves and their 3-dimensional equivalents by rotation; circles and spheres, ellipses and ellipsoids.
- Sub-classes that are included within classes; squares within rectangles, rectangles within parallelograms, parallelograms within quadrilaterals, circles within ellipses, cubes within rectangular prisms.
- Classes that are disjoint, scalene and isosceles triangles, prisms and pyramids.
AO2: Relate three-dimensional models to two-dimensional representations, and vice versa.
This means students will focus on key characteristics of 3-dimensional models (shape and relationship of faces and surfaces, faces joining at edges and vertices) to create 2-dimensional drawings of those models. Drawings of objects can take the form of isometric projections, plan views or nets. Students should also be able to construct a model from given 2-dimensional drawings, e.g. build a model using interlocking cubes from the plan views below.
Students should be able to create nets for simple polyhedral and closed surfaces by visualising the “unwrapping” of those solids, e.g. the net for a dodecahedron.
Position and orientation
AO1: Communicate and interpret locations and directions, using compass directions, distances, and grid references. This means students will apply their understanding of the measurement system, particularly of length and angle. This involves converting the scale on a map to actual measurements and describing direction given the orientation of North.
- Give or interpret the location of a feature on a map using grid references, distances and direction from a landmark, e.g. AA 24 (Street map), 536 721 (topographical map), 160m South-East of the library, Latitude 12o South, 77o East (World map), 2047km South-West of Los Angeles.
- Follow instructions given by others using compass directions, distances and grid references by interpreting a scale map, e.g. Travel from New Plymouth to Tauranga.
Transformation
AO1: Use the invariant properties of figures and objects under transformations (reflection, rotation, translation, or enlargement).
This means students will know invariant properties are those features of a figure that do not change as it is reflected, rotated, translated or enlarged.
Under rotation lengths, areas, angles do not change but orientation does.
Under reflection lengths, areas and angles do not change but orientation does.
Under translation lengths, areas, angles and orientation do not change.
Under enlargement angles and orientation do not change but lengths and areas do.
At Level Four students should be able to use the above invariant properties to create symmetrical patterns such as tessellations, logos and friezes, and to create enlarged copies of graphics.
Level 5
Measurement
AO1: Select and use appropriate metric units for length, area, volume and capacity, weight (mass), temperature, angle, and time, with awareness that measurements are approximate. Further detail on this Achievement Objective will be added shortly.
AO2: Convert between metric units, using decimals. Further detail on this Achievement Objective will be added shortly.
AO3: Deduce and use formulae to find the perimeters and areas of polygons and the volumes of prisms. Further detail on this Achievement Objective will be added shortly.
AO4: Find the perimeters and areas of circles and composite shapes and the volumes of prisms, including cylinders. Further detail on this Achievement Objective will be added shortly.
Click to download a PDF of second-tier material relating to Level 5 Measurement (207KB)
Shape
AO1: Deduce the angle properties of intersecting and parallel lines and the angle properties of polygons and apply these properties. Further detail on this Achievement Objective will be added shortly.
AO2: Create accurate nets for simple polyhedra and connect three-dimensional solids with different two-dimensional representations. Further detail on this Achievement Objective will be added shortly.
Position and orientation
AO1: Construct and describe simple loci. Further detail on this Achievement Objective will be added shortly.
AO2: Interpret points and lines on co-ordinate planes, including scales and bearings on maps. Further detail on this Achievement Objective will be added shortly.
Transformation
AO1: Define and use transformations and describe the invariant properties of figures and objects under these transformations. Further detail on this Achievement Objective will be added shortly.
AO2: Apply trigonometric ratios and Pythagoras’ theorem in two dimensions. Further detail on this Achievement Objective will be added shortly.
Level 6
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.
Attribute | Units |
Length | metre (m), micrometre ( m), millimetre (mm), centimetre (cm), kilometre (km) |
Area | square metre (m2), square millimetre (mm2),square centimetre (cm2), hectare (ha), square kilometre (km2) |
Volume | cubic metre (m3), cubic centimetre (cm3), cubic decimetre (litre),cubic kilometre (km3) |
Capacity | litre (L), millilitre (mL), decilitre (dL) |
Mass | gram (g), microgram (μ g), milligram (mg), kilogram (kg), tonne (t) |
Time | second (s), microsecond (μs), millisecond (ms), minute, hour, day, etc | Temperature | degree Celsius (° C). |
Angle | degree (° ). |
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:
- Angle at centre to any chord is twice the angle at circumference.
- Angles at the circumference to any chord are equal. .
- Angle between a chord and a tangent equals the angle in the opposite segment. .
- For a triangle in a semi-circle the angle at circumference equals 90?..
- 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)
