NZC - Mathematics and statistics Phase 2 (Years 4–6)

During Year 4

During Year 5

During Year 6

During Year 4

During Year 5

During Year 6

Equations and relationships

• Numbers can be compared using “greater than” (>), “less than” (<), and equals (=).
• Applying the same operation to both sides of a number sentence preserves the balance.

• The relative size of expressions involving numbers can be communicated using “greater than” (>), “less than” (<), and equals (=).

• Inequalities can also include 'or equal to' (≤, ≥) to show a relationship that allows for the possibility of equality.

• Checking the truth of number sentences and completing open number sentences involving addition and subtraction (e.g. 8205 − 4721 = 3484, true or false?; 4200- __ = 4001)
• Checking the truth of number sentences and completing open number sentences involving multiplication and division (e.g. 11 × 7 = 78, true or false?; __ ÷ 10 = 12).

• Completing number sentences that involve addition and subtraction by using equality (=) and inequality (<, >) symbols
  (e.g. 2,456 + 203,938 ⬚ 3,456 + 231,930;
  2,456 × 2 ⬚ 1,228 × 4)
• Checking the truth of number sentences and completing open number sentences (e.g. 999,999 − __ = 899,999)

• Developing a rule for a growing pattern in words and making conjectures about further elements in the pattern 
• Checking the truth of and completing open number sentences that involve all four operations and that include the use of inequalities, respecting the order of operations (e.g.
  8 × 7 ≤ 8 × 5 + 4², true or false?)

• Growing patterns can increase or decrease by the addition or subtraction of a constant (arithmetically) or multiplication or division by a constant (geometrically).

• Tables provide a way of organising the positions and elements of a pattern to reveal relationships or rules.
• A coordinate plane is formed when two perpendicular number lines intersect at (0, 0); usually the coordinate plane consists of a horizontal x-axis and a vertical y-axis.
• Coordinates are represented as (x,y), where the x-value represents horizontal movement and the y-value represents vertical movement. Plotting points on a coordinate plane can help to visualise numeric patterns.

• Recognising, continuing, creating, and describing growing patterns (including numerical and non-numerical patterns) that change by adding, subtracting, or multiplying by a constant whole number 
  (e.g. 5, 7, 9, 11, ... 3, 6, 12, 24, ...)

• Recognising, continuing, creating, and describing growing patterns that change by a constant amount 
  (e.g. 3, 4.5, 6, 7.5 ...)

• Developing a rule for a growing pattern in words and making conjectures about further elements in the pattern
• Locating coordinate points on a coordinate plane, including points found on the x- or y-axis
• Generating a table of values from a rule for a growing pattern and plotting these points on a coordinate plane

During Year 4

During Year 5

During Year 6

During Year 4

During Year 5

During Year 6

Measuring

• Different measurement tools and scales use different-sized units; the unit must be recorded with the measurement amount.

• Measurements can be approximated by referencing previously measured benchmark lengths, volumes, or areas.

• Using familiar objects (e.g. body parts) and experiences (e.g. time taken to travel to school, the temperature outside) to create estimation benchmarks
• Using the appropriate tool for measuring length, mass (weight), and capacity in mixed units (e.g. 1 m and 23 cm, 10 kg and 3 g, 2 L and 500 mL)
• Measuring temperature in degrees Celsius

• Accurately measuring length with a ruler, mass (weight) with scales, capacity with measuring jugs, temperature with a thermometer, and duration with a timer, using appropriate metric or time-based units or a combination of units (e.g. 2 hours and 30 minutes)

• Estimating (using benchmarks) length, mass (weight), capacity, temperature, and duration, using appropriate metric or time-based units or a combination of units

• Measurements can be communicated using mixed or decimal units.
• There are 1000 millimetres in a metre, 10 millimetres in a centimetre, 100 centimetres in a metre, and 1000 metres in a kilometre.

• Prefixes are added to base metric units to signify larger or smaller quantities. The prefix:
  • ‘milli–’ (m) signifies a unit one thousand times smaller than the base unit
  • ‘centi–’ (c) signifies a unit one hundred times smaller than the base unit
  • ‘kilo–’ (k) signifies a unit one thousand times larger than the base unit.
• Converting between metric units can involve multiplying and dividing by 10, 100, or 1000.

• Converting metric units of length (m and cm)

• Converting metric units of length (m and cm), mass (g and kg), and capacity (L and mL), including combining mixed units to produce units with up to 2 decimal places (e.g. 10 kg and 500 g = 10.5 kg)

• Volume is a measure of regions in three-dimensional space.
• The areas of rectangles (including squares) can be calculated by multiplication of side lengths.

• The area of a right-angled triangle is equal to half the area of a rectangle with the same base and height.
• The volumes of rectangular prisms can be calculated by multiplication of side lengths.

• Measuring the perimeter of polygons using metric units (mm, cm, and m)
• Measuring the areas of irregular shapes covered with squares and half squares
• Calculating the areas of rectangular figures (including squares) using multiplication of side lengths
• Measuring the volumes of rectangular prisms (cuboids) by filling them with identical 3D blocks

• Approximating the areas of irregular shapes covered with squares, half squares, and partial squares
• Calculating the areas of rectangles (including squares) using multiplication of side lengths
• Measuring the volumes of rectangular prisms (cuboids) filled with centicubes by determining the number of cubes in each layer and then multiplying by the number of total layers
• Calculating the perimeters of regular polygons and other 2D shapes with straight sides
• Recognising that shapes with the same area can have different perimeters, and vice versa

• Calculating, estimating, and comparing the volumes of cubes and rectangular prisms using standard units, including cubic centimetres (cm³) and cubic metres (m³)
• Visualising, estimating, and calculating (using multiplication) the areas of rectangles and right-angled triangles (in cm² and m²) and the volumes of rectangular prisms (in cm³)

• Angles are a measure of turn and can be measured using the unit of degrees; a full turn is 360 degrees, a half turn is 180 degrees, and a quarter turn is 90 degrees.
• Rectangles and squares have four right angles.

• Angle classification can aid the estimation of angle size.
• Angles are classified with reference to the benchmark angles of 90°, 180°, and 360°.
  • Acute angles are less than 90°.
  • Right angles are exactly 90°.
  • Obtuse angles are more than 90° and less than 180°.
  • Reflex angles are more than 180° and less than 360°.

• Angles at a point sum to 360°.
• Angles on a straight line sum to 180°.
• Vertically opposite angles are equal.

• Estimating the size of angles by comparing them to 90, 180, and 360 degrees

• Describing and classifying angles and turns using the terms acute, right, obtuse, straight, and reflex
• Classifying and constructing angles up to 180°, using a protractor

• Classifying, measuring, and constructing angles up to 360°, using a protractor
• Identifying and describing angles at a point, angles on a straight line, and vertically opposite angles, using angle notation
• Reasoning about and finding unknown angles in situations involving angles at a point, angles on a straight line, and vertically opposite angles

• A point in time is typically measured in hours and minutes past midnight.
• Clocks relate seconds to minutes and minutes to hours according to a system based on 60.

• Time and duration arise in a range of situations including solar calendars (e.g. Roman, Gregorian) and lunar calendars (e.g. maramataka Māori, Chinese).
• Timetables can be used to record the time and duration of events.

• Telling the time on analogue and digital clocks to the nearest minute
• Measuring duration in hours, minutes, and seconds, including mixed time units (e.g. 1h and 42mins, 3mins and 21s)
• Finding equivalent durations of time using different units (e.g. 3 weeks is 21 days; 90 seconds = 1.5 minutes; 48 hours = 2 days)

• Telling the time on analogue and digital clocks
• Finding the duration of periods of time involving a.m. and p.m. notation and 24-hour time

• Converting between units of time (h, min, s)
• Measuring duration in both 12- and 24-hour time systems
• Finding elapsed time in minutes across an hour (e.g. the difference between 2:53 pm and 3:28 pm)
• Using and interpreting timetables to calculate the duration of events (e.g. bus and train schedules)

During Year 4

During Year 5

During Year 6

During Year 4

During Year 5

During Year 6

Shapes

• A regular polygon is a two-dimensional shape with all sides of equal length and all interior angles of equal measure.
• Circles have an infinite number of lines of symmetry.

• Parallel lines are always the same distance apart and never meet.
• Perpendicular lines intersect at right angles (90°).
• A prism is a 3D shape with two identical, parallel ends and flat faces.

• Quadrilaterals can be categorised into one or more of the following categories: 
  • a trapezium, which has one pair of parallel sides
  • a parallelogram, which has two pairs of parallel sides
  • a kite, which has two pairs of equal-length adjacent sides
  • a rectangle, which has four right angles
  • a rhombus, which has four equal-length sides
  • a square, which has four right-angles and four equal-length sides.
• The order of rotational symmetry for a given shape is the number of times it appears in an identical orientation when completing a full turn (360°).
• The order of rotational symmetry for a circle is infinite.

• Identifying, classifying, and describing the attributes of regular and irregular polygons of up to 12 sides, using edges, vertices, and angles
• Identifying the number of lines of symmetry in 2D shapes

• Identifying, classifying, and describing the attributes of prisms, using cross sections, faces, edges, and vertices
• Identifying parallel and perpendicular lines, including those forming the sides of polygons 

• Identifying, classifying, and explaining similarities and differences between 2D shapes (including different types of triangles and quadrilaterals) and between prisms and pyramids
• Identifying and describing the interior angles of triangles and quadrilaterals
• Identifying shapes with rotational symmetry and determining their order of rotational symmetry

Spatial reasoning

• Shapes may appear different when viewed from a different perspective.

• A net is a 2D representation of the surfaces of a 3D unfolded shape.

• A tessellation is a pattern made from a repeated shape or combination of shapes that can be rotated or reflected to fit together with no gaps or overlaps.

• Visualising 3D shapes and connecting them with 2D diagrams, verbal descriptions, and the same shapes drawn from different perspectives

• Connecting 3D shapes with nets

• Visualising, creating, and describing 2D geometric patterns and tessellations using rotation, reflection, and translation, and identifying the properties of the shapes that do not change

• A reflection is when a shape is flipped over a line, creating a mirror image.
• A translation is when a shape is slid from one place to another without being turned.
• A rotation is when a shape is turned around a fixed point.

• Performing one-step transformations (reflections, translations, rotations) on 2D shapes

• Describing the transformations performed (reflections, translations, rotations) on 2D shapes

• Predicting the results of two-step transformations (reflections, translations, rotations) on 2D shapes

Pathways

• An alphanumeric grid reference is a system that divides a map into labelled rows (letters) and columns (numbers), so that each square can be identified by combining a letter and a number (e.g. A1, B2).

• Use alphanumeric and general grid references to identify regions and plot positions on a grid map

• Interpreting and creating grid maps to plot positions and pathways, using grid references and directional language, including the four main compass points

• Interpreting and creating grid references and simple scales on maps, using directional language including the four main compass points, turn (in degrees), and distance (in m, km) to locate and describe positions and pathways

During Year 4

During Year 5

During Year 6

During Year 4

During Year 5

During Year 6

Developing knowledge from data

• A variable is an attribute or measurement of the people or objects being studied:
  • categorical variables classify objects or individuals into groups
  • discrete numerical variables are counted
  • continuous numerical variables are measured.

• Measurements for continuous numerical data need to have the place value for rounding specified (e.g. to the nearest centimetre).
• Bivariate data is data in a set that has two variables for each subject (e.g. dislikes and gender for each student).

• Bivariate data includes time-series data with two numerical values, one time-based.
• The answers to a statistical investigative question will vary for different subjects.
• The mean (average) measures the centre of numerical data. The mean is the sum of all the values divided by the total number of values.
• The range measures the spread of numerical data. The range is the difference between the highest and lowest values.

• Collecting numerical data, and, if needed, rounding to an appropriate unit or part of a unit, based on the context (e.g. How many skips can we do in 30 seconds? How long does it take us to run 1000 m?)

• Collecting continuous numerical data by taking measurements, and then applying specified rounding rules
• Collecting bivariate data with two categorical variables (e.g. what students in our class do at lunch time, and their gender)

• Collecting time-series data (e.g. how the mass of a kilogram of carrots varies over 5 days)
• Calculating the mean for numerical data
• Calculating the range for numerical data

Visualisation of data

• Data visualisations are representations of all available values for a variable showing the frequency for each value.
• Data visualisations show patterns, trends, and variations.
• Numerical data can be visualised with dot plots or bar graphs.
• A good data visualisation includes, where appropriate:
  • a title that gives the purpose of the visualisation
  • variable(s) (e.g. labelled on the axis)
  • the group the data is from 
  • units for a numerical variable
  • values or categories
  • frequency, with the scale starting at 0.

• Continuous numerical data can be organised in a table by grouping data into specific ranges of values.
• Paired categorical data can be visualised with a clustered bar graph; one variable is represented on the horizontal axis, the other variable is shown by coloured bars clustered side-by-side, and the colours are explained in a key.

• Time-series data can be visualised with a time-series or line graph formed on a coordinate plane, with the x-axis representing time and the y-axis the second variable.

• Creating dot-plot or bar-graph data visualisations

• Creating tables for continuous numerical data, using groupings (e.g. 0–0.99, 1–1.99, 2–2.99)
• Creating clustered bar graphs for paired categorical data

• Creating time-series graphs
• Choosing and creating an appropriate data visualisation for a given set of data

Interpretation of data

• Interpreting a data visualisation includes describing its variables and their units, the context for the data, and the visualisation’s key features:
  • its shape (e.g. the number of peaks)
  • its middle group(s) (where the middle of the data lies)
  • its spread (how spread the data is from the minimum (lowest) value to the maximum (highest) value).

• Answering questions about the frequency of a particular value in dot plots
• Answering questions about individual values in a dot plot, while referring to the context 
• Interpreting data visualisations
• Distinguishing between when to use a particular value or the frequency for a given value when answering questions about dot plots (e.g. How many pets does the person with the most pets have? What’s the most common number of pets that anyone has?)

• Answering questions about the frequency of particular values or groups of values from a table for continuous numerical data
• Answering questions about bivariate data in which a specific category in one variable appears more frequently than a specific category in another variable
• Interpreting data visualisations

• Identifying whether a time-series graph shows a trend
• Calculating an average and a range for continuous numerical data
• Interpreting data visualisations, including those from contemporary media

During Year 4

During Year 5

During Year 6

During Year 4

During Year 5

During Year 6

Experimental probability

• Situations that involve chance, uncertainty, and randomness are called chance-based situations. Probability can be used to describe such situations.
• A trial is a single run of a chance-based situation that results in one of a set of possible outcomes.
• The possible outcomes for a chance-based situation can be arranged into events.
• The probability of an outcome is the chance of it occurring.
• Probabilities are associated with values between 0 and 1, where:
  • 0 is impossible
  • between 0 and 0.5 or 1—2 ranges from very unlikely to unlikely
  • 0.5 or 1—2 is equally likely
  • between 0.5 or 1—2 and 1 range from likely to very likely
  • 1 is certain.
• Likelihood can be visualised using a number line from 0 to 1.
• The sample space is the set of all possible outcomes of an experiment.
• The probability of an event, if events are believed to be equally likely, is the number of ways the event can happen divided by the total number of possible outcomes.
• The sum of the probabilities of all outcomes is equal to 1.

• Conducting repeated chance experiments or games, identifying the outcomes, and describing differences between them using likelihood vocabulary
• Identifying the likelihood of an everyday event as being impossible, unlikely, even-chance, likely, or certain (e.g. the event ‘the sun will rise tomorrow’ is certain)
• Placing everyday events on a number line according to their likelihood (e.g. placing the event ‘you will eat something later today’ between 1—2 and 1 as ‘likely’ or ‘very likely’)

• Listing the sample space of an event (e.g. the sample space for rolling a die is 1, 2, 3, 4, 5, 6)
• Calculating the probabilities of individual outcomes
• Calculating probabilities using a spinner, where each event is a fraction or combination of fractions on the spinner
• Answering questions about the probability of combinations of outcomes, including checking that the sum of all the probabilities is 1

Number

• approximate
• convert
• decimal
• decimal place
• decimal point
• infinite
• inverse operation
• improper fraction
• mixed number
• multiple
• rename
• scale
• simplest form
• tenth.

• change
• divisor, dividend, quotient, remainder
• factor
• hundredth
• multiple
• negative, positive
• non-unit fraction
• percentage
• product
• proportion
• remainder.

• brackets
• efficient
• simplest form
• square number, cube number
• systematically
• thousandth.

Algebra

• conjecture
• equation
• relationship.

• corresponding element
• equality.

• constant (amount of change)
• coordinate plane, XY graph, x-axis, y-axis
• inequality
• ordered pairs.

Measurement

• angle
• benchmark
• centi–, kilo–
• degree (of angle)
• degrees Celsius
• kilogram
• irregular
• minutes past, minutes to
• right angle
• temperature.

• attribute
• deci–, milli–
• kilometre, millimetre
• acute angle, obtuse, reflex, right, or straight angle
• timetable.

• cubic centimetre (cm³), cubic metre (m³)
• protractor
• square centimetre (cm²), square metre (m²).

Geometry

• diagonal, horizontal, vertical
• grid reference
• parallel line
• perspective
• quadrilateral
• rotation.
• transformation.

• cross section
• net
• perpendicular line
• prism.

• angles at a point, on a straight line, vertically opposite angles
• interior angle
• kite, parallelogram, rhombus, trapezium
• map scale
• right-angled triangle
• rotational symmetry
• tessellation.

Statistics

• discrete numerical, continuous numerical
• interpreting
• spread
• trends
• variation.

• bivariate data
• paired categorical data
• clustered bar graph.

• mean
• range
• time-series data.

Probability

• chance, uncertainty
• chance-based investigation
• equally likely outcome
• evaluate
• event.

• experiment 
• impossible, unlikely, evenly likely, likely, certain
• outcome
• sample space.