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NZC - Mathematics and statistics (Phase 3)

Progress outcome and teaching sequence for Phase 3 (year 7-8) of the Mathematics and statistics Learning Area. From 1 January 2025 this content is part of the statement of official policy relating to teaching, learning, and assessment of Mathematics and statistics in all English medium state and state-integrated schools in New Zealand.

Artwork for NZC, Mathematics and statistics (Years 0-8) (2025)

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NZC – Mathematics and statistics

Years 0‑8

  

NZC – Mathematics and statistics

Phase 1 – Years 0‑3

  

NZC – Mathematics and statistics

Phase 2 – Years 4‑6

  

NZC – Mathematics and statistics

Phase 3 – Years 7‑8

About this resource

This page provides the progress outcome and teaching sequence for Phase 3 (Year 7-8) of the Mathematics and statistics learning area of the New Zealand Curriculum, the official document that sets the direction for teaching, learning, and assessment in all English medium state and state-integrated schools in New Zealand. In mathematics and statistics, students explore relationships in quantities, space, and data and learn to express these relationships in ways that help them to make sense of the world around them. Other parts of the learning area are provided on companion pages.

We have also provided the Maths Years 0-8 curriculum in PDF format. There are different versions available for printing (spreads), viewing online (single page), and to view by phase. You can access these using the icons below. Use your mouse and hover over each icon to see the document description.

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Te Mātaiaho | The New Zealand Curriculum

Mathematics and statistics: Phase 3 – Years 7‑8

Seeing ourselves in the wider world and advocating with and for others
Te aro atu ki te ao whānui me te kōkiri kaupapa hei hāpai tahi i ētahi atu

 

Progress outcome by the end of year 8

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The critical focus of phase 3 is for all students to see themselves in the wider world and to advocate with and for others. Students use logic and reasoning to identify and solve problems, make connections between mathematical and statistical concepts, and investigate patterns and variation. They communicate mathematically and statistically, using notation, conventions, and vocabulary to clearly explain and justify their approaches to solving problems. Students select, use, and adapt representations to visualise and extend their reasoning (e.g., number lines to represent integers, equations to represent linear patterns). They make generalisations and identify unknown quantities (e.g., the size of angles) and use data visualisations to investigate claims and make conjectures.

The phase 3 progress outcome describes the understanding, knowledge, and processes that students have multiple opportunities to develop over the phase.

NZC - Mathematics and statistics Understand-Know-Do diagram showing the three strands weaving together into the learning that matters. Understand is described as patterns and variation, logic and reasoning, visualisation, and application. Know is described as number, algebra, measurement, geometry, statistics, and probability. Do is described as investigating situations, representing situations, connecting situations, generalising findings, and explaining and justifying findings.

The phase 3 progress outcome is found in the table below.

Understand

As students build knowledge through their use of the mathematical and statistical processes, they expand their understanding of the following.

Patterns and variation | Ngā ia auau me ngā rerekētanga

The world is full of patterns and is defined by a multitude of relationships in which change and variation occur. Mathematics and statistics provide structures that are useful for noticing, exploring, and describing different types of patterns and relationships, enabling us to generate insights or make conjectures.

Logic and reasoning | Te whakaaro arorau me te whakaaroaro

By engaging with mathematical concepts, we develop logical reasoning and critical thinking skills that enable us to evaluate information, question assumptions, and present arguments with clarity. Statistical reasoning from observation and theory allows us to differentiate what is probable from what is possible and to draw reliable conclusions about what is reasonable.

Visualisation and application | Te whakakite me te whakatinana

The visualisation of mathematical and statistical ideas profoundly influences how we perceive, understand, and interact with abstract concepts. Application in mathematics and statistics involves creating structures and processes that help us understand complex situations, enabling better decision making and communication of ideas.

Know

Number | Mātauranga tau

By the end of this phase, students know that some numbers have special properties (e.g., primes, composites, squares, square roots, cubes). A fraction can describe a proportional relationship between two amounts. Every fraction can be represented by an infinite set of equivalent fractions that occupy the same point on the number line. Multiplying a fraction by an equivalent form of 1, such as 3—3, results in an equivalent fraction that can be useful for comparing, adding, and subtracting fractions. Decimals continue the place-value system using negative powers of ten. They can be terminating, repeating and infinite, or non-repeating and infinite.

Students come to know that integers are positive and negative whole numbers and include zero. To compare relative magnitude, integers, fractions, and decimals can be represented on a number line. There are real-life situations described by quantities less than zero (e.g., temperature, below sea level, debt), and these quantities can be operated on.

Students also come to know that when evaluating or forming expressions, the order of operations is important. Operations inside brackets (i.e., grouped together) are done first, then powers or exponents. If there are multiplication and division, these are then done in left-to-right order; finally, addition and subtraction are also done in left-to-right order. Division can result in a remainder expressed as a whole number, fraction, or decimal.

Algebra | Taurangi

By the end of this phase, students know that the inverse property applies to addition and multiplication. Inequalities can also include 'or equal to' (≤, ≥) to show a relationship that allows for the possibility of equality. In algebra, a variable can be used to represent an unknown number, a quantity that can vary or change (e.g., y= 3x + 4; A = bh), or a specific unknown value to be solved for (e.g., 3a = 18). In algebra, there are conventional ways of writing multiplication and division.

Students also come to know that linear patterns have a constant increase or decrease and their XY graphs are straight lines. Not all patterns are linear. Algorithms help solve problems in a systematic way. Their instructions are created, tested, and revised.

Measurement | Ine

By the end of this phase, students know that in the metric system there are base measurements, with prefixes added to show the size of units. A measurement can be converted from smaller to bigger units, and vice versa, by multiplying or dividing by powers of 10. Length is a one-dimensional measure, area is a two-dimensional measure, and volume is a three-dimensional measure. This is apparent in the notation of units (e.g., cm, cm², cm³). Shapes can be decomposed or recomposed to help us find their measurements (e.g., their perimeters, areas, volumes).

Geometry | Āhuahanga

By the end of this phase, students know that the spatial properties of simple polygons and polyhedra can also apply to more complex two- and three-dimensional shapes. Properties of two- and three-dimensional shapes that do not change under a transformation are called invariant. Unknown angles can be found using the properties of angles on a straight line, angles at a point, vertically opposite angles, and interior angles in triangles and quadrilaterals. Viewing objects from different angles gives different perspectives, which can be represented in models and diagrams. Position, direction, and pathways can be described using scale, compass points, and environmental features. Coordinate systems and maps can express position, direction, and pathways.

Statistics | Tauanga

By the end of this phase, students know that data collection and use involves a responsibility to protect the rights of people (in relation to data about them) and the ethical use and interpretation of data. People need to understand who they are giving data to and why, before they agree to contribute to a dataset. The statistical enquiry cycle (PPDAC) can be used to conduct data-based investigations about the wider community. There are different types of questions used when undertaking statistical investigations: investigative (summary, comparison, relationship, or time-series), survey, data-collection, interrogative, and analysis questions. Data visualisations show patterns, trends, and variations. Alternative visualisations of the same data can lead to different insights and communicate different information. A distribution is formed from all the possible values of a variable and their frequencies. A relationship investigation looks for a connection between paired numerical or paired categorical variables. Conjectures or assertions may not be reflected in the data, and so may need to be revised or abandoned.

Probability | Tūponotanga

By the end of this phase, students know that a probability experiment involves repeated trials. Results from sets of repeated trials for the same experiment may vary. Some chance-based situations, such as rolling a weighted dice, can only be explored by probability experiments. Estimates of probabilities from experiments should be based on a very large number of trials (the ‘law of large numbers’). The estimated probability of an event from an experiment equals the relative frequency for that event.

Students come to know that if all possible outcomes in a chance-based situation are assumed to be equally likely, the probability of an event equals the number of ways the event can happen divided by the total number of possible outcomes. The statistical enquiry cycle (PPDAC) can be used to conduct probability experiments. For a given situation, probability estimates from experiments and outcomes for theoretical probability models will differ. Probability distributions from experiments and theoretical models will also differ.

Do

Investigating situations | Te tūhura pūāhua

By the end of this phase, students can pose a question for investigation, find entry points for addressing the question, and plan an investigation pathway and follow it in an organised way. They can identify relevant prior knowledge, conditions, assumptions, and relationships. They can monitor and evaluate progress, adjusting the investigation pathway if necessary, and make sense of outcomes or conclusions in light of a given situation and context.

Representing situations | Te whakaata pūāhua

By the end of this phase, students can use representations to find, compare, explore, simplify, illustrate, prove, and justify patterns, variations, and trends. They use representations to learn new ideas, explain ideas to others, investigate conjectures, and support arguments. They select, create, or adapt appropriate mental, oral, physical, virtual, graphical, or diagrammatic representations. They use visualisation to mentally represent and manipulate relationships, objects, and ideas.

Connecting situations | Te tūhono pūāhua

By the end of this phase, students can suggest connections between concepts, ideas, approaches, and representations. They connect new ideas to things they already know. They make connections to ideas in other learning areas and with diverse contexts.

Generalising findings | Te whakatauwhānui i ngā kitenga

By the end of this phase, students can notice and explore patterns, structure, and regularity and make conjectures about them. They identify relationships, including similarities, differences, and new connections. They represent specific instances and look for when conjectures about them might be applied in another situation or always be true. They test conjectures, using reasoning and counterexamples to decide if they are true or not. They use appropriate symbols to express generalisations.

Explaining and justifying findings | Te whakamārama me te parahau i ngā kitenga

By the end of this phase, students can make statements and give explanations inductively based on observations or data. They make deductions based on knowledge, definitions, and rules. They critically reflect on others’ thinking, distinguishing between correct and flawed logic and asking questions to clarify and understand. They use evidence, reasoning, and proofs to explain why they agree or disagree with statements. They develop collective understandings by sharing, comparing, contrasting, critiquing, and building on ideas with others. They present reasoned, coherent explanations and arguments for an idea, solution, or process.

Teaching sequence

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Seeing ourselves in the wider world and advocating with and for others
Te aro atu ki te ao whānui me te kōkiri kaupapa hei hāpai tahi i ētahi atu

 

This section describes how the components of a comprehensive teaching and learning programme for the mathematics and statistics learning area are used during the third phase of learning at school.

Throughout phase 3, demonstrate, highlight, and affirm an attitude of exploration, enthusiasm, and curiosity towards mathematical and statistical endeavour and challenge, holding high expectations for every student. In this phase, students critically reflect on others’ reasoning, evaluating their logic and asking questions for clarification. To promote this, facilitate ongoing discussions and reflections about established expectations for interactions in mathematical and statistical learning, reinforcing that all students will be involved. Support increasing agency for students in making decisions about investigations and problem solving (e.g., while planning their approach, selecting representations, justifying their findings).

Continuously monitor students’ cognitive load, reasoning, questions, and use of representations, and respond quickly to address any issues and misconceptions. Ensure teaching builds on what students already understand, know, and can do.

Explicit teaching 

  • Use worked examples and break down new learning into clearly explained, manageable steps. Show students efficient written and mental methods. Use examples where there may be an error, misconception, or missing step, to support students to develop critical-analysis and reasoning skills. 
  • Plan for students to actively recall learning, practise new procedures and processes, and make connections with prior learning. Provide regular opportunities to practise, so that students maintain their automatic recall of facts and continue to develop procedural fluency and reasoning. Following sufficient blocked practice to achieve proficiency, provide practice opportunities that interleave a mixture of operations or approaches, rather than working on only one concept or procedure in a specific way.

Positive relationships with mathematics and statistics 

  • Provide authentic tasks that reflect students’ experiences, interests, and the wider world. 
  • Demonstrate and teach strategies for perseverance (e.g., trying another way, drawing a diagram, talking about the task with another student).

Rich tasks 

  • Design investigations where students experience rich mathematical situations, as well as investigations where students use their findings to make decisions in their lives (e.g., making a savings plan). When planning an investigation, help students to identify appropriate questions, as well as the mathematical and statistical concepts, procedures, and representations they will need. 
  • Design tasks that have multiple entry and exit points and more than one solution or pathway. 
  • Teach problem-solving and investigation strategies such as:
    • making sense of the problem by drawing a diagram or considering previously solved problems to identify strategies that can be reapplied 
    • trying some sequential numbers, recording the results in a table, and looking for patterns 
    • identifying key information in the problem and connecting it to prior knowledge 
    • translating a word problem into a linear equation, to solve for an unknown quantity 
    • recording calculations in an organised way, using correct mathematical notation 
    • checking the reasonableness of findings.

Communication in mathematics and statistics 

  • Set up opportunities for students to actively listen, reflect, and build on each other’s thinking and learning. Use discourse-based tools and a range of open questions to facilitate productive and thought-provoking discussions. Over the phase, encourage students to convert their observations into a conjecture or claim and to use evidence to justify their claims and findings. Plan to balance ‘teacher talk’ with opportunities for rich, extended student interactions and discussions. 
  • Encourage students to select and use representations that best support the learning purpose, including graphs, tables, and equations. Over the phase, support them to increasingly use equations to represent their reasoning and to visualise situations by drawing a diagram, which can give them a way into a problem. 
  • Teach and use mathematical and statistical vocabulary and concepts. Ensure students connect the correct vocabulary to the learning purpose and problem (e.g., by using the Frayer model’s four quadrants: definition, characteristics, example, non-example). Where possible, draw on students’ first and heritage languages so that they can use their languages as a resource to connect their thinking and learning. 
  • Prompt students to share their thinking when using visualisation to represent and manipulate relationships, shapes, quantities, and data (e.g., to predict or deduce the effect of a transformation; view a solid shape from different perspectives; use coordinate pairs and locations; identify terms in a growing pattern).

Number

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During year 7
Informed by prior learning, teach students to:

During year 8
Informed by prior learning, teach students to:

Teaching considerations

Number structure
  • identify, read, write, compare, and order whole numbers using powers of 10
    (e.g., 10,000 = 104)
  • identify, read, write, compare, and order whole numbers and decimals using powers of 10
    (e.g., 0.01 = 1—100  = 10-2)

Represent and order numbers using place-value (PV) expanders or charts and number lines.
  • find the highest common factor (HCF) of two numbers under 100, and find the least common multiple (LCM) of two numbers under 10
  • use prime factorisation to represent a number and to find the HCF of two numbers.

Represent factors using factor trees, or systematic lists. Connect HCFs to simplifying fractions, and LCMs to renaming fractions.

Generalise conjectures about prime or composite numbers by investigating factors.
  • use exponents to represent repeated multiplication, and identify square roots of square numbers up to at least 100
  • identify and describe the properties of prime and composite numbers up to at least 100 and cube numbers up to at least 125

Investigate and generalise divisibility tests for composite and prime numbers, and connect the results to square and cube numbers and square roots.

Investigate and explain patterns in repeated multiplication and represent them using exponent notation.

Connect prime and composite numbers with factors, and represent a number as a product of its prime factors (prime factorisation).

Operations
  • use rounding and estimation to predict results and to check the reasonableness of calculations
  • use rounding, estimation, and benchmarks to predict results and to check the reasonableness of calculations

Explain efficient methods for supporting estimation (e.g., when adding a long list of numbers, look for numbers that can be grouped and summed to roughly 10, 100, 1000).

Connect operations to benchmarks to make estimates (e.g., 73% is roughly 3—4).

Explain and justify findings, by connecting to estimates and other checking methods such as using the inverse operation.
  • round whole numbers to any specified power of 10, and round decimals to the nearest tenth, hundredth, or whole number
  • round whole numbers to any specified power of 10, and round decimals to the nearest tenth, hundredth, thousandth, or whole number
  • recall multiplication facts to at least 10 × 10 and identify and describe the divisibility rules for 2, 3, 5, 9, and 10
  • identify and describe the divisibility rules for 2–11

Investigate patterns in multiples in 100s boards and multiplication charts to generalise divisibility rules.
  • multiply whole numbers

 

Explain and demonstrate efficient methods using worked examples, including: 

  • the vertical-column method for division and multiplication, ensuring students understand and practise the procedure and connect with place value, known facts, and estimation 
  • making estimates or mental calculations by connecting to place value, partitioning, and known facts.

Investigate, explain, and justify which method (including the use of digital tools) best suits a given situation.

Have students practise decoding and solving word problems and representing them as equations.

Represent and make sense of remainders as fractions, as decimals, and when rounded to the nearest whole number.
  • divide whole numbers by one- or two-digit divisors
    (e.g., 327 ÷ 5 = 65.4 or 65 2—5 )
  • divide whole numbers
    (e.g., 327 ÷ 15 = 21.8 or 21 4—5 )
  • use the order of operations
  • use the order of operations

Use worked examples to demonstrate a step-by-step layout with only one equal sign per line.

Demonstrate how to use the mnemonic GEMA in relation to the order of operations: grouped, exponents, multiplicative (÷ and ×), additive (+ and –).
  • order, compare, and locate integers on a number line and explore adding and subtracting integers
  • order, compare, add, and subtract integers

Generalise that a positive number has an opposite negative number, and that when they are added, the answer is zero (e.g., 4 + –4 = 0).

Explain how to: 

  • find the number of steps between two given numbers on a number line (e.g., –5 and 7) 
  • ‘read’ equations with integers on a number line (e.g., “To solve –9 + 8, start at –9 and move eight numbers in the positive direction.”) 
  • use inequality symbols to compare two integers (e.g., –5 < –3).

Investigate adding and subtracting integers, using number lines and two-sided counters.

Explain the direction of movement on a number line when adding and subtracting integers, and generalise that: 

  • adding a negative number makes the original number smaller
    (e.g., 4 + –3 = 1) 
  • subtracting a negative number makes the original number larger
    (e.g., –7 – (–3) = –4).

Investigate situations where negative integers are used (e.g., temperature, altitude, debt, profit and loss).

Rational numbers
  • identify, read, write, and represent fractions, decimals (to three places), and percentages
  • identify, read, write, and represent fractions, decimals, and percentages

Explain and represent: 

  • percentages using 100s squares, 
  • comparing or ordering fractions, decimals, and percentages using double number lines
  • decimals or percentages as fractions with denominators of tenths or hundredths, and then renamed to their simplest form
  • fractions in equivalent forms to support comparing, ordering, and converting.

Explain and demonstrate converting a fraction to a decimal or percentage by connecting to the understanding of fractions as quotients (e.g., 5—12 = 5 ÷12).

Connect to known benchmarks for comparing and converting (e.g., 7—12  is a little more than 6—12 which is a half or 50%).
  • compare, order, and convert between fractions, decimals (to three places), and percentages
  • compare, order, and convert between fractions, decimals, and percentages
  • multiply and divide numbers by 10, 100, and 1000
  • multiply and divide numbers by powers of 10

Represent decimals using PV expanders or charts, and generalise that multiplying by a power of 10 moves each digit that number of places to the left, and dividing by a power of 10 moves each digit that number of places to the right.
  • find equivalent fractions, simplify fractions, and convert between improper fractions and mixed numbers
  • find equivalent fractions, simplify fractions, and convert between improper fractions and mixed numbers

Explain simplifying fractions and finding equivalent fractions by using HCFs and LCMs.
  • multiply fractions and decimals by whole numbers
  • multiply fractions and decimals by whole numbers

Explain the vertical column method for multiplying decimals, making an estimate before calculating.

Connect to the multiplicative identity to generalise that multiplying a whole number by a decimal less than one results in a product less than the original whole number.
  • find a percentage of a whole number, and find a whole amount, given a simple fraction or percentage (e.g., “25% is $100, what is the total amount?”)
  • find a percentage of a whole
    number, and find a whole
    amount, given a simple fraction
    or percentage (e.g., “75% is $45,
    what is the total amount?”)

Represent situations involving percentages using bar models to show parts of a whole.

Explain how to find a percentage of a whole by using the decimal equivalent to multiply the whole (e.g., 35% of 120 = 0.35 × 120) or by finding 10%, 5%, or 1% of the whole and using operations (e.g., finding 35% of 120 by finding 10%, multiplying this by 3 to get 30%, then adding half of 10% – 12 × 3 + 6 = 42).
  • add and subtract fractions with different denominators of up to a tenth, using equivalent fractions (e.g., 3—4 + 1—3 )
  • add and subtract fractions with different denominators, using equivalent fractions.

Demonstrate and explain renaming fractions, using ideas about equivalence and by finding HCFs and LCMs.
  • add and subtract decimals to three decimal places, with an emphasis on estimating before calculating
  • add, subtract, and multiply decimals, with an emphasis on estimating before calculating

Connect methods for operating on whole numbers with operating on decimals, making an estimate before calculating.

Investigate situations where decimals are compared and the differences between them found (e.g., sporting event times and distances).

Have students practise decoding and solving word problems and representing them as equations.
  • use proportional reasoning to explore multiplicative relationships between quantities (e.g., “If there are 3 red for every 7 blue balls, how many balls are there altogether when there are 18 red balls?”)
  • use proportional reasoning to share with unequal proportions (e.g., “We have 100 stickers to share. For every 1 sticker I get, you get 3. How many do we each get?”)

Investigate proportional reasoning in situations such as mixing paints, cooking from recipes, and sharing resources.

Represent situations involving proportional reasoning using diagrams and comparison bar models.

Connect proportional reasoning to multiplicative thinking and equivalent fractions.

Financial mathematics
  • calculate total cost and change for any amount of money
  • create and compare weekly, monthly, and yearly finance plans (e.g., saving plans, phone plans, budgets, and ‘buy now, pay later’ services)

Explain and justify ‘best deals’, considering personal priorities.

Represent financial plans for practical situations using digital tools such as spreadsheets.

Investigate situations where there are financial percentage losses or gains (e.g., calculating discounts or profits, statistics in the media about growth or decline). Connect the ideas of loss and debt with integers.

Explain, using worked examples, finding a percentage discount by subtracting from the whole or by multiplying the whole by a decimal fraction (e.g., a 35% discount on $180 = $180 – (0.35 × $180), or 0.65 × $180).
  • apply percentage discounts to whole-dollar amounts.
  • apply percentage discounts.

Algebra

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During year 7
Informed by prior learning, teach students to:

During year 8
Informed by prior learning, teach students to:

Teaching considerations

Equations and relationships
  • form and solve one-step linear equations
    (e.g., t + 7 = 12, 2s = 14)
  • form and solve one- or two-step linear equations
    (e.g., 5s + 3 = 18)

Have students practise writing equations to represent word problems.

Demonstrate solving one- or two-step equations and using the inverse operation to check findings.
  • find the value of an expression or formula, given the values of variables
    (e.g., “Calculate w + 12 when w = 4”)
  • find the value of an expression or formula, given the values of variables

Investigate variable values in practical situations with familiar formulae (e.g., for area, volume, speed).

Have students practise substituting measurements or given values into formulae.
  • describe and use the commutative, distributive, and associative properties of operations
    (e.g., a × b = b × a)
  • simplify algebraic expressions involving sums, products, differences, and single brackets
    (e.g., using the distributive property, 2(x + 3) + 1 = 2x + 6 + 1 = 2x + 7)

Represent terms in algebraic expressions using algebra tiles.

Represent algebraic expressions and equations using the conventions of algebra (e.g., 3 × b or b × 3 is written as 3b).

At year 8, explain how to simplify algebraic expressions by collecting like terms together.

At year 8, investigate systematic expansion approaches, including expansion tables, connecting to the distributive property.
  • identify the constant increase or decrease in a linear pattern, use variables and algebraic notation to represent the rule in an equation, and use the rule to make conjectures
  • determine if a pattern is linear and, if it is, write the equation for the pattern and use the equation to make conjectures

Represent a pattern using a table, model, or diagram, and use it to generalise a rule for the pattern. Use the rule and an XY graph to justify a conjecture for another term in the pattern.

Investigate the history, meaning, and structure of growing patterns (e.g., tukutuku, other well-known patterns such as the Fibonacci sequence).

Algorithmic thinking
  • create, test, and revise algorithms involving a sequence of steps and decisions.
  • create, test, revise, and use algorithms to identify, interpret, and explain patterns.

Connect an algorithm with an operation such as the vertical-column method for multiplication or with the procedure for adding fractions.

Represent algorithms using flow charts, numbered step-by-step instructions, or digital tools.

Explore algorithms by investigating: 

  • the formula function of a spreadsheet and the effect of changing the value of a variable in a formula (e.g., hourly wages) 
  • sorting and filtering multivariate data 
  • sorting numbers according to a set of instructions (e.g., the sieve of Eratosthenes) to find prime numbers 
  • situations that can be described and tested (e.g., divisibility, the use of transformations in a shape pattern, converting between units of measurement) 
  • creating, testing, and revising a set of instructions using a digital tool.

Measurement

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During year 7
Informed by prior learning, teach students to:

During year 8
Informed by prior learning, teach students to:

Teaching considerations

Measuring
  • estimate and then measure length, area, volume, capacity, mass (weight), temperature, data storage, time, and angle, using appropriate units
  • estimate and then measure length, area, volume, capacity, mass (weight), temperature, data storage, time, and angle, using appropriate units

Connect to benchmarks to make estimations.

Have students practise the accurate use of rulers, scales, timers, protractors, thermometers, and measuring jugs in practical situations.

Represent all written measurements with their units.

Select appropriate tools and units for a situation, and explain and justify choices.
  • select and use an appropriate base measure (e.g., metre, gram, litre) within the metric system, along with a prefix (e.g., kilo-, centi-) to show the size of units
  • select and use an appropriate base measure within the metric system, along with a prefix to show the size of units
  • convert between metric units of length, mass (weight), and capacity, using whole numbers and decimals to express parts of a unit (e.g., 724 g = 0.724 kg)
  • convert between metric measurement units, including square units

Connect measurement conversions with multiplying and dividing by powers of 10 (e.g., 2.05 L = 2050 mL).

Investigate measurement conversion situations in which all four operations are applied to whole-number and decimal measures.
  • find speed, given distance and time
  • find distance, given speed and time; or time, given distance and speed

Investigate the relationship between speed, distance, and time in practical situations, such as timing how long it takes to walk or run a certain distance.

Have students practise substituting values into the speed formula.

Connect finding the value of variables in the speed formula with solving algebraic equations and multiplication and division operations.
  • read, interpret, and use timetables and charts that present information about duration 
  • convert between units of time, and solve duration problems that involve fractions of time
  • read, interpret, and use timetables, charts, and results that present information about duration 
  • convert times to a common unit, such as seconds or minutes, and use decimal units of time (milliseconds)

Explain how to plan journeys using timetables and charts. Draw on a range of examples, including digital tools.

Explain methods of calculating duration (e.g., subtracting time), using worked examples.

Investigate the use of decimal units (milliseconds) in situations where a more precise measurement is needed (e.g., sporting events).

Perimeter, area, and volume
  • calculate the perimeter and area of composite shapes composed of triangles and rectangles.
  • calculate the volume of triangular prisms and shapes composed of rectangular prisms.

Investigate perimeter, area, and volume, including finding missing lengths, in practical situations. Connect calculations with factors, multiples, and the commutative and associative properties.

Represent working for calculations using a clear layout and by sketching composite shapes to show partitioning.

Generalise the formulae for finding the area of triangles and volume of triangular prisms, and have students practise substituting measurement values into them. Connect the formulae with spatial representations.

Geometry

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During year 7
Informed by prior learning, teach students to:

During year 8
Informed by prior learning, teach students to:

Teaching considerations

Shapes
  • classify and name shapes based on their attributes (e.g., triangles, pyramids)
  • describe triangles, quadrilaterals, and other polygons in relation to their sides, diagonals, and angles

Use and create a range of 2D and 3D shapes, including shapes that draw on tactile materials, diagrams, and digital tools.

Investigate ways of classifying shapes, including by creating algorithms and using Venn diagrams and tables.
  • identify and describe angles at a point, angles on a straight line, and vertically opposite angles
  • reason about unknown angles in situations involving angles at a point, angles on a straight line, vertically opposite angles, and interior angles of triangles and quadrilaterals

Investigate using digital tools and protractors to explore angles.

Investigate unknown angles to generalise the following rules: 

  • the sum of the angles round a point is 360° 
  • the sum of the angles on a straight line is 180° 
  • vertically opposite angles are equal 
  • the sum of the interior angles of a triangle is 180° and of a quadrilateral is 360°.

Represent the value of an unknown angle using an equation and angle notation.

Spatial reasoning
  • visualise, construct, and draw plan views for front, back, left, right, and top views of 3D shapes
  • visualise and draw nets for prisms with a fixed cross section

Represent plan views and nets, using sketches on grid paper, digital tools, and physical models (e.g., blocks, cardboard nets).

Connect to measurement procedures when creating sketches and models.
  • transform 2D shapes, including composite shapes, by resizing by a whole number or unit fraction
  • recognise the invariant properties of 2D and 3D shapes under different transformations

Explain and demonstrate resizing a shape using a centre of enlargement within the shape.

Investigate transforming shapes to generalise which properties (angles, side lengths, area, orientation) do not change under transformation, and test the resulting generalisations using tracing paper, rulers, and protractors.

Investigate the meaning of kōwhaiwhai patterns and other artefacts, and describe the use of transformations in them.

Pathways
  • interpret and communicate the location of positions and pathways using coordinates, angle measures, and the 8 main and halfway compass points (e.g., NE, which is 45° E from N).
  • use map scales, compass points, distance, and turn to interpret and communicate positions and pathways in coordinate systems and grid reference systems.

Use maps of familiar and unfamiliar locations to: 

  • explain and investigate the use of 4-digit grid references 
  • calculate distances using scales 
  • find efficient routes between destinations.

Connect pathways to: 

  • measurement procedures when finding angles and distances 
  • proportional reasoning when using map scales 
  • algorithms to describe routes between two points.

Investigate the navigating techniques of Māori and Pacific voyagers for locating position and finding the direction of travel.

Statistics

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During year 7
Informed by prior learning, teach students to:

During year 8
Informed by prior learning, teach students to:

Teaching considerations

Problem
  • investigate, using multivariate datasets, summary, comparison, time-series, and relationship situations for paired categorical data by:
    • posing an investigative question about a local community matter
    • making conjectures or assertions about expected findings
  • investigate, using multivariate datasets, summary, comparison, time-series, and relationship situations by: 
    • posing an investigative question about a local community matter 
    • making conjectures or assertions about expected findings

Show, with student input, how to pose investigative questions, clearly identifying the variable, the group of interest, and the intent.

Connect investigative questions with conjectures about expected findings.

Plan
  • plan how to collect or source data to answer the investigative question, including: 
    • determining or identifying the variables needed 
    • planning how to collect data for each variable (e.g., how to measure it) or finding out how provided data was collected 
    • identifying the group of interest or who the data was collected from 
    • building awareness of ethical practices in data collection by strategic questioning of data-collection questions or methods

Explain and discuss ethical practices for the collection and use of data.

Represent planning using a planning tool to outline methods of data collection, ‘who’ and what to measure, and how.

Show, with student input, how to pose data-collection and survey questions.

Explain the variables and group or groups of interest in secondary datasets.

Investigate how survey and data collection questions can be misinterpreted, leading to unreliable data.

Data
  • collect primary data or gather information about variables in sourced data, create a simple informal data dictionary, and check for errors
  • collect or source data, including: 
    • checking for errors and following up and correcting them when possible 
    • creating an informal data dictionary with information that will help others know about the context

Show, with student input: 

  • a range of data-collection and recording methods 
  • how to identify errors in data, connecting to the context and explaining why they are errors 
  • how to update primary data when correctable errors are found.

Connect multiple variables for individuals, explaining that most datasets use a table design in which each row focuses on an individual and each column includes the data on multiple individuals for one variable.

Analysis
  • create data visualisations for the investigation
  • make statements about the data, including its features and context, in descriptions of distributions
  • create data visualisations for the investigation, using multiple visualisations to provide different views of the data 
  • make statements about the data, including its features and context, in descriptions of distributions

Show, with student input, how to: 

  • represent data using dot plots, bar graphs, frequency tables, time-series graphs, two-way tables or graphs, scatter plots, fractions, proportions, and percentages, creating them at first by hand and then with digital tools 
  • read the data, read ‘between’ the data, and read ‘behind’ the data 
  • describe what is seen in the data visualisations, recognising that data are numbers with context, and the context includes variables of interest, groups of interest, counts or proportions for categorical variables, and values and units for numerical variables 
  • compare data visualisations of the same variable for different groups by looking at similarities and differences.

Explain how different data visualisations have different features and how to describe them in context (e.g., in relation to the middle, distributional shape, joint and conditional proportions, long-term trends).

Conclusion
  • communicate findings in context to answer the investigative question, using evidence from analysis and comparing findings to initial conjectures or assertions and their existing knowledge of the world
  • communicate findings in context to answer the investigative question, using evidence from analysis, considering possible explanations for findings, and comparing findings to initial conjectures or assertions and their existing knowledge of the world

Show, with student input, how to: 

  • choose the best descriptive statements that answer an investigative question 
  • explore explanations or interpretations of findings that connect to the context of the situation under investigation 
  • prepare and present succinct findings 
  • explain and justify whether or not findings align with initial conjectures or assertions, and if what was found makes sense given what is known about the situation.

Statistical literacy
  • evaluate the findings of others to check if their claims or statements are supported by the data visualisations they use.
  • evaluate the data-collection methods, data visualisations, and findings of others' statistical investigations to see if their claims are reasonable.

Show, with student input, how to: 

  • identify misleading data visualisations, match others’ data visualisations with their statements, and check the claims made by others 
  • explain and justify others’ statements about the findings of statistical investigations and the process for collecting data, using interrogative questions.

Probability

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During year 7
Informed by prior learning, teach students to:

During year 8
Informed by prior learning, teach students to:

Teaching considerations

Probability investigations
  • plan and conduct probability experiments for chance-based situations, including undertaking a large number of trials using digital tools, by: 
    • posing an investigative question 
    • anticipating what outcomes are possible and which of them are more or less likely to occur 
    • identifying and systematically listing possible answers to the investigative question 
    • collecting and recording data 
    • creating data visualisations for the distribution of observed outcomes 
    • describing what these visualisations show 
    • finding the probability estimates for the different outcomes 
    • answering the investigative question 
    • identifying similarities and differences between their findings and those of others 
    • reflecting on anticipated outcomes 
    • comparing findings from the probability experiment and associated theoretical probabilities, as appropriate
  • plan and conduct probability experiments for chance-based situations, including undertaking a large number of trials using digital tools, by: 
    • posing an investigative question 
    • anticipating what outcomes are possible and which of them are more or less likely to occur 
    • identifying and systematically listing possible answers to the investigative question 
    • collecting and recording data 
    • creating data visualisations for the distribution of observed outcomes and for all possible outcomes for theoretical probability models, where they exist 
    • describing what these visualisations show 
    • finding the probability estimates for the different outcomes 
    • proposing possible theoretical outcomes and associated probabilities, for situations where no theoretical model exists 
    • answering the investigative question 
    • identifying similarities and differences between their findings and those of others 
    • reflecting on anticipated outcomes 
    • identifying similarities and differences between findings from the probability experiment and associated theoretical probabilities, as appropriate

Investigate, using the statistical enquiry cycle, games of chance, other everyday chance-based situations, patterns in possible outcomes, and theoretical and experimental distributions.

Represent probability outcomes (theoretical and experimental) using lists, tables, tree diagrams, tally charts, visualisations of distributions, words, numbers, and technology.

Explain how to describe and use probability concepts (e.g., outcomes, events, trials, models; theoretical and experimental probability; with and without replacement; the law of large numbers; probability estimates, probability distributions; chance, randomness, and variation).

Connect anticipated outcomes with theoretical and experimental distributions.

Connect probabilities with proportional reasoning, fractions, and percentages, and with relative frequencies from data investigations.

Critical thinking in probability
  • identify, explain, and check others’ statements about chance-based investigations, referring to evidence.

Show, with student input, how to: 

  • match the results of others’ chance-based investigations with statements 
  • explain and justify the statements made by others about their findings from chance-based investigations, using interrogative questions.
 

The language of mathematics and statistics: Phase 3

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Year 7
Students will know the following words:

Year 8
Students will know the following words:

Number
  • discount 
  • divisibility rule 
  • exponent 
  • highest common factor (HCF) 
  • integer
  • lowest (least) common multiple (LCM) 
  • simplify 
  • square root
  • benchmark fraction 
  • budget 
  • composite number 
  • cube number 
  • financial plan 
  • percentage increase or decrease 
  • powers of 10 
  • prime number 

Algebra
  • coefficient 
  • coordinate 
  • expression 
  • like term 
  • line graph 
  • reciprocal 
  • X axis, horizontal axis 
  • XY plane 
  • Y axis, vertical axis
  • expand 
  • linear relationship 
  • rate of change 
  • substitute

Measurement
  • composite shape 
  • digital 
  • duration 
  • formula 
  • rate
  • speed
  • millisecond 
  • square unit

Geometry
  • complementary or supplementary angle 
  • scale factor
  • cross section 
  • diagonal 
  • exterior angle 
  • grid reference 
  • invariant property

Statistics
  • continuous data 
  • critique 
  • interpret 
  • measure of centre (mean, median, mode)
  • distribution 
  • long-term trend 
  • multivariate data set 
  • time series

Probability
  • dependent, independent
  • event 
  • experiment 
  • experimental or theoretical probability
  • trial
  • distribution 
  • misconception 
  • model 
  • random

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Abstract

Symbolic representation of a concept.

Acceleration

Acceleration or accelerated learning is where students are enabled to learn concepts and procedures more rapidly than the expected rate of progress. Accelerated learning approaches involve teaching students year-level content and experiences and supporting them with appropriate scaffolds to make this work accessible.

Additive identity

Zero will not change the value when added to a number. For example, 16 + 0 = 16.

Algorithm

A set of step-by-step instructions to complete a task or solve a problem.

Algorithmic thinking

Defining a sequence of clear steps to solve a problem.

Argument

Providing an idea or finding that is based on reasoning and evidence.

Associative property

A property of operations when three numbers can be calculated (addition or multiplication) in any order without changing the result. For example, (4 + 3) + 7 = 4 + (3 + 7) because 7 + 7 = 4 + 10, and (4 x 3) x 5 = 4 x (3 x 5).

Assumptions

A proposition (a statement or assertion) which is taken as being true with respect to a given context.

Attribute

A characteristic or feature of an object or common feature of a group of objects —such as size, shape, colour, number of sides.

Base 10

Our number value system with ten digit symbols (0-9); the place value of a digit in a number depends on its position; as we move to the left, each column is worth ten times more, with zero used as a placeholder; to the right, the system continues past the ones’ column, to create decimals (tenths, hundredths, thousandths); the decimal point marks the column immediately to the right as the tenths column.

Benchmarks

A reference point that we can use for comparison or estimation. For example, “My finger is about one centimetre wide.”

Categorical variables

A variable that classifies objects or individuals into groups or categories. For example, hair colour, breed of dog.

Chance

The likelihood that an outcome will occur.

Claim

A statement of what the student believes to be correct.

Commutative property

In addition and multiplication, each number can be operated in any order. For example, 5 + 6 = 6 + 5.

Compose and decompose

Compose is to make a shape using other shapes. Decompose is to break a shape into other shapes. 

Comparison investigative question

An investigative question that compares a variable across two clearly identified populations or groups. For example, “I wonder if girls in our class tend to be taller than boys in our class?”

Conceptual understanding

The comprehension of mathematical concepts, operations, and relations by connecting related ideas, representing concepts in different ways, identifying commonalities and differences between aspects of content, communicating their mathematical thinking, and interpreting mathematical information.

Conjecture

A statement whose truth or otherwise is not yet determined but is open to further investigation.

Constant

A constant term is a fixed value that will not change.

Continuous materials

Models based on relative length or area, such as a number line, fraction wall, bar model.

Data

A collection of facts, numbers, or information; the individual values of which are often the results of an experiment or observations.

Data ethics

The principles behind how data is gathered, protected, and used; at the core of ethical practice is the need to do good and to do no harm.

Data visualisations

A graphical, tabular, or pictorial representation of information or data.

Deduction

Make a conclusion based on knowledge, definitions, and rules.

Digital tools

Digital applications, calculators.

Discrete materials

Separate objects that can be counted and grouped. For example, counters, ice block sticks.

Discrete numerical variables

Variables that can be counted and have a limited range of possibilities. For example, number of students in each team, the result of rolling a die.

Disinformation

False information spread deliberately to deceive others.

Distribution

In mathematics, distribution describes spreading terms out equally across an expression; in statistics, distribution describes how data values are spread across the range of values collected.

Distributive property

An operation is said to be distributive over another operation if it can take priority over the operation used for combination within brackets. For example, 6 × 17 = 6 × (10 + 7) = (6 × 10) + (6 × 7) = 60 + 42 = 102.

Efficient

A procedure is said to be efficient when it is carried out in the most simple and effective way.

Element in a pattern

In a repeating pattern, an element is the repeating core.
In a growing pattern, an element is a section of the pattern. For example, in the Fibonacci sequence 1, 1, 2, 3, 5, 8, the next element is the sum of the previous two elements (5 + 8 = 13).

Equation

A number statement that contains an equal sign. The expressions on either side of the equal sign have the same value (are equal).

Equivalent fraction

Two different fractions that represent the same number are referred to as equivalent fractions. For example, 1—2, 2—4, 3—6, and 4—8 are equivalent fractions because they represent the same number.

Estimate

A rough judgement of quantity, value, or number. In statistics, an assessment of the value of an existing, but unknown, quantity. In probability, the probability of an outcome in an experiment.

Event

One or more outcomes from a probability activity, situation, or experiment.

Evidence

Information, findings, data that support (prove) a statement or argument.

Expression

Two or more numbers, operations or variables connected by operations. Expressions do not include an equal or inequality sign.

Families of facts

A group of equations that use the same numbers. For example, 3 x 2 = 6, 2 x 3 = 6, 6 ÷3 = 2, 6 ÷ 2 = 3.

Generalisation

To recognise and describe patterns in relationships.

Growing pattern

A pattern where there is a constant increase or decrease between each term. For example, 5, 10, 15, 20.

Horizontal method

Representing the operation as an equation across the page, often partitioning one of the numbers into tens and units. For example, 16 + 23 = 16 + 20 + 3.                                                                                                                                          

Inequality

A statement in which one number or expression is greater or less than another.

Inference

Making a conclusion based on evidence and reasoning.

Informal unit

A non-standard unit used to measure. For example, blocks, pens, fingers. The informal units used should all be the same size.

Interpret

To make meaning from something.

Inverse operations

The opposite operation, so addition is inverse to subtraction, and multiplication is inverse to division. They are useful to check calculations. For example, to check 4 x 5 = 20, we can see if 20 ÷5 = 4.

Limitations

Possible missing evidence or information.

Mathematical modelling

An investigation of the relationships and behaviours of quantities in physical, economic, social, and everyday contexts; used to analyse applied situations and make informed decisions, starting the model with forming assumptions.

Misconception

A misunderstanding about a procedure, method or definition in mathematics or statistics. This could be about which procedure is needed, how the procedure or method should be followed or an incorrect definition. For example, thinking that 2.5 is the decimal form of 2/5.

Misinformation

Incorrect information (mistakes).

Multiplicative identity

When a number is multiplied by 1, it does not change its value. For example, 15 x 1 = 15.

Number sentence

An equation or inequality expressed using numbers and mathematical symbols. For example, 10 = 3 + 7 + 5 + 5.

One step transformation

One change in a shape's position or size. For example, a triangle is flipped (reflected).

Ordinal

The numerical position of the element in the sequence. For example, first, second, third.

Orientation

The angle that an object is positioned.

Outcomes

A possible result of a trial of a probability activity or a situation involving an element of chance; could also refer to a result or a finding.

Partition and regroup

Partitioning is the process of "breaking up" numbers. For example, 55 = 50 + 5.
Regroup means to rearrange the formation of the group. For example, 55 = 40 + 15 or 55 = 30 + 25.

Primary data

Data collected first-hand for a specific purpose. For example, a survey, experiment, or interview.

Probabilistic thinking

Considering the likelihood (chance) of an outcome occurring; this is based on logic and reasoning.

Probability experiment

A test that can be carried out multiple times in the same way (trials). The outcome of each trial is recorded.

Procedural fluency

Choosing procedures appropriately and carrying them out flexibly, accurately, and efficiently. It is not the same as memorisation of facts and steps; rather it is being able to activate what you know and when to use it.

Procedure

A sequence of operations carried out in a specific order, such as the procedure to multiply two numbers, or the procedure to measure an object.

Quantifying

Expressing a quantity using numbers.

Rational number

All integers, fractions, and decimals.

Reasoning

Analysing a situation and thinking and working mathematically to arrive at a finding.
Inductive reasoning is where an explanation is made based on observations and data. For example, “18 out of 22 people in our class like grapes, so grapes are the favourite fruit of our class.”
Deductive reasoning is where we apply a known rule or fact to solve a problem. For example, prime numbers have two factors (itself and one), 4 has three factors, so it is not a prime number.

Relational

Symbols that show relationships between elements (terms) in an expression. For example, =, <, >

Repeating pattern

A pattern containing a 'unit of repeat'. For example, red, green, blue, red, green, blue.

Secondary data

Data collected by someone else, or a process, and/or obtained from another source. For example, online, books, other researchers.

Subitise

Instantly recognise the number of items in an arrangement without counting.

Summary investigative question

A question that asks about the overall distribution of the data or what is typical and reflects the population or group. For example, “I wonder how many pets are in our class? I wonder what the heights of the students in the class are?”

Tangible and intangible

Tangible is an object that can be touched. For example, a group of blocks. Intangible is a quality or measurement that cannot be touched. For example, colour or length.

Term in a pattern

One of the numbers in a pattern or sequence. For example, for 2, 4, 6, 8, the second term is 4.

Theoretical probability

A calculation of how likely an event is to occur in a situation involving chance.

Uncertainty

In probability, when the chance of an event occurring is unknown.

Unit of repeat

The part of a repeating pattern that repeats. The part is made up of several elements

Variables (statistics and algebra)

A property that may have different values for different individuals (statistics) or that may have different values at different times (statistics and algebra).

Variation

The differences seen in the values of a property for different individuals or at different times.

Vertical column method

The method for solving an operation by recording numbers in columns according to place value, working down the page.

Visualisation

To mentally represent and manipulate.