Draft Document January 2008
A Continuum of Mathematics Learning
A Conceptual Framework for Teaching, Planning and Assessing in
Mathematics
Years 5 to 8
(Level 3 to Level 5)
Draft Document January 2008
INTRODUCTION:
The Continuum of Learning (Mathematics) has been produced as a resource to support schools in designing their school programs and for teachers to plan, teach and assess mathematics with deep understanding and discipline knowledge.
This document is organised using levels 1 to B6, and associated year levels, the National Numeracy Benchmarks (BM) at Years 3, 5 and 7 and the National Statements of Learning (NSLs) at Years 3, 5, 7 and 9. Where there is a mismatch between levels and years levels, the NSLs have been italicised and underlined.
The current draft of the QCAR Essential Learnings has also been included in the Continuum of Learning (Mathematics).
Note: Students will need to be able to demonstrate Level 5 by mid-Year 9. Students electing to study Mathematics A in the senior phase of learning require Level 6 outcomes and some of the Beyond Level 6 outcomes. Students electing to study Mathematics B or C in the senior phase of learning require outcomes at Beyond Level 6.
FEATURES OF THE CONTINUUM OF LEARNING:
Key Concepts
These are descriptions of the major mathematical underpinnings necessary to understand and teach the mathematical concepts developed in the syllabus and reflected in the Draft QCAR Essential Learnings (Mathematics), the National Numeracy Benchmarks and National Statements of Learning (Mathematics).
They provide information for teachers in terms of the rationale for learning and working mathematically.
Syllabus Outcomes
Currently, these have been taken verbatim from the QSA 1-10 Mathematics Syllabus (2004).
The electronic version of this document allows teachers using different syllabuses to insert other outcome statements. In the event that the Queensland syllabus undergoes rewrite, the new outcome statements can also be added if desired.
Unpacking the outcomes
This section articulates the difference between levels, using the core content components from the syllabus to clarify statements from the core learning outcomes.
Teachers can use this section to place students upon the learning continuum, and thereby inform their planning, teaching and assessment.
The level descriptions in this section are written in terms of what students are able to do, not what teachers are to teach.
This section facilitates the development of multi-level tasks and associated assessment criteria sheets.
Side-by-side “learn about” and “learn through
The “Students learn about” column states the teaching that needs to occur to provide students with the opportunity to demonstrate the core learning outcomes.
The “Ref(erence)” column has been provided for teachers to insert term numbers, dates or resource references. For examples, teachers who are using Diagnostic Mathematical Probe Tasks might reference these. When planning, teachers need to address the National Benchmarks (BM), the NSLs and the Working Mathematically Components.
The “Students learn through Working Mathematically to” column describes the desirable student behaviours aligned with the collective learnings for each topic and level. The working mathematically components that have been referenced are: communicating, reasoning, applying strategies, questioning and reflecting. This section strongly reflects the working Mathematically Strand of the National Statements of Learning, and forms the basis of effective assessment tasks through the higher-order nature of the task stems.
Attention to this section will enhance student achievement in the senior phase of schooling.
Language
Proficiency with the use and interpretation of the language of mathematics is crucial to successful engagement with mathematics.
Mathematics has specific terminology as well as commonplace words that take on a different meaning. There are a multitude of synonyms that are frequently used, and students need to experience all of these, and not be confined to the language used by their teacher or a particular textbook author.
This section in this document endeavours to highlight particular vocabulary and usage relevant to the topic content.
Background Information
This section clarifies anomalies or apparent omissions.
It describes the purpose for particular concepts, developmental progressions and links to other topics and/or KLAs.
Historical and Cultural
This section primarily refers to the use of prefixes and Latin and Greek roots e.g. centi-, -gons. Whilst it would be advantageous for students to be familiar with these sources, it is not envisaged that this would be specifically taught.
Acknowledgements
This document is the result of systemic collaboration and supported by practising teachers and academics. Particular acknowledgement is made of the time and expertise of Independent Schools Queensland and Catholic Education.
Draft Document January 2008
NUMBER: Number Concepts
Key Concepts
· Numbers are used for a variety of purposes i.e. to name a quantity, to label to describe order (ordinal)· Every number can be represented in a variety of ways
· Our number system consists of counting numbers, whole numbers, integers (positive and negative whole numbers), rational numbers (integers and fractions), irrationals number (e.g. √2), real number (rational and irrational) / · Every number has a position relative to other numbers
· Our number system is base ten (places of digits determine the value of the number)
· There is a multiplicative relationship between the places in the base ten system
· Every rational number can be described according to place value (base ten system) / · Addition and subtraction are inverse operations
· Multiplication and division are inverse operations
· Calculations can be undertaken mentally, using written methods, and using technology with choice influenced by the situation
Syllabus Outcomes
N 1.1 Students identify, compare and order small whole numbers, make and match representations of these numbers and identify coins, notes and their uses. / N 2.1 Students compare and order whole numbers to 999, make and match different representations and combinations of whole numbers and of equivalent amounts of money, and identify simple fractions of objects and collections. / N 3.1 Students compare, order and represent whole numbers to 9 999 and common and decimal fractions, calculate cash transactions and describe other methods of payment. / N 4.1 Students compare and order whole numbers and common and decimal fractions of any size, make connections between key percentages and fractions, and describe how a range of factors influence financial decisions. / N 5.1 Students compare and order integers, use and interpret index notation, rates and ratios, and analyse options to make informed financial decisions about saving, credit and debit. / N 6.1 Students compare and order rational numbers, interpret and use scientific notation and analyse options to make informed personal budgeting and other financial decisions. / N DB 6.1a Students interpret and use the various sets of real numbers and integer and unit fractional powers.
N DB 6.1b Students make informed decisions with regard to earning, spending and saving money, with Reference to schedules of government and business charges.
Unpacking the key concepts and conceptual understandings from one level to the next level
Level 1: Year 1Number Concepts / Level 2: Year 2 and 3
Number Concepts / Level 3: Years 4 and 5 Concepts / Level 4: Years 6 and 7 Number Concepts / Level 5: Years 8 and 9
Number Concepts / Level 6: Years 9 and 10 Number Concepts / Beyond Level 6
Subitise to 10. Rote count to100. Rational count (understand quantity) and recognise and use different representations (materials, verbal, symbolic, pictorial, calculator) of numbers to 10.
Money
Use and record combinations of $1 and $2 coins and $5 notes to pay for goods up to $10 in simulated transactions. / Identify, represent, compare, order and describe whole numbers to 999 and common fractions in context (halves and quarters) e.g. a half of an apple is the same as 2 quarters.
Money
Model cash transactions and make amounts of money using different combinations of notes and coins. Record amounts of money as dollars and cents (e.g. $3.45). / Identify, represent, compare, order and describe whole numbers to 9 999, other common fractions in context and decimals in context to two places. Use common fractions to express part-whole relationships in context (e.g. 3 people out of 8 have blue eyes).
Money
Perform cash transactions including giving change and understand about cashless transactions (e.g. EFTPOS). / Identify, represent, compare, order and describe all whole numbers, decimal numbers and common fractions. Make connections between key percentages (100%, 50%, 25%, 20%, 10%, 1%) decimal and common fractions. Use key percentages to express part-whole relationships in context (e.g. 5 people in this class of 25 have blue eyes which is 20% of people in this class have blue eyes). Use and interpret square and cubic notation.
Money
Recognise that financial decisions can be influenced by discounts, advertising, best buys, available budgets. Budget for specific events. Calculate discounts and simple interest based on key percentages. Calculate best buys using proportional thinking. / Identify, represent, compare, order and describe integers (positive and negative whole numbers), whole number indices, all percentages (including fractional and greater than100) and their connections to common and decimal fractions.
Money
Make financial decisions based on credit and debit, charges and fees, short term vs. long term benefits. Calculate simple interest and discounts based on any percentages. / Identify, represent, compare, order and describe rational numbers (positive and negative, whole and common, and decimal fraction numbers), index notation (integer indices) and scientific notation.
Money
Make financial decisions based on budgets, income and borrowing. Calculating compound interest. / Identify, represent, compare, order and describe real numbers (surds), index notation (unit fractional indices).
Money
Consolidate Level 6.
NUMBER: Number Concepts – (including fractions) – Year 5
National Numeracy Benchmarks Year 5 / National Statements of Learning for Mathematics Year 5 / Draft QCARF Essential Learnings Year 5· read, record, compare and order whole numbers to 9 999 in contexts familiar to students
· enter a number presented orally or in written word from on the calculator e.g. given three thousand and forty-eight enter 3048 into the calculator
· compare and order number e.g. compare and order the lengths of the world’s five longest rivers
· compare the relative size of two numbers e.g. say that one river Congo-4670 km is longer than another river Volga-3530 km by more than a thousand kilometres
· use place value knowledge to interpret and model different representations of three-digit and four-digit whole numbers
· recognise different representation of number e.g. say that 740 has the same value as 74 tens
· express numbers in different forms e.g. express 398 as 38 tens and 18ones or express 27 hundreds, 2 tens and 5 ones as two thousand seven hundred and twenty-five
· find a unit fraction of an object or a collection and use the language of common fractions in contexts familiar to students
· find unit fractions limited to ½ , 1/3 , ¼, 1/5 , 1/8, 1/10
· find fractions of continuous objects e.g. break off one-quarter of a stick
· use simple fraction language and record where appropriate
· count forwards and backwards by 2s, 5s and 10s from any number to 100, by ones, tens and hundreds from any number to 1000
· count forwards and backwards by 2s and 5s from any number to 100
· count forwards and backward by 1s, 10s, and 100s from any number to 1000
· use number sense, appropriate strategies, computational skills and key information, within the limits described in the Year 5 Benchmark Standard, to solve one-step word problems involving whole numbers, money and measures in contexts familiar to students
· comparing and ordering decimal fractions with the same number of decimal places as ‘greater than’, ‘less than’ and ‘same’, including the symbols
· comparing and classifying decimal fractions as ‘smaller than 1’, ‘and ‘greater than 1
· comparing decimal fractions using place value
· interpret numbers with decimal fraction on a calculator e.g. know that 2.6 on a calculator display can mean $2.60
· compares money e.g. say that $5.10 is less than $5.25 / · use a variety of manipulatives and other materials to model and compare different representations of whole numbers and decimal fractions
· use place value to compare and order numbers and locate them, relative to zero, on a number line
· identify equal partitions within models of fractions and name the fractions shown
· multiply and divide numbers by 10 and 100 mentally and using technology and describe the changes using models such as a place value chart (e.g. use technology to multiply 1.5 repeatedly by 10, record each change on a place value chart and describe the pattern of the changes)
· use number lines to identify suitable reference points including those for consecutive whole numbers and midpoints to locate numbers involving common fractions with reasonable accuracy (e.g.
· interpret symbolic representations and use concrete representations to compare and order common fractions (e.g. illustrate with diagrams and number lines why given fractions such as are more or less than ) including when fractions are equivalent
· use area, set and linear models such as fraction walls, arrays and number lines as well as simple equivalences to perform mental calculations with common fractions (e.g. work out ; )
· recognise and use fractions in everyday and practical situations (e.g. explain when the third quarter in a game of netball or football will occur, work out that if the third quarter goes for 20 minutes, the whole game should go for 80 minutes).
· recognise different representations of numbers involving decimal fractions (e.g. recognise 2.12 as , 2 + + , 2 + 0.1 + 0.02, two and twelve hundredths) and explore related contexts involving money and measures (e.g. $2.12 and 2.12 m)
· illustrate and explain the connection between whole numbers and decimal fractions using relevant contexts (e.g. use the scale on a tape measure to assist counting forwards by
0.05 m (5 cm) from a given length {0.90 m, 0.95 m, 1.00 m , 1.05 m, 1.10 m …}) and use technology or other appropriate means to support reasoning
· read, record and compare money and measures with the same number of decimal places (up to 2 places) in contexts familiar to students / · whole numbers, simple and decimal fractions and a range of strategies are used to solve problems
· whole numbers (to 9999), decimal fractions (to at least hundredths), and common and mixed fractions have position on a number line
· place value of digits in whole numbers and decimal fractions changes when they are multiplied and divided by 10 and 100 e.g. use calculator to multiply 1.6 repeatedly by 10, record each change on a place value chart and describe the pattern of change
· common and mixed fractions (involving denominators to tenths) can be represented as a collection of objects, number lines and areas to solve problems e.g. if the third quarter of a game goes for 20 minutes, the whole game should go for 1O minutes
· equivalent fractions have easily related denominators that are used to assist mental calculations e.g. 1/2 = 2/4 1/3 = 2/6
· problems are made manageable by using strategies involving estimation
· financial records and simple spending and saving plans are ways to check on available money and income e.g. bank statements; ATM balances; budgets
· money can be saved and borrowed, and interest and fees may apply
NUMBER: Number Concepts – (including fractions) – Year 7