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Approaches to Mathematics

Introduction | Levels 1 and 2 Activites – Which is bigger? | Levels 3 and 4 Actiivites – What's my number? |Levels 5 and 6 Activites – Diophantine equations | Show All

Introduction

This material provides teachers with several sample learning activities related to the standards for the Structure and Working mathematically dimensions of the Mathematics VELS in particular, and other dimensions as applicable.

While Structure does not begin as a dimension with explicit standards until Level 3, the precursor aspects of mathematical structure, that is what students should know and be able to do, in relation to set, logic, function and algebra are naturally embedded in the other dimensions at Levels 1 and 2. As noted in the introduction to the Mathematics standards:

Mathematical reasoning and thinking underpins all aspects of school mathematics, including problem posing, problem solving, investigation and modelling. It encompasses the development of algorithms for computation, formulation of problems, making and testing conjectures, and the development of abstractions for further investigation.

Computation and proof are essential and complementary aspects of mathematics that enable students to develop thinking skills directed toward explaining, understanding and using mathematical concepts, structures and objects. They provide a framework for the development of mathematical skills and techniques exemplified in the use of algorithms for computation and for the development of general case arguments.

The Structure and Working mathematically dimensions of the Mathematics VELS have a central role in developing mathematical reasoning and thinking, and its application in practice and theory.

For each activity the related elements of the standards for the relevant levels have been identified.

Information about the Structure and Working mathematically dimensions and Relationships between the dimensions is available in the Mathematics introduction.

The Structure and Working mathematically dimensions of the Mathematics VELS are closely related to Algebra, function and pattern and Working mathematically curriculum organisers of the National Statements of Learning for Mathematics: MCEETYA Statements of Learning (www.mceetya.edu.au/mceetya/default.asp?id=11893) respectively.

Other useful activities can be found as part of the P-10 mathematics Continuum at: Mathematics developmental continuum P-10 (http://www.education.vic.gov.au/studentlearning/teachingresources/maths/mathscontinuum/default.htm).

Levels 1 and 2 Activites – Which is bigger?

This collection of activities involves students working with sets of objects, such as coloured counters, the natural numbers and their numerals to:

The concept of a one-to-one correspondence underpins both of these processes. To compare the size of two sets (one is smaller, the same size or larger than the other) it is not necessary to be able to count, only to observe which set has ‘left-overs’, if any, when elements of both sets are paired until one set is exhausted. Comparing the size of several sets also leads to the notion of ordering sets by size (even if the actual sizes are not known).

There is also a one-to-one correspondence between numbers, their numerals and their family of representative sets. The size of sets with one, two or three elements can usually be identified visually without the need to count. This correspondence is used for counting, and subsequently arithmetic operations such as addition, subtraction, multiplication and division. In each of these activities it is important that students have access to the relevant concrete materials such as counters, blocks and bundles of icy-pole sticks.

Activity 1

This activity enables the teacher to check student’s ability to determine the size of small sets of objects and their visual representation.

The elements of the standards addressed by this activity are:

Number
At Level 1, students form small sets of objects from simple descriptions … they use one-to-one correspondence to identify when two sets are equal in size and when one set is larger than another. They form collections of sets of equal size.

Working mathematically
At Level 1, students use diagrams and materials to investigate mathematical and real life situations … they test simple conjectures.

Activity 2

This activity extends the previous activity to have students compare the size of two ‘large’ sets of objects without counting by means of a one-to-one correspondence between all of the elements of one set and a subset (some, or possibly all of the elements) of another set.

Students are asked to consider two sets of objects, for example a set of red counters and a set of blue counters, and decide which one of the two sets contains more counters. This can be done by using two fingers of one hand to ‘pair-off’ one red counter with one blue counter until no more pairings are possible and there is a smaller set of only red counters or only blue counters left. The colour of the remaining counters indicates which one of the original sets was larger. By counting the remaining counters, the student can also establish the difference in size of the two sets (for example, there may be four more blue counters than red counters). This activity should be carried out for several different comparisons, including some where the two sets are of equal size.
For sets of size up to 20, students can count the size of the two sets, the difference in size, and express these relationships in terms of addition and subtraction operations, possibly with the assistance of a calculator to represent and check their working

A complementary activity is to ask students to divide a large set of counters (of only one colour) into what seems to be two equal size sets using one partitioning (without counting) and then determine whether they were successful or not.

The elements of the standards addressed by this activity are:

Number
At Level 1, students … use one-to-one correspondence to identify when two sets are equal in size and when one set is larger than another. They form collections of sets of equal size. They use materials to model … subtraction by the … disaggregation (moving apart) of objects.

They add and subtract by counting forward and backward using the numbers from 0 to 20.

Working mathematically
At Level 1, students use diagrams and materials to investigate mathematical and real life situations … they test simple conjectures.

They devise and follow ways of recording computations using the digit keys and +, − and = keys on a four function calculator.

Activity 3

This activity asks students to find the size of a large set (for example coloured counters) by counting in equal sized groups of 2, 5 and 10. This requires students to group and skip-count.

The elements of the standards addressed by this activity are:

Number
At Level 2, students model the place value of the natural numbers from 0 to 1000. They order numbers and count to 1000 by 1s, 10s and 100s. Students skip count by 2s, 4s and 5s from 0 to 100 starting from any natural number. They form patterns and sets of numbers based on simple criteria such as odd and even numbers … they add … two-digit numbers by counting on …

They mentally compute simple addition … calculations involving one- or two-digit natural numbers, using number facts such as complement to 10, doubles and near doubles. They describe and calculate simple multiplication as repeated addition, such as 3 × 5 = 5 + 5 + 5 …

Working mathematically
At Level 2, students … use place value to enter and read displayed numbers on a calculator. They use a four-function calculator, including use of the constant addition function and × key …

Levels 3 and 4 Actiivites – What's my number?

This collection of activities involves students working with sets of natural numbers to solve related combinatorial problems according to given criteria. The underlying problem is to develop a process for determining a specific piece of information, that is, an unknown number (or more precisely, it’s sequence of digits), using a minimum number of questions of a given kind - that is true/false or yes/no questions each of which relates to a single characteristic or proposition. They provide the opportunity to explore binary (that is choices involving only two alternatives) decision procedures as a means of solving a particular type of problem. More sophisticated versions of the informal techniques and strategies students will likely develop lead to many applications in everyday decision and sorting situations where digital technology is involved.

Activity 1

This activity introduces the key elements of the context. The teacher informs students that they have selected a digit from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. Students are asked to guess the number, the teacher repeats the process a suitable number of times, for example 30 times, and the number of guesses required until the number is identified is recorded. This data can then be displayed visually by suitable graph and used to answer related questions such as:

The elements of the standards addressed by this activity are:

Structure
At Level 3, students … list all possible outcomes of a simple chance event … they recognise samples as subsets of the population under consideration …

Working mathematically
At Level 3, students apply number skills to everyday contexts … they recognise the mathematical structure of problems and use appropriate strategies (for example, recognition of sameness, difference and repetition) to find solutions.

Students test the truth of mathematical statements and generalisations. For example, in …

  • number patterns (the patterns of ones digits of multiples, terminating or repeating decimals resulting from division)

… students use calculators to explore number patterns and check the accuracy of estimations. They use a variety of computer software … to organise and present data.

Measurement, chance and data
At Level 3, students … use a column or bar graph to display the results of an experiment (for example, the frequencies of possible categories).

Activity 2

This extends the previous activity and introduces a practical context. Suppose a small island has a small population of less than 100 people, and each person is randomly assigned a different two-digit telephone number, where each digit is from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} and a digit can be repeated. The following table, when completed, will list the set of all possible two digit telephone numbers:

What's my number? - Activity 2

Activity 2 table (Doc - 60KB)

1. To start the activity the teachers assigns each student in the class a (different two) digit number from this table (for example, have each student choose a piece of paper with a two digit number on it from a box) - these numbers are not revealed. For this activity it will be useful to have multiple copies of the completed table Worksheet A (Doc - 88KB) prepared in advance so that students can use these as recording devices.
The class is divided into small groups and students within the groups try to guess each group member's number and record the number of guesses taken in each case until each group member's number is determined. These results are then collated and summarised as before, and the same questions posed as in Activity 1.

2. (Optional) the teacher may at this stage like to show students summary results from a simulation of the guessing process for a much larger number of trials.

3. At this stage the idea of trying to develop a systematic process for determining an unknown number can be introduced. The following criteria is to be applied - any simple question that can be answered by a yes/no  response is permitted. However the question is not permitted to be a compound question (that is a statement that is really asking two or more questions, for example, ' Is the number even and above 50?' is a compound question - the two simple questions are: 'Is the number even?' and 'Is the number above 50?').The teacher selects the unknown number and then invites students to ask questions of the permitted type about the number - some important guidelines for this activity are:

4. Students are then asked to find a systematic process for determining an unknown two digit number and to try out their strategy with several such numbers in small groups (the most efficient strategy is one where questions are devised that halve of the list of possible numbers left at each stage by the response to each question - since 26 = 64, 27 = 128 and 26 < 100 < 27 then at most 7 such yes/no questions will need to be asked. Such questions can be posed using the less than relation '<' .).

5. Discussion of student strategies need not, at this stage, proceed to consideration of which might be the most efficient, although teachers will perhaps wish to include this as part of Activity 3.

The elements of the standards addressed by this activity are:

Structure

At Level 4 students form and specify sets of numbers … according to given criteria and conditions (for example, 6, 12, 18, 24 are the even numbers less than 30 that are also multiples of three). They use … test the validity of statements using the words none, some or all

Students identify relationships between variables and describe them with language and words …

Students recognise that … multiplication and division are inverse operations. They use words and symbols to form simple equations. They solve equations by trial and error.

Working mathematically

At Level 4, use students recognise and investigate the use of mathematics in real … situations …

Students use the mathematical structure of problems to choose strategies for solutions. They explain their reasoning and procedures and interpret solutions.

Students engage in investigations involving mathematical modelling. They use calculators and computers to investigate and implement algorithms … explore number facts and puzzles, generate simulations …

Number

At Level 3, students use place value (as the idea that ‘ten of these is one of those’) to determine the size and order of whole numbers to tens of thousands, and decimals to hundredths. They round numbers up and down to the nearest unit, ten, hundred, or thousand.

They devise and use written methods for …

  • multiplication by single digits (using recall of multiplication tables) and multiples and powers of ten (for example, 5 × 100, 5 × 70 )
  • division by a single-digit divisor (based on inverse relations in multiplication tables).

At Level 4, students comprehend the size and order of … large numbers (to millions) … they recognise and calculate simple powers of whole numbers (for example, 24 = 16).

… they use estimates for computations and apply criteria to determine if estimates are reasonable or not.

Measurement, chance and data

… they present data in appropriate displays … they calculate and interpret measures of centrality (mean, median, and mode) and data spread (range).

Activity 3

This activity is a generalisation from the previous two and provides students with a more open ended opportunity to carry out their own investigations. For example, the problem context could be:

This activity also provides teachers with an opportunity to discuss, in the context of seeking an efficient general strategy, some informal equation solving - to find a natural number solution to a situation that involves an exponential function. This involves the inverse relationship for multiplication between doubling and halving. For example, if students ask questions such that the number of possible combinations of digits is halved each time, for example 100 → 50 → 25 → 12/13 → 6/7 → 3/4 → 2 → 1, then the number of questions needed can be determined by starting with 1 and doubling until the total number of possible combinations, 100 in this case, is exceeded. Thus, 7 doublings are required for 100, since 2 × 2 × 2 × 2 × 2 × 2 × 2 = 27 = 128. In the case of the four digit security number (PIN), there are 10 × 10 × 10 × 10 = 104 = 10 000 possible digit combinations, so the natural number n such that n doublings starting from 1, or 2n  is greater than 10 000, or in symbols 2n >10 000. This value of n can be found using a calculator or spreadsheet.

It should be noted that while this tells us how many questions will be needed, it does not provide an algorithm or systematic mechanical procedure for asking these questions. Teachers will need to develop a description of this with their students. At any stage the question is to ask whether the number sought is less than or greater than the middle value of the set of numbers left at that stage. 

Alternatively, teachers could have students investigate the three digit case for themselves

(210 = 1024) discuss the general result, and then apply the search algorithm in the 4 digit and 8 digit situations.

The elements of the standards addressed by this activity are:

Structure

At Level 4 students form and specify sets of numbers…according to given criteria and conditions … test the validity of statements using the words none, some or all (for example, test the statement ‘all the multiples of 3, less than 30, are even numbers’).

Students construct and use rules for sequences based on the previous term, recursion … and by formula (for example, a term is three times its position in the sequence plus two).

Students identify relationships between variables and describe them with language and words …

Students recognise that … multiplication and division are inverse operations. They use words and symbols to form simple equations. They solve equations by trial and error.

Working mathematically

At Level 4, use students recognise and investigate the use of mathematics in real … situations …

Students develop and test conjectures. They understand that a few successful examples are not sufficient proof and recognise that a single counter-example is sufficient to invalidate a conjecture ...

Students use the mathematical structure of problems to choose strategies for solutions. They explain their reasoning and procedures and interpret solutions. They create new problems based on familiar problem structures.

Students engage in investigations involving mathematical modelling. They use calculators and computers to investigate and implement algorithms … explore number facts and puzzles, generate simulations …

Number

At Level 4, students comprehend the size and order of … large numbers (to millions) …they recognise and calculate simple powers of whole numbers (for example, 24 = 16).

They explain and use mental and written algorithms for the … multiplication and division of natural numbers (positive whole numbers).

… they use estimates for computations and apply criteria to determine if estimates are reasonable or not.

Levels 5 and 6 Activites – Diophantine equations

This collection of activities on Diophantine equations relates in particular to Structure (properties of number systems, variables, validity, quantifiers, linear functions, tables, graphs and equations), Working mathematically (formulation, generalisation, conjecture, deduction and mathematical argument, use of technology) and Number (natural numbers, integers, factors, greatest common divisor, Euclidean division algorithm) dimensions of the Mathematics VELS. Relevant elements of the standards for these dimensions at Level 5 and Level 6 have been noted as applicable for each activity.

Diophantine equations also provide a rich context for historical investigation, which shows certain types of problems that have arisen in varied contexts and different cultures, across centuries and millennia of human mathematical activity and investigation, from the ancient Babylonians through to modern times (see, for example, Stillwell 2002; Stillwell 2003).

Depending on the difficulty of the examples chosen and the approach taken (for example, from finding a solution by any method through to seeking a general form for solutions and justification of this form) these activities, or parts of them, will be appropriate for work at Level 5 (after some work on integers has been covered) and/or Level 6.

A Diophantine equation is one in which only integer coefficients for the terms of the equation and solutions for the variables involved are allowed. That is, coefficients and solutions which are elements of the set Z = {…-3, -2, -1, 0, 1, 2, 3 …}. In practical contexts, solutions to Diophantine equations are often required to be non-negative integers which may also be subject to additional constraints.

A Diophantine equation is a single equation usually with two or more variables, and in one form or another, these equations have been studied since antiquity. The term Diophantine equation arises from the work of Diophantus of Alexandria the Greek mathematician 3rd of the century CE who was one of the first mathematicians in the western tradition to study them in detail. They had also been studied extensively by Chinese and Indian mathematicians from as early as 800 BCE - this is an aspect of the study of such equations that students could be asked to conduct historical research into (see, for example: http://en.wikipedia.org/wiki/Diophantine_equation )

Some well known Diophantine equations are:

Key mathematical questions for Diophantine equations are: do solutions exist? How many solutions are there? How can solutions be found and systematically listed?

These kinds of questions relate naturally to the activities of formulation, conjecture, experimenting and testing, general mathematical argument and deduction that are central to the complementary aspects of Structure and Working mathematically and involve functions, relations and equations.

There is no decision procedure for solving all Diophantine equations. In his address to the 1900 International Congress of mathematicians, the famous German mathematician David Hilbert put forward 23 problems as a challenge to mathematics and mathematicians (see: http://aleph0.clarku.edu/~djoyce/hilbert/problems.html; http://en.wikipedia.org/wiki/Hilbert's_problems). This list was influential in setting a substantial agenda for mathematical research in the 20th Century. Some of these problems have been resolved or partially resolved, others remain open. The tenth problem was:

Given a diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: to devise a process according to which it can be determined by a finite number of operations whether the equation is solvable in rational integers.

This problem was resolved in the negative in 1970 by Yuri Matiyasevich and Julia Robinson. (see http://logic.pdmi.ras.ru/Hilbert10)

Fermat’s last theorem (which involves a generalisation of the Pythagorean equation x² + y² = z²) states that there is no solution to the Diophantine equation xn + yn = zn where n is a natural number greater than 2. Although first stated by Pierre de Fermat in 1637 (with a comment scribbled in the margin of his notebook to the effect that he had found a proof of this conjecture, but that there wasn’t enough space to write it down), the theorem was not proven until 1994 by the English mathematician Andrew Wiles (see: http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Fermat's_last_theorem.html; www.pbs.org/wgbh/nova/proof/wiles.html)

However certain types of Diophantine equation do have general solutions, such as the linear equation ax + by = c. This type of Diophantine equation is suitable for investigation with respect to the Mathematics VELS at Levels 5 and 6. In the following activities all equations are to be considered as Diophantine equations.

Activity 1

Suppose that the ages, in years, of two people, Natalia and Sam, add to a total of 20 years. What could their individual ages be?

Students should be able to determine the solution set to this problem by inspection, where each solution is an ordered pair of integers, and formulate the corresponding equation, x + y = 20, with additional practical restrictions for this context, x > 0 and y > 0.

Technology can be used to draw a graph with a suitable integer scale on the axes and a grid, to visually identify solutions. It can also be used to systematically obtain the solutions in a list or table by generating the ordered pairs (x, 20 - x) for integer values of x from 1 to 19.

This simple introductory activity can be generalised in several ways:

The elements of the standards addressed by this activity are:

Structure
At Level 5 students identify collections of numbers as subsets of natural numbers, integers, rational numbers and real numbers … they test the validity of statements formed by the use of the connectives and, or, not, and the quantifiers none, some and all …

Working mathematically
At Level 5 … students use variables in general mathematical statements. They substitute numbers for variables (for example, in equations, inequalities, identities and formulas).

Activity 2

In this activity the variables are not required to be positive. Students are required to identify a solution, use this or a formula to generate other solutions, and use technology to plot these solutions over a selected domain on a graph.

The elements of the standards addressed by this activity are:

Number
At Level 5, students identify complete factor sets for natural numbers … students use efficient mental and/or written methods for arithmetic computation involving rational numbers, including division of integers …

At Level 6, students comprehend the set of real numbers containing natural, integer … numbers.

… students carry out arithmetic computations involving natural numbers, integers and finite decimals using mental and/or written algorithms.

Structure
At Level 5, they identify the identity element and inverse of rational numbers for the operations of addition and multiplication …

… students use inverses to rearrange simple … formulas, and to find equivalent algebraic expressions…they solve simple equations … using tables, graphs and inverse operations. They recognise and use inequality symbols. They solve simple inequalities … and decide whether inequalities … are satisfied or not for specific values of x and y.

… they represent a function by a table of values, a graph, and by a rule. They describe and specify the independent variable of a function and its domain , and the dependent variable and its range. They construct tables of values and graphs for linear functions.

At Level 6, students classify and describe the properties of the real number system and the subsets of rational … numbers. They identify subsets of these as discrete or continuous, finite or infinite and provide examples of their elements and apply these to functions and relations and the solution of related equations.

Working mathematically
At Level 5, students formulate conjectures and follow simple mathematical deductions …

… students use variables in general mathematical statements. They substitute numbers for variables (for example, in equations, inequalities, identities and formulas).

... they analyse the reasonableness of points of view, procedures and results, according to given criteria, and identify limitations and/or constraints in context.

Students use technology such as graphic calculators, spreadsheets … and computer algebra systems for a range of mathematical purposes including numerical computation, graphing, investigation of patterns and relations for algebraic expressions …

Activity 3

In this activity students are asked to consider the linear equation ax + by = c where a and b are non-zero, and:

Students could initially use an empirical approach, testing a range of equations for selected values of a, b and c and seeing whether solutions exists or not. They could then attempt to construct equations which they know must have a solution or which they believe will not have a solution (See, for example,
www-groups.mcs.st-and.ac.uk/~martyn/teaching/1003/1003linearDiophantine.pdf)

In general, if there are no other restrictions on the variables a linear (Diophantine) equation has either no solution, or infinitely many solutions.

Number
At Level 6, students carry out arithmetic computations involving natural numbers, integers and finite decimals using mental and/or written algorithms.

… students use the Euclidean division algorithm to find the greatest common divisor (highest common factor) of two natural numbers …

Structure
At Level 6, students classify and describe the properties of the real number system and the subsets of rational … numbers. They identify subsets of these as discrete or continuous, finite or infinite and provide examples of their elements and apply these to functions and relations and the solution of related equations.

Students apply the algebraic properties (closure, associative, commutative, identity, inverse and distributive) to computation with number, to rearrange formulas, rearrange and simplify algebraic expressions involving real variables.

Students identify and represent linear … functions by table, rule and graph (all four quadrants of the Cartesian coordinate system) with consideration of independent and dependent variables, domain and range …

They recognise and explain the roles of the relevant constants in ... relationships …

They solve equations … in two variables … using algebraic, numerical … and graphical methods.

Working mathematically
At Level 5, students formulate conjectures and follow simple mathematical deductions …

… Students use variables in general mathematical statements …

… They develop generalisations by abstracting the features from situations and expressing these in words and symbols …

Students use technology such as graphic calculators, spreadsheets … and computer algebra systems for a range of mathematical purposes including numerical computation, graphing, investigation of patterns and relations for algebraic expressions …

At Level 6, students formulate and test conjectures, generalisations and arguments in natural language and symbolic form … they follow formal mathematical arguments for the truth of propositions …

Students choose, use and develop … procedures to investigate and solve problems set in a wide range of practical, theoretical and historical contexts … they generalise from one situation to another, and investigate it further by changing the initial constraints or other boundary conditions. They judge the reasonableness of their results based on the context under consideration.

They select and use technology in various combinations to assist in mathematical inquiry … to analyse functions and carry out symbolic manipulation …

Activity 4

Two variable linear programming problems sometimes involve linear equations with integer coefficients, where non-negative integer solutions are sought (that is, points with whole number coordinates) in terms of boundary conditions with respect to the feasible region and the objective function for the problem at hand. Here the task is often to identify corner points for a region which are non-negative integer solutions to pairs of simultaneous linear equations, and test these for maximising (or minimising) the objective function.

Underpinning this process is the knowledge that a linear equation of the form ax + by = c subject to further (practical) constraints on x and y will have a finite set of solution values. For example, students could be asked to find solutions to problems formulated from the following sorts of contexts:

The elements of the standards addressed by this activity are:

Number
At Level 6, students carry out arithmetic computations involving natural numbers, integers and finite decimals using mental and/or written algorithms …

… students use the Euclidean division algorithm to find the greatest common divisor (highest common factor) of two natural numbers …

Structure
At Level 5 … they use linear … functions … to model various situations.

At Level 6, students classify and describe the properties of the real number system and the subsets of rational … numbers. They identify subsets of these as discrete or continuous, finite or infinite and provide examples of their elements and apply these to functions and relations and the solution of related equations.

Students identify and represent linear … functions by table, rule and … with consideration of independent and dependent variables, domain and range … they use and interpret the functions in modelling a range of contexts.

They solve equations … in two variables … using algebraic, numerical … and graphical methods.

Working mathematically
At Level 5 students use variables in general mathematical statements. They substitute numbers for variables (for example, in equations, inequalities, identities and formulas).

… Students develop simple mathematical models for real situations …

… They predict using interpolation (working with what is already known) and extrapolation (working beyond what is already known). They analyse the reasonableness of points of view, procedures and results, according to given criteria, and identify limitations and/or constraints in context …

At Level 6, students choose, use and develop mathematical models and procedures to investigate and solve problems set in a wide range of practical, theoretical and historical contexts (for example, exact and approximate measurement formulas for the volumes of various three dimensional objects such as truncated pyramids). They generalise from one situation to another, and investigate it further by changing the initial constraints or other boundary conditions. They judge the reasonableness of their results based on the context under consideration.

They select and use technology in various combinations to … analyse functions and carry out symbolic manipulation.

References

Stillwell, J, 2003. Elements of Number Theory. Springer-Verlag. New York.

Stillwell, J, 2002. Mathematics and its History. 2nd edition. Springer-Verlag. New York.

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