Enhancing Maths With Attitude

This page is for teachers and schools which already have Maths With Attitude kits. (If this page is your first experience with the resource, first explore Maths With Attitude.) Tasks and Lessons below can be used to add even more variety to MWA or to extend the time spent on particular units. Tasks invite students to work like a mathematician on their own. Lessons support teachers to model what it means to work like a mathematician. Picture Puzzles and Menu Maths Packs are other Mathematics Centre resources which will help to enrich your mathematics curriculum.   List of additional Tasks   List of additional Maths300 Lessons

Creating a balanced Working Mathematically curriculum.

Tasks

No.	Name	Description	Content	Add To...
12	Matching Cards	The initial challenge is to correctly match given pieces, then to order the correct matches. But how many not-matches are possible? Originally designed as an example of what a task card for Infants (Years K - 2) might look like, this one simple question opens the task to much older students - even students in senior high school.	recognition of attributes and of left and right sides, sorting, classifying and ordering, recording data, informal experience with probability, combinatorial mathematics, sample space and theoretical probability.	S & L, Years 3 & 4 C & M, Years 9 & 10
19	Cookie Count	Place any number of cookies on a plate ready to offer equal shares to guests. The problem is: If I tell you any number of guests can you tell me the share they each receive?	1:1 correspondence, counting, sharing and grouping, partitioning wholes, times tables facts, multiples and factors, fractions of a whole	N & C, Years 3 & 4
21	Tactical	A logic game with clear rules and a spatial dimension. The aim is to force your opponent to take the last counter.	2D spatial perception, logical reasoning, in particular 'if...then' reasoning and working backwards	S & L, Years 5 & 6 S & L, Years 9 & 10
22	Time Together	A task to help students explore the passing of time. The focus is on those moments in a 12 hour cycle when the hands are 'on top of each other'. The task encourages manipulation of the clock hands, estimation and, for the more mathematically mature, precise calculation. The task also offers experience with the meaning of clockwise and opportunity for informal learning related to counting, angles and fractions.	division of time into hours, minutes and seconds, visualising the passing of time, applying understanding of the interconnectedness of the hands of a clock, visualising angles, clockwise & anti-clockwise, counting patterns, proportional reasoning, fractions related to elevenths and twelfths, graphical representation of simultaneous linear equations	C & M, Years 3 & 4 N & C, Years 9 & 10
25	In Between Time	As time passes, both the minute hand and the hour hand move. This task first uses a thirty minute difference (embodying the idea of half of the hourly minute hand journey) to establish that students understand: (a) Time difference can be clockwise or anti-clockwise, and (b) If the minute hand moves half an hour, then the hour hand has moved half of its journey towards the next (or previous) hour. From this basis the students are challenged to set a position for one of the hands and explore where the other hand must, or sometimes could, be.	division of time into hours and minutes, visualising the passing of time, applying understanding of the interconnectedness of the hands of a clock, visualising angles, clockwise & anti-clockwise, before and after a given time, time difference, fractions, proportional reasoning	C & M, Years 3 & 4 C & M, Years 7 & 8
32	Tetrahedron Nets	By making a tetrahedron and unfolding it to make its net, students discover its symmetric properties. The challenge is extended to looking for an alternative net and to discovering that 3D objects can be reflected.	2D representation of 3D objects, nets, axis of symmetry (3D), reflection in 3D, problem solving strategies	See Note below.
50	Flight Departures	An observer gives clues about the take off order of four planes at an airport. The challenge is to decide the actual order in which the planes took off.	logical reasoning, problem solving strategies, combination theory	S & L, Years 3 & 4 S & L, Years 7 & 8
59	13 Away	This is a game situation that has an underlying strategy waiting to be discovered. It is easy to state and easy to start, which is one feature that makes it suitable for younger as well as older students. The starting point is a pile of 13 counters. The person who takes the last one loses. Players take turns to take either 1, 2 or 3 counters on any move. The unwritten challenge is to find a way to always win.	reasoning skills, patterns in number, generalisation, algebraic representation	P & A, Years 3 & 4 P & A, Years 7 & 8
69	Tricubes	A set of tricubes is used to construct a 'building' in a specific order. Each stage is shown as an isometric drawing. The (apparently) simple challenge is to reconstruct the construction sequence.	2D representation of 3D objects, isometric drawing, problem solving strategies	See Note below.
72	Farmyard	A spatial challenge with aspects of area. A farmer wants to divide a field shaped like an L into smaller areas which are all the same size and shape. This task relates to Task 237, Trisquares (see below) and to Task 115, Dividing Shapes which is in S & L, Years 7 & 8.	spatial problem solving, perimeter, area	C & M, Years 5 & 6 P & A, Years 7 & 8
74	Button Sort	A logical challenge that draws attention to the attributes of buttons to sort and classify them. The task informally introduces significant ideas in mathematical logic such as the connectives 'and', 'or' and 'not' and introduces diagrams similar to flow charts as a way of sorting and making decisions.	sorting & classifying, attributes & properties, logic	S & L, Years 5 & 6
88	Rice, Rice, Rice	Set in a story shell, the challenge is to estimate grains of rice without actually counting their number. This is equivalent to the problems faced by many people in practical employment - for example a bricklayer estimating the number of bricks to order to clad a house. Such problems are often guided by 'rules of thumb'. In this task students are experimenting with possibilities so there is no right or wrong answer.	counting, mass, basic arithmetic, volume	C & M, Years 5 & 6
90	Tricube Constructions B	Students are provided with four Tricubes and cards with isometric drawings showing only the outer edges of 'buildings' made from them. The challenge in each case is to discover how the object was made.	3D spatial perception, isometric drawing, representing 3D objects in 2D, problem solving strategies	See Note below.
96	Networks	A game with laminated cardboard tiles which encourages thinking ahead in a spatial situation that is complicated by a growing network of lines. A player can either win by placing the piece which reaches the Finish square, or by placing a piece that forces their opponent to lose.	logical thinking, designing networks	S & L, Years 5 & 6 S & L, Years 7 & 8
104	Building Views	Another task which investigates the link between 3D objects and their 2D representations. Presented with front and side views of a building made from cubes, students have to reconstruct the building and ensure that they have done so with the minimum number of cubes.	3D spatial perception, elevation drawings, representing 3D objects in 2D, problem solving strategies	See Note below.
139	Squound	A square and a circle intersect to form a 'Squound'. The total number of counters is known as is the total in the square and circle. But how many in the Squound? Students must realise that the number in the intersection can't be counted twice and are led to this by an Investigation Guide. Symbolic representation is also encouraged.	addition and subtraction, pattern, generalisation in words and symbols	N & C, Years 3 & 4 P & A, Years 7 & 8
153	Knight Protectors	A famous chess-based puzzle requiring significant patience that results in a beautiful rotationally symmetric solution. The challenge is to place the minimum number of knights on a chess board so that every square is either occupied or attacked.	problem solving skills, spatial perception	S & L, Years 5 & 6 S & L, Years 7 & 8
164	Symmetric Tiles	Using two designs of laminated cardboard tiles, the task encourages exploration of symmetry, where the line of symmetry will most likely be seen as diagonal. One of the shapes is the key to finding many solutions easily, and even when this is realised, the final challenge on the card is not easy to resolve.	line symmetry, problem solving skills	S & L, Years 5 & 6 S & L, Years 7 & 8
170	Equilateral Triangles	Designed to invite younger students to think 'outside the square' this task hints at both algebraic pattern and the idea that three dimensional objects are made from flat surfaces which have familiar names.	constructing triangles, pattern	S & L, Years 3 & 4 P & A, Years 7 & 8
185	Coloured Cubes	This is a classic puzzle requiring a great deal of thought to solve without hint. The four cubes supplied have their faces coloured with four different colours. But each cube has its faces coloured in a different way. The challenge is to place the cubes in a line so that all four colours show down each long surface of the 'square-section rod' that is created.	nets of cubes, problem solving strategies	S & L, Years 5 & 6 S & L, Years 9 & 10
195	Stop At 4	The task is presented as a set of instructions which appear to be a card trick. Indeed, although the procedure could be memorised and used to surprise an audience, further analysis shows that is actually an ingenious device to arrange that the number of cards left in a pile reveal the number on a particular card. Full analysis requires symbolic representation of the instructions.	pattern, number, algebraic representation	N & C, Years 7 & 8
213	Chains	The board is six squares in a line and the equipment is six cubes. The arrangement in which cubes are placed scores points. What is the lowest possible score? The highest? Try boards of different lengths.	number, logical thinking	N & C, Years 5 & 6 N & C, Years 7 & 8
229	Animal Farm	Arithmagon problems (Tasks 188 & 194) are recreated in this task in a more visual way and embedded in a simple story shell. This allows younger and less experienced students to enter the family of problems. In essence the problems are based around three known numbers say a, b & c and three unknown numbers x, y & z. It is also known that x + y = a, y + z = b and x + z = c. The unknown numbers have to be found - but it is not necessary to use algebra to do so.	counting, addition, subtraction, pattern	P & A, Years 3 & 4
230	Pack Up Your Bears	Simply and sweetly this task, which was designed for infants, has children exploring sums of three numbers that add to six (or seven, or eight, or...).	counting, addition, pattern	N & C, Years 3 & 4
231	Flowers In The Field	Also designed for infants, but with a twist that lifts it right up through the school. There are many problems on the card, including ones the students decide for themselves, but they all revolve around having a known number of different flowers and asking how many bunches of a different size can be made.	counting, pattern, combinatoric mathematics	N & C, Years 3 & 4 N & C, Years 7 & 8
234	Growing Tricubes	A Tricube is an L-shape made of three cubes. The task explores different objects that can be made with various numbers of Tricubes and asks for calculations related to their base area, surface area and volume. The greater challenge is making Tricubes from Tricubes and developing a pattern.	3D spatial reasoning, area, surface area, volume, visual and number patterns, algebraic generalisation	Originally designed for: C & M, Years 9 & 10 This task could also be used to enhance: C & M, Years 5 & 6
235	Tables for 25	The familiar context of sitting in table groups in the classroom leads to problems with multiple solutions when the teacher adds conditions related to the size of the table group and the least number of boys (or girls) allowed to be in a group.	multiples, factors, primes, reasoning skills, problem solving strategies - particularly trying every possible case	N & C, Years 3 & 4 N & C, Years 7 & 8
236	Star Numbers	Star Numbers are constructed by adding Triangle Numbers to each side of a Square Number. In this sense, they have to be made as shapes to be understood. As the size of the Star Number increases, so does the number of plugs needed to make it. Students can explore particular cases, thereby applying number skills, or can generalise for any size Star Number.	square and triangle numbers, patterns, arithmetic, generalisation, graphing, substitution & solution of equations, quadratic equations.	N & C, Years 5 & 6 P & A, Years 9 & 10
237	Trisquares	Students explore the creation of new shapes from this simple shape made of three squares joined as an L. This leads to the problem of finding all the new shapes that can be made with just 2 Trisquares and to an investigation of area and perimeter. Each of these aspects of the investigation have a number/computation component.	concept and measurement of area and perimeter, reasoning skills	C & M, Years 3 & 4 N & C, Years 7 & 8
238	Growing Trisquares	The crux of this problem is that 4 Trisquares can be joined to make a new, scaled up, Trisquare. This provides a 'template' for constructing the next size, and the next, and the next... The visual pattern can also be represented as a number pattern and this leads to graphing, equation work and scale factors.	growth pattern, squares, powers of 2, multiples, scale factors, graphing, length/area relationship under enlargement	P & A, Years 5 & 6 P & A, Years 9 & 10
239	Money Charts	Text book exercises on adding and subtracting money come to life in this hands-on logic challenge that is most effectively completed by patient application of if-then reasoning and working backwards. In effect there is a little equation solving going on too.	Australian coin recognition, addition & subtraction of money, making change, concept of minimum, if-then reasoning, working backwards	N & C, Years 3 & 4 N & C, Years 7 & 8
240	Less Than Fractions	Number tiles (1 - 9) allow students to experiment with fractions less than 1 in a non-threatening, open-ended way. Early success is guaranteed because there are 36 possible answers and the obvious one is 1/2. But the greater challenge is to add two fractions (each tile can be used only once) and still get an answer less than one. How many solutions are there? How do you know when you have found them all?	concept of a fraction - especially meaning of numerator and denominator, fractions less than 1, addition of fractions, breaking a problem into smaller parts	N & C, Years 5 & 6 N & C, Years 9 & 10
241	Sicherman Dice	It's easy enough to work out the possible results when two usual cube dice are rolled and the numbers added. It's also not too much of a challenge to work out the distribution of those sums and the associated probability of each outcome. But what happens if I asked you to find other cube dice that could be rolled and summed to get the same probabilities? And what happens if I restrict the dice faces to non-zero whole numbers? These investigations are the focus of Sicherman Dice.	basic number skills, reasoning skills, probability and probability distributions	C & M, Years 5 & 6 C & M, Years 9 & 10

Note

• Task above marked with a 'Note' were originally reserved for the separate kit Points of View. This kit is no longer available, but it does provide a model for creating a Mixed Media unit. Schools which have this kit and Maths300 membership could create their own form of this kit based on the model. It would enhance the Space curriculum in Years 5/6 or 7/8. The kit explores the topic of representing 3D objects in 2D was also used successfully in Year 11 non-academic courses. The relevant Maths300 lesson is Lesson 163, Building Views and its software. Alternatively, or perhaps as well as, if your school has the Picture Puzzles resource, your version of the Mixed Media unit could include the Building Views slide show from the Shape & Space A menu.

Maths300

No.	Name	Description	Content	Add To...
70	Pick A Box	The boxes have been packed, wrapped and numbered. They have been sorted into groups for delivery to different areas. Then - oh no! - a gust of wind and they are scattered and mixed up. Not to worry. Santa's #1 Elf remembers the sorting rules. The groups can be made again just by following the rules. Unfortunately, although old #1 doesn't know it, the rules lead to more than one regrouping. How many ways of grouping the boxes can the class find?	basic number skills, consecutive numbers, odds and evens, problem solving strategies	N & C, Years 5 & 6 N & C, Years 7 & 8
73	Halving Squares	Two students each take a square of the same coloured paper (about 10 cm square), fold it and then cut it (or neatly tear it) in half. There are two main ways to do this. The students now have four pieces between them. Combine these using the rule matching edges have to be the same length. There are several ways to do this and they make an interesting variety of shapes. Challenge questions include: In how many ways can these shapes be combined? and How many different shapes could we make?.	simple numeric relationships such as halves and quarters, Working Mathematically, specific problem solving strategies, 2D shapes and their names, concept of 'different' which involves symmetry	S & L, Years 3 & 4
104	Factorgrams	Factorgrams are an interesting visual way to illustrate the factor relationships that exist within numbers. As well as the practise of division and multiplication facts, there is considerable opportunity for exploring prime factors. In addition several problem solving situations are presented.	factors, multiples & primes, multiplication and division facts, problem solving	N & C, Years 7 & 8
111	Baby In The Car	Newspaper reports show some parents leave their babies locked in cars; sometimes with very sad results. Babies, toddlers and animals must never be locked in cars on a hot day. Why? Because they will quickly dehydrate. While there are physiological reasons for this, one major factor is mathematical; the smaller the volume, the greater the proportional surface area. Mathematical modelling is used in this lesson to inform discussion of a real-life social issue. Concrete materials help students see, understand and explain why the surface area to volume ratio of babies is much greater than that of adults.	surface area, volume, ratio, further development of spatial perception, discussing and writing mathematics in a real-life context, mathematical modelling, collecting personal data (optional), interdisciplinary content such as: challenging sex role stereotypes, child care and parenting, citizenship and links with science	C & M, Years 9 & 10
120	1089	With due flamboyance the teacher announces that this lesson will be special because the answer to all the questions has already been written out. It soon becomes clear that starting with (almost) any 3-digit number, followed by a subtraction and an addition, the answer is always 1089. The young mathematicians in the class are challenged to recognise that this can't be magic, but rather there must be a mathematical explanation for why this happens. The challenge of the lesson is to explain why the answer is always the same and to communicate this to others.	addition and subtraction, place value, Working Mathematically with an emphasis on reasoning and communication, concept of proof, representation of place value in algebraic symbols	N & C, Years 5 & 6 N & C, Years 9 & 10
122	How Many People Can Stand?	How many people can stand? It's a question which engineers and others often have to ask. In this lesson the question is asked in the context of a story shell where the classroom becomes the 'standing room only' section of a venue, for the purposes of encouraging estimation and calculation of number and area and a variety of approaches to solving the problem.	estimation and calculation of number in the context of area, ratio concepts, problem solving strategies, connection to sampling techniques	C & M, Years 5 & 6 C & M, Years 7 & 8
123	Mirror Bounce	An historical story shell questions how artists deep in the Egyptian pyramids were able to find sufficient light to create the elaborate paintings on the walls of the burial chambers. The story generates interest in the properties of light, in particular in the intensity of light and its reflection. Students use mirrors to 'bounce' light around the classroom in a group activity with the challenge of 'hitting' a stated target.	development of spatial perception, symmetry concept of angle, angle of incidence and angle of reflection, cross-curriculum links with history and science, calculations of light intensity, indices	S & L, Years 5 & 6 S & L, Years 9 & 10
126	Make A Moke	Played in a similar way to the parlour game Beetle (Lesson 121), Make A Moke is used to challenge student misconceptions about probability related to a cube dice. In particular the lesson focuses on the misconception that 'the six never comes up for me' and explores the reality behind this idea both empirically and theoretically.	intuitive probability, probability - short term variation and long run frequency, statistics, organising and displaying data including frequency graphs, stem & leaf graphs, mean, median, mode, designing probability experiments, binomial probability distribution	C & M, Years 5 & 6 C & M, Years 9 & 10
130	The Mushroom Hunt	Six people go looking for mushrooms. Each one has a basket. When they return they compare the numbers of mushrooms in their baskets. They discover two things: (a) Adding up the numbers gives a total of 63. (b) Grouping the baskets in different ways gives sub-totals equal to every number from 1 to 63. The problem is, of course, how many mushrooms are in each basket. The problem can be tackled at every level from an exploration of doubling, to an introduction to powers and indices, to the concept of binary numbers, to an investigation of multiplicative (exponential-like) growth.	addition, doubling, powers of 2, binary numbers, multiplicative (exponential) growth, non-linear graphs, problem solving strategies, reasoning and justification	N & C, Years 5 & 6 N & C, Years 9 & 10
132	64 = 65	We all know 64 does not equal 65! But this jigsaw puzzle visually suggests that it is true. This paradox puzzle is one of a genre of 'missing square puzzles'. Cut the 4 pieces from an 8x8 frame and the area is clearly 64. Place the same pieces into a 13x5 frame and the area now appears to be 65. Where did the extra square come from? The search for, and explanation of, the extra square involves many different mathematical tools and suits many year levels. However, noticing that the key numbers involved (3, 5, 8 and 13) are successive terms of the Fibonacci sequence is the starting point for an extended investigation.	area of rectangle calculations, area of parallelogram calculations, visualisation of shapes, Pythagoras Theorem, Fibonacci sequence, gradient or slope calculations, trigonometry calculations, Sine Rule calculations	C & M, Years 5 & 6 C & M, Years 9 & 10
134	Pentagon Triangles	Take a regular pentagon and cut it into three triangles along its diagonals. Easy to state, easy to start and heaps of maths. At one level the three pieces can be used for creating spatial patterns and exploring shapes such as triangles, pentagons and decagons. At another there is work on angles, lengths, areas, fractions, decimals, Fibonacci Numbers and the Golden Ratio. Best of all, the joy of discovering this mathematical content develops in a problem posing and problem solving environment which reflects the work of a professional mathematician.	area of rectangle calculations, area of parallelogram calculations, visualisation of shapes, Pythagoras Theorem, Fibonacci sequence, gradient or slope calculations, trigonometry calculations, Sine Rule calculations	S & L, Years 3 & 4 S & L, Years 9 & 10
145	Estimating Averages	This concept lesson is cleverly constructed to focus on understanding the idea of an average rather than on the algorithmic skill. There are two central features: (a) The use of estimation followed by a discussion of strategy leading to a second round of estimates. Students almost invariable score many more points in the second round. (b) The use of concrete materials in the middle stages of the lesson, involving students with differing numbers of blocks sharing (in silence) until all players have the same number.	whole number skills - addition, multiplication, concept of an arithmetic average, estimation as a strategy	N & C, Years 5 & 6 N & C, Years 7 & 8
146	Division Boxes	Place digits 1 to 9 into three boxes (no repeats) to make a three digit number so that: (a) the number in the first box is divisible by 1, and (b) the number in the first two boxes is divisible by 2, and (c) the whole three digit number is divisible by 3. There are many solutions, eg: 621, however 612 is not a solution since the 61 is not divisible by 2. In this form the puzzle invokes knowledge of divisibility tests. However by adding more boxes and finding all possible solutions, it turns into an extended open-ended investigation which can suit many levels. There are solutions for each string of boxes up to ten. However, for the 10-box problem there is just one unique solution and part of the lesson is the search for this special number.	whole number skills, divisibility tests, combination theory, logic tree diagrams	N & C, Years 5 & 6 N & C, Years 7 & 8
147	Speed Graphs	This is one of the few classroom applications of y = mx which has personal relevance to students. Students go outside to collect data about their personal walking and jogging speeds and graph both sets of data on the same axes. The graph becomes a mathematical model of reality and every feature of it (origin, slope etc.) relates to an aspect of their physical trials. The lesson assumes students can: (a) plot co-ordinates and draw straight lines, (b) recognise linear graphs and deduce the equation from the graph.	recording data, plotting co-ordinates, creating and interpreting a mathematical model, linear graphs, distance vs time (speed) graphs	P & A, Years 9 & 10
150	Fermi Problems	Fermi Problems have the characteristic that most people who encounter them respond that it is a problem that they couldn't solve without recourse to outside information. Students see that through a series of simple steps using only common sense and their experience they can quite often come up with reasonable estimates for the answers. The lesson is structured around a common problem to illustrate the process, followed by group attempts to solve and report on a problem of their choice.	estimation of number and various measures, calculations - four operations, mental computation/ mental arithmetic, problem solving and posing, potential links with a wide variety of mathematical content including measurement of length, area and volume	N & C, Years 5 & 6 N & C, Years 9 & 10
159	Chances With Crosses	Place the digits 1 to 9 in the shape of a 'plus' sign. Now shuffle the digits until the horizontal and vertical arms have the same total. This number puzzle is titled Crosses and is the basis of Lesson 112. However, Chances With Crosses explores a different part of the iceberg of this task. Suppose the digits are placed at random, what is the chance they will form a solution to the puzzle? The results are quite counter-intuitive when compared to students' initial expectations. The companion software allows the chances to be explored at significantly greater depth and allows the empirical results to be compared with theoretical calculations.	basic arithmetic skills, estimating chance and confirming with empirical results, comparing short term variability with long run outcomes in a chance experiment, skill practice in calculating theoretical probabilities	C & M, Years 5 & 6 C & M, Years 9 & 10
162	Multiplication in a Table Format	This very interesting technique for multiplication has a strong theoretical connection to the distributive property and progresses naturally into factorising and expanding algebraic expressions. The technique arises from a concrete array and has strong visual imagery behind the process. It is also very practical, easy to learn and easy to generate understanding of each step.	array concept of multiplication, partitioning arrays to make 'sub-arrays', whole number skills - times tables, place value related to multiplying by 10, 100 etc., Distributive Law, long multiplication, multiplication algorithm, links to area of a rectangle, links to decimal multiplication, links to factorising and expanding algebraic expressions	N & C, Years 5 & 6 N & C, Years 7 & 8
163	Building Views	The first part of the lesson practises the skills of representing a 3D model in front and side (plan) views. A much richer problem develops when the challenge is reversed, ie: given the plan views, students are asked to build all the possible models which could be represented by the views. The companion software extends the problem into an amazing range of possibilities which allow students to set challenges at their own level.	2D/3D spatial ideas, plan views, spatial visual reasoning, problem solving strategies, combination counting theory	Space & Logic kits at any level. Also see Note above.
164	Match Triangles	Match Triangles is a wonderful example of a good task being the tip of an iceberg. The problem is very easy to set up and appears simple to experienced adults - so easy in fact that we can too carelessly dismiss it and overlook the opportunity to 'mine its depth'. However, the multiple intelligences in our mixed ability classroom can surprise with interpretations of the problem which open the door to visual and symbolic algebra, substitution, solving equations, linear and graphical algebra and more.	basic operations with number, visual and number patterns, patterns in tables, linear algebra including: concept of a variable, generalisation, substitution, solving equations/linear functions, domain and range, equivalence	P & A, Years 7 & 8 The lesson was first recorded in the Year 7 & 8, Pattern & Algebra, Maths With Attitude kit, where it is a centre-piece of a fully supported Replacement Unit.
165	Surface Area With Tricubes	The best text books introduce their work on surface area with glossy diagrams of cube-based structures. As pretty as they may be, they are not a substitute for concrete experience and kinaesthetically developed spatial perception. This lesson provides a hands-on problem solving introduction to surface area, base area and volume and in doing so replaces the drudgery of the text book with a willingness to practise and refine newly found skills.	spatial perception, isometric drawing, counting strategies, surface area, volume, problem solving	C & M, Years 5 & 6 C & M, Years 7 & 8
166	Newspaper Cubes & Volume of a Room	This lesson is about visualising volumes of cubes and cuboids. A conceptual notion is developed by making and counting cubes and this provides a firm basis for later understanding and application of the formula V=LxWxH. Additional benefits are experience of structural engineering features such as the need to use triangles to make objects rigid and consequent connections with students' real life experiences of building. The lesson makes the calculation of volume a richer experience than merely multiplying three numbers.	concept of volume in terms of measuring 3D space with cubes, development of spatial perception and mental imagery, estimation of volumes, calculation of volumes from first principles and formula, cube numbers	C & M, Years 5 & 6 C & M, Years 7 & 8
167	Twelve Days of Christmas	Almost everyone knows that on the first day of Christmas my true love gave me a partridge in a pear tree and on and on for twelve days building up a musical pattern. Mathematicians seek and find patterns and teachers seek and find multiple points to encourage students to enter mathematical investigations. This lesson offers students with logical/mathematical, musical, visual/spatial, verbal, kinaesthetic and interpersonal intelligences opportunity to engage in a problem which can focus simply on number patterns, or extend into complex quadratic algebra.	visual and number patterns, natural numbers, triangle numbers, square numbers, generalisation in words and symbols, quadratic formulae, equivalent algebraic expressions - quadratic	P & A, Years 5 & 6 P & A, Years 9 & 10
168	Truth Tiles 2	Easy to state, easy to start and involves heaps of mathematics. What more could you ask for? Students only need to be able to add and subtract single digit numbers to be able to find initial success with this problem which is, given the digits 3 to 7 arrange any four of them to make this equation true ... _ + _ - _ = _ First we find one solution, then another. Then we ask how many there are altogether and how we will know when we have found them all. Can we check our reasoning another way? Then come the What happens if... ? questions.	visual and number patterns, natural numbers, triangle numbers, square numbers, generalisation in words and symbols, quadratic formulae, equivalent algebraic expressions - quadratic	N & C, Years 3 & 4 N & C, Years 7 & 8
169	Human Computer	A group of students become the working parts of a computer - a human computer - which can be taught (programmed) to perform the four basic operations. The team develops their computing talents and then offers to answer challenges from an audience. The human computer shows how the simplicity of the binary system can be used as the basis of electronic computing. Physical involvement and the group co-operation makes it a most enjoyable and productive learning experience. A spin-off is the opportunity for students to revisit the underlying structure of the place value system.	whole number operations, different number systems (binary), place value concepts	N & C, Years 3 & 4 N & C, Years 7 & 8
170	Take A Chance	This lesson is based around a card game which involves risking counters at each play. The game situation initiates interest and the requirement to risk counters is a measure of student understanding of the chances involved. As they experience the game and make judgements, each student develops a notion of 'good chance'. Challenging students to state their clues for this 'good chance' opens a range of possible explorations, including whole class investigation, small group work guided by an Investigation Sheet or a major project.	recognise and use patterns in number, make statements about likelihood, estimate and calculate probabilities, systematically list possible outcomes to deduce probability, test predictions experimentally, connections to combination theory arithmetic calculations - including percentage, collection, organisation and display of data	C & M, Years 5 & 6 C & M, Years 9 & 10
171	Pick's Rule	This easy to run lesson follows the classic Working Mathematically steps of exploring a context leading to data, noticing patterns in the data, which lead to generalisations, theories and conjectures, making and testing predictions from the theory and ultimately the process of proof. Students draw polygons on grid paper and count: the number of border dots (on the edge of the shape), the number of dots inside the shape, the area (number of squares) within the shape. There are underlying patterns and rules to discover that are true for all such shapes.	measurement - area of simple shapes, data - collecting and organising data, algebra - generalising data into algebra rules, space - shapes, triangles and rectangles	C & M, Years 3 & 4 P & A, Years 9 & 10
172	Licorice Factory	This fantasy story introduction to the concept of prime numbers highlights the power of a story shell to facilitate learning. Many students have experienced stretching a piece of licorice so the idea of a factory which has machines to stretch unit pieces to make them 2 times, 3 times, 4 times, ... 100 times longer seems quite natural. In some classes the visualisation of the factory becomes a reality for the students.	counting, multiples, factors, primes, factor trees, multiplicative rather than additive thinking, number facts - especially application of times tables	N & C, Years 3 & 4 N & C, Years 7 & 8
173	Factors	The number 100 has 9 factors. So do 36 and 225. What do these numbers have in common? Why do they all have the same number of factors? There is an (arguably under-appreciated) rule which can tell you how many factors exist for any number. This investigation is designed to both uncover that rule and see the logic behind it.	whole number operations - multiplication, times tables and factors, prime factors, combination theory, generalising patterns into an algebraic rule	N & C, Years 9 & 10
174	Cookie Count	Children have a strong desire to make fair shares. This lesson capitalises on that desire by finding fair ways to make equal shares when presented with various plates of cookies. Challenges occur at a range of levels and the activity provides a reason for discussing fractions. The lesson is perfectly complemented by the children's story The Doorbell Rang, by Pat Hutchins, so the whole experience can become a very rich integrated unit.	whole number operations, counting strategies, multiples, fair shares, division concept - sharing, remainders, fractions, estimating number measuring weight, capacity and time	N & C, Years 3 & 4
175	Dice Cricket	This game simulates real limited-over cricket formats such as one-day cricket or Twenty20. Played as a game between two students, the context provides a rich array of mathematical ideas from a simple starting point. The lesson involves collecting and analysing class data which allows analysis of several aspects of the mathematics. Using the computer simulation of the game, long-term patterns can be explored, and empirical results compared with theoretical expectations.	whole number operations, representation and interpretation of data, averages, simple probability calculations, probability distributions, expected and empirical results, mathematical modelling	N & C, Years 3 & 4 (C & M, Years 5 & 6) C & M, Years 9 & 10
176	Counting Machines	If you want students to understand, apply and interpret a place value system which took thousands of years to refine, then this lesson will contribute. Beginning with physical involvement, the lesson constructs a family of Counting Machines which all operate on a place value system. But only one type works like 'our numbers'. Rather than teaching how this one type works, the students abstract from a wealth of activities over many lessons, the critical elements which make our system work.	basic number skills, place value concepts, addition and subtraction notions, counting, skip counting, group counting	N & C, Years 3 & 4
177	Birth Year Puzzle	This popular puzzle offers many advantages. It shows the mathematical investigative process, has huge opportunity to practise basic skills, can develop creativity, offers 'point-of-need' opportunity for teachers and caters for mixed ability classes. The challenge is to use the four digits of your Birth Year in that order, and with any legitimate mathematical operations, to generate all the numbers from 0 to 100.	whole number skills, order of operations, factors, powers, square roots, factorials, integers, repeating decimals, problem solving strategies	N & C, Years 7 & 8
178	Tables For 25	Working in groups, a context familiar to all students, is the focus of the lesson. Mr. Edwards begins his class of 25 by asking them to organise into groups of 5 at each table, with at least two boys in each group. How the table arrangements work out will depend on how many boys there are in the class altogether, which is what opens the door to a broad investigation and a personal project for each student.	basic number skills, factors and primes, problem solving strategies	N & C, Years 5 & 6 N & C, Years 7 & 8
179	Sporting Finals - NRL Rugby League	This lesson is a version of Sporting Finals, which is based around the Australian Football League (AFL). If Rugby Leagues is your local code then substitute this lesson for Sporting Finals where it appears in Maths With Attitude. As a whole class investigation this is in C & M, Years 9 & 10. As a task, this is the task Final Eight in C & M, Years 5 & 6 and 9 & 10. The cameo for Final Eight includes a playing board for each code.	collecting and organising data, mathematical modelling of a real-life situation, probability and chance, percentage calculations within probability, combination theory analysis	See notes on left.
180	Maths of Lotto	Gambling, in all forms, is a significant social issue. This lesson focuses on the Lotto-style game and takes the approach that if players better understand their chances they are likely to make better choices about how much of their income to commit to the chance of winning. The game is presented as a whole class investigation involving probability, combination theory, statistics and working mathematically. Companion software extends the problem solving possibilities.	probability and expectation, combination theory and simple proportion, statistics, interdisciplinary learning, mathematical modelling, problem-solving strategy: solve a simpler problem	C & M, Years 5 & 6 C & M, Years 9 & 10
181	Natalie's LCM Task	Take a closed question such as What is the lowest common multiple of 45 and 60?, turn into an open question The lowest common multiple is 180, what are the two numbers? and you have an open investigation. The investigation is applicable at many levels and is supported by software.	basic number skills, factors, multiples and primes, combination theory, algebraic modelling, problem solving strategies	N & C, Years 5 & 6 N & C, Years 7 & 8 (N & C, Years 9 & 10)
182	Fractions to Decimals (on a rope!)	This lesson builds a strong conceptual understanding of equivalence when converting fractions to decimals. It offers students concrete, hands-on and visually-based experiences before moving to a formal algorithm. Students use a rope with pegs as markers to first establish ten parts to represent decimals on a number line extending from 0 to 1. Then they fold the rope and explore by estimation how common fractions convert to decimals using the markers on the rope. The lesson features group work, estimation, problem solving and concrete and visual learning.	connecting fractions and decimals as different representations of the same number, exploring fraction families, connecting equivalent fractions and decimals, estimation, problem solving, visual concrete modelling	N & C, Years 5 & 6 N & C, Years 7 & 8
183	Snakes & Ladders	This famous childhood game originated in ancient India around 2 CE and is riddled with opportunities for mathematical investigation. After playing, exploring and analysing the game, students can take on the role of 'mathematical game board designers' and produce their own game boards to meet desired specifications. Using a computer, students can add snakes or ladders wherever they choose and analyse the effects of doing so. Hypotheses can be created and tested, both theoretically and empirically.	data representation and interpretation, averages, simple probability calculations, probability distributions, concepts of expected and empirical results, logic and reasoning	C & M, Years 5 & 6 C & M, Years 9 & 10
184	Making Selections	A standard probability skill exercise in Year 11/12 texts is: "There are 7 students in a team. One of them is the captain, another is the vice-captain. What is the chance that if three students are chosen at random either the captain or vice-captain will be included? " Students can be drilled in the formula, but they often have a limited understanding of why the formula works. In this lesson, the skill-based exercise is converted into an investigation for students from as early as upper primary school. It is designed to aid students' understanding of probability and extend their range of problem-solving strategies.	simple probability, collecting and organising class data, combination theory, probability distributions, expected results/long run frequency, modelling, empirical and theoretical calculations, problem solving strategies	C & M, Years 9 & 10
185	Sicherman Dice	Roll two cube dice and add the two numbers shown. This produces results from 2 to 12, with 7 being the most likely total. The number of ways for each total is, in sequence, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1. Challenge: Is it possible to change the numbers on the two dice (from the standard 1, 2, 3, 4, 5 and 6) to other positive whole numbers and still generate the same table of results? There is just one way of doing this (thought to be unique) - now called Sicherman Dice. The challenge for students is to discover the new numbers to place on the dice. While the lesson uses the context of dice, the major learning is about number combinations, problem solving and reasoning.	simple addition practice, problem solving, data representation, data interpretation, probability distributions, working mathematically, mathematical modelling	C & M, Years 5 & 6 C & M, Years 9 & 10
186	Addo	This lesson involves playing a Bingo-type addition facts game. The game uses two piles of cards. Each pile has eleven cards numbered 0 to 10. Players create their own boards by writing nine different numbers into the squares of a 3 x 3 grid. The game then turns into an investigation to find the 'best' board given there are particular rules for creating an 'addo'. This involves addition facts, then probability ideas and ultimately spatial logic. A computer simulation greatly assists the investigation.	basic addition facts, probability, statistics, spatial ideas, mathematical investigative process	N & C, Years 3 & 4
187	Times 99	Times 99 revolves around the teacher announcing that they can 'multiply by 99' in their head. It is a genre type of lesson where the teacher 'knows the trick' and leads students to discover and then understand why the trick works. It empowers students to use the trick for themselves. Puzzles such as these have always had an interesting place in teaching, but unravelling the trick often involves much reasoning and generalisation. However, the effort often results in a very satisfying sense of achievement.	whole number operations, commutative property, distributive property, order of operations, mental arithmetic, seeing and representing generalisations	N & C, Years 5 & 6 P & A, Years 7 & 8
188	Missing Square Puzzle: 8=9	At one level, missing square puzzles are a fascination or curiosity, often presented with an aspect of 'magic'. Some examples are 168 = 169, 80 = 81 and in this case 8 = 9. They are essentially paradoxes - the eye being deceived to believe something that is logically not true. However, they also provide the opportunity for maths students to investigate and explain with calculations the logic behind the 'paradox'. In this case the calculations exploit areas and can use Pythagoras' Theorem.	spatial perception development, geometric shapes and language, scale diagram, scale factor, area calculations, difference between two squares, Pythagoras' Theorem	C & M, Years 5 & 6 C & M, Years 9 & 10
189	Four Piles Problem	Easy to state and easy to start, this lesson is based on a puzzle about putting blocks into piles. Lurking beneath is a generalisation (most of the time) and the opportunity to ask 'what happens if...?' questions. The challenges can be approached with materials and/or symbolically, which means it offers opportunities for a wide range of abilities in Years 2 to 12. As always the challenge for the teacher is to select from the teaching craft examples embedded in the lesson, those most likely to encourage learning in their own students.	basic arithmetic, number patterns, algebraic rules, negative numbers	N & C, Years 3 & 4 P & A, Years 9 & 10
190	Who Owns The Monkey?	This very popular task provides cards for students to physically manipulate as they follow clues to solve a problem. In this case, given the pets owned by a group of families and other clues, the challenge is 'Who owns the Monkey?'. This type of problem often appears in puzzle books where they are solved using pencil and paper and a grid. Informally teachers reported about one third of their students like doing this type of problem in written form. However, as soon as the task was made concrete by the provision of cards, 100% of students became engaged. Two additional sets of clues are provided, as well as the challenge of designing a set of clues.	problem solving strategies, language of space and order, combination theory, Working Mathematically	S & L, Years 5 & 6 S & L, Years 9 & 10
191	Fractions & Fraction Charts	In this lesson fractions are presented as parts of discrete groups and the several and varied fraction activities have three main purposes: Multiple Representation - deepening understanding of fraction ideas by moving from concrete to physical groupings to diagrams to symbols and particularly use of words. Personal Physical Involvement - students are personally part of the fraction groups created (they are 'their fractions') and therefore more meaningful. Skill Practice - fluency practice of simple fraction calculations supported by software.	simple fractions as proportions of a group, equivalent fractions, fractional and proportional reasoning and calculations	N & C, Years 5 & 6 N & C, Years 7 & 8
192	Number Partitions (Ramanujan's Problem)	One of the most famous of number problems! "I have 5 blocks - in how many ways can I partition them? That is, how many add ups can I make using whole numbers?" The answer is 7 ways - but what if we started with 4 blocks ...or 6 ...or 7 ... or indeed any whole number? Finding a rule or formula for the answer eluded the best mathematicians for many years until a remarkable Indian mathematician, Srinivasa Ramanujan, working with G. H. Hardy at Cambridge University, developed a formula. The problem is so easy to state and easy to get started for quite young grades but it turns into a hugely challenging investigation.	number skills - addition, Working Mathematically process, algebraic reasoning - towards generalising and finding a formula	N & C, Years 5 & 6 P & A, Years 9 & 10