An Amazing Race

THE MATHEMATICAL THE COMMON ASSOCIATION OF VICTORIA DENOMINATOR 2/15 AN AMAZING RACE

Students are totally absorbed in Lumen Christi’s mathematics challenge: The Amazing Race.

INSIDE As children darted from stage to stage eagerly chatting about strategies, encouraging and

praising each other’s mathematical eforts it was clear that this of all school days was one that Pattern and would be remembered. A day from which we could establish interest and engagement, a day structure from which we could raise both the proﬁle of mathematics in our school community and in the

7 minds of the students that we teach. What had started as an ambition to host a competition Problem solving challenges had quickly permeated through the school and taken on a life of its own. Indeed, from little high ability students ideas big things grow or in this case from MAV Games Days whole school engagement grows.

8 Games: In early September 2014, our school, Lumen Christi Catholic Primary School, was fortunate just trivial pursuits? enough to host a Year 6 MAV Games Day. The day was a great success and really helped

11 students apply their knowledge of mathematics in a supportive team environment, where the Mathematica and the focus was less about getting the ‘right’ answer but more about enjoying everything that maths 14Australian Curriculum has to ofer. Continued on page 4 FROM THE PRESIDENT Associate Professor Marj Horne - Faculty of Education, Australian Catholic University

THE COMMON SOAK UP MATHS DURING THE REVISED VCE MATHEMATICS DENOMINATOR SUMMER STUDY IMPLEMENTATION The MAV’s magazine SUPPORT WORKSHOPS published for its members. Applications for the 2016 National ______Mathematics Summer School are now The revised VCE Mathematics study Magazine 254, April 2015 open. design is now available for implementation from 2016 and can be downloaded from The Mathematical Association of Being awarded a place at this summer www.vcaa.vic.edu.au/ Pages/vce/studies/ Victoria, 61 Blyth Street school gives students who love mathematics futuresd.aspx. Brunswick VIC 3056 an opportunity to have fun together as they discuss and explore mathematics from The Victorian Curriculum and Assessment ABN: 34 004 892 755 some very diferent perspectives. Give your Authority (VCAA) in conjunction with Tel: 03 9380 2399 students who love mathematics a chance the Mathematical Association of Victoria Fax: 03 9389 0399 to apply. There is generally a limit of two (MAV) have organised a series of applicants from each school selected, this Professional Learning workshops during [email protected] gives more chance for a variety of schools to May and June 2015 to support teachers www.mav.vic.edu.au be involved. Students from all over Australia in preparation for implementation of the who are in Year 11 in 2015 apply. It is a revised VCE Mathematics study design Ofce hours: two-week program held at the Australian 2016 – 2018. Participants are encouraged 8.30am – 5pm National University in Canberra from 3-16 to take the revised mathematics study Monday to Friday January 2016. Many enduring friendships design and any draft course plans to the are made on this very special summer workshops. Workshops will be provided for President: Marj Horne school, now in its 48th year, and each year each of the following studies: CEO: Simon Pryor FSAE FAIM a few past students return to help run the program. Mathematical Methods: Units 1 - 4

Print Post Approved There is a strong emphasis on solving Specialist Mathematics: Units 1 - 4 Publication No: PP100002988 problems and many students often find ______Further Mathematics: Units 3 & 4 themselves working on problems well into General Mathematics: Units 1 & 2 the night (because they want to)! There Foundation Mathematics: Units 1 & 2 The Common Denominator may contain is also a strong emphasis on recreation, paid advertisements for third party exploration and fun and many activities, products and services. Third party Participants can attend one or more of these advertisements are not endorsements tours and talks are organised. During workshops as applicable. or recommendations by the The afternoon recreation, students are free to WORKSHOP DATES AND VENUES Mathematical Association of Victoria socialise and explore Canberra (popular of the products or services ofered. The activities include ice skating, ten pin The majority of venues have Polycom MAV does not make any representation bowling, tandem bike riding around the access, hence allowing participants to as to the accuracy or suitability of any lake). There is an entertaining dinner and attend at a live venue or participate via of the information contained in those concert on the last night. advertisements. The MAV takes no video conferencing. A main school will responsibility for the content of the ads, Selection is based on the strength of a host the presenter and other schools can promises made, or the quality/reliability student’s written application, a teacher video conference via Polycom into the main of the products or services ofered in all reference, and the creativity and school feed. advertisements. perseverance you show on a problem ALL SESSIONS ARE FREE For more information about advertising solving test. The NMSS contribution is with MAV email Louise Gray, $1200 and an additional travel surcharge [email protected]. (amount to be advised) applies if students/ ______parents want NMSS to organise return travel from Melbourne airport. The closing To find dates and venues refer to the date for applications is 24 July, 2015 and flyer inserted in this edition of Common Denominator. To register, visit the 90-minute selection MAV Test for NMSS will be held on 7 August, 2015 at http://registration.mav.vic.edu.au/reg/. the applicant’s school. For more information contact Helen Haralambous, 9380 2399 or email Application form can be downloaded at [email protected] www.mav.vic.edu.au/activities/student- activities/national-mathematics-summer- school.html. 2 MATHEMATICALLY ACTIVE

Is your school doing amazing things in MATHEMATICS ACTIVE SCHOOLS the area of mathematics education? Applications are now open for MAV’s • Baden Powell College • Milgate Primary School Mathematics Active Schools program. • Killara Primary School • Mount View Primary School • Penleigh and Essendon Grammar • Sacre Coeur Visit www.mav.vic.edu.au/maths active - - School (Primary) • Serpell Primary School school accreditation.html to get more - • Richmond West Primary School • Stonnington Primary School information about the program and details • Sunshine College • St Thomas Moore Primary School on how to apply. • John Monash Science School

MAV Councillor, Max Stephens presents Serpell Primary School with their Mathematics Active Schools Accreditation. MATHS ON ABC SPLASH

MAV has produced several units for maths THE STORY OF MATHS (EXCERPTS) SLOT CAR CHALLENGE on ABC Splash. Check them out. http://splash.abc.net.au/search?keyword Slot car challenge provides an onscreen slot EXPONENTIAL GROWTH =keyword:Marcus%20du%20Sautoy car game with several types of cars (varying power and mass) and several tracks (varying Exponential growth and doubling time FASCINATING PRIME NUMBERS levels of difculty). Students improve utilises video and spreadsheets for performance by interpreting graphs of background and investigations to help http://splash.abc.net.au/res/teacher_ velocity against distance or time. student understand this difcult but res/25-Fascinating-prime-numbers.html important function. CHOOSE YOUR OWN STATISTIC The efect of added mass or power on the ability to round a corner is studied to http://splash.abc.net.au/media/- http://splash.abc.net.au/media/- (almost) simulate real driving. /m/1150634/exponential-growth-and- /m/1520313/statistics-game doubling-time 3 AN AMAZING RACE Cassandra Lowry and Marguerite McGrath - Lumen Christi Catholic Primary School

While the format of rounds one and three were familiar and relatively easy to set up, the middle round the Amazing Race took a little more planning.

The fifteen mathematics stations were set up across the school. Each of these stations required students to complete a task from a diferent area of mathematics. Stations included measuring distance, comparing capacity, matching facts, following directions, ordering numbers, creating shapes and continuing patterns. Like the show, each station involved a choice of two similar challenges. For example at the comparing capacity station (set in the school sand pit) students had the option of finding and ordering the capacity of five diferent containers or calculating the number of cups a much larger container would hold. After choosing and completing one of the tasks, students were rewarded with a yellow letter that would include a clue to the location of the team’s next challenge.

Parent helpers, student leaders and a few teachers helped to supervise each of the 15 stations. Clear instructions and solutions Students absorbed in a puzzle activity. were provided at each station, which allowed station supervisors to encourage, rather than Preparations for this day had seen the show The Amazing Race. Could we host needing to constantly check the accuracy school host an ‘in house’ Year 5 Maths a Maths Amazing Race? A race in which of student responses. The emphasis was on Games day earlier in the year. Following teams worked together to solve a series of trying to problem solve as team rather than a similar format to the MAV Games Day, challenges? The answer came loud and clear being right or wrong. where teams of four students participate ‘of course we can!’. To assist students to complete each in a series of challenges, this event had the The Amazing Race was an easy sell but challenge efciently the option of using added incentive of involving three adult a clue card was incorporated into the teams (two teacher and one parent team). coordinators were conscious of building towards the MAV Games Day too and so Amazing Race. These blue clue cards The additional challenge of facing the adults (adapted from the television series, Blue’s not only helped to ensure the success of both were adapted for the special maths day. Mixed ability teams of four were Clues) could be used by teams at each the day, but really created a buzz around station in order to provide them with either mathematics in the school community. established and the day divided into three rounds. The teams were chosen by teachers additional information or equipment to Even after the buzz of the day had to ensure that teams were as fair as possible help them complete their chosen challenge. subsided, the teachers in other year levels and had a range of mathematical abilities. For example, at a station which required were still talking about hosting their own Maths buddies were already established in students to order the mass of five unlabelled special maths day. It was seen as a valuable the classrooms and discussions were had tins, the clue card gave students the option exercise, one that would raise the profile of as to how teams could work together to of using a measuring balance. mathematics among the students and a way achieve success. Teams began the race at diferent stations of making mathematics more accessible to Round one was similar to the MAV day and and teams were given an hour to make their parents. However, while other levels wanted way through as many stations as possible in to ‘have a go’ the challenge was evident, in involved teams of students working together to solve a series of multiple choice questions order to accumulate points for their team. what format could we achieve maximum Staf members walked the course taking mathematical engagement as well as (adapted from the format used in NAPLAN tests). Round two was the Amazing photos and giving out extra clue cards to any strong parental involvement and make use teams who were demonstrating efective of the whole school? The answer came Race and round three was the final team challenge. (See page 6 for more detail). teamwork, following school rules or were not from a lofty tomb about mathematics simply in need of some assistance. education but rather from the hit TV 4 AN AMAZING RACE

All our stations!

With the stations in place, parent and Clutching their clue, a map and brimming After lunch the children assembled again for student helpers organised and a bucket full with excitement the students took of to round three. This was adapted for each year of enthusiasm the day began. various points in the school to begin their level. The Year 3’s looked at establishing a activities. Roving around it was evident series of numbers in a set while the Year 2’s Round one, which was held in classrooms, that the students were enjoying the tasks. built a boat to specific dimensions that could began in earnest with teams working Automatically they were applying past hold weight. The final hour of any day can through the questions carefully and knowledge to new situations, making be a challenge but engagement remained enthusiastically. It was a sight to behold as connections and working interdependently. high and the children persisted well through the more capable students read questions There were very few occasions in which the tasks. A brain break and run allowed for aloud and the other students, many of whom children had to be reorientated to a task and the helpers to finalise the scores and the find mathematics challenging, answered and in fact there was a general reluctance to ask prizes given out. discussed options, justifying their thinking for a clue card or help. They were, in short, to each other along the way. This method determined to succeed. There were prizes for the team that had of working was not instructed to them. The achieved the highest overall score, the team children established this routine themselves From station to station the students that worked most efectively together, and it was with an enormous sense of pride discussed what mathematics they had used the best mathematician and the most that the classroom teachers looked on. or how they would have done something supportive team member. However, all Following the scoring, all students were diferently. The parent helpers were teams were praised and encouraged for praised and encouraged for team work and impressed and found themselves learning their eforts. given quick positive feedback about being quite a bit. One parent remarked that if she ‘Good Mathematicians’. had learned mathematics in that way during Speaking with students afterwards many school herself that she may have been commented on how much fun the day was The excitement was palpable as the students more confident today - a resounding parent but an equal number said that maths was gathered in the shared learning space for endorsement if ever there was one! As the now one of their favourite subjects and round two. Armed with a yellow piece of announcement was made to return to the that they had learned so much. Most of the paper for all, the coordinating teacher shared area after the hour there were shouts students went home and told their parents handed out the first clue to each group. of joy at completing the challenges and and the photos were shared through the groans of disappointment that it was over. class blogs and newsletter. 5 AN AMAZING RACE

It was wonderful to see team work amongst the students.

This sparked more interest among parents THE BREAKDOWN OF ROUNDS and soon class teachers and co-ordinators were being approached to explain more. Round 1 Round 2 Round 3 The mathematics buzz that had been so Question time The amazing race The final team challenge evident after the MAV Games Day had Similar to the MAV day, teams Each team started at a Modified to best suit the returned once more. of students worked together to diferent station. particular year level, this solve a series of multiple choice Our school had 28 challenge involved teams of Of course lessons were learned throughout questions (adapted from the teams so two teams students working together the day not only by the students format used in NAPLAN tests). started at the majority to complete a specific participating but also by the teachers and After completing each page of of stations. challenge. co-ordinators. Some tasks will need to be questions (3 questions per page; revised for complexity or simplicity and 10 pages in total) one student Teams were The Year 2 final challenge of course a new cohort of students brings from the team would take their encouraged to use involved students with it diferent challenges. However, the solutions to the judging table. their clue cards if they constructing a boat which framework is there and the goal of raising ever ran into difculty not only met the given the profile of mathematics among parents The judge would mark the given and additional clue dimensions, but was also and students was definitely achieved. This is solutions and if any errors were cards were provided to able to hold weight. particularly evident since we have returned made return the page to the teams (by supervisors to school this year, with teachers on yard student so their team could who demonstrated Students were able to use duty being approached with questions like have a second chance. Points supportive teamwork.) their remaining clue cards ‘When is the Amazing Race on?’, ‘Can I be a were awarded to each team, to trade for additional items leader that helps out?’ and comments like ‘I with additional points given if to better support their boat loved that maths day last year, it was one of the team was able to select the design, such as corks and my favourite days’. correct answer on their first small plastic containers. attempt. Clue cards were given In all ways I think we can conclude that this to each team depending on the maths race was indeed amazing. number of points scored.

6 PATTERN AND STRUCTURE

Council for Educational Research, PASA year level and to track individual progress seeks to find out how children think about over time. mathematical ideas underlying tasks; a revolutionary approach that assesses PASA was devised by Joanne Mulligan and fundamental aspects of mathematical Michael Mitchelmore, two mathematics and understanding, not specific skills. teaching experts at Macquarie University’s By knowing how children approach tasks, School of Education, and further developed teachers can plan and scafold individual in collaboration with ACER Senior Research learning experiences. Fellow Andrew Stephanou. It was conceived and developed by Joanne and Michael over As an interview-based individual the course of a series of innovative research assessment, PASA can provide deeper studies on early mathematical development insight into a child’s mathematical spanning more than a decade. Andrew understanding than a group test. It consists applied Rasch measurement principles to Pattern and Structure of three one-on-one assessments designed the PASA study and worked closely with Assessment (PASA) for children in the first three years of formal the developers of the PASA instruments to schooling (Foundation to Year 2). interpret the results of the analysis. An assessment program for early mathematics (F–2) Each assessment consists of approximately 15 high-engagement, play-like child The PASA series includes a teacher’s guide, Australian Council for Educational Research tasks; teachers use the guidelines in the three response booklets (Foundation, Year 1 and Year 2), an assessment materials kit Patterns and their structures underpin all assessment to evaluate and categorise with reusable stimulus materials and a short mathematical ideas. A pattern is some children’s responses to each task. guide for teachers wishing to use PASA regularity observed in a mathematical PASA results have been equated to the with PAT Maths. Scores and reports can be context and the description of this regularity same scale as PAT Maths, ACER’s popular generated by using an Excel spreadsheet is its structure. mathematics assessment, allowing teachers or can be provided by ACER’s Test Scoring Pattern and Structure Assessment (PASA) to compare how young children are thinking and Analysis service. is an innovative way of investigating young mathematically with the skills they are For more information, and to view a video children’s understanding of mathematics. developing. The scale also allows teachers demonstration of the PASA assessment, A new assessment tool from the Australian to compare children’s results regardless of visit: www.acer.edu.au/pasa.

AN INVITATION

Join the MAV and ACER to discuss great outcomes for students Date Venue through efective assessment to guide teacher’s planning and student learning opportunities. Thursday Malvern Central Primary 23 April School 3.45pm registration This event will allow teachers to discuss the use of assessment 4pm - 5pm tools such as the ACER Progressive Achievement Tests in Thursday St Mary’s Refreshments Mathematics (PATMaths) (1-6) and the new Pattern and 30 April Williamstown provided. Structure Assessment (PASA) Maths (F-2) with colleagues and experienced maths education consultants. Thursday ACER 7 May Camberwell You will be able to view samples of these products and discuss with consultants how we can assist with improving teaching and learning outcomes at your school. Thursday Balcombe Grammar 14 May Mt Martha We look forward to meeting you!

THE MATHEMATICAL ASSOCIATION OF VICTORIA 5

PASA invitation 080315_for CD T2.indd 1 19/03/2015 3:10 pm PROBLEM SOLVING CHALLENGES HIGH ABILITY STUDENTS Penny Willoughby

JOHN’S ACHIEVEMENT

I noticed John (Year 4) stealing glances at his neighbour’s work and went over to quietly chat with him.

‘How are you going, John?’

John was in a class with a group of high ability students, selected speciﬁcally for doing challenging maths problem solving with me.

‘Okay,’ he answered cautiously, but then admitted, ‘It’s really hard.’

‘It can be a bit frustrating when you don’t know how to tackle a problem, can’t it?’ I wanted to acknowledge how John was feeling. He’s a clever kid but is used to working out maths questions quickly, without having to think much. This problem was challenging for John because he didn’t immediately know what to do with it. Finding patterns. John nodded, and glanced again at his neighbour’s notes. I did touch base with John a couple of ‘I know it’s really tempting to ﬁnd out how times on that problem to check he wasn’t someone else is doing the problem, John … completely stuck or getting too frustrated. but I wonder how you’d feel if you didn’t get I made some small suggestions to prompt help from anywhere else? If you kept trying him in the right direction, but kept them on your own? Do you think you’d feel pretty light. It was really important that John knew pleased with yourself if you managed to he was doing it on his own. solve it without any help?’ My reward came about ten minutes later, ‘Yeah, that’d be great.’ when John ran over to me calling across the room. ‘I’ve got it! I’ve got it!’ He was ‘How about you move to a diferent table practically bursting. so that you don’t feel tempted to look at anyone else’s workings out? See if you can When I checked, he was right – he had a challenge yourself to do it alone?’ logical solution to the problem.

‘Nuh. I’ll be okay here. I won’t look.’ ‘I did it all by myself!’ John was grinning.

‘Okay, John. It’s your choice,’ I said evenly - I was absolutely thrilled. ‘Hey, how do you I really meant it and wanted him to know. feel about not copying someone else’s work and doing that by yourself? You must feel As I turned away to another student I pretty satisﬁed with that?’ noticed John lean over and look closely at the other child’s writing. I wanted to stop ‘Yeah! I did it all by myself! I can’t believe it!’ him, but it was important to let him make his own learning decisions. After all, that’s my Building patterns. teaching philosophy - to support students to become autonomous learners.

About a minute later, John got up and announced that he was going to sit by himself. I asked him if he’d like any help with the problem, but he said ‘No,’ and seemed determined to ‘go it alone’. 8 Maths and Geospatial Sciences Interactive Presentations Free School Talks RMIT is offering teachers of mathematics the opportunity to have an interactive presentation in Mathematical and Geospatial Sciences hosted at the RMIT City campus. RMIT representatives will run a free session for your students (preferably year 10 and VCE students) on one of the following topics: • Mathematics and statistics in sport • Careers in mathematics and statistics • Careers in mathematical and geospatial sciences

> For further information phone 03 9925 2283 or email [email protected] Be true to you www.rmit.edu.au/mathsgeo/schoolvisits

RMIT3969 SEH UG CommDenom HP.indd 1 20/02/2015 3:16 pm

WHY IS PROBLEM SOLVING SO this is known as low self-efcacy. These EXAMPLE PROBLEMS BENEFICIAL FOR HIGH ABILITY students need regular opportunities to STUDENTS? struggle, such as tackling challenging Here are some examples of problems I maths problems, in a supported use with high ability students when I run I am a firm believer that all children benefit environment, to build and maintain incursions in schools: from doing maths problem solving. their self-efcacy. This enables them Year 2 – Children meet Marty the Martian, However, my specialty is developing high to become resolute and successful explore unusual number patterns based on ability (gifted*) students and helping them learners and grow towards their real engaging stories and hands on activities. move towards achieving their potential. potential. - Here are some compelling reasons why Year 3 – Pulsar Beam Confusion in which • High ability students are equipped problem solving is ideal for these clever students rethink calculator numbers by with brains that are capable of complex minds: focusing on their light bars. and creative thinking. Problem solving • Children with high mathematical ability provides a real opportunity for them Year 4 – Puzzle Me Please provides children often coast their way through most of to exercise and strengthen these with a range of secret number, alphabetic primary school and at least some of capabilities. and other number puzzles. secondary school. Because they coast, they don’t develop the persistence and • When facilitated well, problem Year 5 – Risky Remainders where students self-confidence they need when they solving builds your relationship with solve complex division puzzles with just a later face more complex, mathematical the students. Trust develops and the few clues about the remainders. concepts. Setting problems at the right students learn that you will not make level (neither too easy nor too hard) for it too easy or too hard, nor will your Year 6 – Secret Signs invites students to these students will stop them coasting abandon them when they’re struggling. apply the BODMAS rules to solve mystery and cause them to draw upon and They learn that you are working maths statements. stretch their abilities. towards their proud moments of ‘I did it by myself’ rather than spoon-feeding Year 7 – What’s Your Base? Students • When high ability learners coast, they them through their schooling. This is use non-decimal number bases to solve can lose faith in themselves as learners; what they need from us. problems. 9 PROBLEM SOLVING CHALLENGES HIGH ABILITY STUDENTS (CONT.)

Year 8 – In Prime Suspect Decomposition • Resist giving too much away – Avoid the students use prime decomposition to the temptation to give the answer, how solve number problems. to do something or hints that remove the students’ need to think. Instead TIPS FOR TEACHING PROBLEM ask the students questions to direct SOLVING them back to their plan of action, the Here are my top tips for efective teaching efectiveness of their processes and of maths problem solving: choice of strategies. Have a range of levels, hints and support materials • Pre-assess - Start with a maths available to ofer to students once you problem solving pre-test to assess work out what level of support they students’ levels of problem solving need. competence. This will assist you to pitch the right level of problem to each • Focus on explanations and student. mathematical thinking – Acknowledge correct workings out and • Call-out the challenge - Early on, answers but give more encouragement explicitly discuss with the students (or marks) for justifying an answer, that they may experience frustration, explaining thinking, choosing strategies want to give up and that the goal is wisely and efectively following a logical to develop persistence and derive process. satisfaction from solving something • Don’t panic Expect and accept the challenging. (I have a light ‘Never- - students’ reactions and objections give-up’ story that I start the discussion with.) to problems that at first appear impossible. When a student resists your • Establish a cone of silence - Seek attempts to challenge them, remember the students’ agreement to keep their that it may take time for them to learn answers secret so that everyone gets a to trust you and reach that feeling of chance to do the thinking. (I sometimes immense satisfaction they will get when give the students a scorecard to rate they independently solve their first their independence and satisfaction in difcult problem. solving problems independently. More AUTHOR points when they don’t ask for help and try several strategies or approaches, Penny Willoughby is a teacher and less when they work together or use professional learning facilitator who hints.) is passionate about gifted education, creativity, problem solving and • Teach steps and strategies - Explicitly - teach the steps and strategies for diferentiation. efective maths problem solving and To visit Penny’s website, a wonderful ask students to plan how they will tackle resource on gifted learners for teachers, go each problem, then reflect on how they to www.thinkingoutsidethebox.com.au or went. email [email protected]. Using patterns to calculate. • Do it first - Unless it’s really easy for * In Victoria, the widely accepted definition you, always do the problem yourself of giftedness is adopted from Françoys first to ensure you’re fully prepared to Gagné’s model (2004), where ‘giftedness’ Penny does contract work with the troubleshoot the students’ questions, is understood as outstanding potential. MAV and is available to conduct sticking points and frustrations. I Around 10-15% of people may fall within professional development sessions to sometimes write notes about how I the full range of gifted abilities. (Victorian teachers. worked an especially tricky problem Department of Education & Training, 2015, out, what order I did it in as well as the www.education.vic.gov.au/school/teachers/ To book professional development steps and strategies I used. teachingresources/diversity/pages/gifted. email Jennifer Bowden, jbowden@mav. aspx, Victoria, Australia.) vic.edu.au or telephone 9380 2399.

10 Free resources and support at your fingertips. By Teachers, For Teachers » Australian Curriculum Classroom Activities » Ready-made Assessments and Answer Sheets » Webinars Live and On-demand » On-line Courses

Visit education.ti.com/aus-nz/teachers

GAMES: JUST TRIVIAL PURSUITS? PAUL SWAN PD SESSIONS

Games are often used in mathematics Stonnington Primary School was lucky enough They were really fun because we got to learn to develop ﬂuency. In these sessions Dr to receive various mathematics board games how to play new games about place value. Paul Swan will share some games that he for our Year 1/2 students to use. has created and show how to gain the As the students were learning about place - Year 1 student most from playing a game. Links will be value, these resources aided them to I loved using the place value board games made to the four proﬁciency strands from learn more about place value in a fun and because they were fun games to play. It was the Australian Curriculum while playing challenging way. really easy because all we needed was the the games. Suggestions will be made for These games allowed for diferentiation in board game and nothing else. I also really liked catering for a wide range of abilities while how there were questions on the board games playing games. learning styles and learning abilities it was also easily accessible and could be utilised within that we needed to answer. That made it fun. COME ALONG – IF YOU ARE GAME! our mathematics curriculum, as students only - Year 2 student needed a board game and counters. Jade Lipson DETAILS This meant that students could readily pick up Monday 18 May, 4pm – 5pm Classroom Teacher, Stonnington Primary the games and play in small, unaided groups. School Stonnington Primary School The students really enjoyed playing the games. We asked a few students what they thought of Tuesday 19 May, 4pm – 5pm the resources and here is what they had to say: Cliveden, MAV, Brunswick May 17-23 is Education Week and this years theme is Mathematics: Crack the COST Book online to avoid missing out! code with maths. Visit www.education. $40 each session. You will receive a game www.mav.vic.edu.au/pd. vic.gov.au/about/events/Pages/ of your choice to take back to school. edweek.aspx. 11 GAMES DAYS

In September 2014, Malvern Central School, Ripponlea, Stonnington Primary School, South Yarra Primary School, St Kilda Primary School and Armadale Primary School attended The Stonnington Maths Games Day, organised by the MAV and hosted at Stonnington Primary School.

Some of the things that we did during this very fun and exciting day were Geoshape challenges, such as making 3D shapes out of them and trying to construct 2D letters using the shapes. We also did a pair-challenge where we got into pairs and played games against other pairs from diferent schools. Some other activities involved problem solving and team maths relays.

I would like to thank Clem Newton-Brown, the former State member of Prahran, for coming to present medals to the teams who came in 1st, 2nd and 3rd place and Jennifer Bowden from MAV for hosting this wonderful and exciting day.

My favourite activity was the Pair Challenge because I really enjoyed meeting other people and getting to know them as well as playing against them. Another one of WHAT BETTER WAY TO ENHANCE YOUR STUDENTS’ SKILLS WITH THE my favourite activities was the Geoshape PROFICIENCY STRANDS THAN PARTICIPATING IN A MATHEMATICS challenges because I felt that it was really fun GAMES DAY? to create shapes out of Geoshapes - but it A Maths Games Day is an opportunity for Besides the Statewide Mathematics Games also got a bit tricky! students to develop their mathematical days, the MAV can coordinate and run a Malvern Central School took out 1st and talents and thinking skills in a setting where Maths Games Day to suit your school or 2nd place with only 2 points between maths is regarded as fun and worthwhile a cluster of schools in your area, or we can them. I was in the team that came second with like-minded students from a diverse assist you to run your own games days. range of schools. by 2 points but we missed out on 5 points The MAV is particularly interested in because we forgot to put our name on our It is an ideal way for students to participate hearing from regional school who may like Geoshape Letters challenge which would’ve in mathematical activities without the to host a games day. won us the competition. usual class room pressures and is a very All up this was an extremely fun and tiring efective vehicle for getting young people particularly in the middle years of day of non-stop maths fun! Well done to - If you are interested in learning more about all of the competitors that competed in schooling - excited about maths. Games Days, contact Jennifer Bowden, [email protected] (primary) or the day’s events and congratulations to the Problem solving and mathematical games medal winners. Helen Haralambous, hharalambous@mav. address the AusVELS proﬁciency strands. vic.edu.au (secondary) or - Ben Romeo phone 9380 2399. Year 6 Student, Malvern Central School

MATHS TALENT QUEST AND STATEWIDE GAMES DAY REGISTRATIONS ARE OPEN.

Maths Talent Quest: www.mav.vic.edu.au/activities/student-activities/maths-talent-quest.html

Games Days: www.mav.vic.edu.au/activities/student-activities/student-games-days/student-games-day-registration.html 12

MATHEMATICA AND THE AUSTRALIAN CURRICULUM Carmen Popescu-Rose, Director, Mathematics for Excellence. www.wolfram.com/training/instructors/popescu-rose.html.

This article follows on from article in the YEAR 11 – SPECIALIST Term 1 edition of Common Denominator MATHEMATICS – UNIT 1 – TOPIC where we focussed on Essential 2: VECTORS IN THE PLANE Mathematics and General Mathematics. • DEFINE AND USE THE This article looks at teaching and exploring MAGNITUDE AND mathematical concepts as outlined in the DIRECTION OF A VECTOR Australian Curriculum at Year 11 level for (ACMSM011) Mathematical Methods and Specialist • REPRESENT A SCALAR Mathematics. MULTIPLE OF A VECTOR (ACMSM012) YEAR 11 – MATHEMATICAL • USE THE TRIANGLE RULE METHODS – UNIT 1 – TOPIC1: TO FIND THE SUM AND FUNCTIONS AND GRAPHS DIFFERENCE OF TWO • RECOGNISE FEATURES OF VECTORS. (ACMSM013) 2 2 2 THE GRAPHS OF X +Y =R • USE ORDERED PAIR 2 2 2 Figure 21. Input and output for three AND (X−A) +(Y−B) =R , NOTATION AND COLUMN INCLUDING THEIR diferent sets of values for a, b and r and the corresponding circles. VECTOR NOTATION TO CIRCULAR SHAPES, THEIR REPRESENT A VECTOR CENTRES AND THEIR RADII The commands for plotting the centres of (ACMSM014) (ACMMM020) the circles, Point[{{1, -2}, {0, 0}}], can be • DEFINE AND USE used in order to emphasize the relationship UNIT VECTORS AND This investigation requires the use of between a and b and the coordinates of the THE PERPENDICULAR the function ContourPlot in order to circle. UNIT VECTORS I AND J plot the graph of a circle and explore the (ACMSM015) connections between a, b and r and the TOPIC 3: INTRODUCTION TO centre, the axes intercepts and the radius of DIFFERENTIAL CALCULUS Students could be introduced to the notion the circle. • EXAMINE THE BEHAVIOUR of vector by first using the Mathematica OF THE DIFFERENCE Graphics functionality to construct arrows f ( x h ) f ( x ) QUOTIENT + − in the Cartesian plane between two given h points. The command used to construct the AS h → 0 AS AN INFORMAL vectors in figure 23 is Graphics[{Arrow[{{0, INTRODUCTION TO THE 0}, {1, 1}}], Arrow[{{-1, 0}, {-4, 1}}], CONCEPT OF A LIMIT Arrow[{{4, -3}, {1, -1}}], Arrow[{{-2, -1}, {-1, (ACMMM081) -3}}]}, Axes -> True, AxesLabel -> {x’, ‘y’}, AxesStyle -> Arrowheads[{0.03}]]. This behaviour can be easily examined using one or more of the demonstrations from the Wolfram Demonstrations Project. The snapshots shown in figure 22 (over page) provide some information regarding the features of this manipulation.

Figure 20. Output displaying a circle of radius Students can easily interact with this 1 with centre (0, 0). interactivity by moving the points on the curve. The gradient of the line Figure 23. Four vectors in the Cartesian plane. The command ContourPlot[x^2 + y^2 == connecting the two points is automatically 1, {x, -1.5, 1.5}, {y, -1.5, 1.5}, LabelStyle -> calculated and displayed so the students Depending on the purpose of the Directive[14, Thick], AxesLabel -> {‘x’, ‘y’}, can observe what happens to the value of investigation, it can be further extended by AxesStyle -> Arrowheads[{0.03}], Frame f ( x h ) f ( x ) + − as h → 0 . instructing students to construct polygons -> False, Axes -> True] produces a circle of h using 3, 4, 5 or 6 vectors. radius 1 and centre (0, 0). The next step in this investigation is The following command can be used to calculating the magnitude and the direction generate circles of various radii and centres. of the vectors plotted in previous step. To Students could observe the new positions calculate the angle between the vector in and dimensions of the circles as they change the first quadrant and the horizontal axes the values of a, b and r. use VectorAngle[{1, 0}, {1, 1}]. 14 The first set of values, {1, 0} represents the unit vector in the x-axis and the second set of values represents the vector in the first quadrant.

Figure 24. Input and output for the direction of the vector.

The length of the vector is calculated using the command EuclideanDistance[{0, 1}, {1, 0}] where {0,1} is the horizontal vector component and {1,0} is the vertical vector component.

Figure 25. Input and output for calculating the magnitude of the given vector.

The next step in this exploration is calculating and plotting a scalar multiple of a vector. For ease of calculations deﬁne the vector v = {1,1} and then calculate 3v and 1 v as shown in ﬁgure 26. 2 Next plot the three vectors on the same set of axes. Students could also be instructed to calculate the magnitudes of the two new vectors and check if the magnitudes are 3 1 times larger than the original vector and 2 of the original vector respectively.

Figure 26. (from top to bottom) Deﬁning vector v = {1,1}, Calculation for vector 3v, Calculation for vector v.

Figure 22. Snapshots from the demonstration ‘The Deﬁnition of the Derivative’. 15 MATHEMATICA AND THE AUSTRALIAN CURRICULUM (CONT)

Figure 30. Input and output to determine the unit vector in the direction of vector {1,1}.

CONCLUSION

Mathematical concepts can be taught in various ways and it comes down to the teacher’s creativity and imagination to inspire their students in the learning of Figure 27. Cartesian plane displaying vectors v = {1,1}, 3v and v. Mathematics. One might ask: ‘Why use Mathematica to introduce this topic (or any other topic as a matter of fact)?’ Adding and subtracting vectors in Mathematica is fairly easy to compute. First The answer that comes to mind is ‘Because define the vectors as a = {2,2} and b = {2,-1}, the student can take home a notebook and then simply perform a + b and a - b. full of information, interactive activities, examples and solutions to questions, and hints. They can open it when completing homework or when studying for an assessment task and … it’s all there at their fingertips.’ REFERENCES Figure 29. Graphical representation of (top) Vector addition a + b, (bottom) Vector Australian CURRICULUM, Australian subtraction a - b. Curriculum, Assessment and Reporting Authority (ACARA), Senior Secondary, The command used to represent the Mathematics, Essential Mathematics addition vector is Graphics[{{Blue, www.australiancurriculum.edu.au/ Arrow[{{0, 0}, {2, 2}}]}, {Red, Arrow[{{2, SeniorSecondary/Overview 2}, {4, 1}}]}, { Purple, Arrow[{{0, 0}, {4, 1}}]}}, Axes -> True, AxesLabel -> {‘x’, ‘y’}, Mathematica (Version 9) [Computer AxesStyle -> Arrowheads[{0.03}]]. software], Wolfram, www.wolfram.com/ Figure 28. (top to bottom) Defining vectors a mathematica/ and b, a + b, a - b. The command used to represent the subtraction vector is Graphics[{{Blue, timeanddate.com, www.timeanddate.com/ The geometrical representations of Arrow[{{0, 0}, {2, 2}}]}, {Red, Arrow[{{2, 2}, time/map/ the vector addition a + b and the vector {0, 3}}]},{Purple, Arrow[{{0, 0}, {0, 3}}]}}, subtraction a - b is shown in figure 27. Axes -> True,AxesLabel -> {‘x’, ‘y’},AxesStyle The Definition of the Derivative from ->Arrowheads[{0.03}]]. the Wolfram Demonstrations Project http://demonstrations.wolfram.com/ Mathematica functionality can also be used TheDefinitionOfTheDerivative/ to determine the unit vector of a given Contributed by: Jim Swift. vector by using the command Normalize [x,y]. In figure 28 the two components of the unit vector in the direction of vector The MAV ofer professional {1,1} are given by using the command development advice and workshops on Normalize[{1,1}]. the use of Mathematica. Contact Helen Haralambous, [email protected].

16 TO TEST OR NOT TO TEST?

There will be a lively debate on the topic ‘That national testing NATIONAL TESTING distracts teachers from the business of teaching students mathematics’. Come along and here both sides of the argument. IN TRIAL AT THE OLD DATE: Tuesday 12 May, 2015 TIME: Early evening (exact time TBC) MELBOURNE GAOL VENUE: Old Melbourne Gaol, 377 Russell St, Melbourne. Watch out for further details in MAVlist. A very quick AGM of MAV - FREE DEBATE - members will follow the debate.

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THE MATHEMATICAL ASSOCIATION OF VICTORIA

AGM invitation.indd 1 27/03/2015 11:41 am CAMBRIDGE SENIOR MATHEMATICS AUSTRALIAN CURRICULUM / VCE NEW IN 2015

FROM THE ESSENTIAL MATHEMATICS VCE TEAM

Michael Evans Kay Lipson CONTACT US FOR MORE INFORMATION Peter Jones David Main 1800 005 210 Josian Astruc Kyle Staggard [email protected] Neil Cracknell David Treeby David Greenwood Barbara Tulloch www.cambridge.edu.au/education Douglas Wallace

17 SURVEYING ANZAC TRADITIONS Michelle Brooks - Surveying Taskforce

Students who excel at mathematics may be morning, the sunlight hits an inclined mirror RMIT undergraduate student assignment interested to consider a career in surveying, installed on the pole on the outer walkway. and are freely available for school use. a career wherein maths is at the foundation of the day-to-day work of Surveyors. A That mirror reflects the sunlight up to a Whilst fairly technical given they involve career in surveying can lead to all kinds of space in the roof of the building, where spherical trigonometry, teachers of upper exciting adventures and endeavours, and another mirror directs the sunlight through level mathematics classes are invited to enables participants to help shape the face a space in the ceiling and down onto the contact Rod Deakin at randm.deakin@ and future of the country. The surveying granite Stone of Remembrance. gmail.com for a copy of the assignment and profession also helps safeguard some of our solutions. most esteemed Australian traditions, such as Each year the mirrors must be calibrated Remembrance Day. to ensure the sunlight hits the plaque at Victorian school-leavers can embark upon a the correct time as the mirror in the outer four year undergraduate degree with RMIT The Shrine of Remembrance is heralded walkway is mounted to pavers, which University in surveying, or study for five as Melbourne’s most iconic landmark and move slightly due to age. Further, only on years with Melbourne University (a three one of Australia’s largest war memorials. Remembrance Day morning is the second year Bachelor of Environments followed Many people know that the Shrine of mirror installed which requires precise by a two year Master of Engineering in Remembrance is central to ANZAC and calculation in order for them to align Geomatics). Assistant Surveyors with a Remembrance Day services and that it was correctly. Certificate or Diploma in Surveying can also specifically built to honour Australian men work at a technical level to collect, analyse and women who served in the war. The There are only a small handful of people data and develop plans for land surveyors, Shrine of Remembrance, which houses the who share the knowledge of how to ensure though with reduced capacities to analyse Stone of Remembrance, was built in such the sunlight hits the stone at precisely and apply data. a way that at precisely 11am on the 11th the right time and place. Along with this November a band of light hits the Stone knowledge, a theodolite, small stool and illuminating the word ‘love’, harking back to angular mirror aid the team in their mission. the time and date that the armistice ended ‘In 32 years...You feel as though you’re in the Surveying a is a great example of the war. tradition of the constructors of the pyramids applying mathematics to real world and Stonehenge. You find that all of these situations. If you have students who are What few people know is that the have astronomical features of a similar interested in finding out more about introduction of Daylight Saving in Victoria nature similarity built into them,’ says Frank. surveying as a career, direct them to in 1971 meant that the Remembrance Day www.alifewithoutlimits.com.au. ceremony could not go ahead without some For further information on Frank and clever workarounds. Enter Frank Johnston, his colleagues’ work, a video and article an RMIT surveying lecturer whose elegant published in The Age can be found at solution would save the day - for many years www.theage.com.au/victoria/-11hbkt.html. to come. The mathematics employed for the His solution involved two mirrors, inclined calculation of the dip and strike of the at angles. At 11am on Remembrance Day inclined mirror, have been developed into an 18 While you’re helping secure results for others, let us do the same for your money.

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MAP OF AUSTRALIA

CLOSE YOUR EYES AND IMAGINE A MAP OF AUSTRALIA.

• If the state of Victoria is considered as having an area of one unit, what is your guess for the sizes of the other states? • Which state is bigger? • Which states are the same size? • Is Tasmania the smallest? • If Victoria equals ﬁve cubes what do the other states equal? • Can you make a map using diferent materials to represent your answer? • What is the area of Australia? • What is the area of the diferent states? • How might we estimate our answers? • Is my map correct? • How do we measure area on a map? • What other questions might you ask?

Reference: Lesson 50 Maths300 Country Maps

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