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The potential of posing more challenging mathematics tasks and ways of supporting students to engage in such tasks. Peter Sullivan Sullivan MAT Nov 2013

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Abstract While most students want to work on more challenging mathematics, there are still some who require substantial support. The workshop will explore examples of tasks with low "floors" but high "ceilings" that allow all students to engage with the tasks at some level, but which can be extended productively for those who are ready. A particular lesson structure that supports the work of all students on such tasks will be presented and discussed. Sullivan MAT Nov 2013

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What are the challenges that you are experiencing in teaching mathematics? Sullivan MAT Nov 2013

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Some initial assumptions Planning happens at 4 levels: the school, the year, the unit, the lesson We are starting at the “planning the lesson” end The goal is to improve the experience of students when learning mathematics We will focus on a particular type of lesson structure (that is broadly applicable to many types of tasks) Sullivan MAT Nov 2013

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Even though such investigations can be made realistic and authentic… The maximum gradient of a ramp exceeding 1520mm in length shall be 1:14. Ramps shall be provided with landings at the top and bottom of the ramp and at 9m intervals for a ramp 1:14. The length of landings shall be not less than 1200mm. The gradient of ramps between landings will be consistent. Ramps shall be provided with handrails on both sides which do not encroach on the 1000mm minimum clear width. Angles of approach for ramps, walkways and landings is preferably zero degrees. Sullivan MAT Nov 2013

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Or even extended to Design a ramp for some stairs at the school which do not yet have a ramp And write a report for the School Council Sullivan MAT Nov 2013

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Nor are we focusing on games such as In turn, players roll a 10 sided die (numbered 0 to 9) and, after each roll, write the number rolled in one of the rectangles on a board that looks like ÷ The winner has the answer closest to 100 (for example). Sullivan MAT Nov 2013

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Even though such games can be extended to … How could you place 3, 4, 5 and 6 on a board like this, to make the answer closest to 100 ÷ Sullivan MAT Nov 2013

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And I am assuming that you already know how to structure lessons based on texts Sullivan MAT Nov 2013

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There are plenty of resources of great ways to teach mathematics The Shell Centre Materials – http://www.mathshell.com/ Formative Assessment Lessons and Tasks – http://map.mathshell.org/materials http://map.mathshell.org/materials nrich – http://nrich.maths.org/frontpage http://nrich.maths.org/frontpage transum – http://www.transum.org/ http://www.transum.org/ hotmaths – http://www.hotmaths.com.au/ http://www.hotmaths.com.au/ tarsia - there is not actually a website with this name, but a number that offer software (example below) http://www.tes.co.uk/article.aspx?storyCode=6107407&s_cid= RESads_MathsTarsia http://www.tes.co.uk/article.aspx?storyCode=6107407&s_cid= RESads_MathsTarsia Sullivan MAT Nov 2013

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The following are examples of tasks that exemplify the approach on which we will focus Sullivan MAT Nov 2013

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For year 8 Drawing a single straight line, make two quadrilaterals with the same perimeter Sullivan MAT Nov 2013 12 9 10 11 13 C B A D E

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For Year 1 Basketball scores Parrots106 Galahs97 Sullivan MAT Nov 2013 How much did the Parrots win by? (Work out the answer in two different ways)

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What might make teaching a lesson based on one of those tasks difficult at your school? Sullivan MAT Nov 2013

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Proposition Set 1 More of the same is not a feasible response The pathway to improvement is teaching teams working collaboratively on planning, on teaching and on assessment Each lesson sequence should ideally incorporate a variety of types of lessons, tasks and activities All students need to make time (away from school) to develop their own fluency at the skills being taught (you might need to explain the rationale for this) Sullivan MAT Nov 2013

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Proposition Set 2 Students benefit from working on tasks that they do not already know how to do Students are more likely to connect ideas if they compare and contrast related ideas and build networks of concepts for themselves These connections are the key to remembering and transferring knowledge Asking students to solve and/or represent problems in more than one way helps them to build connections There are risks if we build connections too slowly The goal is that students come to know they can learn mathematics Sullivan MAT Nov 2013

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Should we start easy and wind it up or start at challenging or wind it back? Students can benefit when they move from not knowing how to do something to knowing how to do it. In other words, what they have learned is explicit to them. – This does not necessarily happen if they are working on the “known”. When confronted with a task that they cannot do, students need to explore their existing mental structures and schemes, explore links, build connections and identify aspects that are unknown. Sullivan MAT Nov 2013

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Where does the idea of “challenge” come from? Guidelines for school and system improvement (see, e.g., City, Elmore, Fiarman, & Teitel, 2009) The motivation literature (Middleton, 1995; 1999). Sullivan MAT Nov 2013

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This connects to “mindsets” Dweck (2000) categorized students’ approaches in terms of whether they hold either growth mindset or fixed mindset

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Sullivan MAT Nov 2013 Students with growth mindset: Believe they can get smarter by trying hard Such students – tend to have a resilient response to failure; – remain focused on mastering skills and knowledge even when challenged; – do not see failure as an indictment on themselves; and – believe that effort leads to success.

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Sullivan MAT Nov 2013 Students with fixed mindset: Believe they are as smart as they will even get Such students – seek success but mainly on tasks with which they are familiar; – avoid or give up quickly on challenging tasks; – derive their perception of ability from their capacity to attract recognition.

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Teachers can change mindsets the things they affirm (effort, persistence, co- operation, learning from others, flexible thinking) the way they affirm You did not give up even though you were stuck You tried something different You tried to find more than one answer the types of tasks posed Sullivan MAT Nov 2013

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In the video to follow The first child says something like “when you are confused it means you are learning” The second child says “the best part is being confused because you can think about what you can do” The third child says you “learn from being confused” The fourth child says “you can learn by yourself”

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Proposition Set 3 Posing challenging tasks requires a different lesson structure The lesson should foster the sense of a classroom community to which all students contribute with the intention that students learn from each other The experience of engaging with the task happens before instruction Few rather than many tasks All students are given time to engage sufficiently to participate in the review Sullivan MAT Nov 2013

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This is relevant whether or not the students are grouped by their achievement And is applicable with crowded (and even badly behaved) classrooms Sullivan MAT Nov 2013

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achievement ability

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The conventional mathematics lesson Review homework Explain the concept and model the techniques Students practice the techniques Solutions are corrected (by the teacher) Homework is set Sullivan MAT Nov 2013

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Japanese Lesson Study and Lesson Structure Sullivan MAT Nov 2013

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How many squares? Sullivan MAT Nov 2013

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There are Japanese words for parts of lessons Hatsumon – The initial problem Kizuki – -what you want them to learn Kikanjyuski – Individual or group work on the problem Kikan shido – – thoughtful walking around the desks Neriage – Carefully managed whole class discussion seeking the students’ insights Matome – Teacher summary of the key ideas Sullivan MAT Nov 2013

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A five-component cyclic Chinese lesson structure Reviewing Bridging Variation Summarising, and Reflection/Planning Sullivan MAT Nov 2013

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A revised lesson structure Lappan et al. 2006 Launch Explore Summarise Sullivan MAT Nov 2013

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The summarise phase Smith and Stein (2011) anticipating potential responses monitoring student responses interactively selecting representative responses for later presentation sequencing student responses connecting the students’ strategies with the formal processes that were the intention of the task in the first place. Sullivan MAT Nov 2013

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A further revised lesson structure In this view, the sequence – Launch (without telling) – Explore (for themselves) – Summarise (drawing on the learning of the students) … is cyclical and might happen more than once in a lesson (or learning sequence) Sullivan MAT Nov 2013

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The notion of classroom culture Rollard (2012) concluded from the meta analysis that classrooms in which teachers actively support the learning of the students promote high achievement and effort. Sullivan MAT Nov 2013

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Some elements of this active support : the identification of tasks that are appropriately challenging for most students; the provision of preliminary experiences that are pre-requisite for students to engage with the tasks but which do not detract from the challenge of the task; the structuring of lessons including differentiating the experience through the use of enabling and extending prompts for those students who cannot proceed with the task or those who complete the task quickly; Sullivan MAT Nov 2013

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the potential of consolidating tasks, which are similar in structure and complexity to the original task, with which all students can engage even if they have not been successful on the original task; the effective conduct of class reviews which draw on students’ solutions to promote discussions of similarities and differences; holistic and descriptive forms of assessment that are to some extent self referential for the student and which minimise the competitive aspects; and finding a balance between individual thinking time and collaborative group work on tasks. Sullivan MAT Nov 2013

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Getting started “zone of confusion” “four before me” representing what the task is asking in a different way such as drawing a cartoon or a diagram, rewriting the question … choosing a different approach to the task, which includes rereading the question, making a guess at the answer, working backwards … asking a peer for a hint on how to get started looking at the recent pages in the workbook or textbook for examples. Sullivan MAT Nov 2013

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The lessons consist of One or more challenging task(s) One or more consolidating task(s) (see Dooley, 2012) preliminary experiences that are pre-requisite but which do not detract from the challenge of the tasks supplementary tasks that offer the potential for differentiating the experience through the use of – enabling prompts (see Sullivan, et al., 2009) which can reduce the number of steps, simplify the complexity of the numbers, and vary the forms of representation for those students who cannot proceed with the task; – extending prompts for students who complete the original task quickly which often prompt abstraction and generalisation of the solutions. Sullivan MAT Nov 2013

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But if I try this, … I will not have enough time for the rest of this topic I do not have time to prepare lessons like this My students will not persist enough to engage with the task I am not sure I will be able to control the class My students will not learn the mathematics by themselves. I need to tell them. … Sullivan MAT Nov 2013

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Epmc perimeter Sullivan MAT Nov 2013

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A primary example Sullivan MAT Nov 2013

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There are many ways to find the difference between two numbers Sullivan MAT Nov 2013

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Basketball scores Parrots106 Galahs97 Sullivan MAT Nov 2013 How much did the Parrots win by? (Work out the answer in two different ways)

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Basketball scores Wombats26 Possums18 Sullivan MAT Nov 2013 How much did the Wombats win by? (Work out the answer in two different ways)

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Enabling prompt(s) Sullivan MAT Nov 2013

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Basketball scores Eels18 Carp13 Sullivan MAT Nov 2013 How much did the Eels win by? (Work out the answer in two different ways)

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Basketball scores Cats8 Dogs3 Sullivan MAT Nov 2013 How much did the Cats win by? (Work out the answer in two different ways)

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Extending prompt Sullivan MAT Nov 2013

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Darts scores Parrots1005 Galahs988 Sullivan MAT Nov 2013 How much did the Parrots win by? (Work out the answer in two different ways)

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Consolidating task(s) Sullivan MAT Nov 2013

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Football scores Seagulls63 Kingfishers55 Sullivan MAT Nov 2013 How much more did the Seagulls score? (Work out the answer in two different ways)

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A junior secondary lesson Southern Adelaide Region

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Surface area = 22 A rectangular prism is made from cubes. It has a surface area of 22 square units. Draw what the rectangular prism might look like? Southern Adelaide Region

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Enabling prompt: Arrange a small number of cubes into a rectangular prism, then calculate the volume and surface area. Southern Adelaide Region

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Extending prompt: The surface area of a closed rectangular prism is 94 cm 2. What might be the dimensions of the prism? Southern Adelaide Region

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A consolidating task The surface area of a closed rectangular prism is 46 cm 2. What might be the dimensions of the prism? Southern Adelaide Region

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What does that task do? Southern Adelaide Region

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What does the curriculum say? Southern Adelaide Region

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YEAR 7 Establish the formulas for areas of rectangles, triangles and parallelograms and use these in problem solving Calculate volumes of rectangular prisms YEAR 8 Choose appropriate units of measurement for area and volume and convert from one unit to another volume Find perimeters and areas of parallelograms, trapeziums, rhombuses and kites Investigate the relationship between features of circles such as circumference, area, radius and diameter. Use formulas to solve problems involving circumference and area Develop the formulas for volumes of rectangular and triangular prisms and prisms in general. Use formulas to solve problems involving volumevolume Southern Adelaide Region

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Year 9 to solve problems involving circumference and area of circles and part circles, and the surface area and volume of prisms and cylinders Southern Adelaide Region

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Describing the proficiencies Understanding – (connecting, representing, identifying, describing, interpreting, sorting, …) Fluency – (calculating, recognising, choosing, recalling, manipulating, …) Problem solving – (applying, designing, planning, checking, imagining, …) Reasoning – (explaining, justifying, comparing and contrasting, inferring, deducing, proving, …) Southern Adelaide Region

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Connecting to the proficiencies: YEAR 7 – Understanding includes making connections between representations … – Fluency includes …calculating areas of shapes and volumes of prisms – Problem Solving includes formulating and solving authentic problems using Measurements – Reasoning includes investigating strategies to perform calculations efficiently YEAR 8 – Understanding includes explaining measurements of perimeter and areaperimeter – Fluency includes evaluating perimeters, areas of common shapes and their volumes and three dimensional objects – Problem Solving includes formulating, and modeling practical situations involving … areas and perimeters of common shapes, – Reasoning justifying the result of a calculation or estimation as reasonable Southern Adelaide Region

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The achievement standards: By the end of Year 7, students use formulas for the area and perimeter of rectangles and calculate volumes of rectangular prisms.perimeter By the end of Year 8, students … solve problems relating to the volume of prisms … convert between units of measurement for area and volume, … perform calculations to determine perimeter and area of parallelograms, rhombuses and kites.volume perimeter Southern Adelaide Region

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The lessons consist of One or more challenging task(s) One or more consolidating task(s) (see Dooley, 2012) preliminary experiences that are pre-requisite but which do not detract from the challenge of the tasks supplementary tasks that offer the potential for differentiating the experience through the use of – enabling prompts (see Sullivan, et al., 2009) which can reduce the number of steps, simplify the complexity of the numbers, and vary the forms of representation for those students who cannot proceed with the task; – extending prompts for students who complete the original task quickly which often prompt abstraction and generalisation of the solutions. Sullivan MAT Nov 2013

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A probability task Sullivan MAT Nov 2013

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First do this task On a train, the probability that a passenger has a backpack is 0.6, and the probability that a passenger as an MP3 player is 0.7. How many passengers might be on the train? How many passengers might have both a backpack and an MP3 player? What is the range of possible answers for this? Represent each of your solutions in two different ways. Sullivan MAT Nov 2013

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Assume we have 10 people 12345678910 BP MP3 Sullivan MAT Nov 2013

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Assume we have 10 people 12345678910 BP MP3 Sullivan MAT Nov 2013

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Assume we have 10 people 12345678910 BP MP3 Sullivan MAT Nov 2013

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Assume we have 10 people 12345678910 BP MP3 Sullivan MAT Nov 2013

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A consolidating task On a train, the probability that a passenger has a backpack is 0.65, and the probability that a passenger as an MP3 player is 0.57. How many passengers might be on the train? What is the maximum and minimum number of possibilities for people who have both a backpack and an MP3 player? Represent each of your solutions in two different ways. Sullivan MAT Nov 2013

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An enabling prompt On a train, there are 10 people. Six of the people have a backpack, and 7 of the people have an MP3 player. How many people might have both a backpack and an MP3 player? What is the smallest possible answer for this? What is the largest possible answer? Sullivan MAT Nov 2013

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An extending prompt On a train, the probability that a passenger has a backpack is 2/3, and the probability that a passenger has an MP3 player is 2/7.How many passengers might be on the train? How many passengers might have both a backpack and an MP3 player? What is the range of possible answers for this? Represent each of your solutions in two different ways. Sullivan MAT Nov 2013

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Student preferences for teaching approaches

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A container and 3 eggs weighs 170 grams. The same container and 5 eggs weighs 270 grams. What is the weight of the container?

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PRE TEST AnswerResponse% 7 g216% 50 g9829% 30 g5516% 20 g16749% Total341100

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POST TEST AnswerResponse% 7 g 83% 50 g 6324% 30 g 3312% 20 g 16361% Total 267100%

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I prefer … much harder than the egg. I prefer …about as hard as the egg. I prefer … much easier than the egg. I prefer …egg question through working by myself 47675119 I prefer … egg question through working with other students 2212527174 I prefer …egg question by listening to explanations from the teacher before I work on the question 4251847 Total 7321750340

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The “Advanced” (generic) statements Understanding – I chose, used and showed relevant ideas and connected them together. I used mathematical words correctly Fluency – My working out was complete with no errors, I used appropriate formulas if they were needed, and I presented calculations efficiently, incorporating relevant shortcuts PEP Nov 25 symposium

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Problem solving – I explained clearly how I planned and solved the problem, my method was creative and I checked that my solution(s) had no errors. Reasoning – The steps I took are shown, and I used examples to explain and justify my thinking. PEP Nov 25 symposium

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You need a MYKI card before you can travel on public transport in Melbourne. It costs $4 to buy a MYKI card and you need to put extra cash on the card to travel. If each journey costs $2.50, what is the total cost of 6 journeys?

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AnswerResponse% $1551157% $6.50647% $1928932% $10283% Total892100% PRE TEST

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AnswerResponse% $1522847% $6.50265% $1920943% $10194% Total482100% POST TEST

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I prefer …much harder than MYKI I prefer … about as hard as MYKI I prefer … be much easier than MYKI I prefer …questions like the MYKI question working by myself 1958411290 I prefer …MYKI question working with other students 388114133 I prefer …the MKYI question by listening to explanations from the teacher before I work on the question 9261247 Total 24219137470

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Missing number multiplicaiton Patterns with remainders Jigsaw SA = 22 2/3 and 201/301 Sullivan MAT Nov 2013

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Curriculum documentation should presumably inform … School planning The planning of the program for the year in mathematics by level The planning of units of work (lesson sequences) The planning of teaching of lessons and assessment of student learning PEP Nov 25 symposium

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The Curriculum includes content descriptions, such as … Represent events in two-way tables and Venn diagrams and solve related problems (ACMSP292) (ACMSP292) PEP Nov 25 symposium

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An example of a task that can be made into a lesson On a train, the probability that a passenger has a backpack is 0.6, and the probability that a passenger as an MP3 player is 0.7. How many passengers might be on the train? How many passengers might have both a backpack and an MP3 player? What is the range of possible answers for this? Represent each of your solutions in two different ways. PEP Nov 25 symposium

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… and proficiencies Problem Solving includes … using two-way tables and Venn diagrams to calculate probabilities Reasoning includes justifying the result of a calculation or estimation as reasonable, deriving probability from its complement …probability PEP Nov 25 symposium

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… and achievement standards By the end of Year 8, students … model authentic situations with two-way tables and Venn diagrams. They choose appropriate language to describe events and experiments …describe Students … determine complementary events and calculate the sum of probabilities PEP Nov 25 symposium

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Some people came for a sports day. When the people were put into groups of 3 there was 1 person left over. When they were lined up in rows of 4 there were two people left over. How many people might have come to the sports day? SA Sullivan and Aulert 2013

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Some “enabling” prompts Some people came for a sports day. When they were lined up in rows of 4 there were two people left over. How many people might have come to the sports day? Some people came for a sports day. When the people were put into groups of 3 there was no- one left over. When they were lined up in rows of 4 there was no-one left over. How many people might have come to the sports day? SA Sullivan and Aulert 2013

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An extending prompt Some people came for a sports day. When the people were put into groups of 3 there was 1 person left over. When they were lined up in rows of 4 there was 1 person left over. When they were lined up in columns of 5 there was 1 person left over. How many people might have come to the sports day? SA Sullivan and Aulert 2013

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The “consolidating” task I have some counters. When I put them into groups of 5 there was 2 left over. When they were lined up in rows of 6 there was the same number in each column and none left over. How many counters might I have? SA Sullivan and Aulert 2013

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How does that lesson connect to algebra? SA Sullivan and Aulert 2013

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Multiplication content descriptions Year 4: Develop efficient mental and written strategies and use appropriate digital technologies for multiplication and for division where there is no remaindermultiplication remainder Year 5: Solve problems involving division by a one digit number, including those that result in a remaindernumber remainder Year 6: Select and apply efficient mental and written strategies and appropriate digital technologies to solve problems involving all four operations with whole numbers SA Sullivan and Aulert 2013

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Patterns Explore and describe number patterns resulting from performing multiplication (ACMNA081)number (ACMNA081) Solve word problems by using number sentences involving multiplication or division where there is no remainder (ACMNA082)numbermultiplication remainder(ACMNA082) SA Sullivan and Aulert 2013

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Connecting to the proficiencies Year 5: Understanding includes making connections between representations of numbers, … Fluency includes … using estimation to check the reasonableness of answers to calculations Problem Solving includes formulating and solving authentic problems using whole numbers … Reasoning includes investigating strategies to perform calculations efficiently, continuing patterns … SA Sullivan and Aulert 2013

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Year 6: Understanding includes describing properties of different sets of numbers, … and making reasonable estimations Fluency includes … using brackets appropriately, Problem Solving includes formulating and solving authentic problems … Reasoning includes explaining mental strategies for performing calculations, describing results for continuing number sequencesnumber SA Sullivan and Aulert 2013

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The achievement standards: Year 5: By the end of Year 5, students solve simple problems involving the four operations using a range of strategies. They check the reasonableness of answers using estimation and rounding. rounding Year 6: By the end of Year 6, students …solve problems involving all four operations with whole numbers. They write correct number sentences using brackets and order of operations.numberorder of operations SA Sullivan and Aulert 2013

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Where is the “ceiling”? The lowest possible (though not necessarily correct) solution is 6 (Largest option out of 4 plus 2 and 3 plus 1) so we know the solution must be 6 or larger. Consider integers from 6 onwards ie 6+n (n is an integer larger than or equal to zero) The lowest number of people at sports day will be the lowest value of n for which (6+n-1)/3 AND (6+n-2)/4 are both integers (6+n-1)/3 = (5+n)/3 and (5+n)/3 is an integer when n=1, 4, 7… (6+n-2)/4 = (4+n)/4 and (4+n)/4 is an integer when n=0, 4, 8… Therefore n=4 and the lowest number of people at sports day is 10. I’ve tested this approach on a couple of other examples and seems OK so could be expressed in general terms if we really wanted to… Sullivan MAT Nov 2013

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AVAILABLE TO DOWNLOAD FREE FROM http://research.acer.edu.au/aer/13/ aer Sullivan MAT Nov 2013

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