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Using Open Ended Tasks in the Mathematics Classroom Glenroy West Primary School 19 th May, 2010 Northern Metropolitan Region, DEECD Adrian Berenger

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Part 1: Types of Questions Closed Questions are those which require an answer or response to be given from memory. Open Questions require students to think more deeply and to give a response which involves more than recalling a fact or reproducing a skill. Usually more than one possible answer.

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Mother’s Day Bill $711 was the total bill Mum & Dad, Brother and his Wife and 2 children, Sister and her Boyfriend, and me Who pays what?

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Buying Shoes You go with a friend to buy shoes. Each pair cost $80 The sale offers $100 off if you buy 3 pairs. Your friend gets 2 pairs and you get only one. Who pays what?

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Making Shapes Minimum Properties of Shapes Construct a quadrilateral that has two pairs of adjacent sides equal. What shapes are possible?

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Closed to Open 12 cm 3 cm Calculate the Perimeter and Area of the rectangle. I want to make a garden in the shape of a rectangle. I have 30 metres of fence for my garden. What might be the area of the garden?

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Garden paving Peter wants to pave an area using 16 square paving stones and 16 straight border pieces. Each stone is 1m 2 and each border piece is 1m. What different arrangements are possible? On the way home, Peter carelessly breaks several of the paving stones. What is the maximum number of broken stones possible in order to still be able to use all his 16 border pieces? Draw all possible arrangements.

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‘Our goals in education are for our students to think, to learn, to analyse, to criticise and to be able to solve unfamiliar problems.’ Peter Sullivan and Doug Clarke

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Bloom’s Taxonomy Can the student GENERATE new products, ideas or ways of viewing things? Can the student JUSTIFY a decision or course of action? Can the student DIFFERENTIATE between constituent parts? Can the student USE the new knowledge in another familiar situation? Can the student EXPLAIN ideas or concepts? Can a student RECALL information?

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Why do we use Open Tasks? To draw out misconceptions To encourage deeper thinking To develop problem solving To cater for mixed ability levels

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Creating Open-Ended Tasks ‘Any closed questions can be reformulated to create an open ended questions using one of two methods’. (Sullivan & Lilburn 1997)

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There are 11 more students enrolled at the school this term. What’s possible? How many whole number pairs sum to 11? Misconceptions with = 8 + 3 = ? + 4 7 because 8 and 3 equals 11 and so does 4 and 7 7 because 4 is one more than 3 so ? must be one less than 8 11

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Totals 11 + 0 10 + 1 9 + 2 8 + 3 7 + 4 6 + 5 10 + 0 9 + 1 8 + 2 7 + 3 6 + 4 5 + 5 9 + 0 8 + 1 7 + 2 6 + 3 5 + 4 8 + 0 7 + 1 6 + 2 5 + 3 4 + 4 7 + 0 6 + 1 5 + 2 4 + 3 6 + 0 5 + 1 4 + 2 3 + 3 11 678910

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A maths teacher always likes to have his class working together in groups but no matter if he suggests his students work in groups of 2, 3 or 4 there is always one student on their own. How many students might there be in this teacher’s class? Other Questions with Numbers

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What scores are possible from 4 darts? 10731

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Method 1 TRADITIONAL 249 + 173 Omit enough information so that, although the answer remains the same, the digits required to achieve the answer becomes variable. OPEN-ENDED 2** + *7* 422

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Method 1 (continued) TRADITIONAL Two fifths of 250 students borrow books from the library each day. Calculate the number of students who borrow books each day. Find the missing angle on this trapezoid 40◦ 140◦ OPEN-ENDED Two fifths of the students in a school borrow books from the library each day. How many students might there be in the school and how many of them borrow books each day? What might the angles on this trapezoid be?

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Method 2 Work backwards from the answer. Begin with a closed task. Calculate the answer, then work backwards and using the context of the question, create a question that would allow multiple responses to achieve the same answer. TRADITIONAL The following numbers represents the temperature of 5 consecutive days in Melbourne: 44 ◦ C, 42 ◦ C, 36 ◦ C, 22 ◦ C, 29 ◦ C. Find the average temperature. OPEN-ENDED The average temperature over five consecutive days in Melbourne was 35 ◦ C. The highest temperature was 44 ◦ C. What might the temperature have been on the other days?

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Method 2 (continued) TRADITIONAL What is the volume of the cylinder? 35.0 X 0.5 4cm 6 cm OPEN-ENDED The answer is 17.5 What might the question be? What might the dimensions of a cylinder prism that has a volume of 300 cm 3 ?

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Summing Up! Open Ended Maths Activities 2 nd Edition Peter Sullivan and Pat Lilburn Oxford Resources Thinking Tools for the Mathematics Classroom. Sue Gunningham Hawker Brownlow Education

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Part 2: Creating Open-Ended Questions 8 closed questions to open questions in pairs Hot dots

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Video: 7X13=28

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