1. Planning engaging and inclusive mathematics lessons Peter Sullivan mtant 2013

2. Planning engaging and inclusive mathematics lessons This presentation will focus on structuring lessons that engage students by allowing them to build connections between ideas for themselves, which also extend student who are ready and support students who need it. Using content and proficiencies from the Australian Curriculum: Mathematics, examples from both primary and secondary level lessons will be presented, and processes for assessing the learning discussed. mtant 2013

3. Why challenge? Learning will be more robust if students connect ideas together for themselves, and determine their own strategies for solving problems, rather than following instructions they have been given. Both connecting ideas together and formulating their own strategies is more complex than other approaches and is therefore more challenging. It is potentially productive if students are willing to take up such challenges. mtant 2013

4. 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. mtant 2013

5. This week Yesterday, on the literacy and numeracy panel 47% of NT students are Indigenous 29% do not have English as their first language If you could say one thing to the chief minister (about numeracy teaching), what would it be? (do not ask for anything that will cost extra money) mtant 2013

6. This week On one hand … mtant 2013

8. On the other hand … mtant 2013

10. How might we write it … 70 = 50 + 20 mtant 2013

11. I have 50c in my hand … What might it look like … mtant 2013

12. Football scores Saints105 Bombers98 mtant 2013 How much are the Saints winning by? (Work out the answer in two different ways)

13. Elizabeth is 202 years old Debbie is 97 years old How much older is Elizabeth? mtant 2013

14. 20297 mtant 2013

15. 20297200100 3 2 mtant 2013

16. Enabling prompt mtant 2013

17. Football scores Saints27 Bombers19 mtant 2013 How much are the Saints winning by? (Work out the answer in two different ways)

18. Football scores Saints18 Bombers13 mtant 2013 How much are the Saints winning by? (Work out the answer in two different ways)

19. Basketball scores Cats8 Dogs3 mtant 2013 How much did the Cats win by? (Work out the answer in two different ways)

20. Darts scores Parrots1005 Galahs988 mtant 2013 How much did the Parrots win by? (Work out the answer in two different ways)

21. Race to 10 Start at 0 You can add either 1 or 2 Person who says 10 is the winner mtant 2013

22. Race to $1 Start at 0 You can add either 1 or 2 Person who says 10 is the winner mtant 2013

25. 12345678910 111214 151617181920 21222324252627282930 313233343537 383940 41424344454647484950 515253 55565758 60 61626364656667686970 71727374757677787980 Which ones are wrong? mtant 2013

26. What might be the numbers on the L shaped piece that has been turned over? (I know that one of the numbers is 65) mtant 2013

27. An enabling prompt What might be the missing numbers on this piece? 65 mtant 2013

28. As a consolidating task The numbers 62 and 84 are on the same jigsaw piece. Draw what might that piece look like? mtant 2013

30. What might be the missing numbers on this piece? 650 mtant 2013

31. 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. mtant 2013

32. Starting from the content descriptions mtant 2013

33. Reading the content description(s) to identify the key ideas Represent events in two-way tables and Venn diagrams and solve related problems (ACMSP292) (ACMSP292) mtant 2013

34. Reading the content description(s) to identify the key ideas Represent events in two-way tables and Venn diagrams and solve related problems (ACMSP292) (ACMSP292) mtant 2013

35. Reading the content description(s) to identify the key ideas Represent events in two-way tables and Venn diagrams and solve related problems (ACMSP292) (ACMSP292) mtant 2013

36. Reading the content description(s) to identify the key ideas Represent events in two-way tables and Venn diagrams and solve related problems (ACMSP292) (ACMSP292) mtant 2013

37. Reading the content description(s) to identify the key ideas Represent events in two-way tables and Venn diagrams and solve related problems (ACMSP292) (ACMSP292) mtant 2013

38. What would we say to the students are the learning goals/intentions? Devising for ourselves different ways of representing categorical data mtant 2013

39. Assume we have 10 people 12345678910 BP MP3 mtant 2013

40. Assume we have 10 people 12345678910 BP MP3 mtant 2013

41. Assume we have 10 people 12345678910 BP MP3 mtant 2013

42. Assume we have 10 people 12345678910 BP MP3 mtant 2013

44. Two way tables back pack No back pack MP3 player 34 No MP3 player 30 mtant 2013 back pack No back pack MP3 player 61 No MP3 player 03

45. Venn diagrams mtant 2013 Back pack MP3 player 3 3 0 4 Back pack MP3 player 0 6 3 1

46. What about the students who cannot get started? 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? mtant 2013

47. 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. mtant 2013

48. 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. mtant 2013

49. (relevant) Year 8 Proficiencies Understanding includes … Fluency includes … 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 mtant 2013

50. Year 8 Achievement Standard By the end of Year 8, students solve everyday problems …. Students model authentic situations with two-way tables and Venn diagrams. They choose appropriate language to describe events and experiments. …solvedescribe Students determine complementary events and calculate the sum of probabilities. mtant 2013

51. Now This is a plan of paths in a park. Each path goes from one node to another. The vertical paths are twice as long as the horizontal paths. The triangle at the top is equilateral. I know one of the paths is 1 km long but I do not know which one. What might be the total length of the paths? (there are three different answers) Tassie Numeracy Leadership Day 6

52. Special offer THREE PAIRS FOR THE PRICE OF TWO The free pair is the cheapest one Special offer THREE PAIRS FOR THE PRICE OF TWO The free pair is the cheapest one Jenny and Carly go shopping for shoes. Jenny chooses one pair for $110 and another for $100. Carly chooses a pair that cost $160. When they go to pay, the assistant says that there is a sale on, and they get 3 pairs of shoes for the price of 2 pairs. Give two options for how much Jenny and Carly should each pay? Explain which option is fairer. mtant 2013

53. Representing the situation mtant 2013

54. $160 $110 $100 Carly Jenny mtant 2013

55. $160 $110 $100 Carly Jenny This pair is free mtant 2013

56. The sharing option They have to pay $270 So Jenny pays $180 and Carly pays $90 mtant 2013

57. They save $100 If they share the saving equally, – Then Jenny pays $210 - $50 = $160 – Carly pays $160 - $50 = $110 The Saving Option mtant 2013

58. Jenny and Carly go shopping for shoes. Jenny chooses one pair for $110 and another for $100. Carly chooses a pair that cost $60. When they go to pay, the assistant says that there is a sale on, and they get 3 pairs of shoes for the price of 2 pairs (the cheapest pair becomes free). Give two options for how much Jenny and Carly should each pay? Explain which is the fairer. Explain in what ways the fairer solution depends on the cost of Carly’s shoes. A Consolidating Task mtant 2013

59. Kerry and Kathy are twins and can share shoes. Kerry chooses one pair for $20. Kathy chooses a pair that costs $40. How much should they each pay? Enabling Prompt: mtant 2013

60. What are enabling prompts? Enabling prompts can involve slightly varying an aspect of the task demand, such as – the form of representation, – the size of the numbers, or – the number of steps, so that a student experiencing difficulty, if successful, can proceed with the original task. This approach can be contrasted with the more common requirement that such students – listen to additional explanations; or – pursue goals substantially different from the rest of the class. mtant 2013

61. An extending task Today only FIVE SHIRTS FOR THE PRICE OF THREE The free ties are the cheaper ones Bert, Bob and Bill are shopping for shirts. Bill chooses a shirt costing $30 and another for $50. Bob chooses one shirt for $60. Bert chooses one shirt for $30 and another for $40. When they go to pay, the assistant says that there is a sale on, and they get 5 shirts for the price of 3. Give two options for how much Bill and Bert and Bob should each pay? Explain which is the fairest. mtant 2013

62. What would be the point of asking a question like that? mtant 2013

63. Year 5 Money and financial mathematics – Create simple financial plans (ACMNA106)(ACMNA106) Number and place value – Use estimation and rounding to check the reasonableness of answers to calculations (ACMNA099) (ACMNA099) – Use efficient mental and written strategies and apply appropriate digital technologies to solve problems (ACMNA291)(ACMNA291) mtant 2013

64. Year 6 Money and financial mathematics – Investigate and calculate percentage discounts of 10%, 25% and 50% on sale items, with and without digital technologies (ACMNA132)(ACMNA132) Number and place value – Select and apply efficient mental and written strategies and appropriate digital technologies to solve problems involving all four operations with whole numbers (ACMNA123)(ACMNA123) mtant 2013

65. Proficiencies At year 5: Understanding includes making connections between representations … Fluency includes … using estimation to check the reasonableness of answers to calculations Problem Solving includes formulating and solving authentic problems using whole numbers and creating financial plans Reasoning includes investigating strategies to perform calculations … At year 6: Understanding includes … making reasonable estimations Fluency includes … calculating simple percentages Problem Solving includes formulating and solving authentic problems Reasoning includes explaining mental strategies for performing calculations, mtant 2013

66. Achievement Standards 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. … They explain plans for simple budgets.solve reasonablenessexplain By the end of Year 6, students … solve problems involving all four operations with whole numbers.solve mtant 2013

67. A probability task mtant 2013

68. 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. mtant 2013

69. Starting from the content descriptions mtant 2013

70. Reading the content description(s) to identify the key ideas Represent events in two-way tables and Venn diagrams and solve related problems (ACMSP292) (ACMSP292) mtant 2013

71. Reading the content description(s) to identify the key ideas Represent events in two-way tables and Venn diagrams and solve related problems (ACMSP292) (ACMSP292) mtant 2013

72. Reading the content description(s) to identify the key ideas Represent events in two-way tables and Venn diagrams and solve related problems (ACMSP292) (ACMSP292) mtant 2013

73. Reading the content description(s) to identify the key ideas Represent events in two-way tables and Venn diagrams and solve related problems (ACMSP292) (ACMSP292) mtant 2013

74. Reading the content description(s) to identify the key ideas Represent events in two-way tables and Venn diagrams and solve related problems (ACMSP292) (ACMSP292) mtant 2013

75. What would we say to the students are the learning goals/intentions? Devising for ourselves different ways of representing categorical data mtant 2013

76. Assume we have 10 people 12345678910 BP MP3 mtant 2013

77. Assume we have 10 people 12345678910 BP MP3 mtant 2013

78. Assume we have 10 people 12345678910 BP MP3 mtant 2013

79. Assume we have 10 people 12345678910 BP MP3 mtant 2013

81. Two way tables back pack No back pack MP3 player 34 No MP3 player 30 mtant 2013 back pack No back pack MP3 player 61 No MP3 player 03

82. Venn diagrams mtant 2013 Back pack MP3 player 3 3 0 4 Back pack MP3 player 0 6 3 1

83. What about the students who cannot get started? 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? mtant 2013

84. 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. mtant 2013

85. 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. mtant 2013

86. (relevant) Year 8 Proficiencies Understanding includes … Fluency includes … 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 mtant 2013

87. Year 8 Achievement Standard By the end of Year 8, students solve everyday problems …. Students model authentic situations with two-way tables and Venn diagrams. They choose appropriate language to describe events and experiments. …solvedescribe Students determine complementary events and calculate the sum of probabilities. mtant 2013

88. Missing number multiplication mtant 2013

89. Our goal Sometimes solving multiplication and division problems is about finding patterns. In this case look for numbers that when multiplied have an answer that ends in 0. mtant 2013

90. Missing digit This number has a digit missing __ 4 What might be the number? mtant 2013

91. The question I did a multiplication question correctly for homework, but my printer ran out of ink. I remember it looked like 2 _ x 3 _ = _ _ 0 What might be the digits that did not print? (Give as many answers as you can) mtant 2013

92. If you are stuck What might be the missing digits __ × __ = __ 0 mtant 2013

93. If you are finished What might be the missing digits? _ x _ 0 x 3 _ = _ _ 0 mtant 2013

94. Now do this I did a multiplication question correctly for homework, but my printer ran out of ink. I remember it looked like 1 __ × 4 __ = __ __ 2 What might be the digits that that did not print? (give as many answers as you can) mtant 2013

95. We said our goal was … Sometimes solving multiplication and division problems is about finding patterns. In this case look for numbers that when multiplied have an answer that ends in 0. mtant 2013

96. 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? mtant 2013

97. For the students: If you are given the surface area of a rectangular prism, you are able to work out what the prism might look like. mtant 2013

98. Introductory task: What is the surface area and volume of a cube that is 2 cm × 2 cm × 2 cm? mtant 2013

99. Enabling prompt: Arrange a small number of cubes into a rectangular prism, then calculate the volume and surface area. mtant 2013

100. Extending prompt: The surface area of a closed rectangular prism is 94 cm 2. What might be the dimensions of the prism? mtant 2013

101. The surface area of a closed rectangular prism is 46 cm 2. What might be the dimensions of the prism? mtant 2013

102. 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. We interpret this to refer to ways that teachers can support students in engaging with the challenge of the task, and in maintaining this challenge as distinct from minimising it. mtant 2013

103. 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; mtant 2013

104. 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. mtant 2013

105. Missing number multiplication mtant 2013

106. Our goal Sometimes solving multiplication and division problems is about finding patterns. In this case look for numbers that when multiplied have an answer that ends in 0. mtant 2013

107. Missing digit This number has a digit missing __ 4 What might be the number? mtant 2013

108. The question I did a multiplication question correctly for homework, but my printer ran out of ink. I remember it looked like 2 _ x 3 _ = _ _ 0 What might be the digits that did not print? (Give as many answers as you can) mtant 2013