Planning engaging and inclusive mathematics lessons

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Planning engaging and inclusive mathematics lessons

Peter Sullivan

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.

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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.

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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.

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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)

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This week

• On one hand …

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On the other hand …

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How might we write it …

• 70 = 50 + 20

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I have 50c in my hand …

• What might it look like …

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Football scores

Saints 105Bombers 98

How much are the Saints winning by?

(Work out the answer in two different ways)

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• Elizabeth is 202 years old• Debbie is 97 years old• How much older is Elizabeth?

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20297

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20297 200100

3 1002

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Enabling prompt

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Football scores

Saints 27Bombers 19



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Football scores

Saints 18Bombers 13



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Basketball scores

Cats 8Dogs 3

How much did the Cats win by?


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Darts scores

Parrots 1005Galahs 988

How much did the Parrots win by?


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Race to 10

• Start at 0• You can add either 1 or 2• Person who says 10 is the winner

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Race to $1

• Start at 0• You can add either 1 or 2• Person who says 10 is the winner

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1 2 3 4 5 6 7 8 9 10

11 12 14 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 37 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 53 55 56 57 58 58 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

Which ones are wrong?

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What might be the numbers on

the L shaped piece that has been turned

over? (I know that one of the numbers is 65)

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An enabling prompt

• What might be the missing numbers on this piece?

65

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As a consolidating task

• The numbers 62 and 84 are on the same jigsaw piece.

• Draw what might that piece look like?

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• What might be the missing numbers on this piece?

650

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

Starting from the content descriptions

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Reading the content description(s) to identify the key ideas

• Represent events in two-way tables and Venn diagrams and solve related problems (ACMSP292)

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http://www.australiancurriculum.edu.au/Curriculum/ContentDescription/ACMSP292


What would we say to the students are the learning goals/intentions?

• Devising for ourselves different ways of representing categorical data

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Assume we have 10 people1 2 3 4 5 6 7 8 9 10

BP BP BP BP BP BP

MP3 MP3 MP3 MP3 MP3 MP3 MP3

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Two way tablesback pack

No back pack

MP3 player 3 4

No MP3 player 3 0

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back pack

No back pack

MP3 player 6 1

No MP3 player 0 3

Venn diagrams

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Back pack MP3 player

3 3 04


0 6 31

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?

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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.

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A consolidating task


• 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?


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(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, …

http://www.australiancurriculum.edu.au/Glossary?a=M&t=Probability

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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. …

• Students determine complementary events and calculate the sum of probabilities.

http://www.australiancurriculum.edu.au/Glossary?a=&t=Solve

http://www.australiancurriculum.edu.au/Glossary?a=&t=Describe

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

Special offer

THREE PAIRS FOR THE PRICE

OF TWO

The free pair is the cheapest one

Special offerTHREE PAIRS FOR THE

PRICE OF TWOThe 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.

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Representing the situation

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$160

$110

$100

Carly

Jenny

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$160

$110

$100

CarlyJenny

This pair is free

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The sharing option

They have to pay $270• So Jenny pays $180 and Carly pays $90

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They save $100• If they share the saving equally,– Then Jenny pays $210 - $50 = $160– Carly pays $160 - $50 = $110

The Saving Option

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• 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

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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:

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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

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An extending task

Today onlyFIVE SHIRTS FOR THE PRICE OF THREEThe 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.

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What would be the point of asking a question like that?

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Year 5

Money and financial mathematics– Create simple financial plans (ACMNA106)

Number and place value– Use estimation and rounding to check the

reasonableness of answers to calculations (ACMNA099)

– Use efficient mental and written strategies and apply appropriate digital technologies to solve problems (ACMNA291)

http://www.australiancurriculum.edu.au/Curriculum/ContentDescription/ACMNA106



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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)

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)



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ProficienciesAt 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,

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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.

• By the end of Year 6, students … solve problems involving all four operations with whole numbers.


http://www.australiancurriculum.edu.au/Glossary?a=&t=Reasonableness

http://www.australiancurriculum.edu.au/Glossary?a=&t=Explain


A probability task

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First do this task


• 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

Starting from the content descriptions

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Reading the content description(s) to identify the key ideas

• Represent events in two-way tables and Venn diagrams and solve related problems (ACMSP292)

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What would we say to the students are the learning goals/intentions?

• Devising for ourselves different ways of representing categorical data

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Assume we have 10 people1 2 3 4 5 6 7 8 9 10

BP BP BP BP BP BP

MP3 MP3 MP3 MP3 MP3 MP3 MP3

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Two way tablesback pack

No back pack

MP3 player 3 4

No MP3 player 3 0

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back pack

No back pack

MP3 player 6 1

No MP3 player 0 3

Venn diagrams

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3 3 04


0 6 31

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?

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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?


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A consolidating task


• 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?


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(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, …

http://www.australiancurriculum.edu.au/Glossary?a=M&t=Probability

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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. …

• Students determine complementary events and calculate the sum of probabilities.


http://www.australiancurriculum.edu.au/Glossary?a=&t=Describe

Missing number multiplication

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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.

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Missing digit

This number has a digit missing__ 4

What might be the number?

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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)

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If you are stuck

• What might be the missing digits

__ × __ = __ 0

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If you are finished

• What might be the missing digits?

_ x _ 0 x 3 _ = _ _ 0

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Now do this


1 __ × 4 __ = __ __ 2• What might be the digits that that did not

print?• (give as many answers as you can)

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We said our goal was …



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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?

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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.

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Introductory task:

• What is the surface area and volume of a cube that is 2 cm × 2 cm × 2 cm?

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Enabling prompt:

• Arrange a small number of cubes into a rectangular prism, then calculate the volume and surface area.

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Extending prompt:

• The surface area of a closed rectangular prism is 94 cm2.

• What might be the dimensions of the prism?

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• The surface area of a closed rectangular prism is 46 cm2.

• What might be the dimensions of the prism?

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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. 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.

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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;

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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.

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Missing number multiplication

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Our goal



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Missing digit

This number has a digit missing__ 4

What might be the number?

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The question


2 _ x 3 _ = _ _ 0• What might be the digits that did not print?

(Give as many answers as you can)

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Planning engaging and inclusive mathematics lessons

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