Task

Kāhui Ako – Maths Series

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Kayla de Lange and Sam Kordan

(KaylaD@chbc.school.nz, SamK@chbc.school.nz)

Series 1 – Supporting ākonga who find maths difficult

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Episode 1 – Using the bar model to support ākonga understanding of fractions

Supporting ākonga who find maths difficult

1) Teach concepts explicitly but provide plenty of opportunity for interaction and peer-to-peer discussion

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2) Teach in small steps to avoid cognitive overload

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3) Use materials and images to help ākonga visualise the abstract nature of mathematics

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4) Develop an understanding of mathematical vocabulary

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5) Provide small group intervention to compliment high quality teaching

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Session Outcomes

• To understand the difficulties ākonga have with fractions

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• To learn how the bar model can be used to support the teaching of fractions – both calculation and problem solving - to ākonga of all abilities

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This session is aimed at Levels 3-4 of the NZ National Curriculum (but the principles can be adapted for use with Levels 1 and 2).

Why do ākonga struggle �with fractions?

They don’t fully understand what they are

Some of the rules don’t seem intuitive

They are hard to visualise, unlike whole numbers

They have a complex language

What is a fraction?

Part of a set

Part of a whole

A quantity

Pirie and Kieren’s Recursive Theory

of Mathematical Understanding (1994)

What are bar models?

• A pictorial representation which helps students ‘see’ what’s happening.

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• Not a method that performs the calculation itself, although can support the understanding behind the calculation.

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• A problem solving strategy which uses bars to represent quantities.

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A Consistent Picture

Supporting calculation

Draw bar models to show the following:

• ¼ + ¾
• ¾ - ¼
• ¼ x 7
• 2 ÷ ½
• ¾ of 20
• Write 3 ½ as an improper fraction

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Which other representations can we use to visualise fractions?

What are the limitations of using a bar model with fractions?

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Leading to a generalisation

• Bar models can help ākonga to see what’s happening, leading to generalisations and in some cases, algorithms.

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• One recommendation is to use the abstract alongside the visual.

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• The end result is that they should calculate without these, however, some will need to use these for longer than others.

Model

Calculations

300g

60

300 ÷ 5 =

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180

3 x 60 = 180

60

60

60

60

60

Bar Model with Fractions

Supporting �problem solving

Draw bar models to help solve these additional challenges…

3/5 of a cost of a car is $10,000. What is the total cost of the car?

Extending to percentages and ratios

Key Points

• Make it clear that a whole bar represents the whole amount

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• Use squared paper initially so that the bar is easier to split into equal parts. Mini-whiteboards are also a great tool.

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• Show the abstract calculations alongside the model where relevant

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• When solving problems, label the parts of the bar which are of interest

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• Tell ākonga that bar models help learners of all ages, including those at CHBC, and also adults.

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Jake

Pete

Lucy

42 – 2 = 40

40 ÷ 10 = 4

Each bar = 4 years

Jake is 3 × 4 = 12 years

Lucy is 16 years

Pete is 12 + 2 = 14 years

+2

Next Steps

Short term (by the end of Term 2)

• Trial using the bar model with one of your classes at the earliest opportunity
• Collect (student feedback), reflect (consider what you might adapt) and respond (make adjustments)
• With colleagues, discuss the effectiveness of using a bar model

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Longer term (by the end of Term 4)

• Highlight parts of your scheme of learning which can be supported using the bar model – the more often it is used, the more confident ākonga will be at using them