Number sense to algebra

Julia Crawford

crawfordj@cognitioneducation.com

cognitioneducation.com

cognitioneducation.com

Take a quick look … what do you see?

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What did you see?

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Shared by Jo Boaler, Waikato 2019

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24 x 5

What ways can you show the answer to 24 x 5?

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• Dot paper
• Squared paper
• Equipment
• Coloured pencils/highlighters
• Pen and paper

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Arrays to Areas

Phases of experiences:

• Materials
• Pictures of dots in arrays
• Pictures of squares (to replace dots)
• Regions (area model)
• Whole number
• Decimals
• Generalise: algebra

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Areas to algebra

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Multiplying decimals

1

1

1 x 1 = 1

= 100 hundredths

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Multiplying decimals

0.4 x 0.6

0.4 x 0.6 = 24 hundredths = 0.24

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What do you notice? What do you wonder?

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Does it always work?

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Guiding strategy development

1. Use simple story problems designed so that students design a strategy to solve it.

• Share and discuss multiple strategies
• Write new strategies on the board… make poster… name them.

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• Don’t expect to have a strategy introduced and understood with just one word problem or one exposure.
• Children need lots of opportunity to make a strategy their own.

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Guiding strategy development

2. Integrate computation with place value and fact development

• Strong understanding of number
• Hundreds charts, tens frames, patterns, base-ten blocks give visual references to the underlying structure of our number system

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Guiding strategy development

3. Three part lesson format

Using a worthwhile task:

• Before Phase: activate prior knowledge and/or engagement
• During Phase: let go! listen actively, cautiously provide hints
• After phase: productive discussion, listen activity without evaluation, summarise

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Guiding strategy development

4. Record students’ processes

• Support students’ thinking by recording each step
• Use arrows, lines, statements
• Show the ‘split’ in the numbers
• Model an empty number line

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• Numeracy Project was not about teaching a sequence of increasingly sophisticated strategies…
• The framework is a developmental structure to support teachers understand what they see as students learn to calculate

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The goal

To be able to use strategies efficiently and flexibly…

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… Not to know lots of strategies!

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Using ‘calculation tricks’

• Does it work?
• What about with bigger numbers?
• Why does it work?

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Does it always work?

62²

62 – 50 = 12

100 ( 25 + 12 ) = 3700

3700 + 12²

Answer 63² = 3844

Why does it work?

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Role of the calculator?

• Supports the exploration of ‘why it works’
• Check the solution
• Consider that is the goal of the lesson is

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Thoughts and queries?

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Where to find resources

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