Scaffolding Numeracy in the Middle Years

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Zone 3: Sensing

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How to use this resource

There are eight zones in the LAF. If your students are across several zones, you should access information for each of the zones where they are located.

Where there is a reference to:

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Key characteristics

Demonstrates intuitive sense of proportion and partitioning. For example:

  • Butterfly House part F (partial solution)
  • Missing Numbers part B.

Works with ‘useful’ numbers such as 2 and 5, and strategies such as doubling and halving. For example:

  • Packing Pots part B
  • Pizza Party part C.

May list all options in a simple Cartesian product situation but cannot explain or justify solutions. For example, Canteen Capers part B.

Uses ab breviated methods for counting groups. Uses doubling and doubling again to find 4 groups of, or repeated halving to compare simple fractions. For example, Pizza Party part C.

Beginning to work with larger whole numbers and patterns but tends to rely on count all methods or additive thinking to solve problems. For example:

  • Stained Glass Windows parts A and B
  • Tiles, Tiles, Tiles part C.

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Teaching implications – consolidation and establishment

Consolidate or establish the ideas and strategies introduced or developed in the previous zone.

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Teaching implications – introduction and development

Introduce and develop the following strategies.

Place-value based strategies

Problems involving single-digit by two-digit multiplication are informally solved, either mentally or in writing. For example:

3 twenty-eights, think: 3 by 2 tens, 60 and 24 more, 84

Initial recording to support place-value for multiplication facts

For more information, see:

More efficient strategies for solving number problems involving simple proportion

For example, recognising that problems involve several steps:

  • What do I do first? Find value for common amount
  • What do I do next? Determine multiplier/factor and apply
  • Why?

Rename number of groups

Demonstrate how to rename number of groups. For example:

  • 6 fours, think: 5 fours and 1 more four

Practice by using tasks such as Multiplication Toss (PDF - 49Kb).

Re-name composite numbers in terms of equal groups. For example:

  • 18 is 2 nines, 9 twos, 3 sixes, 6 threes

Cartesian product

For each idea use concrete materials and relatively simple problems such as:

  • 3 tops and 2 bottoms: how many outfits?
  • how many different types of pizzas, given choice of small, large, medium and 4 varieties.

Discuss how to:

  • recognise problems of this type
  • keep track of the count such as draw all options, make a list or a table.

Tree diagrams appear to be too difficult at this zone and these are included in Zone 5.

How to interpret problem situations and solutions relevant to context

Students should ask, for example:

  • What operation is needed?
  • Why?
  • What does it mean in terms of original question?

Interpretation of remainders

Introduce simple, practical division problems that require the interpretation of remainders relevant to context.

Practical sharing situations

Names for simple fractional parts beyond the halving family (for example, thirds for 3 equal parts/shares, sixths for 6 equal parts etc) are introduced through practical sharing situations.

These situations will also help build a sense of fractional parts, for example:

  • 3 sixths is the same as a half and 50%
  • 7 eighths is nearly 1
  • “2 and 1 tenth” is close to 2.

Use a range of continuous and discrete fraction models including mixed fraction models.

Thirding and fifthing partitioning strategies

Apply thinking involved to help children create their own fraction diagrams (regions) and number line representations through:

  • paper folding (kinder squares and streamers)
  • cutting plasticine ‘cakes’ and ‘pizzas’
  • sharing collections equally (counters, cards etc).

For more information, please see Partitioning – The Missing Link in Building Fraction Knowledge and Confidence (PDF - 200Kb).

Focus on making and naming parts in the thirding and fifthing families (for example, 5 parts, fifths). This includes:

  • mixed fractions (for example, “2 and 5 ninths”)
  • informal recording (for example, 4 fifths), no symbols.

Revisit key fraction generalisations from Zone 2. Include:

  • whole to part models (for example, partition to show 3 quarters)
  • part to whole (for example, if this is 1 third, show me the whole).

Use diagrams and representations to rename related fractions.

Extend partitioning strategies

Construct number line representations by extending partitioning strategies. Use multiple fraction representations.

Key fraction generalisations

Use strategies that support the key fraction generalisation that greater the number of parts, the smaller they are, and conversely, the fewer the parts the larger they are.

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Learning plans

The following learning plans have been developed for this zone, to support targeted teaching after students have been assessed and located on the LAF.

Zone 3 – Learning Plans Introducing (PDF - 114Kb)

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More information

Using the resources on this site, you can also: