Scaffolding Numeracy in the Middle Years

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Zone 7: Connecting

The information below describes the key characteristics and teaching implications for Zone 7 of the Learning and Assessment Framework for Multiplicative Thinking (LAF), including:

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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

Able to solve and explain one-step problems involving multiplication and division with whole numbers using informal strategies and/or formal recording. For example:

  • Filling the Buses part a
  • Fencing the Freeway part d
  • Packing Pots part d.

Can solve and explain solutions to problems involving simple patterns, percent and proportion. For example:

  • Fencing the Freeway part c
  • Swimming Sports part b
  • Butterfly House part g
  • Tables & Chairs parts g and l
  • Speedy Snail part c
  • Tiles, Tiles, Tiles parts b and c
  • School Fair part a
  • Stained Glass Windows part a
  • Computer Game part b
  • How Far part b.

May not be able to show working and/or explain strategies for situations involving larger numbers. For example:

  • Tables & Chairs parts m and k
  • Tiles, Tiles, Tiles part c.

May not be able to show working and/or explain strategies for less familiar problems. For example:

  • Adventure Camp part b
  • School Fair part b
  • How Far part c.

Locates fractions using efficient partitioning strategies. For example, Missing Numbers part a.

Beginning to make connections between problems and solution strategies and understand how to communicate this mathematically.

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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

Comparing, ordering, sequencing, counting forwards and backwards in place-value parts

Introduce strategies for:

  • comparing, ordering, sequencing, counting forwards and backwards in place-value parts
  • renaming large whole numbers, common fractions, decimals and integers.

For example:

a 3 to 4 metre length of rope, appropriately labelled number cards and pegs could be used to sequence numbers from 100 to 1,000,000, from -3 to +3, from 2 to 5 and so on

The metaphor of a magnifying glass can be used to locate numbers involving hundredths or thousandths on a number line as a result of successive tenthing.

For more information, see Siemon (2004) Partitioning – The Missing Link in Building Fraction Knowledge and Confidence (PDF - 200Kb).

Inverse and identity relations

Build an appreciation of inverse and identity relations through strategies such as recognising which number when added leaves the original number unchanged (zero) and how inverses are determined in relation to this. For example:

the inverse of 8 is -8 as -8 + 8 = 0 and 8 + -8 = 0

In a similar fashion, recognise that 1 is the corresponding number for multiplication, where the inverse of a number is defined as its reciprocal. For example:

the inverse of 8 is 1/8

Index notation

Introduce strategies for representing multiplication of repeated factors. For example:

5 × 5 × 5 × 5 × 5 × 5 = 56

Place-value

A more generalised understanding of place-value and the structure of the number system in terms of exponentiation should be developed. For example:

10-3, 10-2, 10-1, 100, 101, 102,103 …

Multiplication and division in a broader range of situations

Strategies to recognise and apply multiplication and division in a broader range of situations can include:

  • ratio
  • proportion
  • unfamiliar, multiple-step problems.

See the Orange Juice (PDF - 25Kb) task for an example.

Formal recognition and description of number patterns

Develop an awareness of how to recognise and describe number patterns more formally. For example:

  • triangular numbers
  • square numbers
  • growth patterns.

See the ‘Garden Beds’ task from Maths 300 and Super Market Packer (PDF - 30Kb) task for examples.

Notation to support general arithmetic

Notation for simple algebra or general arithmetic can be used, for example, to recognise and understand the meaning of expressions such as:

x+4, 3x, 5x2 , or x - 1/3

Ratio as the comparison of any two quantities

For example:

the comparison of the number of feral cats to the size of the national park

Recognise that ratios can be used to compare measures of the same type. For example:

the number of feral cats compared to the number of feral dogs

Recognise that within this, two types of comparison are possible. For instance, one can compare:

  • the parts to the parts (for example, cats to dogs), or
  • the parts to the whole (for example, cats to the total number of cats and dogs).

Ratios can be also used to compare measures of different types. In other words, they are generally described as a rate. For example:

the number of feral cats per square kilometre

Ratios are not always rational numbers. For example:

the ratio of the circumference of a circle to its diameter

Proportion problems involving larger numbers and/or fractions

Introduce strategies for recognising and representing proportion problems involving larger numbers and/or fractions. For example:

  • problems involving scale such as map calculations, increasing/reducing ingredients in a recipe
  • simple problems involving derived measures such as volume, density, speed and chance.

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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. See assessment materials.

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More information

Using the resources on this site, you can also: