Zone 8: Reflective Knowing

The information below describes the key characteristics and teaching implications for Zone 8 of the Learning and Assessment Framework for Multiplicative Thinking (LAF), including:

• How to use this resource
• Key characteristics
• Teaching implications – consolidation and establishment
• Teaching implications – introduction and development
• Learning plans
• More information

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:

• an assessment task, the title is italicised, and can be found in the Assessment Materials
• an Authentic Task or task from a Learning Plan, the title is linked so you can download the relevant document

Key characteristics

Can use appropriate representations, language and symbols to solve and justify a wide range of problems involving unfamiliar multiplicative situations including fractions and decimals. For example:

• Adventure Camp part b
• Speedy Snail part b.

Can justify partitioning. For example, Missing Numbers part b.

Can use and formally describe patterns in terms of general rules. For example, Tables and Chairs, parts m and k.

Beginning to work more systematically with complex, open-ended problems. For example:

• School Fair part b
• Computer Game part c.

 

Teaching implications – consolidation and establishment

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

Teaching implications – introduction and development

Expand the range of multiplicative situations

Introduce a broader range of multiplicative situations. For example, problems involving:

• the calculation of area or volume
• derived measures and rates
• variation
• complex proportion.

Multiple step problems involving large whole numbers, decimals and fractions should also be introduced. For example:

Find the volume of a cylinder 4 cm in diameter and 9 cm long.

Find the surface area of a compound shape.

Foreign currency calculations.

Determine the amount of water lost to evaporation from the Hume Weir during the summer.

Simplifying expressions

Strategies for simplifying expressions include:

• adding and subtracting like terms
• justifying and explaining the use of cancellation techniques for division through the use of common factors.

For example:

^42a / ₇ = 6a because ^42a / ₇ = ^{7 × 6a} / ₇ and ⁷ / ₇ = 1

Algebraic reasoning and representation strategies

To solve problems involving multiplicative relationships, students will require algebraic reasoning and representation strategies. For example:

If 2 T-shirts and 2 drinks cost $44 and 1 T-shirt and 3 drinks cost $30, what is the price of each?

5 locker keys are returned at random to the students who own them. What is the probability that each student will receive the key that opens their locker?

A scientist has a collection of beetles and spiders. The sensor in the floor of the enclosure indicated that there were 174 legs and the infra-red image indicated that there were 26 bodies altogether. How many were beetles and how many were spiders?

365 is an extraordinary number. It is the sum of 3 consecutive square numbers and also the sum of the next 2 consecutive square numbers. Find the numbers referred to.

Numbers and operations expressed in exponent form

Introduce strategies for working with numbers and operations expressed in exponent form. For example:

Why 2³ × 2⁶ = 2⁹.

More abstract problem solving situations

These situations require an appreciation of problem solving as a process and an awareness of the value of recognising problem type.

Student will be required to develop a greater range of strategies and representations, including the manipulation of symbols. For example:

• tables
• symbolic expressions
• rule generation
• testing.

Learning plans

The following learning plan has been developed for this zone, to support targeted teaching after students have been assessed and located on the LAF. <link to Assessment Materials>

Zone 8 – Learning Plans Introducing (PDF - 101Kb)

More information

Using the resources on this site, you can also:

• compare across the eight zones of the LAF
• find out more about multiplicative thinking
• read about how the LAF was developed through a research project
• use the assessment materials with your class.