# LAF Zone 6: Startegy Extending

The information below describes the key characteristics and teaching implications for Zone 6 of the Learning and Assessment Framework for Multiplicative Thinking (LAF), including:

## 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 work with Cartesian Product idea to systematically list or determine the number of options. For example:

*Canteen Capers*part b*Butterfly House*parts i and h.

Can solve a broader range of multiplication and division problems involving two digit numbers, patterns and/or proportion. For example:

*Tables & Chairs*part h*Butterfly House*part f*Stained Glass Windows*parts b and c*Computer Game*parts a and b.

However, may not be able to explain or justify solution strategy. For example:

*Fencing the Freeway*parts b and d*Swimming Sports*part b*How Far*part b*Speedy Snail*part b.

Able to rename and compare fractions in the halving family (for example, *Pizza Party* part c) and use partitioning strategies to locate simple fractions (for example, *Missing Numbers* part a).

Developing sense of proportion (for example, sees relevance of proportion in *Adventure Camp* part a *and Tiles, Tiles, Tiles* part b), but unable to explain or justify thinking.

Developing a degree of comfort with working mentally with multiplication and division facts.

## Teaching implications – consolidation and establishment

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

## Teaching implications – introduction and development

**Hundredths as a new place-value part**

Introduce strategies for hundredths as a new place-value part by making/representing, naming and recording ones, tenths and hundredths (see Booker et al, 2004).

Consolidate by comparing, ordering, sequencing counting forwards and backwards in place-value parts and renaming.

**Explain and justify solution strategies**

For problems involving multiplication and division, introduce strategies that show students how to explain and justify solutions. This is particularly important in relation to interpreting decimal remainders appropriate to context. For example:

How many buses will be needed to take 594 students and teachers to the school Speech night, assuming each bus hold 45 passengers and everyone must wear a seatbelt?”

For more information, see the multiplication workshop From Additive to Multiplicative Thinking – The Big Challenge of the Middle Years.

**Proportion problems**

Introduce more efficient and systematic processes that can be generalised for dealing with proportion problems.

For example:

- use of the ‘for each’ idea
- formal recording
- the use of fractions
- percent to justify claims.

For example:

Jane scored 14 goals from 20 attempts. Emma scored 18 goals from 25 attempts. Which girl should be selected for the school basketball team and why?

6 girls share 4 pizzas equally. 8 boys share 6 pizzas equally. Who had more pizza, the girls or the boys?

35 feral cats were found in a 146 hectare nature reserve. 27 feral cats were found in a 103 hectare reserve. Which reserve had the biggest feral cat problem?

Orange juice is sold in different sized containers: 5L for $14, 2 L for $5, and 500mL for $1.35. Which represents the best value for money?

**Multiplication and division involving larger numbers based on sound place-value ideas**

Introduce more efficient strategies and formal processes for working with multiplication and division involving larger numbers based on sound place-value ideas.

For example:

3486 × 21 can be estimated by thinking about 35 hundreds by 2 tens, 70 thousands, and 1 more group of 35 hundred, ie, 73,500

or

3486 × 21 can be calculated by using factors of 21, ie, 3486 × 3 × 7

Two digit multiplication can be used to support the multiplication of ones and tenths by ones and tenths. For example:

for 2.3 by 5.7, rename as tenths and compute as 23 tenths by 57 tenths, which gives 1311 hundredths hence 13.11

Consider a broader range of problems and applications. For example,

Average gate takings per day over the World Cricket cup Series.

Matt rode around the park 8 times. The odometer on his bike indicated that he had ridden a total of 15 km. How far was it around the park?

After 11 training sessions, Kate’s average time for 100 metres butterfly was 61.3 seconds. In her next 2 trials, Kate clocked 61.21 and 60.87 seconds. What was her new average time?

**Integers**

Use real-world examples to explain integers, such as:

- heights above and below sea-zone
- temperatures above and below zero
- simple addition and difference calculations.

**Notion of variable and pattern recognition and description**

Introduce strategies for understanding the notion of variable and how to recognise and formally describe patterns involving all four operations.

Use the Max’s Matchsticks (pdf - 27.6kb) task to explore how patterns may be viewed differently leading to different ways of counting and forms of representation.

## 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 6 – Learning Plans Introducing (pdf - 49.06kb)

## More information

Using the resources on this site, you can also:

- look at the key characteristics and teaching implications for Zone 7
- 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.

Last Update: 04 June 2020