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:
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: