The information below describes the key characteristics and teaching implications for Zone 1 of the Learning and Assessment Framework for Multiplicative Thinking (LAF), including:
How to use this resource
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:
Can solve simple multiplication and division problems involving relatively small whole numbers. For example,
Butterfly House parts a and b.
Tends to rely on drawing, models and count-all strategies. For example, draws and counts all pots for part a of
May use skip counting (repeated addition) for groups less than 5. For example, to find number of tables needed to seat up to 20 people in
Tables and Chairs.
Can make simple observations from data given in a task. For example,
Adventure Camp a.
Can reproduce a simple pattern. For example,
Tables and Chairs, parts a to e.
Multiplicative Thinking (MT) not really apparent as there is no indication that groups are perceived as composite units, dealt with systematically, or that the number of groups can be manipulated to support a more efficient calculation.
Teaching implications – consolidation and establishment
Trusting the count for numbers to 10
For 6, as an example, this involves working with mental objects of 6 without having to model and/or count-all.
Use flash cards to develop subitising (the ability to say how many without counting) for numbers to 5 initially and then to 10 and beyond using part-part-whole knowledge. For example:
8 is 4 and 4, or 5 and 3 more, or 2 less than 10
Simple skip counting
Use simple skip counting to determine how many are in a collection and to establish numbers up to 5 as countable objects. For example:
count by twos, fives and tens, using concrete materials and a 0-99 Number Chart.
Mental strategies for addition and subtraction facts to 20
Count on from larger. For example:
for 2 and 7, think: 7, 8, 9.
Double and near doubles. Use ten-frames and a 2-row bead-frame to show that 7 and 7 is 10 and 4 more, 14.
Make-to-ten. For example:
for 6 and 8, think: 8, 10, 14, scaffold using open number lines
Explore and name mental strategies to solve subtraction problems such as 7 take 2, 12 take 5, and 16 take 9.
2 digit place-value
Work flexibly with ones and tens by making, naming, recording, comparing, ordering, counting forwards and backwards in place-value parts, and renaming (see
Booker et al, 2004 ).
Play the Place-Value Game (pdf - 16.36kb)
Teaching implications – introduction and development
Doubling (and halving) strategies
Use doubling and halving strategies for 2-digit numbers that do not require renaming. For example:
34 and 34, half of 46
Build to numbers that require some additional thinking. For example:
to double 36, think: double 3 tens, double 6 ones, 60 and 12 ones, 72
Extended mental strategies for addition and subtraction
Use efficient, place-value based strategies. For example:
37 and 24, think: 37, 47, 57, 60, 61
Scaffold thinking with open number lines.
Efficient and reliable strategies for counting large collections
Use strategies for large collections, for example:
counting a collection of 50 or more by 2s, 5s or 10s
Focus on how to organise the number of groups to facilitate the count. For example, by arranging the groups systematically in lines or arrays and then skip counting.
Make, name and use arrays or regions
Explain how to solve simple multiplication or sharing problems using concrete materials and skip counting. For example:
1 four, 2 fours, 3 fours …
Lead up to more efficient counting strategies based on reading arrays in terms of a consistent number of rows. For example:
4 rows of anything, that is, 4 ones, 4 twos, 4 threes, 4 fours …
3 digit place-value
Work flexibly with tens and hundreds by making with MAB , naming, recording, comparing, ordering, counting forwards and backwards in place-value parts, and renaming (see
Booker et al, 2004 ).
Strategies for unpacking and comprehending problem situations
Read, re-tell and ask questions such as ‘What is the question asking?’ or ‘What do we need to do?’.
Use realistic word problems to explore different ideas for multiplication and division. For example:
3 rows, 7 chairs in each row, how many chairs (array)?
Mandy has three times as many…as Tom…, how many … does she have (scalar idea)?
24 cards shared among 6 students, how many each (partition)?
Lollipops cost 5c each, how much for 4 (‘for each’ idea)?
How to explain and justify
Present strategies for developing a solution orally and in writing through words and pictures. This is important for mathematical literacy.
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 1 – Learning Plans Consolidating (pdf - 165.15kb)
Zone 1 – Learning Plans Introducing (pdf - 41.49kb)
Using the resources on this site, you can also: