Scaffolding Numeracy in the Middle Years: Learning Plans

Learning plans have been designed to help teachers with the development of multiplicative thinking in students. After using the assessment materials to locate students on the Learning and Assessment Framework for Multiplicative Thinking (LAF), teachers can use these activities and explicit teaching to target particular groups of students.

To use the learning plans with your students:

find out how the plans were developed
consult the Learning and Assessment Framework
download the learning plans
follow up with more information

How the plans were developed

The learning plans were developed as part of the Scaffolding Numeracy in the Middle Years (SNMY) research project by small teams of teachers working collaboratively across schools and within clusters.

Each learning plan gives a clear description of two to three highly focussed activities or targeted teaching that can help scaffold learning in one or two of the specific aspects of multiplicative thinking from a particular zone of the LAF to the next.

They are a guide rather than a complete list of lessons at any one zone. They can be used as they stand, adapted or added to depending on the requirements of students.

While they describe teacher actions, they are deliberately referred to as learning plans because the specified learning outcomes should be recognised and achieved by the target group of students.

Learning & Assessment Framework for Multiplicative Thinking	Learning Plans
Zone 1 – Primitive Modelling Can solve simple multiplication and division problems involving relatively small whole numbers (eg, Butterfly House parts a and b), but tends to rely on drawing, models and count-all strategies (eg, draws and counts all pots for part a of Packing Pots). May use skip counting (repeated addition) for groups less than 5 (eg, 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 (eg, Adventure Campa) and reproduce a simple pattern (eg, Tables and Chairs a* to e) Multiplicative thinking (MT) not really apparent as 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	Zone 1 – Learning Plans Consolidating (pdf - 165.15kb) Subitising – Seeing it all Trusting the Count Magic Bean Toss Exploring Part-Part-Whole Using Part-Part-Whole Picking the Plugs Bead Frame Doubles Ten-Frame Race Dice Double Number Line Jump Searching for Straws Modelling MAB Puzzling Place Value Zone 1 – Learning Plans Introducing (pdf - 41.49kb) Chicken Scramble Array Play Hurray for Arrays
Zone 2: Intuitive Modelling Trusts the count for groups of 2 and 5, that is, can use these numbers as units for counting (eg, Tables & Chairs j,Butterfly House d), counts large collections efficiently, systematically keeps track of count (for instance may order groups in arrays or as a list) but needs to ‘see’ all groups (eg, Tiles, Tiles, Tiles a, or for Butterfly House e, may use list and/or doubling as follows: 2 butterflies 5 drops 4 butterflies 10 drops 6 butterflies 15 drops … 12 butterflies 30 drops) Can share collections into equal groups/parts (eg, Pizza Party a and b). Recognises small numbers as composite units (eg, can count equal groups, skip count by twos, threes and fives) Recognises multiplication is relevant (eg, Packing Pots c, Speedy Snail a) but tends not to be able to follow this through to solution Can list some of the options in simple Cartesian Product situations (eg, Canteen Capers a) Orders 2 digit numbers (eg, partially correct ordering of times in Swimming Sports a) Some evidence of multiplicative thinking as equal groups/shares seen as entities that can be counted systematically	Zone 2 – Learning Plans Introducing (pdf - 541.69kb) Lotsa Lids Paint Spill Multiplication Toss Exploring Facts A Cup Cake Collection Painting Proportions Folding Fractions A Tale of Two Spreads
Zone 3: Sensing Demonstrates intuitive sense of proportion (eg, partial solution to Butterfly House f) and partitioning (eg, Missing Numbers b) Works with ‘useful’ numbers such as 2 and 5, and strategies such as doubling and halving (eg, Packing Pots b, and Pizza Party c) May list all options in a simple Cartesian product situation (eg, Canteen Capers b), but cannot explain or justify solutions Uses abbreviated methods for counting groups, eg, doubling and doubling again to find 4 groups of, or repeated halving to compare simple fractions (eg, Pizza Party c) Beginning to work with larger whole numbers and patterns but tends to rely on count all methods or additive thinking to solve problems (eg, Stained Glass Windows a and b, Tiles, Tiles, Tiles b)	Zone 3 – Learning Plans Introducing (pdf - 114.39kb) Multiplication Strategies Explained Double Trouble Multiplying Mentally Card Multiplication Game Think Board Multiplication & Place Value Ideas School Rubbish If I Know The Magician’s Costumes
Zone 4: Strategy Exploring Solves more familiar multiplication and division problems involving two-digit numbers (eg, Butterfly House c and d, Packing Pots c, Speedy Snail a) Tend to rely on additive thinking, drawings and/or informal strategies to tackle problems involving larger numbers and/or decimals and less familiar situations (eg, Packing Pots d, Filling the Buses a and b, Tables & Chairs g and h, Butterfly House h and g, Speedy Snail c, Computer Game a, Stained Glass Windows a and b). Tend not to explain their thinking or indicate working Able to partition given number or quantity into equal parts and describe part formally (eg Pizza Party a and b), and locate familiar fractions (eg, Missing Numbers a) Beginning to work with simple proportion, eg, can make a start, represent problem, but unable to complete successfully or justify their thinking (eg, How Far a, School Fair a and b)	Zone 4 – Learning Plans Introducing (pdf - 63.28kb) Identifying Fractions Sharing Pizzas Fraction Action Continuous Quantities Exploring Halving Strategies Exploring Thirding Strategies Naming Fractions A Recipe for Fruit Salad
Zone 5: Strategy Refining Systematically solves simple proportion and array problems (eg, Butterfly House e, Packing Pots a, How Far a) suggesting multiplicative thinking. May use additive thinking to solve simple proportion problems involving fractions (eg, School Fair a, Speedy Snail b) Able to solve simple, 2-step problems using a recognised rule/relationship (eg, Fencing the Freeway a) but finds this difficult for larger numbers (eg, Tables & Chairs k and l, Tiles, Tiles, Tiles c, Stained Glass Windows c) Able to order numbers involving tens, ones, tenths and hundredths in supportive context ( Swimming Sports a) Able to determine all options in Cartesian product situations involving relatively small numbers, but tends to do this additively (eg, Canteen Capers a, Butterfly House l and i) Beginning to work with decimal numbers and percent (eg, Swimming Sports a and b, Computer Game b) but unable to apply efficiently to solve problems Some evidence that multiplicative thinking being used to support partitioning (eg, Missing Numbers b) Beginning to approach a broader range of multiplicative situations more systematically	Zone 5 – Learning Plans Introducing (pdf - 77.05kb) Dashing Decimals Decimal Comparisons Factor Find Chocolate Partitioning Diverse Dimensions How Many Wholes? Branching out with Tree Diagrams
Zone 6: Strategy Extending Can work with Cartesian Product idea to systematically list or determine the number of options (eg Canteen Capers b, Butterfly House i and h) Can solve a broader range of multiplication and division problems involving two digit numbers, patterns and/or proportion (eg, Tables & Chairs h, Butterfly House f, Stained Glass Windows b and c, Computer Game a and b) but may not be able to explain or justify solution strategy (eg, Fencing the Freeway b, Fencing the Freeway d, and Swimming Sports b, How Far b, Speedy Snail b) Able to rename and compare fractions in the halving family (eg, Pizza Party c) and use partitioning strategies to locate simple fractions (eg, Missing Numbers a) Developing sense of proportion (eg, sees relevance of proportion in Adventure Camp b, Tiles, Tiles, Tiles b), but unable to explain or justify thinking Developing a degree of comfort with working mentally with multiplication and division facts	Zone 6 – Learning Plans Introducing (pdf - 49.06kb) Block Pattern for a Quilt Multiplying with Graph Paper Square Numbers
Zone 7: Connecting Able to solve and explain one-step problems involving multiplication and division with whole numbers using informal strategies and/or formal recording (eg, Filling the Buses a, Fencing the Freeway d, Packing Pots d) Can solve and explain solutions to problems involving simple patterns, percent and proportion (eg, Fencing the Freeway c, Swimming Sports b, Butterfly House g, Tables & Chairs g and l, Speedy Snail c, Tiles, Tiles, Tiles b and c, School Fair a, Stained Glass Windows a, Computer Game b, How Far b). May not be able to show working and/or explain strategies for situations involving larger numbers (eg, Tables & Chairs m and k, Tiles, Tiles, Tiles c) or less familiar problems (eg, Adventure Camp b, School Fair b, How Far c) Locates fractions using efficient partitioning strategies (eg, Missing Numbers a) Beginning to make connections between problems and solution strategies and how to communicate this mathematically	Zone 7 – Learning Plans Introducing (pdf - 52.87kb) Multiple Patterns Working out ‘Value for Money’ Patterns and Solutions Combining Speeds
Zone 8: Reflective Knowing Can use appropriate representations, language and symbols to solve and justify a wide range of problems involving unfamiliar multiplicative situations including fractions and decimals (eg, Adventure Camp b, Speedy Snail b) Can justify partitioning (eg, Missing Numbers b) Can use and formally describe patterns in terms of general rules (eg, Tables and Chairs, m and k) Beginning to work more systematically with complex, open-ended problems (eg, School Fair b, Computer Game c)	Zone 8 – Learning Plans Introducing (pdf - 100.89kb) Anno’s Mysterious Jar Working Groups Problem Solving Recognising Problem Type

Learning & Assessment Framework for Multiplicative Thinking

Learning Plans

Zone 1 – Primitive Modelling

Can solve simple multiplication and division problems involving relatively small whole numbers (eg, Butterfly House parts a and b)*, but tends to rely on drawing, models and count-all strategies (eg, draws and counts all pots for part a of Packing Pots). May use skip counting (repeated addition) for groups less than 5 (eg, 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 (eg, Adventure Campa) and reproduce a simple pattern (eg, Tables and Chairs a to e)
Multiplicative thinking (MT) not really apparent as 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

Zone 1 – Learning Plans Consolidating (pdf - 165.15kb)

Subitising – Seeing it all
Trusting the Count
Magic Bean Toss
Exploring Part-Part-Whole
Using Part-Part-Whole
Picking the Plugs
Bead Frame Doubles
Ten-Frame Race
Dice Double
Number Line Jump
Searching for Straws
Modelling MAB
Puzzling Place Value

Zone 1 – Learning Plans Introducing (pdf - 41.49kb)

Chicken Scramble
Array Play
Hurray for Arrays

Zone 2: Intuitive Modelling

Trusts the count for groups of 2 and 5, that is, can use these numbers as units for counting (eg, Tables & Chairs j,Butterfly House d), counts large collections efficiently, systematically keeps track of count (for instance may order groups in arrays or as a list) but needs to ‘see’ all groups (eg, Tiles, Tiles, Tiles a, or for Butterfly House e, may use list and/or doubling as follows:

2 butterflies 5 drops

4 butterflies 10 drops

6 butterflies 15 drops

…

12 butterflies 30 drops)

Can share collections into equal groups/parts (eg, Pizza Party a and b). Recognises small numbers as composite units (eg, can count equal groups, skip count by twos, threes and fives)
Recognises multiplication is relevant (eg, Packing Pots c, Speedy Snail a) but tends not to be able to follow this through to solution
Can list some of the options in simple Cartesian Product situations (eg, Canteen Capers a)
Orders 2 digit numbers (eg, partially correct ordering of times in Swimming Sports a)
Some evidence of multiplicative thinking as equal groups/shares seen as entities that can be counted systematically

Zone 2 – Learning Plans Introducing (pdf - 541.69kb)

Lotsa Lids
Paint Spill
Multiplication Toss
Exploring Facts
A Cup Cake Collection
Painting Proportions
Folding Fractions
A Tale of Two Spreads

Zone 3: Sensing

Demonstrates intuitive sense of proportion (eg, partial solution to Butterfly House f) and partitioning (eg, Missing Numbers b)
Works with ‘useful’ numbers such as 2 and 5, and strategies such as doubling and halving (eg, Packing Pots b, and Pizza Party c)
May list all options in a simple Cartesian product situation (eg, Canteen Capers b), but cannot explain or justify solutions
Uses abbreviated methods for counting groups, eg, doubling and doubling again to find 4 groups of, or repeated halving to compare simple fractions (eg, Pizza Party c)
Beginning to work with larger whole numbers and patterns but tends to rely on count all methods or additive thinking to solve problems (eg, Stained Glass Windows a and b, Tiles, Tiles, Tiles b)

Zone 3 – Learning Plans Introducing (pdf - 114.39kb)

Multiplication Strategies Explained
Double Trouble
Multiplying Mentally
Card Multiplication Game
Think Board
Multiplication & Place Value Ideas
School Rubbish
If I Know
The Magician’s Costumes

Zone 4: Strategy Exploring

Solves more familiar multiplication and division problems involving two-digit numbers (eg, Butterfly House c and d, Packing Pots c, Speedy Snail a)
Tend to rely on additive thinking, drawings and/or informal strategies to tackle problems involving larger numbers and/or decimals and less familiar situations (eg, Packing Pots d, Filling the Buses a and b, Tables & Chairs g and h, Butterfly House h and g, Speedy Snail c, Computer Game a, Stained Glass Windows a and b). Tend not to explain their thinking or indicate working
Able to partition given number or quantity into equal parts and describe part formally (eg Pizza Party a and b), and locate familiar fractions (eg, Missing Numbers a)
Beginning to work with simple proportion, eg, can make a start, represent problem, but unable to complete successfully or justify their thinking (eg, How Far a, School Fair a and b)

Zone 4 – Learning Plans Introducing (pdf - 63.28kb)

Identifying Fractions
Sharing Pizzas
Fraction Action
Continuous Quantities
Exploring Halving Strategies
Exploring Thirding Strategies
Naming Fractions
A Recipe for Fruit Salad

Zone 5: Strategy Refining

Systematically solves simple proportion and array problems (eg, Butterfly House e, Packing Pots a, How Far a) suggesting multiplicative thinking. May use additive thinking to solve simple proportion problems involving fractions (eg, School Fair a, Speedy Snail b)
Able to solve simple, 2-step problems using a recognised rule/relationship (eg, Fencing the Freeway a) but finds this difficult for larger numbers (eg, Tables & Chairs k and l, Tiles, Tiles, Tiles c, Stained Glass Windows c)
Able to order numbers involving tens, ones, tenths and hundredths in supportive context ( Swimming Sports a)
Able to determine all options in Cartesian product situations involving relatively small numbers, but tends to do this additively (eg, Canteen Capers a, Butterfly House l and i)
Beginning to work with decimal numbers and percent (eg, Swimming Sports a and b, Computer Game b) but unable to apply efficiently to solve problems
Some evidence that multiplicative thinking being used to support partitioning (eg, Missing Numbers b)
Beginning to approach a broader range of multiplicative situations more systematically

Zone 5 – Learning Plans Introducing (pdf - 77.05kb)

Dashing Decimals
Decimal Comparisons
Factor Find
Chocolate Partitioning
Diverse Dimensions
How Many Wholes?
Branching out with Tree Diagrams

Zone 6: Strategy Extending

Can work with Cartesian Product idea to systematically list or determine the number of options (eg Canteen Capers b, Butterfly House i and h)
Can solve a broader range of multiplication and division problems involving two digit numbers, patterns and/or proportion (eg, Tables & Chairs h, Butterfly House f, Stained Glass Windows b and c, Computer Game a and b) but may not be able to explain or justify solution strategy (eg, Fencing the Freeway b, Fencing the Freeway d, and Swimming Sports b, How Far b, Speedy Snail b)
Able to rename and compare fractions in the halving family (eg, Pizza Party c) and use partitioning strategies to locate simple fractions (eg, Missing Numbers a)
Developing sense of proportion (eg, sees relevance of proportion in Adventure Camp b, Tiles, Tiles, Tiles b), but unable to explain or justify thinking
Developing a degree of comfort with working mentally with multiplication and division facts

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

Block Pattern for a Quilt
Multiplying with Graph Paper
Square Numbers

Zone 7: Connecting

Able to solve and explain one-step problems involving multiplication and division with whole numbers using informal strategies and/or formal recording (eg, Filling the Buses a, Fencing the Freeway d, Packing Pots d)
Can solve and explain solutions to problems involving simple patterns, percent and proportion (eg, Fencing the Freeway c, Swimming Sports b, Butterfly House g, Tables & Chairs g and l, Speedy Snail c, Tiles, Tiles, Tiles b and c, School Fair a, Stained Glass Windows a, Computer Game b, How Far b). May not be able to show working and/or explain strategies for situations involving larger numbers (eg, Tables & Chairs m and k, Tiles, Tiles, Tiles c) or less familiar problems (eg, Adventure Camp b, School Fair b, How Far c)
Locates fractions using efficient partitioning strategies (eg, Missing Numbers a)
Beginning to make connections between problems and solution strategies and how to communicate this mathematically

Zone 7 – Learning Plans Introducing (pdf - 52.87kb)

Multiple Patterns
Working out ‘Value for Money’
Patterns and Solutions
Combining Speeds

Zone 8: Reflective Knowing

Can use appropriate representations, language and symbols to solve and justify a wide range of problems involving unfamiliar multiplicative situations including fractions and decimals (eg, Adventure Camp b, Speedy Snail b)
Can justify partitioning (eg, Missing Numbers b)
Can use and formally describe patterns in terms of general rules (eg, Tables and Chairs, m and k)
Beginning to work more systematically with complex, open-ended problems (eg, School Fair b, Computer Game c)

Zone 8 – Learning Plans Introducing (pdf - 100.89kb)

Anno’s Mysterious Jar
Working Groups
Problem Solving
Recognising Problem Type

Download the learning plans

Learning plans can be used to scaffold student learning from one zone of the LAF to the next. You can download the learning plan for a specific zone below, or look across all of the zones in the LAF and the associated learning plans using the table above.

More information

access authentic tasks designed to encourage students to find ways of solving rich mathematical tasks themselves
find out more on the eight zones within the Learning and Assessment Framework
read about the Scaffolding Numeracy in the Middle Years project background.