Introduction
Welcome to Multiplication Architects!
Welcome to our multiplication adventure! In this learning journey, you will explore how multiplication works using arrays, area models, games, and real-world design challenges. Across four exciting lessons, you will learn how to break large numbers into smaller parts, use visual models to solve problems, and understand how the standard multiplication algorithm really works.
You will use digital tools such as Mathigon Polypad and PhET Area Builder to build arrays, investigate area models, and solve interactive multiplication challenges. You will also work like a real “Sustainable Playground Architect,” designing playground spaces and using multiplication strategies to solve real-world problems.
Throughout this WebQuest, you will explain your thinking, compare strategies with classmates, and discover efficient ways to multiply large numbers. Get ready to explore, design, collaborate, and become a confident multiplication expert!
Task
Welcome, Multiplication Architects!
In this WebQuest, your challenge is to use your understanding of multiplication to design, explore, and solve real-world problems like a professional architect.
Your mission is to investigate how large multiplication problems can be broken into smaller, easier parts using arrays, area models, and efficient mental strategies. You will learn how multiplication is not just a set of steps, but a way of thinking about numbers, space, and structure.
Across this learning journey, you will:
- build and explore multiplication using visual models (arrays and area grids)
- use digital tools to test and check your thinking
- compare different strategies to find the most efficient way to solve problems
- explain and justify your mathematical thinking using clear language
- apply your knowledge to real-life design challenges
In the final task, you will take on the role of a Sustainable Playground Architect. You will design a playground space and use multiplication to calculate areas, justify your design choices, and explain why your strategy is efficient. You will present your thinking using digital tools and communicate your ideas clearly to others.
By the end of this WebQuest, you will not only be able to solve multiplication problems confidently, but also understand why the methods work and when to use them in real life.
Get ready to explore, design, and think like a mathematician!
Process
Lesson 1: Building Multiplication with Arrays
🎯 Aim
In this lesson, you will explore how multiplication can be represented using arrays. You will use hands-on materials, drawings, and digital tools to understand how multiplication shows equal groups and area.
You will also learn how to explain your mathematical thinking using correct mathematical language.
📖 Description
Have you ever arranged chairs in rows, organised cupcakes into trays, or seen tiles arranged in a rectangle? These are all examples of arrays!
An array is a way of arranging objects into equal rows and columns. Arrays help us understand multiplication visually instead of only memorising times tables.
For example:
- 3 rows of 4 counters = 12
- 4 columns of 3 counters = 12
Both represent multiplication, even though they are arranged differently.
In this lesson, you will build arrays, break numbers into smaller parts, and explain how multiplication works using visual models.
🎥 Watch and Learn
Watch a short video that explains how arrays represent multiplication in real life.
👉 https://youtu.be/QphXFi30aFk
While watching, think about:
- What do rows and columns represent?
- How does an array help us solve multiplication problems?
- How can arrays help us work with bigger numbers?
🧮 Understanding Arrays
Example 1
If we build an array for:
3 × 4 = 12
we can arrange:
- 3 rows
- 4 counters in each row
This gives a total of 12 counters.
Example 2
Now look at:
13 × 12
This is a much larger multiplication problem.
Instead of counting one by one, we can:
- split 13 into 10 + 3
- split 12 into 10 + 2
This helps us break the problem into smaller, easier parts.
🧱 Activities and Resources
Hands-on Array Building
Use counters, blocks, graph paper, or drawings to build arrays such as:
- 4 × 5
- 6 × 3
- 13 × 12 (challenge)
As you build:
- count rows and columns carefully
- look for equal groups
- discuss patterns you notice
Online Interactive Practice
🟦 Array Builder Game
👉 https://au.splashlearn.com/maths/multiplication-with-arrays-games?
Use this game to:
- build digital arrays
- match arrays with equations
- practice multiplication visually
💬 Think and Talk (Mathematical Communication)
Discuss these questions with a partner:
- What does each row represent?
- What does each column represent?
- How does an array help you solve multiplication faster?
- Can one array represent different multiplication equations?
Try using sentence starters:
- “I notice that…”
- “The array shows…”
- “I solved it by…”
💻 Technology Integration
In this lesson, technology helps you:
- build arrays digitally
- test different multiplication ideas
- receive instant feedback
- communicate your thinking visually
You may:
- take photos of your arrays
- annotate your work on a tablet
- compare physical and digital models
🧠 Quick Quiz
Complete a short online quiz to check your understanding.
Focus questions:
- identifying rows and columns
- matching arrays to equations
- understanding equal groups
Lesson 2: Exploring Multiplication with Area Models
🎯 Aim
In this lesson, you will learn how to use an area model (grid method) to solve larger multiplication problems. You will explore how numbers can be partitioned into tens and ones to make multiplication easier to understand and calculate.
You will also explain your mathematical thinking using mathematical language and visual models.
📖 Description
In Lesson 1, you used arrays to represent multiplication visually. In this lesson, you will extend that understanding by using area models.
An area model helps us break large multiplication problems into smaller parts called partial products. Instead of solving one large multiplication problem at once, we can split numbers into tens and ones and calculate smaller sections separately.
For example:
24 × 16
can be broken into:
- 20 + 4
- 10 + 6
This creates four smaller multiplication sections that are easier to solve.
Area models help mathematicians understand:
- place value
- distributive property
- multiplication structure
🎥 Watch and Learn
Watch a short video explaining how area models work in multiplication.
👉 https://www.youtube.com/watch?v=MVZRD4Fa1OY
While watching, think about:
- How are the numbers being split apart?
- What does each box in the grid represent?
- How do smaller products combine to make the final answer?
🧮 Understanding Area Models
Example 1
Solve:
24 × 16
Step 1: Partition the numbers
- 24 = 20 + 4
- 16 = 10 + 6
Step 2: Multiply each section
- 20 × 10 = 200
- 20 × 6 = 120
- 4 × 10 = 40
- 4 × 6 = 24
Step 3: Add all partial products
- 200 + 120 + 40 + 24 = 384
Final Answer:
24 × 16 = 384
🧱 Activities and Resources
1. Build Digital Area Models
Use an online area model tool to create multiplication grids.
🟩 Mathigon Polypad
👉 https://polypad.amplify.com/p
Use the tool to:
- split numbers into tens and ones
- build multiplication grids
- colour-code partial products
- explore different multiplication strategies
Try these examples:
- 24 × 16
- 32 × 14
- 45 × 12 (challenge)
2. Interactive Multiplication Practice
🎮 Area Model Multiplication Games
👉 https://au.splashlearn.com/maths/multiplication-games
Use the game to:
- practise grid multiplication
- identify partial products
- receive instant feedback
💬 Think and Talk (Mathematical Communication)
Discuss these questions with a partner:
- Why do we split numbers into tens and ones?
- What does each box in the area model represent?
- How do partial products help solve large multiplication problems?
- Is there more than one way to partition a number?
Use sentence starters:
- “I partitioned the number because…”
- “This section represents…”
- “The partial products show…”
💻 Technology Integration
Technology in this lesson helps you:
- visualise multiplication structure
- manipulate numbers dynamically
- test different strategies
- communicate mathematical reasoning visually
You may:
- annotate screenshots of your area model
- colour-code sections digitally
- compare multiple strategies using online tools
🧠 Quick Quiz
Quizizz or Wordwall Challenge
Lesson 3: Connecting Area Models to the Standard Algorithm
🎯 Aim
In this lesson, you will connect the area model (grid method) to the standard vertical multiplication algorithm. You will learn how both methods represent the same mathematical thinking and how place value helps us multiply larger numbers accurately.
You will also explain and justify your strategies using mathematical language.
📖 Description
In Lesson 2, you used area models to break multiplication problems into smaller parts called partial products. In this lesson, you will discover that the standard multiplication algorithm is simply a more compact and efficient way of organising the same thinking.
Mathematicians do not just follow steps—they understand why the steps work.
For example:
24 × 16
can be solved using:
- an area model
- partial products
- the standard vertical algorithm
All three methods represent the same multiplication.
This lesson will help you:
- connect visual models to symbolic methods
- understand place value alignment
- explain the meaning of regrouping and zero placeholders
🎥 Watch and Learn
Watch this video that explains the standard multiplication algorithm.
👉 https://www.youtube.com/watch?v=ItXOu8o5Lbs
While watching, think about:
- What does each row in the algorithm represent?
- Where do the partial products appear?
- Why is place value important?
- How is this connected to the area model?
🧮 Understanding the Standard Algorithm
Example 1 — Area Model Connection
Solve:
24 × 16
Using an area model:
- 20 × 10 = 200
- 20 × 6 = 120
- 4 × 10 = 40
- 4 × 6 = 24
Total:
- 200 + 120 + 40 + 24 = 384
Example 2 — Standard Algorithm
Now solve the same problem using the vertical algorithm.
Students should notice:
- the first row represents multiplying by 6
- the second row represents multiplying by 10
- place values shift when multiplying tens
- both methods produce the same answer
Final Answer:
24 × 16 = 384
🧱 Activities and Resources
Compare Two Methods
Solve each problem using BOTH:
- area model
- standard algorithm
Examples:
- 23 × 15
- 34 × 12
- 41 × 16 (challenge)
As you solve:
- label partial products
- highlight place value changes
- compare efficiency between methods
Interactive Algorithm Practice
🎮 Multi-digit Multiplication Game
Use the game to:
- practise the standard algorithm
- check place value alignment
- receive instant feedback
Error Detective Challenge
Look at incorrect multiplication examples and identify mistakes.
Questions to consider:
- Was place value aligned correctly?
- Was a partial product missing?
- Was regrouping completed accurately?
Explain:
- what went wrong
- how to fix it
- why the correction works
💬 Think and Talk (Mathematical Communication)
Discuss with a partner:
- How is the standard algorithm connected to the area model?
- Why does place value matter in multiplication?
- Which method helps you understand multiplication better?
- Is the standard algorithm always the best strategy?
Use sentence starters:
- “The algorithm works because…”
- “This row represents…”
- “The area model helps me understand…”
Lesson 4: Real-World Multiplication Challenge: Playground Architects
🎯 Aim
In this lesson, you will apply your multiplication skills to real-world design problems. You will use area models, mental strategies, and the standard algorithm to calculate area, compare strategies, and justify your mathematical thinking.
You will also use digital tools to design and present your ideas like a real architect.
📖 Description
In the previous lessons, you explored multiplication using arrays, area models, and the standard algorithm. Now it is time to apply these skills to a real-world situation.
Imagine you are a playground architect designing a new play space for your school. Architects and designers use multiplication and area every day to plan safe, organised, and efficient spaces.
In this lesson, you will:
- design playground zones
- calculate area using multiplication
- compare different solving strategies
- explain and justify your decisions
You will discover that mathematics is not just about getting answers—it is about solving meaningful problems and communicating ideas clearly.
🎥 Watch and Learn
Watch this video about how multiplication and area are used in real-life situations.
👉 https://www.youtube.com/watch?v=1PXa0TdOS_w
While watching, think about:
- Where is multiplication used in real life?
- Why is area important in design and construction?
- How do people use measurements to plan spaces?
- How could these ideas help you design a playground?
🧮 Understanding Area in Real Life
Example 1 — Playground Zone
A playground sandpit measures:
12 × 8
Area:
12 × 8 = 96
The sandpit covers 96 square metres.
Example 2 — Sports Area
A basketball zone measures:
15 × 18
Students may solve using:
- area model
- mental strategy
- standard algorithm
This shows that different strategies can still lead to the same answer.
🏗️ Activities and Resources
Playground Design Challenge
🟩 PhET Area Builder
👉https://phet.colorado.edu/en/simulations/area-builder?
Use the simulation to:
- design playground zones
- test different dimensions
- calculate area visually
- explore how changing dimensions changes area
Create:
- a sandpit
- a sports area
- a seating area
You must:
- calculate each area
- label dimensions clearly
- explain your strategy
Real-World Design Task
The school council gives you this requirement:
The playground must include a total space measuring 15m × 18m.
Your task:
- create TWO different playground layouts
- calculate area using different strategies
- compare efficiency of methods
- explain which design is best and why
Remember:
👉 There is more than one possible solution.
💬 Think and Talk (Mathematical Communication)
Discuss with a partner:
- Which multiplication strategy was most efficient?
- How does multiplication help architects and designers?
- Why is area important in real life?
- How did technology help you test your ideas?
- Which design would you choose and why?
Use sentence starters:
- “I chose this design because…”
- “My strategy was efficient because…”
- “The area changes when…”
💻 Technology Integration
Technology in this lesson helps you:
- model real-world situations
- test mathematical ideas visually
- compare multiple strategies
- communicate mathematical thinking clearly
You may:
- take screenshots of your playground design
- annotate dimensions digitally
- record a Flip video explaining your design
- compare designs using digital tools
🧠 Quick Quiz
Kahoot or Quizizz Challenge
🌟 Final Reflection
Reflect on your learning journey:
- How has your understanding of multiplication changed?
- Which strategy helps you most and why?
- How did technology support your learning?
- Where might you use multiplication in real life?
Evaluation
The evaluation of this technology-enhanced multiplication unit is designed to capture both student learning outcomes and the effectiveness of digital pedagogy in developing multiplicative thinking. Assessment is embedded throughout the sequence and includes ongoing formative evidence, a culminating performance task, and structured student reflection.
Student learning is assessed continuously through digital and physical artefacts produced across the lessons, including annotated arrays, Mathigon Polypad area models, PhET Area Builder explorations, and explanations recorded during the final playground design task. These artefacts provide rich evidence of students’ conceptual understanding, strategy use, and ability to move between representations. This aligns with relational understanding, where emphasis is placed on understanding why mathematical strategies work rather than memorising procedures (Skemp, 1976).
Technology plays a key role in evaluation by enabling real-time observation of student thinking. Digital tools such as Polypad and PhET Area Builder allow students to construct, test, and adjust mathematical models, while also allowing teachers to monitor reasoning processes as they occur. This supports more responsive assessment practices by identifying misconceptions and understanding development during learning rather than after instruction.
The summative assessment is the “Sustainable Playground Architect” task, where students apply multiplication to design and justify a real-world playground layout. Students are assessed on accuracy in calculating area, selection and justification of efficient strategies, and clarity of mathematical communication. This task demonstrates the application of multiplicative thinking in authentic problem-solving contexts aligned with the Australian Curriculum.
Student voice is incorporated through a structured digital reflection survey, which allows learners to evaluate their understanding of multiplication strategies, use of technology, and overall learning experience. This promotes metacognitive awareness and provides valuable feedback on engagement and conceptual growth. The survey is accessible at: https://form.typeform.com/to/kQdz21xe
Overall, this evaluation framework integrates formative assessment, summative performance-based tasks, digital learning evidence, and student reflection. It ensures that assessment captures not only correct answers but also reasoning, communication, and conceptual development, while highlighting the role of ICT in supporting both learning and assessment.
Conclusion
RATIONALE:
This sequence of four lessons is designed to develop students’ multiplicative thinking through a technology-enhanced learning environment that promotes meaningful social interaction, conceptual understanding, and mathematical reasoning. The unit focuses on a coherent progression from concrete representations of multiplication using arrays, to visual models through area representations, and finally to abstract symbolic methods using the standard algorithm and real-world application. This progression is informed by the work of Jerome Bruner (1966), moving from enactive to iconic to symbolic representations, ensuring that students build deep and connected understandings of multiplication.
A central feature of this resource is the use of generative and open-ended task design to promote higher-order thinking and reasoning. Across the lessons, students are consistently engaged in tasks that require them to create, compare, and justify multiple solutions rather than follow a single prescribed method. For example, students construct different arrays representing the same product, generate multiple area models through flexible partitioning, and compare the efficiency of various multiplication strategies. In the final lesson, students apply their understanding in a real-world playground design task, where they interpret and justify mathematical decisions. These open-ended tasks encourage students to explore mathematical relationships, identify patterns, and generalise their understanding, thereby fostering critical thinking and problem-solving skills. This approach aligns with contemporary views of mathematics learning as an active and constructive process.
Mathematical communication is explicitly embedded throughout the unit to support the development of precise mathematical language and reasoning. Students are provided with structured discussion prompts and sentence stems to articulate their thinking, such as explaining how arrays represent equal groups or how partial products contribute to the total. Key vocabulary, including factor, product, partition, and area, is consistently reinforced across lessons. Collaborative “Think and Talk” activities promote peer discussion, allowing students to negotiate meaning and refine their ideas. This approach establishes strong socio-mathematical norms, where explanations are valued over answers, multiple strategies are explored, and reasoning must be justified, consistent with the work of Paul Cobb (1994). The emphasis on social interaction is also supported by Lev Vygotsky (1978), who highlights the role of social discourse and scaffolding in supporting learning within the Zone of Proximal Development.
Technology is purposefully integrated throughout the unit to enhance learning and support the construction of meaningful mathematical ideas. A range of digital tools is incorporated, including interactive games (e.g., SplashLearn), construction and modelling software (e.g., Mathigon Polypad), and simulation tools (e.g., PhET Area Builder). These technologies enable students to dynamically construct arrays and area models, manipulate numbers, and receive immediate feedback on their thinking. The use of such tools transforms traditional mathematics instruction by making abstract concepts visible and allowing students to experiment with different representations in real time. For instance, students can visually explore how partitioning numbers affects the structure of an area model, or how changing dimensions impacts the total area in a simulated environment. This demonstrates the effective use of ICT in supporting conceptual understanding, engagement, and exploratory learning.
The integration of technology also allows students to construct and communicate personally meaningful mathematical ideas. By using digital tools to create, annotate, and share their work, students are able to externalise their thinking and reflect on their learning. The combination of physical and digital representations supports multimodal learning and caters to diverse learners, including those who benefit from visual and interactive approaches. Compared to traditional paper-based methods, technology provides a more flexible and responsive learning environment, enabling students to test ideas, receive feedback, and refine their understanding in a way that promotes deeper learning.
The unit also promotes the development of relational understanding, where students learn not only how to perform multiplication but also why mathematical procedures work. Students are encouraged to make connections between different representations, such as arrays, area models, and the standard algorithm, and to justify the efficiency of different strategies. This aligns with the distinction between instrumental and relational understanding proposed by Richard Skemp(1976), emphasising the importance of conceptual understanding in achieving long-term mathematical proficiency.
Overall, this resource demonstrates how technology can be used effectively to enhance traditional mathematics teaching by promoting engagement, interaction, and deep conceptual learning. The combination of open-ended tasks, structured mathematical communication, and purposeful use of digital tools creates a rich learning environment in which students actively construct knowledge, engage in collaborative reasoning, and develop confidence in their mathematical thinking. The unit aligns with the Australian Curriculum through its focus on understanding, fluency, problem-solving, and reasoning, and provides a coherent and detailed sequence of lessons that support the development of multiplicative thinking in upper primary students.
The Misconceptions Lab: Investigating Student Thinking
This section identifies common student misconceptions across the four-lesson sequence and outlines targeted strategies to address them. Misconceptions are viewed as indicators of developing understanding and are used to guide instruction toward relational thinking (Skemp, 1976).
| Lesson | Misconception | Why it Happens | Teaching Strategy (Response) |
| Lesson 1: Arrays and Equal Groups (Enactive Stage) | Students count objects one-by-one instead of using equal groups (e.g., counting all items individually in a 24-counter array). | Students count objects one-by-one instead of using equal groups (e.g., counting all items individually in a 24-counter array). | Use physical materials (MAB blocks) to construct arrays and emphasise grouping. Prompt students with questions such as “Is it faster to count individually or by groups?” Encourage use of larger units (e.g., 10s or 100-flats) to shift thinking from counting to structure. |
| Lesson 2: Area Models and Partitioning (Iconic Stage) | Students calculate only 20×10 and 4×6 when solving 24×16, producing incorrect results (e.g., 224 instead of 384). | Students partially apply the distributive property and omit cross-products (20×6 and 4×10), reflecting incomplete structural understanding. | Use colour-coded area models in Mathigon Polypad to visually separate all four sections of the rectangle. Teacher questioning focuses attention on completeness: “Have all parts of the area been included?” Emphasise that each section contributes to the total product. |
| Lesson 3: Standard Algorithm (Symbolic Stage) | Students add a zero in the second line of the algorithm without understanding its meaning (“magic zero” rule). | Students treat the algorithm as a set of procedures rather than a representation of place value and structure. | Explicitly link the algorithm to the area model by matching each row to a partial product. Use colour coding to show place value shifts. Reinforce that the zero represents multiplication by tens, not a rule to be memorised. |
| Lesson 4: Real-World Application (Strategic Competence) | Students confuse area and perimeter, adding side lengths instead of multiplying dimensions. | Students focus on the boundary (edge) of a shape rather than the space inside it, leading to conceptual confusion in measurement. | Use PhET Area Builder to physically fill shapes with unit tiles. Emphasise the difference between “covering space” (area) and “measuring edges” (perimeter). Use prompts such as “What is inside the space we are covering?” |
Credits
References
Australian Curriculum, Assessment and Reporting Authority. (2022). Australian Curriculum: Mathematics Version 9.0. https://v9.australiancurriculum.edu.au
Bruner, J. S. (1966). Toward a theory of instruction. Harvard University Press.
Cobb, P. (1994). Where is the mind? Constructivist and sociocultural perspectives on mathematical development. Educational Researcher, 23(7), 13–20.
Polya, G. (1957). How to solve it (2nd ed.). Princeton University Press.
Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77, 20–26.
Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes. Harvard University Press.
Digital Tools / ICT Resources
Ample Education. (n.d.). Mathigon Polypad. https://mathigon.org/polypad
PhET Interactive Simulations. (n.d.). Area Builder. University of Colorado Boulder. https://phet.colorado.edu/en/simulations/area-builder
SplashLearn. (n.d.). Multiplication with arrays games. https://www.splashlearn.com
Wayground (Quizizz). (n.d.). Mathematics quizzes and flashcards. https://wayground.com
Math Games. (n.d.). Multi-digit multiplication practice. https://www.mathgames.com
Video Resources
YouTube. (n.d.). Multiplication with arrays [Video]. https://youtu.be/QphXFi30aFk
YouTube. (n.d.). Area model multiplication [Video]. https://www.youtube.com/watch?v=MVZRD4Fa1OY
YouTube. (n.d.). Standard algorithm multiplication [Video]. https://www.youtube.com/watch?v=ItXOu8o5Lbs
YouTube. (n.d.). Multiplication in real-life contexts [Video]. https://www.youtube.com/watch?v=1PXa0TdOS_w
Teacher Page
Teachers' Corner: Multiplicative Thinking & Spatial Reasoning
Welcome to this professional resource for Year 6 Mathematics, centered on the Australian Curriculum: Mathematics Version 9.0 . This unit is designed to shift students from additive "answer-getting" toward a deep, structural understanding of multiplication through the lens of architectural design.
Curriculum Alignment
This unit is designed for Year 6 students within the Number strand of the Australian Curriculum: Mathematics Version 9.0. It directly aligns with the content description AC9M6N05, which requires students to solve problems involving multiplication of larger numbers using efficient mental and written strategies. The learning sequence is intentionally structured to develop both conceptual understanding and procedural fluency, ensuring that students can not only perform multiplication accurately but also understand and justify the strategies they use.
The unit is explicitly aligned with the four proficiency strands of the curriculum: Understanding, Fluency, Problem-Solving, and Reasoning. Understanding is developed through the use of arrays and area models, which allow students to visualise multiplication as structured groups and spatial representations. Fluency is developed through repeated engagement with mental strategies and the standard algorithm, enabling students to compute efficiently. Problem-solving is embedded in authentic contexts such as the playground design task, where students must apply multiplication to real-world constraints. Reasoning is consistently developed through activities that require students to explain, compare, and justify the efficiency and validity of different strategies.
Learning Outcomes
By the end of this learning sequence, students will demonstrate the ability to apply relational thinking by decomposing multi-digit numbers using place value understanding. They will use the area model as a conceptual bridge between concrete representations (arrays) and abstract symbolic forms (standard algorithm), demonstrating an understanding of how mathematical structures are connected across representations.
Students will also be able to justify the steps of the vertical multiplication algorithm by explicitly linking each step to partial products generated through area models. This ensures that procedural fluency is grounded in conceptual understanding rather than rote memorisation. In addition, students will demonstrate strategic competence by selecting efficient and appropriate multiplication strategies when solving real-world problems, particularly in the context of designing and calculating areas for a playground space.
Prior Knowledge and Informal Beginnings
This unit builds upon prior learning from Year 4 and Year 5 of the Australian Curriculum, ensuring that tasks are appropriately sequenced within students’ Zone of Proximal Development as described by Lev Vygotsky (1978). In Year 4 (AC9M4N06), students developed foundational skills in multiplication involving single-digit numbers and problem-solving without remainders. In Year 5 (AC9M5N06), students extended this understanding to include multiplication of larger numbers, including one- and two-digit factors.
The unit also draws upon students’ informal mathematical knowledge, including intuitive understandings of repeated groups, doubling and halving strategies, and early forms of partitioning. These informal strategies are used as a foundation for developing more formal mathematical representations, supporting a smooth transition from informal reasoning to structured mathematical thinking.
Pedagogical Framework
The design of this unit is grounded in Learning Trajectories, which position mathematical learning as a developmental pathway from informal to formal understanding. This ensures that students progress through increasingly sophisticated representations of multiplication in a coherent and structured manner.
The sequence is also informed by Jerome Bruner’s E–I–S model of representation, moving from enactive (hands-on manipulation of materials such as MAB blocks), to iconic (visual representations such as arrays and digital grids), and finally to symbolic (formal multiplication algorithms). This progression ensures that abstract mathematical ideas are first grounded in concrete and visual experiences.
In addition, the unit establishes socio-mathematical norms that value explanation, justification, and reasoning over correct answers alone. Students are encouraged to compare strategies, evaluate efficiency, and articulate their thinking using precise mathematical language. This aligns with the work of Paul Cobb (1994), who emphasises the importance of classroom norms in supporting mathematical discourse and reasoning.
Assessment and Differentiation
Assessment within this unit is embedded in a technology-enhanced learning environment that supports ongoing formative feedback. Students generate digital artefacts such as annotated arrays, colour-coded area models created in Mathigon Polypad, and recorded “design pitches” that explain their reasoning. These artefacts provide teachers with continuous insight into students’ developing understanding and allow for timely instructional adjustments.
Differentiation is achieved through carefully designed scaffolding and extension opportunities. For students requiring additional support, enabling prompts are provided in the form of structured grids, visual cues, and sentence stems such as “I partitioned the factor into…”. These reduce cognitive load while maintaining conceptual demand. For students requiring extension, tasks are designed to promote generalisation and abstraction, including multiplication involving three-digit numbers or decimal dimensions such as 12.5 × 14. These extensions encourage students to apply relational understanding to more complex and unfamiliar contexts.
Overall, the integration of assessment, differentiation, and technology ensures that all learners are supported to engage meaningfully with multiplicative thinking while progressing toward deep conceptual understanding and strategic mathematical competence.