Array Models and Area: Concrete Ways to Teach Multiplication and Division
If you've ever taught multiplication, you've probably seen this happen.
A student confidently recalls their times tables but struggles to explain why 7 × 8 equals 56. Ask them to solve 56 ÷ 8 and they hesitate. Introduce the area of a rectangle and suddenly the multiplication facts they seemed to know no longer feel familiar.
These challenges often have the same underlying cause.
Students have learned multiplication as a collection of facts to remember rather than as a mathematical relationship to understand.
When educators look for ways to teach multiplication with arrays, they are often searching for more than another classroom activity. They want an approach that helps students build lasting understanding - one that naturally connects multiplication, division and area.
Arrays provide exactly that foundation.
Seeing the Structure Behind Multiplication
Multiplication is much more than finding the answer to a number sentence.
It describes equal groups, organised rows and columns, patterns and relationships.
When students only memorise multiplication facts, they often rely on recall alone. If they forget a fact, they have very little to support their thinking.
Arrays change this.
Instead of seeing 4 × 6 as simply "24", students see four rows of six or six columns of four.
The answer is no longer an isolated fact.
It is something they can visualise.
This simple shift helps students recognise that multiplication is organised, predictable and connected.
Why Arrays Matter
When we teach multiplication with arrays, we give students an opportunity to build understanding before expecting fluency.
Arrays help students:
- recognise equal groups organised into rows and columns
- understand that multiplication can be represented visually
- discover the commutative property naturally
- connect multiplication and division through fact families
- prepare for learning area, fractions and algebra.
Rather than memorising individual facts, students begin recognising patterns across the multiplication table.
Those patterns become the foundation for future mathematical thinking.
Building Understanding Through the I-CRAVE Maths® Methodology
Within the I-CRAVE Maths® Methodology, arrays fit naturally within the progression from concrete experiences to abstract reasoning.
Identify
Students first identify what the mathematics is describing.
Are we finding the total number of objects?
How many rows are there?
How many objects are in each row?
Beginning with these questions encourages students to think mathematically before calculating.
Concrete
Students physically build arrays using manipulatives such as Integer Blocks, counters or connecting cubes.
To represent 3 × 5, students build three rows containing five objects.
They move the materials.
They count them.
They compare different arrangements.
The mathematics becomes something they can experience rather than simply imagine.
Representation
Once students can confidently build an array, they begin drawing it.
The concrete model becomes a visual representation.
Students quickly notice that turning an array changes its orientation but not the total number of objects.
Three rows of five and five rows of three both contain fifteen.
Without needing formal definitions, students begin understanding the commutative property.
They also begin recognising multiplication and division as connected ideas rather than separate topics.
Abstract
Only after students understand the array do they record it symbolically.
They connect:
Three rows of five
↓
Array of 15
↓
3 × 5 = 15
↓
15 ÷ 5 = 3
↓
15 ÷ 3 = 5
Students begin recognising these as a connected fact family rather than separate equations to memorise.
Verbal
Students explain what they notice.
"I have four rows with six in each row."
"There are twenty-four altogether."
"If I know four groups of six, I also know twenty-four divided into four groups gives six."
These conversations strengthen mathematical language while giving teachers valuable insight into student thinking.
Explicit
Throughout the lesson, instruction remains clear, structured and carefully sequenced.
The teacher models each step.
Students practise with support.
Responsibility is gradually transferred as understanding develops.
This reduces misconceptions while building confidence.
Arrays Create a Natural Bridge to Area
One of the greatest strengths of arrays is that they prepare students for much more than multiplication.
They prepare students to understand area.
When students later investigate a rectangle measuring 6 centimetres by 8 centimetres, they don't need to memorise a new formula.
They already recognise the rectangle as an array.
Six rows.
Eight columns.
Forty-eight equal square units.
The formula for area now has meaning because students understand where it comes from.
Area is no longer a separate topic.
It becomes another application of multiplication.
This understanding also supports later learning in fractions, algebra and proportional reasoning because students continue recognising mathematical structure across different concepts.
What This Looks Like in Practice
Imagine introducing the 7 times tables.
Rather than beginning with flashcards or chanting, students first build arrays using counters or Integer Blocks.
They create seven rows of four.
Then four rows of seven.
They compare the arrangements.
They draw the arrays.
They record the multiplication and related division facts.
Next, they cover part of the array and predict how many counters are hidden.
They notice patterns across neighbouring multiplication facts.
They reason that if 5 × 7 equals 35, then adding two more groups of seven gives 49.
Students are no longer relying solely on memory.
They are reasoning mathematically.
This same sequence works equally well in homeschool settings, where children often benefit from spending additional time building and discussing concepts before moving to symbolic notation.
Why This Approach Works
Research across mathematics education and cognitive science consistently supports building conceptual understanding before expecting procedural fluency.
Cognitive Load Theory reminds us that working memory has limited capacity. Arrays reduce unnecessary cognitive load because students can see mathematical relationships instead of trying to hold them entirely in memory.
Explicit Instruction supports this process through careful modelling, guided practice and a logical sequence that gradually develops independence.
The Science of Learning also tells us that durable learning occurs when students connect concrete materials, visual representations, mathematical language and symbolic notation.
Within a Response to Intervention (RTI) framework, arrays are particularly valuable for students experiencing difficulty with multiplication and division.
Rather than asking students to memorise more facts, teachers can revisit the underlying structures that give those facts meaning.
Once understanding is secure, purposeful retrieval practice helps students develop fluent recall.
What Changes for Teachers
Teaching with arrays changes more than student achievement.
It changes how teachers observe learning.
Misconceptions become visible.
Assessment becomes more meaningful.
Planning becomes more purposeful.
Instead of wondering whether students have memorised a multiplication fact, teachers can see whether students genuinely understand the relationship behind it.
For classroom teachers, intervention specialists and homeschool educators, arrays provide a consistent way to build strong mathematical foundations while reducing the guesswork that often accompanies intervention.
Most importantly, students become more confident because the mathematics is visible.
Helping Students See the Mathematics
Multiplication is not simply about recalling facts.
It is about recognising patterns, relationships and mathematical structure.
When we teach multiplication with arrays, students begin seeing mathematics instead of simply trying to remember it.
That understanding extends well beyond multiplication.
It strengthens division.
It gives meaning to area.
It supports future learning in fractions, algebra and proportional reasoning.
Most importantly, it helps students understand why mathematics works.
When students can see the mathematics, they no longer rely on memorisation alone.
They reason.
They make connections.
They solve problems with confidence.
That is the kind of mathematical understanding that lasts.
Maths Australia provides practical, research-informed training that shows educators exactly how to teach multiplication with arrays and other evidence-based strategies using the I-CRAVE Maths® Methodology.
If you're working with students who are disengaged, stuck or missing foundational concepts, explore our educator training and accreditation pathways.
Learn more at: https://mathsaustralia.com.au/training