Educators all over the world face a common challenge: making multiplication accessible and engaging for every type of learner, especially those who think visually. 

Traditional methods of teaching multiplication can seem abstract to visual thinkers and disconnected from real-world applications. Incorporating visual thinking strategies into the teaching process, however, can take multiplication from a rote memory exercise and turn it into an intuitive, visual experience that makes sense for every type of learner.  

Multiplication is often a stumbling block for many students–including visual thinkers–but using imagery, spatial relationships, and hands-on exploration can help make multiplication’s underlying concepts more easily understood and relatable. 

The following strategies and exercises demonstrate just how impactful it can be to incorporate a visual-thinking approach into your lesson planning. 

Why Visual Thinking Is Effective for Learning Multiplication

Visual thinking isn’t just about prettying up numbers—it’s a fundamental way of processing information that can change how students understand mathematics. 

For visual thinkers, the ability to “see” a problem is often the first step to solving it. This approach taps into the brain’s capacity for visual processing, turning conceptual numerical relationships into concrete, memorable images.

When applied to multiplication, visual thinking strategies offer several distinct advantages:

1. Concept Visualization: Visual thinking strategies allow students to “see” what multiplication means so they can move beyond the stale routine of mechanical drill and practice.
2. Pattern Recognition: Visual representations help students identify and understand multiplication patterns more easily.
3. Memory Enhancement: Visual cues create stronger memory hooks, improving recall of multiplication facts.
4. Problem-Solving Skills: Visual approaches often reveal multiple ways to solve a problem, which expands a student’s capacity for critical thinking.

By integrating these strategies, we’re not just teaching multiplication—we’re developing a visual language for mathematics that can benefit all students throughout their academic journey.

Teaching Single-Digit Multiplication to Visual Thinkers

For visual thinkers, conquering single-digit multiplication is all about creating clear, engaging visual models. Here are some effective techniques:

1) Array Adventures

Use colorful digital tools or objects like Demme Learning’s Integer Block Kit to create arrays. For instance,  represent 4 x 3 as three rows of four items each. This visually demonstrates how the first factor (4) represents what is being counted, and the second factor (3) shows how many times it’s being counted.

2) Multiplication Murals

Encourage students to create “multiplication murals” where they illustrate each math fact with a unique picture. For example, 5 x 2 could be represented as two groups of five stars or two hands with five fingers.

3) Number Line Hops

Visualize multiplication on a number line, showing how 4 x 3 is three hops of four spaces each. This reinforces the concept of the first factor (4) being what we’re counting (the distance of each hop) and the second factor (3) being how many times we’re counting it (the number of hops).

4) Pattern Blocks

Use pattern blocks to represent multiplication facts by leveraging the unique properties of different shapes. For example:

To show 4 x 3 using squares:

• Arrange 3 blocks in a row.
• Each square has 4 sides, so this visually represents 4 + 4 + 4 (three groups of four).
• The total of 12 blocks demonstrates the product of 4 x 3.

To illustrate the commutative property:

• Use 3 squares to represent 4 x 3 (as above)
• Then, use  4 triangles to represent 3 x 4 (each triangle has 3 sides, so 4 triangles represent 3 + 3 + 3 + 3 (four groups of three).
• Both arrangements result in a total of 12 sides, showing 4 x 3 = 3 x 4.

These hands-on activities allow students to physically manipulate the blocks. They also reinforce two key concepts: that multiplication is repeated addition and that the order of factors doesn’t change the product (the commutative property).

The goal is to help students visualize multiplication as a concept, not just as numbers to be memorized. These visual thinking strategies lay a strong foundation for understanding more complex mathematical ideas down the road.

Teaching Multiple-Digit Multiplication to Visual Thinkers

As we venture into multiple-digit multiplication, visual representation becomes even more important. Here’s how to break down these complex problems using visual aids:

1) Grid Power

Introduce an area model using grid paper. For 23 x 14, create a grid with 14 rows and 23 columns, then divide it into manageable sections (10 x 20, 10 x 3, 4 x 20, 4 x 3). This visually demonstrates the distributive property of multiplication while maintaining the concept that the first factor (23) represents what is being counted (the number of columns in each row), and the second factor (14) represents how many times it’s being counted (the number of rows).

Here’s how to break it down:

• 10 rows of 20: 10 x 20 = 200
• 10 rows of 3: 10 x 3 = 30
• 4 rows of 20: 4 x 20 = 80
• 4 rows of 3: 4 x 3 = 12

Total: 200 + 30 + 80 + 12 = 322

This method allows students to visualize how 14 groups of 23 can be broken down into more manageable parts while still maintaining the original multiplication concept.

2) Place Value Palettes

Use Demme Learning’s Integer Block Kit to visually represent multiplication of two-digit numbers. For example, when multiplying 32 x 21:

• Use red blocks for hundreds (100-blocks)
• Use blue blocks for tens (10-blocks)
• Use orange blocks for units (1-blocks)

Build the problem as follows:

• 30 x 20 = 600 (represented by 6 red 100-blocks)
• 30 x 1 = 30 (represented by 3 blue 10-blocks)
• 2 x 20 = 40 (represented by 4 blue 10-blocks)
• 2 x 1 = 2 (represented by 1 orange 2-block)

This method visually breaks the multiplication into partial products:

• The red rectangle (600) shows the product of the tens digits
• The blue rectangles (30 and 40) show the cross-products of tens and units
• The orange rectangle (2) shows the product of the ones digits

The final sum of 672 is clearly represented by the total blocks: 6 hundreds, 7 tens, and 2 units.

This approach reinforces place value concepts and the distributive property of multiplication and allows students to see how each part of the multiplication problem contributes to the solution.

3) Expanded Form Exploration

1. Show how 46 x 23 can be visualized as (40 + 6) x (20 + 3), creating a visual “box” method that breaks the problem into smaller, manageable parts.

For 46 x 23, expand the numbers (46 = 40 + 6, 23 = 20 + 3) and create a box made up of two columns and two rows. Label the rows with 20 and 3 respectively, and the columns with 40 and 6, respectively. Multiply these numbers to fill each cell (40 x 20 = 800, 40 x 3 = 120, 6 x 20 = 120, 6 x 3 = 18).

Problem: 46 x 23	40	6
20	800	120
3	120	18

The sum of these products (800 + 120 + 120 + 18) gives the final result (1058).

4) Digital Sketch Pads

Utilize digital drawing tools to create dynamic, color-coded representations of multiple-digit multiplication problems, allowing for easy manipulation and correction.

When designing lesson exercises, focus on problems that lend themselves well to these visual methods. Encourage students to explain their visual process and demonstrate their deeper understanding of the concepts at play.

Integrating Visual Thinking Strategies and Manipulatives into Your Lesson Plan

Effectively incorporating visual thinking strategies into your multiplication lessons can be done with a variety of approaches:

1) Visual Vocabulary

Start each lesson by building a “visual vocabulary” for the concept at hand. Here are some examples to get you started:

• Arrays: Serves as visual representations of objects arranged in rows and columns.
• Groups: Identifies sets of items clustered together.
• Over: A dimensional term that indicates horizontal arrangement or length in multiplication. For example, in 4 x 3, “over” refers to the horizontal dimension of the resulting rectangle, which is 4 units long. It represents how many items in each group or how far the array extends horizontally.
• Repeated addition: Demonstrates how multiplication is a shortcut for adding the same number multiple times.
• Scaling: Demonstrates how one quantity increases in proportion to another.
• Up: A dimensional term that indicates vertical arrangement or height in multiplication. In 4 x 3, “up” refers to the vertical dimension of the resulting rectangle, which is 3 units high. It represents how many groups or how tall the array is vertically.

2) Manipulative Stations

Set up stations with different manipulatives (integer blocks at one station, beads at another, grid paper at another, and so on) and challenge students to represent the same multiplication problem using various methods.

3) Digital Integration

Use digital whiteboards or tablets to create dynamic visual models that students can manipulate together in real time. Our Virtual Manipulatives are a great way to experience multiplication visually in an online format. 

4) Art Integration

Collaborate with other teachers or parents to create cross-curricular projects that represent multiplication concepts through various artistic mediums. Drawing representations of the physical block models is an important intermediate step in moving from physical models to the abstract problem and is part of the concrete-representational-abstract process.

5) Drawing Representations

Incorporate drawing as a key component of the learning process. One way to do this is to set up drawing stations where students can sketch their understanding of multiplication problems. This could include drawing arrays, groups, or number lines to visualize multiplication facts.

6) Peer Teaching

Encourage visual thinkers to create their own visual explanations of multiplication concepts so they can share them with classmates.

When assessing understanding and math mastery, look for students’ ability to create and explain visual representations of multiplication problems. Can they translate between visual models and numerical equations? Use this feedback to refine your approach and provide targeted support.

Empowering Visual Thinkers for Mathematical Success

By embracing visual thinking strategies in multiplication instruction, we’re not just teaching math—we’re cultivating a way of thinking that can benefit students across learning preferences. These methods develop spatial reasoning, enhance problem-solving skills, and help create a deeper, more intuitive understanding of mathematical relationships.

Every visual thinker’s mind works uniquely. Offering a diverse toolkit of visual approaches to multiplication can ensure each student finds a method that resonates with their individual way of processing information.

Whether you’re an educator looking to liven up your math lessons or a parent seeking to support your visual thinker at home, visual-thinking strategies offer a path to take multiplication from a dreaded chore and turn it into an engaging adventure. 

So, grab those manipulatives, fire up those digital tools, and get ready to watch your visual thinkers multiply their mathematical potential!

Unlock Your Child’s Visual Math Potential with Our Tools and Resources!

For an inspiring look at the power of visual thinking, check out this insightful episode of The Demme Learning Show, where we chat with Temple Grandin, one of the most famous visual thinkers of our time. For more helpful multiplication resources, check out The Guild for free access to activity sheets, webinars, sample lessons, and content