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Learning how to teach multiplication is easier when you help your child understand what multiplication means before expecting them to memorize facts. Start with concrete objects, move to pictures and arrays, and introduce abstract notation only after they understand the concept. The same progression can continue as multiplication becomes more complex.
Read on for an in-depth look at the best, evidence-based practice for helping your child master multiplication in a way that sticks.
Multiplication can feel like a big leap. Your child has spent years learning to count, add, and subtract, and suddenly you are asking her to learn a multiplication table filled with dozens of new facts.
Most math programs place a heavy emphasis on memorizing facts without providing a solid foundation for understanding why number relationships work the way they do.
No wonder multiplication can become a stumbling block.
If your student has memorized 3 x 4 as 12 but doesn't understand that the math fact represents 3 groups of 4 objects that combine to equal one group of 12, she may be able to answer, "What is 3 x 4?" but she may struggle to use that math in more advanced problem solving. She may be able to answer, "what is 3 x 4," but may lack the foundational concept. Without understanding the concept, she won't know how to solve a story problem such as:
There are 12 children and 3 tables; divide them equally among the tables. How many children will be at each table?
The foundation of multiplication is not the times table. It is a child's ability to make sense of numbers: to see quantities, compose and decompose them, recognize equal groups, and reason about how those groups relate. Those early understandings need to be mastered so that multiplication becomes the next logical step in the progression toward proficiency rather than feeling like a whole new universe. When children understand the connections, multiplication starts making mathematical sense.
All About® Math uses a Concrete → Representational → Abstract progression (CRA). Your child first experiences a new concept with physical objects he can touch and move, followed by a visual representation of that concept. For example, using an array to lay out 4 rows of 3 dots, representing 4 groups of 3 or 4 x 3. The final step is an abstract version of the concept – writing out 4 x 3.
The Concrete-Representational-Abstract approach is an evidence-based framework for math instruction. A recent meta-analysis of 30 studies found a statistically significant positive effect on students' math achievement with CRA-based instruction.1
This sequence helps your student connect what mathematical concepts mean to the procedures and symbols he eventually uses to solve problems.
Let's begin at the beginning.
Before asking your child to solve 3 × 4 = 12 on paper or defaulting to flash cards, we build the equal groups:
Give her a dozen counters, snap cubes, counting bears, base-ten units, buttons, coins, or other small objects. Ask her to make three groups with four objects in each group. Make it simple by drawing three boxes or circles for her to put each group of objects into.
As you work, talk about what your child is doing:
Continue to reinforce this concept with more examples, and occasionally throw in an odd number of objects, so they don't make equal groups. For example, give her 10 cubes and 3 boxes, with instructions to place the same number of cubes in each box.
She will end up with 2 groups of 3 and one of four. Now you can discuss why this does not make equal groups, which helps reinforce the difference between equal and unequal groups. Follow up by asking her what she can do to make equal groups. She might remove one cube from the final box or grab two more cubes and put 4 in each box.
This combination of seeing, touching, moving, hearing, and talking about mathematics is one reason a multisensory approach, using Orton-Gillingham principles, is so effective. All About Math incorporates the applicable principles of Orton-Gillingham instruction, including multisensory, sequential, incremental, cumulative, individualized, and explicit teaching.
Your child's first encounter with multiplication isn't a worksheet. It's a hands-on representation of the concept they can create themselves.
Arrays help students move from concrete to abstract thinking. They begin by building arrays with physical objects, then represent those arrays with drawings, and eventually use numbers and symbols to solve the same multiplication problem.
This explicit, step-by-step progression helps the child transition from concrete objects to visual representations. By relying on their drawing rather than physical objects, they learn to hold mathematical relationships, such as 3 groups of 4 objects, in their short-term memory.
This is an important intermediate step most math programs ignore, but it is critical to helping a child develop the mental muscles needed to work with abstract equations. The drawing acts as a scaffolding to support the concrete understanding of equal groups. As mastery of abstract concepts grows, eventually the scaffolding is reduced and finally removed entirely.
An array is a way of arranging objects into equal rows and columns.
Start by using the example problem above and building the problem using two-color counters in an array.
For example, you could arrange 12 counters into:
3 rows of 4
Now he can see 3 × 4.
He can also turn the array and see:
4 rows of 3
The quantity hasn't changed. There are still 12 objects.
After he is comfortable with building arrays, have him draw a representation of the array. In the abstract stage, he will have an improved understanding that the concrete model of equal groups is the same as the abstract written problem, thanks to the bridge created by drawing a representation of the array.
The goal is to help him see the pattern created and visualize multiplication as repeated addition, or "skip counting". One row of 3, plus another row equals 6, plus a third row is 9, plus a fourth row is 12.
See for yourself how All About Math uses arrays in lesson 19 of the sample teacher's manual. Students learn to create arrays, draw a representation of an array, and then write equations to solve the equation. Later, they connect multiplication and division to groups, drawings, and related equations.
Skip counting helps your child notice another multiplication pattern. As she becomes comfortable using arrays to represent equal groups, she can begin using more efficient strategies to determine the total. Rather than counting each object in an array individually, your student can count by the number of objects in each row or group. This introduces skip counting as a natural extension of her understanding of equal groups and prepares her to use multiplication more efficiently.
Count by 2s:
2, 4, 6, 8, 10...
Then by 5s:
5, 10, 15, 20, 25...
And by 10s:
10, 20, 30, 40, 50...
Now connect the pattern back to equal groups.
If your child has four groups of five counters, she can count each counter individually. But she can also count the groups:
5, 10, 15, 20.
Four groups, with five in each group, total 20.
So, 4 × 5 = 20.
The goal isn't simply to memorize a skip-counting chant. Connect the words to quantities your child can see. That helps her understand why the pattern works.
As your child builds and draws multiplication problems, connect what he sees to mathematical notation. Three groups of four counters can also be written as:
3 × 4 = 12
The numbers and symbols are abstract, but they now represent a relationship your child already understands. Instead of asking him to memorize what 3 × 4 means, you have given meaning to the symbols.
By taking time to teach each step of CRA separately and asking him to demonstrate his understanding, you are building mastery of concepts that provides a solid, stress-free path to math proficiency.
Once your child understands what multiplication represents, mathematical properties can make learning the facts much less intimidating.
The commutative property of multiplication tells us that changing the order of the factors doesn't change the product:
5 × 7 = 7 × 5
Both equal 35
An array makes this wonderfully obvious. Build five rows of seven counters, then turn the array. You now have seven rows of five without changing the total number of counters.
Teaching the commutative property means your child doesn't have to treat 5 × 7 and 7 × 5 as two unrelated facts, and it cuts the work of learning all the facts in half.
The identity property of multiplication tells us that any number multiplied by 1 remains the same:
1 × 8 = 8
One group of eight still contains eight objects.
The zero property of multiplication tells us that any number multiplied by zero equals zero:
6 × 0 = 0
Six groups containing zero objects still contain zero objects altogether.
This is where multiplication starts becoming proficiency.
Suppose your child doesn't remember:
6 × 7 = ?
But she does know:
6 × 6 = 36
Instead of treating 6 × 7 as an entirely new fact, build on what she knows.
Seven groups of six are one more group than six groups of six:
6 × 6 = 36
36 + 6 = 42
Therefore:
6 × 7 = 42
Now we can see exactly why mastering equal groups and arrays is an important early step. Once those concepts are mastered, using known facts to find an unknown fact becomes second nature, not an exercise in frustration. Your child isn't helpless if an answer doesn't immediately pop into her head. She can use number relationships to find it.
You can encourage this thinking with simple questions:
Eventually, many multiplication facts will become automatic. But automaticity built on understanding gives your child a backup plan when memory fails.
You may have been taught the classic 9s finger trick that works for 9 x 1 through 9 x10:
Hold up all 10 fingers, palms facing you, and number them mentally from 1 to 10, left to right.
For 9 × 7:
So 9 × 7 = 63.
Another example: for 9 × 5, bend the 5th finger. You have 4 fingers on the left and 5 on the right, or 45
It's a fun trick, and some children enjoy discovering why it works. But treat it as a calculation shortcut, not as your child's introduction to multiplication or even a way to solve unfamiliar problems.
Striving for math fluency and working to learn challenging concepts rather than relying on tricks will serve him best in the long run when math proficiency for higher-level mathematics is needed.
A trick can help your child find 9 x 7. It doesn't automatically teach what multiplication means, why the answer is 63, or how multiplication connects to other mathematical ideas.
Build the understanding first. Then tricks can be fun tools to add to the toolbox.
Eventually, multiplication grows beyond facts such as 6 × 7.
Suppose your child encounters:
23 × 14
This can look dramatically different from the multiplication he already know. But the underlying concept hasn't changed.
This is where place-value understanding and visual models become especially valuable.
First, use base ten cubes and make sure he sees:
23 = 20 + 3
and
14 = 10 + 4
It would take a long time to make models of 23 groups of 14, but it would also be too abstract to jump right into the traditional "stacked" problem you may be used to seeing. This model helps you show your student how to break each number down by place value and make sure they multiply each part of 14 by each part of 23.
This helps avoid common problems like skipping steps when multiplying. Here's what to do:
Now, have your student use his fingers to trace down from 20 and across from 10 until they meet in the empty box. Have him fill in that box with the product of 10 x 20. Then do the same with each of the remaining numbers to fill in the other boxes.
Now your child can see the four partial products:
Add them:
200 + 30 + 80 + 12 = 322
Therefore:
23 × 14 = 322
This visual model makes the distributive property visible. Your child isn't just following instructions about which digits to multiply. He can see why all four partial products are necessary.
Once your child understands the place-value relationships underneath the calculation, and is proficient at the area model for multidigit multiplication, you can introduce the standard algorithm for multiplication as a more efficient way of recording the same mathematical work.
The standard algorithm isn't a mysterious new form of multiplication. It is the traditional "stacked" calculation you learned in school.
It's a compact notation for mathematical relationships he already understands because you taught the foundational concepts first. That's an important principle when learning how to teach standard algorithm multiplication. Don't make the standard way of writing a problem carry the entire burden of teaching the concept.
If your child forgets a step in the algorithm but understands place value and partial products, he has something to reason from. If he has memorized only the steps, forgetting one can bring the entire process to a halt.
The same Concrete → Representational → Abstract progression that helped your child understand 3 × 4 can therefore help when multiplication becomes more complex:
build the quantities → represent the relationships visually → use abstract notation efficiently
The same concrete-to-abstract progression applies when your child begins multiplying fractions. For example, fraction tiles clearly show that three groups of 1/4 equal 3/4 before your child writes:
3 × 1/4 = 3/4
Area models can also help her visualize what it means to multiply two fractions before learning the procedure of multiplying the numerators and denominators. The goal remains the same: help her understand what the numbers represent before relying on an abstract rule.
When multiplication expands to decimals, a strong understanding of place value becomes especially important. The base-10 blocks, used previously for showing tens and ones, can be used to show decimals of tenths and hundredths. Visual models can help your child see the quantities involved, while estimation helps her predict whether an answer is reasonable.
For example, before solving 2.5 × 1.2, she can estimate that the answer should be somewhere around 3. After multiplying, she can check decimal placement by counting the total decimal places across both factors and using that many in the product. If she calculates 300 or 0.3, she has a reason to stop and reconsider her work rather than simply trusting the procedure.
As multiplication becomes more advanced, continue pairing efficient procedures with an understanding of why they work.
Once your child understands the multiplication concepts you are teaching, regular practice helps those skills become easier and more automatic.
Short games used weekly will help strengthen skills without the stress of worksheets.
Games provide repeated exposure to facts without making practice feel boring or frustrating. They can also give your child opportunities to recognize patterns and apply strategies.
You can make practice fun with games using dice, cards, counters, dominoes, and other materials you probably already own.
Periodic cumulative reviews are valuable, but games add an element of fun. The emotional connection between fun and math generates dopamine in the brain that eventually connects "doing math" with something pleasurable.
Teach understanding first. Use games to reinforce fluency.
Roll two dice and multiply the numbers. Or turn over two playing cards and find their product.
You can adjust the game to match his current lessons instead of practicing everything at once.
This is the secret to turning abstract math into a useful tool! A common complaint from students is, "When will I ever use this?" By pointing out where multiplication is used in everyday life, you connect the numbers on the page with real-world applications that matter.
These questions help your child connect multiplication facts to real-life situations.
Children are commonly introduced to multiplication around second or third grade, typically between ages 7 and 9. But age isn't the best way to decide when your child is ready.
A better question is:
Does my child have the mathematical foundation needed to understand multiplication?
Before beginning formal multiplication instruction, look for readiness skills such as:
A child with these foundations is prepared to understand what multiplication represents.
A child who doesn't may find multiplication facts confusing even if she has reached the "right" age according to a grade-level chart.
This is one advantage of mastery-based learning. You don't have to push forward because the calendar says your 9-year-old should be working on multiplication. You can strengthen the foundation first and introduce the next concept when he is ready.
All About Math levels are based on skill mastery rather than grade labels. If you're unsure where your child should begin, you can find the right starting level with the free placement tests.
All About Math teaches multiplication as part of a larger, carefully sequenced understanding of numbers and operations.
Your child doesn't begin by staring at a multiplication table and trying to lodge dozens of unrelated facts into memory. He works with mathematical relationships through hands-on materials, visual representations, discussion, practice, and eventually abstract notation.
Throughout All About Math, your child learns to:
As students advance, the same foundation supports increasingly sophisticated mathematics. All About Math Level 5 includes multiplication and division of fractions and decimals, along with work involving scaling, the distributive property, and the associative property.
The complexity changes. The basic teaching principle doesn't have to.
When a new idea becomes too abstract, give it something your child can see, touch, draw, discuss, or connect to what he already knows.
If you're looking for a multisensory math curriculum that teaches multiplication as part of this step-by-step progression, explore All About Math and see how hands-on instruction, visual models, games, and mastery-based lessons work together.
Begin with the meaning of multiplication rather than memorization. Let your child build equal groups with manipulatives, connect multiplication to repeated addition, represent facts with arrays, and use known facts to derive new ones. Once she understands the relationships, games and regular practice can help facts become automatic.
Think understanding first, automaticity second.
There isn't one correct timeline. Some children develop automatic recall quickly, while others need much more time and repeated exposure.
Instead of measuring progress by how many weeks your child has spent on multiplication, look at what she can do. Does she understand equal groups? Can she represent multiplication with an array? Does she recognize patterns and related facts? Can she use a strategy when she doesn't immediately remember an answer?
Use the All About Math progress chart to track and celebrate each intermediate step your child masters on the way to conquer multiplication! Each small win, when acknowledged and celebrated, helps her buildconfidence to tackle the next step.
Mastery is more useful than completing a curriculum in one school year.
Go back to the connection between the two.
Build three groups of four objects and write: 4 + 4 + 4 = 12.
Then write: 3 × 4 = 12.
Ask your child to explain how the two expressions are related.
Next, build or draw an array showing three rows of four. Connect that representation to the same equation.
Your child may not need more multiplication drills. She may need the missing conceptual bridge between addition she already understands and multiplication that still feels abstract.
If your child continues to struggle, step back and determine which prerequisite skills are solid before moving forward. The free All About Math placement tests can help you find the right starting level based on your child's current skills rather than age or grade.
Learning multiplication is a journey, not a single unit to conquer. Your child may begin by physically placing four counters into three groups. Eventually, he will recognize 3 × 4 = 12 without needing the counters at all. Later, that same understanding of multiplication will expand into larger numbers, fractions, decimals, area, ratios, algebra, and countless real-world applications.
The path from those first counters to abstract computation becomes much easier when each new step has something solid underneath it.
That's how multiplication becomes more than a table of facts your child once memorized. It becomes a mathematical tool he understands well enough to apply it in new situations with fluency and proficiency.
1Ebner, S., MacDonald, M. K., Grekov, P., & Aspiranti, K. B. (2025). A meta-analytic review of the concrete-representational-abstract math approach. Learning Disabilities Research & Practice, 40(1). https://doi.org/10.1177/09388982241292299
