If the standard way of multiplying feels cramped or confusing, partial products gives you room to see the math. You break each factor into place-value parts, multiply those smaller parts, and add the results at the end.
That extra writing helps because each step stays visible on the page. If you are learning this method yourself, or helping a child with homework, the process below will show you how it works and why it makes sense.
Why partial products makes multiplication easier to see
Partial products is a method built on one simple idea: large numbers are made of smaller place-value parts. When you write those parts separately, you can multiply them one at a time instead of trying to manage the whole problem at once.
Under the method is the distributive property. When you rewrite a number such as 21 as 20 + 1, and 14 as 10 + 4, you are showing the value of each digit. Then you multiply every part of one number by every part of the other number.
In partial products, every place-value part of one factor multiplies every place-value part of the other factor.
This method often helps when regrouping in the standard algorithm causes mistakes. You do not need to carry numbers across several steps right away. Instead, you write each smaller product clearly, then combine them at the end.
Expanded form shows what each digit means
Expanded form means writing a number as the sum of its place values. For example, 21 becomes 20 + 1. A three-digit number such as 155 becomes 100 + 50 + 5.
That matters because the digits in a multi-digit number do not all mean the same thing. A 5 in the ones place is different from a 5 in the tens place. When you separate the number into expanded form, you can see that difference without guessing.
You also reduce a common source of confusion. Instead of seeing 21 and 14 as two solid blocks, you see tens and ones. That shift makes the multiplication more concrete.
Place value comes before speed
Place value is the skill that holds this method together. You need to know whether a digit means ones, tens, or hundreds before you can multiply accurately with partial products.
For that reason, students who already understand the box method video for multiplication often move into partial products with less trouble. The thinking is nearly the same. You still break numbers apart by place value and multiply each part. The main difference is the layout on the page.
At first, the method may look longer than standard multiplication. However, the extra space is often the reason it works so well.
Solve 21 x 14 with partial products
The problem 21 x 14 is a good place to start because it shows the full pattern without too many pieces. You will use expanded form first, then multiply each part.
Break both numbers into expanded form
Write each factor in terms of tens and ones:
- 21 = 20 + 1
- 14 = 10 + 4
Now you have four smaller multiplication facts hidden inside the original problem. The 4 must multiply by both parts of 21, and the 10 must also multiply by both parts of 21.
Multiply each piece and add the results
Start with the ones in 14, which is 4.
- 4 x 1 = 4
- 4 x 20 = 80
Then move to the tens in 14, which is 10.
3. 10 x 1 = 10
4. 10 x 20 = 200
Now add the partial products:
4 + 80 + 10 + 200 = 294
So, 21 x 14 = 294.
This example also shows a shortcut many students use when multiplying by tens. For 4 x 20, you can multiply 4 x 2 = 8 and place the zero back to make 80. The same idea works for 10 x 20. You can multiply 1 x 2 = 2 and place both zeros back to get 200.
That shortcut works because 20 means 2 tens, and 10 means 1 ten. You are still using place value, even if the work looks shorter.
If you are teaching this method, keep returning to that point. The zero is not decoration. It shows the size of the number.
Solve 155 x 13 by keeping place values separate
A larger problem does not change the method. It only gives you more parts to multiply. Once you see that pattern, three-digit by two-digit multiplication becomes much less intimidating.
Set up the expanded form
Begin by breaking both numbers into place values.
| Number | Expanded form |
| 155 | 100 + 50 + 5 |
| 13 | 10 + 3 |
You do not add a hundreds part to 13 because there is no hundreds digit. That is why 13 stays as 10 + 3.
Now the 3 must multiply by 5, 50, and 100. After that, the 10 must also multiply by 5, 50, and 100. The structure is the same as the smaller example. You are still pairing every part with every part.
Work through all six products
Start with 3:
- 3 x 5 = 15
- 3 x 50 = 150
- 3 x 100 = 300
Then move to 10:
- 10 x 5 = 50
- 10 x 50 = 500
- 10 x 100 = 1000
Now add all six partial products:
15 + 150 + 300 + 50 + 500 + 1000 = 2,015
So, 155 x 13 = 2,015.
This is the point where some students notice that the method involves more writing. That is true, but the writing has a purpose. Each line keeps the place values visible, and that reduces errors.
For example, 10 x 50 = 500 may look simple, yet it is easy to lose a zero when you rush. Partial products slows the work enough for you to see that 1 ten times 5 tens gives 5 hundreds.
The same is true for 10 x 100 = 1000. You can use the shortcut of multiplying 1 x 1 and writing the three zeros back, but the place-value meaning still matters. One ten times one hundred equals one thousand.
Move from a full setup to a shorter written method
Once you understand the expanded version, you can shorten the layout. You do not need to rewrite every number in expanded form forever. Over time, you can keep the same thinking and write the partial products under the original problem.
List the partial products under 43 x 24
Take the example 43 x 24.
In expanded form, you have:
- 43 = 40 + 3
- 24 = 20 + 4
Now find the four partial products:
- 4 x 3 = 12
- 4 x 40 = 160
- 20 x 3 = 60
- 20 x 40 = 800
Then add them:
12 + 160 + 60 + 800 = 1,032
So, 43 x 24 = 1,032.
At this stage, you may no longer need to write the expanded form above the problem each time. You can look at 43 and know that it means 40 and 3. You can look at 24 and know that it means 20 and 4. The products can go straight underneath the original problem, one line at a time.
That shorter layout is useful because it keeps the method efficient without dropping the place-value thinking that makes it work.
Practice until the pattern feels natural
At first, this method takes more writing. After enough practice, the writing turns into a pattern you recognize quickly.
The goal is not speed on the first attempt. The goal is accurate thinking. Once the pattern becomes familiar, you can move between the full expanded setup and the shorter list of products with much more confidence.
This is also why partial products is a helpful bridge for students who struggle with the standard algorithm. You still multiply the same numbers, but you do it in a way that keeps each place value visible.
If you want extra help beyond worked examples, Beth at MyTutoringBee offers one-to-one online tutoring for math students.
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If you are comparing methods, it is worth reviewing both partial products and the box method. The visual structure in one often strengthens your understanding of the other.
A clearer path to multi-digit multiplication
When multiplication feels messy, partial products gives you a cleaner way to organize the work. You separate the number by place value, multiply each smaller part, and add the results when you are done.
That structure helps you see why the answer makes sense. With practice, you do not only get the product, you understand where it comes from.
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