How To Teach The Standard Algorithm for Multiplication So All Your Students ‘Get It’
The standard algorithm for multiplication can be very difficult to teach in 5th and 6th grade, as anyone who has taught upper elementary school before will know.
Despite best intentions, there will always be a few students who are either unsure of the methods they used for multi-digit multiplication before or who are not secure on their multiplication facts.
If this academic year will be your first time teaching the standard multiplication algorithm, you have all of this to look forward to, but don’t despair – it happens every year.
- What is the standard algorithm for multiplication?
- The standard algorithm in the curriculum
- What is the standard algorithm?
- The standard algorithm step by step
- How to do the standard algorithm step by step
- How cognitive science has affected my teaching of the standard algorithm
- How to teach the standard algorithm
- The standard algorithm method: Lesson 1
- Conclusion of your first standard algorithm lesson!
- The standard algorithm examples
- Standard algorithm questions
What is the standard algorithm for multiplication?
Before you learn the standard algorithm for multiplication, you typically understand multiplying multi-digit numbers in parts and adding the partial products at the end. In the standard algorithm, it is more of a shorthand way of multiplying in parts, while doing some of the addition at the same time as multiplication. It is often referred to as multi-digit multiplication and is the recommended method to use when multiplying larger numbers.
Before tackling the standard algorithm, students should ideally be confident with their multiplication facts and understand key terms like the multiplicand and the multiplier.
- The multiplicand is the number you are starting with for the multiplication
- The multiplier is how many groups of these you need; how many times you’re going to multiply the multiplicand by.
The standard algorithm in the curriculum
4th Grade students may be introduced to multi-digit multiplication by multiplying 3 digit numbers by a 1-digit number and multiplying two 2-digit numbers together. They will do this through methods such as area models and partial products. As students’ abilities develop, 5th and 6th Grade students will multiply multi-digit whole numbers using the standard algorithm.
It is crucial that students become fluent in the method to enable them to access more challenging math as they progress to middle school and beyond. In middle school, students will build on this skill and multiply multi-digit decimals. When I say fluent this is what I mean:
‘Fluency is the process of retrieving information from long-term memory with no effort on our working memory, freeing up valuable space in our working memory to give attention to other things.’
Read more: Fluency, Reasoning and Problem Solving
What is the standard algorithm?
The formal standard algorithm is a step by step method of supporting children to understand conceptually and practically how to multiply multi-digit numbers, and is typically introduced when students begin to multiply large numbers.
The standard algorithm step by step
Here is the standard algorithm broken down step by step:
How to do the standard algorithm step by step
Example: 124 x 26
- Rewrite the question vertically
- Remember to start the process of multiplication on the right in the ones place
- Multiply 6 by 4
- Write the answer down correctly – including any carrying
- Multiply 6 by 2
- Add anything that you have carried from the previous multiplication.
- Multiply 6 by 1
- Add anything that you have carried from the previous multiplication and write the answer down correctly
- Start the next partial product line by placing a 0 in the ones place since we are now multiplying in the tens place
- Multiply 2 by 4
- Write the answer correctly
- Multiply 2 by 2
- Write the answer down
- Multiply 2 by 1
- Write the answer correctly
- Add the two answers up together correctly
That’s a total of 16 steps that children need to become fluent in when learning this new process to get to the final answer. Bearing in mind the limits of our working memory, this is a lot to take on and can quite easily overwhelm and prevent this information from being encoded.
So the answer to the question: How to do the standard algorithm? is to simply follow the steps!
But this is missing a crucial stage of learning – moving from the procedural to the conceptual understanding of what’s going on.
The rest of this article explains how to teach the standard algorithm to have the biggest impact for your class. It includes links to the standard algorithm worksheets to provide you with lots of practice.
How cognitive science has affected my teaching of the standard algorithm
Two lessons from cognitive science have massively changed the way I approach teaching the standard algorithm.
1. Long and short term memory
The first has been understanding that we have a long-term memory that is near limitless in the information that it can store; and working memory, where we make our conscious thoughts.
Important to note is that the space in our working memory is limited, many researchers put it at between 4 or 7 items. Oliver Caviglioli has graciously sketched a wonderful poster that shows this process.
From the model we can see that the person uses their attention to take things from the environment into the working memory. We then attempt to encode this information into our long-term memory, but some information may be forgotten for a myriad of reasons.
When that information is in our long-term memory we can bring it back to the forefront of our working memory to use it. If those memories remain dormant for too long however (that is we don’t recall those memories for a long period of time) they too can be forgotten.
Read more: Learning and memory in the classroom
2. Cognitive load theory
The other lesson from cognitive science that has impacted my teaching has been that of cognitive load theory. Cognitive load theory attempts to explain why it is that we fail to encode new information from our working memory into our long-term memory.
This could be due to many reasons, such as: the work being too complicated; being covered too quickly; too many distractions in the environment; not having prior knowledge of the topic (we will come back to this later) etc.
How does this help us teach the standard algorithm? Well, let’s be clear about something first.
My outcome for the first lesson or two will be to give my students confidence in learning the method. Only then will we move on to the rest.
How to teach the standard algorithm
Essential precursor multiplication knowledge
Before we start work on the standard algorithm I will always check which members of my class have already struggled with multiplication in earlier grades.
If a child is not secure in their multiplication facts then you need to stage an intervention to get them up to speed – contrary to opinion learning multiplication facts is important, and while you may be able to teach multiplication facts for instant recall at earlier ages, by upper elementary school it’s very difficult to find the time.
You may also like: 35 multiplication fact games suitable for home and school – choose one or two each week for home learning if your students still need to build consistency.
How to make the standard algorithm easier
It has certainly been my experience that those students who know they are fluent in 3 or 4 digit by 1-digit multiplication have an easier time working with larger numbers.
This makes sense, as if they are fluent in these areas they are effectively reducing what their working memory needs to attend to. Assuming fluency in these two things, what they need to learn is reduced from 16 to 4-6 things.
A child who is not secure in multiplication is likely to use so much of their working memory on solving the multiplication part of the question that all the other steps, as we saw in the model earlier, are forgotten.
This is an important point for teachers to recognize: it’s not that one child has an innate ability to do the standard algorithm and one child does not. It’s that one child has simply retained the crucial knowledge needed to be successful and therefore can make the connection to prior knowledge to drastically reduce what they need to actively work out.
As Ausubel said, “The most important single factor influencing learning is what the learner already knows. Ascertain this and teach them accordingly”
The standard algorithm method: Lesson 1
No matter what students’ starting point is, there are still things we can do in the classroom to help them all get to grips with the procedure of the standard algorithm. As I mentioned earlier, my aim for the first couple of lessons is to build confidence in the method.
To do that, I ensure that our first multiplier is 11. By making the second factor 11 all that is required here is to multiply by one. I have yet to come across a child, who may struggle with their multiplication, who doesn’t know their 1’s multiplication facts.
This significantly reduces the cognitive load on and helps free up all their working memory to learn the procedure of the standard algorithm. Of course, these students will still have to learn their multiplication facts, but this just helps break down those barriers and helps them become successful.
Now all of sudden, the procedure looks like this:
The step-by-step process to solve the problem is the same as the example above but we have dramatically cut down the strain on working memory.
This makes it far more likely that the procedure will be remembered, as students can focus all their attention on understanding the procedure and not on the multiplication. Again, I would like to stress that the purpose of this is so students can get to grips with the procedure so it can be internalized.
Step 1 – Establishing prior multiplication knowledge
To start the lesson I would have several 4 by 1-digit questions on the board for the class to make their way through independently, making sure I get around to all the students who I believe may struggle at this and ascertain what they are struggling with – is it the multiplication or the procedure? If it was the former I would assist them with their multiplication tables and if it was the latter, I would go through an example with them.
After sufficient time has passed, I would go through the questions on the board to check for understanding both of the procedure and their knowledge of ‘multiplication’:
- What is the multiplicand and multiplier? (i.e. ‘the top number’ and ‘the bottom number’)
- What is the result of ___ multiplied by____?
- What happens if the product is greater than a single digit?
- What place value do I start multiplying at?
Students’ responses to these questions will help plan future interventions. In my experience, I have not come across many students whose prior-attainment means they cannot set out the column method of multiplication correctly.
If you do need to track back to establish a more solid base in multiplication then, there is a bootcamp for multiplication or a more comprehensive guide to teaching multiplication to different grades in elementary school. If parents want to support their children with multiplication, then this article gives a straightforward summary: What is the standard algorithm?
Step 2 – Introducing new multi-digit multiplication problems
During this next part of the lesson, I would show an example of the type of question they would be expected to answer by the end of the unit – in this case it would be a 4 by 2-digit multiplication with any digit, using the standard algorithm.
I would very quickly ask them to spend 30 seconds discussing with each other to see what is different about this question than the one that they did at the start of the lesson.
Once they have picked up that there is a double digit number as the multiplier, I would then solve this silently at a normal pace – the reason for this is to show how effortless it can be and to give them the confidence that this is something that they do not need to struggle with.
I would then show them another example, this time with 11 as the multiplier – this would be on the same slide as the previous example.
I would then ask: ‘Thumbs up for yes, thumbs down for no. Has the way I have set up the problem changed when the multiplier has two digits?”
I would then hope to see all thumbs down. If a child has put their thumbs up, I would engage in a whole-class dialogue to see why this is the case and refer to the example that is on the board.
Step 3 – Setting up the standard algorithm
My next step is to write the problem out for the standard algorithm.
My next instruction to the class would be: ‘For the starter, we looked at examples where the multiplier was a one-digit number. That number would be in the ‘ones’ place value. So with the number that is in the ‘ones’ in this two-digit number, we do exactly the same.’
To ensure everyone is participating I would ask them to show me using fingers or a mini-whiteboard the answer to the multiplication questions – not because I think they don’t know it but to keep their working memory firmly on the math at hand.
On the board I now have:
Now we are onto the new piece of information we want students to learn, so I would slow down and explain what is happening here, using this moment again to reinforce place value.
“So far everything that has happened before is not new to us. Now we have a brand new step. To understand what happens we need to activate our knowledge of place value. The first digit in the multiplier is in the ones and it is worth one.
The second digit is in the tens place so it is worth 10. This means we have 10 multiplied by 3. To show that we are multiplying by 10, we can place a zero in the ones place to act as a placeholder.”
Then I would write the zero in the correct place.
“We can then multiply the numbers in the multiplicand as if we were multiplying them by 1.”
Next, I would call upon all students to solve the multiplication, again showing me on their fingers to ensure participation.
Finally, I would ask students to look at the other worked example on the board and to tell their partner what the final step would be –the addition of the two products. The class would do this with me, showing the answers with their fingers/mini-whiteboard.
That will leave us with the finished product of:
Step 4 – Repeated examples of the standard algorithm
Repeat the above process with 2 more examples.
As you go through each example, get the students to do more of the explaining, particularly when it comes to the dropping of the zero and reminding one another to add the two products together. If you find children struggling, stop and rehearse this to ensure the correct language is being embedded.
Insist on correct answers in full sentences and correct language. When students are unable to do this, I ask for a volunteer who I have picked out who can do this to give a model answer, and then get the original students who were unable to answer at first to repeat what was said.
Step 5 – Students’ turn with the standard algorithm
I would then provide two standard algorithm questions that I would ask the students to complete independently. During this time, I will observe and support as required.
In previous blogs, I have mentioned being aware of learning vs performing and this is no different. Despite hearing students give really articulate answers during step 2 or getting both questions right in step 3, I am still very much aware that although these students are performing well, nothing has changed in their long-term memory as they are merely repeating what has been shown to them.
Depending on the outcome of step 3, I will either need to: go over more examples and alter my explanations, or continue onto step 4.
Step 6 – Students’ repeated practice of the standard algorithm
Happy that students are able to copy the process and understand it, I would now provide a standard algorithm worksheet for them to complete.
There is no need to differentiate the worksheet; every child will have equal access to the work.
To differentiate the worksheet would only lead to an increase in the achievement gap. The differentiation will come from additional instruction that I may give during this time.
The worksheet that I would give would not be 20 questions of the same topic. Here I would make use of interweaving. 10 questions of what I have taught would be on the sheet in random order, the other 10 questions would be made up from previous taught content.
Again, these would be allocated in a random order so that students have to switch between what has been taught in this moment and strengthening the retrieval of previously learned content. This continuous switching helps the encoding process.
Where possible make the content relatable to what has been taught; for example, as I have taught multiplication I would have some division questions from the previous year’s objectives in there to reinforce that division is the inverse of multiplication.
The last multiplication question would also have a different multiplier than 11 to see if students could apply the process when the demand on working memory is greater.
As this happens, I would be circulating the room to gauge how students are doing – not only on the questions from this lesson but previous content too. Students are free to skip over questions that they are not sure of.
Step 7 – Shared marking
In this step, students will be called on to give answers and the whole class can mark as they hear the answer. If some of them disagree with an answer we can discuss it as a class until the correct answer is found.
Step 8 – Diagnostic questions
Diagnostic questions and diagnostic assessment in general is an incredibly effective way to gauge students’ understanding of a concept. They work by posing a question and giving 4 possible answers.
While one answer is correct, the other three distractors will be carefully planned to show a specific misconception.
An example of the one I would use in this lesson is below.
Which standard algorithm question shows the correct answer?
In this example each wrong answer shows the misconception in play.
- A is correct but you can see how each other answer could be an error a child could make:
- In B they have dropped a zero when multiplying by the ones.
- In C they forgot to drop the zero when multiplying by the tens column
- In D they forgot to add on the one that had been carried over when they added 8 to 6.
It is having this selection of incorrect answers that makes diagnostic questions so powerful; they clearly identify what the student is thinking, and can provide you with immediate feedback on performance which you can correct based on the answer given.
When doing this in lessons, I assign each letter a number so A=1, B=2 etc which corresponds with the number of fingers I want them to hold up. I then give the command ‘think’. Students will think about what the correct answer is.
I will then say ‘hide’ and they will cover the fingers they wish to show on one hand with the other. Finally, I will say ‘show’ and the students show me the corresponding finger and I can quickly look around the classroom to see the answers they have given.
The other benefit of diagnostic questions is to discuss the wrong answers and understand why they are wrong. These make for fantastic discussion points and really get the class thinking and looking to find the errors.
If you’re interested in trying out more diagnostic questions, you can download a free set of math diagnostic tests for 4th and 5th Grade.
Conclusion of your first standard algorithm lesson!
Hopefully the gradual progressive structure of the lesson – or it may be two or three, depending on your class – shows how the standard algorithm can be taught with confidence.
It is worth repeating again that the main aims for the first lesson are to build student confidence and begin to learn this method of multiplication.
As their confidence grows and the process is embedded further, the multiplier can be changed and reasoning and problem solving questions can be introduced and answered with greater independence.
The standard algorithm examples
Here are two standard algorithm examples set out for you.
Example 1: 6321 x 15 = 94,815
Example 2: 6321 x 25 = 158,025
Standard algorithm questions
Here are a few standard algorithm questions and answers to get you started:
- 1543 x 11 = 16,973
- 2,374 x 13 = 30,862
- 4,537 x 27 = 122,499
- 8,983 x 37 = 332,371
- 9,452 x 48 = 453,696
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The content in this article was originally written by primary school lead teacher Neil Almond and has since been revised and adapted for US schools by elementary math teacher Jaclyn Wassell.