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## The Long Multiplication Method: How To Teach Long Multiplication So All Your KS2 Pupils ‘Get It’

Neil Almond

The long multiplication method can be very difficult to teach in Years 5 and 6, as anyone who has taught upper KS2 before will know.

Despite best intentions, there will always be a few pupils who are either unsure of the simpler 4 by 1-digit approach or who are not secure on their times tables.

If this academic year will be your first time teaching Year 6, you have all of this to look forward to but don’t despair – it happens every year.

## What is long multiplication?

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Long multiplication is a written multiplication method used when multiplying two or three digit numbers by another number of two or more digits. It is often referred to as column multiplication.

In Year 5 and Year 6 at primary school, long multiplication means multiplying a number that is at least three digits by one that is two digits or more.

Before tackling long multiplication at KS2 children should ideally be confident with their times tables and understand key terms including 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.

Not sure if your pupils are ready to jump into long multiplication? Grid method multiplication is a great ‘stepping stone’ to the column method .

In the maths national curriculum for England, the formal long multiplication method is mentioned in both in Year 5 and Year 6.

- In the Year 5 objectives for multiplication and division, it states that ‘pupils should be taught to multiply numbers up to 4 digits by a one- or two-digit number using a formal written method, including long multiplication for two-digit numbers.’
- In the Year 6 objectives for multiplication and division, it says that, ‘pupils should be multiply multi-digit numbers up to 4 digits by a two-digit whole number using the formal written method of long multiplication.’

The appendix at the end of the year group objectives gives us an insight into what this looks like:

This nicely sets out a progression model for teachers once the class are comfortable with multiplying 3 or 4-digit numbers by a 1-digit number.

A quick look through the 2019 SATs arithmetic paper shows there are 4 marks up for grabs for getting the long multiplication questions correct. There are also many examples where pupils will need to select this method in the two reasoning papers as it is the most time efficient method – important to be able to get through the paper!

Therefore, it is crucial that pupils become fluent in the method. When I say fluent, this is what I mean:

‘Fluency is the process of retrieving information from out of 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

The formal long multiplication method is a step by step method of supporting children to understand conceptually and practically how to multiply one three or four digit number by another two digit number or greater.

Here is the long multiplication method broken down step by step using the second example from the national curriculum appendix:

## Example: 124 x 26

- Set the question in the formal method, making sure the ones and tens digits, and so on, line up
- Remember to start the process of multiplication with the ones
- Multiply 6 by 4
- Write the answer down correctly – including any regrouping
- Multiply 6 by 2
- Add anything that you have regrouped from the previous multiplication.
- Write the answer down correctly
- Drop a zero as we are now multiplying with 10s
- Multiply 2 by 4
- Write the answer correctly
- Multiply 2 by 2
- Write the answer down
- Multiply 2 by 1
- Add the two answers (sometimes known as partial products) up together correctly

That’s a total of 16 steps that children need to become fluent in 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 it. This will prevent this information from being encoded.

So one answer to the question: How to do long multiplication? is to simply follow the steps.

But this is missing out a crucial stage of learning – moving from a procedural to a conceptual understanding of what’s going on.

The rest of this article explains how to teach long multiplication to develop a conceptual understanding, which will have the biggest impact for your class. It includes links to multiplication worksheets to provide you with lots of practice.

Two lessons from cognitive science have massively changed the way I approach teaching the long multiplication method.

## 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 show this process.

From the model, we can see that the person uses their attention to take things from the environment into the working memory. The person then attempts to encode this information into their long-term memory, but some information may be forgotten for a myriad of reasons.

When that information is in their long-term memory, they will be able to bring it back to the forefront of their working memory to use it. If those memories remain dormant for too long (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 the role of cognitive load theory in the classroom . 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 down to many reasons, such as:

- the work being too complicated;
- being covered too quickly;
- too many distractors in the environment;
- not having prior knowledge of topic (we will come back to this later);

How does this help us teach the long multiplication method? Well let’s be clear about something first.

My outcome for the first lesson or two will be to give my pupils confidence in learning the method. Only then will we move on to the rest.

## Essential precursor multiplication knowledge

Before we start work on long multiplication, I will always check which members of my class have already struggled with multiplication in Year 3 or 4.

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 popular opinion, learning multiplication facts is important, and while you may be able to teach times tables for instant recall at earlier ages, by upper KS2 it’s very difficult to find the time.

You may also like: 35 times tables games suitable for home and school – choose one or two each week for home learning if your pupils still need to build consistency.

## Read more: Commutative property of multiplication

How to make long multiplication easier.

It has been my experience that pupils who are fluent in their multiplication tables have an easier time working with larger numbers, such as 3 or 4 digit by 1-digit multiplication.

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 recognise: it’s not that one child has an innate ability to do long multiplication 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”

KS2 Long Multiplication Worksheets

Give your pupils a head start on practising their long multiplication skills with this free pack of multiplication worksheets.

No matter what pupils’ starting point is, there are still things we can do in the classroom to help them all get to grips with the procedure of long multiplication. 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 factors 11, all that is required here is to multiply by one. I have yet to come across a child, even those who may struggle with their multiplication, who doesn’t know the 1 times table.

## See also: Lowest common multiple , Highest common factor & What is a multiple

This significantly reduces the cognitive load on and helps free up all their working memory to learn the procedure of long multiplication. Of course, these pupils will still have to learn their multiplication facts but this just helps break down those barriers and helps them become successful.

Now, 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 pupils 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 pupils can get to grips with the procedure so it can be internalised.

## 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. During this, I’d make sure that I got around to all the pupils 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’)
- How do I write this in the column method?
- 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?

Pupils’ responses to these questions will help plan future interventions. In my experience, I have not come across many pupils 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 each year group throughout KS2. Keeping your pupils engaged whilst helping them develop their multiplication skills is also important, so we created a blog on the best multiplication games to play at KS1 and KS2.

## Step 2 – Introducing the new idea of long multiplication

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 long multiplication method.

I would very quickly ask them to spend 30 seconds discussing with each other to see what is different about this question from the one that they did at the start of the lesson.

Once they have noticed 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, the example would be 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 out the calculation in the column method 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 out the long multiplication method

My next step is to write the calculation out in the column method for long multiplication.

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 2-digit number, we do exactly the same.’

To ensure everyone is participating, I would ask them to show me, using fingers or mini-whiteboards, the answer to the multiplication questions – not because I think they don’t know it but to keep their working memory firmly on the maths at hand.

On the board, I now have:

Now we are onto the new piece of information we want pupils 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 place holder. ”

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 pupils to solve the multiplication, again showing me on their fingers or mini-whiteboards to ensure participation.

Finally, I would ask pupils 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 or on mini-whiteboards.

That will leave us with the finished product of:

## Step 4 – Repeated examples of long multiplication method

Repeat the above process with 2 more examples.

As you go through each example, get the pupils 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 pupils 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 pupils who were unable to answer at first to repeat what was said.

## Step 5 – Pupils’ turn with long multiplication method

I would then provide two long multiplication questions that I would ask pupils 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 pupils give really articulate answers during step 2 or getting both questions right in step 3, I am still very much aware that although these pupils 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 – Pupils’ repeated practice of long multiplication

Happy that pupils are able to copy the process and understand it, I would now provide a long multiplication 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 work sheet would only lead to an increase in the attainment 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 interleaving. 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.

## Read more: 8 Differentiation Strategies For Your Classroom To Use Across The Attainment Gap

Again, these would be allocated in a random order so that pupils have to switch between what has been taught in this moment and strengthening the retrieval of previously learnt 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.

When revising for SATs, you may want to interleave long multiplication problems with long division problems to further reinforce the relationship between the two.

The last multiplication question would also have a different multiplier than 11 to see if pupils could apply the process when the demand on working memory is greater.

As this happens, I would be circulating the room to gauge how pupils are doing – not only on the questions from this lesson but previous content too. Pupils are free to skip over questions that they are not sure of.

## Step 7 – Shared marking

In this step, pupils 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 question s

Diagnostic questions and diagnostic assessment in general is an incredibly effective way to gauge pupils’ 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 long multiplication 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 the added 8 to 6.

It is having this selection of incorrect answers that makes diagnostic questions so powerful; they clearly identify what the pupil 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’. Pupils 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 pupils 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 through the wrong answers and get to the bottom of 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 maths diagnostic tests for Year 5 and 6 or visit the Third Space Learning Maths Hub for a large number of diagnostic assessments on every KS2 maths curriculum topic.

Hopefully the gradual progressive structure of the lesson – or it may be two or three, depending on your class – shows how the long multiplication method can be taught with confidence and learnt by most Year 5s and Year 6s.

It is worth repeating again that the main aims for the first lesson are to build pupil 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.

If you need more long multiplication examples, Third Space Learning’s White Rose lesson slides and worksheets for Year 6 Four Operations gives you more opportunities to work through the stages step by step.

Here are two long multiplication examples set out for you.

Example 1: 6321 x 15 = 94,815

Example 2: 6321 x 25 = 158,025

Here are a few long multiplication 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

## Read more: Teaching multiplication KS2

If you’re looking for more questions and long multiplication worksheets then register for more primary maths resources here.

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## Related articles

What Is Subitising And Why Is It Important? Tips For Parents & Teachers

What Is The Column Method Of Addition And Subtraction? Explained For Primary School

What Is BIDMAS: Explained for Primary School

What Is Order Of Operations: Explained For Primary School

## FREE Long Division Worksheets for KS2

3 ready to use worksheets for your class that will help them with all aspects of long division from 1-digit numbers through to working out multiples!

One worksheet covering division with 1-digit numbers, one covering 2-digit numbers and one for working out multiples.

## Privacy Overview

## How to do long multiplication

Multiplying large numbers is easy when breaking down the problem into parts!

Author Amber Watkins

Published November 14, 2023

Published Nov 14, 2023

- Key takeaways
- Long multiplication makes multiplying large numbers easy by breaking them down into parts
- Mastering long multiplication takes practice, but it’s a very important skill that will help with maths more widely

Table of contents

## What is long multiplication?

- Practice questions

Are you encountering large numbers in maths? They can seem a bit scary at first! But luckily, there’s a silly saying that can help: ‘How to eat an elephant? One bite at a time!’. It teaches us that if you have a large task, the best way to do it is in parts.

The same can be true with multiplication. When we are given large numbers to multiply, instead of trying to do the problem all at once in our heads, we can multiply those numbers in parts.

When we multiply large numbers in parts, then add those parts together, it’s called long multiplication.

Long multiplication is the steps you follow to multiply larger numbers in an easy way. Long multiplication allows you to find partial answers and add them together to find the final product.

For example, instead of multiplying the numbers 64 x 32 as they are, you can break up the number 32 into two parts: 30 and 2, then multiply those parts by 64. It would look like this:

(64 x 2) + (64 x 30) 128 + 1,920 You would get a total of 2,048. Multiplying in parts, and then adding the products together, makes multiplying large numbers easy!

Unlock unlimited maths questions

Put your learning into practice with fun exercises + games that are proven to boost ability!

## How to set up a long multiplication problem

When doing long multiplication problems in a column method, you first line up the numbers you’re multiplying in columns.

For example, would we set up the problem 64 x 32 using the column method?

- The number 62 would be written above the number 32. The equal sign will be represented with a line underneath
- You will also have two or more rows beneath. This is where you write the partial products. The first partial product is written in the first row, the second partial product is written in the second row, and so on.

Let’s keep this in mind when reviewing the steps for how to do long multiplication .

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## Shape, space and measure

Number and place value, addition and subtraction, multiplication and division, operations (asmd), shape/geometry, ratio and proportion, probability, sample questions, long multiplication methods: column method.

Let’s learn the use what’s known as the column method to solve the following problem:

What is 33 x 21?

1. Line up the numbers in a column format.

2. Multiply each top digit by the last digit in the bottom number. Place each answer in the first row from right to left. You should have the number 33 in the first product row.

3. Once each of the top digits is multiplied by that number, cross it off. 4. Next add a zero as a place value holder in the second row to represent already multiplying by the digit in that place value.

5. Multiply each top digit by the first digit in the bottom number. You will have the number 660 in the second partial product row.

6. Finally add the two products 31 and 660 to get the final answer of 693.

## Explore long multiplication with DoodleMaths

DoodleMaths is an award-winning app that’s filled with thousands of questions and games exploring multiplication, division and more!

Designed by teachers, it creates each child a unique work programme tailored to their needs, doubling their progression with just 10 minutes of use a day. Try it for free !

## Carry-over rule while doing long multiplication

If you multiply two digits and the answer is in the double digits, the carry-over rule says you must write the second digit in the partial product line, and the first digit above the next number you will need to multiply. That way it carries over.

Let’s see how these long multiplication steps and the carry-over rule work

## Long multiplication method: horizontal method

There’s also another way to do long multiplication – the horizontal method. The horizontal method allows us to break up the second number in parts and multiply those parts by the first number. Let’s learn how to do long multiplication with the horizontal method. Let’s look at this example.

Multiply 43 x 65 using the horizontal method 1. Write the second number 65 in Expanded form. Those two numbers will be the parts we multiply the first number 43 by. 65 in Expanded form is 60 + 5. 2. Begin by multiplying 43 by the first part, 60. This can be done by multiplying 43 x 6, then adding a zero to the answer.

43 x 6 is 258.

Then add a zero, so it would be 258 0 . 3. Next we will multiply 43 by the second part 5.

43 x 5 is 215. 4. Finally, we add the two partial products together to get the final answer.

2580 + 215 is 2,795.

## Long multiplication practice questions

Click on the boxes below to see the answers!

- Write 72 and 24 in columns.
- Multiply 2 x 4 and 7 x 4 and write the answers in the first row.
- Cross off the 4 and add a zero placeholder in the second row.
- Multiply 2 x 2 and 7 x 2 and write the answers in the second row.
- Add both columns together to get 1,728 .
- Write 48 and 62 in columns.
- Multiply 8 x 2 and 4 x 2 and write the answers in the first row.
- Cross off the 2 and add a zero placeholder in the second row.
- Multiply 8 x 6 and 4 x 6 and write the answers in the second row.
- Add both columns together to get 2,976
- Write the second number 54 in expanded form: 50 + 4.
- Multiply 98 times the first part 50. Multiply 98 x 5 and add a zero to the answer: 4,900.
- Next multiply 98 times the second part 4: 392.
- Finally, add those two partial products together: 4,900 + 392 = 5,292

## FAQs about long multiplication

You do long multiplication by multiplying numbers in parts. You multiply each digit in the top number, by each digit in the bottom number. Finally, you add the partial products to get the final answer.

Long multiplication helps make multiplication with large numbers easy. The more you practice long multiplication, the easier these problems will be.

The long multiplication method is often called the column method. This is because the numbers you multiply are written above and below one another in columns.

You begin learning long multiplication in Year 5 and learn to multiply even larger numbers in Year 6.

Lesson credits

Amber Watkins

Amber is an education specialist with a degree in Early Childhood Education. She has over 12 years of experience teaching and tutoring. "Knowing that my work in math education makes such an impact leaves me with an indescribable feeling of pride and joy!"

Amber is an education specialist with a degree in Early Childhood Education. She has over 12 years of experience teaching and tutoring . "Knowing that my work in math education makes such an impact leaves me with an indescribable feeling of pride and joy!"

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The best order to learn times tables

We outline the best (and easiest!) order to learn the 1-12 multiplications in

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## Long Multiplication Worksheets

This page includes Long Multiplication worksheets for students who have mastered the basic multiplication facts and are learning to multiply 2-, 3-, 4- and more digit numbers. Sometimes referred to as long multiplication or multi-digit multiplication, the questions on these worksheets require students to have mastered the multiplication facts from 0 to 9.

There are a variety of strategies for completing long multiplication including the classic paper and pencil methods, lattice multiplication (which we feature on this page), mental strategies, manipulative use, technology, and various other paper and pencil algorithms. Multi-Digit multiplication can be a frustrating experience for many students. Try to teach multi-digit multiplication using more than one strategy.

## Most Popular Long Multiplication Worksheets this Week

Long multiplication practice worksheets including a variety of number sizes and options for different number formats.

Two-Digit multiplication is a natural place to start after students have mastered their multiplication facts. The concept of multiplying two-digit numbers requires a knowledge of place and place value, especially if students are to fully understand what they are accomplishing with the various strategies they use. A question such as 24 × 5 can be thought of as (20 + 4) × 5. Mentally, this becomes much easier as students multiply 20 by 5 then 4 by 5 and add the two products. A good way to build understanding of place value is with base ten blocks. These manipulatives also translate very well into paper and pencil and mental math strategies.

An extra digit can throw off some students but add an extra challenge to others. Always ensure that students are ready for three-digit multiplication or both you and your student will be frustrated. Three-digit multiplication worksheets require a mastery of single-digit multiplication facts and a knowledge of a multi-digit multiplication strategy that will enable students to both understand the question and get the correct answer. Four-digit multiplication was invented in 350 B.C. as a way of punishing children who stole bread from the market. Just kidding! It's actually a great challenge for students who have experienced success with their multiplication facts and have a good handle on a long multiplication strategy. What do you give students who have mastered their multiplication facts and long multiplication and who love a challenge? Look no further than five- to eight-digit multiplication. Enjoy!

There are no thousand separators in the numbers on the first worksheets. It makes it a little more difficult to read the numbers, but sometimes it is better not to have too many things in the way when students are learning long multiplication. The answer keys include answers with the steps shown, so students and teachers can diagnose any problems in the steps they took to answer the questions. The answers use a paper and pencil algorithm that is commonly used in the U.S. and other countries.

- Long Multiplication Worksheets up to 3-Digit Numbers 2-Digit by 1-Digit Multiplication ✎ 2-Digit by 2-Digit Multiplication ✎ 3-Digit by 1-Digit Multiplication ✎ 3-Digit by 2-Digit Multiplication ✎ 3-Digit by 3-Digit Multiplication ✎
- 4-Digit Long Multiplication Worksheets 4-Digit by 1-Digit Multiplication ✎ 4-Digit by 2-Digit Multiplication ✎ 4-Digit by 3-Digit Multiplication ✎ 4-Digit by 4-Digit Multiplication ✎
- 5-Digit Long Multiplication Worksheets 5-Digit by 1-Digit Multiplication 5-Digit by 2-Digit Multiplication 5-Digit by 3-Digit Multiplication 5-Digit by 4-Digit Multiplication 5-Digit by 5-Digit Multiplication
- 6-Digit Long Multiplication Worksheets 6-Digit by 1-Digit Multiplication 6-Digit by 2-Digit Multiplication 6-Digit by 3-Digit Multiplication 6-Digit by 4-Digit Multiplication 6-Digit by 5-Digit Multiplication 6-Digit by 6-Digit Multiplication
- 7-Digit Long Multiplication Worksheets 7-Digit by 1-Digit Multiplication 7-Digit by 2-Digit Multiplication 7-Digit by 3-Digit Multiplication 7-Digit by 4-Digit Multiplication 7-Digit by 5-Digit Multiplication 7-Digit by 6-Digit Multiplication 7-Digit by 7-Digit Multiplication
- 8-Digit Long Multiplication Worksheets 8-Digit by 1-Digit Multiplication 8-Digit by 2-Digit Multiplication 8-Digit by 3-Digit Multiplication 8-Digit by 4-Digit Multiplication 8-Digit by 5-Digit Multiplication 8-Digit by 6-Digit Multiplication 8-Digit by 7-Digit Multiplication 8-Digit by 8-Digit Multiplication
- Large Print Long Multiplication Worksheets up to 3-Digit Numbers 2-digit by 1-digit Multiplication ( Large Print ) 2-digit by 2-digit Multiplication ( Large Print ) 3-digit by 1-digit Multiplication ( Large Print ) 3-digit by 2-digit Multiplication ( Large Print ) 3-digit by 3-digit Multiplication ( Large Print )
- Large Print 4-Digit Long Multiplication Worksheets 4-digit by 1-digit Multiplication ( Large Print ) 4-digit by 2-digit Multiplication ( Large Print ) 4-digit by 3-digit Multiplication ( Large Print ) 4-digit by 4-digit Multiplication ( Large Print )
- Large Print 5-Digit Long Multiplication Worksheets 5-digit by 1-digit Multiplication ( Large Print ) 5-digit by 2-digit Multiplication ( Large Print ) 5-digit by 3-digit Multiplication ( Large Print ) 5-digit by 4-digit Multiplication ( Large Print ) 5-digit by 5-digit Multiplication ( Large Print )
- Large Print 6-Digit Long Multiplication Worksheets 6-digit by 1-digit Multiplication ( Large Print ) 6-digit by 2-digit Multiplication ( Large Print ) 6-digit by 3-digit Multiplication ( Large Print ) 6-digit by 4-digit Multiplication ( Large Print ) 6-digit by 5-digit Multiplication ( Large Print ) 6-digit by 6-digit Multiplication ( Large Print )

Commas are included as thousands separators for the numbers on the next worksheets. Commas are used in the U.S. and other English-speaking countries as a way of making numbers easier to read. As with the other long multiplication worksheets on this page, the answer keys include the steps.

- Long Multiplication Worksheets up to 3-Digit Numbers with Comma-Separated Thousands 2-Digit by 2-Digit Multiplication (Comma-Separated Thousands) 3-Digit by 1-Digit Multiplication (Comma-Separated Thousands) 3-Digit by 2-Digit Multiplication (Comma-Separated Thousands) 3-Digit by 3-Digit Multiplication (Comma-Separated Thousands)
- 4-Digit Long Multiplication Worksheets with Comma-Separated Thousands 4-Digit by 1-Digit Multiplication (Comma-Separated Thousands) 4-Digit by 2-Digit Multiplication (Comma-Separated Thousands) 4-Digit by 3-Digit Multiplication (Comma-Separated Thousands) 4-Digit by 4-Digit Multiplication (Comma-Separated Thousands)
- 5-Digit Long Multiplication Worksheets with Comma-Separated Thousands 5-Digit by 1-Digit Multiplication (Comma-Separated Thousands) 5-Digit by 2-Digit Multiplication (Comma-Separated Thousands) 5-Digit by 3-Digit Multiplication (Comma-Separated Thousands) 5-Digit by 4-Digit Multiplication (Comma-Separated Thousands) 5-Digit by 5-Digit Multiplication (Comma-Separated Thousands)
- 6-Digit Long Multiplication Worksheets with Comma-Separated Thousands 6-Digit by 1-Digit Multiplication (Comma-Separated Thousands) 6-Digit by 2-Digit Multiplication (Comma-Separated Thousands) 6-Digit by 3-Digit Multiplication (Comma-Separated Thousands) 6-Digit by 4-Digit Multiplication (Comma-Separated Thousands) 6-Digit by 5-Digit Multiplication (Comma-Separated Thousands) 6-Digit by 6-Digit Multiplication (Comma-Separated Thousands)
- 7-Digit Long Multiplication Worksheets with Comma-Separated Thousands 7-Digit by 1-Digit Multiplication (Comma-Separated Thousands) 7-Digit by 2-Digit Multiplication (Comma-Separated Thousands) 7-Digit by 3-Digit Multiplication (Comma-Separated Thousands) 7-Digit by 4-Digit Multiplication (Comma-Separated Thousands) 7-Digit by 5-Digit Multiplication (Comma-Separated Thousands) 7-Digit by 6-Digit Multiplication (Comma-Separated Thousands) 7-Digit by 7-Digit Multiplication (Comma-Separated Thousands)
- 8-Digit Long Multiplication Worksheets with Comma-Separated Thousands 8-Digit by 1-Digit Multiplication (Comma-Separated Thousands) 8-Digit by 2-Digit Multiplication (Comma-Separated Thousands) 8-Digit by 3-Digit Multiplication (Comma-Separated Thousands) 8-Digit by 4-Digit Multiplication (Comma-Separated Thousands) 8-Digit by 5-Digit Multiplication (Comma-Separated Thousands) 8-Digit by 6-Digit Multiplication (Comma-Separated Thousands) 8-Digit by 7-Digit Multiplication (Comma-Separated Thousands) 8-Digit by 8-Digit Multiplication (Comma-Separated Thousands)
- Large Print Long Multiplication Worksheets up to 3-Digit Numbers with Comma-Separated Thousands 2-Digit by 2-Digit Multiplication ( Large Print ) (Comma-Separated Thousands) 3-Digit by 1-Digit Multiplication ( Large Print ) (Comma-Separated Thousands) 3-Digit by 2-Digit Multiplication ( Large Print ) (Comma-Separated Thousands) 3-Digit by 3-Digit Multiplication ( Large Print ) (Comma-Separated Thousands)
- Large Print 4-Digit Long Multiplication Worksheets with Comma-Separated Thousands 4-Digit by 1-Digit Multiplication ( Large Print ) (Comma-Separated Thousands) 4-Digit by 2-Digit Multiplication ( Large Print ) (Comma-Separated Thousands) 4-Digit by 3-Digit Multiplication ( Large Print ) (Comma-Separated Thousands) 4-Digit by 4-Digit Multiplication ( Large Print ) (Comma-Separated Thousands)
- Large Print 5-Digit Long Multiplication Worksheets with Comma-Separated Thousands 5-Digit by 1-Digit Multiplication ( Large Print ) (Comma-Separated Thousands) 5-Digit by 2-Digit Multiplication ( Large Print ) (Comma-Separated Thousands) 5-Digit by 3-Digit Multiplication ( Large Print ) (Comma-Separated Thousands) 5-Digit by 4-Digit Multiplication ( Large Print ) (Comma-Separated Thousands) 5-Digit by 5-Digit Multiplication ( Large Print ) (Comma-Separated Thousands)
- Large Print 6-Digit Long Multiplication Worksheets with Comma-Separated Thousands 6-Digit by 1-Digit Multiplication ( Large Print ) (Comma-Separated Thousands) 6-Digit by 2-Digit Multiplication ( Large Print ) (Comma-Separated Thousands) 6-Digit by 3-Digit Multiplication ( Large Print ) (Comma-Separated Thousands) 6-Digit by 4-Digit Multiplication ( Large Print ) (Comma-Separated Thousands) 6-Digit by 5-Digit Multiplication ( Large Print ) (Comma-Separated Thousands) 6-Digit by 6-Digit Multiplication ( Large Print ) (Comma-Separated Thousands)

Separating thousands with spaces avoids any confusion with commas and periods. Various number formats in different countries and languages use commas and periods for both decimals and thousand separators, but a space is only ever used as a thousands separator. It is more common in some countries such as Canada and France, but it is being adopted more in other parts of the world.

- Long Multiplication Worksheets up to 3-Digit Numbers with Space-Separated Thousands 2-Digit by 2-Digit Multiplication (Space-Separated Thousands) 3-Digit by 1-Digit Multiplication (Space-Separated Thousands) 3-Digit by 2-Digit Multiplication (Space-Separated Thousands) 3-Digit by 3-Digit Multiplication (Space-Separated Thousands)
- 4-Digit Long Multiplication Worksheets with Space-Separated Thousands 4-Digit by 1-Digit Multiplication (Space-Separated Thousands) 4-Digit by 2-Digit Multiplication (Space-Separated Thousands) 4-Digit by 3-Digit Multiplication (Space-Separated Thousands) 4-Digit by 4-Digit Multiplication (Space-Separated Thousands)
- 5-Digit Long Multiplication Worksheets with Space-Separated Thousands 5-Digit by 1-Digit Multiplication (Space-Separated Thousands) 5-Digit by 2-Digit Multiplication (Space-Separated Thousands) 5-Digit by 3-Digit Multiplication (Space-Separated Thousands) 5-Digit by 4-Digit Multiplication (Space-Separated Thousands) 5-Digit by 5-Digit Multiplication (Space-Separated Thousands)
- 6-Digit Long Multiplication Worksheets with Space-Separated Thousands 6-Digit by 1-Digit Multiplication (Space-Separated Thousands) 6-Digit by 2-Digit Multiplication (Space-Separated Thousands) 6-Digit by 3-Digit Multiplication (Space-Separated Thousands) 6-Digit by 4-Digit Multiplication (Space-Separated Thousands) 6-Digit by 5-Digit Multiplication (Space-Separated Thousands) 6-Digit by 6-Digit Multiplication (Space-Separated Thousands)
- 7-Digit Long Multiplication Worksheets with Space-Separated Thousands 7-Digit by 1-Digit Multiplication (Space-Separated Thousands) 7-Digit by 2-Digit Multiplication (Space-Separated Thousands) 7-Digit by 3-Digit Multiplication (Space-Separated Thousands) 7-Digit by 4-Digit Multiplication (Space-Separated Thousands) 7-Digit by 5-Digit Multiplication (Space-Separated Thousands) 7-Digit by 6-Digit Multiplication (Space-Separated Thousands) 7-Digit by 7-Digit Multiplication (Space-Separated Thousands)
- 8-Digit Long Multiplication Worksheets with Space-Separated Thousands 8-Digit by 1-Digit Multiplication (Space-Separated Thousands) 8-Digit by 2-Digit Multiplication (Space-Separated Thousands) 8-Digit by 3-Digit Multiplication (Space-Separated Thousands) 8-Digit by 4-Digit Multiplication (Space-Separated Thousands) 8-Digit by 5-Digit Multiplication (Space-Separated Thousands) 8-Digit by 6-Digit Multiplication (Space-Separated Thousands) 8-Digit by 7-Digit Multiplication (Space-Separated Thousands) 8-Digit by 8-Digit Multiplication (Space-Separated Thousands)
- Large Print Long Multiplication Worksheets up to 3-Digit Numbers with Space-Separated Thousands 2-Digit by 2-Digit Multiplication ( Large Print ) (Space-Separated Thousands) 3-Digit by 1-Digit Multiplication ( Large Print ) (Space-Separated Thousands) 3-Digit by 2-Digit Multiplication ( Large Print ) (Space-Separated Thousands) 3-Digit by 3-Digit Multiplication ( Large Print ) (Space-Separated Thousands)
- Large Print 4-Digit Long Multiplication Worksheets with Space-Separated Thousands 4-Digit by 1-Digit Multiplication ( Large Print ) (Space-Separated Thousands) 4-Digit by 2-Digit Multiplication ( Large Print ) (Space-Separated Thousands) 4-Digit by 3-Digit Multiplication ( Large Print ) (Space-Separated Thousands) 4-Digit by 4-Digit Multiplication ( Large Print ) (Space-Separated Thousands)
- Large Print 5-Digit Long Multiplication Worksheets with Space-Separated Thousands 5-Digit by 1-Digit Multiplication ( Large Print ) (Space-Separated Thousands) 5-Digit by 2-Digit Multiplication ( Large Print ) (Space-Separated Thousands) 5-Digit by 3-Digit Multiplication ( Large Print ) (Space-Separated Thousands) 5-Digit by 4-Digit Multiplication ( Large Print ) (Space-Separated Thousands) 5-Digit by 5-Digit Multiplication ( Large Print ) (Space-Separated Thousands)
- Large Print 6-Digit Long Multiplication Worksheets with Space-Separated Thousands 6-Digit by 1-Digit Multiplication ( Large Print ) (Space-Separated Thousands) 6-Digit by 2-Digit Multiplication ( Large Print ) (Space-Separated Thousands) 6-Digit by 3-Digit Multiplication ( Large Print ) (Space-Separated Thousands) 6-Digit by 4-Digit Multiplication ( Large Print ) (Space-Separated Thousands) 6-Digit by 5-Digit Multiplication ( Large Print ) (Space-Separated Thousands) 6-Digit by 6-Digit Multiplication ( Large Print ) (Space-Separated Thousands)

In some places, periods are used as thousands separators and commas are used as decimals. This is very confusing to people who are used to U.S. formatted numbers.

- Long Multiplication Worksheets up to 3-Digit Numbers with Period-Separated Thousands 2-Digit by 2-Digit Multiplication (Period-Separated Thousands) 3-Digit by 1-Digit Multiplication (Period-Separated Thousands) 3-Digit by 2-Digit Multiplication (Period-Separated Thousands) 3-Digit by 3-Digit Multiplication (Period-Separated Thousands)
- 4-Digit Long Multiplication Worksheets with Period-Separated Thousands 4-Digit by 1-Digit Multiplication (Period-Separated Thousands) 4-Digit by 2-Digit Multiplication (Period-Separated Thousands) 4-Digit by 3-Digit Multiplication (Period-Separated Thousands) 4-Digit by 4-Digit Multiplication (Period-Separated Thousands)
- 5-Digit Long Multiplication Worksheets with Period-Separated Thousands 5-Digit by 1-Digit Multiplication (Period-Separated Thousands) 5-Digit by 2-Digit Multiplication (Period-Separated Thousands) 5-Digit by 3-Digit Multiplication (Period-Separated Thousands) 5-Digit by 4-Digit Multiplication (Period-Separated Thousands) 5-Digit by 5-Digit Multiplication (Period-Separated Thousands)
- 6-Digit Long Multiplication Worksheets with Period-Separated Thousands 6-Digit by 1-Digit Multiplication (Period-Separated Thousands) 6-Digit by 2-Digit Multiplication (Period-Separated Thousands) 6-Digit by 3-Digit Multiplication (Period-Separated Thousands) 6-Digit by 4-Digit Multiplication (Period-Separated Thousands) 6-Digit by 5-Digit Multiplication (Period-Separated Thousands) 6-Digit by 6-Digit Multiplication (Period-Separated Thousands)
- 7-Digit Long Multiplication Worksheets with Period-Separated Thousands 7-Digit by 1-Digit Multiplication (Period-Separated Thousands) 7-Digit by 2-Digit Multiplication (Period-Separated Thousands) 7-Digit by 3-Digit Multiplication (Period-Separated Thousands) 7-Digit by 4-Digit Multiplication (Period-Separated Thousands) 7-Digit by 5-Digit Multiplication (Period-Separated Thousands) 7-Digit by 6-Digit Multiplication (Period-Separated Thousands) 7-Digit by 7-Digit Multiplication (Period-Separated Thousands)
- 8-Digit Long Multiplication Worksheets with Period-Separated Thousands 8-Digit by 1-Digit Multiplication (Period-Separated Thousands) 8-Digit by 2-Digit Multiplication (Period-Separated Thousands) 8-Digit by 3-Digit Multiplication (Period-Separated Thousands) 8-Digit by 4-Digit Multiplication (Period-Separated Thousands) 8-Digit by 5-Digit Multiplication (Period-Separated Thousands) 8-Digit by 6-Digit Multiplication (Period-Separated Thousands) 8-Digit by 7-Digit Multiplication (Period-Separated Thousands) 8-Digit by 8-Digit Multiplication (Period-Separated Thousands)
- Large Print Long Multiplication Worksheets up to 3-Digit Numbers with Period-Separated Thousands 2-Digit by 2-Digit Multiplication ( Large Print ) (Period-Separated Thousands) 3-Digit by 1-Digit Multiplication ( Large Print ) (Period-Separated Thousands) 3-Digit by 2-Digit Multiplication ( Large Print ) (Period-Separated Thousands) 3-Digit by 3-Digit Multiplication ( Large Print ) (Period-Separated Thousands)
- Large Print 4-Digit Long Multiplication Worksheets with Period-Separated Thousands 4-Digit by 1-Digit Multiplication ( Large Print ) (Period-Separated Thousands) 4-Digit by 2-Digit Multiplication ( Large Print ) (Period-Separated Thousands) 4-Digit by 3-Digit Multiplication ( Large Print ) (Period-Separated Thousands) 4-Digit by 4-Digit Multiplication ( Large Print ) (Period-Separated Thousands)
- Large Print 5-Digit Long Multiplication Worksheets with Period-Separated Thousands 5-Digit by 1-Digit Multiplication ( Large Print ) (Period-Separated Thousands) 5-Digit by 2-Digit Multiplication ( Large Print ) (Period-Separated Thousands) 5-Digit by 3-Digit Multiplication ( Large Print ) (Period-Separated Thousands) 5-Digit by 4-Digit Multiplication ( Large Print ) (Period-Separated Thousands) 5-Digit by 5-Digit Multiplication ( Large Print ) (Period-Separated Thousands)
- Large Print 6-Digit Long Multiplication Worksheets with Period-Separated Thousands 6-Digit by 1-Digit Multiplication ( Large Print ) (Period-Separated Thousands) 6-Digit by 2-Digit Multiplication ( Large Print ) (Period-Separated Thousands) 6-Digit by 3-Digit Multiplication ( Large Print ) (Period-Separated Thousands) 6-Digit by 4-Digit Multiplication ( Large Print ) (Period-Separated Thousands) 6-Digit by 5-Digit Multiplication ( Large Print ) (Period-Separated Thousands) 6-Digit by 6-Digit Multiplication ( Large Print ) (Period-Separated Thousands)

## Other Long Multiplication Strategies

Lattice, or sieve, multiplication is a great strategy for students to use to calculate long multiplication problems on pencil and paper. We've made the first step of preparing a lattice easy as the worksheets below have them pre-drawn. With a little practice, students can use graph paper or draw their own lattices freehand. The first factor is separated by place value along the top of the lattice, giving each place value its own column. The second factor is separated in the same way, but along the right side with one place value per row. The single digit column and row numbers are multiplied together and their product is written in the corresponding box, separating the tens and ones places on either side of the diagonal. Finally, the diagonal "rows" are summed and regrouped starting with the diagonal in the lower right hand corner which will only have a singl-digit in it. The answer keys we've provided should give you a good idea of how to accomplish lattice multiplication like a pro. Once students have a little practice, you might find that this is their preferred method for calculating the products of large numbers. This method is highly scalable, which means it is a straight-forward task to multiply a 10-digit by a 10-digit number, etc.

- Various-Digit Lattice Multiplication Worksheets With Lattices Included 2-Digit × 2-Digit Lattice Multiplication 2-Digit × 3-Digit Lattice Multiplication 3-Digit × 2-Digit Lattice Multiplication 3-Digit × 3-Digit Lattice Multiplication 4-Digit × 2-Digit Lattice Multiplication 4-Digit × 3-Digit Lattice Multiplication 4-Digit × 4-Digit Lattice Multiplication 4-Digit × 5-Digit Lattice Multiplication 5-Digit × 4-Digit Lattice Multiplication 5-Digit × 5-Digit Lattice Multiplication

The distributive property of multiplication allows students to "split" factors into addends to make the multiplication easier. This also helps students learn to complete multiplication mentally rather than using paper and pencil methods. Often times, the numbers are split up by place value, so 123 becomes 100 + 20 + 3. If one wanted to multiply 123 by 4, you could multiply 100 by 4, 20 by 4 and 3 by 4 to get 400, 80 and 12 which sums to 492. If you multiply a multi-digit number by a multi-digit number, you can split both numbers into place value addends. For example, 938 × 74 = (900 + 30 + 4) × (70 + 4) = (900 × 70) + (900 × 4) + (30 × 70) + (30 × 4) + (8 × 70) + (8 × 4) = 63000 + 3600 + 2100 + 120 + 560 + 32 = 69412. All of the worksheets in this section have a model question included. Of course, it might be easier to complete questions with larger numbers using box multiplication.

- Multiplication Worksheets For Learning The Distributive Property Of Multiplication 2-Digit × 1-Digit Multiplication Using the Distributive Property ✎ 3-Digit × 1-Digit Multiplication Using the Distributive Property ✎ 4-Digit × 1-Digit Multiplication Using the Distributive Property ✎ 5-Digit × 1-Digit Multiplication Using the Distributive Property ✎ 2-Digit × 2-Digit Multiplication Using the Distributive Property ✎ 3-Digit × 2-Digit Multiplication Using the Distributive Property ✎

Multiplying on graph paper helps students "line up" their numbers when completing long multiplication questions. These worksheets include custom grids that have the right amount of room for one question.

- Multiplication with Grid Support 2-Digit × 1-Digit Multiplication with Grid Support ✎ 2-Digit × 2-Digit Multiplication with Grid Support ✎ 3-Digit × 1-Digit Multiplication with Grid Support ✎ 3-Digit × 2-Digit Multiplication with Grid Support ✎ 3-Digit × 3-Digit Multiplication with Grid Support ✎ 4-Digit × 1-Digit Multiplication with Grid Support ✎ 4-Digit × 2-Digit Multiplication with Grid Support ✎ 4-Digit × 3-Digit Multiplication with Grid Support ✎ 4-Digit × 4-Digit Multiplication with Grid Support ✎ 5-Digit × 1-Digit Multiplication with Grid Support ✎ 5-Digit × 2-Digit Multiplication with Grid Support ✎ 5-Digit × 3-Digit Multiplication with Grid Support ✎ 5-Digit × 4-Digit Multiplication with Grid Support ✎ 5-Digit × 5-Digit Multiplication with Grid Support ✎
- Multiplication with Grid Support (No Regrouping Boxes) 2-Digit × 1-Digit Multiplication with Grid Support (No Regrouping Boxes) ✎ 2-Digit × 2-Digit Multiplication with Grid Support (No Regrouping Boxes) ✎ 3-Digit × 1-Digit Multiplication with Grid Support (No Regrouping Boxes) ✎ 3-Digit × 2-Digit Multiplication with Grid Support (No Regrouping Boxes) ✎ 3-Digit × 3-Digit Multiplication with Grid Support (No Regrouping Boxes) ✎ 4-Digit × 1-Digit Multiplication with Grid Support (No Regrouping Boxes) ✎ 4-Digit × 2-Digit Multiplication with Grid Support (No Regrouping Boxes) ✎ 4-Digit × 3-Digit Multiplication with Grid Support (No Regrouping Boxes) ✎ 4-Digit × 4-Digit Multiplication with Grid Support (No Regrouping Boxes) ✎ 5-Digit × 1-Digit Multiplication with Grid Support (No Regrouping Boxes) ✎ 5-Digit × 2-Digit Multiplication with Grid Support (No Regrouping Boxes) ✎ 5-Digit × 3-Digit Multiplication with Grid Support (No Regrouping Boxes) ✎ 5-Digit × 4-Digit Multiplication with Grid Support (No Regrouping Boxes) ✎ 5-Digit × 5-Digit Multiplication with Grid Support (No Regrouping Boxes) ✎

In case you or your students want to make up your own questions, these blanks should expedite the process.

- Multiplication with Grid Support Blanks 2-Digit × 1-Digit Multiplication with Grid Support Blanks ✎ 2-Digit × 2-Digit Multiplication with Grid Support Blanks ✎ 3-Digit × 1-Digit Multiplication with Grid Support Blanks ✎ 3-Digit × 2-Digit Multiplication with Grid Support Blanks ✎ 3-Digit × 3-Digit Multiplication with Grid Support Blanks ✎ 4-Digit × 1-Digit Multiplication with Grid Support Blanks ✎ 4-Digit × 2-Digit Multiplication with Grid Support Blanks ✎ 4-Digit × 3-Digit Multiplication with Grid Support Blanks ✎ 4-Digit × 4-Digit Multiplication with Grid Support Blanks ✎ 5-Digit × 1-Digit Multiplication with Grid Support Blanks ✎ 5-Digit × 2-Digit Multiplication with Grid Support Blanks ✎ 5-Digit × 3-Digit Multiplication with Grid Support Blanks ✎ 5-Digit × 4-Digit Multiplication with Grid Support Blanks ✎ 5-Digit × 5-Digit Multiplication with Grid Support Blanks ✎

The halving and doubling strategy is accomplished very much in the same way as its name. Simply halve one number and double the other then multiply. In many cases, this makes the multiplication of two numbers easier to accomplish mentally. This strategy is not for every multiplication problem, but it certainly works well if certain numbers are involved. For example, doubling a 5 results in a 10 which most people would have an easier time multiplying. Of course, this would rely on the other factor being easily halved. 5 × 72, using the halving and doubling strategy (doubling the first number and halving the second in this case) results in 10 × 36 = 360. Practicing with the worksheets in this section will help students become more familiar with cases in which this strategy would be used.

- Halving and Doubling Multiplication Strategy Worksheets Halving and Doubling Strategy with Easier Numbers Halving and Doubling Strategy with Harder Numbers

Multiplying numbers in number systems other than decimal numbers including binary, quaternary, octal, duodecimal and hexadecimal numbers.

- Multiplying in Other Base Number Systems Worksheets Multiplying Binary Numbers (Base 2) Multiplying Ternary Numbers (Base 3) Multiplying Quaternary Numbers (Base 4) Multiplying Quinary Numbers (Base 5) Multiplying Senary Numbers (Base 6) Multiplying Octal Numbers (Base 8) Multiplying Duodecimal Numbers (Base 12) Multiplying Hexadecimal Numbers (Base 16) Multiplying Vigesimal Numbers (Base 20) Multiplying Hexatrigesimal Numbers (Base 36) Multiplying Various Numbers (Various Bases)

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## Long Multiplication

Long multiplication is considered a special method of multiplying large numbers that are 2-digits and more. The method to multiply numbers more than 10 is known as the long multiplication method. For this method, knowledge of the multiplication table from 1-10 is needed. In this section, we will learn about long multiplication by understanding the multiplication of large numbers, the column method of multiplication, and how to apply them while solving problems.

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## What is Long Multiplication?

Long multiplication is a method of multiplying 2 or more numbers together. Consider that we have to multiply any two numbers greater than 10 or 100, we usually perform long multiplication. The other name for long multiplication is the column method of multiplication as numbers can be multiplied in a column as well. Usually, finding the product of two numbers cannot be simple, that is when we use the long multiplication method.

Let us look at this example, consider 31 × 49. Here, we are multiplying 31 with 49 directly by writing one of these numbers in expanded form i.e. 31 = 30 + 1. 30 is the tenth part and 1 is the unit part. So 31 × 49 can be written as 30 × 49 + 1 × 49. First, we are multiplying 49 with 30 and then we are multiplying 49 with 1, and then we are adding them. So, instead of direct multiplication, we have performed long multiplication, which makes the process simple and accurate.

## Long Multiplication Column Method

The column method of multiplication is almost as same as the long multiplication method, the only difference is here in the long multiplication method we perform multiplication horizontally, and in the column method of multiplication, we perform multiplication vertically. Just like the long multiplication, the column method also has a step-wise procedure. They are:

- Step 1: Arrange the numbers in a column format according to their place value. The larger number is usually written on top.
- Step 2: Once arranged, start by multiplying the bottom number in the one's place with the top number.
- Step 3: Always remember to move right to left, hence when the result is obtained arrange it below the two numbers. Start multiplying the ten's number in the bottom number with the top number. Place the result by leaving the one's place empty or placing a zero.
- Step 4: Once the numbers are obtained, using the method of addition to arrive at the final solution.

Let us look at an example for a better understanding. Multiply 52 × 11.

Step 1: Vertically arrange the numbers as shown below.

Step 2: First multiply 52 with 1.

Step 3: Now multiply 52 with the 1 at the tenth place, here we are actually multiplying 52 with 10.

Step 4: Now add 52 and 520.

Therefore, 52 × 11 = 572.

## Long Multiplication With Decimals

The long multiplication method can be used in decimal numbers as well. Let us look at an example: Multiply 4.1 × 2.7.

While performing the multiplication, keep the smaller number to the right-hand side.

Step 1: Remove the decimal and convert the decimal number into a fraction.

4.1 × 2.7 = 41/10 × 27/10

Step 2: Now keep the 10s in the denominator.

41/10 × 27/10 = (41 × 27) / (10 × 10)

Step 3: Perform the long multiplication on numerators and keep the denominator aside for a while.

Step 4: Now divide the result of the multiplication by the denominator that we kept aside. To derive at a decimal number we convert 1107 by considering the two zeros from the denominator and counting the decimal point from the last number i.e. 7 towards the next number i.e. 0. Hence, the decimal point is placed two numbers after the last number.

(1107) / (10 × 10) = 1107 / 100 = 11.07

Therefore, 4.1 × 2.7 = 11.07

## Long Multiplication Horizontal Method

In long multiplication, one of the methods apart from the column method is the horizontal method. This method is mostly used among single-digit numbers and 2-digit numbers. Let us look at the step-wise procedure of solving long multiplication in the horizontal method:

- Step 1: Arrange the numbers horizontally next to each other in the normal multiplication format.
- Step 2: Start by multiplying the first number at the one's place with the other number.
- Step 3: Always move from right to left in long multiplication. Once the first number is done, multiply the number in the ten's place with the other number.
- Step 4: Once the result is obtained and written in a column format, use the method of addition to find the final solution.

Let us use the above example for a better understanding of multiplying 2-digits. Multiply 31 × 49

Step 1: Arrange the numbers horizontally and begin with multiplying 49 with 1.

Step 2: Now, multiply 49 with 3, and put a cross sign just below 9 (the unit place of the number 49), this cross sign represents a 0.

Step 3: Write zeros before the number 49 so that it will cover till the number 1 of 147, write the numbers just below another number so that the addition becomes easy.

Step 4: Add these two numbers 0049 and 1470.

Hence, 31 × 49 = 1519. In long multiplication, the cross represents a zero. Also, while multiplying numbers up to 3-digits, we have to add two crosses while we multiply to a hundred place number.

## Long Multiplication With Negative Numbers

The long multiplication on negative numbers follows the same rule as positive numbers, the only difference is the sign convention. We have to remember the sign conventions while multiplying the numbers. When a positive number is multiplied by a negative number, the solution is a negative number. Whereas, when two negative numbers are multiplied with each other, the solution is a positive number.

## Examples of Long Multiplication

Example 1: An NGO wants to create wall paintings on all the metro stations in a city. If a city has a total of 23 metro stations and the NGO decided to run the campaign in 19 such cities. How many metro stations can they cover?

Solution: Given,

Metro stations in a city = 23

Number of cities = 19

By using the horizontal method, we arrange the numbers 23 and 19 horizontally and start multiplying 9 with 23 first and move on to 1 with 23.

Total metro stations covered by the NGO will be:

Therefore, the NGO will paint a total of 437 metro stations.

Example 2: Help Paul to perform the multiplication 101 × 34

Solution: We can solve 101 × 34 by the long multiplication horizontal method.

Therefore, 101 × 34 = 3434

Example 3: Help Tim to multiply 3214 × 4561 by column method of long multiplication.

Solution: We can solve 3214 × 4561 by the column method of long multiplication.

Therefore, 3214 × 4561 = 14659054

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## Practice Questions on Long Multiplication

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## FAQs on Long Multiplication

What is meant by long multiplication.

Long multiplication is a method of multiplication used for multiplying numbers up to 2-digits and more. Long multiplication can be done in two ways - numbers written in a horizontal manner and numbers written in a column manner. Large numbers with more than 3-digits are multiplied using the long multiplication method. Multiplication of two numbers is also called the product of two numbers. Long multiplication can also be known as the column method of multiplication.

## What are the Steps for Long Multiplication Horizontal Method?

There 4 steps required to perform long multiplication on large numbers, they are:

- Arrange the numbers in a horizontal manner.
- Start by multiplying the first number at the one's place with the other number.
- Always move from right to left in long multiplication. Once the first number is completed, multiply the number in the ten's place with the other number.
- Once the result is obtained and written in a column format, use the method of addition to find the final solution.

## What are the Steps for Long Multiplication Column Method?

The steps for column-wise long multiplication are very similar to the normal long multiplication. The column method is done vertically and the 4 steps are:

- Arrange the numbers in a column format according to their place value.
- Start by multiplying the bottom number in the one's place with the top number.
- Once the result is obtained arrange it below the two numbers. Start multiplying the ten's number in the bottom number with the top number.
- Once the numbers are obtained, using the method of addition to arrive at the final solution.

## Can Long Multiplication be Used for Decimal Numbers?

Yes, the long multiplication method can be used for decimal numbers as well. While multiplying decimal numbers, always keep the smaller number to the right-hand side. For decimal numbers, both the horizontal and the vertical long multiplication methods can be used.

## Can Long Multiplication be Used for Negative Numbers?

Yes, the long multiplication method can be used for negative numbers with the help of a few rules such as:

- A positive number multiplied with a positive number will result in a positive number.
- A negative number multiplied with a positive number or vice versa will result in a negative number.
- A negative number multiplied by a negative number will result in a positive number.

## How to Solve Long Multiplication Problems?

Long multiplication problems can be solved using two methods - the long multiplication method and the column method. The long multiplication method requires the numbers to be written horizontally whereas the column method requires the numbers to be written vertically. Both methods help in solving problems with large numbers.

## How to do Long Multiplication Easily?

The long multiplication method can be performed easily and in a quick manner if we keep a few things in mind. For multiplication, it is always better to relate to addition as it is used to arrive at the final solution. Always remember the multiplication table from the numbers 1-10. For long multiplication, it is always helpful in breaking down the problem into steps for better understanding and arriving at the result faster.

## Long Multiplication: Definition with Examples

What is long multiplication, how to multiply using long multiplication, long multiplication column method, solved examples on long multiplication, practice problems on long multiplication, frequently asked questions on long multiplication.

Long multiplication is a method of multiplying large numbers easily. It simplifies the process of multiplication of two numbers that are not otherwise easy to multiply.

For example, we can easily find the product of $55 \times 20$ by multiplying 55 by 2 and then adding a 0 at the rightmost place of the answer.

$55 \times 2 = 110$ and $55 \times 20 = 1100$

However, sometimes, finding the product is not this easy. In such cases, we use the method of long multiplication. It is often termed as the long hand multiplication (long multiplication by hand).

## Long Multiplication: Definition

Long multiplication is a method of multiplication using which multiplying two large numbers that have two or more digits becomes easier.

But how to multiply large numbers? Many times, finding the product is not this easy. In such cases, we use the long method of multiplication.

For example: $47 \times 63 =$ ?

We can find the answer in simple steps using long multiplication. Let’s discuss the method in detail in the next section.

Let’s discuss the steps for multiplying a two-digit number by a two-digit number using the long multiplication method.

Let us multiply 47 by 63 using the long multiplication method .

Step 1: First, write the two numbers, one below the other, such that their place values are aligned. We generally write the bigger number on top and a multiplication sign on the left and draw a line below the numbers as shown below.

Important Note:

- In the long multiplication method, the number on the top (63) is called the multiplicand . The number by which it is multiplied, that is, the number at the bottom (47), is called the multiplier .
- If you write the small number on the top and perform the multiplication, the answer will still be the same since the order does not matter while multiplying two numbers.

Step 2: Multiply the ones digit of the top number by the ones digit of the bottom number.

Write the product as shown in the image. Don’t forget the carryover that goes in the next place!

Step 3: Multiply the tens digit of the top number by the ones digit of the bottom number. Add the carryover.

This is our first partial product which we got by multiplying the top number by the ones digit of the bottom number.

Step 4: Now, we place a 0 below the ones digit as shown. This is because we will now be multiplying the digits of the top number by the tens digit of the bottom number.

Step 5: Multiply the ones digit of the top number by the tens digit of the bottom number.

Step 6: Multiply the tens digit of the top number by the tens digit of the bottom number.

This is the second partial product obtained by multiplying the top number by the tens digit of the bottom number.

Step 7: Now, add the two partial products.

Let’s summarize.

We follow the above long multiplication steps only for multiplying numbers greater than two digits.

## Related Worksheets

Long multiplication is also known as the column method of multiplication since we perform the multiplication vertically or column wise.

Let’s take another example to understand this better.

Multiply 321 by 23.

Take a look at the long multiplication chart that shows the long multiplication step by step. You can refer to it to avoid mistakes when solving long multiplication problems.

## Multiplying Decimals Using Long Multiplication

Let’s understand how to multiply decimals with the help of the long multiplication method.

Example: Multiply $3.6 \times 5.5$

Step 1 : First, we place the smaller number out of the two on the right-hand side and change the decimal number to a fraction.

$5.5 \times 3.6 = \frac{55}{10} \times \frac{36}{10}$

Step 2 : Then, we multiply the numerators using the steps of the long multiplication method. We leave the denominator as it is for now.

Step 3 : Now, we divide the answer (that we got in step 2) by the denominator to get the final answer.

$\frac{1980}{100} = 19.80$

## Long Multiplication Using the Horizontal Method

Let’s now understand how to use the horizontal method with the help of a long multiplication example. As the name suggests, we multiply the numbers horizontally but follow the same steps as we discussed earlier.

Example: Multiply 48 by 24.

Step 1 : First, we write the numbers beside each other and place them horizontally.

$48 \times 24$

Step 2 : Now, we multiply the second number (number on the right side) with the ones digit of the first number.

Here, we multiply 24 with 8, since 8 is at the ones place in 48.

$24 \times 8 = 192$

Step 3 : Now, we multiply the second number with the tens digit of the first number.

Here, we multiply 24 with 40, since 4 is at the tens place in 48.

$24 \times 40 = 960$

Step 4 : Finally, we add the partial products obtained from steps 2 and 3 to get the final answer.

Thus, $48 \times 24 = 1152$

## Long Multiplication with Negative Numbers

The multiplication of negative numbers using the method of long multiplication follows the same steps as mentioned above. However, we have to carefully consider the signs of the two numbers in order to decide the sign of the product.

For instance, when we multiply two negative numbers with each other, the sign of the result remains positive. On the other hand, it becomes negative when a negative number and a positive number are multiplied.

The figure below shows how the sign changes upon multiplication with negative numbers:

## Fun Facts about Long Multiplication

- Long multiplication is also known as the column method of multiplication.
- Whenever any number is multiplied by zero, the answer is also zero.
- When the number 9 is multiplied by any other number, the sum of digits in the product is always 9.

For example, $9 \times 25 = 225$ and $2 + 2 + 5 = 9$;

$9 \times 9 = 81$ and $8 + 1 = 9$.

- Long multiplication can also be referred to as repeated addition because multiplying any number is just an alternative to adding it repeatedly.
- Whenever any even number from 0 to 9 is multiplied by 6, the product has the same even number in the ones digit.

For example, $6 \times 4 = 24,\; 6 \times 2 = 12$

- Whenever two large numbers (let’s say three-digit or four-digit numbers) are multiplied, the process of long multiplication remains the same. For example:

Long multiplication is a method that simplifies the multiplication of large numbers, including two-digit, three-digit numbers, etc., with each other. Let’s solve a few examples and practice problems based on long multiplication!

1. Multiply 38 by 91 using the column method.

Solution:

2. Multiply 72 by 44 using the column method.

Solution :

3. Multiply – 58 by 30.

The sign of the product when we calculate $(\;-\;58) \times (30)$ will be negative.

Since 58 has a negative sign, the final answer will be $ – 1740$.

$(\;-\;58) \times (30) = \;-\;1740$

4. Alia bought 42 notebooks, each worth $\$9.6$ . How much did she spend?

To find the amount Alia spent, multiply 9.6 by 42.

$9.6 \times 42 = \frac{96}{10} \times 42$

Now, we divide 4032 by 10 to get the final answer, which is $403.2$.

$9.6 \times 42 = 403.2$

Thus, Alia spent $403.2 on notebooks.

5. Multiply $-70$ and $-15$ .

Let’s find $(70) \times (15)$.

Since both the numbers are negative, the final answer will remain positive, $1050$.

$(70) \times (15) = 1050$

Attend this quiz & Test your knowledge.

## Multiply 95 by 13 using the column method.

## Multiply the decimals 1.2 and 6.7.

Multiply the negative numbers $(-\;89)$ and $(-\;61)$..

## Multiply 45 by 21 using the horizontal method.

Multiply 530 by 22..

What are negative numbers?

Negative numbers are those whose value is less than zero and thus have a negative sign (-) before them.

What are positive numbers?

Positive numbers are those whose value is greater than zero.

What is a decimal?

A decimal number represents the number that separates a whole number from a fractional number. It is represented by a dot (.) sign.

What is the meaning of the horizontal method of multiplication?

The horizontal method refers to a method wherein numbers are arranged in a horizontal line from left to right.

What is the meaning of the column method of multiplication?

The column method refers to a method wherein numbers are placed one below the other from top to bottom.

What is short multiplication?

Short multiplication is the method of multiplication commonly used when multiplying a three-digit or larger number with a single-digit number.

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## Solve Problems with Multiplication-Reasoning and Problem Solving

## Solve Problems with Multiplication - Reasoning and Problem Solving

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## Long multiplication

Subject: Mathematics

Age range: 7-11

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## Developing Logical Mathematical Intelligence

How to unlock your problem-solving potential

Cynthia Vinney, PhD is an expert in media psychology and a published scholar whose work has been published in peer-reviewed psychology journals.

- Characteristics and Examples
- Develop Logical Mathematical Intelligence
- Tips and Strategies

Logical mathematical intelligence is one of eight intelligences that Howard Gardner, a professor of cognition and education at Harvard University, proposed in his theory of multiple intelligences , which he outlined in his book Frames of Mind . Gardner's theory posited these types of intelligence:

- Visual-spatial
- Linguistic-verbal
- Logical-mathematical
- Body-kinesthetic
- Interpersonal
- Intrapersonal
- Naturalistic

According to Rebecca Mannis, PhD and founder and learning specialist at Ivy Prep, for a long time American psychologists viewed intelligence as a single factor, typified by Lewis Terman in the early to mid-1900s, who referred to intelligence as “g.” Similarly, Kimberly Nix Berens, PhD , founder of Fit Learning and author of "Blind Spots: Why Students Fail and the Science That Can Save Them," explains that intelligence is often viewed as an innate ability, but there is a lack of scientific evidence to support this.

Instead, Gardner theorized that there are a number of kinds of intelligence that we each may have, that we may vary in the degree we have of each one, and that we may have more or less of each of these throughout our lifetimes. Logical mathematical intelligence is one of the intelligences he described, and it represents the ability to use numbers effectively, to reason well, and to recognize and solve problems using logical patterns.

In this article, we’ll review the characteristics of logical mathematical intelligence, discover how to develop it, explore strategies to enhance it, and look at the benefits of this kind of intelligence.

Anchiy / E+ / Getty

## Characteristics and Examples of Logical Mathematical Intelligence

People who have logical mathematical intelligence solve problems using logic, can quickly calculate math problems , and like when things are categorized in a rational way. They’re also good at understanding patterns, the relationships between things, and understanding complex ideas.

Thus, logical-mathematical intelligence encompasses the following:

- Being good with numbers
- Understanding logical concepts
- Having good reasoning skills
- Enjoying experiments
- Enjoying solving puzzles and mysteries
- Being good at manipulating numbers and operations
- Being good at understanding and applying scientific principles

According to Berens, examples of logical mathematical intelligence include “fluently solving multi-step equations, [solving] language-based math problems , interpreting and analyzing scientific findings, and designing experiments.”

Mannis explains that mastery of basic facts and concepts that leads to the ability to adapt those facts and concepts into more complex tasks and to new settings is a hallmark of logical mathematical intelligence.

For instance, Mannis gives the example of a third grader with strong logical mathematical intelligence who not only knows how to distinguish between perimeter and area but can also use those formulas for other things. The child can apply “that information to a math challenge to design a playground… where they are tasked with determining what the area of a complex shape would be if there were sections of semicircles or squares within that figure ‘cut out.’ They would also be able to then identify the cost of paving the entire play area given a particular per foot fee.”

For those of us who are more inclined toward verbal linguistics or another type of intelligence, this may seem beyond our reach, but it is possible to build your strength in the more mathematical areas of intelligence.

## How to Develop Logical Mathematical Intelligence

Decades of research indicate that higher-level skills, such as those seen in logical mathematical intelligence, require the mastery of foundational skills, says Berens. As a result, “Young learners require explicit instruction and repeated practice to fluency in core math skills such as numeracy, basic computation, fractions, decimals, percents, and ratios," she says. "They also require fluent reading skills and fluent comprehension, problem-solving, and critical thinking skills.”

Mannis explains that some children are better at memorizing facts and drawing connections between concepts than others, but there are ways to develop these skills.

These methods include:

- Teacher education: According to Mannis, teachers must have a “strong understanding of how children develop these skills and methods [by] first teaching this content systematically and then gradually ‘scaffolding’ or adding complexity and integration."
- Focusing on basic math: As Berens indicates students must attain fluency in the basics before they can begin applying these core skills for mastery of highly complex skills, including algebra, geometry, and calculus.
- Engagement beyond the classroom: Encouraging children to see how mathematical concepts can be part of their real lives can help develop logical mathematical intelligence skills.

Mannis provides an example of the last point: “A child usually reads a book in four days and would like to borrow the series to read during the three weeks between the end of the school year and sleep away camp. How can they estimate how many books they will get through? How does being free of homework shift their estimate? That is an example of living math that offers a chance to systematize, use concepts such as ratios and estimating, and also encourage them to engage through creating a system to make their estimate.”

## Strategies for Enhancing Logical Mathematical Intelligence

To enhance logical mathematical intelligence you first have to learn the basics, so explicit instruction and repeated practice in math skills is essential, says Berens.

Moreover, per Mannis, to enhance skills in logical mathematical intelligence further:

- Provide opportunities to create systems and patterns, and solve logic problems
- Encourage ‘metacognitive awareness,’ or being aware of how you think, and talk through this approach
- Balance learning facts and math operations with real-world problem-solving

For example, Mannis speaks of a middle school class she consulted with that timed its geometry unit so it was right before the school carnival. “After completing the basics of the course, the students were put in charge of designing, creating, and manning some of the carnival stations using” what they learned. This allowed them to use the skills they got from the course and enhance their logical mathematical intelligence.

## Benefits of Logical Mathematical Intelligence

People with logical mathematical intelligence are good at rational thinking, analyzing problems logically, and thinking about issues scientifically. “Not only does mastery of high-level math skills produce long-term academic success," says Berens, "but it also gives learners access to careers in science, technology, and engineering.”

We have a lot of complex problems in these fields to solve, such as climate change, and we need people with logical mathematical intelligence to solve them, she says.

While some people may have more innate ability with logical mathematical intelligence, anyone can enhance their abilities. Developing math skills, engaging in strategy games and logic problems, explaining your thinking, and using your skills in the real world can help develop your logical mathematical intelligence and have you on your way to unlocking your problem-solving potential.

Gardner H. Frames of Mind: The Theory of Multiple Intelligences . 10th anniversary ed. BasicBooks; 1993.

Arani HK, Mobarakeh SD. Metacognitive strategies and logical/mathematical intelligence in EFL context: Investigating possible relationships. TPLS . 2012;2(2):304-313. doi:10.4304/tpls.2.2.304-313

Šafranj J. Logical/mathematical intelligence in teaching English as a second language . Procedia - Social and Behavioral Sciences . 2016;232:75-82. doi: 10.1016/j.sbspro.2016.10.019

By Cynthia Vinney, PhD Cynthia Vinney, PhD is an expert in media psychology and a published scholar whose work has been published in peer-reviewed psychology journals.

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## Reasoning skills of large language models are often overestimated

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When it comes to artificial intelligence, appearances can be deceiving. The mystery surrounding the inner workings of large language models (LLMs) stems from their vast size, complex training methods, hard-to-predict behaviors, and elusive interpretability.

MIT's Computer Science and Artificial Intelligence Laboratory (CSAIL) researchers recently peered into the proverbial magnifying glass to examine how LLMs fare with variations of different tasks, revealing intriguing insights into the interplay between memorization and reasoning skills. It turns out that their reasoning abilities are often overestimated.

The study compared “default tasks,” the common tasks a model is trained and tested on, with “counterfactual scenarios,” hypothetical situations deviating from default conditions — which models like GPT-4 and Claude can usually be expected to cope with. The researchers developed some tests outside the models’ comfort zones by tweaking existing tasks instead of creating entirely new ones. They used a variety of datasets and benchmarks specifically tailored to different aspects of the models' capabilities for things like arithmetic, chess, evaluating code, answering logical questions, etc.

When users interact with language models, any arithmetic is usually in base-10, the familiar number base to the models. But observing that they do well on base-10 could give us a false impression of them having strong competency in addition. Logically, if they truly possess good addition skills, you’d expect reliably high performance across all number bases, similar to calculators or computers. Indeed, the research showed that these models are not as robust as many initially think. Their high performance is limited to common task variants and suffer from consistent and severe performance drop in the unfamiliar counterfactual scenarios, indicating a lack of generalizable addition ability. The pattern held true for many other tasks like musical chord fingering, spatial reasoning, and even chess problems where the starting positions of pieces were slightly altered. While human players are expected to still be able to determine the legality of moves in altered scenarios (given enough time), the models struggled and couldn’t perform better than random guessing, meaning they have limited ability to generalize to unfamiliar situations. And much of their performance on the standard tasks is likely not due to general task abilities, but overfitting to, or directly memorizing from, what they have seen in their training data. “We’ve uncovered a fascinating aspect of large language models: they excel in familiar scenarios, almost like a well-worn path, but struggle when the terrain gets unfamiliar. This insight is crucial as we strive to enhance these models’ adaptability and broaden their application horizons,” says Zhaofeng Wu, an MIT PhD student in electrical engineering and computer science, CSAIL affiliate, and the lead author on a new paper about the research. “As AI is becoming increasingly ubiquitous in our society, it must reliably handle diverse scenarios, whether familiar or not. We hope these insights will one day inform the design of future LLMs with improved robustness.” Despite the insights gained, there are, of course, limitations. The study’s focus on specific tasks and settings didn’t capture the full range of challenges the models could potentially encounter in real-world applications, signaling the need for more diverse testing environments. Future work could involve expanding the range of tasks and counterfactual conditions to uncover more potential weaknesses. This could mean looking at more complex and less common scenarios. The team also wants to improve interpretability by creating methods to better comprehend the rationale behind the models’ decision-making processes. “As language models scale up, understanding their training data becomes increasingly challenging even for open models, let alone proprietary ones,” says Hao Peng, assistant professor at the University of Illinois at Urbana-Champaign. “The community remains puzzled about whether these models genuinely generalize to unseen tasks, or seemingly succeed by memorizing the training data. This paper makes important strides in addressing this question. It constructs a suite of carefully designed counterfactual evaluations, providing fresh insights into the capabilities of state-of-the-art LLMs. It reveals that their ability to solve unseen tasks is perhaps far more limited than anticipated by many. It has the potential to inspire future research towards identifying the failure modes of today’s models and developing better ones.” Additional authors include Najoung Kim, who is a Boston University assistant professor and Google visiting researcher, and seven CSAIL affiliates: MIT electrical engineering and computer science (EECS) PhD students Linlu Qiu, Alexis Ross, Ekin Akyürek SM ’21, and Boyuan Chen; former postdoc and Apple AI/ML researcher Bailin Wang; and EECS assistant professors Jacob Andreas and Yoon Kim.

The team’s study was supported, in part, by the MIT–IBM Watson AI Lab, the MIT Quest for Intelligence, and the National Science Foundation. The team presented the work at the North American Chapter of the Association for Computational Linguistics (NAACL) last month.

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## COMMENTS

Reasoning and Problem Solving Multiply 3 Digits by 2 Digits Reasoning and Problem Solving Multiply 3 Digits by 2 Digits Developing 1a. Omar is correct. 302 x 21 = 6,342 2a. Various possible answers where the total is less than 5,064, for example: 332 x 12 = 3,984 3a. Harold is correct. 130cm x 61cm = 7,930cm² so Derek will need to buy 2 tins ...

Long multiplication method: guide for teaching your Year 5 and Year 6 pupils long multiplication using the long multiplication method. ... 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.

Reasoning and Problem Solving Step 2: Multiply 2 Digits 1 National Curriculum Objectives: Mathematics Year 5: (5C6a) Multiply and divide numbers mentally drawing upon known facts Mathematics Year 5: (5C7a) Multiply numbers up to 4 digits by a one- or two-digit number using a formal written method, including long multiplication for two-digit numbers

Reasoning and Problem Solving Step 5: Multiply 4-Digits by 2-Digits National Curriculum Objectives: ... Multiply numbers up to 4 digits by a one- or two-digit number using a formal written method, including long multiplication for two-digit numbers Differentiation: Questions 1, 4 and 7 (Problem Solving) Developing Find missing numbers in a ...

7a. 6,425 x 37 = 237,725 8a. An explanation that recognises Philip has not added on the 2 hundreds which were carried over from the multiplication of the 4 tens and the 6 ones. 9a. Various answers, for example: 4,735 x. 14 = 66,290 and 4,726 x 14 = 66,164. Reasoning and Problem Solving Multiply 4-Digits by 2-Digits.

Begin by multiplying 43 by the first part, 60. This can be done by multiplying 43 x 6, then adding a zero to the answer. 43 x 6 is 258. Then add a zero, so it would be 2580. 3. Next we will multiply 43 by the second part 5. 43 x 5 is 215. 4. Finally, we add the two partial products together to get the final answer.

Reasoning and Problem Solving - Multiply 2 Digits by 2 Digits - Teaching Information. 1a.Multiply 2 digits by 2 digits using the cards below to create an odd number. 5. x. D. 1b. Multiply 2 digits by 2 digits using the cards below to create an even number. 4. x.

A year 5 long multiplication lesson pack on multiplying 3- and 4-digits by 2-digits, using the formal method of long multiplication. ... reasoning and problem-solving activities. This lesson supports the year 5 national curriculum aim 'Multiply numbers up to 4 digits by a two-digit number using a formal written method, including long ...

This page includes Long Multiplication worksheets for students who have mastered the basic multiplication facts and are learning to multiply 2-, 3-, 4- and more digit numbers. Sometimes referred to as long multiplication or multi-digit multiplication, the questions on these worksheets require students to have mastered the multiplication facts from 0 to 9.

Let us look at an example: Multiply 4.1 × 2.7. While performing the multiplication, keep the smaller number to the right-hand side. Step 1: Remove the decimal and convert the decimal number into a fraction. 4.1 × 2.7 = 41/10 × 27/10. Step 2: Now keep the 10s in the denominator.

Step 1: First, we place the smaller number out of the two on the right-hand side and change the decimal number to a fraction. 5.5 × 3.6 = 55 10 × 36 10. Step 2: Then, we multiply the numerators using the steps of the long multiplication method. We leave the denominator as it is for now. Step 3: Now, we divide the answer (that we got in step 2 ...

Math explained in easy language, plus puzzles, games, quizzes, videos and worksheets. For K-12 kids, teachers and parents. Long Multiplication Worksheets Worksheets » Long Multiplication Worksheet Example; Easy : 6 × 15: Simple : 11 × 44: Intermediate : 15 × 67: Hard : 34 × 129:

Expected Solve the word problems given in a real-life context. Multiplying 4-digits by 2-digits with up to 3 exchanges. Greater Depth Solve the word problems given in a real-life context. Multiplying 4-digits by 2-digits with multiple exchanges where numbers are represented in numerals and words. Questions 3, 6 and 9 (Reasoning and Problem Solving)

These long multiplication word problems are great for testing your KS2 children's knowledge of multiplications sums. The differentiated worksheets will help your KS2 class understand the concept of long worded problems and how to pick out the important information they need, to work out the sum. Answers are included to make your life easier!

This fantastic Diving into Mastery teaching pack has been written to complement version 3.0 of the White Rose Maths scheme of learning for year 5 spring term block 1 small step 3 'Multiply a 2-digit number by a 2-digit number'. The pack contains a variety of fluency, reasoning and problem-solving questions all giving children the opportunity to ...

Long multiplication Year 5. Subject: Mathematics. Age range: 7-11. Resource type: Worksheet/Activity. Mrs SCM. 4.56 82 reviews. ... docx, 25.75 KB docx, 102.91 KB docx, 36.34 KB docx, 25.75 KB. Green- fluency Red- reasoning Purple- problem solving. Creative Commons "Sharealike" Reviews. 4 Something went wrong, please try again later. Brommie. 4 ...

Reasoning and Problem Solving Multiply 2 Digits by 1 Digit 1 Reasoning and Problem Solving Multiply 2 Digits by 1 Digit 1 Developing 1a. Destiny is correct. Rehan has added 2 to each number instead of multiplying. 2a. 12; 2 x 12 = 24 3a. 23 x 3 = 69 Expected 4a. Brody is correct. Rose has added the numbers in the ones column. 5a. 11; 8 x 11 ...

Long Multiplication Worksheet. Math explained in easy language, plus puzzles, games, quizzes, videos and worksheets. For K-12 kids, teachers and parents.

Year 5 Diving into Mastery: Step 1 Multiply up to a 4-Digit Number by a 1-Digit Number Teaching Pack 5 reviews. Year 6 Formal Long Multiplication Homework Tasks with Adult Support Activity Pack. KS2 Reasoning Test Practice Missing Number Calculations 2-Digit by 2-Digit Multiplication Resource Pack 1 review.

multiplying. Practice Questions. Parallel and Perpendicular Lines (graphs) Practice Questions. The Corbettmaths Practice Questions on Multiplication.

This worksheet includes a range of reasoning and problem solving questions for pupils to practise the main skill of Solve Problems with Multiplication. ... Long Term Plan; Autumn Block 1 (Ready to Write) Autumn Block 2 (Punctuating Sentences) ... Solve Problems with Multiplication reasoning and problem solving worksheet

Age range: 7-11. Resource type: Worksheet/Activity. File previews. pdf, 18.87 KB. pdf, 276.47 KB. pdf, 245.04 KB. pdf, 15.6 KB. This selection of activities and worksheets will give children some varied practice to help embed the concept. Creative Commons "Sharealike".

This comprehensive Diving into Mastery teaching pack supports the White Rose Maths Y6 small step 7: 'Multiply up to a 4-digit number by a 2-digit number'. Included in the pack is an easy-to-follow PowerPoint that contains fluency, reasoning and problem-solving activities for your children to work through together. The accompanying activity sheets are the ideal resource for children to work ...

According to Rebecca Mannis, PhD and founder and learning specialist at Ivy Prep, for a long time American psychologists viewed intelligence as a single factor, typified by Lewis Terman in the early to mid-1900s, who referred to intelligence as "g." Similarly, Kimberly Nix Berens, PhD, founder of Fit Learning and author of "Blind Spots: Why Students Fail and the Science That Can Save Them ...

It turns out that their reasoning abilities are often overestimated. The study compared "default tasks," the common tasks a model is trained and tested on, with "counterfactual scenarios," hypothetical situations deviating from default conditions — which models like GPT-4 and Claude can usually be expected to cope with.