Math Antics - Multi-Digit Multiplication Pt 2

Summary

In this Math Antics video, viewers are guided through the advanced concept of multiplying multi-digit numbers. The tutorial begins with a refresher on single-digit multiplication and progresses to a detailed explanation of multiplying a three-digit number by a two-digit number. The process involves breaking down the problem into multiple steps, multiplying each digit of the bottom number by the top number, and then combining the results. Emphasis is placed on the importance of carrying over and aligning digits correctly. The video uses a humorous approach to explain the method, making it accessible and engaging. The tutorial concludes with a practical example, multiplying 324 by 46, and encourages practice for mastery.

Takeaways

• 📚 The video introduces the method for multiplying a multi-digit number by another multi-digit number, building upon the previously learned technique of multiplying by a one-digit number.
• 🔢 The process involves breaking down the multiplication into a series of steps, one for each digit of the bottom number, treating each as if it were a separate problem with a one-digit bottom number.
• 📝 For a two-digit bottom number, you perform the multiplication twice, once for each digit, resulting in two different intermediate answers.
• 😯 The different intermediate answers are not errors but represent the results of the multiplication procedure for each bottom digit separately.
• 🧮 To handle the multiplication of a multi-digit bottom number, you multiply each digit of the bottom number by each digit of the top number, similar to the process used for a one-digit bottom number.
• ✏️ When multiplying, carry over any digits as necessary, and remember to shift the position of the carry to align with the next digit of the top number.
• 🔄 It's crucial to account for the place value of the bottom number's digits; for example, a digit in the tens place represents a value ten times greater, requiring an additional zero in the multiplication result.
• 📈 The video uses the example of multiplying 324 by 46 to demonstrate the step-by-step process, highlighting the importance of carrying and aligning digits correctly.
• 📝 After completing the multiplication steps for each digit of the bottom number, you add the two intermediate answers together to get the final result.
• 🔍 The final step is to verify the result using a calculator to ensure accuracy, emphasizing the importance of checking work in mathematical computations.

Q & A

What is the main topic of the Math Antics video?

The main topic of the Math Antics video is learning how to multiply a multi-digit number by another multi-digit number.

What is the procedure for multiplying a multi-digit number by a one-digit number?

The procedure involves breaking the problem into a series of multiplication steps, one for each of the top digits, starting with the ones place and working left until all digits are multiplied.

How does the multiplication process change when the bottom number has more than one digit?

When the bottom number has more than one digit, the same procedure is performed for each bottom digit, resulting in multiple steps and potentially different intermediate answers.

Why are there two different answers when multiplying a two-digit bottom number by a multi-digit top number?

There are two different answers because the process involves treating the multiplication as if it were two separate problems, one for each digit of the bottom number.

What is the significance of the zeros placed in the answer line when starting the second set of multiplication steps?

The zeros are placed to account for the value of the bottom digit being in the tens place, making the result 10 times bigger than it would be if the digit were in the ones place.

What is the purpose of carrying over digits in the multiplication process described in the video?

Carrying over digits is necessary when the product of a multiplication step is a two-digit number, to ensure the correct place value is maintained in the final answer.

How do you handle the carrying over of digits from the first answer line to the second in the video's example?

Before carrying over digits to the second answer line, all carried digits from the first answer line are crossed out to avoid double counting.

What is the final step in solving the multiplication problem presented in the video?

The final step is to add the two separate answer lines together to obtain the final result of the multiplication problem.

Why is it recommended to put the number with the fewest digits on the bottom when multiplying?

Placing the number with the fewest digits on the bottom simplifies the multiplication process, as it reduces the number of steps and carrying over of digits required.

What resource is mentioned for additional practice and examples in the video?

The video mentions 'Prob with Rob' video examples and exercises on the Math Antics website for additional practice.

Outlines

00:00

📚 Introduction to Multi-Digit Multiplication

The video begins with a recap of basic multi-digit multiplication, focusing on multiplying a multi-digit number by a one-digit number. It then introduces the concept of multiplying a multi-digit number by another multi-digit number. The presenter explains that the process involves breaking down the problem into a series of multiplication steps for each digit of the bottom number. The video uses the example of multiplying 324 by 46 to illustrate the process, emphasizing the need to multiply each digit of the bottom number by each digit of the top number, resulting in two separate answers that are then added together. The presenter also highlights the importance of carrying over digits and shifting the place value appropriately when dealing with multi-digit numbers.

05:01

🔢 Completing the Multiplication Process

This paragraph continues the explanation of the multiplication process, focusing on the second bottom digit and its impact on the multiplication. The presenter demonstrates how to multiply the second bottom digit (4 in the example) by each digit of the top number (324), taking into account the place value of the bottom digit. The video shows the importance of carrying over digits and adjusting the place value by adding a zero to the beginning of the second answer line to account for the tens place of the bottom digit. The presenter then completes the multiplication steps, resulting in two separate answer lines. The final step is to add these two answers together to obtain the final result of the multiplication problem. The video concludes with a reminder to practice the procedure and offers additional resources for subscribers, including extra problems and exercises.

Keywords

💡Multi-digit multiplication

Multi-digit multiplication refers to the process of multiplying numbers that have more than one digit. In the video, this concept is central as it explains how to multiply a multi-digit number by another multi-digit number, building upon the simpler concept of multiplying a multi-digit number by a one-digit number. The script uses the example of multiplying 324 by 46 to illustrate the step-by-step process, emphasizing the importance of handling each digit of the bottom number separately.

💡Carrying

Carrying is a technique used in addition and multiplication where a value exceeding the base (usually 10 in the decimal system) is added to the next higher place value. In the script, carrying is mentioned as an essential step when the product of a multiplication step is two digits or more, such as when multiplying 6 by 4 to get 24, where the '2' is carried over.

💡Place value

Place value is the numerical value of a digit based on its position within a number. The video script explains that when multiplying a multi-digit number, each digit of the bottom number is treated as if it were in a different place value, which affects the multiplication process. For instance, the second digit of the bottom number (which is in the tens place) is multiplied as if it were 40, not just 4.

💡One-digit bottom number

The term 'one-digit bottom number' is used in the script to contrast with the main focus of the video, which is multiplying by a multi-digit bottom number. It refers to a simpler multiplication scenario where the bottom number in a multiplication problem has only one digit, making the process easier as it requires fewer steps.

💡Multiplication steps

Multiplication steps are the individual multiplication operations performed on each digit of the bottom number when multiplying multi-digit numbers. The script outlines a series of these steps, such as multiplying each digit of 46 by each digit of 324, which results in a series of partial products that are then added together to get the final result.

💡Partial products

Partial products are the intermediate results obtained during the multiplication of multi-digit numbers. In the script, after multiplying each digit of the bottom number by the top number, the results are referred to as partial products, which are then added together to get the final answer. For example, when multiplying 324 by 46, there are partial products for 6×324 and 4×324.

💡Zero-padding

Zero-padding in the context of the video refers to the practice of adding zeros to the beginning of a number to increase its place value. This is done when starting the multiplication steps for the second or subsequent digits of the bottom number, as seen when the script instructs to put a zero in the first spot for the second set of multiplication steps involving the digit '4' from the bottom number.

💡Columnar multiplication

Columnar multiplication, also known as long multiplication, is a method of multiplying numbers where the process is written out in columns. The video script describes this method in detail, showing how each digit of the bottom number is multiplied by each digit of the top number, with the results aligned in columns and then summed to find the final product.

💡Digit by digit

The phrase 'digit by digit' is used in the script to emphasize the process of multiplying each digit of the bottom number by the entire top number in turn. This is a key aspect of columnar multiplication, where the multiplication is broken down into smaller steps for each digit, as demonstrated in the multiplication of 324 by 46.

💡Final answer

The 'final answer' in the context of the video is the result obtained after completing all the multiplication steps and adding up the partial products. The script guides the viewer through the process of finding this final answer, which in the example given is 14,904, after adding the partial products from the multiplication of 324 by 46.

Highlights

Introduction to multi-digit multiplication, building on one-digit multiplication techniques.

Explanation of the process for multiplying a multi-digit number by another multi-digit number.

Breaking down the multiplication into a series of steps for each bottom digit.

Multiplying each digit of the bottom number separately to get different intermediate answers.

The concept of treating the problem as two separate multiplications with one-digit bottom numbers.

Procedure for carrying over in multiplication when dealing with two-digit results.

The importance of adding the results from each multiplication step to get the final answer.

Practical example of multiplying 324 by 46, demonstrating the step-by-step process.

Detailing the multiplication steps for the first digit of the bottom number (6 in 324).

Carrying over and adding the carried value to the next multiplication step.

Explanation of the need to shift the second answer line for the second bottom digit due to its place value.

Procedure for the second set of multiplication steps involving the second bottom digit (4 in 324).

The rule of shifting the answer line by adding zeros for each additional bottom digit.

Final step of adding the two intermediate answers to obtain the final result of the multiplication.

Verification of the multiplication result using a calculator for accuracy.

Encouragement for practice and re-watching the video for better understanding.

Invitation for subscribers to access additional problem-solving videos for practice.

Transcripts

Hi! Welcome to Math Antics. In our last video, we learned the basics of multi-digit multiplication.

We learned how to multiply a multi-digit number by a one-digit number.

But in this video, we’re gonna take it to the next level.

We’re gonna learn how to multiply a multi-digit number by another multi-digit number.

You remember the procedure for multiplying when we have a one-digit bottom number, right?

You break up the problem into a series of multiplication steps, one for each of the top digits.

And in each step, you just multiply the bottom digit by a top digit,

starting with the ones place and working your way left until you’ve multiplied all of the digits.

Well, when the bottom number of your multiplication problem also has more than one digit,

you have to do that same procedure we learned for EACH bottom digit.

For example, if you have a two-digit bottom number,

you multiply the first digit by each top digit, and then you multiply the second digit by each top digit.

That means you’re gonna have twice as many steps to do,

AND it means you’re gonna get two different answers!

What?!! - How can we have two different answers for the same problem?

Now don’t panic.

The different answers are just what you get from doing the multiplication procedure for each digit of the bottom number separately.

In fact, it’s kind of like we’re pretending we have two SEPARATE multiplication problems that each have a one-digit bottom number

…which is nice, cuz I kinda like pretending…

No Luke… I AM your father!

NO! That’s IMPOSSIBLE!

Ha Ha Ha!

Ahh… wait…no…

Nooooooooooo…..

Ha Ha Ha Ha Ha…

Err… well… I mean… it’s nice because we already know how to multiply when we have a one-digit bottom number.

But then what will we do with the two different answers we’re gonna get?

Well, it turns out that all we have to do is add them together once we’re finished doing all of our multiplication steps.

Are you ready to see an example? It should make a lot more sense when you see the procedure in action.

So let’s multiply 324 (a three-digit number) by 46 (a two-digit number).

Now remember, we’re gonna do the same procedure that we did in the last video for each of the bottom digits.

And since our top number has three digits, that means there’ll be

three multiplication steps for the first digit AND three steps for the second digit.

Fortunately, we can start the same way we would if the bottom had only one digit

by just ignoring the second digit until we finish the first three steps.

Alright, so our first multiplication step is 6 × 4 which is 24

And since 24 has two-digits, we can leave the ‘4’ in our answer line,

but we need to carry the ‘2’ and put it above the next top digit that we’re gonna multiply.

So the next step is 6 × 2 which is 12.

But we have to add in the ‘2’ that we carried, so 12 + 2 gives us 14.

That’s another two-digit answer,

so we leave the ‘4’ in our answer line, and carry the ‘1’ up above the next digit that we’re gonna multiply.

And our third step is 6 × 3 which is 18. And then we add in the ‘1’ that we carried and we get 19.

This time we can leave both digits of the 19 in our answer line, because there’s no more multiplication steps to do.

Well… at least there’s no more steps for the FIRST digit.

Remember, we still have that other bottom digit that we’ve been ignoring.

NOW we have to multiply IT by each of the top digits also, which means we have three more steps to do.

It also means that we’ll get a second answer.

And because we’ll get a second answer, we need to start a second answer line for the next set of steps.

We’re gonna put our new answer just below the first one.

So this answer line comes from our first bottom digit, and this answer line will come from our second bottom digit.

At the very end, after we’re all done multiplying, we’re gonna add the two answers together.

But for now, let’s continue with the second set of multiplication steps.

Oh… and I almost forgot to tell you…

there’s something VERY important that you need to do when you start the second set of steps.

Because the second digit of the bottom number is in the TENS place,

that means that even though the digit is only a ‘4’, it’s value is really 40.

That’s 10 times bigger… so the answer we get should also be 10 times bigger.

So before we start multiplying, we need to put a ZERO in the first spot of our answer line so it’s 10 times bigger.

That means that all the other answer digits we put there are shifted to the next bigger number place.

And (just so you know) if we happened to have a third bottom digit,

we would get a third answer line, and we’d need to shift the third answer over by TWO zeros, cuz it would be 100 times bigger.

And if we had a fourth digit, there would be a fourth answer line shifted over by THREE zeros.

And if we had a fifth bottom digit, there would be a fifth answer line shifted over by FOUR zeros!

See the pattern?

Boy am I glad we’ve only got two bottom digits!!

And now you see why we always put the number with the fewest digits on the bottom when we’re multiplying.

But let’s continue with our problem…

Let’s do the first step for our second bottom digit.

We multiply that digit (4) by the ones place digit of the top number (which is also a ‘4’).

4 × 4 gives us 16, and that goes in our second answer line, right next to the extra zero we put there.

Now remember, because 16 is a two-digit answer, we have to carry.

And we always put the digit we carry above the next top digit that we will multiply.

But before we can carry it up to that place,

we need to cross out all the digits that we carried from our first answer line, because we’ve already used them.

We don’t want to accidentally add them to our second answer line.

There… so now we can carry our ‘1’ to the top of the tens place, which means it will go in the column above this ‘2’,

because that’s the next top digit that we’ll multiply with our bottom digit.

Now we can do the next multiplication step.

4 × 2 gives us 8, and then we’ll add the ‘1’ that we carried and we get 9.

Great! …finally a one-digit answer, so we don’t have to carry this time.

We just write the ‘9’ in the next place of our answer line and move on to the next step.

The next (and last) multiplication step is 4 × 3 which is 12.

And since there’s no more steps, we can write both digits in our second answer line.

All right… we’re finally done with all our multiplication steps.

We multiplied each bottom digit by each top digit, just like we were supposed to.

But, now what do we do? We have two answers lines, but this is just one multiplication problem.

Well, remember…

the reason we have two answer lines is that we’re pretending that we’re doing two separate multiplication problems.

We treated it like it was 6 × 324 and 4 × 324.

But since the ‘4’ was in the tens place, we had to put an extra zero in our second answer since it would really be the answer from 40 × 324.

Now as I mentioned earlier, all we have to do to get the final answer is add those two answers together.

And the great news is that those answers are already stacked up like an addition problem should be,

so we can just draw a line below them and stick a plus sign on the left side.

Now we can add them, column by column, starting from ones place, just like we did in the multi-digit addition video.

4 + 0 = 4

4 + 6 = 10, so we carry the ‘1’

1 + 9 + 9 = 19, so we carry the ‘1’ again.

1 + 1 + 2 = 4, and then our last answer digit is just 1.

There, our final answer is a pretty big number: 14,904.

Let’s double check with a calculator to make sure we got it right.

Yep, that’s exactly what you get when you multiply 324 by 46.

Okay, I know that procedure is kinda complicated, so don’t get frustrated if you don’t get it right away.

It just takes some time and practice to really get the hang of it.

And you can always re-watch this video if you need to.

And if you’re a Math Antics subscriber, be sure to check out the extra problems I work in the “Prob with Rob” video examples.

After that, you can try some of the exercises on your own.

Thanks for tuning into Math Antics and I’ll see you next time.

Learn more at www.mathantics.com