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# 1.1: Operations with Whole Numbers

Difficulty Level: At Grade Created by: CK-12

## Introduction

Feeding Time at the Zoo

Jonah is a student volunteer at the city zoo. He is working with the seals. Jonah loves his job, especially because he gets to help feed the seals who live at the zoo. There are 25 female and 18 male seals. One day, Ms. Guttierez, Jonah’s supervisor at the zoo, asks him to help her place the order for the week’s seafood. Jonah begins to do some calculations.

Each seal eats an average of 11 lbs. of seafood each day.

The seafood comes in 25 lb. buckets.

Jonah is puzzled. He doesn’t know how much food to order for one week. He doesn’t know how many buckets will be delivered.

Jonah needs help.

In this lesson, you will learn how to help Jonah figure out his fish problem.

Here are a few questions to keep in mind:

How many seals are there altogether?

How many pounds of seafood will they need to feed all of the seals for one week?

If the seafood comes in 25 lb. buckets, how many buckets will they need?

Is this the correct amount of food? Will there be any extra?

Later in this chapter we will return to Jonah and help him fix his problem, but first we need to learn and practice the skills to do so.

What You Will Learn

In this lesson, you will learn the following skills:

• Subtracting Whole Numbers
• Multiplying Whole Numbers
• Dividing Whole Numbers

Teaching Time

Let’s start with something that you have been doing for a long time. You have been adding whole numbers almost as long as you have been in school. Here is a problem that will look familiar.

Example

\begin{align*}4 + 5 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}

In this problem, we are adding four and five. We have four whole things plus five whole things and we get an answer of nine.

This first problem was written horizontally or across.

In the past, you may have seen them written vertically or up and down.

Now that you are in the sixth grade, you will need to write your problems vertically on your own.

How do we do this?

We can add whole numbers by writing them vertically according to place value.

Do you remember place value?

Place value is when you write each number according to the value that it has.

Millions Hundred Thousands Ten Thousands Thousands Hundreds Tens Ones
1 4 5 3 2 2 1

This number is 1,453,221. If we used words, we would say it is one million, four hundred and fifty-three thousand, two hundred and twenty-one.

What does this have to do with adding whole numbers?

Well, when you add whole numbers, it can be less confusing to write them vertically according to place value.

\begin{align*}4+5=9\end{align*}

If we wrote that vertically, we would line up the numbers. They both belong in the ones column.

\begin{align*}& \quad 4\\ & \ \underline{+5}\\ & \quad 9\end{align*}

What happens when we have more digits?

Example

\begin{align*}456 + 27 = \underline{\;\;\;\;\;\;\;\;}\end{align*}

When you have more digits, you can write the problem vertically by lining up each digit according to place value.

\begin{align*}& \quad 456\\ & \ \underline{+ \ 27}\end{align*}

Now we can add the columns.

Here are a few problems for you to try on your own:

1. \begin{align*}3,456 + 87 =\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
2. \begin{align*}56, 321 + 7, 600 =\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
3. \begin{align*}203,890 + 12, 201 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}

Take a minute and check your work with a peer.

II. Subtracting Whole Numbers

Just like you have been adding whole numbers for a long time, you have been subtracting them for a long time too. Let’s think about what it means to subtract.

Subtraction is the opposite of addition.

Hmmm.... What does that mean exactly?

It means that if you can add two numbers and get a total, then you can subtract one of those numbers from that total and end up with the other starting number.

In other words,

Subtraction is the opposite of addition.

When you add two numbers you get a total, when you subtract two numbers, you get the difference.

Let’s look at an example.

Example

\begin{align*}15-9=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}

This is a pretty simple example. If you have fifteen of something and take away nine, what is the result?

Think about how we can do this problem.

First, we need to rewrite the problem vertically, just like we did when we were adding numbers.

Remember to line up the digits according to place value.

Example

\begin{align*}15\\ \underline{ - \ 9}\\ 6\end{align*}

This could probably be completed using mental math.

Example

\begin{align*}12, 456 - 237 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

Our first step is to line up these digits according to place value.

Let’s look at what this will look like in our place value chart.

Ten Thousands Thousands Hundreds Tens Ones
1 2 4 5 6
2 3 7

Wow! This problem is now written vertically. We can go ahead and subtract.

Example

\begin{align*}12,456\\ \underline{ \ - \ \ 237}\end{align*}

To successfully subtract these two values, we are going to need to regroup.

What does it mean to regroup?

When we regroup we borrow to make our subtraction easier.

Look at the ones column of the example.

We can’t take 7 from 6, so we borrow from the next number.

The next number is in the tens column, so we can “borrow a 10” to subtract.

If we borrow 10, that makes the 5 into a 4.

We can make the 6 into 16 because \begin{align*}10 + 6 = 16\end{align*}. There’s the 10 we borrowed.

Let’s put that into action.

Example

Be careful-be sure you subtract according to place value. Don’t let the regrouping mix you up.

Here are a few problems for you to try on your own:

1. \begin{align*}674 - 59 =\underline{\;\;\;\;\;\;\;}\end{align*}
2. \begin{align*}15,987 - 492 =\underline{\;\;\;\;\;\;\;}\end{align*}
3. \begin{align*}22,456 - 18, 297 =\underline{\;\;\;\;\;\;\;}\end{align*}

Take a minute and check your work with a peer.

III. Multiplying Whole Numbers

Now that we have learned about addition and subtraction, it is time for multiplying whole numbers.

Hmmm... What does that mean exactly?

Well to explain it, let’s look at an example.

Example

\begin{align*}5 \times 6 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}

You can use your times tables to complete this problem using mental math, but let’s look at what we MEAN when we multiply five by 6.

\begin{align*}5 \times 6\end{align*} means that we are going to need five groups of six.

@@@@@@ ######  &&&&&& ******

We could think of this another way too.

We could add 5 six times.

\begin{align*}5 + 5 + 5 + 5 + 5 + 5 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}

Wow, that is a lot of work.

It is easier to use our times tables.

Example

\begin{align*}5 \times 6 = 30\end{align*}

When multiplying larger numbers, it will help you to think of multiplication as just a short cut for addition.

5 and 6 are factors in this problem.

What is a factor?

A factor is the name of each of the two values being multiplied.

30 is the product of the factors 5 and 6.

What does the word product mean?

The product is the answer to a multiplication problem.

Now let’s take what we have learned and look at how to apply it to a few more challenging problems.

Example

\begin{align*}567 \times 3 = \underline{\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

That is a lot of work, so let’s use our multiplication short cut.

First, let’s line up our numbers according to place value.

Example

To complete this problem, we take the digit 3 and multiply it by each digit of the top number.

The three is called the multiplier in this problem because it is the number being multiplied. Since 7 is the first number in the upper row, we start by multiplying it by our multiplier, 3:

\begin{align*}7 \times 3 = 21\end{align*}

We can put the 1 in the ones place and carry the 2 (which is really two tens) to the next column over, where it can be added to the other tens after the next multiplication step.

Example

\begin{align*}& \quad 5^267\\ & \underline {\times \quad \ \ 3}\\ & \qquad \ \ 1\end{align*}

Next, we multiply the 3 by 6 and add the two we carried.

\begin{align*}6 \times 3 = 18 + 2 = 20\end{align*}

Leave the 0 in the tens place and carry the two.

Example

\begin{align*}& \quad ^25^267\\ & \underline {\times \quad \quad 3}\\ & \quad \quad \ 0 1\end{align*}

Next, we multiply the 3 by 5 and add the two we carried.

Example

\begin{align*}& \quad ^25^267\\ & \underline {\times \qquad 3}\\ & \quad 1,701\end{align*}

Our product is 1,701.

In this first problem, we multiplied three digits by one digit.

What about three digits by two digits?

Example

\begin{align*}234 \times 12 =\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}

First, we need to line up the digits according to place value.

Example

\begin{align*}& \qquad 234\\ & \underline {\times \quad \ \ 12}\end{align*}

Our multiplier here is 12.

12 has two digits. We need to multiply each digit of the top number by each digit of the number 12.

Example

\begin{align*}& \qquad 234\\ & \ \underline {\times \quad \ 12}\\ & \qquad 468 \quad Here \ is \ the \ result \ of \ multiplying \ the \ first \ digit \ of \ the \ multiplier.\end{align*}

Next, we multiply the 1, which is in the tens place, by each digit.

Because we are multiplying by a number in the "tens" place, we start the second row of numbers with a zero so that the answer to the multiplication is kept in the correct place value for the addition we will do next.

Here is what this looks like.

Example

Our product is 2,808.

What about when we multiply three digits by three digits?

Wow! That may seem like a lot of work, but if you follow each step, you will end up with the correct answer.

First, you will multiply the first digit of the multiplier by each of the three digits of the top number.

Second, you will multiply the second digit of the multiplier by all three digits of the top number. Don’t forget that placeholder zero!

Third, you will multiply the third digit of the multiplier by all three digits of the top number. Use two zeros since you are now multiplying by a number in the "hundreds" place.

Let’s look at an example

Example

\begin{align*}& \qquad \quad \ \ 214\\ & \ \ \underline {\times \qquad \ 362}\\ & \qquad \quad \ \ 428 \quad Here \ is \ the \ result \ of \ multiplying \ by \ 2.\\ & \qquad \ \ 12840 \quad Here \ is \ the \ result \ of \ multiplying \ by \ 6. \ Notice \ we \ had \ to \ carry \ and \ add \ in \ a \ zero.\\ & \underline {+ \qquad 64200} \quad Here \ is \ the \ result \ of \ multiplying \ by \ 3. \ Notice \ we \ had \ to \ add \ in \ two \ zeros.\\ & \qquad \ 77,468\end{align*}

Our product is 77,468.

You could multiply even more digits by more digits.

You just need to remember two things.

1. Multiply each digit of the multiplier one at a time.
2. Add in a zero for each digit that you have already multiplied.

Here are a few problems for you to try on your own:

1. \begin{align*}456 \times 9 = \underline{\;\;\;\;\;\;\;}\end{align*}
2. \begin{align*}321 \times 18 = \underline{\;\;\;\;\;\;\;}\end{align*}
3. \begin{align*}562 \times 248 = \underline{\;\;\;\;\;\;\;}\end{align*}

Take a minute and check your work with a peer.

IV. Dividing Whole Numbers

Our final operation is division. First, let’s talk about what the word “division” actually means.

To divide means to split up into groups.

Since multiplication means to add groups of things together, division is the opposite of multiplication.

Let’s look at an example.

Example

\begin{align*}72 \div 9 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}

In this problem, 72 is the number being divided, it is the dividend.

9 is the number doing the dividing, it is the divisor.

We can complete this problem by thinking of our multiplication facts and working backwards. Ask yourself "What number multiplied by 9 equals 72?" If you said "8", you're right! 9 x 8 = 72, so 72 can be split into 8 groups of 9.

The answer to a division problem is called the quotient.

Sometimes, a number won’t divide evenly.

When this happens, we have a remainder.

Example

\begin{align*}15 \div 2 =\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}

Hmmm. This is tricky, fifteen is not an even number. There will be a remainder here.

Example

We can use an “\begin{align*}r\end{align*}” to show that there is a remainder.

We can also divide larger numbers. We can use a division box to do this.

Example

\begin{align*}8 \overline{)825 \;}\end{align*}

Here we have a one digit divisor, 8, and a three digit dividend, 825.

We need to figure out how many 8’s there are in 825.

To do this, we divide the divisor 8 into each digit of the dividend.

Example

\begin{align*}& 8 \overline{)825 \;} \qquad “How \ many \ 8’s \ are \ there \ in \ 8?”\\ & \qquad \qquad \ \ The \ answer \ is \ 1.\end{align*}

We put the 1 on top of the division box above the 8.

Example

\begin{align*}& \overset{\ 1}{8\overline{ ) 825}}\\ & \underline{-8} \Bigg \downarrow\\ & \quad 02\end{align*}

We multiply 1 by 8 and subtract our result from the dividend. Then we can bring down the next number in the dividend.

Then, we need to look at the next digit in the dividend.

“How many 8’s are there in 2?”

We put a 0 into the answer next to the 1.

Example \begin{align*}& \overset{\ 10}{8\overline{ ) 825}}\\ & \underline{-8} \;\; \Bigg \downarrow\\ & \quad \ 025\end{align*}

Because we couldn’t divide 8 into 2, now we can bring down the next number, 5, and use the two numbers together: 25

“How many 8’s are in 25?”

The answer is 3 with a remainder of 1.

Example

\begin{align*}& \overset{\ 103rl}{8\overline{ ) 825 \;}}\\ & \ \underline{ -8 \ \ }\\ & \ \ \ 025\\ & \ \ \underline{-24}\\ & \qquad 1\end{align*} We can check our work by multiplying the answer by the divisor.

\begin{align*}& \qquad 103\\ & \ \underline {\times \quad \ \ 8 \ }\\ & \qquad 824 + r \ \text{of} \ 1 = 825\end{align*}

Let’s look at an example with a two-digit divisor.

Example

\begin{align*}& \overset{\ \hspace{2 mm} 2}{12\overline{ ) 2448}} && “How \ many \ 12’s \ are \ in \ 2? \ None.”\\ & \ \underline{-24} \Bigg \downarrow && “How \ many \ 12’s \ are \ in \ 24? \ Two. \ So \ fill \ that \ in.”\\ & \qquad \ 4 && \ Now \ bring \ down \ the \ "4".\\ \\ & \overset{\ \hspace{4 mm} 20}{12\overline{) 2448}} && “How \ many \ 12’s \ are \ in \ 4? \ None, \ so \ we \ add \ a \ zero \ to \ the \ answer.”\\ && &“How \ many \ 12’s \ are \ in \ 48?”\\ && &Four\\ && &There \ is \ not \ a \ remainder \ this \ time \ because \ 48 \ divides \ exactly \ by \ 12.\\ \\ &\overset{\ \hspace{6 mm} 204}{12\overline{ ) 2448}}\end{align*}

We check our work by multiplying: \begin{align*}204 \times 12\end{align*}.

\begin{align*}& \qquad \quad 204\\ & \ \underline {\times \qquad \ 12}\\ & \qquad \quad 408\\ & \ \underline {+ \quad \ 2040}\\ & \qquad \ 2448\end{align*}

We can apply these same steps to any division problem even if the divisor has two or three digits.

We work through each value of the divisor with each value of the dividend.

We can check our work by multiplying our answer by the divisor.

Here are a few problems for you to try on your own:

1. \begin{align*}4\overline{ ) 469 \;}\end{align*}
2. \begin{align*}18\overline{ ) 3678 \;}\end{align*}
3. \begin{align*}20\overline{ ) 5020 \;}\end{align*}

Take a minute and check your work with a peer.

## Real Life Example Completed

Feeding Time at the Zoo

Remember Jonah?

Now, you have learned all that you need to learn to help Jonah.

First, let’s revisit the problem.

Jonah is a student volunteer at the city zoo. He is working with the seals. Jonah loves his job, especially because he gets to help feed the seals who live at the zoo. There are 25 female and 18 male seals. One day, Ms. Guttierez, Jonah’s supervisor at the zoo, asks him to help her place the order for the week’s seafood. Jonah begins to do some calculations.

Each seal eats an average of 11 lbs. of seafood each day.

The seafood comes in 25 lb. buckets.

Jonah is puzzled. He doesn’t know how much food to order for one week. He doesn’t know how many buckets will be delivered.

Jonah needs help.

Like many real world math problems, you will need to perform several different operations to help Jonah with his dilemma.

First, let’s underline anything that seems important in the problem. You will see that this has been done for you in the paragraph above.

We need to help Jonah figure out the total number of seals that he needs to feed.

Words like “total”, “altogether” and “in all” let us know that we need to add.

Let’s look back at the problem to find the part about the number of seals.

This has been underlined in the paragraph:

There are 25 female and 18 male seals.

Next, we write the addition problem.

\begin{align*}25 + 18 = 43 \ \text{seals}\end{align*}

Jonah needs to feed 43 seals.

We know from the problem that each seal eats an average of 11 lbs of seafood per day.

We could do repeated addition here, add 11 forty-three times once for each seal.

Boy that is a lot of work. When we have a repeated addition problem, our short cut is to multiply.

\begin{align*}& \qquad \quad 43\\ & \ \underline {\times \qquad 11}\\ & \qquad \quad 43\\ & \ \underline {+ \quad \ 430}\\ & \qquad \ 473\end{align*}

Jonah needs 473 pounds of seafood to feed all of the seals for one day.

That's great, but we need to feed all the seals for ONE WEEK!

Once again, we could use repeated addition, but multiplication is so much quicker.

There are 7 days in one week, so we can multiply 7 by the total pounds of seafood for one day.

\begin{align*}& \quad \ \ \ \ \ 473\\ & \ \underline {\times \quad \ \ \ \ \ 7}\\ & \quad \ \ \ \ 3311 \quad \text{pounds of seafood for one week}\end{align*}

Okay, we have helped Jonah with half of his problem. Now we know how much seafood he needs for one week.

Now we can help him complete the order.

If the seafood comes in 25 lb. buckets, how many buckets will he need?

To complete this problem, we need to divide the number of pounds of seafood by the number of pounds in a bucket.

Notice, that we divide pounds by pounds. The items we are dividing have to be the same.

Let’s set up the problem.

\begin{align*}& \overset{\ \ \ \hspace{2 mm} 132}{25\overline{) 3311 \;}}\\ & \ \ \underline{-25}\\ & \quad \ \ 81\\ & \quad \underline{-75 \ }\\ & \qquad \ 61\\ & \quad \ \ \underline{-50}\\ & \qquad \ \ 11\end{align*}

Uh oh, we have a remainder. This means that we are missing 11 pounds of fish. One seal will not have enough to eat if Jonah only orders 132 buckets.

Therefore, Jonah needs to order 133 buckets. There will be extra fish, but all the seals will eat.

## Vocabulary

Here are the vocabulary words used in this lesson. Remember you can find them in italics throughout the lesson.

Sum
Horizontally
across
Vertically
up and down
Difference
the answer to a subtraction problem
Regroup
when you need to borrow from the next column in subtraction
Factor
the numbers being multiplied in a multiplication problem
Product
the answer to a multiplication problem
Multiplier
the number you multiply with
Dividend
the number being divided
Divisor
the number doing the dividing
Quotient
the answer to a division problem
Remainder
the value left over if the divisor does not divide evenly into the dividend

## Technology Integration

Here are some places on the web where you can learn more about operations with whole numbers.

1. http://www.gamequarium.org/cgi-bin/search/linfo.cgi?id=7543 – This site covers two-digit addition with estimation also included.
2. http://www.teachertube.com/members/viewVideo.php?video_id=163933&title=Long_Multiplication_The_Video_ - You will need to register with this website. This video covers multiplication with two digits and goes through all of the steps.
3. http://www.gamequarium.org/cgi-bin/search/linfo.cgi?id=7635 – This site looks at division in a very interactive way. This is a fun way for you to learn about division.

## Time to Practice

Directions: Use what you have learned to solve each problem. Remember, you will be adding, subtracting, multiplying and dividing.

1. \begin{align*}56 + 123 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

2. \begin{align*}341 + 12 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

3. \begin{align*}673 + 127 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

4. \begin{align*}549 + 27 =\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

5. \begin{align*}87 + 95 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

6. \begin{align*}124 + 967 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

7. \begin{align*}1256 + 987 =\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

8. \begin{align*}2345 + 1278 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

9. \begin{align*}3100 + 5472 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

10. \begin{align*}3027 + 5471 =\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}

11. \begin{align*}56 - 21 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}

12. \begin{align*}50 - 23 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

13. \begin{align*}267 - 19 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

14. \begin{align*}345 - 127 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

15. \begin{align*}560 - 233 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

16. \begin{align*}1600 - 289 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

17. \begin{align*}5400 - 2334 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

18. \begin{align*}8990 - 7865 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

19. \begin{align*}12340 - 3456 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

20. \begin{align*}23410 - 19807 =\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

21. \begin{align*}34 \times 8 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

22. \begin{align*}67 \times 12 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

23. \begin{align*}34 \times 87 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

24. \begin{align*}124 \times 9 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

25. \begin{align*}345 \times 11 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

26. \begin{align*}6721 \times 9 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

27. \begin{align*}8723 \times 31 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

28. \begin{align*}9802 \times 22 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

29. \begin{align*}345 \times 123 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

30. \begin{align*}617 \times 234 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

31. \begin{align*}12 \div 6 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

32. \begin{align*}13 \div 4 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

33. \begin{align*}132 \div 7 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

34. \begin{align*}124 \div 4 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

35. \begin{align*}1244 \div 40 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

36. \begin{align*}248 \div 18 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

37. \begin{align*}3264 \div 16 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

38. \begin{align*}4440 \div 20 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

39. \begin{align*}7380 \div 123 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

40. \begin{align*}102000 \div 200 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

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