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LESSON: Operations with Whole Numbers

Site: MN Partnership for Collaborative Curriculum
Course: Mathematics Essentials Q1
Book: LESSON: Operations with Whole Numbers
Printed by: Guest user
Date: Thursday, November 21, 2024, 4:10 PM

Description

Operations with Whole Numbers

Table of contents

READ: Adding Whole Numbers

Adding Whole Numbers

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

4 + 5 = \underline{\;\;\;\;\;\;\;\;\;\;}

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.


The numbers that we are adding are called addends. The answer to an addition problem is the sum.


This first problem was written horizontally or across. In the past, you may have seen them written vertically or up and down. You will need to be able 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.


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?

When you add whole numbers, you write them vertically according to place value.

Think about the example we had earlier.


4+5=9


If we wrote that vertically, we would line up the numbers. They are both ones.


& \quad 4\ & \ \underline{+5}\ & \quad 9


What happens when you have more digits?


Example

456 + 27 = \underline{\;\;\;\;\;\;\;\;}


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

& \quad 456\ & \ \underline{+ \ 27}


Now we can add the columns.


EXAMPLE 1


EXAMPLE 2


READ: Subtracting Whole Numbers

Subtracting Whole Numbers

Subtraction is the opposite of addition.

This 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 where you started.


When you add two numbers you get a total (or sum), when you subtract two numbers, you get the difference.


Example

15-9=\underline{\;\;\;\;\;\;\;\;\;\;}

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

15\ \underline{ - \ 9}\ 6


What about if you had more digits?

Example

12, 456 - 237 = \underline{\;\;\;\;\;\;\;\;\;}

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

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


12,456\ \underline{ \ - \ \ 237}


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 10 + 6 = 16. There’s the 10 we borrowed.

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


EXAMPLE 1


EXAMPLE 2


EXAMPLE 3


EXAMPLE 4


EXAMPLE 5


READ: Multiplying Whole Numbers

Multiplying Whole Numbers

Addition and multiplication are related. When you are multiplying larger numbers, it will be help you to think that multiplication is just a short cut for addition.

Example

5 \times 6 = \underline{\;\;\;\;\;\;\;\;\;\;}

What does it MEAN when we multiply five times 6? 5 \times 6 means that we are going to need five groups of six.

We could add 5 six times.

5 + 5 + 5 + 5 + 5 + 5 = \underline{\;\;\;\;\;\;\;\;\;\;}

However, it is easier to use our times tables.

5 \times 6 = 30


What is a factor?

A factor is the name 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 in a multiplication problem.


Example

567 \times 3 = \underline{\;\;\;\;\;\;\;\;\;\;\;}

If you think about this like addition, we have 567 added three times. 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.


To complete this problem, we take the digit 3 and multiply it times each digit of the top number. The three is called the multiplier in this problem because it is the number being multiplied.

7 \times 3 = 21

We can put the 1 in the ones place and carry the 2 (a ten) to the next column.

& \quad 5^267\ & \underline {\times \quad \ \ 3}\ & \qquad \ \ 1

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

6 \times 3 = 18 + 2 = 20


& \quad ^25^267\ & \underline {\times \quad \quad 3}\ & \quad \quad \ 0 1


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


& \quad 5^26^27\ & \underline {\times \qquad 3}\ & \quad 1,701

Our product is 1, 701.


What about three digits by two digits?

Example

234 \times 12 =\underline{\;\;\;\;\;\;\;\;\;\;}


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


Example

& \qquad 234\ & \underline {\times \quad \ \ 12}


Our multiplier here is 12. 12 had two digits. We need to multiply each digit of the top number with each digit of the number 12. We can start with the 2 of the multiplier.


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


Next, we multiply the 1 by each digit. Because we already multiplied one digit, we start of the second row of numbers with a zero to hold the place of the one number we already multiplied. Here is what this looks like.

Our product is 2,808.


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.


EXAMPLE 1


EXAMPLE 2


EXAMPLE 3


READ: Dividing Whole Numbers

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. If multiplication means to add groups of things, then division is the opposite of multiplication.

Example

72 \div 9 = \underline{\;\;\;\;\;\;\;\;\;\;}

In this problem, 72 is the number being divided, it is the dividend. 9 is the number doing the dividing, it is the divisor. The answer in a division problem is called the quotient. We can complete this problem by thinking of our multiplication facts.


Sometimes, a number won’t divide evenly. When this happens, we have a remainder.

Example

15 \div 2 =\underline{\;\;\;\;\;\;\;\;\;\;}

Fifteen is not an even number. There will be a remainder here.

We can use an “r” to show that there is a remainder. Our solution to 15 \div 2 =\underline{\;\;\;\;\;\;\;\;\;\;} is 7 r. 2 (or 7 remainder 2)


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

Example

8 \overline{)825 \;}

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.

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

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

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

We multiply 1 times 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?” The answer is 0. We put a 0 into the answer next to the 1.

& \overset{\ 10}{8\overline{ ) 825}}\ & \underline{-8} \;\; \Bigg \downarrow\ & \quad \ 025

Because we couldn’t divide 8 into 2, now we can look at 25. “How many 8’s are in 25?” The answer is 3 with a remainder of 1. We can add this into our answer.

& \overset{\ 103rl}{8\overline{ ) 825 \;}}\ & \ \underline{ -8 \ \ }\ & \ \ \ 025\ & \ \ \underline{-24}\ & \qquad 1

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

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

Our answer checks out.


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


Example

& \overset{\ 2}{12\overline{ ) 2448}} && “How \ many \ 12’s \ are \ in \ 2? \ None.”\ & \ \underline{-24} \Bigg \downarrow && “How \ many \ 12’s \ are \ in \ 24?”\ & \qquad \ 4 && \ Two\ \ & \overset{20}{12\overline{) 2448}} && “How \ many \ 12’s \ are \ in \ 4? \ None, \ so \ we \ add \ in \ a \ zero.”\ && &“How \ many \ 12’s \ are \ in \ 48?”\ && &Four\ && &There \ in \ not \ a \ remainder.\ \ &\overset{204}{12\overline{ ) 2448}}

We check our work by multiplying 204 \times 12.

& \qquad \quad 204\ & \ \underline {\times \qquad \ 12}\ & \qquad \quad 408\ & \ \underline {+ \quad \ 2040}\ & \qquad \ 2448

Our answer checks out.


We can apply these same steps to any addition 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 times the divisor too.

EXAMPLE 1


EXAMPLE 2


EXAMPLE 3


EXAMPLE 4