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Module by: First Last. E-mail the author

Summary: By the end of this section, you will be able to:

• Model addition of whole numbers
• Add whole numbers without models
• Translate word phrases to math notation
• Add whole numbers in applications

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## Note:

Before you get started, take this readiness quiz.

1. What is the number modeled by the base-10base-10 blocks?

If you missed this problem, review (Reference).
2. Write the number three hundred forty-two thousand six using digits?
If you missed this problem, review (Reference).

A college student has a part-time job. Last week he worked 33 hours on Monday and 44 hours on Friday. To find the total number of hours he worked last week, he added 33 and 4.4.

The operation of addition combines numbers to get a sum. The notation we use to find the sum of 33 and 44 is:

3+43+4

We read this as three plus four and the result is the sum of three and four. The numbers 33 and 44 are called the addends. A math statement that includes numbers and operations is called an expression.

To describe addition, we can use symbols and words.

Operation Notation Expression Read as Result
Addition ++ 3+43+4 three plus four the sum of 33 and 44

### Example 1

#### Problem 1

Translate from math notation to words:

1. 7+17+1
2. 12+1412+14

##### Solution: Solution
• The expression consists of a plus symbol connecting the addends 7 and 1. We read this as seven plus one. The result is the sum of seven and one.
• The expression consists of a plus symbol connecting the addends 12 and 14. We read this as twelve plus fourteen. The result is the sum of twelve and fourteen.

### Note:

#### Exercise 1

Translate from math notation to words:

1. 8+48+4
2. 18+1118+11
##### Solution
• eight plus four; the sum of eight and four
• eighteen plus eleven; the sum of eighteen and eleven

### Note:

#### Exercise 2

Translate from math notation to words:

1. 21+1621+16
2. 100+200100+200
##### Solution
1. twenty-one plus sixteen; the sum of twenty-one and sixteen
2. one hundred plus two hundred; the sum of one hundred and two hundred

## Model Addition of Whole Numbers

Addition is really just counting. We will model addition with base-10base-10 blocks. Remember, a block represents 11 and a rod represents 10.10. Let’s start by modeling the addition expression we just considered, 3+4.3+4.

Each addend is less than 10,10, so we can use ones blocks.

 We start by modeling the first number with 3 blocks. Then we model the second number with 4 blocks. Count the total number of blocks.

There are 77 blocks in all. We use an equal sign (=)(=) to show the sum. A math sentence that shows that two expressions are equal is called an equation. We have shown that. 3+4=7.3+4=7.

### Example 2

#### Problem 1

##### Solution: Solution

2+62+6 means the sum of 22 and 66

Each addend is less than 10, so we can use ones blocks.

 Model the first number with 2 blocks. Model the second number with 6 blocks. Count the total number of blocks There are 88 blocks in all, so 2+6=8.2+6=8.

Model: 3+6.3+6.

### Note:

#### Exercise 4

Model: 5+1.5+1.

##### Solution

When the result is 1010 or more ones blocks, we will exchange the 1010 blocks for one rod.

### Example 3

#### Problem 1

##### Solution: Solution

5+85+8 means the sum of 55 and 8.8.

 Each addend is less than 10, se we can use ones blocks. Model the first number with 5 blocks. Model the second number with 8 blocks. Count the result. There are more than 10 blocks so we exchange 10 ones blocks for 1 tens rod. Now we have 1 ten and 3 ones, which is 13. 5 + 8 = 13

Notice that we can describe the models as ones blocks and tens rods, or we can simply say ones and tens. From now on, we will use the shorter version but keep in mind that they mean the same thing.

### Note:

#### Exercise 6

##### Solution

Next we will model adding two digit numbers.

### Example 4

#### Problem 1

##### Solution: Solution

17+2617+26 means the sum of 17 and 26.

 Model the 17. 1 ten and 7 ones Model the 26. 2 tens and 6 ones Combine. 3 tens and 13 ones Exchange 10 ones for 1 ten. 4 tens and 3 ones 40+3=4340+3=43 We have shown that 17+26=4317+26=43

## Add Whole Numbers Without Models

Now that we have used models to add numbers, we can move on to adding without models. Before we do that, make sure you know all the one digit addition facts. You will need to use these number facts when you add larger numbers.

Imagine filling in Table 6 by adding each row number along the left side to each column number across the top. Make sure that you get each sum shown. If you have trouble, model it. It is important that you memorize any number facts you do not already know so that you can quickly and reliably use the number facts when you add larger numbers.

Table 6
+ 0 1 2 3 4 5 6 7 8 9
0 0 1 2 3 4 5 6 7 8 9
1 1 2 3 4 5 6 7 8 9 10
2 2 3 4 5 6 7 8 9 10 11
3 3 4 5 6 7 8 9 10 11 12
4 4 5 6 7 8 9 10 11 12 13
5 5 6 7 8 9 10 11 12 13 14
6 6 7 8 9 10 11 12 13 14 15
7 7 8 9 10 11 12 13 14 15 16
8 8 9 10 11 12 13 14 15 16 17
9 9 10 11 12 13 14 15 16 17 18

Did you notice what happens when you add zero to a number? The sum of any number and zero is the number itself. We call this the Identity Property of Addition. Zero is called the additive identity.

### Note: Identity Property of Addition:

The sum of any number aa and 00 is the number.

a+0=a0+a=aa+0=a0+a=a

### Example 5

#### Problem 1

Find each sum:

1. 0+110+11
2. 42+042+0

##### Solution: Solution
 ⓐ The first addend is zero. The sum of any number and zero is the number. 0+11=110+11=11 ⓑ The second addend is zero. The sum of any number and zero is the number. 42+0=4242+0=42

### Note:

#### Exercise 9

Find each sum:

1. 0+190+19
2. 39+039+0

##### Solution
1. 0+19=190+19=19
2. 39+0=3939+0=39

### Note:

#### Exercise 10

Find each sum:

1. 0+240+24
2. 57+057+0

##### Solution
1. 0+24=240+24=24
2. 57+0=5757+0=57

Look at the pairs of sums.

 2+3=52+3=5 3+2=53+2=5 4+7=114+7=11 7+4=117+4=11 8+9=178+9=17 9+8=179+8=17

Notice that when the order of the addends is reversed, the sum does not change. This property is called the Commutative Property of Addition, which states that changing the order of the addends does not change their sum.

### Note: Commutative Property of Addition:

Changing the order of the addends aa and bb does not change their sum.

a+b=b+aa+b=b+a

### Example 6

#### Problem 1

1. 8+78+7
2. 7+87+8
##### Solution: Solution

Did you notice that changing the order of the addends did not change their sum? We could have immediately known the sum from part just by recognizing that the addends were the same as in part , but in the reverse order. As a result, both sums are the same.

### Note:

#### Exercise 11

##### Solution

9+7=16; 7+9=169+7=16; 7+9=16

### Note:

#### Exercise 12

##### Solution

8+6=14; 6+8=148+6=14; 6+8=14

### Example 7

#### Problem 1

##### Solution: Solution

To add numbers with more than one digit, it is often easier to write the numbers vertically in columns.

 Write the numbers so the ones and tens digits line up vertically. 28 +61____28 +61____ Then add the digits in each place value. Add the ones: 8+1=98+1=9Add the tens: 2+6=82+6=8 28 +61____8928 +61____89

32+54=8632+54=86

### Note:

#### Exercise 14

##### Solution

25+74=9925+74=99

In the previous example, the sum of the ones and the sum of the tens were both less than 10.10. But what happens if the sum is 1010 or more? Let’s use our base-10base-10 model to find out. Figure 1 shows the addition of 1717 and 2626 again.

When we add the ones, 7+6,7+6, we get 1313 ones. Because we have more than 1010 ones, we can exchange 1010 of the ones for 11 ten. Now we have 44 tens and 33 ones. Without using the model, we show this as a small red 11 above the digits in the tens place.

When the sum in a place value column is greater than 9,9, we carry over to the next column to the left. Carrying is the same as regrouping by exchanging. For example, 1010 ones for 11 ten or 1010 tens for 11 hundred.

1. Step 1. Write the numbers so each place value lines up vertically.
2. Step 2. Add the digits in each place value. Work from right to left starting with the ones place. If a sum in a place value is more than 9,9, carry to the next place value.
3. Step 3. Continue adding each place value from right to left, adding each place value and carrying if needed.

### Example 8

#### Problem 1

##### Solution: Solution
 Write the numbers so the digits line up vertically. 43 +69____43 +69____ Add the digits in each place.Add the ones: 3+9=123+9=12 Write the 22 in the ones place in the sum.Add the 11 ten to the tens place. 413 +69____2413 +69____2 Now add the tens: 1+4+6=111+4+6=11Write the 11 in the sum. 413 +69____112413 +69____112

### Note:

#### Exercise 15

##### Solution

35+98=13335+98=133

### Note:

#### Exercise 16

##### Solution

72+89=16172+89=161

### Example 9

#### Problem 1

##### Solution: Solution
 Write the numbers so the digits line up vertically. Add the digits in each place value.Add the ones: 4+6=104+6=10Write the 00 in the ones place in the sum and carry the 11 ten to the tens place. Add the tens: 1+2+8=111+2+8=11Write the 11 in the tens place in the sum and carry the 11 hundred to the hundreds Add the hundreds: 1+3+5=91+3+5=9Write the 99 in the hundreds place.

### Note:

#### Exercise 17

##### Solution

456+376=832456+376=832

### Note:

#### Exercise 18

##### Solution

269+578=847269+578=847

### Example 10

#### Problem 1

##### Solution: Solution
 Write the numbers so the digits line up vertically. 1,683 +479______1,683 +479______ Add the digits in each place value. Add the ones: 3+9=12.3+9=12.Write the 22 in the ones place of the sum and carry the 11 ten to the tens place. 1,6813 +479______21,6813 +479______2 Add the tens: 1+7+8=161+7+8=16Write the 66 in the tens place and carry the 11 hundred to the hundreds place. 1,61813 +479______621,61813 +479______62 Add the hundreds: 1+6+4=111+6+4=11Write the 11 in the hundreds place and carry the 11 thousand to the thousands place. 1,61813 +479______1621,61813 +479______162 Add the thousands 1+1=21+1=2.Write the 22 in the thousands place of the sum. 1,61813 +479______2,1621,61813 +479______2,162

When the addends have different numbers of digits, be careful to line up the corresponding place values starting with the ones and moving toward the left.

### Note:

#### Exercise 19

##### Solution

4,597+685=5,2824,597+685=5,282

### Note:

#### Exercise 20

##### Solution

5,837+695=6,5325,837+695=6,532

### Example 11

#### Problem 1

##### Solution: Solution
 Write the numbers so the place values line up vertically. 21,357 861 +8,596_______21,357 861 +8,596_______ Add the digits in each place value. Add the ones: 7+1+6=147+1+6=14Write the 44 in the ones place of the sum and carry the 11 to the tens place. 21,3517 861 +8,596_______ 421,3517 861 +8,596_______ 4 Add the tens: 1+5+6+9=211+5+6+9=21Write the 11 in the tens place and carry the 22 to the hundreds place. 21,32517 861 +8,596_______ 1421,32517 861 +8,596_______ 14 Add the hundreds: 2+3+8+5=182+3+8+5=18Write the 88 in the hundreds place and carry the 11 to the thousands place. 21,132517 861 +8,596_______ 81421,132517 861 +8,596_______ 814 Add the thousands 1+1+8=101+1+8=10.Write the 00 in the thousands place and carry the 11 to the ten thousands place. 211,132517 861 +8,596_______ 0814211,132517 861 +8,596_______ 0814 Add the ten-thousands 1+2=31+2=3.Write the 33 in the ten thousands place in the sum. 211,132517 861 +8,596_______ 30,814211,132517 861 +8,596_______ 30,814

This example had three addends. We can add any number of addends using the same process as long as we are careful to line up the place values correctly.

### Note:

#### Exercise 21

##### Solution

46,195+397+6,281=52,87346,195+397+6,281=52,873

### Note:

#### Exercise 22

##### Solution

53,762+196+7,458=61,41653,762+196+7,458=61,416

## Translate Word Phrases to Math Notation

Earlier in this section, we translated math notation into words. Now we’ll reverse the process. We’ll translate word phrases into math notation. Some of the word phrases that indicate addition are listed in Table 16.

Table 16
Operation Words Example Expression
sum
increased by
more than
total of
11 plus 22
the sum of 33 and 44
55 increased by 66
88 more than 77
the total of 99 and 55
1+21+2
3+43+4
5+65+6
7+87+8
9+59+5
4+64+6

### Example 12

#### Problem 1

Translate and simplify: the sum of 1919 and 23.23.

##### Solution: Solution

The word sum tells us to add. The words of 1919 and 2323 tell us the addends.

 The sum of 1919 and 2323 Translate. 19+2319+23 Add. 4242 The sum of 1919 and 2323 is 42.42.

### Note:

#### Exercise 23

Translate and simplify: the sum of 1717 and 26.26.

##### Solution

Translate: 17+2617+26; Simplify: 4343

### Note:

#### Exercise 24

Translate and simplify: the sum of 2828 and 14.14.

##### Solution

Translate: 28+1428+14; Simplify: 4242

### Example 13

#### Problem 1

Translate and simplify: 2828 increased by 31.31.

##### Solution: Solution

The words increased by tell us to add. The numbers given are the addends.

 2828 increased by 31.31. Translate. 28+3128+31 Add. 5959 So 2828 increased by 3131 is 59. 59.

### Note:

#### Exercise 25

Translate and simplify: 2929 increased by 76.76.

##### Solution

Translate: 29+7629+76; Simplify 105105

### Note:

#### Exercise 26

Translate and simplify: 3737 increased by 69.69.

##### Solution

Translate 37 + 69; Simplify 106

## Add Whole Numbers in Applications

Now that we have practiced adding whole numbers, let’s use what we’ve learned to solve real-world problems. We’ll start by outlining a plan. First, we need to read the problem to determine what we are looking for. Then we write a word phrase that gives the information to find it. Next we translate the word phrase into math notation and then simplify. Finally, we write a sentence to answer the question.

### Example 14

#### Problem 1

Hao earned grades of 87,93,68,95,and8987,93,68,95,and89 on the five tests of the semester. What is the total number of points he earned on the five tests?

##### Solution: Solution

We are asked to find the total number of points on the tests.

 Write a phrase. the sum of points on the tests Translate to math notation. 87+93+68+95+8987+93+68+95+89 Then we simplify by adding. Since there are several numbers, we will write them vertically. 837936895+89____432837936895+89____432 Write a sentence to answer the question. Hao earned a total of 432 points.

Notice that we added points, so the sum is 432432 points. It is important to include the appropriate units in all answers to applications problems.

### Note:

#### Exercise 27

Mark is training for a bicycle race. Last week he rode 1818 miles on Monday, 1515 miles on Wednesday, 2626 miles on Friday, 4949 miles on Saturday, and 3232 miles on Sunday. What is the total number of miles he rode last week?

##### Solution

He rode 140 miles.

### Note:

#### Exercise 28

Lincoln Middle School has three grades. The number of students in each grade is 230,165,and325.230,165,and325. What is the total number of students?

##### Solution

The total number is 720 students.

Some application problems involve shapes. For example, a person might need to know the distance around a garden to put up a fence or around a picture to frame it. The perimeter is the distance around a geometric figure. The perimeter of a figure is the sum of the lengths of its sides.

### Example 15

#### Problem 1

Find the perimeter of the patio shown.

##### Solution: Solution
 We are asked to find the perimeter. Write a phrase. the sum of the sides Translate to math notation. 4+6+2+3+2+94+6+2+3+2+9 Simplify by adding. 2626 Write a sentence to answer the question. We added feet, so the sum is 2626 feet. The perimeter of the patio is 2626 feet.

### Note:

#### Exercise 29

Find the perimeter of each figure. All lengths are in inches.

##### Solution

The perimeter is 30 inches.

### Note:

#### Exercise 30

Find the perimeter of each figure. All lengths are in inches.

##### Solution

The perimeter is 36 inches.

## Key Concepts

• Addition Notation To describe addition, we can use symbols and words.
Operation Notation Expression Read as Result
Addition ++ 3+43+4 three plus four the sum of 33 and 44
• The sum of any number aa and 00 is the number. a+0=aa+0=a 0+a=a0+a=a
• Changing the order of the addends aa and bb does not change their sum. a+b=b+aa+b=b+a.
1. Step 1. Write the numbers so each place value lines up vertically.
2. Step 2. Add the digits in each place value. Work from right to left starting with the ones place. If a sum in a place value is more than 9, carry to the next place value.
3. Step 3. Continue adding each place value from right to left, adding each place value and carrying if needed.

### Practice Makes Perfect

In the following exercises, translate the following from math expressions to words.

#### Exercise 31

5+25+2

##### Solution

five plus two; the sum of 5 and 2.

6+36+3

#### Exercise 33

13+1813+18

##### Solution

thirteen plus eighteen; the sum of 13 and 18.

15+1615+16

#### Exercise 35

214+642214+642

##### Solution

two hundred fourteen plus six hundred forty-two; the sum of 214 and 642

#### Exercise 36

438+113438+113

In the following exercises, model the addition.

2+42+4

2+4=62+4=6

5+35+3

8+48+4

8+4=128+4=12

5+95+9

14+7514+75

14+75=8914+75=89

15+6315+63

16+2516+25

16+25=4116+25=41

#### Exercise 44

14+2714+27

In the following exercises, fill in the missing values in each chart.

1. 0+130+13
2. 13+013+0
1. 13
2. 13

#### Exercise 52

1. 0+5,2800+5,280
2. 5,280+05,280+0

1. 8+38+3
2. 3+83+8
1. 1111
2. 1111

1. 7+57+5
2. 5+75+7

45+3345+33

7878

37+2237+22

71+2871+28

9999

43+5343+53

26+5926+59

8585

38+1738+17

64+7864+78

142142

92+3992+39

168+325168+325

493493

247+149247+149

584+277584+277

861861

175+648175+648

832+199832+199

1,0311,031

775+369775+369

#### Exercise 69

6,358+4926,358+492

6,8506,850

#### Exercise 70

9,184+5789,184+578

#### Exercise 71

3,740+18,5933,740+18,593

22,33322,333

#### Exercise 72

6,118+15,9906,118+15,990

#### Exercise 73

485,012+619,848485,012+619,848

##### Solution

1,104,8601,104,860

#### Exercise 74

368,911+857,289368,911+857,289

#### Exercise 75

24,731+592+3,86824,731+592+3,868

29,19129,191

#### Exercise 76

28,925+817+4,59328,925+817+4,593

#### Exercise 77

8,015+76,946+16,5708,015+76,946+16,570

101,531101,531

#### Exercise 78

6,291+54,107+28,6356,291+54,107+28,635

Translate Word Phrases to Math Notation

In the following exercises, translate each phrase into math notation and then simplify.

#### Exercise 79

the sum of 1313 and 1818

13+18=3113+18=31

#### Exercise 80

the sum of 1212 and 1919

#### Exercise 81

the sum of 9090 and 6565

##### Solution

90+65=15590+65=155

#### Exercise 82

the sum of 7070 and 3838

#### Exercise 83

3333 increased by 4949

33+49=8233+49=82

#### Exercise 84

6868 increased by 2525

#### Exercise 85

250250 more than 599599

##### Solution

250+599=849250+599=849

#### Exercise 86

115115 more than 286286

#### Exercise 87

the total of 628628 and 7777

##### Solution

628+77=705628+77=705

#### Exercise 88

the total of 593593 and 7979

#### Exercise 89

##### Solution

915+1,482=2,397915+1,482=2,397

#### Exercise 90

In the following exercises, solve the problem.

#### Exercise 91

Home remodeling Sophia remodeled her kitchen and bought a new range, microwave, and dishwasher. The range cost $1,100,$1,100, the microwave cost $250,$250, and the dishwasher cost $525.$525. What was the total cost of these three appliances?

#### Exercise 98

Home sales Emma is a realtor. Last month, she sold three houses. The selling prices of the houses were $292,540,$505,875,$292,540,$505,875, and $423,699.$423,699. What was the total of the three selling prices?

In the following exercises, find the perimeter of each figure.

#### Exercise 99

##### Solution

The perimeter of the figure is 44 inches.

#### Exercise 101

##### Solution

The perimeter of the figure is 56 meters.

#### Exercise 103

##### Solution

The perimeter of the figure is 71 yards.

#### Exercise 105

##### Solution

The perimeter of the figure is 62 feet.

### Everyday Math

#### Exercise 107

Calories Paulette had a grilled chicken salad, ranch dressing, and a 16-ounce16-ounce drink for lunch. On the restaurant’s nutrition chart, she saw that each item had the following number of calories:

Grilled chicken salad – 320320 calories
Ranch dressing – 170170 calories
16-ounce16-ounce drink – 150150 calories

What was the total number of calories of Paulette’s lunch?

##### Solution

The total number of calories was 640.

#### Exercise 108

Calories Fred had a grilled chicken sandwich, a small order of fries, and a 12-oz12-oz chocolate shake for dinner. The restaurant’s nutrition chart lists the following calories for each item:

Grilled chicken sandwich – 420420 calories
Small fries – 230230 calories
12-oz12-oz chocolate shake – 580580 calories

What was the total number of calories of Fred’s dinner?

#### Exercise 109

Test scores A students needs a total of 400400 points on five tests to pass a course. The student scored 82,91,75,88,and70.82,91,75,88,and70. Did the student pass the course?

##### Solution

Yes, he scored 406 points.

#### Exercise 110

Elevators The maximum weight capacity of an elevator is 11501150 pounds. Six men are in the elevator. Their weights are 210,145,183,230,159,and164210,145,183,230,159,and164 pounds. Is the total weight below the elevators’ maximum capacity?

### Writing Exercises

#### Exercise 111

How confident do you feel about your knowledge of the addition facts? If you are not fully confident, what will you do to improve your skills?

### Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

After reviewing this checklist, what will you do to become confident for all objectives?

## Glossary

sum:
The sum is the result of adding two or more numbers.

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