m81244 Add Whole Numbers By the end of this section, you will be able to: Use addition notation Model addition of whole numbers Add whole numbers without models Translate word phrases to math notation Add whole numbers in applications 8069044b-6fb1-49bf-b03a-64988f9b1ddd Before you get started, take this readiness quiz. What is the number modeled by the base-10 blocks? If you missed this problem, review . 215 Write the number three hundred forty-two thousand six using digits? If you missed this problem, review . 342,006

A college student has a part-time job. Last week he worked 3 hours on Monday and 4 hours on Friday. To find the total number of hours he worked last week, he added 3 and 4. The operation of addition combines numbers to get a sum. The notation we use to find the sum of 3 and 4 is: 3+4 We read this as three plus four and the result is the sum of three and four. The numbers 3 and 4 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+4 three plus four the sum of 3 and 4 Translate from math notation to words:ⓐ 7+1ⓑ 12+14 ⓐ 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. Translate from math notation to words: ⓐ 8+4ⓑ 18+11 ⓐ eight plus four; the sum of eight and fourⓑ eighteen plus eleven; the sum of eighteen and eleven Translate from math notation to words: ⓐ 21+16ⓑ 100+200 ⓐ twenty-one plus sixteen; the sum of twenty-one and sixteenⓑ one hundred plus two hundred; the sum of one hundred and two hundred

Addition is really just counting. We will model addition with base-10 blocks. Remember, a block represents 1 and a rod represents 10. Let’s start by modeling the addition expression we just considered, 3+4. Each addend is less than 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 7 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.Doing the Manipulative Math Worksheets activity “Model Addition of Whole Numbers” will help you develop a better understanding of adding whole numbers. Model the addition 2+6. 2+6 means the sum of 2 and 6Each 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 8 blocks in all, so 2+6=8. Model: 3+6. Model: 5+1. When the result is 10 or more ones blocks, we will exchange the 10 blocks for one rod. Model the addition 5+8. 5+8 means the sum of 5 and 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. Model the addition: 5+7. Model the addition: 6+8. Next we will model adding two digit numbers. Model the addition: 17+26. 17+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=43 We have shown that 17+26=43 Model the addition: 15+27. Model the addition: 16+29.

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 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. + 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. The sum of any number a and 0 is the number.a+0=a0+a=a Find each sum: ⓐ 0+11ⓑ 42+0 ⓐ The first addend is zero. The sum of any number and zero is the number. 0+11=11 ⓑ The second addend is zero. The sum of any number and zero is the number. 42+0=42 Find each sum: ⓐ 0+19ⓑ 39+0 ⓐ 0+19=19ⓑ 39+0=39 Find each sum: ⓐ 0+24ⓑ 57+0 ⓐ 0+24=24ⓑ 57+0=57 Look at the pairs of sums. 2+3=5 3+2=5 4+7=11 7+4=11 8+9=17 9+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. Changing the order of the addends a and b does not change their sum.a+b=b+a Add: ⓐ 8+7ⓑ 7+8 ⓐ Add. 8+7 15 ⓑ Add. 7+8 15 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. Add: 9+7 and 7+9. 9+7=16; 7+9=16 Add: 8+6 and 6+8. 8+6=14; 6+8=14 Add: 28+61. 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____ Then add the digits in each place value. Add the ones: 8+1=9Add the tens: 2+6=8 28 +61____89 Add: 32+54. 32+54=86 Add: 25+74. 25+74=99 In the previous example, the sum of the ones and the sum of the tens were both less than 10. But what happens if the sum is 10 or more? Let’s use our base-10 model to find out. shows the addition of 17 and 26 again.

When we add the ones, 7+6, we get 13 ones. Because we have more than 10 ones, we can exchange 10 of the ones for 1 ten. Now we have 4 tens and 3 ones. Without using the model, we show this as a small red 1 above the digits in the tens place. When the sum in a place value column is greater than 9, we carry over to the next column to the left. Carrying is the same as regrouping by exchanging. For example, 10 ones for 1 ten or 10 tens for 1 hundred. Write the numbers so each place value lines up vertically. 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. Continue adding each place value from right to left, adding each place value and carrying if needed. Add: 43+69. Write the numbers so the digits line up vertically. 43 +69____ Add the digits in each place.Add the ones: 3+9=12 Write the 2 in the ones place in the sum.Add the 1 ten to the tens place. 413 +69____2 Now add the tens: 1+4+6=11Write the 11 in the sum. 413 +69____112 Add: 35+98. 35+98=133 Add: 72+89. 72+89=161 Add: 324+586. Write the numbers so the digits line up vertically. Add the digits in each place value.Add the ones: 4+6=10Write the 0 in the ones place in the sum and carry the 1 ten to the tens place. Add the tens: 1+2+8=11Write the 1 in the tens place in the sum and carry the 1 hundred to the hundreds Add the hundreds: 1+3+5=9Write the 9 in the hundreds place. Add: 456+376. 456+376=832 Add: 269+578. 269+578=847 Add: 1,683+479. Write the numbers so the digits line up vertically. 1,683 +479______ Add the digits in each place value. Add the ones: 3+9=12.Write the 2 in the ones place of the sum and carry the 1 ten to the tens place. 1,6813 +479______2 Add the tens: 1+7+8=16Write the 6 in the tens place and carry the 1 hundred to the hundreds place. 1,61813 +479______62 Add the hundreds: 1+6+4=11Write the 1 in the hundreds place and carry the 1 thousand to the thousands place. 1,61813 +479______162 Add the thousands 1+1=2.Write the 2 in the thousands place of the sum. 1,161813 +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. Add: 4,597+685. 4,597+685=5,282 Add: 5,837+695. 5,837+695=6,532 Add: 21,357+861+8,596. Write the numbers so the place values line up vertically. 21,357 861 +8,596_______ Add the digits in each place value. Add the ones: 7+1+6=14Write the 4 in the ones place of the sum and carry the 1 to the tens place. 21,3517 861 +8,596_______ 4 Add the tens: 1+5+6+9=21Write the 1 in the tens place and carry the 2 to the hundreds place. 21,32517 861 +8,596_______ 14 Add the hundreds: 2+3+8+5=18Write the 8 in the hundreds place and carry the 1 to the thousands place. 21,132517 861 +8,596_______ 814 Add the thousands 1+1+8=10.Write the 0 in the thousands place and carry the 1 to the ten thousands place. 211,132517 861 +8,596_______ 0814 Add the ten-thousands 1+2=3.Write the 3 in the ten thousands place in the sum. 211,132517 861 +8,596_______ 30,814This 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. Add: 46,195+397+6,281. 46,195+397+6,281=52,873 Add: 53,762+196+7,458. 53,762+196+7,458=61,416

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 . Operation Words Example Expression Addition plussumincreased bymore thantotal ofadded to 1 plus 2the sum of 3 and 45 increased by 68 more than 7the total of 9 and 56 added to 4 1+23+45+67+89+54+6 Translate and simplify: the sum of 19 and 23. The word sum tells us to add. The words of 19 and 23 tell us the addends. The sum of 19 and 23 Translate. 19+23 Add. 42 The sum of 19 and 23 is 42. Translate and simplify: the sum of 17 and 26. Translate: 17+26; Simplify: 43 Translate and simplify: the sum of 28 and 14. Translate: 28+14; Simplify: 42 Translate and simplify: 28 increased by 31. The words increased by tell us to add. The numbers given are the addends. 28 increased by 31. Translate. 28+31 Add. 59 So 28 increased by 31 is 59. Translate and simplify: 29 increased by 76. Translate: 29+76; Simplify 105 Translate and simplify: 37 increased by 69. Translate 37 + 69; Simplify 106

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. Hao earned grades of 87,93,68,95,and89 on the five tests of the semester. What is the total number of points he earned on the five tests? 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+89 Then we simplify by adding. Since there are several numbers, we will write them vertically. 837936895+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 432 points. It is important to include the appropriate units in all answers to applications problems. Mark is training for a bicycle race. Last week he rode 18 miles on Monday, 15 miles on Wednesday, 26 miles on Friday, 49 miles on Saturday, and 32 miles on Sunday. What is the total number of miles he rode last week? He rode 140 miles. Lincoln Middle School has three grades. The number of students in each grade is 230,165,and325. What is the total number of students? 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. Find the perimeter of the patio shown. We are asked to find the perimeter. Write a phrase. the sum of the sides Translate to math notation. 4+6+2+3+2+9 Simplify by adding. 26 Write a sentence to answer the question. We added feet, so the sum is 26 feet. The perimeter of the patio is 26 feet. Find the perimeter of the figure. All lengths are in inches. The perimeter is 30 inches. Find the perimeter of the figure. All lengths are in inches. The perimeter is 36 inches. Adding Two-Digit Numbers with base-10 blocks Adding Three-Digit Numbers with base-10 blocks Adding Whole Numbers

Addition Notation To describe addition, we can use symbols and words. Operation Notation Expression Read as Result Addition + 3+4 three plus four the sum of 3 and 4 Identity Property of Addition The sum of any number a and 0 is the number. a+0=a 0+a=a Commutative Property of Addition Changing the order of the addends a and b does not change their sum. a+b=b+a. Add whole numbers. Write the numbers so each place value lines up vertically. 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. Continue adding each place value from right to left, adding each place value and carrying if needed.

Use Addition Notation In the following exercises, translate the following from math expressions to words. 5+2 five plus two; the sum of 5 and 2. 6+3 13+18 thirteen plus eighteen; the sum of 13 and 18. 15+16 214+642 two hundred fourteen plus six hundred forty-two; the sum of 214 and 642 438+113 Model Addition of Whole Numbers In the following exercises, model the addition. 2+4 2+4=6 5+3 8+4 8+4=12 5+9 14+75 14+75=89 15+63 16+25 16+25=41 14+27 Add Whole Numbers In the following exercises, fill in the missing values in each chart. In the following exercises, add. ⓐ 0+13ⓑ 13+0 ⓐ 13 ⓑ 13 ⓐ 0+5,280ⓑ 5,280+0 ⓐ 8+3ⓑ 3+8 ⓐ 11ⓑ 11 ⓐ 7+5ⓑ 5+7 45+33 78 37+22 71+28 99 43+53 26+59 85 38+17 64+78 142 92+39 168+325 493 247+149 584+277 861 175+648 832+199 1,031 775+369 6,358+492 6,850 9,184+578 3,740+18,593 22,333 6,118+15,990 485,012+619,848 1,104,860 368,911+857,289 24,731+592+3,868 29,191 28,925+817+4,593 8,015+76,946+16,570 101,531 6,291+54,107+28,635 Translate Word Phrases to Math Notation In the following exercises, translate each phrase into math notation and then simplify. the sum of 13 and 18 13+18=31 the sum of 12 and 19 the sum of 90 and 65 90+65=155 the sum of 70 and 38 33 increased by 49 33+49=82 68 increased by 25 250 more than 599 599+250=849 115 more than 286 the total of 628 and 77 628+77=705 the total of 593 and 79 1,482 added to 915 915+1,482=2,397 2,719 added to 682 Add Whole Numbers in Applications In the following exercises, solve the problem. Home remodeling Sophia remodeled her kitchen and bought a new range, microwave, and dishwasher. The range cost $1,100, the microwave cost $250, and the dishwasher cost $525. What was the total cost of these three appliances? The total cost was $1,875. Sports equipment Aiden bought a baseball bat, helmet, and glove. The bat cost $299, the helmet cost $35, and the glove cost $68. What was the total cost of Aiden’s sports equipment? Bike riding Ethan rode his bike 14 miles on Monday, 19 miles on Tuesday, 12 miles on Wednesday, 25 miles on Friday, and 68 miles on Saturday. What was the total number of miles Ethan rode? Ethan rode 138 miles. Business Chloe has a flower shop. Last week she made 19 floral arrangements on Monday, 12 on Tuesday, 23 on Wednesday, 29 on Thursday, and 44 on Friday. What was the total number of floral arrangements Chloe made? Apartment size Jackson lives in a 7 room apartment. The number of square feet in each room is 238,120,156,196,100,132, and 225. What is the total number of square feet in all 7 rooms? The total square footage in the rooms is 1,167 square feet. Weight Seven men rented a fishing boat. The weights of the men were 175,192,148,169,205,181, and 225 pounds. What was the total weight of the seven men? Salary Last year Natalie’s salary was $82,572. Two years ago, her salary was $79,316, and three years ago it was $75,298. What is the total amount of Natalie’s salary for the past three years? Natalie’s total salary is $237,186. Home sales Emma is a realtor. Last month, she sold three houses. The selling prices of the houses were $292,540,$505,875, and $423,699. What was the total of the three selling prices? In the following exercises, find the perimeter of each figure. The perimeter of the figure is 44 inches. The perimeter of the figure is 56 meters. The perimeter of the figure is 71 yards. The perimeter of the figure is 62 feet.

Calories Paulette had a grilled chicken salad, ranch dressing, and a 16-ounce drink for lunch. On the restaurant’s nutrition chart, she saw that each item had the following number of calories: Grilled chicken salad – 320 calories Ranch dressing – 170 calories 16-ounce drink – 150 calories What was the total number of calories of Paulette’s lunch? The total number of calories was 640. Calories Fred had a grilled chicken sandwich, a small order of fries, and a 12-oz chocolate shake for dinner. The restaurant’s nutrition chart lists the following calories for each item: Grilled chicken sandwich – 420 calories Small fries – 230 calories 12-oz chocolate shake – 580 calories What was the total number of calories of Fred’s dinner? Test scores A student needs a total of 400 points on five tests to pass a course. The student scored 82,91,75,88,and70. Did the student pass the course? Yes, he scored 406 points. Elevators The maximum weight capacity of an elevator is 1150 pounds. Six men are in the elevator. Their weights are 210,145,183,230,159,and164 pounds. Is the total weight below the elevator’s maximum capacity? Yes, the total weight is 1091 pounds.

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? Answers will vary. How have you used models to help you learn the addition facts?

ⓐ 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?

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