Wednesday, 10 June 2020

Add Whole Numbers


Learning Objectives
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


Be Prepared 1.1

Before you get started, take this readiness quiz.



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





If you missed this problem, review Example 1.2.


Be Prepared 1.2



Write the number three hundred forty-two thousand six using digits?
If you missed this problem, review Example 1.7.
Use Addition Notation

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:

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.

Addition Notation

To describe addition, we can use symbols and words.

OperationNotationExpressionRead asResultAddition + 3+4 three plus four the sum of 3 and 4


Example 1.11



Translate from math notation to words:
ⓐ 7
ⓑ 12


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.


Try It 1.21



Translate from math notation to words:
ⓐ 8
ⓑ 18


Try It 1.22



Translate from math notation to words:
ⓐ 21
ⓑ 100
Model Addition of Whole Numbers

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.

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.

Manipulative Mathematics

Doing the Manipulative Mathematics activity “Model Addition of Whole Numbers” will help you develop a better understanding of adding whole numbers.

Example 1.12



Model the addition 2+6.


Solution

2+6 means the sum of 2 and 6

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

When the result is 10 or more ones blocks, we will exchange the 10 blocks for one rod.

Example 1.13



Model the addition 5+8.


Solution

5+8 means the sum of 5 and 8.

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.


Try It 1.25



Model the addition: 5+7.


Try It 1.26



Model the addition: 6+8.

Next we will model adding two digit numbers.

Example 1.14



Model the addition: 17+26.


Solution

17 means the sum of 17 and 26.


Try It 1.27



Model each addition: 15+27.


Try It 1.28



Model each addition: 16+29.
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 1.1 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.

+01234567890 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


Table 1.1

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.

Identity Property of Addition

The sum of any number a and 0 is the number.

Example 1.15



Find each sum:
ⓐ 0
ⓑ 42


Solution

ⓐ 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



Try It 1.29



Find each sum:
ⓐ 0
ⓑ 39


Try It 1.30



Find each sum:
ⓐ 0
ⓑ 57

Look at the pairs of sums.

2 3
4 7
8 9


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.

Commutative Property of Addition

Changing the order of the addends a and b does not change their sum.

Example 1.16



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.

Example 1.17



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____
Then add the digits in each place value.
Add the ones: 8
Add the tens: 2 28+61____89


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. Figure 1.10 shows the addition of 17 and 26 again.


Figure 1.10

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.

How To
Add whole numbers.
Step 1. Write the numbers so each place value lines up vertically.
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.
Step 3. Continue adding each place value from right to left, adding each place value and carrying if needed.

Example 1.18



Solution

Write the numbers so the digits line up vertically. 43+69____
Add the digits in each place.
Add the ones: 3
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
Write the 11 in the sum. 413+69____112


Example 1.20



Add: 1,683+479.


Solution

Write the numbers so the digits line up vertically. 1,683+479______
Add the digits in each place value.
Add the ones: 3
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
Write the 6 in the tens place and carry the 1 hundred to the hundreds place. 1,61813+479______62
Add the hundreds: 1
Write the 1 in the hundreds place and carry the 1 thousand to the thousands place. 1,61813 +479______162
Add the thousands 1.
Write the 2 in the thousands place of the sum. 1,61813 +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.


Try It 1.39



Add: 4,597+685.


Try It 1.40



Add: 5,837+695.

Example 1.21



Add: 21,357+861+8,596.


Solution

Write the numbers so the place values line up vertically. 21,357861+8,596_______
Add the digits in each place value.
Add the ones: 7
Write the 4 in the ones place of the sum and carry the 1 to the tens place. 21,3517861 +8,596_______4
Add the tens: 1
Write the 1 in the tens place and carry the 2 to the hundreds place. 21,32517861 +8,596_______14
Add the hundreds: 2
Write the 8 in the hundreds place and carry the 1 to the thousands place. 21,132517861 +8,596_______814
Add the thousands 1.
Write the 0 in the thousands place and carry the 1 to the ten thousands place. 211,132517861 +8,596_______0814
Add the ten-thousands 1.
Write the 3 in the ten thousands place in the sum. 211,132517861 +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.


Try It 1.41



Add: 46,195+397+6,281.


Try It 1.42



Add: 53,762+196+7,458.
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 1.2.

OperationWordsExampleExpressionAddition plus
sum
increased by
more than
total of
added to 1 plus 2
the sum of 3 and 4
5 increased by 6
8 more than 7
the total of 9 and 5
6 added to 4 1+2
3+4
5+6
7+8
9+5
4+6


Table 1.2

Example 1.22



Translate and simplify: the sum of 19 and 23.


Solution

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
Add. 42
The sum of 19 and 23 is 42.



Try It 1.43



Translate and simplify: the sum of 17 and 26.


Try It 1.44



Translate and simplify: the sum of 28 and 14.

Example 1.23



Translate and simplify: 28 increased by 31.


Solution

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

28 increased by 31.
Translate. 28
Add. 59
So 28 increased by 31 is 59.



Try It 1.45



Translate and simplify: 29 increased by 76.


Try It 1.46



Translate and simplify: 37 increased by 69.
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 1.24



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?


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
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.


Try It 1.47



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?


Try It 1.48



Lincoln Middle School has three grades. The number of students in each grade is 230,165,and325. What is the total number of 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 1.25



Find the perimeter of the patio shown.




Solution

We are asked to find the perimeter.
Write a phrase. the sum of the sides
Translate to math notation. 4
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.



Try It 1.49



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




Try It 1.50



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



Section 1.2 Exercises
Practice Makes Perfect

Use Addition Notation

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

Model Addition of Whole Numbers

In the following exercises, model the addition.

Add Whole Numbers

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



74.





76.





78.



In the following exercises, add.



80.

ⓐ 0
ⓑ 5,280



99.

3,740+18,593



100.

6,118+15,990



101.

485,012+619,848



102.

368,911+857,289



103.

24,731+592+3,868



104.

28,925+817+4,593



105.

8,015+76,946+16,570



106.

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.



107.

the sum of 13 and 18



108.

the sum of 12 and 19



109.

the sum of 90 and 65



110.

the sum of 70 and 38



111.

33 increased by 49



112.

68 increased by 25



113.

250 more than 599



114.

115 more than 286



115.

the total of 628 and 77



116.

the total of 593 and 79



117.

1,482 added to 915



118.

2,719 added to 682

Add Whole Numbers in Applications

In the following exercises, solve the problem.



119.

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?



120.

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?



121.

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?



122.

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?



123.

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?



124.

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?



125.

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?



126.

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.



128.





130.





132.





134.


Everyday Math



135.


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?



136.


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?



137.

Test scores A students 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?



138.

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?
Writing Exercises



139.

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?



140.

How have you used models to help you learn the addition facts?
Self Check

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

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