1.2 Add Whole Numbers

Learning objectives

Learning objectives

• Use addition notation

• Model addition of whole numbers

• Add whole numbers without models

• Translate word phrases to math notation

• Add whole numbers in applications

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.

Example 1.11

Solution

1. Identify the operation and addends. Each expression contains a plus sign, which names addition, and the two numbers written on either side are its addends.
2. Read the first expression. The expression $7+1$ is read as seven plus one, and its result is called the sum of seven and one.
3. Read the second expression. The expression $12+14$ is read as twelve plus fourteen, and its result is called the sum of twelve and fourteen.
4. State both translations clearly. The operation and both addends must be named in words for each expression before the resulting sum is identified.

Takeaway: An addition expression is read from its addends and operation, so $7+1$ is seven plus one and $12+14$ is twelve plus fourteen.

Model Addition of Whole Numbers

Addition is really just counting. We will model addition with $\mathrm{base-10}$ 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 $\mathrm{(=)}$ to show the sum. A math sentence that shows that two expressions are equal is called an equation. We have shown that. $3+4=7.$

Example 1.12

Solution

1. Model the first addend. Draw $2$ ones blocks to represent the first addend in $2+6$. The first diagram shows these two blocks above the label $2$.
2. Model the second addend. Draw $6$ more ones blocks beside the first group to represent $6$. The next diagram shows both groups, labeled $2$ and $6$.
3. Combine the blocks. Count all $2+6=8$ ones blocks in the final diagram. Because the total is fewer than $10$, no exchange for a tens rod is needed.

Takeaway: A model of addition combines equal-size groups, and ten ones blocks can be exchanged for one tens rod.

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

Example 1.13

Solution

1. Model the first addend. Use $5$ ones blocks to represent the first number in $5+8$; the first diagram labels this group with $5$.
2. Model the second addend. Place a group of $8$ ones blocks alongside the group of $5$, as shown in the second diagram labeled $5$ and $8$.
3. Combine and regroup. The combined model contains $13$ ones blocks. Regroup one group of $10$ ones into a tens rod, leaving $3$ ones, so the final diagram represents $1$ ten and $3$ ones.

Takeaway: When combining base-ten blocks, regroup ten ones into one ten whenever the ones total reaches $10$ or more.

Next we will model adding two digit numbers.

Example 1.14

Solution

1. Model $17$. Use $1$ tens rod and $7$ ones blocks, matching the first diagram that shows a group of ten and a group of seven.
2. Model $26$. Add $2$ tens rods and $6$ ones blocks, producing the second diagram with two groups of ten and six individual ones.
3. Combine by place value. The model now has $3$ tens rods and $13$ ones blocks. The diagram marks $10$ of the ones for regrouping.
4. Regroup and count. Exchange $10$ ones for $1$ tens rod. This leaves $4$ tens rods and $3$ ones, so the total represented is $40+3=43$ and $17+26=43$.

Takeaway: Adding tens and ones separately keeps place values aligned and makes regrouping visible before the total is counted.

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.

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.

Example 1.15

Solution

1. Recall the additive identity. Adding zero to a whole number leaves that number unchanged, regardless of whether zero is the first or second addend.
2. Add the first pair. Since the first addend in $0+11$ is zero, the sum is the other addend, so $0+11=11$.
3. Add the second pair. Since the second addend in $42+0$ is zero, the sum is the other addend, so $42+0=42$.
4. Compare the two results. Both calculations follow the same identity rule, so adding zero never changes the nonzero addend in this example.

Takeaway: The additive identity is $0$, so adding zero to either side of a sum leaves the other addend unchanged: $0+11=11$ and $42+0=42$.

Look at the pairs of sums.

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.

Example 1.16

Solution

1. Add the first ordered pair. Combine $8$ and $7$ to obtain $15$, so $8+7=15$.
2. Add the reversed ordered pair. Combine $7$ and $8$ to obtain the same total, so $7+8=15$.
3. Compare the addends and sums. The two expressions contain the same two addends in opposite orders, and both sums equal $15$.
4. Apply the commutative property. Reversing the order of the addends in addition does not change the result.

Takeaway: Addition is commutative because reversing the addends does not change the result, so $8+7$ and $7+8$ both equal $15$.

Example 1.17

Solution

1. Align the addends vertically. Write $28$ above $61$ so that the ones digits $8$ and $1$ share one column and the tens digits $2$ and $6$ share the next column.
2. Add the ones digits. The ones column gives $8+1=9$, so write $9$ in the ones place of the sum.
3. Add the tens digits. The tens column gives $2+6=8$, so write $8$ in the tens place of the sum.
4. Read the completed sum. The result $89$ matches $28+61$, because each column has been added by place value.

Takeaway: Line up digits by place value and add within each column, so $28+61$ has ones sum $8+1=9$ and tens sum $2+6=8$, giving $89$.

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 $\mathrm{base-10}$ model to find out. Figure 1.10 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.

Example 1.18

Solution

1. Align the addends vertically. Write $43$ above $69$ so that $3$ and $9$ are ones digits and $4$ and $6$ are tens digits.
2. Add the ones and carry. The ones digits give $3+9=12$, so write $2$ in the ones place and carry $1$ ten to the tens column.
3. Add the tens and carry. Including the carried ten, the tens column gives $1+4+6=11$, so write $1$ in the tens place and carry $1$ hundred.
4. Write the hundreds place. Put the carried $1$ in the hundreds place of the result to complete the sum.
5. Read the final result. The completed addition shows that $43+69=112$, with both carries included correctly.

Takeaway: When a ones-column sum reaches $12$, write $2$ and carry $1$ ten; then $1+4+6=11$ in the tens column gives $43+69=112$.

Example 1.19

Solution

1. Line up the addends. Write $324$ above $586$ so the ones, tens, and hundreds digits align in their corresponding columns.
2. Add the ones. The ones digits give $4+6=10$. Write $0$ in the ones place and carry $1$ ten to the tens column, as shown by the carried digit above the tens place.
3. Add the tens. Include the carry, so $2+8+1=11$. Write $1$ in the tens place and carry $1$ hundred to the hundreds column.
4. Add the hundreds. The hundreds digits and carried hundred give $3+5+1=9$. Thus $324+586=910$, matching the completed addition in the final diagram.

Takeaway: When a column sum reaches $10$, write the ones digit and carry the remaining ten to the next place on the left.

Example 1.20

Solution

1. Align the place values. Write $1,683$ above $479$ with the rightmost digits aligned, so $3$ and $9$ are ones, $8$ and $7$ are tens, and $6$ and $4$ are hundreds.
2. Add the ones and carry. The ones column gives $3+9=12$, so write $2$ and carry $1$ ten into the tens column.
3. Add the tens and carry. Including the carry, the tens column gives $1+8+7=16$, so write $6$ and carry $1$ hundred.
4. Add the hundreds and carry. Including that carry, the hundreds column gives $1+6+4=11$, so write $1$ and carry $1$ thousand.
5. Add the thousands. The thousands column gives $1+1=2$, so write $2$ in the thousands place.
6. Read the completed sum. The digits $2,162$ form the answer, so $1,683+479=2,162$.

Takeaway: Align addends by place value and carry each ten or hundred leftward as needed, so $1,683+479=2,162$.

Example 1.21

Solution

1. Align all three addends. Write $21,357$, $861$, and $8,596$ vertically so matching ones, tens, hundreds, thousands, and ten-thousands digits form columns.
2. Add the ones and carry. The ones column gives $7+1+6=14$, so write $4$ and carry $1$ ten into the tens column.
3. Add the tens and carry. Including the carry, the tens column gives $1+5+6+9=21$, so write $1$ and carry $2$ hundreds.
4. Add the hundreds and carry. Including the carry, the hundreds column gives $2+3+8+5=18$, so write $8$ and carry $1$ thousand.
5. Add the thousands and carry. The thousands column gives $1+1+8=10$, so write $0$ and carry $1$ ten-thousand.
6. Finish and read the sum. The ten-thousands column gives $1+2=3$, so the completed result is $30,814$ and $21,357+861+8,596=30,814$.

Takeaway: Any number of addends can be added column by column when their place values align, giving $21,357+861+8,596=30,814$.

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.

Table

1.2

Example 1.22

Solution

1. Identify the operation word. The word sum in the phrase tells us to perform addition, while $19$ and $23$ are the two addends.
2. Translate the phrase. Put the addends in the stated order and join them with a plus sign to obtain $19+23$.
3. Simplify the expression. Add the ones digits to get $9+3=12$, write $2$, and carry $1$ ten; then $1+1+2=4$ in the tens column.
4. State the result. The sum of $19$ and $23$ is $42$, so the translated expression simplifies to $19+23=42$.

Takeaway: The phrase the sum of $19$ and $23$ translates to $19+23$, and adding the two addends gives $42$.

Example 1.23

Solution

1. Interpret the phrase. The words increased by describe an addition, so $28$ is the starting amount and $31$ is the amount added.
2. Translate to notation. Write the starting amount first and the increase second, giving the expression $28+31$.
3. Add the ones digits. The ones column gives $8+1=9$, so write $9$ in the ones place of the result.
4. Add the tens digits and state the result. The tens column gives $2+3=5$, so $28$ increased by $31$ is $59$.

Takeaway: The phrase increased by signals addition, so $28$ increased by $31$ becomes $28+31=59$.

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

Solution

1. Identify the requested quantity. The word total asks for the sum of all five test scores, so every grade must be included once.
2. Write the addition expression. Translate the five scores into math notation as $87+93+68+95+89$.
3. Align and add the ones. The ones digits give $7+3+8+5+9=32$, so write $2$ in the ones place and carry $3$ tens.
4. Add the tens with the carry. The tens digits give $8+9+6+9+8=40$, and adding the carried $3$ tens gives $43$, so write $3$ in the tens place and carry $4$ hundreds.
5. Finish and include the unit. The hundreds place receives the carried $4$, giving $432$ points, so Hao earned a total of $432$ points.

Takeaway: To find Hao's total test points, add all five scores, $87+93+68+95+89=432$, and report the result in points.

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

Solution

1. Identify what perimeter means. The perimeter of the patio is the sum of the lengths of all six sides shown in the diagram.
2. List the side lengths. Read the measurements around the polygon as $4$, $6$, $2$, $3$, $2$, and $9$ feet.
3. Write the sum. Combine all six side lengths in the expression $4+6+2+3+2+9$, keeping the unit feet throughout.
4. Simplify carefully. Adding the lengths gives $4+6=10$, $2+3=5$, and $2+9=11$, for a total of $10+5+11=26$ feet.
5. State the perimeter in context. The perimeter of the patio is $26$ feet, because it measures the distance around the entire patio.

Takeaway: Perimeter is the sum of all side lengths, so the patio's perimeter is $4+6+2+3+2+9=26$ feet.

Section 1.2 Exercises

Section 1.2 Exercises

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.

In the following exercises, add.

Translate Word Phrases to Math Notation

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

Add Whole Numbers in Applications

In the following exercises, solve the problem.

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

Everyday Math

Writing Exercises

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?

1. Translate from math notation to words:

1. ⓐ $8+4$

2. ⓑ $18+11$

2. Translate from math notation to words:

1. ⓐ $21+16$

2. ⓑ $100+200$

3. Find each sum:

1. ⓐ $0+19$

2. ⓑ $39+0$

4. Find each sum:

1. ⓐ $0+24$

2. ⓑ $57+0$

1. Add: $9+7$ and $7+9.$

2. Add: $8+6$ and $6+8.$

3. Add: $32+54.$

4. Add: $25+74.$

1. Add: $35+98.$

2. Add: $72+89.$

3. Add: $456+376.$

4. Add: $269+578.$

1. Add: $4,597+685.$

2. Add: $5,837+695.$

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

4. Add: $53,762+196+7,458.$

1. Translate and simplify: the sum of $17$ and $26.$

2. Translate and simplify: the sum of $28$ and $14.$

3. Translate and simplify: $29$ increased by $76.$

4. Translate and simplify: $37$ increased by $69.$

1. 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?

2. Lincoln Middle School has three grades. The number of students in each grade is $230,165,\mathrm{and}\,325.$ What is the total number of students?

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

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

1. $5+2$

2. $13+18$

3. $214+642$

4. $2+4$

1. $8+4$

2. $14+75$

3. $16+25$

4. 1. ⓐ $0+13$

2. ⓑ $13+0$

1. 1. ⓐ $8+3$

2. ⓑ $3+8$

2. $45+33$

3. $71+28$

4. $26+59$

1. $64+78$

2. $168+325$

3. $584+277$

4. $832+199$

1. $6,358+492$

2. $3,740+18,593$

3. $485,012+619,848$

4. $24,731+592+3,868$

1. $8,015+76,946+16,570$

2. the sum of $13$ and $18$

3. the sum of $90$ and $65$

4. $33$ increased by $49$

1. $250$ more than $599$

2. the total of $628$ and $77$

3. $1,482$ added to $915$

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

1. 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?

2. 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?

3. Salary Last year Natalie’s salary was $\text{\textdollar 82,572}.$ Two years ago, her salary was $\text{\textdollar 79,316},$ and three years ago it was $\text{\textdollar 75,298}.$ What is the total amount of Natalie’s salary for the past three years?

4. Calories Paulette had a grilled chicken salad, ranch dressing, and a $\mathrm{16-ounce}$ 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 $\mathrm{16-ounce}$ drink – $150$ calories

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

1. Test scores A student needs a total of $400$ points on five tests to pass a course. The student scored $82,91,75,88,\mathrm{and}\,70.$ Did the student pass the course?

2. Elevators The maximum weight capacity of an elevator is $1150$ pounds. Six men are in the elevator. Their weights are $210,145,183,230,159,\mathrm{and}\,164$ pounds. Is the total weight below the elevator’s maximum capacity?

3. 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?