Whole Numbers: Learn It 4

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Whole Numbers: Learn It 4

Adding Whole Numbers

A college student has a part-time job. Last week they worked [latex]3[/latex] hours on Monday and [latex]4[/latex] hours on Friday. To find the total number of hours they worked last week, they must add [latex]3[/latex] and [latex]4[/latex] together. The operation of addition combines numbers to get a sum. The notation we use to find the sum of [latex]3[/latex] and [latex]4[/latex] is:

[latex]3+4[/latex]

We read this as three plus four and the result is the sum of three and four. The numbers [latex]3[/latex] and [latex]4[/latex] 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.

Operation	Addition
Notation	[latex]+[/latex]
Expression	[latex]3+4[/latex]
Read as	three plus four
Result	the sum of [latex]3[/latex] and [latex]4[/latex]

You may see other words in problems besides “plus” that indicate the need for addition. Below are other addition words you might come across.

Operation	Words	Example	Expression
Addition
	plus	[latex]1[/latex] plus [latex]2[/latex]	[latex]1+2[/latex]
	sum	the sum of [latex]3[/latex] and [latex]4[/latex]	[latex]3+4[/latex]
	increased by	[latex]5[/latex] increased by [latex]6[/latex]	[latex]5+6[/latex]
	more than	[latex]8[/latex] more than [latex]7[/latex]	[latex]7+8[/latex]
	total of	the total of [latex]9[/latex] and [latex]5[/latex]	[latex]9+5[/latex]
	added to	[latex]6[/latex] added to [latex]4[/latex]	[latex]4+6[/latex]

After talking to her counselor, Esmeralda needs to determine how many classes she has taken so far at Blue Sky College. Last year she took [latex]6[/latex] classes, and this year will complete [latex]8[/latex] classes. She needs to determine the sum of [latex]6[/latex] and [latex]8[/latex].

First translate the sentence into math notation and then calculate the results.

In the previous example, the sum of the ones was less than [latex]10[/latex]. But what happens if the sum is [latex]10[/latex] or more? Let’s use our base – [latex]10[/latex] model to find out. The graphic below shows the addition of [latex]17[/latex] and [latex]26[/latex].

An image containing two groups of items. The left group includes 1 rod with 10 blocks and 7 individual blocks 2 rods with 10 blocks each and 6 individual blocks. The label to the left of this group of items is 17 + 26. The right group contains four rods of 10 blocks each and 3 individual blocks. This is labeled showing 17+26=43.

When we add the ones, [latex]7+6[/latex], we get [latex]13[/latex] ones. Because we have more than [latex]10[/latex] ones, we can exchange [latex]10[/latex] of the ones for [latex]1[/latex] ten. Now we have [latex]4[/latex] tens and [latex]3[/latex] ones. Without using the model, we show this as a small red [latex]1[/latex] above the digits in the tens place. When the sum in a place value column is greater than [latex]9[/latex], we carry over to the next column to the left. Carrying is the same as regrouping by exchanging. For example, [latex]10[/latex] ones for [latex]1[/latex] ten or [latex]10[/latex] tens for [latex]1[/latex] hundred.

Adding numbers together may seem confusing, but it can be easy and fun with the right problem-solving strategies! When adding two numbers together, try the following steps.

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 [latex]9[/latex], carry to the next place value.
Continue adding each place value from right to left, adding each place value and carrying if needed.

Add: [latex]43+69[/latex].

When adding whole numbers together we have to keep a couple of properties in mind.

properties of addition

Identity Property of Addition

The sum of any number [latex]a[/latex] and [latex]0[/latex] is the number.

[latex]\begin{array}{}\\ a+0=a\\ 0+a=a\end{array}[/latex]

Commutative Property of Addition

Changing the order of the addends [latex]a[/latex] and [latex]b[/latex] does not change their sum.

[latex]a+b=b+a[/latex]

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.

Hao earned grades of [latex]87,93,68,95,\text{and }89[/latex] on the five tests of the semester. What is the total number of points he earned on the five tests?

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.

This is an image of a perimeter of a patio. There are six sides. The far left side is labeled 4 feet, the top side is labeled 9 feet, the right side is short and labeled 2 feet, then extends across to the left and is labeled 3 feet. From here, the side extends down and is labeled 2 feet. Finally, the base is labeled 6 feet.

Write the numbers so the digits line up vertically.	[latex]\begin{array}{c}\hfill 43\\ \\ \hfill \underset{\text{____}}{+69}\end{array}[/latex]
Add the digits in each place. Add the ones: [latex]3+9=12[/latex]
Write the [latex]2[/latex] in the ones place in the sum. Add the [latex]1[/latex] ten to the tens place.	[latex]\begin{array}{c}\hfill \stackrel{1}{4}3\\ \hfill \underset{\text{____}}{+69}\\ \hfill 2\end{array}[/latex]
Now add the tens: [latex]1+4+6=11[/latex] Write the [latex]11[/latex] in the sum.	[latex]\begin{array}{c}\hfill \stackrel{1}{4}3\\ \hfill \underset{\text{____}}{+69}\\ \hfill 112\end{array}[/latex]

Write a phrase.	the sum of points on the tests
Translate to math notation.	[latex]87+93+68+95+89[/latex]
Then we simplify by adding.
Since there are several numbers, we will write them vertically.	[latex]\begin{array}{}\\ \\ \\ \stackrel{3}{8}7\hfill \\ 93\hfill \\ 68\hfill \\ 95\hfill \\ \underset{\text{+____}}{89}\hfill \\ 432\hfill \end{array}[/latex]
Write a sentence to answer the question.	Hao earned a total of [latex]432[/latex] points.

We are asked to find the perimeter.
Write a phrase.	the sum of the sides
Translate to math notation.	[latex]4+6+2+3+2+9[/latex]
Simplify by adding.	[latex]26[/latex]
Write a sentence to answer the question.
We added feet, so the sum is [latex]26[/latex] feet.	The perimeter of the patio is [latex]26[/latex] feet.