Download Commutative Property of Addition with Decimals

Commutative Property of
Addition with Decimals
Jen Kershaw
Say Thanks to the Authors
Click http://www.ck12.org/saythanks
(No sign in required)
To access a customizable version of this book, as well as other
interactive content, visit www.ck12.org
CK-12 Foundation is a non-profit organization with a mission to
reduce the cost of textbook materials for the K-12 market both
in the U.S. and worldwide. Using an open-content, web-based
collaborative model termed the FlexBook®, CK-12 intends to
pioneer the generation and distribution of high-quality educational
content that will serve both as core text as well as provide an
adaptive environment for learning, powered through the FlexBook
Platform®.
Copyright © 2014 CK-12 Foundation, www.ck12.org
The names “CK-12” and “CK12” and associated logos and the
terms “FlexBook®” and “FlexBook Platform®” (collectively
“CK-12 Marks”) are trademarks and service marks of CK-12
Foundation and are protected by federal, state, and international
laws.
Any form of reproduction of this book in any format or medium,
in whole or in sections must include the referral attribution link
http://www.ck12.org/saythanks (placed in a visible location) in
addition to the following terms.
Except as otherwise noted, all CK-12 Content (including CK-12
Curriculum Material) is made available to Users in accordance
with the Creative Commons Attribution-Non-Commercial 3.0
Unported (CC BY-NC 3.0) License (http://creativecommons.org/
licenses/by-nc/3.0/), as amended and updated by Creative Commons from time to time (the “CC License”), which is incorporated
herein by this reference.
Complete terms can be found at http://www.ck12.org/terms.
Printed: November 20, 2014
AUTHOR
Jen Kershaw
www.ck12.org
Chapter 1. Commutative Property of Addition with Decimals
C HAPTER
1
Commutative Property of
Addition with Decimals
Here you’ll learn to identify and apply the commutative and associative properties of addition in decimal operations
with numerical and variable expressions.
Have you ever trained for a running event?
While training, Travis had a daily routine. On Monday, began his day with a 3.2 mile run followed by 1.2 miles of
sprints on the track at school. On Tuesday, he began with the sprints and then ran 3.2 miles.
Except for the order, was his training any different?
Which mathematical property does this represent?
This Concept is about identifying the commutative and associative properties of addition in decimal operations. By the end of it, you will know which property is illustrated in this situation.
Guidance
Previously we worked on decimals. Now we need to examine decimal addition through the lens of the two mathematical properties which pertain to addition: the Commutative Property of Addition and the Associative Property
of Addition.
In the examples that we worked with in the Decimal Addition Using Front-End Estimation Concept, we added three
decimal addends in the order in which they appeared. What would happen if we added them in a different order?
Would the sum change?
The Commutative Property of Addition states that the order of the addends does not change the sum .
Let’s test the property using simple whole numbers.
4 + 5 + 9 = 18
5 + 4 + 9 = 18
9 + 5 + 4 = 18
4 + 9 + 5 = 18
5 + 9 + 4 = 18
9 + 4 + 5 = 18
As you can see, we can add the three addends (4, 5, and 9) in many different orders. The Commutative Property of
Addition works also works for four, five, six addends, and it works for decimal addends, too.
What about the Associative Property of Addition?
The Associative Property of Addition states that the way in which addends are grouped does not change the
sum . The Associative Property uses parentheses to help us with the grouping. When parentheses are inserted into
a problem, according to the order of operations, you perform that operation first. What happens if you move the
parentheses? This is where the Associative Property comes in.
Once again, let’s test the property using simple whole numbers.
(4 + 5) + 9 = 18
(5 + 4) + 9 = 18
(9 + 5) + 4 = 18
Clearly, the different way the addends are grouped has no effect on the sum. The Associate Property of Addition
works for multiple addends as well as decimal addends.
1
www.ck12.org
These two properties are extremely useful for solving equations. An equation is a mathematical statement that
two expressions are equal. For example, we can say that 15 + 7 = 24 − 2 since both sides of the equation equal 22.
A variable equation is an equation such as 12 + x = 14. It is called a variable equation because it includes an
algebraic unknown, or a variable.
Find the value for x in the following equation, 71.321 + 42.29 = x + 71.321
Here is where we can use the Commutative Property of Addition. We know that one side of the equation is equal to
the other side of the equation therefore we just substitute the given value in for x since the second addend did not
change
Our answer is x is equal to 42.29.
We can use the Associative Property for both expressions and equations. Let’s look at an expression.
Find the value when x is 2 and y is 3. (x + 4) − y
This is a variable expression. Notice that there are two given values for x and y, but that those values could be any
numbers. The key here is that we could move the parentheses and the two variable expressions would have the same
value. We can find the value of this expression by substituting 2 and 3 into the expression.
(2 + 4) − 3
The value of this variable expression is 3.
What happens if we move the parentheses?
x + (4 − y)
We substitute the given values in for x and y.
2 + (4 − 3)
The value of this expression is 3.
Because of the Associative Property, both variable expressions have the same value regardless of the groupings.
Now it’s time for you to identify which property is being used in the following examples.
Example A
3.4 + 3.2 + 5.6 = 5.6 + 3.4 + 3.2
Solution: Commutative Property of Addition
Example B
(3.4 + 3.2) + 5.6 = 3.4 + (3.2 + 5.6)
Solution: Associative Property of Addition
Example C
True or false. When using the Associative property the order of the numbers always changes.
Solution: False, only the position of the grouping symbols changes.
Here is the original problem once again.
2
www.ck12.org
Chapter 1. Commutative Property of Addition with Decimals
While training, Travis had a daily routine. On Monday, began his day with a 3.2 mile run followed by 1.2 miles of
sprints on the track at school. On Tuesday, he began with the sprints and then ran 3.2 miles.
Except for the order, was his training any different?
Which mathematical property does this represent?
When you look at Travis’ training schedule, you will see that the components are exactly the same. He ran the same
distance on and off the track. The order of the elements did change.
This is an illustration of the Commutative Property of Addition.
Vocabulary
Decimal
a part of a whole represented by digits to the right of a decimal point.
Estimate
to find an approximate solution to a problem.
Rounding
a method of estimating where you rewrite a decimal or whole number according to the place value that it is
closest to.
Commutative Property of Addition
states that the order in which you add numbers does not impact the sum of those numbers.
Associative Property of Addition
states that the grouping of addends does not impact the sum of the addends. The groupings are completed
using parentheses.
Guided Practice
Here is one for you to try on your own.
Hassan’s recipe for potato soup calls for 14.25 kilograms of potatoes. He already has 6.12 kilograms. How many
more kilograms of potatoes does he need to buy?
Answer
In this problem, we are trying to find the number of kilograms of potatoes Hassan needs to buy.
Let’s call that value x.
We know how many kilograms he already has, and we know the total kilograms he needs, so we can write the
equation
6.12 + x = 14.25.
Now we can solve for the value of x using mental math. x needs to be a value that, when added to 6.15, equals 14.25.
Let’s begin with the whole numbers. 6 + 6 = 12; 6 + 8 = 14; the whole number portion of x is 8.
Now let’s look at the decimals. .12+? = .25. The decimal parts differ in both tenths and hundredths, but if we think
of them as whole numbers it is easier to see their relationship 12 + 13 = 25, so .12 + .13 = .25.
Combining the whole number portion of x and the decimal portion of x(8 + .13), we get 8.13.x = 8.13. Remember
to put the units of measurement in your answer!
The answer is 8.13 kilograms.
3
www.ck12.org
Video Review
MEDIA
Click image to the left or use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/54784
Khan Academy Video on Associative Law of Addition
MEDIA
Click image to the left or use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/54785
Khan Academy Video on Commutative Law of Addition
Explore More
Directions: Identify the property illustrated in each number sentence.
1. 4.5 + (x + y) + 2.6 = (4.5 + x) + y + 2.6
2. 3.2 + x + y + 5.6 = x + 3.2 + y + 5.6
3. 1.5 + (2.3 + y) + 5.6 = (1.5 + 2.3) + y + 5.6
4. 3.2 + 5.6 + 1.3 + 2.6 = 3.2 + 2.6 + 5.6 + 1.3
5. 4.5 + 15.6 = 15.6 + 4.5
6. (x + y) + 5.6 = x + (y + 5.6)
7. 17.5 + 18.9 + 2 = 2 + 17.5 + 18.9
8. (x + y) + z = x + (y + z)
Directions: Find the value of x in the following equations.
9. .8603 + .292 = x + .8603
10. (2.65 + x) + 19.35 = 22 + 2.115
11. .306 + 1.076 = (.782 + x) + .294
12. 6.174 + 76.41 = 76.41 + x
Directions: Solve each problem.
13. Lamont measures the amount of water he drinks. His results for four consecutive days were as follows: 0.6
liters, 0.72 liters, 0.84 liters, 0.96 liters. If the pattern continues, how much water will Lamont drink on the fifth day?
14. Barbara is starting a jewelry making enterprise. At the supply store, she spent $19.19 on beads and $6.81 on
wire and clasps. If she left the store with $24.50, how much did she start with?
4
www.ck12.org
Chapter 1. Commutative Property of Addition with Decimals
15. A playground has a length of 18.36 yards and a width of 12.24 yards. What is the perimeter of the playground?
(Remember: P = l + l + w + w.)
5