The Commutative Property: Everything You Need to Know

Here is Everything You Need to Know About the Commutative Property

What is the commutative property in math and what does it look like?

Welcome to this free lesson guide that accompanies this Commutative Property Explained! video lesson, where you will learn the answers to the following key questions and information:

• What is the commutative property of addition?

• What is the commutative property of multiplication?

• Commutative Property Examples

• Commutative Property Proof

This Complete Guide to the Commutative Property includes several examples, a step-by-step tutorial, an animated video mini-lesson, and a free worksheet and answer key.

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*This lesson guide accompanies our animated Commutative Property Tutorial on YouTube.

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What is the Commutative Property of Addition?

The Commutative Property of Addition states that for any real numbers a and b:

Notice that the terms are in reverse order!

If you replace the a and b terms with real numbers, like a = 2 and b =8 as shown below:

You can see from this example that, even though the terms are in reverse order on each side of the equal sign, that each side is equal to 10.

This is an example of why the commutative property holds under addition!

Does the Commutative Property work with subtraction?

Now that you understand the commutative property of addition, what about subtraction?

Is this true???

For example, if you replace the a and b terms with real numbers, like a = 2 and b =8 as shown below:

View fullsize

You can see from this example that, even though the terms are in reverse order on each side of the equal sign, that each side is NOT equal ( -6 does not equal 6)

This is an example of why the commutative property does NOT hold under addition!

Summary: Addition is commutative, but Subtraction is not!

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What is the Commutative Property of Addition?

The Commutative Property of Multiplication states that for any real numbers a and b:

Notice that the terms are in reverse order!

For example, if you replace the a and b terms with real numbers, like a = 4 and b =8 as shown below:

You can see from this example that, even though the terms are in reverse order on each side of the equal sign, that each side is equal to 32.

This is an example of why the commutative property holds under multiplication!

Does the Commutative Property work with division?

Now that you understand the commutative property of multiplication, what about division?

Is this true???

For example, if you replace the a and b terms with real numbers, like a = 4 and b =8 as shown below:

You can see from this example that, even though the terms are in reverse order on each side of the equal sign, that each side is NOT equal ( one-half does not equal 2)

This is an example of why the commutative property does NOT hold under division!

Summary: Multiplication is commutative, but Division is not!

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Final Word: Commutative Property Definitions

In summary, the commutative property only works with addition and multiplication. It does not work with subtraction and division.

For all real numbers a and b:

Commutative Property of Addition Definition: a + b = b + a

Commutative Property of Multiplication Definition: (a)(b) = (b)(a)

The terms are the same, but the order is reversed!

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Commutative Property Explained: Video Tutorial

Still confused? Check out the animated video lesson below:

Check out the video lesson below to learn more about the commutative property and to see more commutative property examples:

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Extra Practice: Free Commutative Property Worksheet

Free Worksheet!

Are you looking for some extra practice? Click the links below to download your free worksheets and answer key:

Identifying The Commutative Property:

CLICK HERE TO DOWNLOAD YOUR FREE WORKSHEET

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By Anthony Persico

Anthony is the content crafter and head educator for YouTube's MashUp Math. You can often find me happily developing animated math lessons to share on my YouTube channel . Or spending way too much time at the gym or playing on my phone.

How to Convert Improper Fractions to Mixed Numbers Explained!

Converting Improper Fractions to Mixed Numbers: Your Complete Guide

How can you convert an improper fraction to a mixed number?

Welcome to this free lesson guide that accompanies this Converting Improper Fractions to Mixed Numbers video lesson, where you will learn the answers to the following key questions:

• What is a mixed number?

• What is a proper fraction?

• What is an improper fraction?

• What is the difference between a proper fraction and an improper fraction?

• How do you convert improper fractions to mixed numbers?

This Converting Improper Fractions to Mixed Numbers: Complete Guide includes several examples, a step-by-step tutorial, an animated video mini-lesson, and a free worksheet and answer key.

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*This lesson guide accompanies our animated Converting Improper Fractions to Mixed Numbers Tutorial on YouTube.

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Before you learn to convert improper fractions to mixed numbers you need to understand some key vocabulary first.

What is a Mixed Number?

A mixed number is a number consisting of an integer, like 3, and a proper fraction, like (2/5), as seen in the example below:

Figure 1

Note that a mixed number is ALSO equal to the sum of the integer and the fraction. The numbers shown in Figure 1 and Figure 2 both represent the same mixed number. The plus sign is not usually included, however, understanding this relationship will help you later on!

Figure 2

What is a Proper Fraction and What is an Improper Fraction?

What is the difference between a proper fraction and an improper fraction?

A proper fraction is a fraction where the value of the numerator is less than the value of the denominator.

The value of a proper fraction is less than one. For example, (1/4) or one fourth is a proper fraction.

An improper fraction is a fraction where the value of the numerator is greater than the value of the denominator.

The value of a proper fraction is greater than one. For example, (5/4) or five fourths is an improper fraction.

Notice that the value of an improper fractions is greater than one!

Now you’re ready to start converting improper fractions to mixed numbers, starting with the above example of (5/4).

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How To Convert Improper Fractions to Mixed Numbers

Converting Improper Fractions to Mixed Numbers Example 1

Example 1: Convert five fourths into a mixed number

Let’s start by expressing (5/4) or five fourths as the sum of 5 one fourths as follows:

Since (1/4) is equal to one quarter, we can think about this example in terms of money and replace each (1/4) with one-quarter coins as follows:

And since four quarters equals one whole dollar…

In terms of money, five quarters equals one whole dollar and one quarter.

In terms of values, five quarters (or five fourths) equals one whole and one quarter. (or one fourth)

Answer:

(5/4) expressed as a mixed number…

You can think about this situation in a more visual way by observing Figure 3 below. You will use this kind of thinking to solve future problems involving converting improper fractions to mixed numbers!

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Converting Improper Fractions to Mixed Numbers Example 2

Example 2: Convert thirteen sixths into a mixed number.

*Start by noting that 13 divided by 6 is equal to 2 with a remainder of 1, or 2R1.

Keep this fact in mind for now, as we’ll come back to it later!

(13/6) equals two with a remainder of 1

Next, let’s visualize what (13/6) looks like using fraction charts:

By combining these values (finding the sum): 1 + 1 + (1/6), we are left with a mixed number.

Answer:

(13/6) expressed as a mixed number…

This should make sense considering that we knew that 13 divided by 6 equaled 2 with a remainder of 1.

In this case, two wholes (six sixths) with a remainder of one sixth as shown in Figure 4 below.

Figure 4

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Converting Improper Fractions to Mixed Numbers: Video Tutorial

Still confused? Check out the animated video lesson below:

Check out the video lesson below to learn more about converting improper fractions to mixed numbers and for more free practice problems:

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Extra Practice: Free Converting Improper Fractions to Mixed Numbers Worksheet

Free Worksheet!

Are you looking for some extra practice? Click the links below to download your free worksheets and answer key:

Practice Converting Improper Fractions to Mixed Numbers:

CLICK HERE TO DOWNLOAD YOUR FREE WORKSHEET

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Keep Learning with More Free Lesson Guides:

Have thoughts? Share your thoughts in the comments section below!

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By Anthony Persico

Anthony is the content crafter and head educator for YouTube's MashUp Math. You can often find me happily developing animated math lessons to share on my YouTube channel . Or spending way too much time at the gym or playing on my phone.

Absolute Value Calculator Basics: Everything You Need to Know

Absolute Value Calculator Basics: How to Graph and Solve Absolute Value Equations

How can you solve and graph absolute value equations with a calculator?

Welcome to this free lesson guide that accompanies this Absolute Value Calculator Basics YouTube video, where you will learn the following skills:

• Absolute Value Calculator Skill #1: How to use the absolute value function on your graphing calculator (TI-83, TI-84, or TI-84 plus).

• Absolute Value Calculator Skill #2: How to solve basic absolute value equations and expressions on your graphing calculator (TI-83, TI-84, or TI-84 plus).

• Absolute Value Calculator Skill #3: How to graph absolute value equations and expressions on your graphing calculator (TI-83, TI-84, or TI-84 plus).

This Absolute Value Calculator Basics: Complete Guide includes several examples, a step-by-step tutorial, an animated video mini-lesson, and a free worksheet and answer key.

Are you ready to get started!?

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*This lesson guide accompanies our animated Absolute Value Calculator Lesson on YouTube.

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Absolute Value Calculator Skill 01:

How to use the absolute value function on your graphing calculator.

Note that the graphing calculators referred to in this guide are the common TI-83, TI-84, and TI-84 plus.

To use the Absolute Value function on your calculator, use the following steps:

Press: MATH > NUM > abs{

Figure 1

Once you hit enter on abs( , as seen in figure one, you can use the absolute value function on your calculator to solve the next two examples.

Solving Absolute Value Example 01

Example 01: Use your graphing calculator to find | -8 | = ?

Answer: | -8 | = 8

Solving Absolute Value Example 02

Example 02: Use your graphing calculator to find -| 27 | = ?

*Notice that the negative sign is outside of the absolute value bars.

Answer: - | 27 | = -27

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Absolute Value Calculator Skill 02:

How to solve basic absolute value equations and absolute value expressions on your graphing calculator.

Now that you know how to use the absolute value function on your graphing calculator ( press: MATH > NUM > abs{ ), you can use it to solve basic expressions and equations like in the following examples:

Solving Absolute Value Example 01

Example 01: Use your graphing calculator to find the value of x when x = | 3 -14 + 6 | = ?

Answer: x = 5

Solving Absolute Value Example 02

Example 02: Use your graphing calculator to find the value of x when -2 | -8(11) - 3 | = x

*Make sure you enter the -2 before you input the absolute value

Answer: x = -182

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Graphing Absolute Value Equations

Absolute Value Calculator Skill 03:

How to graph absolute value equations using a graphing calculator.

Your graphing calculator is a powerful tool for graphing absolute value equations.

(Note: you can watch a video tutorial on how to solve the following examples by clicking here—skip to minute 3:08)

To graph Absolute Value Equations on your calculator, use the following steps:

Figure 2

Once you hit enter on Y=, as seen in figure 2, you will be able to enter your absolute value equation or absolute value function. Then you just have to press graph to see the graph of your equation on the screen as in the following examples:

Graphing Absolute Value Example 01

Example 01: Use your calculator to find the graph of y = | 2x + 3 |

After you input the function into Y=, press the graph button:

The graph of y = | 2x + 3 |

Press: 2nd > GRAPH to access the function table

Graphing Absolute Value Example 02

Example 02: Use your calculator to find the graph of y = -3| x - 1 |

Remember to input the -3 before you input the absolute value portion.

The graph of y = -3 | x - 1 |

Press: 2nd > GRAPH to access the function table

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Free Absolute Value Calculator (Online Version)

If you don’t have a graphing calculator, this free online absolute value calculators are great options:

The website www.mathpapa.com offers a very useful (and free) absolute value calculator that you can use for both graphing absolute value equations AND solving absolute value equations.

This free absolute value calculator is a great tool for learning more about graphing and solving absolute value functions and checking your answers.

Observe figure 3 (solving absolute value equations) and figure 4 (graphing absolute value equations) to see examples of how the free online absolute value calculator works.

Figure 3

Figure 4

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Absolute Value Calculator Basics: Video Tutorial

Still confused? Check out the animated video lesson below:

Check out the video lesson below to learn more about multiplying fractions and for more free practice problems:

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Extra Practice: Free Absolute Value Worksheets

Free Worksheet!

Are you looking for some extra practice? Click the links below to download your free worksheets and answer key:

Solving Absolute Value Equations:

CLICK HERE TO DOWNLOAD YOUR FREE WORKSHEET

Graphing Absolute Value Equations:

CLICK HERE TO DOWNLOAD YOUR FREE WORKSHEET

Tags:  Multiplying Fractions by Whole Numbers, Multiplying Fractions and Whole Numbers, Multiplying Fractions by Whole Numbers Practice, Multiplying Fractions by Whole Numbers Examples, Simplifying Fractions

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Have thoughts? Share your thoughts in the comments section below!

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By Anthony Persico

Anthony is the content crafter and head educator for YouTube's MashUp Math. You can often find me happily developing animated math lessons to share on my YouTube channel . Or spending way too much time at the gym or playing on my phone.

April Fools' Day Math Puzzle for Grades 1-6

They’re Here! Are Your Kids Ready for These April Fools’ Day Math Challenges?

April Fools' Day is a perfect time to add a little silliness into your lesson plans.

Whether you’re teaching your kids at home, remotely, or in the classroom, you don’t want to miss this opportunity to make your April 1st math lesson extra special and memorable.

You can start by sharing a few of these 11 Super Cute and Funny Math Jokes and Puns for Kids.

And you can keep the fun going with 3 April Fools’ Day Math Puzzles!

These kinds of activities are best used for warm-ups (anticipatory sets), cooldowns and exit tickets, and transitions. They are especially good for helping your students develop problem-solving skills, mathematical reasoning, and applying the order of operations.

I like to share fun math puzzles like today’s at least once per week to keep my lessons fresh and exciting, and my students love them!

Today’s April Fools’ Day Math Challenges for grades 1-6 are sample puzzles from the best-selling workbook: The Big Book of Super Fun Math Puzzles for Grades 1-6.

The activities are tiered by difficulty using the following system:

Pink Level (Grades 1-2) | Blue Level (Grades 3-4) | Green Level (Grades 5-6+)

April Fools’ Day Math Puzzles for Grade 1 through 6+

Note that:

(answer keys to follow)

Pink Level Puzzle (for Grades 1-2)

Extra Resources: Which One Doesn't Belong? Free Math Activities for Grades 1-8

Blue Level Puzzle (for Grades 3-4)

Free Math Resources: Free Math Coloring Worksheets for 3rd and 4th Grade

Green Level Puzzle (for Grades 5-6+)

Bonus! Would you like over 300 more math puzzles like todays?

Click here to take a closer look at our best-selling The Big Book of Math Puzzles for Students in Grades 1-6 and access more free sample puzzles with answer keys!

And click here to get your copy of The Big Book of Super Fun Math Puzzles for Grades 1-6 as a PDF download.

And here are the answers to the April Fools’ Day Math Puzzles:

Pink Level (Grades 1-2): Jester=5, Disguise=2, Monkey=8, ?=13

Blue Level (Grades 3-4): Jester=10, T-Rex=7, Monkey=5, Disguise=6, ?=84

Green Level (Grades 5-6+): T-Rex=53, Monkey=53, Jester=11, Disguise=66, ?=51

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Sharing fun math puzzles with your students is just one effective strategy for improving student engagement. Subscribe to our mailing list here to get more free daily resources, lesson plans, ideas, and insights for K-12 math teachers in your inbox every week.

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By Anthony Persico

Anthony is the lead educator and founder of Mashup Math. He lives in Denver, Colorado and is also a YouTube for Education partner. Follow him on Twitter at @mashupmath.

Multiplying Fractions by Whole Numbers: Your Complete Guide

Complete Guide: Multiplying Fractions by Whole Numbers

Key Question: How do you multiply fractions and whole numbers?

Learn how to solve these kinds of problems.

Welcome to this free lesson guide where you will learn and easy two-step process for multiplying fractions by whole numbers AND multiplying whole numbers by fractions.

This complete guide to multiplying fractions by whole numbers includes several examples, an animated video mini-lesson, and a free worksheet and answer key.

Let’s get started!

Multiplying Fractions by Whole Numbers: Multiplication Review

Before we explore how to multiply fractions, let’s do a super quick review of how to multiply a fraction by a fraction (understanding how to apply the rule below will make multiplying fractions and whole numbers much easier for you!)

Multiplying Fractions Rule: Whenever multiplying fractions together, multiply the numerators together, then multiply the denominators together as follows…

Example of the Rule:

What is (3/4) x (1/2) ?

Notice that the fraction (3/8) can not be simplified (since 8 and 3 do not have a common divisor)

Answer: (3/4) x (1/2) = 1/8

Looking for More Help With Multiplying a Fraction by a Fraction? Check out this free guide

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How to Multiply a Fraction by a Whole Number (and Vice Versa)

Now that you are familiar with the rule for multiplying a fraction by a fraction, you can use it to help you easily multiply a fraction by a whole number.

Let’s start with an example:

Multiplying Fractions by Whole Numbers: Example 1

Example 1: What is (2/7) x 3 ?

Start by rewriting the whole number (3 in this example) as a fraction, (3/1) as follows…

(You can do this because any number divided by one is always equal to itself)

Now, because you are multiplying a fraction by a fraction, you can apply the rule and solve as follows…

And since (6/7) can not be simplified, you can conclude that:

Answer: (2/7) x 3 = (6/7)

Wait! What would happen if the answer could be simplified? Let’s address the situation in the next example…

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Multiplying Fractions by Whole Numbers: Example 2

Example 1: What is 5 x (9/10) ?

Start by rewriting the whole number (5 in this example) as the fraction (5/1)…

Then apply the rule as follows…

In this example, (45/10) is not the final answer because it can be simplified.

Since the greatest common factor (GCF) of 45 and 10 is 5, you can simplify by dividing both the numerator and the denominator by 5 as follows…

And since (9/2) can not be simplified any further, you can conclude that:

Answer: 5 x (9/10) = (9/2)

Still confused? Check out the animated video lesson below:

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Video: Multiplying Fractions by Whole Numbers Explained!

Check out the video lesson below to learn more about multiplying fractions and for more free practice problems:

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Multiplying Fractions by Whole Numbers Worksheets

Free Worksheet!

Are you looking for some extra practice multiplying fractions by whole numbers? Click the links below to download your free worksheets and answer key:

CLICK HERE TO DOWNLOAD YOUR FREE WORKSHEET

Tags:  Multiplying Fractions by Whole Numbers, Multiplying Fractions and Whole Numbers, Multiplying Fractions by Whole Numbers Practice, Multiplying Fractions by Whole Numbers Examples, Simplifying Fractions

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Keep Learning:

Have thoughts? Share your thoughts in the comments section below!

(Never miss a Mashup Math blog--click here to get our weekly newsletter!)

By Anthony Persico

Anthony is the content crafter and head educator for YouTube's MashUp Math. You can often find me happily developing animated math lessons to share on my YouTube channel . Or spending way too much time at the gym or playing on my phone.