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Foundation of Rational Numbers

Learning Module

Abstract

This learning module provides 7th grade students with the knowledge and skills needed to add, subtract, multiply, and divide rational numbers. Each update provides students with online resources to learn and apply what they have learned. Students will also have the opportunity to share their understanding with their classmates. At the end of the learning module, a peer reviewed project will require students to take what they have learned about rational numbers and create their own real-world word problem.

Keywords

Rational number, Integer, Additive inverse, Opposites, Arrow diagram, Counting on, Multiply, Divide, Add, Subtract

Introduction

This learning module will teach students about rational numbers by engaging students in math through the use of interactive activities, videos, and diagrams. In each update, students will participate in some technology based activities that will teach students how to add, subtract, multiply, and divide rational numbers. The goal of this module is for students to be able to solve rational number problems and how rational numbers are used in their everyday lives. At the end of the learning module, students will be able to use their understanding of rational numbers to create their own real-life situation word problem that relates to their interest. Students will provide their peers with feedback on how to improve their work and students will record or present a live teaching to their classmates of how to solve their word problem.

As a 7th grade math teacher, I have taught the operations of rational numbers for four years and have noticed that it is a very short unit yet, it is a very important skill students use for the rest of their educational and personal life. Students begin learning about number sense from an early age and develop their skills of this concept throughout their educational careers. It is important to begin the learning of rational numbers with pictures and diagrams to be able to fully grasp the concept and understanding of how rational numbers work (Viseu, Pires, Menezes, and Costa 2021). My school uses the Eureka Math curriculum which has great techniques to help students learn about rational numbers but it is a very bland curriculum that doesn’t offer much engagement or visuals for the student other than solving problems in a workbook. This learning module is teaching students about rational numbers using the same standards, techniques, and skills but through an interactive approach that involves both collaboration and independent application.

The assessment that will be used in this learning module will be in the creation of a final product that will require both student and teacher feedback. Students will use the skills and concepts they have learned about rational numbers in this learning module to create their final product that shows their understanding and application to real life experiences (Cope and Kalantzis, 2010).

 

Objectives and Standards

For the Students

At the end of this learning module students will be able to:

  • describe the absolute value of an integer
  • understand additive inverse
  • add integers
  • add rational numbers
  • subtract integers
  • subtract rational numbers
  • multiply and divide integers
  • multiply and divide rational numbers
  • apply operations of rational numbers to real world situations

For the Instructor

At the end of this learning module students will be able to:

  • describe the absolute value of an integer
  • understand additive inverse
  • add integers
  • add rational numbers
  • subtract integers
  • subtract rational numbers
  • multiply and divide integers
  • multiply and divide rational numbers
  • apply operations of rational numbers to real world situations

 

Standards:

CCSS.MATH.CONTENT.7.NS.A.1

CCSS.MATH.CONTENT.7.NS.A.2

CCSS.MATH.CONTENT.7.NS.A.3

 

Supplies Needed:

  • Kami
  • Screencastify or seesaw 
  • Integer Playing Cards
  • Anchor Charts 
  • Visuals 
  • Video Resources
  • Paper Handouts (optional)

 

Update 1: Real Number System

For the Students

Learning Objectives:

-I can describe the absolute value of an integer.
-I can understand additive inverse.

 

Introduction to Integers Video:

Media embedded October 10, 2021

PBS Math Club. (2013, October 7). What is an Integer? | PBSMathClub. YouTube. https://www.youtube.com/watch?v=x0E4vxLydNY

 

What are real numbers?

  • Natural numbers are positive whole numbers.
  • Whole numbers are numbers without a fractional part.
  • Integers are whole numbers that can’t be written with a fractional part and can be positive or negative.
  • Rational numbers are numbers that can be written with a fractional part or repeating decimal.
  • Irrational numbers are numbers that can not be written as a fraction.

 

Refer to the diagram below for examples of real numbers:

Shields, J. (2021, September 9). What Are Real Numbers? HowStuffWorks. https://science.howstuffworks.com/math-concepts/real-numbers.htm

 

 What is absolute value?

Absolute value describes the distance from zero that a number is on the number line without considering direction. The absolute value of a number is never negative because distance can’t be negative.

Watch the video below to learn how absolute value on a number line:

Media embedded September 25, 2021

Math with Mr. J. (2020, January 12). What is Absolute Value? | Absolute Value Examples | Math with Mr. J. YouTube. https://www.youtube.com/watch?v=LnfhdtjcpVY

 

What is the additive inverse?

Watch the video below to learn what the additive inverse is:

Media embedded September 25, 2021


Nutshell, P. (2013, November 5). Math Shorts Episode 1 - Additive Inverse. YouTube. https://www.youtube.com/watch?v=p8OsA062OPY&feature=youtu.be

 

The additive inverse is also known as opposites.

 

 

Your work:

Now that you have learned about real numbers, you will use what you have learned about absolute value and the additive inverse to play the integer game below.

  1. Find a group of three peers to play with
  2. Grab a deck of integers cards
  3. Open the Integer Game Directions
  4. Have fun!

 

On your own:

Comment: After you are done playing the integer game, leave a comment explaining how you used the concepts of absolute value and the additive inverse during the game. Were there any tricks or strategies you used or discovered while playing the integer game?

Update: Create an update explaining one way you have used integers in your life.

Comment on two peers updates: Leave a positive comment on your peers' update about their real life experience with using integers.

 

 

For the Instructor

Purpose: The purpose of this first update is provide students with the foundation of understanding different types of real numbers and how a number line works. In this update students will watch an introduction to integers that will help them create a connection to integers and real life situations that they experience everyday. The pre-survey will help get a better understanding of students' prior knowledge of integers and what they already know before beginning the learning module. Students will play the integer card game which will introduce them to adding and subtracting integers using the additive inverse. 

 

Learning Objectives:

-Students will be able to-

  • describe the absolute value of an integer
  • Understand and apply the additive inverse.

 

Duration of update: 60 minutes

 

Resources:

Integer Pre-Check Survey 

Real Numbers Graphic Organizer 

Integer Playing Cards

 

Extra Practice or extension:

Have students who need extra practice play a blooket practicing absolute value and additive inverse. Students who need an extension can work in khan academy to practice adding and subtracting absolute values. 

 

Standards:

​CCSS.MATH.CONTENT.7.NS.A.1.A

CCSS.MATH.CONTENT.7.NS.A.1.B

CCSS.MATH.CONTENT.7.NS.A.1.C

 

 

 

Update 2: Adding Integers on a Number Line

For the Students

Learning Objectives:

-I can add integers on a number line.

 

Before you get started:

-Complete the integers pre-check survery to show what you already know about integers. 
 

How do I use a number line to add integers?

When adding integers on a number line we will use the arrow diagram. On the number line start at zero. If the integer is positive, move your arrow to the right and if the integer is negative, move the arrow to the left. Move the arrow in the direction of the first integer in the problem and stop. Then use the second integer in the problem and move the arrow in that direction. The number you land on will be your answer.

 

The diagram below is an example for reference:

Online Math Help and Learning Resources. (2021). Adding Integers using the Number Line. Onlinemathlearning.com. https://www.onlinemathlearning.com/add-integers.html

 

Your work:

Watch the video below:

Media embedded September 26, 2021

Math with Mr. J. (2021, January 4). Adding Integers Using a Number Line | Math with Mr. J. YouTube. https://www.youtube.com/watch?v=3LUTYhmltQY&feature=youtu.be

  1. Complete: adding integers on a number line practice
  2. Play: line jumper

 

On your own:

Comment: Leave a comment explaining what patterns you noticed when adding integers on a number line using the arrow diagram.

Update: Create an update by creating your own example of adding integers. 

Comment on two peers updates: Leave a comment answering your peers problem and use a number line and the arrow diagram to support your answer. 

 

For the Instructor

Purpose: The purpose of this update is provide students with the skill of using an arrow diagram to add integers using a number. Since students do not know the rules of adding integers yet, using a number will help with understanding why the interger opeations rules work in tomorrow's update. Students will use the videos and diagrams to help them complete the learning tasks and answer the question in the update. 

 

Learning Objectives:

-Students will be able to add integers using a number line. 
 

Duration of update: 60 minutes

 

Resources:

Pre-Check Survery

Number line

Whiteboard

Kami

Video 

 

Extra Practice or extension:

Students who need extra practice will grab a number line and whiteboard and practice more problems using the arrow diagram with the teacher. Students who do not need the number line can complete more problems and try to figure out the rule of adding integers. 

 

Standards:

CCSS.MATH.CONTENT.7.NS.A.1

 

Update 3: Adding Rational Numbers

For the Students

Learning Objectives:

-I can use the rules of integer operations in addition to adding rational numbers.


 

Integer Operation Rules of Addition:

-Adding numbers with the same sign:

  • Add the number together and keep the sign the same.

-Adding numbers with different signs:

  • Subtract the numbers and take the sign of the larger number.

 

Use the diagram below for reference:

Carter, H. (2019). Integer Rules Positives & negatives. Slide Player. https://slideplayer.com/slide/14849161/


 

How do I add rational numbers?

  1. If you are adding fractions, change to improper fractions and create common denominators.
  2. If you are adding decimals, remember to line up your decimals by place value.
  3. Use the rules discussed above to add rational numbers.
  4. Remember to simplify your answer.

 

Watch the video below to learn how to add rational numbers:

Media embedded September 25, 2021

Anywhere Math. (2014, September 19). Adding Rational Numbers. YouTube. https://www.youtube.com/watch?v=yZYu0debFtE&feature=youtu.be

 

Your work:

  1. Work with your partner to complete: Adding Rational Numbers Matching Activity
  2. Practice: Adding rational numbers dreambox game

 

On your own:

Comment: Leave a comment discussing the differences and similarities between adding rational numbers and adding integers. Use numbers, words, and pictures to explain your thinking.

Update: Create an update discussing any patterns or tricks you discovered when adding rational numbers.

Comment on two peers updates: Leave a comment by agreeing or disagreeing to the patterns and tricks your peers discovered. 

 

For the Instructor

Purpose: The purpose of this update is to teach students the rules of integer operations for adding. Students will use the rules of adding integers and their prior knowledge or adding decimals and fractions to add rational numbers. 

 

Learning Objectives:

-Students will be able to add rational numbers using the rules of addition of integer operations.
 

Duration of update: 60 minutes

 

Resources:

Integer Addition Rules Anchor Chart

Whiteboard

Kami

Video

Coloring Supplies (optional)

 

Intervention:

Students who need practice with adding fractions and decimals will receive a reteaching from the teacher to build the foundation to be able to add rational numbers.

 

Standards:

CCSS.MATH.CONTENT.7.NS.A.1.D

 

Update 4: Subtracting Integers on a Number Line

For the Students

Learning Objectives:

-I can subtract integers using a number line.

 

How do I use a number line to subtract integers?

Start at zero on the number line and draw an arrow moving in the direction of the first number in the problem. Since you are subtracting, move the arrow to the left the number of units of the second integer in the problem. The number you land on will be your answer.

 

Use the diagram below for reference:

Subtracting Integers. (2020). Softschools.Com. https://www.softschools.com/math/topics/subtracting_integers/

 

Your work:

Watch the video below:

Media embedded September 25, 2021

Math with Mr. J. (2021b, January 5). Subtracting Integers Using a Number Line | Math with Mr. J. YouTube. https://www.youtube.com/watch?v=Dfytkh_lYME&feature=youtu.be

 

  1. Play: Subtracting Intgers Dreambox Game
  2. With your partner complete: Subtracting Integers Practice

 

On your own:

Comment: Leave a comment on what you have learned about subtracting integers on a number line and how they compare to adding integers on a number line.

Update: Create an update explaining what you noticed when you subtract a negative integer on a number line. How did you figure this out? Use numbers, words, and pictures to explain your thinking.

Comment on two peers updates: Leave a positive comment sharing your thoughts about your peers update. 

For the Instructor

Purpose: The purpose of this update is for students to investigate how to subtract integers on a number line and that the direction of the arrow diagram is connected to the integer itself. This update will scaffold students to learn what happens when subtracting a negative. Students do not know the rules of subtracting integers yet but, through this update they will be able to make a connection between subtracting and adding integers. 

 

Learning Objectives:

-Students will be able to use a number line to subtract integers.

 

Duration of update: 60 minutes

 

Resources:

Number line

Whiteboard

Kami

Video

 

Intervention:

Students who need practice will work with the teacher to review adding integers on a number line and discuss how it compares and differs from subtracting integers on a number line.

 

Standards:

CCSS.MATH.CONTENT.7.NS.A.1

 

Update 5: Subtracting Rational Numbers

For the Students

Learning Objective:

-I can use the rules of integer operations in subtraction to subtracting rational numbers.

 

Integer Operation Rules of Subtraction:

1) Golden Rule- Use KFC:

  • K: Keep the first number the same
  • F: Flip addition to subtraction
  • C: Change to opposite sign

2) Once you have changed to addition, use the rules of adding integers:

-Adding numbers with the same sign:

  • Add the number together and keep the sign the same.

-Adding numbers with different signs:

  • Subtract the numbers and take the sign of the larger number.

 

Use the diagram below for reference:

Maisonet. (2021). Subtracting Integers. Maisonet Math. https://maisonetmath.com/integers/quizzes/352-subtracting-integers


 

How do I subtract rational numbers?

  1. If you are subtracting fractions, change to improper fractions and create common denominators.
  2. If you are subtracting decimals, remember to line up your decimals by place value.
  3. Use KFC to change the subtraction problem to an addition problem.
  4. Use the rules discussed above to add rational numbers.
  5. Remember to simplify your answer.

 

Your work:

Watch the video below:

Media embedded September 25, 2021

Anywhere Math. (2014b, September 22). Subtracting Rational Numbers. YouTube. https://www.youtube.com/watch?v=VDmSK1W5BM8&feature=youtu.be

 

  1. Play Rational Numbers Card Play with your partner. Card a deck of cards to start the game.
  2. Play: Add/Subtract Rational Numbers Game
  3. Extra practice: Play Rational Numbers Jeopardy with a partner.

 

On your own:

Comment: Leave a comment explaining what you have learned about subtracting rational numbers. Why is using KFC helpful when subtracting integers and rational numbers? Use numbers, words, and pictures to explain your thinking.

Update: Create an update by creating one addition and one subtraction rational numbers problem. 

Comment on two peers updates: Leave a comment by answering your peers math problem in their update. 

 

For the Instructor

Purpose: The purpose of this update is for students to learn the rules of subtracting integers. Students will use their prior knowledge of subtracting fractions and decimals and what they have learned in this update to subtract rational numbers. Students will be able to use today's update and connect it back to yesterday's update of subtracting positive and negative numbers and why the rules make sense in relevance to a number line.

 

Learning Objectives:

-Students will be able to use the rules of integer operations in subtraction to subtract rational numbers.

 

Duration of update: 60 minutes

 

Resources:

Subtracting Integers Anchor Chart

Whiteboard

Kami

Video

 

Intervention:

Students who need extra help with subtracting fractions and decimals will work with the teacher on practicing the concept of KFC and building the foundation needed for this update.

 

Standards:

CCSS.MATH.CONTENT.7.NS.A.1.D

 

 

 

Update 6: Multiplying and Dividing Integers

For the Students

Learning Objective:

-I can multiply and divide integers using the rules of operations.

 

How do you multiply and divide integers?

-Multiply and divide integers with like signs:

  • Multiply and divide the integers and keep the sign the same.

-Multiply and divide numbers with unlike signs:

  • Multiply and divide the integers and make the answer negative.

 

Use the diagram below for reference:

Multiplying & Dividing Integers. (2021). Onlinemathlearning.Com. https://www.onlinemathlearning.com/multiplying-integers.html

 

Your work:

Watch the video below:

Media embedded September 25, 2021


Simplify The Middle. (2020, August 30). Multiply & Divide Integers. YouTube. https://www.youtube.com/watch?v=b5rK3XzHhTM&feature=youtu.be

 

  1. Complete: Integers Operations Practice
  2. Play Multiply and Divide Integers Dreambox Game

 

On your own:

Comment: Leave a comment explaining what you have learned about multiplying and dividing integers. How would the signs of the integers affect the answer when multiplying more than two numbers? Use pictures, words, and numbers to explain your thinking.

Update: Create an update by creating either a multiplication or division problem containing at least three or more integers. 

Comment on two peers updates: Leave a comment on your peers update by finding the solution to their math problem. *Pay attention to the signs of the integers*

 

For the Instructor

Purpose: The purpose of this update is for students to learn the rules of multiplying and dividing integers. Students will use what they learn in this update to investigate how the signs would change when multiplying and dividing more than two numbers. In this update, it is important for students to be able to understand how and why the rules of multiplication and division work in order to be able to distinguish between the rules of addition/subtraction and multiplication/division. 

 

Learning Objectives:

-Students will be able to multiply and divide integers using the rules of integer operations.

 

Duration of update: 60 minutes

 

Resources:

Multiply and Divide Integers Anchor Chart

Whiteboard

Kami

Video

 

Extension:

Students who are ready to move on can apply what they have learned in this update and apply it to solving multi step problems in khan academy.

 

Standards:

CCSS.MATH.CONTENT.7.NS.A.2

 

Update 7: Multiplying and Dividing Rational Numbers

For the Students

Learning Objectives:

-I can multiply and divide rational numbers using the rules of integer operations.

 

How do I multiply and divide rational numbers?

Multiplying Rational Numbers:

  • Multiplying rational numbers do not require common denominators. Multiply the numerators together and then multiply the denominators together.
  • Follow the rules of multiplying integers to find the sign of your answer.

 

Divide Rational Numbers:

-Use KFC:

  • K: Keep the first fraction the same
  • F: Flip division to multiplication
  • C: Change the second fraction to the reciprocal (flip the fractions numerator and denominator)

-Use the rules of multiplying rational numbers to solve

 

Watch the video below to learn about multiplying and dividing rational numbers:

Media embedded September 25, 2021

Anywhere Math. (2014c, September 24). Multiplying & Dividing Rational Numbers. YouTube. https://www.youtube.com/watch?v=sy0YSJHIvtk&feature=youtu.be

 

Your work:

Choose from the games linked below. You should try at least two games.

Game 1: Fraction Contraption

Game 2: Hugo and the Balloon Quest

Game 3: Area Model Multiply and Divide Decimals

Game 4: Multiply and Divide Rational Numbers

 

Own your own:

Comment: Leave a comment explaining what you have learned about multiplying and dividing rational numbers. What are some shortcuts you can take when multiplying and dividing rational numbers? Use numbers, words, and pictures to explain your thinking.

Update: Create an update by creating either a multiplication or division rational number problem. 

Comment on two peers updates: Leave a comment on your peers update by finding the solution to their math problem. 

 

For the Instructor

Purpose: The purpose of this update is for students to apply the rules of multiplying and dividing integers to solve problems with rational numbers. Students will use their prior knowledge of multiplying and dividing fractions and decimals as the foundation of this update. 

 

Learning Objectives:

-Students will be able to multiply and divide rational numbers using the rules of integer operations.

 

Duration of update: 60 minutes

 

Resources:

Multiply and Divide Integers Anchor Chart

Multiplying and Dividing Fractions Anchor Chart

Whiteboard

Kami

Video

 

Intervention:

Students who need extra help will work with the teacher in a small group to practice multiplying and dividing fractions and decimals and apply this to rational numbers.

 

Standards:

CCSS.MATH.CONTENT.7.NS.A.2.A

CCSS.MATH.CONTENT.7.NS.A.2.B

CCSS.MATH.CONTENT.7.NS.A.2.C

 

 

Update 8: Word Problems

For the Students

Learning Objectives:

-I can use integer operations rules to solve real life word problems with rational numbers.

 

How do I solve word problems with rational numbers?

  • Highlight or underline important information.
  • Use keywords to decide which type of operation will be used.
  • Use the information in the problem to decide if the rational numbers will be positive or negative.
  • Solve the problem using the rules of integer operations.
  • Write your answer in a complete sentence with appropriate labels.

 

Use the diagram below for reference when solving word problems:

Steps for Solving Word Problems Chart. (2021). Pinterest. https://www.pinterest.com/pin/465700417691392418/


 

Watch the video below to learn about solving word problems with rational numbers:

Media embedded September 25, 2021

Khan Academy. (2011, November 21). Rational number word problem with decimals | Pre-Algebra | Khan Academy. YouTube. https://www.youtube.com/watch?v=fFdOr8U4mnI&feature=youtu.be

 

Your work:

  1. With a partner complete Rational Numbers Word Problems Practice. Use a QR reader to check your answers when you are done.
  2. Complete Rational number word problems on Khan Academy.


 

On your own:

Comment: Leave a comment explaining how you use rational numbers in your everyday life. You can describe a specific situation or you can describe ways you have used rational numbers in your life.

Update: Create an update explaining why it is important to know how to add, subtract, multiply, and divide rational numbers without a calculator. Use pictures, words, and numbers to explain your thinking.

Comment on two peers updates: Leave a positive comment sharing your thoughts about your peers update. 

 

For the Instructor

Purpose: The purpose of this update is for students to use everything they have learned about adding, subtracting, multiplying, and dividing rational numbers to solve word problems or real life situations. This update will help prepare students for their peer reviewed project at the end of the learning module.

 

Learning Objectives:

-Students will be able to use the integer operations rules to solve real life word problems with rational numbers.

 

Duration of update: 60 minutes

 

Resources:

Solving Word Problems Anchor Chart

Whiteboard

Kami

Video

Highlighters

 

Intervention:

Students who need extra practice with solving word problems will work with the teacher to practice the skills of highlighting, underlining, and defining parts or word problems. This will help students who are struggling grasp the tools needed to help solve problems.

 

Standards:

CCSS.MATH.CONTENT.7.NS.A.3

 

 

Peer Reviewed Project

For the Students

Your project for this learning module is to use what you have learned about rational numbers to create your own real-life word problem and record yourself teaching how to solve the word problem.

 

Below you find the steps to complete your project:

1) Create your word problem. Your word problem must include:

  • A real life situation
  • At least three numbers are used with two of them being rational numbers.
  • Multi-step problem
  • Must include at least two different operations of your choice

2) Recording or Live Presentation:

  • You will have the option to record yourself teaching others how to solve the word problem you created or teach how to solve your word problem live in front of the class. Your recording or live presentation should be clear in explaining step by step how to solve the world problem and explaining the final answer in a complete sentence. You can use drawings, sketches, and visuals to help support your explanation if needed.

3) Peer Review:

  • You will review one peer’s word problem and provide feedback for improvements and analyze the method used to find the solution.
  • Use the Peer Rubric to review your peer’s work.
  • Your work will be reviewed with feedback and given back to you by one peer and your teacher. Make any changes needed to your word problem and recording before you turn in your final product. Your goal is to be in the green area of the rubric.

 

Project Options:

  • You can create your word problem on kami
  • You can create your word problem on a poster board
  • Use coloring supplies to catch the attention of the audience

 

Peer Review Rubric:

Peer Review Feedback

Grading:

  • After completing the peer review and making revisions on your work, your teacher will use the rubric below to grade your project. This project is worth 40 points. 
Grading Rubric

 

For the Instructor

Purpose: The purpose of this peer reviewed project is for students to use what they have learned about solving problems with rational numbers to create their own real-life situation word problem and teach how to solve the word problem to others. This gives students the chance to create a word problem using operations of their choice and about a topic that is of interest to them. Students will peer review one classmates’ work to provide feedback for improvement and analyze the method of solution used to solve the problem. Students will be able to use the feedback to make any updates before turning in their final product for a grade. 

 

Learning Objectives:

-Students will be able to use the rules of integer operation to create a multi-step real life situation word problem.

 

Duration of update: 60 minutes

 

Resources:

Anchor Charts

Kami

Videos

Coloring Supplies

Posters

Screencastify or Seesaw

Peer Rubric

Grading Rubric

 

Extra Support:

Students who have been struggling in this learning module will create a one step word problem using the same criteria in the rubric. This will help support and encourage students to use what they have learned but at a level that they are at.

 

Standards:

CCSS.MATH.CONTENT.7.NS.A.3


 

References

Cope, B. & Kalantzis, M. (2010). The Teacher As Designer: Pedagogy In The New Media Age. E-Learning and Digital Media.

Mathematics Standards Common Core State Standards Initiative. (2021). Common Core State Standards Initiative. http://www.corestandards.org/Math/

Shields, J. (2021, September 9). What Are Real Numbers? HowStuffWorks. https://science.howstuffworks.com/math-concepts/real-numbers.htm

Online Math Help and Learning Resources. (2021). Adding Integers Using The Number Line. Onlinemathlearning.com. https://www.onlinemathlearning.com/add-integers.html

Carter, H. (2019). Integer Rules Positives & Negatives. Slide Player. https://slideplayer.com/slide/14849161/

Subtracting Integers. (2020). Softschools.Com.

https://www.softschools.com/math/topics/subtracting_integers/

Maisonet. (2021). Subtracting Integers. Maisonet Math. https://maisonetmath.com/integers/quizzes/352-subtracting-integers

Multiplying & Dividing Integers. (2021). Onlinemathlearning.Com. https://www.onlinemathlearning.com/multiplying-integers.html

Steps For Solving Word Problems Chart. (2021). Pinterest. https://www.pinterest.com/pin/465700417691392418/

Math with Mr. J. (2020, January 12). What Is Absolute Value? | Absolute Value Examples | Math with Mr. J. YouTube. https://www.youtube.com/watch?v=LnfhdtjcpVY

Nutshell, P. (2013, November 5). Math Shorts Episode 1 - Additive Inverse. YouTube. https://www.youtube.com/watch?v=p8OsA062OPY&feature=youtu.be

Math with Mr. J. (2021, January 4). Adding Integers Using a Number Line | Math with Mr. J. YouTube. https://www.youtube.com/watch?v=3LUTYhmltQY&feature=youtu.be

Line Jumper. (2021). Funbrain. https://www.funbrain.com/games/line-jumper
Anywhere Math. (2014, September 19). Adding Rational Numbers. YouTube. https://www.youtube.com/watch?v=yZYu0debFtE&feature=youtu.be

Dreambox. (2021). Interactive Fraction And Decimal Sums. Dreambox Learning. https://play.dreambox.com/student/dbl/TeacherTool_MiniGolf?atype=2&back=http%3A%2F%2Fwww.dreambox.com%2Fteachertools&eng=Intermediate&ie_skin=paperfrenzy

Math with Mr. J. (2021b, January 5). Subtracting Integers Using A Number Line | Math with Mr. J. YouTube. https://www.youtube.com/watch?v=Dfytkh_lYME&feature=youtu.be

Dreambox. (2021). Interactive Subtracting Integers. Dreambox Learning. https://play.dreambox.com/student/dbl/Decimals_AddSubtract_PirateShip_LevelForty?atype=1&back=http%3A%2F%2Fwww.dreambox.com%2Fteachertools&eng=Intermediate&ie_skin=paperfrenzy

Anywhere Math. (2014b, September 22). Subtracting Rational Numbers. YouTube. https://www.youtube.com/watch?v=VDmSK1W5BM8&feature=youtu.be

Math Games. (2021). Add And Subtract Rational Numbers. Math Games. https://www.mathgames.com/skill/7.86-add-and-subtract-rational-numbers

Math Playground. (2021). 7th Grade Numbers And Operators Jeopardy Game. Math Playground. https://www.math-play.com/7th-Grade-Numbers-and-Operations-Jeopardy/7th-grade-numbers-and-operations-jeopardy-game_html5.html

Simplify The Middle. (2020, August 30). Multiply & Divide Integers. YouTube. https://www.youtube.com/watch?v=b5rK3XzHhTM&feature=youtu.be

Cognitive Cardio Math. (2014). Integers Operations. Teachers Pay Teachers. https://www.teacherspayteachers.com/Store/Cognitive-Cardio-Math#

Anywhere Math. (2014c, September 24). Multiplying & Dividing Rational Numbers. YouTube. https://www.youtube.com/watch?v=sy0YSJHIvtk&feature=youtu.be

Legends of Learning. (2021). Fraction Contraption. Legends of Learning. https://app.legendsoflearning.com/join/YXNzaWdubWVudC03NzE1MjI=

Legends of Learning. (2021b). Hugo And The Balloon Quest. Legends of Learning. https://app.legendsoflearning.com/join/YXNzaWdubWVudC03NzE1MjA=

Legends of Learning. (2021a). Area Model. Legends of Learning. https://app.legendsoflearning.com/join/YXNzaWdubWVudC03NzE1MjY=


Math Games. (2021). Multiply and Divide Rational Numbers. Math Games

https://www.mathgames.com/skill/7.88-multiply-and-divide-rational-numbers

Khan Academy. (2011, November 21). Rational Number Word Problem With Decimals | Pre-Algebra | Khan Academy. YouTube. https://www.youtube.com/watch?v=fFdOr8U4mnI&feature=youtu.be

Virginia Department of Education. (2018). Solving Problems Involving Operations With Rational Numbers. https://doe.virginia.gov/testing/sol/standards_docs/mathematics/2016/mip/gr7/mip-7-2-prob-ops-rat-num.pdf

Khan Academy. (2021). Rational Number Word Problems (Practice). https://www.khanacademy.org/math/cc-seventh-grade-math/cc-7th-fractions-decimals/cc-7th-multistep-word-probs/e/multistep-equations-without-variables

PBS Math Club. (2013, October 7). What Is An Integer? | PBSMathClub. YouTube. https://www.youtube.com/watch?v=x0E4vxLydNY

Viseu, F., Pires, A. L., Menezes, L., & Costa, A. M. (2021). Semiotic Representations In The Learning Of Rational Numbers By 2nd Grade Portuguese Students. International Electronic Journal of Elementary Education, 13(5), 611–624. Retrieved from https://www.iejee.com/index.php/IEJEE/article/view/1453