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Standard 6.EE addresses sixth graders ability to make sense of and solve equations and inequalities with one variable. Grasping the concept of variables and mastering the techniques to solve for them forms the cornerstone of algebra and mathematical problem-solving. Learning how to solve for variables and understand what they represent was the first concept that I remember finding exciting and beautiful in my own math career, so I thought I would explore it for this project. The standard is made up of four sub-standards. The first addresses the way a variable functions as a puzzle piece that completes the picture of an equation/inequality. The second emphasizes that variables are representative of unknown numerical values, and can be used as a placeholder or label for a missing value. In real-world problems, variables often represent unknown values such as distance, time or cost. Being able to write, reason, and solve equations (in real-world andmathematical contexts) of the form  and  is the third sub-standard. Lastly, the fourth sub-standard tackles inequalities in which variables have infinitely  many solutions. Students are expected to understand that inequalities of the form  and  have endless solutions and how to represent them on number line diagrams. Overall, Standard 6.EE offers a clear framework for sixth graders to master the essential skills of working with variables, solving equations, and understanding inequalities, laying a solid foundation for future algebraic thinking.

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Khan Academy: variables, expressions, & equations

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Khan Academy: variables, expressions, & equations

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I would be remiss if I did not include khan academy as a strong external resource. This is interactive for students, clear, and easy to understand

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Variables, expressions, & equations (video) | Khan Academy

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Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

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Maneuvering the Middle: Teaching one and two step inequalities

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Maneuvering the Middle: Teaching one and two step inequalities

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This resource has some beautiful bright visuals and great advice for teachers which discuss ways to address students who struggle with this standard

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What I love so much about inequalities are the infinite solutions. For so much of students' previous math experience, there is one exact answer. When it comes to inequalities, it is fun to push students to think of their answers beyond "x<4" and brainstorm all of the infinite solutions that x can be and then connect it to the graph.

Below, I have outlined a few ideas and things to consider when planning to teach one- and two-step inequalities.

Let's take a look at the standards, shall we?

Many of my recommendations for solving one or two-step inequalities are the same as solving one and two-step equations. This post talks about how using algebra tiles is my favorite practice.

Something I did not learn until I was teaching is the why behind changing the sign when you divide or multiply by a negative number. As

You can do whatever you want to an inequality as long as it is done to both sides, it will remain a true statement.

This is true of inequalities with one exception. If you multiply or divide by a negative number, the inequality becomes untrue.

While this mistake still persisted, I found that my students did much better than previous groups in remembering to flip the sign.

Able to Read an Inequality Statement

Another big misconception that I have found to be true is the "alligator." Sure, "the alligator eats the bigger number" works when you are comparing 3 and 7, but what about when it's 3 and -7 or 4x and 6? Students need to be able to read an inequality statement and explain what it means in terms of the numbers around it. They need to feel confident choosing a number for x to test the inequality.

I have found the students struggle identifying "<" as less than and ">" as greater than, so when they do solve an inequality and they are left with x<4, they don't actually know what that symbol means. 1<4 is something that a student can say but replace one of the numbers with a variable and the inequality becomes unclear. This is definitely a trick, but showing students how the less than symbol can be slightly rotated to be an L will allow them to actually state, "x is less than 4."

Using Number Lines

The graph can seem like one more thing to do, but the solution(s) and the thinking lies within the graph. The number line representation is a perfect visual to actually get students talking about the potential solutions. Some questions to get students thinking:

What is a value of x that is in the solution set? Is there another?

Why is _____ not a solution to inequality? What other numbers would not be a solution?

Describe the process for graphing the solution.

Given the graph, what inequality statement best describes it?

When we are given a graph and are trying to figure out what inequality is the match, I avoid the trick that the arrows should match the inequality sign. Instead I might ask:

What are the potential solutions?

Are the potential solutions greater than or less than the given amount?

When I substitute a solution into X, is the original inequality true?

Common Misconceptions

1. Forgetting to change the inequality sign when dividing by a negative number

2. General confusion when the answer is "4 > x" or written with the constant on the left

3. Unclear of what number is considered a solution

4. Unable to make connections between the inequality and the actual solution on the number line

5. Writing inequality statements from situations when the terms "less than" or "greater than" are not included

Anchor Chart Ideas

Anchor charts are fabulous ways to showcase the content in a visual manner for students to reference. They can easily be created before the lesson or as you are teaching, depending on the content. This example includes the emphasis on vocabulary, as students tend to struggle with writing inequalities.

The real-life example that I think students have the most understanding of is related to movie ratings and/or height requirements for roller coasters. These very real-life experiences will help students understand the terms minimum means greater than or equal to. The minimum height you can be to ride this roller coaster is 48 inches. That means you can be 48, 49, 50 ... inches to ride. Similarly, a person must be the minimum age of 17 to purchase an R-rated movie ticket.

Maximum was also another term that confused my students. Again, relating it to something that students experience daily seemed to do the trick. I would use seat belts as an example. I would ask students how many seat belts were typically in a car (or their car). Then I would say that the maximum number of people that could safely ride in said car was, let's go with 5, which means 5 people or fewer could ride in the car. Maximum meant less than or equal to.

IDEAS FOR STRUGGLING STUDENTS

1. Bring out the manipulatives - algebra tiles, pattern blocks, etc - have students group like items

2. Break down the steps into a simple checklist

3. Go back to positive whole numbers to see if students are struggling with the concept or the mathematical skills

4. Give students a possible solution and ask them to work backwards

Hopefully, this gives you some ideas for teaching one- and two-step inequalities or even insight as to what knowledge your students are coming with. I would love to hear other great activities or ideas you have used! Feel free to share in the comments.

Be sure to check out these resources for inequalities:

How do you teach inequalities?

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How to Teach One- and Two-Step Inequalities

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What I love so much about inequalities are the infinite solutions. For so much of students' previous math experience, there is one exact answer. When it comes to inequalities, it is fun to push students to think of their answers beyond "x<4" and brainstorm all of the infinite solutions that x can be and then connect it to the graph.

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NMPED: Algebra Reasoning with Equations and Inequalities

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NMPED: Algebra Reasoning with Equations and Inequalities

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This chart was an excellent read. It pointed out some important aspects of teaching this standard that are often overlooked, such as, "Culturally and Linguistically Responsive Instruction" and "Multi-Layered System of Supports."

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This resource is excellent for students practicing solving for inequalities. Instead of having to make worksheets, students can generate a wide variety of inequality problems to practice with! It saves on paper and is fun to use!

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I stumbled upon this resource in my SED 414 class during my undergrad, and I thought it was brilliant. While it does not explicitly use variables (or even just one variable), it is an engaging way to introduce students to the concepts of solving for missing pieces to create a balanced product!

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Mathematics Teaching in the Middle School: Prove it to Me!

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Mathematics Teaching in the Middle School: Prove it to Me!

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I really enjoyed all of the articles we read in class from this particular journal, so I found another one that investigates how students reason algebraically and what teachers can do to help them do so!

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The first and most notable thing that I learned while setting up this padlet portfolio is that finding and sorting through resources takes time and investigation. I did not include every article or resource I found while looking for them, but I went down quite a few rabbit holes trying to find relevant information which addressed standard 6.EE. There are so many math education resources out there! While it is absolutely incredible that the internet hold infinite accessible knowledge about how to teach math, it is also a bit daunting. With excess of information also comes the anxiety of choice. I do feel though, after exploring on my own for this assignment, that I am better prepared, and more interested in exploring all of the fantastic resources that exist for teachers of math!

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In making my own activities and resources for this portfolio, I was reminded how important it is to look at math from different perspectives. I have been doing high level math for so long, understanding one variable inequalities and equations is like second nature. For students learning this standard for the first time, this is not the case. If I want to be an effective and empathetic teacher of this standard, I must practice seeing this concept through the eyes of a student who may not understand. As I set up my activities, I challenged myself to think outside the box, and be creative with my explanations of concepts. Articles I linked in my resources such as Prove it to Me! and Maneuvering the Middle: Teaching One and Two Step Inequalities pointed out some useful misconceptions and quotes of student thinking that helped me understand students' perspective.

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When I was writing my introduction of the standard, I challenged myself to reiterate each substandard in my own words so that I better understood it. Writing about each individual aspect of the overall standard helped me focus on what I needed to research as I looked for external sources and created my own! I think writing about the standards cemented my understanding and mental organization of them. I would like to do the same with the other standards at some point in my life.

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I had never heard of or used padlet before this assignment, and I have come to really enjoy it as an educational resource! I can see myself making a padlet profile for my future classes and posting supplemental resources to it. This way, students who may need extra help have a curated resource to find it! I can also use it to post answer keys, announcements, practice problems etc. I most enjoyed that all of the information was on one page. I sometimes find canvas hard to navigate because things can easily get buried in links and pages, but padlet has everything laid out very clearly.

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Standard 6.EE: Reason about and solve one-variable equations and inequalities.

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Standard 6.EE: Reason about and solve one-variable equations and inequalities.

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1. Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

2. Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

3. Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.

4. Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams

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I stumbled upon this site after submitting my Portfolio, but I thew it in because it is so cool! This website holds tons of interactive games that help students practice solving for equations and inequalities!

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Reason about and solve one-variable equations and inequalities | 6th Grade Math | Math Chimp

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Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

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Here is a worksheet I created for students to use to practice their solving of equations and inequalities of one variable. I have noticed that many middle and high schoolers love the New York Times Games, so I am hoping by putting it in this context, it will be fun and engaging for students!