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On an instructional level, I learned that the best way to introduce 6.EE 1-4 to sixth grade students is by introducing the purpose and function of variables before teaching them how to solve for variables. Furthermore, a lot of the content covered in 6.EE is reliant on a strong understanding of equivalence and simplification. Beginning with a thorough review of the definition of equivalence and giving students situations where variables would be useful will set students up for success in later algebra lessons. Using counting materials, real-world applications, and connections to past material will help students understand why solidifying understanding of equivalence is important and will help them better understand variables. The materials provided in this portfolio do just that to prepare students for more challenging algebra lessons.

Beginning this portfolio, I thought it would be much easier to assemble a diverse collection of resources to help teachers who are planning on teaching the content covered in 6.EE topics 1-4. However, I was surprised to see the lack of resources online to support this content. Most resources online were resources to support students learning how to find the values of variables, not students who are learning what variables are and how to work with them in the first place. Most online materials were assembled to teach a class of eighth graders as a review, not for sixth graders with little to no algebra experience. However, if you search the internet thoroughly enough, you will be able to find resources that teach the first four topics of 6.EE even if they are few and far between.

Algebra has a very negative connotation around it as a subject. Teachers can easily prepare students for future algebra lessons with the right preparation and engagement. Part of this preparation comes from lessons surrounding 6.EE 1-4. Because these first few topics are so important to student success in algebra, it was disappointing to see the lack of resources online to support this learning. When teaching this topic, creating your own materials to support student learning may be more necessary than other topics. I advise any middle school teacher to take their time with these topics because they are the backbone of most algebra lessons. Introducing new content to students is always the hardest part of learning, but if done correctly and thoroughly, providing a strong foundation of conceptual understanding before introducing new content will help students accept the new content easier.

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Below is a collection of activities designed by the creator of this portfolio. Activity 1 is a practice worksheet to strengthen understanding of equivalence and variables. Activity 2 is an activity to strengthen student understanding of converting English sentences into equations. Neither requires a strong understanding of how to solve for variables, but Activity 2 might be more challenging for students who struggle to solve for variables in equations.

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Activity 2: Experimenting with Variables 

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Activity 2: Experimenting with Variables 

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https://docs.google.com/document/d/1FPrQUkHzd0ZVtPoCU7sP-FmgZkh9NO2VZ_GvHN7-u0E/edit?usp=sharing

In this second activity, I wanted to allow students to think about algebra in a context other than a worksheet. The "mystery" behind what object creates motivation to create expressions with variables and solve for the variables. To mainly address the first four topics in 6.EE, this activity would focus more on creating equations than solving for the values. Ideally, this activity would have more objects, more difficult and intricate clues, and more varying units from object to object. This activity could also be altered by giving students the object and verbal statements about the length and weight, then having them find the weight and mass on their own using the appropriate tools. This alternative method of incorporating this activity would be more hands-on if students are looking for a more interactive lab. I have included some more side notes on how this activity would be ideally done on the second page of the document.

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Activity 1: Variables and Equivalence Problem Set

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Activity 1: Variables and Equivalence Problem Set

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https://docs.google.com/document/d/1Dj5rObH28D79sYG-Z4GoVIyb9hUO6dXpLzfidxluuNs/edit?usp=sharing

After viewing the content covered in 6.EE 1-4, I noticed most of the content requires a great understanding of equivalence and the function of variables. Because of this, I decided to focus one of my self-made activities on practice with these concepts. In order for students to succeed in algebra, they must be confident in their own definitions of equivalence. I ensured this confidence would begin early on by offering early practice with algebra and equivalence before introducing the notions of solving for variables. This ties into the idea of showing the reasoning behind the procedures before introducing the procedure itself. If students have a strong understanding of equivalence, they will be able to understand why it's important to keep equations "balanced" and how to simplify expressions. In summary, this problem set is provided to give students ample opportunities to practice problems with equivalence to set students up for success in future algebra lessons.

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Posted are resources to support any teacher trying to teach content listed in 6.EE topics 1-4. These resources range from games students can play to readings teachers can take advice from in their lessons. References are listed at the bottom of this page for each resource.

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Resources to support teaching CCSSM 6.EE 1-4

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Resources to support teaching CCSSM 6.EE 1-4

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Video: "Combining like terms introduction" Khan Academy

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Video: "Combining like terms introduction" Khan Academy

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https://www.youtube.com/watch?v=CLWpkv6ccpA

Khan Academy is a great resource for just about any topic in math, but I found this video great because of the way it introduces variables and the act of combining like terms. The video introduces combining like terms by example first instead of using variables right off the bat. Using Chuck Norris and plums instead of x and y shows students the meaning of the process of combining like terms before telling them to simply add their coefficients together. The content covered in this video tackles the topics shared in 6.EE.4 quite nicely and would be a great resource to offer students who are struggling to understand how to combine like terms.

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Combining like terms introduction | Introduction to algebra | Algebra I | Khan Academy

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BYJU'S: "Variables and Constants in Algebraic Expressions"

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BYJU'S: "Variables and Constants in Algebraic Expressions"

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https://byjus.com/maths/variable/#:~:text=In%20Maths%2C%20a%20variable%20is,9%20and%204%20are%20constants.

Understanding the function of variables in an algebraic expression is exactly what CCSSM 6.EE.2a covers. Introducing letters into mathematics is intimidating and may be extremely confusing for students. This resource not only tackles any confusion students may have about variables, but it also covers definitions of mathematical terminology, such as operation, coefficient, constants, and many others. In order to set students up for success in algebra, they must have strong foundational knowledge to work with. This foundational knowledge includes vocabulary. Allowing students to reference this resource for reminders or additional help would be a great way to ensure students are set up for success before handing them algebraic problems to solve.

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In Maths, a variable is an alphabet or term that represents an unknown number or unknown value or unknown quantity. The variables are specially used in the case of algebraic expression or algebra. For example, x+9=4 is a linear equation where x is a variable, where 9 and 4 are constants.

The variable is a quantity that can be changed or which is not fixed according to the mathematical operation performed. Usually, in algebra, we express an unknown number using the term 'x' and 'y'. But this is not particular, and we can use any alphabet.

Learn more: Variables And Constants In Algebraic Expressions

In this article, we are going to discuss what is a variable in Math, types of variables such as dependent variable and independent variable with examples.
Table of contents:
Definition

A variable in Mathematics is defined as the alphabetic character that expresses a numerical value or a number. In algebraic equations, a variable is used to represent an unknown quantity.

These variables can be any alphabets from a to z. Most commonly, 'a','b','c', 'x','y' and 'z' are used as variables in equations.

Making arithmetical calculations with variables as though they were express numbers permits one to take care of scope of issues in a solitary calculation. An ordinary illustration is a quadratic recipe, which allows one to explain each quadratic condition by just substituting the numerical estimations of the coefficients of the offered condition to the variables that speak to them.

The idea of a variable is significant in math. Regularly, a function y = f(x) includes two variables, y and x, speaking to individually the quality and the contention of the function. The expression "variable" originates from the way that when the argument (additionally called the "variable of the capacity") shifts, then the quality changes accordingly.

Parts of Equation

The equations in Mathematics are formed of different parts. They are:

Variable

Constant

Coefficient

Operators

Term

For example, in equation, 3x+9 = 21,
x → Variable
3 → coefficient of x
9 and 21 → constants
'+' → operator

Operator: An operator is an arithmetic symbol that is used to perform various operations on the values. These operators are (+, -, ×, ÷, etc.).

Term: A term is a single expression of an equation. It can be a number or a variable or a number multiplied by a variable.

Examples of Variables

In the above examples, x and y are called variables.

Constant in Math

The constants in math are the values which cannot be changed or which are fixed. It is a contradiction to the variable. In algebraic expressions, the constant are terms which define themselves.

For example, if 9x+2=15 is an equation, then;

x is the variable, which can have different values and 9 is the coefficient of x.

But numbers 2 and 15 are the constant here.

Variable in Statistics

In statistics, the variables are used in terms of real-life scenarios. It is also said to be an attribute here. They are used to representing people, places, things, etc. For example, the colour of the hair of a person is a variable. Since the hair-colour could vary from person to person like for some it is blond, for some, it is black, etc.

Types of variables in Math

The variables can be classified into two categories, namely

Dependent Variable

Independent Variable

Let's discuss the two types of variables in detail.

Dependent Variables

The dependent variable is characterized as the variable whose quality depends on the estimation of another variable in its condition. That is, the estimation of the word variable is dependably said to be reliant on the free variable of math condition.

For instance, consider the condition y = 4x + 3. In this condition, the estimation of the variable 'y' changes as per the adjustments in the estimation of 'x'. In this manner, the variable 'y' is said to be a reliant variable. A portion of the cases that include subordinate variables is talked about in point of interest as beneath with their answers.

Independent Variables

In an algebraic equation, an independent variable describes a variable whose values are independent of changes. If x and y are two variables in an algebraic equation and every value of x is linked with any other value of y, then 'y' value is said to be a function of x value known as an independent variable, and 'y' value is known as a dependent variable.

Example: In the expression y = x ², x is an independent variable and y is a dependent variable.
Thus, we can summarise the variables mathematically as given below:
In mathematics, we can define variables with the help of a function; In simple words, a function is a rule for taking a number or set of numbers as input and producing an output which may also be a number or set of numbers. The most common character used to denote the input is x, and the result is y. Thus, the function is written in the form of y = f(x). Here, the character that stands for a random input is called an independent variable, i.e. x, whereas a symbol that stands for an arbitrary output is called a dependent variable; in this case, it is y.
Apart from these dependent and independent variables, there exist other types of variables such as:

Random variables

Categorical and continuous variables

Intervening variables

Moderator variables

Control variables

Extraneous variables

Free variables

Bound variables

Application of Variables

Variables are not used in the field of Mathematics but also in various other fields for different purposes, such as:

Programming

Research

Science

Statistics

For a better understanding of the concepts, register with BYJU'S - The Learning App and watch the interactive videos to learn with ease.

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Variable (Mathematics) | Dependent and Independent variables

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Variables in Maths are used to represent unknown quantities. Variables are also used as symbols sometimes to denote vectors, functions and matrices. In the equation, 3x+8 = 0, x is the variable.

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https://www.thoughtco.com/understanding-equivalent-equations-4157661

This page offers quite a few useful examples of when equations are equivalent to one another, a topic covered in 6.EE.3 and 6.EE.4. A strong understanding of equivalence among equations is something necessary for students to be successful in algebra. Because of this, students need plenty of examples and explanations behind the meaning of equivalence, especially when variables are involved. The examples and non-examples provided in this page are straightforward, practical, and increase in difficulty as the page continues. While I think asking a student to read this material on their own could lead to some confusion, it could be a great teaching aid. Using the examples available on the website and using the student-friendly vocabulary offered on the page would be great inspiration for a lesson plan. My biggest critique is that it's not the most student-friendly. If there were more images depicting the algebraic expressions and a greater use of student-friendly language, this resource would be great to offer to students to strengthen their understanding of equivalence.

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Equivalent equations are systems of equations that have the same solutions. Identifying and solving equivalent equations is a valuable skill, not only in algebra class but also in everyday life. Take a look at examples of equivalent equations, how to solve them for one or more variables, and how you might use this skill outside a classroom.

Key Takeaways

Linear Equations With One Variable

The simplest examples of equivalent equations don't have any variables. For example, these three equations are equivalent to each other:

Recognizing these equations are equivalent is great, but not particularly useful. Usually, an equivalent equation problem asks you to solve for a variable to see if it is the same (the same root) as the one in another equation.

For example, the following equations are equivalent:

In both cases, x = 5. How do we know this? How do you solve this for the "-2x = -10" equation? The first step is to know the rules of equivalent equations:

Adding or subtracting the same number or expression to both sides of an equation produces an equivalent equation.

Multiplying or dividing both sides of an equation by the same non-zero number produces an equivalent equation.

Raising both sides of the equation to the same odd power or taking the same odd root will produce an equivalent equation.

If both sides of an equation are non-negative, raising both sides of an equation to the same even power or taking the same even root will give an equivalent equation.

Putting these rules into practice, determine whether these two equations are equivalent:

To solve this, you need to find "x" for each equation. If "x" is the same for both equations, then they are equivalent. If "x" is different (i.e., the equations have different roots), then the equations are not equivalent. For the first equation:

x + 2 = 7

x + 2 - 2 = 7 - 2 (subtracting both sides by same number)

x = 5

For the second equation:

2x + 1 = 11

2x + 1 - 1 = 11 - 1 (subtracting both sides by the same number)

2x = 10

2x/2 = 10/2 (dividing both sides of the equation by the same number)

x = 5

So, yes, the two equations are equivalent because x = 5 in each case.

Practical Equivalent Equations

You can use equivalent equations in daily life. It's particularly helpful when shopping. For example, you like a particular shirt. One company offers the shirt for $6 and has $12 shipping, while another company offers the shirt for $7.50 and has $9 shipping. Which shirt has the best price? How many shirts (maybe you want to get them for friends) would you have to buy for the price to be the same for both companies?

To solve this problem, let "x" be the number of shirts. To start with, set x =1 for the purchase of one shirt. For company #1:

For company #2:

Price = 7.5x + 9 = (1)(7.5) + 9 = 7.5 + 9 = $16.50

So, if you're buying one shirt, the second company offers a better deal.

To find the point where prices are equal, let "x" remain the number of shirts, but set the two equations equal to each other. Solve for "x" to find how many shirts you'd have to buy:

6x + 12 = 7.5x + 9

6x - 7.5x = 9 - 12 (subtracting the same numbers or expressions from each side)

-1.5x = -3

1.5x = 3 (dividing both sides by the same number, -1)

x = 3/1.5 (dividing both sides by 1.5)

x = 2

If you buy two shirts, the price is the same, no matter where you get it. You can use the same math to determine which company gives you a better deal with larger orders and also to calculate how much you'll save using one company over the other. See, algebra is useful!

Equivalent Equations With Two Variables

If you have two equations and two unknowns (x and y), you can determine whether two sets of linear equations are equivalent.

For example, if you're given the equations:

You can determine whether the following system is equivalent:

To solve this problem, find "x" and "y" for each system of equations. If the values are the same, then the systems of equations are equivalent.

Start with the first set. To solve two equations with two variables, isolate one variable and plug its solution into the other equation. To isolate the "y" variable:

Now, plug "y" back into either equation to solve for "x":

Working through this, you'll eventually get x = 7/3.

To answer the question, you could apply the same principles to the second set of equations to solve for "x" and "y" to find that yes, they are indeed equivalent. It's easy to get bogged down in the algebra, so it's a good idea to check your work using an online equation solver.

However, the clever student will notice the two sets of equations are equivalent without doing any difficult calculations at all. The only difference between the first equation in each set is that the first one is three times the second one (equivalent). The second equation is exactly the same.

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Here's How Equivalent Systems of Equations Work

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Equivalent equations are systems of equations that have the same solutions. Identifying and solving equivalent equations is a valuable skill, not only in algebra class but also in everyday life. Take a look at examples of equivalent equations, how to solve them for one or more variables, and how you might use this skill outside a classroom.

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https://solveme.edc.org/mobiles/

This resource was introduced to me through another class at OSU. To summarize, it uses the balance mechanics of a mobile to visually represent equivalence between two sets of shapes. Some of the shapes have value or weight to them, others are unknown. Through this activity, students must solve for the unknown values of the shapes. There are pre-made puzzles already available on the website, or teachers and students have the option to create their own puzzles. The visual aspect of these puzzles makes solving for the weights or values of the shapes less like an algebraic equation and more like a fun puzzle for students to enjoy. Oftentimes, students see math problems and are instantly disinterested. This model of algebra provides a new and exciting way of viewing algebraic expressions without the daunting form of an equation. The added option for students to make their own puzzles also allows students to experiment with equivalence, thus practicing CCSSM 6.EE.3 and 6.EE.4. They can edit the weight and amount of shapes added as well as the number of mobiles on the screen. They can make problems as simple or complicated as they would like, which makes this website an enjoyable way for anyone at any level of understanding to explore algebraic expressions.

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Video: "Algebra Basics: What is Algebra - Math Antics"

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Video: "Algebra Basics: What is Algebra - Math Antics"

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https://www.youtube.com/watch?v=NybHckSEQBI

This video introduces the overall concept of algebra and how we will use it when covering the topics in 6.EE. The video does a great job of comparing algebra to the notions of arithmetic, something students are already comfortable with. It also provides a look at what students will eventually be able to accomplish with what they've learned from algebra: creating linear functions to describe certain scenarios. What I particularly liked about what Math Antics did in this video is how they talked about variables. It's intimidating to switch from expressions containing only numbers to expressions that contain letters and numbers. However, Math Antics described the purpose of the variable as a placeholder, much like how in the expression 1 + 2= ? the ? is the placeholder for the answer, or 3. Overall, when introducing the use of variables in equations to students, this video could be a valuable teaching resource.

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Algebra Basics: What Is Algebra? - Math Antics

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The first four topics of 6.EE tackle the introductory ideas of algebraic expressions and equivalence. Students must learn to view expressions as versatile statements and use the properties they know to manipulate them as needed. These few topics will heavily emphasize the commutative and distributive properties. Students must also be able to create algebraic expressions out of real-world situations or verbal statements. For example, if we were to ask students to take three away from two times x, they should recognize this is equivalent to saying 2x - 3. Expression equivalence is covered in this topic as well. Treating portions of the equation as whole units, using properties they've learned along the way, and manipulating equations as needed are crucial skills students will gain from learning the content covered in 6.EE.

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During my time as a tutor, I have had many conversations with students from grades 6 to 12 about algebraic operations. I've discovered that students commonly struggle with the list of topics covered in 6.EE 1-4. When I saw this topic in the list, I was determined to explore this topic further by using my past experience to make sense of 6.EE 1-4 myself and solidify my teaching methods for my future students.

I also remember learning this content myself and having trouble manipulating equations correctly. My first experiences learning algebra were difficult because it was so different from what we had been learning previously. I would like to use the experiences of me and my students to create a collection of resources to clearly and smoothly introduce algebraic expressions to sixth-grade students.

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CCSSM 6.EE (Expressions and Equations) 1-4

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CCSSM 6.EE (Expressions and Equations) 1-4

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Apply and extend previous understandings of arithmetic to algebraic expressions.

1. Write and evaluate numerical expressions involving whole-number exponents

2. Write, read, and evaluate expressions in which letters stand for numbers

2a. Write expressions that record operations with numbers and with letters standing for numbers.

2b. Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity.

2c. Evaluate expressions at specific values of their variables. Include expressions that arise from formulas uased in real-world problems. Perform arithmetic operations, including those involving whole-number exponsents, in the conventional order when there are no perentheses to specify a particular order (Order of Operations).

3. Apply the properties of operations to generate equivalent expressions.

4. Identify when two expressions are equivalent (i.e. , when the two expressions name the same number regardless of which value is substituted into them)