Introducing Variables and Expressions

How to introduce variable expressions to algebra students

Are you ready to introduce variables to your class? Thinking about diving right into the textbook and starting students with the printed problems on the page? They are just letters and symbols that represent unknowns so this should be no problem, right?! For some students, yes, that is probably enough of an explanation to get them started. But, if you want your students to start making sense of what it means to combine variables and create expressions to represent real-world situations, you might want to give them some different opportunities to engage with what it means to move from expressions with integers to expressions with variables.

A simple way to start is to use real-world problems with some physical representations they can use to manipulate to combine variables and create expressions. One strategy I have had success with while working with the students in my classroom is using picture tiles.

I always start small and build up the concepts so that students can take small steps toward the end goal. I typically start with word problems with one variable. Then, I build on a number term. Next, I would throw two separate variables into an expression. Finally, I would end with two variables and a number term. Here is an example of a sequence introducing terms, variables, coefficients and expressions.

One Variable Expressions

Given the problem “Janice bought two chocolate cupcakes. Show me what Janice has with picture tiles.” Your students would get two cupcake tiles. These pictures could be represented with the expression . Talk to your students about what this expression represents. We are talking about a purchase so 2C probably represents what Janice paid for two cupcakes. Introduce formal vocabulary here. The variable is , the coefficient of the variable is , and  a term. Have your student complete a few more expressions with only one variable. Talk about what each expression represents. They will begin to see the connection between the coefficient of the variable and the variable itself. As they complete a few more examples, have your students identify the variable, coefficient, and the term. This will help them to pair visuals with the definitions.  

One Variable Expressions with a Number Term

Next, introduce a number term with a variable.  You can start with the same problem starter used to introduce a one variable expression and add a dollar value. Given the problem “Janice bought two chocolate cupcakes and still had . Show me what Janice has with picture tiles.” Your students would get two cupcake tiles and 3 dollar tiles. These pictures could be represented with the expression . Again, discuss what the expression represents. Let them know that both  and  are expressions. Expressions can have one term, like . Expressions can have more than one term like . Have your students complete a few more examples of one variable expressions with a number term. Then, start the discussion about why the number of cupcakes and the dollars are kept separate in the expression. Each term is representing a different item. We don’t know the cost of a cupcake, so we represent that cost with . What would happen if we had an additional item with an unknown value?

Two Variable Expressions

At this point you can introduce an additional unknown so that students can see how that changes the look of an expression. Given the problem “Janice bought four salads and two cupcakes. Show me what Janice has with picture tiles.” Your students would get four salad tiles and two cupcake tiles. These pictures could be represented with the expression . Now we can discuss why it is important to use a different variable to represent each different type of item in the expression. The different variables represent different unknown values. Since buying a salad is different than buying a cupcake and using different variables is how we distinguish that difference without words. 

Two Variable Expressions with a Number Term

Now you can introduce an additional unknown so that students can see how that changes the look of an expression. Given the problem “Janice bought four salads and two cupcakes. She still had $3. Show me what Janice has with picture tiles.” Your students would get four salad tiles, two cupcake tiles, and three dollar tiles. These pictures could be represented with the expression . At this point, you can extend the problem. How would the expression change if she had purchased seven salads instead of four? How would the expression change if she still had $12?

Final thoughts on introducing variable expressions

By introducing the definitions of terms, variables, coefficients, and expressions with a visual representation we can deepen student understanding. Picture tiles are low-prep, easy-to-use visuals that help students to engage with a new concept. Here is a free printable to help get you started with introducing variables and expressions.

If you’d like step-by-step guided notes on this topic, please check out our Algebra 1 Introduction to Variables, Terms, and Expressions Product.