Engaging students: Introducing variables and expressions

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Christine Gines. Her topic, from Pre-Algebra: introducing variables and expressions.

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As we all know, introducing variables in a Mathematics class often intimidates students. As teachers, we can minimize this by creating activities where students are eased into the new topic in a fun and educational environment.  This can be achieved through the following activity that introduces variables:

In this activity, students discover “the value of words.” On notebook paper, have students write the letters of the alphabet in order down the left side of the paper. Down the right side of the notebook paper, have them write the numbers from 0 to 25. The letters should corresponding to the numbers. The numbers are the values to each letter, or variable.

To begin, you could have your students find the value of their own name and last name.

Ex. Chris –> C=2, H=7, R=17, I=8, S=18

= 2+7+17+8+18 = 52

You could ask the following questions:

  • Which has a higher value – first or last name?
  • What is the difference in the values of your first and last names?
  • Find words whose values are equal to 25, 36, or 100.
  • What is the three-letter word with the greatest value?
  • Are the greatest values always associated with words that contain the most letters?

You could also pair your students and have them write codes to each other. Furthermore, challenge them to write their code with value restrictions and allowing them to *,/,+,-.

This activity develops algebraic thinking in a concrete manner students can understand without presenting them with an overwhelming amount of new information. It is a very flexible activity in which you could make it your own and get the kids excited about it. For example, the activity could even be competitive by challenging students to write an expression for CAT where the value would equal 2 (C+A*T = 2+0*10). This is definitely something I would use to introduce variables.

More about this game can be found here:  http://illuminations.nctm.org/Lesson.aspx?id=1156

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A variable expression is a combination of variables, numbers and operations. The only new information being presented is the unknown represented as variables and how to solve for that variable. Students don’t know this, but it’s quite similar to what they have been doing in school for years. Take 2x=4 for example. We know x=2 because 4/2=2. This expression is equivalent to just writing 4/2=_, which is a simple division problem that students have seen time and time again.

Variable expression are not always given, though. Students will learn how to construct them by analyzing word problems for key clues. This is where the vocabulary students have been working with comes into play.  Common words that they will see are sum, difference, quotient, product, etc.

A key rule to +/- fractions is “Whatever you do to the top, you have to do to the bottom.” This theme directly correlates with solving expression with respect to the left and right side of the equal sign. Therefore, we can conclude that variable expressions are a combination of skills that students have learned previously with the exception of written variables.

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With the fast growth of  technology, more and more useful sources are becoming readily available to us and it’s important to take advantage of this. Math Play is a website that provides a variety of interactive online games organized by content and all grade levels.

One game in particular, Algebraic Expressions Millionaire Game, serves perfectly as an introduction to constructing variable expressions. The game has the theme of “Who wants to be a millionaire?” and challenges students to chose an equivalent representation of an expression written in words. The problems increase in difficulty as you progress, using clues such as less than, difference, sum, product, quotient, etc.  This Algebraic Expressions Millionaire Game can be played online alone or in two teams. The link to this game can be found below:


This game is a great way for students to develop a conceptual idea of what variable expressions represent. It also builds a foundation for solving and constructing word problems. Try pairing students to compete against each other to add motivation. You could even hold a tournament!


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