Big Idea 10: Mathematical situations and structures can be translated and represented abstractly using variables, expressions, and equations.

Mathematical variables are letters or symbols that represent numbers that can change. They are often used in algebra expressions and equations. Equations include an equals sign (e.g. 2x+5=25) whereas expressions do not (e.g. 2n-1). Although variables are used in algebra, the concept should be introduced much earlier while students are in elementary school.

Some problem types lend themselves to using variables. Problems types with the start unkown or the change unknown can be written with variables. Here are some examples:

Separate, start unknown: Amanda had some oranges. She gave 2 oranges to Jonathon. Now Amanda has 10 oranges. How many oranges did Amanda have to start with? This can be written as x-2=10. Younger students may use an empty box or blank instead of a letter variable. They may see the equation written as __-2=10.

Separate, change unknown: Carlos had 25 stickers. He gave some of the stickers to Lucille. Now Carlos has 10 stickers left. How many stickers did Carlos give to Lucille? This can be written as the equations 25-x=10 or 25-__=10.

In upper elementary grades, particularly sixth grade, students learn to use letter variables instead of boxes or other symbols. They can learn that the equations or expressions can be solved if given quantities for the variables. For example, to solve the expression 2x+1 if x=2, then students substitute 2 for x in the problem. In this case 2(2)+1 which equals 5. Equations can be solved the same way when given values for the variables. For example, to solve the equation y=3x-1 when x=2, then students again substitute 2 for x in the problem. In this case, y=3(2)-1=5. In this problem students are solving for the variable y instead of a quantity as in the expression example.

One activity that helps students learn how variable work is in and out boxes. Students have data that shows when a certain number is put into the box, a certain different number comes out. For example, consider this chart:

In Out

2 4

3 6

1 2

Students use the data to discover the rule about how the number changes within the box. In this case, the numbers put in the box are multiplied by 2. This can be written using a variables x (for the number put in) and y (for the number that comes out) as 2x=y. By writing the rule this way, students learn that they can put any value for x and solve for y. In other words, when given a quantity to put in the box, they can solve the equation and find out what quantity would come out.

Lesson Plans:

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Understanding Variables(5th grade). This lesson uses cards with symbols to create expressions and equations with variables.

http://www.uen.org/Lessonplan/preview.cgi?LPid=6161

What’s My Function (5th grade). This lesson uses in and out charts to create equations with variables.

http://www.uen.org/Lessonplan/preview.cgi?LPid=6160

Missing Numbers (1st grade) This lesson uses symbol cards as variables in simple addition and subtraction equations. Students solve for the variable.

http://www.uen.org/Lessonplan/preview.cgi?LPid=10713

Games:

There are three variable games at this website. In the **variable solving game**, students are given an equation with a variable and they must calculate the value of the variable. In the **evaluating an expression with one variable **and **evaluating an expression with two variables** games, students substitute the given quantity for the variables and solve the expressions.

http://www.gamequarium.com/prealgebra.html

Other helpful websites:

http://www.aaastudy.com/equ.htm

http://www.math.com/school/subject2/lessons/S2U2L3DP.html