Variable Expressions Part 2

Variable Expressions

                                     ¹

Now that students have an understanding of algebra and the language involved we can look at understanding how to work with a variable expression.

An algebraic expression is a combination of numbers and variables together with at least one mathematical operation.
Examples:    x + 7   
               
                     8f - 6

                     7y + 2

Khan Academy

I am not a huge fan of Khan Academy.  I think they do good work, it's more of a personal thing.  Sal Kahn's voice gets to me and the way he tends to work back and forth bothers me after a while too. However, they have accumulated a great library of mathematical lessons that are relevant and informative.  I think I would ask my class if they find Khan Academy helpful before I started using them consistently.  If other students are affected by Mr. Kahn's mannerisms like I am then there is no point in showing several important ideas presented by Khan Academy.

Why all the letters in algebra?

https://www.khanacademy.org/math/algebra/introduction-to-algebra/overview-hist-alg/v/why-all-the-letters-in-algebra

This video is short, 3:03, and gives a brief explanation of why we use letters in algebra.  It seems harmless enough and I'd use it to start this lesson.

Why aren't we using a multiplication sign?  (4:57)

https://www.khanacademy.org/math/algebra/introduction-to-algebra/alg1-intro-to-variables/v/why-aren-t-we-using-the-multiplication-sign

This video explains why we don't use a multiplication sign when using variables.  It's a very important topic that students must understand in order to be able to "read" algebraic expressions.  We do a lot of multiplying in algebra and the letter X is a popular variable.  The expression 2 x X + 9 looks crazy and would be very difficult to figure out.
The video explains that we have several ways to show multiplication at this point in our mathematical learning:

     5 * x, in this case the dot is the symbol for multiplication. 
   
     5 (x), in this case the parentheses show multiplication.
   
     5 x, we can remove the parentheses and place the 5 next to the x, 5x,  because it is easy to see that the two items are different and do not go together.  Students would need to learn that when they see 5x or any number sitting next to a variable the operation between them is multiplication.  This will be important later when we need to move, or undo, that 5 coefficient from the x variable. 

Evaluating Variable Expressions

To begin working with variable expressions we need one more word for our word wall.

Substitution - The act of replacing one thing with another thing.  
         * In this case we will replace the variable (letter) with a number.

How do we begin?  When we have an expression given to us, and we are asked to evaluate that expression, we are normally give the number that we are going to use.  Take a look at the example:

Evaluate the expression below, where Y = 3:

Y + 7:  Students should understand that Y is the variable and 7 is a constant. 

They are asked to evaluate the expression if Y = 3.
The next step would be to substitute the 3 for the Y in the expression.    3 + 7
Then do the math.  3 + 7 = 10

This lesson would probably continue with a few more examples, either with the class, in small groups or individual work and then a worksheet could be added so they could practice what they learned.  However, I found algebra tiles in my research for this topic!  The next great step in understanding this concept would allow the students time to work with a manipulative and hopefully help to make these ideas more concrete. 

Manipulative Help

I love the idea of algebra tiles.  They remind me of the previous work that most students will have done with base ten blocks and ten frames.  Algebra tiles are a tactile way of helping with algebraic manipulation. 

 

This set of tiles contains red and green rectangles and red and yellow squares.  The rectangles represent the variable and the small squares represent the constant or the number.  The color red represents a negative number.  The green rectangle is a positive variable and the yellow square is a positive number. 

Video:  Learn to evaluate expressions using algebra tiles

https://www.youtube.com/watch?v=f2o8EI0iOYg

The video is 7:01 in length.  It demonstrates how to use the tiles.  At 4:13, they explain how to model algebraic expressions and solve for the variable. 

Algebra tiles would be a great way for students to practice working with variable expressions.  Once they have mastered the objective in a visual, tactile way, they are ready to move to paper.  Students who feel they have mastered the concept can work on a standard worksheet.  Those that aren't ready to leave the shapes and colors completely, can work with a modified worksheet that provides them the familiar help they need.  It is also possible for yet another group of students to complete the worksheet while still using the algebra tiles.

Common Core Standard

CCLS - Math:  6.EE.2.c

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

Conclusion

Wow!  I feel like I didn't really get anywhere with this, but I made a few discoveries.  I have a better understanding about how difficult it is to break down an idea into manageable chunks for young learners to understand.  I can see there are a lot of resource available, but you need time to sort through them to make sure they are appropriate for your students and helping to advance the lesson in the way that you want it to go.  I can understand not that it would be easy to lose track of the mathematical objectives as you sort through all the information.  I am really impressed with the algebra tiles and think I'll devote the next blog to them.  I'll pick up with variable expressions after I explore how helpful the tiles will be for young learners.   

Sources

1 - http://www.mathfunny.com/images/mathpics-mathjoke-haha-humor-pun-mathmeme-meme-joke-math-problems-harry-variable-equation-xyz.jpg

2 - https://www.engageny.org/ccls-math/6ee2c