Equality Matters When Solving Linear Equations

     When I initially studied algebra in high school, linear equations were one of the few algebraic expressions that made sense to me. These equations create a straight line that can be graphed. To understand linear equations, four very important properties of equality needed to be understood first:

Addition Property:	For all real numbers x, y, and z, if x = y, then x + z = y + z.
Subtraction Property:	For all real numbers x, y, and z, if x = y, then x – z = y – z.
Multiplication Property:	For all real numbers x, y, and z, if x = y, then xz = yz.
Division Property:	For all real numbers x, y, and z, if x = y, and z ≠ 0, then x/z = y/z.

 http://hotmath.com/hotmath_help/topics/properties-of-equality.html

In essence, whatever has to be done to one side of the equation in order to isolate the unknown quality has to be done to the other side of the equation to maintain equality. Here is an example:

       Solve the equation: 3x + 5 = 11

1. Isolate 3x on one side by subtracting 5 from both sides using the subtraction property of equality.

          3x + 5 – 5 = 11 - 5

          3x = 6

2. Next, divide both sides of the equation by 3 to isolate the value for x. This step uses the division property of equality.

           3x/3 = 6/3

         X = 2

3. The last step involves checking you work by placing the value of x into the original equation.

           3(2) + 5 = 11

         6 + 5 = 11

The value of x (2) makes the statement true. We have found our answer.

     Knowing the properties of equality enables us to solve more complex equations as well. When thinking about these properties and their relation to linear equations, I picture two platters that need to remain equal. In looking at the equation above, if I take 5 cookies off of the first platter, I need to take 5 cookies off of the second platter to keep them equal. If I divide one platter by 3 to reduce the amount of cookies further, I need to do the same to the second platter to keep them identical.

Here is another look at solving linear equations:

     When solving linear equations, remember that the properties of equality are essential in finding the unknown value. Equality matters! Here is a great game that can be used to practice solving linear equations:

http://www.mathplayground.com/SaveTheZogs/SaveTheZogs.html

Sharing With the Distributive Property

Sharing With the Distributive Property

     Sharing can be a difficult procedure for a young preschooler. When I first encountered the distributive property two years ago, I found the same difficulty in learning how to “share” the number outside of the parenthesis with the numbers inside the parenthesis. I had been away from college since 1995 and many of the math concepts were foreign to me. Thankfully, I was introduced to an explanation like the one here from http://www.northstarmath.com/sitemap/DistributiveProperty.html:

 Definition of Distributive Property

• Distributive property states that the product of a number  and a sum is equal to the sum of the individual products of addends and the number.

                     That is: a(b + c) = ab + ac

At first, I copied down this definition and simply applied it to every problem I needed to solve for the assignment. After working with the property, I started to make other connections that helped me understand this property.

    Look at the problem 4 (9 + 5) = x.  Imagine that 9 and 5 are friends of our dear number 4. Let's imagine that number 4 wants to spend time equally with both friends. Number 4 will visit number 9 and multiply. Then, number 4 will visit number 5 and multiplies. Like the example in Danica McKellar's book, the number outside of the parenthesis, 4, needs to split it's time fairly with number 9 and number 5 in terms of multiplication. Here is the solution:

            4 (9 + 5) = x

            36 + 20 = x

             56 = x

           

Here is a video that also addresses the distributive property:

     

     After tackling problems in algebra that utilized the distributive property, I was glad I had a good understanding of the basic concept. I happen to be someone who loves to share, so this helped solidify this concept in my mind. Teaching this concept may also require some creativity:

It may also require some stories concerning sharing and fairness that I have learned from the preschoolers in my life. After all, fairness is very important to a preschooler as well as to the distributive property!

Here are some additional resources and games:

http://www.freemathhelp.com/distributive-property.html

http://www.quia.com/ba/15357.html

http://www.glencoe.com/sec/math/brainpops/00112041/00112041.html

Playing Along a Number Line

Here it is… the best board game I ever played on family game night.

 

http://www.geekalerts.com/taking-pac-man-tetris-to-the-next-dimension/

This is Pac- Man, the board game. I remember not only trying to outwit the ghosts that my siblings controlled but also strategically gobbling up marbles. With the roll of the two dice, I moved my Pac-Man along the rows as well as the ghost of my choosing. Despite the fact that my brother loved to win and coulddn't stand losing, we always had a great time gobbling up marbles. Not only did the fate of my brothers rested in the luck of my roll, but I was also learning an important strategy in adding and subtracting.

     This game and other board games that use movement in terms of counting and moving pieces a specific number of spaces have a cousin in the number line. When I first encountered the number line in college, it felt like an old friend. In terms of board games, if my starting point is 5 and I am moving (adding) 6 spaces, I end up at 11. If I was unfortunate enough to move 3 spaces back from 5, I know that I will end at the number 2. 

http://www.youtube.com/watch?v=wzEZ1m9GUgM

The number line also helps students see the addition and/ or subtraction in action. Here is a video that addresses addition along a number line.

 

     This strategy not only assists students by providing a neat activity, but also helps students with mental math as they see the spaces traveled between numbers. Students can also move along a number line taped to the floor and "feel" the number of spaces moved. Moving from the number 5 to the number 11 as we add 6 spaces feels very different as we manipulate tokens or our feet along the number line. Playing with the number line has been fun for me as a college student and I see how students benefit from working with it. Here are some additional resources and games:

Resources:

http://www.math-aids.com/Number_Lines/

http://www.math.com/school/subject1/lessons/S1U1L11GL.html

 

 Games:

http://www.funbrain.com/linejump/

http://www.brainpop.com/games/battleshipnumberline/

http://www.sheppardsoftware.com/mathgames/earlymath/fruit_shoot_NumberLine.htm