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Introducing Algebra - Unknown Variables
At the end of the school year, I had a new fifth grade tutee tell me that the letters in math problems just confused his world. He was referring to variables within in equations such as 3y = 12. He was confused because he was not able to take the unfamiliar and attach it to something familiar in his world until I shared and showed him the following.
In everyday life, when you meet an adult woman her name is unknown to you. For instance before you learned your teacher’s name, you may have referred to her as Miss, Ms., or Mrs. to get her attention. Later when you learned her name, you were able to be more specific. For instance, you now call me Mrs. Mackie.
Likewise for those letters you see in equations. When you see a number and a letter side by side, it means to multiply the number by another number. However, we don’t know the other number, so we use a letter. Letters, such as “X”, “Y”, or any other letter used in equations are generally used as placeholders for an unknown number. The unknown number is called a variable. Thus, read “3y = 12” as 3 times some number equals twelve.
When you figure out which number it represents, the letter gets a specific value. You may say something like, “Oh, y = 4.” Is that true? Let’s see. Since we’ve given “y” a specific value, substitute ‘4’ in place of ‘y’. 3 x 4 = 12; Is that a true statement? Yes! Remember the ‘=’ sign demands that both sides of the equation must mean the same. Notice, I didn’t say both sides have to look the same. In this problem, one side consists of a two numbers and a math operation, and the other side only consist of a number. However when you do the math, they mean the same. Also, you may have a nickname. Nickname = you. This means you will answer if I call you by your real name or your nickname. In other words…different names, same person.
Now, go back to the world of math. What if my tutee had no idea on how to find the value of ‘y’ in the above equation? Well, I knew my tutee was familiar with fact families. Fact families show the relationship between two families using addition and subtraction or multiplication and division. Thus, there are four facts included in each fact family. So, let’s list the fact family of 3Y=12. (note: “/” means division)
1) 3 x Y = 12
2) Y x 3 = 12
3) 12 / Y = 3
4) 12 / 3 = Y
Which equation helps to find y? the fourth equation, y = 12 / 3; it tells us to divide 12 by 3 to find the value of y. ***Alert*** okay, to solve 3y = 12, we can divided 12 by 3, but remember whatever we do to one side of the equal sign, we must do to other side to keep both sides the same. I guess you could think of an equation like a scale.
So again, to solve 3y = 12, divide both sides by 3.
3y / 3 = 12 / 3
3y / 3; thus 3/3 = 1; thus “1 x y” is just “Y” … in Algebra, we say “3/3 cancels out.”
Therefore, Y = 12 / 3
Final answer, Y = 4
Finally, “the light bulb came on”; my fifth grade friend, future middle school student was able to make a connection, and he was very proud that he understood some Algebra. I just love to see that look of hope and pride on my students’ faces. It’s awesome!!
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