by Josh Rappaport
It's that time of year again when children are studying exponents and logarithms. That being the case, I'd like to present a concept that helps students work with these concepts.
I call it the "Whole-Part-Equality Principle" (the WPE Principle). But if I have enough air in my lungs, I sometimes like to call it the: "Peter-Paul-Pennies-in-the-Pocket Principle."
Here's how it works. We know two people: Peter and Paul. Peter and Paul possess no money except pennies, and they transport their pennies only in their right and left pants pockets (if anyone can propose a way to pack this story with even more p's, please let me know).
I almost forgot … we have three additional facts:
1) The number of pennies Peter transports equals the number of pennies that Paul transports. (i.e., Peter and Paul have the same total number of pennies!)
2) Peter and Paul each have three pennies in their right pants pockets.
3) Peter and Paul carry pennies NOWHERE but in their pants pockets.
QUESTION: What can we conclude about the number of pennies that Peter and Paul have in their left pants pockets?
ANSWER: While we don't know how many pennies Peter and Paul are transporting in their left pants pockets (it could be any number, right?), it is nevertheless obvious that they must have the same number of pennies in their left pants pockets.
WHY? View it like this … If the wholes are the same (the total number of pennies that Peter and Paul each has), and if one of two key parts are the same (the number of pennies that Peter and Paul have in their right pants pockets), then the other parts must also be equal (the number of pennies they have in their left pants pockets).
Why am I bringing this up? To point to an important principle.
Here’s how I would summarize this principle: Suppose you have two quantities that are equal: Q1 and Q 2. Suppose further that each quantity has two parts, A and B. Finally suppose you know that Part A of Q1 is equal to Part A of Q2. With this being given, you can be 100% certain that Part B of Q1 is equal to Part 2 of Q2.
This principle can be used to solve many problems in math, including problems involving exponents and logs. A few examples …
EXAMPLE 1: Suppose you’re given this equation: log x = log 7.2. What can we conclude? Well, the wholes are equal (meaning the left and right sides of this equation), and the bases are equal (logs are always base 10 unless another base is stated), therefore the remaining parts, the 'arguments,' also must be equal. The 'argument' is the term after the word 'log.' In this equation the arguments are x and 7.2. Using the principle, we conclude that x and 7.2 must be equal: x = 7.2.
EXAMPLE 2: Suppose you have the equation: log 2^x = log 16. Again, the wholes are equal, and the logs have the same base, so the arguments must be equal. That means that 2^x = 16. Since 2^4 = 16, x = 4, and that's the answer.
EXAMPLE 3: Suppose you have the equation: a^log x = a^log 12.9. Since the wholes (the left and right sides of the equation) are equal, and since the bases are equal since they are both 'a,' therefore the only remaining parts, the exponents, must also be equal. So this means that log x = log 12.9. Following the same logic as we used in Examples 1 and 2, this means that x = 12.9.
This same principle can also be used in earlier math problems. For example, if you know that: x + a = x + 12, can you conclude anything? Well yes. Since the left side of the equation equals the right side, and since the x’s must be equal because they are the same variable, it stands to reason that the remaining parts, the “a” and the 12.
Once your children learn this principle, encourage them to use it, not only in math, but also in real-life situations, such as the situation involving the pennies. It’s fun, and it builds logical thinking skills.
Josh Rappaport is the author of the Algebra Survival Guide and Workbook, which together comprise an award-winning program that makes algebra do-able! Josh also is the author of PreAlgebra Blastoff!, an engaging, hands-on approach to working with integers. All of Josh's books, published by Singing Turtle Press, are available on Amazon.com
It's that time of year again when children are studying exponents and logarithms. That being the case, I'd like to present a concept that helps students work with these concepts.
I call it the "Whole-Part-Equality Principle" (the WPE Principle). But if I have enough air in my lungs, I sometimes like to call it the: "Peter-Paul-Pennies-in-the-Pocket Principle."
Here's how it works. We know two people: Peter and Paul. Peter and Paul possess no money except pennies, and they transport their pennies only in their right and left pants pockets (if anyone can propose a way to pack this story with even more p's, please let me know).
I almost forgot … we have three additional facts:
1) The number of pennies Peter transports equals the number of pennies that Paul transports. (i.e., Peter and Paul have the same total number of pennies!)
2) Peter and Paul each have three pennies in their right pants pockets.
3) Peter and Paul carry pennies NOWHERE but in their pants pockets.
QUESTION: What can we conclude about the number of pennies that Peter and Paul have in their left pants pockets?
ANSWER: While we don't know how many pennies Peter and Paul are transporting in their left pants pockets (it could be any number, right?), it is nevertheless obvious that they must have the same number of pennies in their left pants pockets.
WHY? View it like this … If the wholes are the same (the total number of pennies that Peter and Paul each has), and if one of two key parts are the same (the number of pennies that Peter and Paul have in their right pants pockets), then the other parts must also be equal (the number of pennies they have in their left pants pockets).
Why am I bringing this up? To point to an important principle.
Here’s how I would summarize this principle: Suppose you have two quantities that are equal: Q1 and Q 2. Suppose further that each quantity has two parts, A and B. Finally suppose you know that Part A of Q1 is equal to Part A of Q2. With this being given, you can be 100% certain that Part B of Q1 is equal to Part 2 of Q2.
This principle can be used to solve many problems in math, including problems involving exponents and logs. A few examples …
EXAMPLE 1: Suppose you’re given this equation: log x = log 7.2. What can we conclude? Well, the wholes are equal (meaning the left and right sides of this equation), and the bases are equal (logs are always base 10 unless another base is stated), therefore the remaining parts, the 'arguments,' also must be equal. The 'argument' is the term after the word 'log.' In this equation the arguments are x and 7.2. Using the principle, we conclude that x and 7.2 must be equal: x = 7.2.
EXAMPLE 2: Suppose you have the equation: log 2^x = log 16. Again, the wholes are equal, and the logs have the same base, so the arguments must be equal. That means that 2^x = 16. Since 2^4 = 16, x = 4, and that's the answer.
EXAMPLE 3: Suppose you have the equation: a^log x = a^log 12.9. Since the wholes (the left and right sides of the equation) are equal, and since the bases are equal since they are both 'a,' therefore the only remaining parts, the exponents, must also be equal. So this means that log x = log 12.9. Following the same logic as we used in Examples 1 and 2, this means that x = 12.9.
This same principle can also be used in earlier math problems. For example, if you know that: x + a = x + 12, can you conclude anything? Well yes. Since the left side of the equation equals the right side, and since the x’s must be equal because they are the same variable, it stands to reason that the remaining parts, the “a” and the 12.
Once your children learn this principle, encourage them to use it, not only in math, but also in real-life situations, such as the situation involving the pennies. It’s fun, and it builds logical thinking skills.
Josh Rappaport is the author of the Algebra Survival Guide and Workbook, which together comprise an award-winning program that makes algebra do-able! Josh also is the author of PreAlgebra Blastoff!, an engaging, hands-on approach to working with integers. All of Josh's books, published by Singing Turtle Press, are available on Amazon.com
