This post is really for a friend of mine who is very much math aware and capable, but does not teach math. He has twin sons who are absolutely amazeballs smart, and through talking with him one day I espoused my approach to algebra. He was intrigued and wanted more info. This is my attempt at more info. I figured that others may find value in it, so I am publically posting it for all to see (and read, hopefully.) Please rip it to shreds if you feel I am in error or made a mistake. I want to do better, and I can’t if I am doing something wrong.
So let me begin with where this came from. At the NCTM Las Vegas regional conference in 2013 I was introduced by an elementary teacher to what she called 13 Rules that Expire.
There are some definite Algebra concepts on here like number 2) Use keywords to solve word problems and number 3) you can’t take a bigger number from a smaller number or what really kills me is number 8) multiply everything inside the parenthesis by the number outside the parenthesis.
Yea, right. Try that with f(x)=2x+5.
And from there we have Nix the Tricks. A more rigorous treatment of the stupid things we as math teachers do to mess up learners in the guise of teaching them to get an answer instead of understand the mathematics.
So I sat back and tried to come up with some rules that did not expire. Some essential rules that always work, that always build understanding and not destroy it. I ended up with 3.
1. When solving an equation, you can do absolutely anything you want, as long as you do it to all terms of the equation.
2. When working with expressions, you can only change it by adding or subtracting 0 (by using additive inverses), or when solving equations, you create zeros by adding or subtracting 0 (by using additive inverses).
3. When working with expressions, you can only change it by multiplying by 1 (in any form), or when solving equations, you create 1 by using the inverse functions/operations.
That’s it. When I am teaching math, I stress the idea of creating a zero or using a zero. When demanding written explanations, I demand they say that 5+-5=0 in their explanation. So, a nice short example.
Find the function that is the inverse of y = 1/3x – 4
Work Why did I do what I did?
x = 1/3y – 4 Because the first step in finding the inverse is switching the x & y
+ 4 +4 Add 4 to both sides because –4 + 4 = 0 (additive inverses = identity)
x + 4 = 1/3y result
3(x+4) = 3(1/3)y Mult by 3 because 3(1/3) = 1, (multiplicative inverses = identity)
3(x+4) = y finished, but should check it by …….
So there is an example of the work I require. I want them to be using the language of inverses and identity. Why does the square and the square root cancel each other? Because the exponent of 1/2 (the square root) and the exponent of 2 (the square) when multiplied equals 1, the identity.
Along with this, there are some forbidden words and phrases in my classroom. One is above, “Cancel”. I do not allow my learners to use it. At all. Ever.
Why? Because I have seen all of the following described by the word: -5 + 5 cancels to make 0. 3/3 cancels to make 1. Log10^4 cancels to make 4, sqrt(5^2) cancels to make 5 and on and on and on.
If those all “cancel” then that word means nothing, and it does not mean anything at all. It is just a word used to hid the mathematical knowledge of inverses and identities.
Another phrase that I will not allow is the common answer to the question, “Why did we add 4 in the first step above?” They typical answer I get is, “Because we want the y by itself.”
Huh? That is not why we added 4. We could do anything we want in the world. We could have added 6, or subtracted 3, or taken both left and right side and made them exponents with a base of 7.5. We added 4 because –4 + 4 = 0. That zero is important. I also write it down. I think most teachers do not. Or they draw a line through the –4 and 4. What does that line mean? I tell them it means a 1, and did we make a one? No, we made a zero.
It takes some time to unlearn the bad habits, but this is a Nix the Tricks kind of endeavor. In the end, I think I am helping the learners understand math better and more deeply. At least, I hope I am. Only time will tell.
Glenn,
Thank you.
We are at an age with the children where they are actively using the subject-driven vocabulary of the topics they are learning.
I need them to start thinking mathematically vs operationally. Or, better still, I need them to start correctly using the language of the discpline to describe how they are solving the problem.
My desire for correct, accurate terminology is not limited to mathematics, mind you. My “expertise” – biology and biochemistry – requires specific syntax when describing everything from the top and bottom of a frog (dorsal and ventral) to orientation of atoms in an organic molecule (cis- and trans- isomers; http://en.wikipedia.org/wiki/Cis%E2%80%93trans_isomerism).
I am more than happy to use euphamism, imagery, sock puppets and cartooms to get them to understand difficult subjects. But ultimately, introducing the young ones to the specific language of the subject (mathematics, philosophy, theater, art, sciences, etc) furthers their knowledge of the subject.
Thanks again for putting this to the written, or at least typed, page.
Regards.
Hi Glenn,
Great post, as usual. I’ve always liked showing the steps as an algebraic proof rather than just operations to remember. Maybe just a slight change … Should “Find the inverse function to y = 1/3x – 4” be “Find the inverse function to x = 1/3y – 4”?
Thanks,
Laurie
No, I wanted to say, find the function that is the inverse of y =. I will change the wording to be more precise.
Thank you, I appreciate the feedback.
I really like your rules, especially the last two, as they do a great job clearly explaining to me how equations and expressions are both using identities to accomplish something, just in different ways.
Depending on how you interpret the first rule (When solving an equation, you can do absolutely anything you want, as long as you do it to all terms of the equation.), I worry that students will try to turn 1 + 2 = 3 into 1^2 + 2^2 = 3^2. Same applies to roots, logs, and trig functions. Is it too generic to say you can do anything you want to the entire side of an equation only if you do the same thing to the entire other side?
I think you are right Andy, that is a better way to phrase it and keeps from having the mistakes you pointed out. I will make that change to the things I have written down for my classroom immediately! Thank you!
I’m curious if I lose anything in this rewording:
Expression rules (problems with NO =, >, , <, etc):
3. You can do anything you want to the entire side of an equation only if you do the same thing to the entire other side. Usually this means using the inverse operation of each existing operation around the variable you are trying to isolate.
Part of the last comment was cut off:
The comment kept failing, so feel free to delete, but I was wondering if you think this rewording accurately says everything you were going for without leaving anything out:
Expression rules (problems with NO =, >, <, etc): 1. You can change it by adding or subtracting 0 (by using additive inverses). 2. You can change it by multiplying by 1 (in any form). Inequality / Equation rule (problems with =, >, <, etc): 3. You can do anything you want to the entire side of an equation only if you do the same thing to the entire other side. Usually this means using the inverse operation of each existing operation around the variable you are trying to isolate.
Pingback: First day! | Insert Clever Math Pun Here
Pingback: 3 Essential Rules of Math for Solving Equations | Math = Love