Algebra 2 is a completely different beast of a course compared to algebra 1, especially at a school like Debakey High. The pace is faster and the math problems require more practice than understanding. One can go through algebra 1 and be a good listener and possibly get away with an A without much practice, especially if all the tests are multiple choices. [This strategy is flawed because the student will not retain much algebra 1 and usually that surfaces when the student takes algebra 2].
Anyway, in algebra 2, if you don't practice, you can forget about the A.
A student's success in this class depends on two major things: practice and know the graphs
1) practice
The best kind of practice is one without other's help. Every time someone helps you, it's a good and a bad thing. It's good that you're progressing in your understanding, but it's bad that someone now has taken away a practice problem from you. And to fix this, you now have to go find/make a new problem to work on. If you don't makeup for the taken practice, then it might come back and haunt you down the road.
Understanding math isn't the same as being able to do it.
Students who study math only by reading solutions from the textbook, the teacher, a classmate or by watching videos, are not getting the practice they need. This tactic can work in middle school math where everything is modeled. However, in high school, the math classes are more abstract and require critical thinking. Students are required to apply their knowledge, not just model problems done in homework or in class. To have enough knowledge to apply it, practice is necessary. Why? When you practice with math, you discover the right things to do and the wrong things to do. You're exposed to all kinds of problems. And struggling is part of the learning process (to fully understand). If you give up and just ask the teacher or a classmate how to do a problem, you may gain the understanding, but never the ability to solve problems on your own. You may be able to solve that particular problem the next time you see it, but what if it's modified? You have a higher chance of solving never-before-seen problems when you work on your problem solving skills rather than understanding of a particular problem.
Now, I'm not saying to never ask me questions about a problem. But your philosophy should always be this: every time you see a problem, you need to think about it first, before you ask about it. Too often, students stop me at every step when I'm working out a 4 step problem. The right approach is to think about why the next step happens, then rethink why not, then rethink again if necessary. In other words, if you don't see it right away, keep thinking. Because if you don't see it right away and you get the next step fed to you, the end result is that you were able to follow the steps [but not able to learn to solve a problem--which is really what all math classes are all about! Solving problems!] Have you ever noticed I walk you guys through a problem completely the first time I cover it, but all the subsequent problems, I want you to think and rethink about the steps more by yourselves? There's a reason for that!
2) know the graphs
Typically students know how to plot points using (x, y). They know the relation between x and y, mathematically. They know to plug in x values to get y values, and can make a x-y table. They know how to solve for y. They know their definitions of function, domain, range, etc. They treat all of these as different problems.
To be a true master student in algebra 2, you should approach every problem in a holistic way. It pays to link all these "separate" problems and see all of them as part of one large problem. What is that large problem?
If you are given a problem in algebra 2, whether it's a graphing problem or a problem that deals with expressions of x and y mixed, you should try to see the other part of the larger problem at the same time. In other words, if it's a graphing problem, then TRY to figure out the equation that yields the graph. If it's an equation or inequality of variables, then TRY to visualize the corresponding graph. The key word here is "try", as most of the time you're not going to be able to do it. But by trying, you can at least gain insight into the original problem. That's where the advantage comes in! See the following comparison of two different students' approach to the same problem in quiz 2 of cycle 1:
x^2 + y^2 = 25. Is this relationship a function? Find the domain and range:
Student 1 sees the problem:
x^2 + y^2 = 25.
To decide if it's a function, the student starts with the question: does each input goes to exactly one output?
If x=0, then student can quickly conclude y=5 and y=-5. That's one input that goes to 2 outputs. It's not a function!
To find domain and range, the student is not going to have fun here...and may take up to 5 minutes just to find the wrong answer. [Luckily on quiz 3, you were asked to find domain/range only if something's a function].
Student 2 sees the problem and figures out the graph:
x^2 + y^2 = 25 is a circle (students should have learned this in geometry by the way). If the student was given a graph of a circle from the start, that student would've said "not a function" within 5 seconds, since it fails the vertical line test. The graph is to the right. From the graph, the student can also find the domain and range within 30 seconds.
Domain: -5 < x < 5 inclusive
Range: -5 < y < 5 inclusive
Quick check on the graph against the function defintion:
If x=0, y=5 and y=-5 FAILS FUNCTION DEFINITION
FINAL RESULT: Student 2 finishes the problem under 1 minute.
-----------------------------------------------------------------------------------
Everything that you will learn this year, will be some offspring function of some parent function! Know your parent transformations well (both by graph and by equation), that's the key!
If you keep this in mind throughout the entire algebra 2 year, you have a good shot at solving a problem someone just randomly toss at you. It is something we teachers use all the time whenever a random student come in during tutorial and ask some random math question. We see each math question as an equational AND a graphical problem. When we get stuck, often times we think about how the graph looks. From there, we figure out what to do next.
If you can see the connection between parent functions and equations on every single problem, you will find precalculus class easier, AP calculus easier, and college math classes easier.
Anyway, in algebra 2, if you don't practice, you can forget about the A.
A student's success in this class depends on two major things: practice and know the graphs
1) practice
The best kind of practice is one without other's help. Every time someone helps you, it's a good and a bad thing. It's good that you're progressing in your understanding, but it's bad that someone now has taken away a practice problem from you. And to fix this, you now have to go find/make a new problem to work on. If you don't makeup for the taken practice, then it might come back and haunt you down the road.
Understanding math isn't the same as being able to do it.
Students who study math only by reading solutions from the textbook, the teacher, a classmate or by watching videos, are not getting the practice they need. This tactic can work in middle school math where everything is modeled. However, in high school, the math classes are more abstract and require critical thinking. Students are required to apply their knowledge, not just model problems done in homework or in class. To have enough knowledge to apply it, practice is necessary. Why? When you practice with math, you discover the right things to do and the wrong things to do. You're exposed to all kinds of problems. And struggling is part of the learning process (to fully understand). If you give up and just ask the teacher or a classmate how to do a problem, you may gain the understanding, but never the ability to solve problems on your own. You may be able to solve that particular problem the next time you see it, but what if it's modified? You have a higher chance of solving never-before-seen problems when you work on your problem solving skills rather than understanding of a particular problem.
Now, I'm not saying to never ask me questions about a problem. But your philosophy should always be this: every time you see a problem, you need to think about it first, before you ask about it. Too often, students stop me at every step when I'm working out a 4 step problem. The right approach is to think about why the next step happens, then rethink why not, then rethink again if necessary. In other words, if you don't see it right away, keep thinking. Because if you don't see it right away and you get the next step fed to you, the end result is that you were able to follow the steps [but not able to learn to solve a problem--which is really what all math classes are all about! Solving problems!] Have you ever noticed I walk you guys through a problem completely the first time I cover it, but all the subsequent problems, I want you to think and rethink about the steps more by yourselves? There's a reason for that!
2) know the graphs
Typically students know how to plot points using (x, y). They know the relation between x and y, mathematically. They know to plug in x values to get y values, and can make a x-y table. They know how to solve for y. They know their definitions of function, domain, range, etc. They treat all of these as different problems.
To be a true master student in algebra 2, you should approach every problem in a holistic way. It pays to link all these "separate" problems and see all of them as part of one large problem. What is that large problem?
- You have y that depends on x. When you plot the relationship, it generates a graph. Sometimes, there are certain parts of the graph that are "special": points that do not exist (not part of the domain), pointy parts, jumps, gaps, holes, steps, vertex, max/min, and inflection points. Sometimes, there are end behaviors toward the infinities. Sometimes it cycles or oscillates. Sometimes, there are boundaries on the graph: at certain points, or at the infinities. Sometimes it fails the vertical line test.
If you are given a problem in algebra 2, whether it's a graphing problem or a problem that deals with expressions of x and y mixed, you should try to see the other part of the larger problem at the same time. In other words, if it's a graphing problem, then TRY to figure out the equation that yields the graph. If it's an equation or inequality of variables, then TRY to visualize the corresponding graph. The key word here is "try", as most of the time you're not going to be able to do it. But by trying, you can at least gain insight into the original problem. That's where the advantage comes in! See the following comparison of two different students' approach to the same problem in quiz 2 of cycle 1:
x^2 + y^2 = 25. Is this relationship a function? Find the domain and range:
Student 1 sees the problem:
x^2 + y^2 = 25.
To decide if it's a function, the student starts with the question: does each input goes to exactly one output?
If x=0, then student can quickly conclude y=5 and y=-5. That's one input that goes to 2 outputs. It's not a function!
To find domain and range, the student is not going to have fun here...and may take up to 5 minutes just to find the wrong answer. [Luckily on quiz 3, you were asked to find domain/range only if something's a function].
Student 2 sees the problem and figures out the graph:
x^2 + y^2 = 25 is a circle (students should have learned this in geometry by the way). If the student was given a graph of a circle from the start, that student would've said "not a function" within 5 seconds, since it fails the vertical line test. The graph is to the right. From the graph, the student can also find the domain and range within 30 seconds.
Domain: -5 < x < 5 inclusive
Range: -5 < y < 5 inclusive
Quick check on the graph against the function defintion:
If x=0, y=5 and y=-5 FAILS FUNCTION DEFINITION
FINAL RESULT: Student 2 finishes the problem under 1 minute.
-----------------------------------------------------------------------------------
Everything that you will learn this year, will be some offspring function of some parent function! Know your parent transformations well (both by graph and by equation), that's the key!
If you keep this in mind throughout the entire algebra 2 year, you have a good shot at solving a problem someone just randomly toss at you. It is something we teachers use all the time whenever a random student come in during tutorial and ask some random math question. We see each math question as an equational AND a graphical problem. When we get stuck, often times we think about how the graph looks. From there, we figure out what to do next.
If you can see the connection between parent functions and equations on every single problem, you will find precalculus class easier, AP calculus easier, and college math classes easier.