Algebra 2 Topics for Not-So-Strong Math Students?

[yeongil]

What are the MUST COVER topics in Algebra 2?

I said this in the "Should calculus be taught in high school?" thread:

I mentioned in mathwonk's "Teaching Calculus Today in College" thread about some of the incredible errors that my precalculus students make, and these errors were in algebra. (I'm wondering if it's because our high school math books these days are so packed with material that in the teachers' attempts to cover as much as possible students aren't getting enough practice in many concepts.)

I've mentioned before that I'm a HS teacher (math and music) in a Catholic all-girls school in the US. This school year I taught Honors Algebra 1 and Honors Precalculus. Next year I'm dropping most of the music and will be essentially teaching Math full time, so I've been given 2 sections of Algebra 2, which I've not taught before. Here is the list of chapters and sections in our book, and as you can see, it's very long:

Chap. 1: Equations and Inequalities
1.1 Apply Properties of Real Numbers
1.2 Evaluate and Simplify Algebraic Expressions
1.3 Solve Linear Equations
1.4 Rewrite Formulas and Equations
1.5 Use Problem Solving Strategies and Models
1.6 Solve Linear Inequalities
1.7 Solve Absolute Value Equations and Inequalities

Chap. 2: Linear Equations and Functions
2.1 Represent Relations and Functions
2.2 Find Slope and Rate of Change
2.3 Graph Equations of Lines
2.4 Write Equations of Lines
2.5 Model Direct Variation
2.6 Draw Scatter Plots and Best-Fitting Lines
2.7 Use Absolute Value Functions and Transformations
2.8 Graph Linear Inequalities in Two Variables

Chap. 3: Linear Systems and Matrices
3.1 Solve Linear Systems by Graphing
3.2 Solve Linear Systems Algebraically
3.3 Graph Systems of Linear Inequalities
3.4 Solve Systems of Linear Equations in Three Variables
3.5 Perform Basic Matrix Operations
3.6 Multiply Matrices
3.7 Evaluate Determinants and Apply Cramer's Rule
3.8 Use Inverse Matrices to Solve Linear Systems

Chap. 4: Quadratic Functions and Factoring
4.1 Graph Quadratic Functions in Standard Form
4.2 Graph Quadratic Functions in Vertex or Intercept Form
4.3 Solve x2 + bx + c = 0 by Factoring
4.4 Solve ax2 + bx + c = 0 by Factoring
4.5 Solve Quadratic Equations by Finding Square Roots
4.6 Perform Operations with Complex Numbers
4.7 Complete the Square
4.8 Use the Quadratic Formula and the Discriminant
4.9 Graph and Solve Quadratic Inequalities
4.10 Write Quadratic Functions and Models

Chap. 5: Polynomials and Polynomial Functions
5.1 Use Properties of Exponents
5.2 Evaluate and Graph Polynomial Functions
5.3 Add, Subtract, and Multiply Polynomials
5.4 Factor and Solve Polynomial Equations
5.5 Apply the Remainder and Factor Theorems
5.6 Find Rational Zeros
5.7 Apply the Fundamental Theorem of Algebra
5.8 Analyze Graphs of Polynomial Functions
5.9 Write Polynomial Functions and Models

Chap. 6: Rational Exponents and Radical Functions
6.1 Evaluate nth Roots and Use Rational Exponents
6.2 Apply Properties of Rational Exponents
6.3 Perform Function Operations and Composition
6.4 Use Inverse Functions
6.5 Graph Square Root and Cube Root Functions
6.6 Solve Radical Equations

Chap. 7: Exponential and Logarithmic Functions
7.1 Graph Exponential Growth Functions
7.2 Graph Exponential Decay Functions
7.3 Use Functions Involving e
7.4 Evaluate Logarithms and Graph Logarithmic Functions
7.5 Apply Properties of Logarithms
7.6 Solve Exponential and Logarithmic Equations
7.7 Write and Apply Exponential and Power Functions

Chap. 8: Rational Functions
8.1 Model Inverse and Joint Variation
8.2 Graph Simple Rational Functions
8.3 Graph General Rational Functions
8.4 Multiply and Divide Rational Expressions
8.5 Add and Subtract Rational Expressions
8.6 Solve Rational Equations

Chap. 9: Quadratic Relations and Conic Sections
9.1 Apply the Distance and Midpoint Formulas
9.2 Graph and Write Equations of Parabolas
9.3 Graph and Write Equations of Circles
9.4 Graph and Write Equations of Ellipses
9.5 Graph and Write Equations of Hyperbolas
9.6 Translate and Classify Conic Sections
9.7 Solve Quadratic Systems

Chap. 10: Counting Methods and Probability
10.1 Apply the Counting Principle and Permutations
10.2 Use Combinations and the Binomial Theorem
10.3 Define and Use Probability
10.4 Find Probabilities of Disjoint and Overlapping Events
10.5 Find Probabilities of Independent and Dependent Events
10.6 Construct and Interpret Binomial Distributions

Chap. 11: Data Analysis and Statistics
11.1 Find Measures of Central Tendency and Dispersion
11.2 Apply Transformations to Data
11.3 Use Normal Distributions
11.4 Select and Draw Conclusions from Samples
11.5 Choose the Best Model for Two-Variable Data

Chap. 12: Sequences and Series
12.1 Define and Use Sequences and Series
12.2 Analyze Arithmetic Sequences and Series
12.3 Analyze Geometic Sequences and Series
12.4 Find Sums of Infinite Geometric Series
12.5 Use Recursive Rules with Sequences and Functions

Chap. 13: Trigonometric Ratios and Functions
Chap. 14: Trigonometric Graphs, Identities, and Equations

(I know that I'm not going to cover Chapters 13-14, so I didn't list the sections.)

Unfortunately, our school does not attract the strongest math students. In past years our students only got through the first six (!) chapters. I'm determined to cover more, maybe the first eight chapters. But in case I'm pressed for time, are there sections within the first eight chapters that could be skipped? Assume that the students who are interested in majoring in math/science in college would be taking the Honors Algebra 2 course, which covers chapters 1-9 and bits of chapters 10 & 12. Note also that our students use their graphing calculators a lot (too much, in my opinion). At the moment, the math sequence is Algebra 1-Algebra 2-Geometry, though I've submitted a proposal to change it to Algebra 1-Geometry-Algebra 2, though, if approved, this won't take affect until the following school year. TIA.01

 

[symbolipoint]

Almost all of that book's material is the essential content for Algebra 2. Try a careful comparison between that book and your school's PreCalculus book. The Algebra portion of PreCalculus extends beyond the content of Algebra 2, but contains all of Algebra 2.

What you choose to include in the course as you design it is your choice, but a guide on how may be like this: Chapter 1 and parts of chapter 2 are a review of some content from Algebra 1. Any review of previous material would likely be beneficial since students forget skills during weeks of recess. As said, you will want to include almost the whole book. You could go light in chapter 5 about polynomial functions (but go heavier regarding quadratic functions). Go light in chapter 8 Rational Functions. Maybe go light in chapter 10 Counting & Probablility, and light in chapter 11 Data Analysis & Statistics. Omit chapters 13 and 14 because Trigonometry is either for a dedicated course or for the PreCalculus course.

Another source of guidance would be your state's or district's content standards for Algebra 2.

 

[Moonbear]

Based on only the titles, I think you could skip parts of Chapter 3. I didn't see matrices until college, and it wasn't any hindrance to my learning for that to be the first time I saw them (I was a bio major thinking I was going to double major in chemistry at the time, so linear algebra was required for the chem major, not bio). So, assuming these not-so-strong math students will not be going into engineering or comp sci or physics majors in college, they don't need to learn matrices if you need to cut things out.

I would suggest trying to fit in at least part of Chapter 10 if you can. I think a general understanding of probability is important in life. The kids who are bad in math are the ones who might think they can beat the odds gambling in casinos or playing the lottery. And, if you teach it near the end of the term, you can have a lot of fun with it at a time when the students are more interested in staring out the windows wanting to be out in the sun, and possibly hold their interest for those last couple of weeks with rolling dice and card games.

 

[qntty]

I think a lot of students continue to make algebraic mistakes because the mistakes can be missed easily and so they aren't pointed out enough for the student to stop making the mistakes. To help students to realize what operations are erroneous it might be helpful to put up a poster of the most common ones. a list is here http://www.math.vanderbilt.edu/~schectex/commerrs/#Signs

 

[Pinu7]

First of all, it seems like quite the advanced book. You can probably go through most of the book , and the students will understand if you give A LOT of homework. Practice makes purpose and you can cut down the time spent on a subject. Make sure to review the HW each day.

Also, calculators are a burden for learning, especially this early. Perhaps this is why the students have poor math abilities.

Take away the graphing calculators(which they are just going to use to play with) and ban ANY calculators on the tests.

3.5-3.8,5.6 can be skipped
Chapter 11 should be abandoned
Chapter 10&12 should be compromised a bit.

Also, teach trigonometry! It is good to introduce it early.

 

[JazzFusion]

I am mainly with Pinu on this one (except I would keep 5.6).

Skip matrices. Chapter 11 is better handled in a Science lab course (if at all).

As far as trigonometry - try to cover the basics of right-angle trigonometry at least. It will be impossible for your students to succeed in any physics course without it.

 

[snipez90]

I think if you could cover the first 8 chapters, give or take chapter 3, then that would be really strong. Conic problems can force you to think, but they don't necessarily improve geometric intuition or algebraic fortitude. I don't see much coherency of subject matter after chapter 8 but some of those topics may be worth covering provided that the first 7 or 8 chapters are not compromised.

 

[symbolipoint]

The Chapter 9 Quadratic Relations needs to be included because those are part of the characteristic content of "Intermediate" Algebra, and they lead directly to applied problems. Conic sections also involve Completion of the Square and graphing.

 

[icannotstaysi]

First of all, it seems like quite the advanced book. You can probably go through most of the book , and the students will understand if you give A LOT of homework. Practice makes purpose and you can cut down the time spent on a subject. Make sure to review the HW each day.

Also, calculators are a burden for learning, especially this early. Perhaps this is why the students have poor math abilities.

Take away the graphing calculators(which they are just going to use to play with) and ban ANY calculators on the tests.

3.5-3.8,5.6 can be skipped
Chapter 11 should be abandoned
Chapter 10&12 should be compromised a bit.

Also, teach trigonometry! It is good to introduce it early.

I was searching for something on the internet, and I accidentally came across this thread. Interested, I started reading through the responses, getting more interested as I read. Until I came to the above quoted reply. I had to register just to post my oppinion on this single reply.
You are absolutely WRONG in that way of thinking. As a university student that regrets not paying any attention to the advanced maths course in high school, I feel like I have to give you an idea of how the world actually works, not as 99.99% of the maths teachers work.
Rewind to a few years before. I got 85% and 90% most of the time on tests. Then along came "Integration". "Integration is not difficult, it's actually very easy.It's the reverse of differentiation." said my maths teacher. Great I thought, another topic to get pretty good grades in. All was going well, I (and the rest of the class) was happy with it. Until we hit the 5 part exercises. 50%. Then the 10 part exercises.40%. Then the next thing you know, a whole page for a simple solution to an exercise...15% (yes that is actually true) Why? (hold on to that thought, continued below)
Fast forward to my first uni year. In a maths tutorial I was having a lot of difficulty with finding the determinant of a 6x7 matrix. That's what the exercise was asking for. I raise my hand, the professor comes over, I show him the exercise. Frankly,looking at his expression, I was frightened. "Are you out of your mind?" he barked. I was puzzled. What did I do wrong? "A 6x7 matrix? Use a computer" he replied. The exercise clearly did not ask for a computer solution. When I told him this, he replied "well, people can still try to light a fire by rubbing two pieces of wood together, I prefer to use a match".
A couple days later at another tutorial, on a different subject, we were having a discussion on why students aren't doing well in maths. The professors said a little story on the subject, based on his own learning years, what amazed me was, and I quote: "he spent 2 hours explaining what that was and how to get it. The *beep* did NOT mention anything on the use of that, that is how to use it. I could be on my deathbed, cursing at the 2 hours wasted to learn something I did not use my entire life. I could simply have a computer do that math for me. Imagine, on my deathbed, cursing at those 2 hours!"
Yet again, a couple days later, one of my maths professors said something along the lines of "I hate mathematicians. I really hate them. They have magical powers. They can see things I cannot see. The only problem is that they cannot explain to me what they see". Yes he is my math professor.

If you have read through my entire post, surely you should come to a logical, non math's teacher conclusion.1x1=2 . A book of 62 pages was written on this. Imagine what the benefit of the world was if the poor guy did not spend a few days writting that book. The cure for cancer, the solution to humanity's hunger problem could be found.

Bottom line: Use a calculator next time you have to find what 348249824792x12387129312783 is. Don't spend your time solving thousands and thousands of exercises just (in a math's teacher's view) "get the idea of how it works". You take the second one and multiply it by the first. End of story.
My personal oppinion (and quite frankly I strongly believe the oppinion of the next 20 generations) is that there is no need to constantly re-invent the wheel. If a person, or group of persons spent his/their entire lives writting a computer program to do maths no human brain can do in his lifetime, there is absolutely NO reason why every other human after them should spend his entire lifetime reworking on that mathematical solution.
Students are not doing well in maths because they cannot see the use of spending hours to learn something. I certainly don't see the use of it and I'm doing an engineering degree.
Don't get me wrong. People can come to the conclusion that I think that maths should be banned from schools. That is not what I'm trying to put across here. What's I'm trying to say is that if math books were a LOT smaller, covering less topics, instead of rushing through 30 chapters a year, of which only about 4 stay in student's brains, students would get all As in their maths courses. If a student is really really really interested in maths, he can follow them in his uni degree. Until then, a calculator can show him what 1x1 is.

 

[thrill3rnit3]

^^ then we should just teach students how to use the TI-84 and they should be set for life

If a student is really really really interested in maths, he can follow them in his uni degree. Until then, a calculator can show him what 1x1 is.

Too bad knowing how to plug in 1x1 in the calculator is not going to get you a math degree in college.

 

[niklaus]

Fast forward to my first uni year. In a maths tutorial I was having a lot of difficulty with finding the determinant of a 6x7 matrix. That's what the exercise was asking for. I raise my hand, the professor comes over, I show him the exercise. Frankly,looking at his expression, I was frightened. "Are you out of your mind?" he barked. I was puzzled. What did I do wrong? "A 6x7 matrix? Use a computer" he replied. The exercise clearly did not ask for a computer solution. When I told him this, he replied "well, people can still try to light a fire by rubbing two pieces of wood together, I prefer to use a match"

While I agree that computing determinants of large matrices is a big waste of time, maybe the excercise in question wanted to check if you understood that determinants are only defined for square matrices? Also, if your professor needs a computer to figure that out, maybe you should switch classes.

 

[iRaid]

The first few chapters are basically algebra 1 and the matrices aren't really needed because of the graphing calculators out there.