This year, my math department has changed radically from last year. Among the many changes is a revamping of the Algebra II curriculum. Not only are we adding new topics, but we’ve removed a whole bunch of what we traditionally taught — pushing it to precalculus, I suppose. The course is totally reordered. For our first quarter, we are covering:
Unit 1: Number Lines, Intervals, and Sets
1.Set notation and interval notation (along with union, intersection, and subset)
2.Linear inequalities –graph on a number line
4.Absolute value inequalities
Unit II: Algebraic Manipulation: Rational Expressions and Exponents
1.Factoring two, three, and four term polynomials
2.Review of basic exponent rules and simplification
3.Polynomial addition, subtraction, multiplication, and division
4.Rational expression addition, subtraction, multiplication, and division
Unit III: Radical Equations
1.Review properties of radicals (integer exponents)
2.Simplifying radicals with exponents under them
3.Solving radical equations
This seems very hodge-podgy to me, now that we’re going through it for the first time. One day we’re talking about sets and subsets, the next how to solve . We are doing a lot without the textbook (which I’m fine with), but every so often we’re turning to the middle of the textbook to cover a topic (e.g. compound inequalities and absolute value inequalities come at the end of chapter 3). It just feels fractured. Why am I concerned?
I don’t know. I can’t articulate it. It’s just a whole bunch of thoughts running through my head…
We haven’t started graphing, and that makes me nervous. I’m now feeling like we’ve cut out too much of the curriculum in quarters three and four to cover this stuff. I don’t see the natural flow in this beginning material, as I saw the natural flow in our old curriculum (functions and lines; quadratics; polynomials; rational functions; exponentials and logarithms; trigonometry).
However I’m hoping that the past month and a half — which seems like a lot of vamping before we get to the good stuff — is worth it, because it does force students to practice their basic algebra skills.
If there are any other Algebra II teachers out there: how do you start the course? And do you like it (equivalently put: does it work)?
In non-AP calculus — since I get to cover what I like at the pace I like — I focused the past month and a half specifically on the skills that my students last year had difficulty with: visualizing basic functions including logs and exponents, solving logarithmic and exponential equations, solving trigonometric equations, and knowing trig values at special angles. This too is totally different than what I did last year, where as a new teacher I just forged through our book. But let me tell you — unlike with Algebra II — I am certain that all this review is going to do some good, because I remembered some of the wounds my students suffered last year, and I am applying extra padding in those same areas to my students this year.
I haven’t taught Alg II (unless pre-calc counts :-) ), but your comment about not having started graphing yet concerns me too. I really think multiple representations (graph, table, equation, and situation) should be embedded throughout each topic. Why can’t graphing be done in the context of solving absolute value inequalities? And factoring polynomials seems like a natural place to talk about the zeros of the graph.
@Jackie: I agree with you, and those are my concerns! We will do what you write about soon. I can’t wait until we start the next two chapters (functions & quadratics) we will be doing lots of graphing. And there we’ll talk about the relationship between the graph / table / equation, and of course we’ll talk about the zeros of the graph.
If you’ve seen my blog, you know that Algebra 2 is my baby this year. I’ve also reordered things a bit, to front-load all of the new numbers and operations, so that we can use them more fluidly later on when solving and graphing (roots, rational exponents, raising to a reciprocal power, logs, absolute value, complex numbers). We are just finishing unit 2, and so far, it seems to be going well. We also haven’t done any graphing yet, but we are about to start our functions unit. After that, each unit will be using multiple representations. Here is my skills syllabus:
Unit 1: Real Numbers
– Real number system
– Nth roots and rational exponents (plus review of zero and negative exponents)
– Definition of logarithm, using Big-L notation (solve for x in any of the 3 locations)
– Solving basic power and exponential equations (when to use roots vs. logs)
– Definition of absolute value (geometric and algebraic)
– Solving absolute value equations
Unit 2: Complex Numbers
– Imaginary number line as 90° rotation of real number line
– Simplifying powers of i
– Plotting complex numbers
– Expanding the real number system Venn Diagram
– Simplifying radicals with negative radicands (focus on square roots only)
– Operations with complex numbers
Unit 3: Functions
– Functions and relations – definitions, multiple representations (table, graph, arrow map, equation, set of ordered pairs)
– Function notation
– Composite functions (3 forms: algebra, tables, graphs)
– Operations on functions (3 forms, no algebraic division)
– Domain and range (interval notation)
– Solving equations and inequalities graphically
– Translating and transforming functions
Unit 4: Systems of Equations and Inequalities
– Graphing standard and slope-intercept form linear functions (focus on finding x- and y-intercepts)
– Graphing absolute value functions
– Solving 2 x 2 systems of equations using elimination (including word problems)
– Graphing linear inequalities and systems of inequalities (focus on determining if a point is a solution)
– Solving 3 x 3 systems of equations using elimination
Unit 5: Polynomials
– Classification by degree and number of terms
– Polynomial long division
– Generate 2nd and 3rd degree polynomial graphs by piecewise multiplication of linear graphs
– End behavior
– Graphing factored polynomials
– Factoring 2nd and 3rd degree polynomials including difference of squares, sum/difference of cubes, trinomial factoring, factoring by grouping
– Use area/volume problems for factoring/multiplication practice
Unit 6: Quadratics
– Factoring and graphing quadratic functions
– Solving quadratic equations by factoring and with the quadratic formula
– Graphing quadratic functions in vertex form
– Solving to find zeros in vertex form by working backwards
– Locating maximum/minimum and zeros
– Translating vertex form quadratic functions
– Basic completing the square
Unit 7: Logarithms and Exponentials
– Exponential growth and decay
– Using logarithms to solve exponential equations
– Properties of logarithms
Unit 8: Rationals
– Property of exponents review
– Simplifying, multiplying, and dividing rationals (monomials only)
– Simplifying, multiplying, and dividing rationals (polynomials included)
– Generate hyperbola graphs by piecewise division of linear graphs
– Understanding asymptotic behavior – vertical and horizontal
– Graphing rational functions
– Domain and range of rational functions
Unit 9: Inverses
– Inverses of linear, quadratic, exponential, and rational functions
This plan covers 50 out of 65 benchmark questions (77%). It leaves out conics, combinatorics, probability & statistics, and sequences & series topics (15 of 65).
Students should be able to do 39 out of the 48 released questions (81%).
Sam – The start of your course is very similar to mine. This is the first time I’m teaching it, so I really don’t have much to compare to. I chose to adapt another (successful) teacher’s notes and assignments (ok, at times copy them verbatim) because next year we start a new curriculum and will have to reinvent the wheel anyway.
I would just suggest that when you get into functions and quadratics, it will be a great opportunity to bring back the old stuff, keep folding the new stuff into what you’ve already done, help them make the connections.
@Dan: Thanks! I have been checking out your blog aperiodically and love looking at your pdfs. I’m really intrigued that you introduce logs so early. Does it work? How do you motivate them? (I am assuming you avoid graphing them until Unit 7.)
@Kate: It’s nice to to know that you’re doing something similar.
I’m so intrigued by how different Alg II curricula are — in terms of topics and ordering — as well the real freedom there is in deciding where and how to start the course. I guess coming off of Geometry, instead of Algebra I, we get that freedom.
(…but with freedom comes great responsibility…)
The idea to introduce logs so early is a new one I’m trying this year. With the normal method, I’ve found that students get overwhelmed with the concept, and it all falls apart as soon as you start working with log properties. So I decided to front load it this year; that way, we can review it periodically throughout the year, and when we get to the log unit, it will be easier to manage the hard stuff (graphing, solving, using properties of logs). At this point of the year, I’ve been focusing on log as operation, as something you can do to both sides of an equation, just like a root. And, in fact, I’ve been trying to draw lots of parallels between logs and roots. Using the “big L” notation has also facilitated this, because it is so similar to the radical symbol. So, right now, my students can solve basic exponential equations in the form a(b)^x + c = d.
Whether or not this will work is still unknown. They have been doing well on the skills tests, but those are things they can retake. I am giving the first midterm tomorrow, where all of unit 1 and 2 comes together, so we’ll see how they do.