Today was a good day in Algebra 1–my students are on the way to factoring simple trinomials of the form x2+bx+c. Students did an incredible amount of math today, much more than normal. They seem to like it when I prepare a series of worksheets and they get to work at their own pace through them. We ran out of time so not all of them were proficient by the end of class, but we will definitely work on it more next Monday.
But, learning about factoring was not my original plan for today’s Algebra 1 class. I had originally planned to teach students how to find the point of intersection between two lines in a plane (algebraically and graphically). I had to scuttle my lesson plan because I realized just how hopelessly behind we are. My students are probably going to do very badly on the Algebra 1 California Standards Test (CST). There is just so much content in Algebra 1, and there is not enough time to get to it all. Even I feel overwhelmed about how much content there is to learn in Algebra 1.
Here is what we’ve learned so far in Algebra 1 this year:
- arithmetic operations on positive and negative numbers
- arithmetic commutative, associative, distributive properties (Standard 1.0)
- simplifying an algebraic expression by collecting like terms, and using the distributive property (Standard 4.0)
- substituting values into expressions and evaluate them using correct order of operations
- determining if a point is on a line (Standard 7.0)
- graphing linear functions (Standard 6.0), identifying positive, negative and zero slope
- knowing what the slope of a line means and how to calculate the slope of a line from a graph (but I’m not sure they know how to calculate the slope between two points)
- looking at a table of numbers in a linear pattern and being able to predict terms in the pattern and describe it with words and algebra (sort of related to Standard 7.0)
- making connections between a pattern, a graph, an equation and a table of numbers
- solving linear equations (Standard 5.0)
- expanding the product of binomials, maybe polynomials too (Standard 10.0)
- graphing quadratic functions (Standard 21.0)
- square roots, identifying the perfect squares up from 1 to around 144
- solving quadratic equations using the quadratic formula, but right now the quadratic formula is just something that came out of thin air (Standard 22.0)
- factoring simple trinomials (Standard 11.0)
- and, we also have spent time learning how to answer multiple-choice questions well
Here is what my students have yet to learn to be able to do well on the Algebra 1 CST.
- finding the reciprocal or additive opposite of a number; calculating roots, raising numbers to a fractional powers; understanding and using the rules of exponents (Standard 2.0)
- solve equations and inequalities involving absolute values (Standard 3.0)
- solving multi-step word problems (Standard 5.0)
- sketching the region defined by linear inequality (Standard 6.0)
- understanding the concepts of parallel lines and perpendicular lines and how those slopes are related; finding the equation of a line perpendicular to a given line that passes through a given point (Standard 8.0)
- solving a system of two linear equations in two variables algebraically (Standard 9.0)
- more generalized factoring: finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials (Standard 11.0)
- simplifying rational expressions by factoring (Standard 12.0)
- add, subtract, multiply, and divide rational expressions and functions, meaning that you have to know how to find the common denominator when adding two rational expressions (Standard 13.0)
- solving rate problems, work problems, and percent mixture problems (Standard 15.0)
- understanding the concepts of a relation and a function, determine whether a given relation defines a function, determining domain and range of a function (Standard 16.0, 17.0)
- determining whether a relation defined by a graph, a set of ordered pairs, or a symbolic expression is a function (Standard 18.0)
- using the quadratic formula and being familiar with its proof by completing the square (Standard 19.0)
- knowing that the roots of quadratic functions are its x-intercepts on a graph and being able to find them (Standard 21.0)
- using the quadratic formula or factoring techniques or both to determine whether the graph of a quadratic function will intersect the x-axis in zero, one, or two points (Standard 22.0)
- applying quadratic equations to physical problems, such as the motion of an object under the force of gravity (Standard 23.0)
- knowing the simple aspects of logical arguments: difference between inductive and deductive reasoning, identifying hypothesis and conclusion in a statement, using and recognizing counterexamples (Standard 24.0)
- using properties of the number system to judge the validity of results, to justify each step of a procedure, and to prove or disprove statements (Standard 25.0, not really sure what this standard means)
As I look at the complete list of state standards in Algebra 1, I feel completely overwhelmed. No wonder students find Algebra 1 so difficult! And as I look at the list of topics yet to be learned this year, I feel totally depressed. There is no way we can get to all of these topics by the end of the year, let alone in the remaining two weeks before my students take the CST. All I can do at this point is just try to find the topics that appear most frequently the CST and tackle those. That is why today I skipped finding the point of intersection between two lines–since the CST is a multiple-choice test, for now I will just tell them to take the four choices and determine if each point is on both lines (they already know how to do that). If the point is on both lines, it must be a point of intersection. On the other hand, factoring seems to appear quite a bit on the test, so that is why I have changed suddenly to the topic of factoring polynomials.
When you these lists of what my students know and don’t yet know, it will seem like we’ve been playing around in our Algebra 1 class. But, I promise you that I’ve been very efficient with my instructional time. We are doing math most every minute of every class, except for short breaks. And yet, we are so far from where we need to be. How did this happen?
Excuse #1. I’m a new teacher and I suck at this. My pacing is all wrong. I’m very reluctant to move on until most of my students are proficient with something–perhaps I need to move on to the next topic sooner and hope instead that accumulated review will solidify students’ skills and understanding after the initial exposure to a topic. Also, as I wrote before, I still have a lot to learn about differentiating instruction. I need to switch topics faster and find other ways to help students who need more time and practice.
Excuse #2. We’ve spent time on prerequisite skills and concepts that aren’t on the state standards, maybe too much time. For example, we’ve spent time developing fluency with positive and negative numbers, order of operations, learning how to the area and perimeter of a rectangle, calculating the slope of a line, substituting values into expressions, knowing what a square root is–these are things that aren’t on the Algebra 1 standards.
Excuse #3. We’re using the CPM curriculum for Algebra 1 in this school. Our district’s periodic assessments don’t match up with the curriculum, and the curriculum includes things that aren’t explicitly on the state standards: focusing on patterns, making connections between graphs and patterns. CPM encourages students to construct their own solution methods and concepts before the teacher tells it to them–this takes more time than direct instruction. (I firmly believe more of my students are becoming proficient because of the student-centered instruction, however.)
Excuse #4. We lost a whole month of instruction at the beginning of the year. I’m really wishing for that extra month right now.
A good friend of mine gave me some advice: Have the courage to intentionally skip topics that are on the state standards. She’s not saying that we should throw out the state standards (even though I’m not crazy about them). I think what she is saying is that I should trust my intuition on what I think is best for my students and not feel bound this list of topics. If I try to cover all of the material, we would be going so fast that most of my students would probably not get it (at least not with my current teaching skills) and so I’d be setting them up for failure. I should instead focus on a smaller set of topics and make sure my students really know it well and feel confident about themselves.
Argh! There is so much that I would do differently if I had the chance to start the year over.
One positive thing–last year, only 2% of students at this school were deemed “Proficient” by the state in Algebra 1. There’s no way my students will do worse than that, I’m confident of that.
By the way, who in California decided that the pinnacle of knowledge in Algebra 1 is knowing how to derive the quadratic formula by completing the square? When you look at the released questions for the CST, it certainly feels that way.