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Why Is Algebra 2 So Hard?

For those students who have survived the early days of the math curriculum gauntlet [integer arithmetic, fraction arithmetic, exponents, geometry, pre-algebra, Algebra 1], Algebra 2 is the next in the math misery creator. Most college programs require that students pass Algebra 2 in order to earn at least an Associate’s Degree and most certainly a Bachelor’s degree regardless of the major [there are a few majors that are an exception]. This makes Algebra 2 a course that’s labeled as a “gatekeeper” / “rite of passage”. Passing Algebra 2 opens up the gates to pre-Calculus which in turn opens up the degree options. Science degrees / careers become a possibility with Calculus.

In high school, Algebra 2 is usually a one year course preceded by at least a year of Algebra 1. And Algebra 1 is often introduced in various forms early in childhood math education. Most college programs will look at standardized test scores or high school credentials to determine whether a student is eligible to place out of various courses. For those attending a two-year college, Algebra 2 is often a required course if one wants to transfer into a four-year college. And so begins the resentment, angst, bitterness, and all things curmudgeony surrounding Algebra 2.

“Why do I have to take this class if I’ll never use it?”
“I don’t know how to buy a house, but I know how to factor polynomials.”
“I could’ve been an engineer, but I couldn’t pass Algebra 2.”

These are some of the “classic” complaints one hears about Algebra 2. But what actually makes Algebra 2 such a difficult course?

There are several things. First usual culprits

  • Students had been shuffled up through Algebra with weak Arithmetic [fractions, exponents] and algebraic manipulation skills.
  • Algebra 2 in college is typically a one semester course rather than one year as in high school.
  • Compounding the above, the contact hours in college is probably around 45 hours with many hours going towards testing [the test itself, exam review, exam postmortem, quizzes]. Thus, the total instructional time can be low.
  • Additionally, since most students have taken Algebra 2 in high school, they have a false sense of competence and tend to underestimate what they do and don’t know.
  • Somewhat ironically, professors also sometimes take some knowledge as assumed for students.

However, often forgotten or missed is that Algebra 2 is pretty much the first time that students are asked, on a regular basis, to “undo” things, to combine multiple concepts, and to recognize abstract forms.

Here are a few examples:

Complete the square for \(y = 3x^{2} + 2x -9\). This is not a “solve for \(x\)” type of problem. Nor is it a “plug in” some number for \(x\) and get an output for \(y\). This problem requires some massaging. There are several ways to answer this question, but they are all multi-stepped, even if one of the steps requires some form “plugging in”.

Solve the quadratic \(x^{2} + 4x – 12 = 0\). The idea that two solutions can exist, baffles students. And when both solutions are complex (\(a + bi\)), sometimes there is open revolt to the idea.

Given points \((0,8),(1,16),(2,32)\), find the exponential formula in the form of \(y = C\cdot b^{x}\) that generates those points. This is like finding the equation of the line, but now it’s the equation of the exponential. Some students have a difficult time identifying where / how there is non-linearity. In addition, problems like these require students to be able to pick up on the fact that the \(y\) coordinates are doubling for every unit increase in \(x\). This can be a bit of a foreign concept.

Reduce (Simplify) \(\frac{x^{2}-4}{x^2-5x+6}\). This problem requires the student to recognize certain forms and their factorization. In essence, this is just an extension of reducing fractions from arithmetic. But given the aforementioned fraction woes, many students find themselves up the creek. This is also not a “solve for \(x\)” type problem. Note that reducing something like \(\frac{48}{15}\) students can find easy, but reducing \(\frac{48x}{15x}\) becomes incredibly difficult because of the presence of the \(x\). As far as many students are concerned, \(x\) is to be solved for.

Solve for \(x\) in \(\frac{x^{2}-4}{x^2-5x+6} = 2\). Oops. This is just like the previous problem but now requires a more general view of algebraic manipulation as well as closer attention to the domain — a concept generally lost on many students.

Rationalize the denominator in \(\frac{3}{\sqrt{x} -7}\). Understanding the special technique of working with the conjugate becomes as a bit of a head scratcher. Is it \(\sqrt{x} + 7 \)? \(-\sqrt{x} – 7\)? \(7 – \sqrt{x}\)? \(\sqrt{x}\)?

Then there’s ye old Radical Misery

Solve for \(x\) in \(5^{-2x} = 5^{3x^{2}-9}\). This requires that the student first understand how exponentials and logarithms work, then it requires the student to apply some solution techniques for quadratics.

Solve for \(x\) in \(2^{3x} = 3^{x-2}\). This problem requires a careful application of laws of logarithms, laws of exponents, and the ability to algebraically manipulate expressions involving \(\ln\) (or \(\log\)).

There is also more math specific vocabulary: vertex, parabola, exponential, logarithm, zero (as in “zeros of the quadratic”), asymptote, conjugate, imaginary, axis of symmetry, etc that becomes another hurdle.

Then there are the various formulae and laws. Laws of exponents, laws of logarithms, quadratic formula, formula for the \(x\)-coordinate of the vertex of a parabola, etc. And when instructors begin to gimmickify this material because of time and testing constraint (ironic), any hope for logical cohesion gets lost.

Examples of gimmicks:

  • Sing the quadratic formula song.
  • And now we “scooch” the power out of the logarithm: \(\log(3^{x}) = x\log(3)\) where the \(x\) was scooched out.
  • The “snail” trick for understanding the relationship between logarithms and exponentials.

Also, while many students can probably skate through Algebra 1, Algebra 2 requires a little more attention to detail and a little bit more of a time commitment than what students are used to, especially in college. There are a lot of little nuances to problems. Algebra 2 also begins a sequence of mechanical courses where one of the primary goals is to “get used to” mathematical writing and manipulation of equations. And this takes practice.

Finally, while many of the topics are generally tied together, they often feel disjointed for students. At one moment it’s talk of radicals, at the next it’s quadratics, and then all of a sudden it’s exponentials and logarithms. It is in the course planning and discussion that the instructor should take care to build an anticipatory process for students. And for students, it’s a course with many new topics that build heavily on previous coursework. Thus building fluency needs more than simply working through problems.

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