ALEKS-based Algebra: Course Structure, Key Topics, and Quick-Recall Notes

Course structure and ALEKS layout

  • ALEKS separates content into two parts: green portion = actual homework (algebra skills, factoring, radicals); blue portion = learned portion (not required for homework but accessible).
  • Green homework: contains a set number of questions; you earn credit per question you answer correctly; you can save work and come back later before submitting.
  • Do not submit the entire assignment until you’ve completed as much as possible; due dates are in the system.
  • Homework availability and extensions: if not finished by the due date tonight, a two-day extension can be granted for those not yet done; next module is due the following Tuesday; course typically follows one module per week with lessons twice a week.
  • Course rhythm: two days per week (e.g., Tuesday and Thursday) to cover modules, with a major lesson on Thursdays to prepare for module one and address questions from earlier material.
  • Final module flow: after fall break, proceed through four modules, one more module before the exam, then finish the course.
  • Grades: the ALEKS gradebook in ALEKS or Canvas may not reflect the final course grade; the course content is completed separately and then recorded in the college gradebook later.
  • Initial knowledge check may show a different percentage of topics known than later in the course; topics can be removed or added as you progress.
  • Both blue and green portions map to related topics (e.g., factoring and radicals often appear in both). Green is the track you’re graded on for homework credit; blue is the reference/learning track.

Key topics and common ideas (quick map)

  • Nth roots and real numbers
    • If the index n is even and the radicand is negative, the root is not a real number.
    • If n is odd, the root of a negative number is negative (real).
    • Example:
      oot 4 rom{81}=3; with a minus outside the radical: -
      oot 4 rom{81}=-3.
    • Negative inside a radical with even index is not a real number.
  • Radicals and signs
    • When multiplying negatives, two negatives make a positive; signs outside vs inside radicals affect the result.
  • Exponents and exponent rules (recap)
    • Power of a power: (am)n=amn(a^m)^n = a^{mn}
    • Product of powers: aman=am+na^m a^n = a^{m+n}
    • Negative exponents: a−n=1ana^{-n} = \frac{1}{a^n}
    • Example: (24)−2=24⋅(−2)=2−8=1256(2^4)^{-2} = 2^{4\cdot(-2)} = 2^{-8} = \frac{1}{256}
    • If you have separate bases with the same base, exponents add/subtract: 24⋅2−2=24−2=22=42^4 \cdot 2^{-2} = 2^{4-2} = 2^2 = 4
  • Limits and units in applied problems
    • Volume: V=πr2hV=\pi r^2 h
    • Surface area of a cylinder: S=2πrh+2πr2S=2\pi r h + 2\pi r^2
    • Volumes use cubic units; surface areas use square units; plugging in radius and height with correct units is essential.
  • Rational expressions (quotients of polynomials)
    • Addition/subtraction requires a common denominator.
    • For fractions with denominators $x$ and $x+5$, a common denominator is x(x+5)x(x+5).
    • Multiply numerators and denominators to achieve the common denominator:
    • First fraction: multiply top and bottom by $(x+5)$.
    • Second fraction: multiply top and bottom by $x$.
    • Then combine the numerators and simplify.
    • Multiplication/division of rational expressions: do not require a common denominator to multiply; division uses the reciprocal.
  • Simplification and cancellation rules
    • You can cancel common factors in numerator and denominator only if the factor is exactly the same in both (fully factored form).
    • Cautions about canceling terms like a standalone x: you can cancel only when a common factor appears in both numerator and denominator.
  • Factoring strategies (quick reminders)
    • Greatest Common Factor (GCF) first: example
    • For 25m2+15fm25m^2 + 15fm, GCF is 5m5m, giving 5m(5m+3f)5m(5m+3f).
    • Grouping and FOIL factoring for quadratics; AC method for trinomials: find two numbers that multiply to $ac$ and sum to $b$.
    • AC method (brief): for $ax^2+bx+c$, find integers $m$ and $n$ such that $mn=ac$ and $m+n=b$, rewrite, and factor by grouping.
  • Practice mindset with reference sheets
    • Use provided references to plug formulas rather than memorizing every detail; you’ll often have to insert numbers into formulas.

Specific examples highlighted in class (concepts only)

  • Example: factoring by GCF
    • Given 25m2+15fm25m^2 + 15fm
    • Factor out 5m5m: 5m(5m+3f)5m(5m + 3f)
  • Example: AC method (conceptual)
    • For a quadratic such as ax2+bx+cax^2+bx+c, look for two numbers that multiply to acac and add to bb, then factor by grouping.
  • Example: adding fractions with different denominators
    • To add fractions with denominators $x$ and $x+5$, convert to a common denominator x(x+5)x(x+5) by multiplying the first numerator by $(x+5)$ and the second numerator by $x$, then combine and simplify.
  • Example: combining exponents with parentheses
    • If you have $(a^m)^n$, simplify to $a^{mn}$; if multiplying like bases, add exponents; if raising a power with a negative outside, apply the exponent rule correctly.

Quick reminders for exam-ready recall

  • Real vs not-real roots depending on parity of index and sign of radicand.
  • Cylinder formulas and unit analysis for geometry problems.
  • Remember the difference between subtracting/adding fractions and the need for a common denominator.
  • Use AC method and GCF clearly before attempting more complex factoring.
  • Use the two-day extension policy if you’re close to the deadline; keep track of module-by-module progression.
  • The green ALEKS homework can be attempted multiple times; you earn credit per correct answer, not per attempt; don’t rely on guessing.
  • When in doubt, revert to the reference materials and formulas to guide problem solving rather than trying to memorize everything.