Wednesday, August 27, 2014

A different way of looking at Algebra

I accidentally stumbled across this old doc I made a couple years ago.  It breaks down the concepts of algebra into a few different types of categories and suddenly seems more useful than I remember it being when I made it.  I would love some feedback / pushback / thoughts on this way of classifying the subject.  Thanks to Kate @nerdypoo for her monster comment that has helped me improve this by the time you read it -- but more comments and refinement are welcome!

Modeling scenarios:
  • Every known value can be represented with a number
  • Every unknown value can be represented with a variable
  • An expression combines multiple values (2*boys - girls, 3x^2, etc.)
  • An equals sign creates a balance/equivalence between two or more expressions
  • When there is a single variable among numbers in an equation, it is possible to find the numeric value of the variable
  • When there are multiple unknown variables, a relationship between the variables can be created to see many possible solutions

Classifying 2D equations:
  • One variable mono/polynomial; arithmetic growth:
    • linear y = 3x
    • quadratic y = 3x^2, x = 3y^2 -3x + 2
    • higher order polynomial y = x^3 - 3x^2 + 4
  • Inverse of one variable mono/polynomial:
    • square root function y = sqrt(x - 3)
    • nth root function y = 5 root (4-x)
  • Two squared variables in conics:
    • circle x^2 + y^2 = 4
    • ellipse 3x^2 + y^2 = 4
    • hyperbola 3x^2 - y^2 = 4
  • Geometric growth/decay:
    • exponential y = 2^x, y = e^(2x)
    • logarithmic y = log(3x), y = ln(x - 2)
  • Trig:
    • cyclical waves y = sin(x), y = cos(2x)
    • cyclical waves divided y = tan(2x), y = 2cot(3x)
    • inverse of cyclical waves y = csc(x), y = sec(x)
  • Recursive: a0 = 1, a(n) = 3*a(n-1) + 4

Analyzing number types:
  • Real numbers: any number that can be placed on the number line
    • Rational numbers: using fractions to represent parts of numbers
      • Integers: counting with a way to go positive and negative using whole numbers
        • Whole numbers: counting with a way to represent nothing (0,1,2,3,4,5...), and natural numbers (no zero: 1,2,3,4,5...) [US definition]
    • Irrational numbers: numbers that can’t be captured as a fraction and never end
  • Complex numbers: makes 1D numbers become 2D with a real and imaginary component
    • Imaginary numbers: any real number multiplied by the square root of -1, a useful construct for dealing with the common problems caused by negative square roots

Analyzing operations:
  • Addition:
    • moving left/right on a number line
    • with fractions, keep common denominator (factor out the denominator and add the numerators)
    • with vectors/complex numbers, add each component separately (factor out the x-components or i-components and add leftovers)
    • with polynomials/radicals, treat each power of x or each type of radical as a separate component and add each separately (factor out the common sqrt(5) or x^3 and add leftover coefficients)
    • subtraction = addition with second value as a negative
  • Multiplication:
    • finding area on a grid
    • cancel pairs of negatives (a non-obvious rule)
    • think “repeated addition”
    • with fractions, multiply numerators AND denominators separately
    • division = multiplication with second value as an inverse
  • Exponentials:
    • think “repeated multiplication”
    • with fractions, raise numerator AND denominator to a power separately as you would when multiplying
    • logarithms = undo an exponential equation to isolate the power term

Equation / function graph transforms [ assuming a form similar to (y-k)/b = (x-h)/a ]
  • Translate:
    • Move right by subtracting directly from x, move left by adding directly to x
    • Move up by subtracting directly from y, move down by adding directly to y
  • Reflect:
    • Reflect over the y=x line by finding the inverse function (switching x and y)
    • Reflect over the y-axis by taking the opposite of all x’s
    • Reflect over the x-axis by taking the opposite of all y’s
    • Reflect over any vertical or horizontal line by doing a translation, then an axis reflection
  • Scale/stretch from origin:
    • Divide one side of equation to stretch out along that axis
  • Rotate:
    • On 90 degree rotations, the x and y coordinates change places and one value becomes negative depending on direction.
    • For 0-89 degree rotations, it may be easiest to change to polar coordinates x=rcos(t) and y=rsin(t), then add or subtract from t.

Forms of equations:
  • Function:
    • solve for y, y = mx + b, y = ax^2 + bx + c
    • purpose -- make it easy to find an output given an input, easy to create x-y table
  • Expose an important point:
    • Point-slope form for linear, y - y1 = m(x - x1)
    • Vertex form for quadratic, y = a(x - x1)^2 + y1
    • Standard form for circles, (x - x1)^2 + (y - y1)^2 = r^2
  • Highlight zeros:
    • factored form for polynomials, y = a(x - r1)(x - r2)
    • purpose -- make it easy to cancel terms in rational expressions, make it easy to find where graph crosses x-axis (useful in well-designed applications)

Using the coordinate plane:
  • Goal: make a problem easier to understand or solve by mapping two different quantities (complex numbers, (x,y) coordinates, latitude/longitude, vectors) onto a visual (requires 2D space)
  • How: notice differences between a single answer (a coordinate), a relationship (an equation), and a region of acceptable solutions (an inequality or set of inequalities)
  • Use for non-equations: map shapes onto a coordinate plane to more easily find distances and angles

Parameterizing scenarios:
  • Functions:
    • take an input (usually 1 variable called x)
    • have restrictions on what the input can be
    • create an output (usually 1 variable called y)
    • the output can be classified by the bounds it fits within
    • accomplishes something specific
  • Domain is the set of input restrictions
  • Range is the set of output limitations
  • End behavior:
    • Purpose: to know which term dictates the long-term behavior and in which direction
    • In rational expressions, the highest power term dominates towards positive and negative infinity
    • In some expressions, there is an asymptotic line that the output approaches

Accumulation and rates of change:

  • Physics:
    • Acceleration is rate that the velocity changes (wrt time)
    • Velocity is the rate that the position changes (wrt time)
    • Position is where something is at a given time
  • Rate of change is found using a tangent line of a function at a given point/time
  • Accumulation over a period of time is found by measuring the area bounded between a curve and the horizontal-axis.
  • Starting with one of the 3 physics graphs, rate of change allows you to move towards acceleration and accumulation allows you to move towards position.


  1. wow, that is extensive and impressive! i'm curious though -- do you intend this for your own clarifying and planning purposes, or is this something you would share with students?

    i have semi-extensive comments on the section on analyzing number types, which is part of my first unit with my students, so i work with this a lot. the first thing i notice is that the way you break it up into categories can be a little confusing for students, because while it is clear that whole numbers is a subset of integers, it's not at all clear that it's also a subset of real numbers. the way it's broken up it almost seems like integers aren't real.

    i worry that your definition of integers is too broad because couldn't you lose, for example, $2.50? wouldn't that imply that -2.5 is an integer? i think it needs to be clearly stated that integers are positive and negative *whole* numbers. (btw, as an interesting aside, internationally the set of natural numbers includes the number 0; there is no set called whole numbers. if you want what americans call natural numbers, that is Z+ or positive integers. i think it's nice to see how arbitrary things can be!)

    i think the concept of real numbers is actually a bit tricky so an actual definition of them is needed: any number that can be placed somewhere (even approximately) on the number line. that always begs the question what *can't* be put on the number line, which counterexamples, yay!

    the definition of a rational number is similarly challenging for students because they need to develop what kinds of numbers can be represented as a ratio, for example 7-->7/1 (not 7/7!!!), -11-->-11/1 (yes, it works for negatives too!), 0.75-->3/4, 0.3333->1/3 (hey, what about 0.4444, etc?), sqrt4-->2/1, -1.9--> -19/10. the question is therefore what kinds of decimal numbers are rational numbers? and therefore what kinds are irrational numbers? always nice to get an example like 0.1011011101111 as well as pi and sqrt7.

    i hope the above is helpful and not annoying. :)

    on another note, with regards to transformational geometry, for translations it seems like you're thinking of translating a function, ie translate y = 2x +5 to the right 3 units gives y = 2(x-3)+5 since you say subtract from x, whereas if you just had a point (17,4) and translate to the right 3 units, you'd add to get (20,4). but then when you talk about scale/stretch, it seems as if you're talking about points because if you have (4,3) and want to stretch horizontally by a factor of 2, you'd do as you say and multiply to get (8,3), but if you're working with a function and talking about scale, you don't want to multiply by the scale factor but rather divide. for example, if you want to stretch horizontally by a factor of 2, you'd take your equation y = 7x + 5 and do y = 7(x/2) + 5. so i found that a little confusing in your explanation because it seems like you switched midsection which things (points or functions) you were talking about. that also explains why y = 2x^2 is "thinner" -- it was stretched vertically [(y/2)=x^2--> y = 2x^2] and why y = 0.5x^2 is "fatter" -- it's been stretched horizontally by a scale factor of sqrt2 (woah, tough stuff!!). also with rotations, i think "take the negative of one of them" is pretty unspecific -- which one? how will the students know?

    all of my comments depend on what your plans for this are, of course. i'm super impressed with what you've done and the comments are only meant to help you develop it further! :)

  2. Thank you! I did my best to fix some of the issues you noticed by reorganizing the number classification and creating a template form for transformations (and not working with points at all). That helped a lot. For clarification on purpose, I wrote it for future curriculum planning as a different way to think about the course with no intention of directly using it with students, but I'm not opposed to doing so if there is a good place to use it.