Let’s take a look at the course syllabus together.
EDS 212: Day 1, Lecture 1
Course intro, algebra refresher
August 4th, 2025
- Instructor: Ruth Oliver (rutholiver@ucsb.edu)
- Teaching assistant: Alessandra Vidal Meza
- Course hours: 10am - 4:30pm PST
- Location: NCEAS 1st Floor Classroom
- Units: 2
- Grading: Satisfactory/Unsatisfactory
- Description: Quantitative skills are critical when working with, understanding, analyzing and gleaning insights from environmental data. In the intensive EDS 212 course, students will refresh fundamental skills in math (algebra, uni- and multivariate functions, units and unit conversions), summary statistics and basic probability theory, derivative and differential equations, linear algebra, and reading, writing and evaluating logical operations.
- Day 1: Course introduction & math basics refresher
- Day 2: Derivatives
- Day 3: Differential equations, intro to linear algebra
- Day 4: Linear algebra, summary statistics
- Day 5: Basic probability theory, Boolean algebra
or,
Why do I have to know math when a computer will do it for me?
MEDS students have diverse work & academic histories
This course (re)introduces math concepts and tools that:
- Focus on applications to environmental data science
- Are specifically relevant for MEDS projects, coursework
- Provide an entryway into building computational skills
- Refresh quantitative thinking skills generally
- Refresh essentials like units, conversions, notation, language
- Algebra blitz
- Exponentials and logarithms
- Common units and unit conversions
- Functions
- Understanding graphs
- Interpreting equations
- Algebra blitz
- Exponentials and logarithms
- Common units and unit conversions
- Functions
- Understanding graphs
- Interpreting equations
You can get far with a few rules:
- Order of operations
- Equations are already solved (but sometimes we need them in a different format)
- Do whatever you want but do the same thing to both sides
P - Parentheses
E - Exponents
M/D - Multipication / division
A/S - Addition / subtraction
Simplify the following: \((12 - 2)/5 + 5(3+2)/6\)
Simplify the following: \(\frac{4-6}{2}(3+1)-\frac{1+2*4}{3}\)
Simplify the following: \(3x+4(8x-6x) -(2y-5)+\frac{2x(1-3)}{2}\)
- \((12 - 2)/5 + 5(3+2)/6\)
- \((10/5) + 5 \times 5 / 6\)
- ANSWER: \(2 + 25/6\)
- \(\frac{(4-6)}{(2)}(3+1)-\frac{(1+2*4)}{(3)}\)
- \(\frac{(-2)}{(2)}(4)-\frac{(9)}{(3)}\)
- \(-1 \times 4 - 3\)
- \(-4-3\)
- ANSWER: \(-7\)
- \(3x+4(8x-6x) -(2y-5)+\frac{2x(1-3)}{2}\)
- \(3x+4 \times 2x - 2y + 5 + \frac{2x-6x}{2}\)
- \(3x+4 \times 2x - 2y + 5 + \frac{2x}{2}-\frac{6x}{2}\)
- \(3x+4 \times 2x - 2y + 5 + x - 3x\)
- \(11x - 2y + 5 - 2x\)
- ANSWER: \(9x - 2y + 5\)
Simplify the following: \(6 \div 3(4+2)\)
What would be a harder-to-misinterpret way to write this?
ANSWER: 12
A harder to misinterpret way to write this: \((6 \div 3) \times (4+2)\)
–
Being readable & hard to incorrectly interpret is often as important as being technically “correct”
When designing things, it’s important to consider the different ways that users might misuse or misunderstand it - then build in safeguards to help them use it correctly. Clear communication and user-centered design is critical in environmental data science.
\[2x-5y+3.9=8x^2-100.7x\]
Provides solutions for the questions:
- “What is the value of \(2x-5y+3.9\)?” and
- “What is the value of \(8x^2-100.7x\)?”
In the equation on the previous slide (shown below), we might want to solve for y:
\[2x-5y+3.9=8x^2-100.7x\]
The one rule to rule them all:
You can do whatever you want to an equation, as long as you do the exact same thing to both sides. That includes ensuring that you are applying something entirely to each side.
Example:
Apply the same operation to each side of the following equation step-by-step to isolate \(x\) on one side. Write out all steps.
\[4x+8=5-2x\]
- Start with: \[4x+8=5-2x\]
- Add \(2x\) to each side: \(6x + 8 = 5\)
- Subtract \(8\) from both sides: \(6x = -3\)
- ANSWER: \(x = -\frac{1}{2}\)
Example:
Apply the same operation to each side of the following equation step-by-step to isolate \(a\) on one side. Write out all steps.
\[\frac{2(a+1)}{3a}+4=6\]
ENDING NOTE: We might encounter complex expressions for how values are related to each other. While the equation is already solved, it can be helpful to use some basic algebra to rearrange things to make our equations easier to understand.
- Start with: \[\frac{2(a+1)}{3a}+4=6\]
- Subtract 4 from both sides: \[\frac{2a+2}{3a}=2\]
- Multiply both sides by \(3a\): \[2a+2=6a\]
- Subtract \(2a\) from both sides: \[2=4a\]
- Divide each side by 4; ANSWER: \(a = \frac{1}{2}\)
- Algebra blitz
- Exponentials and logarithms (this afternoon)
- Common units and unit conversions
- Functions
- Understanding graphs
- Interpreting equations
\[x^n=x\times x\times\ x \times x...(n\space times)\]
Evaluate the following to find a value for \(y\):
\(y = 12-2^4\)
\(2y + 30=y+3^3\)
Sometimes things are related to each other in an exponential way, so we’re going to remind ourselves how they work using numbers.
- \(y = 12-2^4\)
- \(y = 12 - (2 \times 2 \times 2 \times 2)\)
- \(y = 12 - 26\)
- ANSWER: \(y = -4\)
- \(2y + 30=y+3^3\)
- \(2y + 30 = y + (3 \times 3 \times 3)\)
- \(2y + 30 = y + 27\)
- \(y + 30 = 27\)
- ANSWER: \(y = -3\)
- \(x^ax^b=x^{a+b}\) ; Example: \(x^5x^3=x^{5+3}=x^8\) ; Example: \(2^2 \times 2^2\) = \((2 \times 2) \times (2 \times 2)\) = \(2^4\)
. . .
- \(\frac{x^a}{x^b}=x^{a-b}\) ; Example: \(\frac{z^5}{z^3}=z^{5-3}=z^2\) ; Example: \(\frac{2^3}{2^2}\) = \(\frac{2 \times 2 \times 2}{2 \times 2}\) = \(2^1\)
. . .
- \(\frac{1}{x^a}=x^{-a}\) ; Example: \(b^{-4x}=\frac{1}{b^{4x}}\)
. . .
- \((x^a)^b=x^{ab}\) ; Example: \((2^3)^2=2^{3*2}=2^6=64\)
. . .
- \((\frac{x}{y})^a=\frac{x^a}{y^a}\) ; Example: \((\frac{y}{2^2})^2=\frac{y^2}{(2^2)^2}=\frac{y^2}{2^4}=\frac{y^2}{16}\)
. . .
- \((xy)^a=x^ay^a\) ; Example: \((3x)^2=3^2x^2\)
Simplify the following expressions using the rules of exponents:
1. \(3x^5x^8x^{-11}\)
2. \(\frac{-8x^6}{2x^4}+7x^2\)
3. \(\frac{3x}{x^5}-3.8x^4\frac{x^3}{x^6}+8.1x-11.2\)
- \(3x^5x^8x^{-11}\)
- ANSWER: \(3x^2\)
- \(\frac{-8x^6}{2x^4}+7x^2\)
- \(-4x^2 + 7x^2\)
- ANSWER: \(3x^2\)
- \(\frac{3x}{x^5}-3.8x^4\frac{x^3}{x^6}+8.1x-11.2\)
- \(3x^{-4}-3.8x^4 \times x^{-3} + 8.1x - 11.2\)
- \(3x^{-4}-3.8x^1 + 8.1x - 11.2\)
- ANSWER: \(3x^{-4}+4.3x - 11.2\)
First, Outside, Inside, Last
Example:
\[(2x+5)(x-3)\]
. . .
\[= (2x \times x) (2x \times -3) (5 \times x) (5 \times -3)\]
\[= 2x^2 - 6x + 5x - 15\]
\[=2x^2-x-15\]
- Algebra blitz
- Exponentials and logarithms
- Common units and unit conversions
- Functions
- Understanding graphs
- Interpreting equations
Think about these statements, which all contain the same value of 4:
There are four in the refrigerator.
There are four burritos in the refrigerator.
There are four roaches in the refrigerator.
There are four million dollars in the refrigerator.
We cannot responsibly work with data without knowing the units of each variable we’re working with.
That means we need to always familiarize ourselves with metadata, carefully check units and any unit conversions, and understand how units combine into the units of a dependent variable.
In dimensional analysis, we multiply initial units by a sequence of conversion factors to arrive at the final desired units.
For example, to convert \(100 \frac{g}{cm^3}\) into units of \(\frac{kg}{in^3}\), given that 1 cm3 = 0.061 in3.
\[100\frac{g}{cm^3}*\frac{1kg}{1000g}*\frac{1cm^3}{0.061in^3}=1.639\frac{kg}{in^3}\]
Practice dimensional analysis to perform the following conversions:
- Convert \(8.1\frac{km}{s}\) to miles per hour, given that 1 km = 0.621 miles.
- Convert a mass flux of \(3.2\frac{g}{min\cdot m^2}\) to \(\frac{mg}{s\cdot cm^2}\).
\(8.1\frac{km}{s} \times \frac{60s}{min} \times \frac{60min}{hr} \times \frac{0.621 miles}{1km} = 18,108.36 \frac{miles}{hr}\)
\(3.2\frac{g}{min\cdot m^2} \times \frac{1min}{60s} \times \frac{1000 mg}{g} \times \frac{m}{100cm} \times \frac{m}{100cm}\)
- Algebra blitz
- Exponentials and logarithms
- Common units and unit conversions
- Functions
- Understanding graphs
- Interpreting equations
Functions are mathematical expressions that tell us how input values are related to output values.
For example, \(y = 3x-5\) is a function that tells us the value of y at any value of x. In this scenario, we would probably say y is a function of x.
Could you also rewrite it and say x is a function of y? Here, with no knowledge of what’s an input and what’s an output, sure - but usually in environmental data science we specify the input variable(s), and the output variable(s) carefully. What follows is the expression in the format of: “[output variable(s)] is/are a function of [input variable(s)]”.
For the following combinations of related variables, which do you expect would be the input and the output in a function describing how they are related? Say your answer in a sentence, e.g. “Evapotranspiration is a function of air temperature.”
fuel (biomass) / slope / wildfire severity / windspeed / air temperature
wind speed / power generated by wind turbine
soil C:N ratio / bacterial biomass / soil water content / leaf litter decomposition rate
- wildfire severity is a function of fuel and slope and windspeed and air temperature
- power generated by wind turbine is a function of wind speed
- leaf litter decomposition rate is a function of soil C:N ratio and bacterial biomass and soil water content
Single variable (univariate) function:
\(f(x) = [expression\space containing \space x]\)
Multivariate function:
\(g(a,T,z)=[expression\space containing \space a, T, \space and \space z]\)
For continuous functions, we evaluate them by plugging in variable values.
Example:
Evaluate \(g(x,t)=2.4x+0.5t^2\) at \(x = 3\) and \(t = 10\)
\(g(3, 10) = 2.4(3) + 0.5(10^2)= 57.2\)
- \(g(3,10) = 2.4 \times 3 + 0.5 \times 10^2\)
- \(g(3,10) = 7.2 + 50 = 57.2\)