Quantitative Reasoning

1015SCG

Lecture 1


Quantitative Reasoning

Welcome!


People

• Gold Coast

  • Convenor/Lecturer: Dr Juan Carlos Ponce Campuzano
  • Workshop demonstrator: Jordan Holdorf

• Nathan

  • Main Convenor: Dr Michelle Ward
  • Lecturer: Dr Amal Succarie
  • Workshop demonstrator: Llewyn Randall


Navigating the course site


Navigating the course site

Do especially check:




How the course works

  • Pre-recorded videos
  • Lecture Notes
  • In-person lectures
  • Face-to-face workshops (in even weeks, mandatory and marked)





What if you need help?

Course stuff

  • Ask questions during workshops
  • Lectures
  • Course Teams Channel
  • Email - for private matters only

Bigger stuff





Assessment

  • No Exam 😃
  • 50% of total marks to pass
  • Extensions/Deferred assessment/Special consideration:
    Apply via the Official Channel





📝 Workshop Tasks - 25%

  • Workshops in Even weeks only - 6 in total
  • Report writing task throughout all workshops - marked component
  • The marks will only be granted during the workshop
  • If you have a valid reason to miss the workshop apply for deferred assessment until next workshop within 3 days and provide supporting documentation (NOT the extension). Do the task at home and present it for marking to your demonstrator during the next workshop.



🤔 What do you need for workshops?






📆 Workshops schedule

  • Workshop Group 1
    • Thursdays: Weeks 2, 6, 8, 10, and 12 (12pm - 2pm, G01_3.55)
    • Friday: only in Week 4‼️ (10am - 12pm, G01_3.55)
  • Workshop Group 2
    • Fridays: Weeks 2, 4, 6, 8, 10, and 12 ( 1pm - 3 pm, G31_3.14)


Employability Task A and B - 5%

📝 You have to write a short reflection on how mathematics is likely to play a role in your career ambitions.

  • Part A: Identify vital Maths skills and key topics
          Draft → week 3
  • Part B: Evidence your skills and plan the way forward
          Completed → week 9

Full details in Canvas → Modules → Employability Tasks



Maths & Inference and Scientific Critiquing Tasks

  • ♾️ Maths & Inference Task - 30% (week 7)

    Full details in Canvas → Modules → Maths and Inference Task


  • 🔬 Scientific Critiquing Task - 35% (week 11)

    Full details in Canvas → Modules → Scientific Critiquing Task





Assessment: Summary

Item Weight Due
Workshop (best 5) 25% Ongoing
Employability A 5% Week 3
Maths and Inference 30% Week 7
Employability B 5% Week 9
Scientific Critiquing 35% Week 11
🎉 NO FINAL EXAM!! 🎉


Schedule Part I

Week 1:

  • Course intro


Week 2:

  • Data and Numbers
Workshop 1

Week 3:

  • Errors, Best Estimates, Correlation, Regression
Employability Task A

Week 4:

  • Regression, Hypothesis Testing, Basic Functions

Workshop 2

Week 5:

  • Functions


Week 6:

  • Critical Thinking and Problem Solving
Workshop 3


Schedule Part II

Week 7:

  • Logical Arguments
Maths and inference Task

Week 8:

  • The power of approximation - Fermi Problems
Workshop 4

Week 9:

  • The Structure of Scientific Writing
Employability Task B

Week 10:

  • Communicating Scientific Data

Workshop 5

Week 11:

  • Big Data


Scientific Critic Task

Week 12:

  • Final week


Workshop 6


What is QR?

🤔

What is QR?

Is this just another Maths course? Not really!

🥱 🙂

Not just another Maths course

  • Relate the use of mathematics to your own scientific studies and interests

Image source: National Academies



Not just another Maths course

  • Use computational tools to perform mathematical calculations

Image source: Mapple Soft.



Not just another Maths course

  • Construct and critique quantitative scientific arguments

Image source: An Introduction to Scientific Argumentation



Not just another Maths course

  • Communicate scientific arguments effectively in writing

Image source: Writing a scientific article: A step-by-step guide for beginners



Not just another Maths course

  • Construct suitable mathematical models of real-world phenomena
\(T(t)=\left(T_0-T_m\right) e^{-kt} + T_m\) $\ds P(t) = \frac{\theta P_0 e^{rt}}{\theta-P_0+P_0e^{rt}}$


Not just another Maths course

  • Solve problems related to mathematical models of real-world phenomena

Fluid simulation by Amanda Ghassaei    


Not just another Maths course

  • Relate the use of mathematics to your own scientific studies and interests
  • Use computational tools to perform mathematical calculations
  • Construct and critique quantitative scientific arguments
  • Communicate scientific arguments effectively in writing
  • Construct suitable mathematical models of real-world phenomena
  • Solve problems related to mathematical models of real-world phenomena
🚀

Why QR?

🤔

Example - attendance vs grades



Example - attendance vs grades

Question: Is there a relationship between the course attendance and the grades?

  • How would you approach answering this questions?
  • What do we need to answer it?


Example - attendance vs grades

Question: Is there a relationship between the course attendance and the grades?

  • we need data, but what kind? where do we get it? is data reliable?
  • how do I analyze the data?
  • what do we see from the data? correlation? causation?
  • what factor can affect our results?
  • other studies - are they comparable? are they reliable? what are the results? do they agree? why/why not? how they differ?
  • what can we add to the topic?
  • how can we present our results?

Example - attendance vs grades

Research results: Better attendance leads to better success.

Source: Kassarnig V, Bjerre-Nielsen A, Mones E, Lehmann S, Lassen DD (2017) Class attendance, peer similarity, and academic performance in a large field study. PLoS ONE 12(11): e0187078. doi.org/10.1371/journal.pone.0187078


Why quantitative reasoning?

To answer scientific questions

  • Quantitative: the arguments involve numbers
  • Reasoning: we want to take a scientific approach

In this course, you will learn

  • Why and how mathematics is important to seemingly non-mathematical sciences
  • How to use mathematical methods to answer scientific questions
  • How to present those answers


Quantitative Reasoning

Source: Elrod, Susan. 2014. Quantitative Reasoning: The Next “Across the Curriculum” Movement. Peer Review 16 (3).



Examples

  • Vaccines 💉

Image source: mRNA vaccines - What they are and what they are not



Examples

  • Climate change 🥵

Data Visualization of Temperature Anomalies:
Global and Hemispheric Monthly Means and Zonal Annual Means

Reference:
GISTEMP Team, 2025: GISS Surface Temperature Analysis (GISTEMP), version 4. NASA Goddard Institute for Space Studies. Dataset accessed 2025-11-05 at https://data.giss.nasa.gov/gistemp/.


Examples

  • Green energy 🌏🌲✅

Examples

  • Elections 🗳️

Examples

  • Artificial Intelligence 🤖

DALL-E, Jan 2023: Hierarchical Text-Conditional Image Generation with CLIP Latents



Examples

  • Artificial Intelligence 🤖

ChatGPT, Jul 2025



Examples

🌏 Global:

  • Vaccines 💉
  • Climate change 🥵
  • Green energy 🌏🌲✅
  • Elections 🗳️
  • Artificial Intelligence 🤖




📌 Local:

  • Is it better to go to uni by public transport 🚞 or by car 🚗?
  • How much toilet paper 🧻 does my household use in a year?
  • Which company offers the best price for Wi-Fi?
  • How much could I save 💰 if I made my own coffee instead of buying it every day?


Course Topics

  • Maths: Numbers and data, Inferring from data, Functions (weeks 2-5)
  • Critical Thinking, Logical Arguments, Approximations (weeks 6-8)
  • Communicating science and scientific writing (weeks 9-10)
  • (Big) Data Science (week 11)




What will we look at? - Maths

  • Numbers and data
  • Ratios, related rates, linear relations
  • Inference from data sets
  • Functions we use to represent data

What will we look at? - Science

  • Communicating science:
    • How do we structure a scientific argument? (how do I write up my own scientific findings)
    • How do we critique a scientific argument? (how do I work out whether I agree with someone else's)
    • How do we present scientific data? (why are pie charts so terrible?)

Image source: Wikipedia


What will we look at? - Science

  • Problem posing and problem solving:
    • Converting word problems into mathematical problems
    • Getting approximate answers (e.g. how much water do we flush down the toilet each year?)

What will we look at? - Science

  • Data Science:
    • Working with large data sets
    • Deeper inference from data sets




Image source: Wikipedia

Some nice references


What is science?

🤔

What is science?

... a system of knowledge covering general truths or the operation of general laws especially as obtained and tested through scientific method

Merriam-Webster Dictionary (2025)

🧐

What is science?

... a system of knowledge covering general truths or the
operation of general laws especially as obtained and tested through scientific method

-Merriam-Webster Dictionary (2025)

The use of evidence to construct testable explanations and predictions of natural phenomena, as well as the knowledge generated through this process.

The National Academy of Sciences (2008)

Science alone of all the subjects contains within itself the lesson of the danger of belief in the infallibility of the greatest teachers in the preceding generation... As a matter of fact, I can also define science another way: Science is the belief in the ignorance of experts.

Richard Feynman (1968)

The net of science covers the empirical universe: what is it made of (fact) and why does it work this way (theory).

Stephen Jay Gould (1997)

Science is not an encyclopedic body of knowledge about the universe. Instead, it represents a process for proposing and refining theoretical explanations about the world that are subject to further testing and refinement. But, in order to qualify as ‘scientific knowledge,’ an inference or assertion must be derived by the scientific method. Proposed testimony must be supported by appropriate validation—i.e., ‘good grounds,’ based on what is known. In short, the requirement that an expert’s testimony pertain to ‘scientific knowledge’ establishes a standard of evidentiary reliability

The US Supreme Court (1993) in Daubert v. Merrell

Source: Hohenberg, P.C. What is Science?


What are properties of science?
(Source: Hohenberg, P.C. What is Science?)

  • Science is collective, public knowledge
  • Science is universal
  • Science is the cumulation of (on-going) scientific endeavor
  • Science is bathed in ignorance and subject to change




What lies outside of science?
(and this course)

  • Moral issues
  • Existential issues
  • Political issues




The Scientific Method


Comments on the Scientific Method

Observation:
  • Distinguish facts from assumptions
  • Avoid (un)consciously favoring expected result (confirmation bias)
  • Use other available data, beyond what we find ourselves
  • Extraordinary claims require extraordinary evidence





Comments on the Scientific Method

Explanation:
  • Correlation does not mean causation
  • Different forms of explanation
    • Cause
    • Causal mechanism
    • Laws
    • Underlying processes
  • Occam's Razor - we prefer explanations that introduce the fewest new questions/assumptions
  • Experimentation - design tests to minimize false confirmation and false rejection
  • Negative results are part of the Scientific Method
  • The Scientific Method isn't always linear


Quantitative Reasoning and the Scientific Method

👉 Maths/stats to evaluate quantities and provide data 💻 📈 📊

Statistical approaches to infer...

  • ...how related are two quantities?
  • ... how likely is an outcome given an underlying theory?


Mathematical models to investigate causation

  • Can we explain/predict results using a mathematical model of underlying processes?


Doing maths in the 21st century

In QR, we will focus on understanding how to set up mathematical problems.

  • We can outsource solving those calculations to machines 🤖 where possible 😃 💻
  • We'll use online calculators (wolframalpha.com ), mathematical software (Excel)
  • Yes, we will use Excel a lot! 😃 💻 📈 📊 📉
🦾


Assumed knowledge

📖

Assumed knowledge

  • Algebra (add, subtract, multiply, divide) with fractions and negative numbers
  • Work with positive, negative and fractional powers
  • Find absolute values
  • Work out whether one number is greater or less than another



Addition and Multiplication

\[ \begin{eqnarray*} 3+2 &=& \\ 43+12 &=& \\ 11 + 433 &=& \\ 7 + 7 + 7 + 7 + 7 + 7 &=& \end{eqnarray*} \] \[ \begin{eqnarray*} 3 \times 2 &=& \\ 12 \times 21 &=& \\ 40 \times 2 &=& \\ 2 \times 40 &=& \\ \end{eqnarray*} \]
\[ \begin{eqnarray*} 3+2 &=& 5 \\ 43+12 &=& 55 \\ 11 + 433 &=& 444\\ 7 + 7 + 7 + 7 + 7 + 7 &=& 42 \end{eqnarray*} \] \[ \begin{eqnarray*} 3 \times 2 &=& 6\\ 12 \times 21 &=& 252\\ 40 \times 2 &=& 80\\ 2 \times 40 &=& 80\\ \end{eqnarray*} \]



Properties of Addition and Multiplication

Commutative law:

\(a+b \) \( = b+a\qquad \qquad\) \(a\times b \) \( = b \times a\)

Associative law:

\((a+b) + c \) \(= a + ( b + c)\quad \quad \) \( (a\times b)\times c = a\times (b\times c) \)

Distributive law:

\( a\times (b+ c) \) \( = a\times b + a \times c\)



Inverse Operations

Subtracting is the inverse of adding:

\( 7 = 4+ 3 \) \(\quad \Ra \quad 7- 3 = 4 \)

Dividing is the inverse of multiplying:

\( 8 = 2\times 4 \) \(\quad \Ra \quad 8 ÷2 = 4 \)

Question: Are these operations commutative? 🤔




Various Kinds of Numbers

Negative Numbers and Fractions

Subtracting a negative number is the same as adding that number, e.g.

\(5-(-4) \) \(=5+4 \) \(=9\)

Multiplying and dividing fractions:

\(\dfrac{a}{b}\times \dfrac{c}{d} \) \(=\dfrac{ac}{bd}\)

\(\dfrac{a}{b} ÷\dfrac{c}{d} \) \( = \dfrac{a}{b} \times \dfrac{d}{c} \) \(= \dfrac{ad}{bc}\)

Adding and subtracting fractions:

\(\dfrac{a}{b}+ \dfrac{c}{d} \) \( =\dfrac{ad+ bc}{bd}\)

\(\dfrac{a}{b}- \dfrac{c}{d} =\dfrac{ad- bc}{bd}\)


Exponents (aka Powers or Indices)

Repeated multiplication:

\(\large 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2 \times 2\)

\(\large \underbrace{2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2 \times 2}_{10-\text{times}}\)

\(\large \underbrace{2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2 \times 2}_{10-\text{times}} = 2^{10}\)

If you multiply two numbers with exponents, you add the exponents:

\(\large 3^{10}\times 3^{5}\) \(\large = 3^{10+5}\) \( \large = 3^{15}\)

If you exponent twice, you multiply the exponents:

\(\large \left(3^{10}\right)^{3}\) \(\large = 3^{10}\times 3^{10}\times3^{10}\) \( \large = 3^{10+10+10}\) \( \large = 3^{30}\)


Roots as Rational Exponents

One inverse of exponents is taking roots:

\(\Large 2^{3}=8\quad \) \( \Large \Ra \quad \sqrt[3]{8}=2\)

Taking roots can be considered as 1/exponent:

\(\Large \sqrt{3}\) \( \Large = \sqrt[2]{3}\) \( \Large = 3^{\frac{1}{2}}\)

\(\Large \sqrt[3]{8}\) \(\Large = 8^{\frac{1}{3}}\)

\(\Large \sqrt[5]{7}\) \(\Large = 7^{\frac{1}{5}}\)

⚠️ Careful! This is only really well-defined for positive numbers and odd roots of negative numbers


Logarithms

The other inverse of exponents is taking logarithms: It tells us what power we need to raise a base to in order to get a certain number.


Logarithms

  • \( \log_2 8 = 3\; \) because \( \;2^3 = 8 \)
  • \( \log_{10} 1000 = 3 \;\) because \( \;10^3 = 1000 \)

The natural logarithm has base $e$ and is always denoted "$\ln$"

Sometimes "$\log$" on its own means "$\log_{10}$", or sometimes "$\ln$"

👉 always check‼️



Absolute value

The absolute value of a number is:

  • that number again if it is positive

  • minus that number if it is negative.

\[|3|=3\]

\[|-5|=5\]





Equations & Inequalities

This is an equation:

\(x+3 = 8.\)

To solve for $x$ we have:

\(\Ra x+3-3 = 8-3\)

\(\Ra x = 8-3\) \(=5\)


This is an inequality:

\(x \gt 3.\)

We can perform the common operations:

\(\Ra x-3\gt 0\)

\(\Ra 3x\gt 9\)

\(\Ra x^3\gt 27\)

\(\Ra \log_{10}x\gt \log_{10}3\)



Practice 📝

  • Is \( - 3 > 5 \)?
  • Compute \( 17 - (-12)\times( -3+7) \)
  • What is \( \dfrac{17}{32} \) as a percentage?
  • Compute \( \dfrac{1}{8} - \dfrac{1}{7} \)
  • Solve \( e^x = 40. \)
  • Solve \( 3x - 5 = 2x + 6. \)
  • Solve \( x^2 + 2x - 6 = 0. \)

Simplify but use only positive powers:

  • \( x^3\times x^2\)
  • \(y^5 ÷ y^8\)
  • \( 4^{-1/2} \)
  • \( \dfrac{a^2b^3}{ b^2a^9} \)
  • \( 27^{1/3} \)
  • \( 5^{-3} \)
  • \( 343^{-1/3} \)

Practice 📝

  • Is \( - 3 > 5 \)? No
  • Compute \( 17 - (-12)\times( -3+7) \) = 65
  • What is \( \dfrac{17}{32} \) as a percentage? 53.125%
  • Compute \( \dfrac{1}{8} - \dfrac{1}{7} \) \(=-\dfrac{1}{56}\)
  • Solve \( e^x = 40. \) \(\; x = \ln(40) \approx 3.689\)
  • Solve \( 3x - 5 = 2x + 6. \) \(x = 11\)
  • Solve \( x^2 + 2x - 6 = 0. \) \(x = -1 \pm \sqrt{7}\)

Simplify but use only positive powers:

  • \( x^3\times x^2\) \(=x^5\)
  • \(y^5 ÷ y^8\) \(=y^{-3} = \dfrac{1}{y^3}\)
  • \( 4^{-1/2} \) \(=\frac{1}{2}\)
  • \( \dfrac{a^2b^3}{b^2a^9} \) \(=\frac{b}{a^7}\)
  • \( 27^{1/3} \) \(=3\)
  • \( 5^{-3} \) \(=\frac{1}{125}\)
  • \( 343^{-1/3} \) \(=\frac{1}{7}\)

That's all for today!

See you in Week 2!

😶