MAT133: Calculus and Linear Algebra for Commerce
Project #4 Instructions Booklet
Introduction
Project 4 is an interface between mathematics and the arts. There are two main components:
. You’ll use mathematics to investigate the beneits of engaging in the arts, by analysing survey data. Linear regression is the tool you’ll use to analyse this data. In your Week 21 Tutorial and in the individual portion of the project, you’ll use multivariable calculus to derive and make sense of this tool.
. You’ll also use art to improve your understanding of a mathematical idea from the course. In particular, you’ll create a conceptual metaphor for a mathematical idea from MAT133 of your choice, then construct an art piece to represent this metaphor.
This project is based primarily on the learning goals from Weeks 15 - 20, but also may touch on any learning goal from MAT133. There are several pieces to the project: the pod contract, the individual part, the pod project, the presentation, and the relection. We strongly recommend that all students inish the individual part before starting the pod project. This document outlines the instructions for accessing and submitting each part.
. The individual part and the pod contract is due on Tuesday, March 19, at 10pm.
. Your irst draft for the pod project report will be due on Tuesday, March 26, at 10pm. . The inal report (including your art piece) is due on Tuesday, April 2, at 10pm.
. You will present your art pieces in a gallery during your tutorial on April 4-5. You are not required to submit presentation slides this time. Your Hive TA will leave feedback on the art piece as submitted in your main report.
. The relection questions are to be included with your regular weekly relections, to be released on Friday, April 5 and due on Tuesday, April 9, at 10pm.
Extended late deadlines are provided in Gradescope to accommodate for technical issues without penalty. No late submissions will be accepted via email.
IMPORTANT: When you submit your pod project report, you will be asked to select which pages correspond to which section. It is very important that you select these pages; otherwise your report will be very difficult for us to grade. You may lose points if you skip this step.
1 Individual Submission
Along with your solutions to the individual problems, you’ll need to include an academic integrity statement.
1.1 The Code and the OK List
The UofT upholds high standards of academic integrity. It is your responsibility to read and understand the Code of Behaviour on Academic Matters and to adhere to the list of “OK” resources below.
The OK List
This OK list is a closed list of allowed resources, not just a list of examples. If you are unsure of what you are allowed to use, do not hesitate to ask on Ed.
OK: Collaboration with other members of your pod – this is encouraged for all parts of this project and required for the Pod Project portion.
OK: You may discuss the project with other students outside of your pod if you are in Drop-in Hours or
Tutorials, as long as you write up your own analysis and report without copying from anyone else.
OK: Anything that can be found on the MAT133 Quercus page
OK: Your own and your pod member’s previous MAT133 work, including any projects, tests, or homework
OK: Your own and your pod members’notes from this course and your other courses
OK: Any other textbook (online or physical) you have access to
OK: Online learning videos (e.g. Khan Academy)
OK: Data sources including, but not limited to, those listed in Appendix A
OK: General advice on the MAT133 Ed Discussion page (e.g. “Can someone please help me understand RREF?”)
OK: wolframalpha.com, desmos.com, math3d.org, any calculator
OK: Anything else declared as OK in a written announcement by the course coordinator
Examples of Not OK Things
Here are some examples of things that are not OK. These are just examples. Unless something is on the OK list, it is not OK.
NO: Communicating about the project with anyone not in your hive. This means, for example, that you must not use group chats to share project content if these group chats involve anyone who is not in your hive.
NO: Asking for answers on Ed (e.g. ”how do you solve Question 2...”)
NO: Accessing or posting on so-called “tutoring” websites or services like chegg.com or Easy 4.0
NO: Using online forums like stackexchange
NO: Entering the question text into a search engine
NO: Using generative AI systems like ChatGPT
An important note on group chats: If you have administration privileges for any online chat that involves anyone not in your hive and in which assignment content is shared, you must delete any non-authorized content from that group chat as soon as you see it (if this is technologically possible on that platform). Otherwise, you are considered to have helped someone cheat and therefore committed an academic ofence.
As part of your individual project submission, you’ll include an abbreviated statement to Gradescope that implies the following.
On the front page of your individual submission, you’ll be asked handwrite and sign the abbreviated academic integrity statement, as follows:
“I have read the project instructions. I know the Code. I didn’t cheat. This is my work. I only used “OK” aids. I
pledge upon my honour that I have not violated the Code during this assessment.”
Include your student ID number, the date, and your signature at the bottom. Here is an example:
The individual problems are to be completed by each student individually. You are allowed and encouraged to discuss these problems with other members of your hive, but you must write your own solution for submission.
Please note the following general policies for the individual problems:
. You must handwrite directly into the problem template, using either a tablet or printer and scanner.
. Check that the boxes in your submission roughly line up with the boxes the template when you submit to Gradescope.
. Your solutions should written clearly and concisely in a linear fashion. Do not submit messy scrapwork. . Explain all your steps. Your solution should be easy to understand by any other student in the class.
. When applicable, please state your inal answer in the form of a sentence, including units.
The individual problems will be accessed and submitted on gradescope.ca.
The pod contract is your mutual agreement on how you will productively collaborate on your project. It is essential to plan your work in advance, to ensure that you’ll have enough time to meet and work together, give each other feedback, and make improvements. If a dispute arises in your pod, you should refer back to your contract to guide your resolution. Please access the pod contract in Gradescope.
Note: Even if you have decided to work individually (as a pod of 1) for this project, please still submit a pod contract. This will help us to know that you plan to work on your own and it will also help you to map out your work on Project 4.
Please access and submit the pod contract in Gradescope.
. Handwrite directly into the contract template, using either a tablet or printer and scanner.
. Check that the boxes in your submission roughly line up with the boxes the template when you submit to Gradescope.
3 Pod Project: Mathematics and Art
3.1 Introduction to the Project
Project 4 is an interface between mathematics and the arts. There are two main components:
. Part I: You’ll use mathematics to investigate the beneits of engaging in the arts, by analysing survey data. Linear regression is the tool you’ll use to analyse this data. (In the Week 21 Tutorial and in the individual part of the project, you used multivariable calculus to derive and make sense of this tool.)
. Part II: You’ll also use art to improve your understanding of a mathematical idea from the course. In particular, you’ll create a conceptual metaphor for a mathematical idea from MAT133 of your choice, then construct an art piece to represent this metaphor.
3.2 Part I: Investigating the Impacts of Engaging in the Arts
The HEartS Survey 2019 collected data on demographics, engagement in a variety of art forms, as well as health and social data, from a representative sample of 5,338 adults in the UK. For more information about this data set, please see Williamon, Aaron et al. (2021). HEartS Survey 2019: Charting the health, economic, and social impact of the ARTs [Dataset]. Dryad. https://doi.org/10.5061/dryad.3r2280gdj.
The data set linked in the Project 4 page comes from HEartS Survey 2019. The MAT133 Teaching Team has curated, renamed, and colour-coded some of the variables to make them easier for you to read, but we’ve generally preserved the order of the variables, so that you can easily refer back to the original data set and compare if needed. If you have questions about the data set, start by checking out the HEartS Survey 2019 webpage.
In the individual part of the project, you used linear regression to investigate a relationship between one independent (input) variable and one dependent (output) variable. In this part of the project, you’ll investigate a relationship between two independent variables (x and y) and one dependent variable (z). Each data point now becomes a point (x,y, z) in 3-space. To investigate the relationship, we’ll use linear regression ind a plane of best it for the data set. This plane minimizes the sum of squares of vertical distances between your data points and the plane.
Luckily, Excel will be able to take care of the bulk of the numerical calculations for you! Your main job will be to make decisions about what to investigate, how to interpret and communicate your results. The instructions below will guide you through the process.
3.2.1 Using LINEST for Creating a Plane of Best Fit
LINEST is a linear regression tool in Excel that you’ll be using for this project. To help orient you with LINEST, we’ve created a worksheet for Excel that generates random data sets and provides sample uses of LINEST.
Start by reading at the top of the sheet. Follow the instructions.
. The green rows contain instructions or questions.
. The blue cells contain numbers that represent slopes or intercepts. You can change these numbers in order to generate a new random data set below.
. The yellow cells contain LINEST outputs. You can click on the leftmost cell to see the LINEST formula that generated these outputs.
It might take around 10 minutes or so to work through the sheet and make sense of it. After you’re inished, take note:
. The formula used for the 2-input case was =LINEST(C36:C46 ,A36:B46,TRUE,FALSE).
– The irst argument, C36:C46, represents the column of outputs.
– The second argument, A36:B46, represents the two columns of inputs.
– The third argument, TRUE, means that the vertical intercept b should be estimated. (If FALSE were instead entered here, Excel would assume that b = 0.)
– The fourth argument, FALSE, means that we are not asking for LINEST to share additional regression statistics. (We won’t use them for this project. You might learn more about these statistics if you take a statistics course in the future! You can feel free to change this variable to TRUE if you are curious and would like to learn more.)
. IMPORTANT: Take note of the order of the yellow outputs at the bottom of the page! The two slopes mx and my appear in the opposite order from what you might expect!
For the 2-input case, the output of LINEST determines the equation for a plane of best it,
z = mxx + my y + b
Please note that mx and my are constants that represent the slopes of this plane in the directions of the positive x and y-axes respectively. The subscripts are only a label; they do not represent partial derivatives.
You are not required to submit this LINEST worksheet with your report. This worksheet is only provided to help you learn how to use and read the output from LINEST as you complete the project.
The HEartS Survey 2019 Variables ile contains a list of all variables in the Hearts Survey 2019 data set for Project 4, which comes from HEartS Survey 2019. The MAT133 Teaching Team has curated, renamed, and colour-coded some of the variables to make them easier for you to read, but we’ve generally preserved the order of the variables, so that you can easily refer back to the original data set and compare if needed. If you have questions about the data set, start by checking out the HEartS Survey 2019 webpage.
You will be using linear regression with LINEST to investigate possible relationships between these variables. As you read through the HEartS Survey 2019 Variables, what questions come to your mind about how these variables are related to each other? Start by writing down a few of your own questions before discussing them with your pod.
In particular, please note that each person in your pod will create a contour map representing the plane of best it for three variables in the data set. For each case, you’ll think of one variable as dependent on two independent variables.
Examples of variables related to engaging in artistic activities
. artspend represents the amount of money spent (in GBP) on arts/culture in the past month.
. Participants were asked if and with what frequency they engage in a variety of participatory art activities (such as writing, painting, or playing musical instruments) and receptive art activities (such as attending art galleries, theatres, or live music events); variables related to these questions are highlighted in grey. Their overall engagement in the arts is measured by their art overall score, which represents the total number of artistic activities that participants engage in (with any nonzero frequency), as well as their art participation score and art reception score, which represent the total numbers of participatory and receptive artistic activities that participants engage in.
Examples of variables related to health
. The Mental Health Continuum SF score, mchcscore, is an overall measure of mental health well-being, based on a 40-item survey. This score is broken down into emotional, psychological, and social components, in the variables mhchedonic, mhcsocialw and mhcpsycholw.
. The depression scale variable represents the Centre for Epidemiologic Study Depression scale (CES-D) short form 8-item scale.
. social connectedness is a score between 0 to 75 on the Social Connectedness Scale. The number is found by adding up the scores on 15 items (highlighted in grey), then subtracting 15.
. loneliness UCLA and loneliness djgare two diferent measurements of loneliness, on the UCLA Loneliness Scale and the De Jong Gierveld Loneliness Scale.
The lists above are not exhaustive. Please see the variable list and data set for more variables.
Categorical variables for iltering
There are several variables in the data set that are categorical or take a small number of values. For example, the art-profession variable is equal to 1 if the participant’s profession is in the arts and 0 otherwise. The education category variable indicates the participant’s educational level in the UK. You might wish to use variables like this as flters, to explore the behaviours and health outcomes of certain groups. In Excel, you can use the Sort & Filter function to ilter the data set, to only view participants with variables that take a certain value – for example, you could ilter the art-profession variable to view only the data corresponding to art professionals. However, please take note before iltering:
. CAUTION: Select the entire data set before iltering or sorting. Otherwise, you risk mismatching the data to the participants.
. CAUTION: Be mindful of the iltered sample size. Whenever you ilter the data set, you reduce the number of participants under consideration. Small samples are more prone to show patterns that are not representative of the overall population. Please ensure that you always have at least 50 rows in any iltered data set that you run a linear regression on.
. CAUTION: If you ilter the data, please be sure to copy and paste the iltered data, including all three variables, to a new spreadsheet before analyzing it.
3.2.3 Research Question or Theme
After viewing the variables, meet with your pod to decide on what you want to learn from this data set when you ind your planes of best it. As a pod, you must have a coherent research question or theme that guides the choices for the variables used in your planes of best it.
Your pod members’variable choices need not be totally disjoint. For example:
. Each person in your pod could investigate the same variables, but for diferent segments of the population (by iltering).
. Each person in your pod could investigate the same dependent (z) variable as a function of diferent indepen- dent variables.
. Each person in your pod could use the same independent (x, y) variables, but investigate diferent outputs.
There are many other possibilities, not only these! Our goal in listing these examples is to clarify that you don’t each need to choose completely separate variables. The important thing is that your pod has a common research question or theme, which is coherently addressed by each of your planes of best it.
3.2.4 Analysis and Contour Maps
Before starting your analysis, make sure you’ve completed the brief LINEST worksheet described in Section 3.2.1.
Each member of the pod should carry out a linear regression using LINEST and create a contour map, as described below.
Once you’ve chosen the variables (with or without ilters) that you wish to investigate, we recommend copy/pasting this data into a fresh spreadsheet. Then carry out your linear regression using LINEST. (Remember that the two slopes in the output are in the opposite order to what you’d probably expect.) With your pod, verbally discuss these results. What do they mean? Do they match your intuition? (If you’re working alone, discuss with a classmate or just write down your thoughts informally.)
Use the output of LINEST to write down a formula for the plane of best it. The formula should be of the form,
z = mxx + my y + b,
where the constants my , mx and b are the numbers that come from the output of LINEST. The variables x,y and z should be replaced with your independent variables and dependent variable respectively.
Create a contour map to graphically represent your plane of best it. You may use Desmos or another tool. Please make sure that the viewing window (i.e. the range of values for each variable) is appropriate to the variables you’ve chosen. Label your axes, including numerical scales, words, and units (if appropriate). Label your contour values as well. Ensure that there is a consistent spacing between the z-values of your contours.
Your report should include the following sections.
1. Introduction. Write about 1-2 paragraphs to introduce the reader to your project. Your introduction should at least address the following (in any order):
. What is your research question? Explain why you are interested in this research question, including any relevant context.
. Briely summarize how you will use the HEartS 2019 Survey data in order to address your research question.
. Cite the original dataset as “Williamon, Aaron et al. (2021). HEartS Survey 2019: Charting the health, economic, and social impact of the ARTs [Dataset]. Dryad. https://doi.org/10.5061/dryad.3r2280gdj”
2. Analysis and Discussion. Your analysis and discussion section should include a segment written by each pod member. (For example, if you have three pod members, there should be three segments.) Within each pod member’s segment, please ensure that the following elements are included:
. Names and descriptions for each of the three variables under study. Indicate which variables you are considering to independent and which you are considering to be dependent.
. Outline the results of your linear regression using LINEST. Express your results in the form of a linear equation involving the three variables (with the appropriate variable names, not x, y and z).
. A well-labeled contour map for a plane of best it, as described in Section 3.2.4. Ensure that the viewing window is appropriate, the axes are well-labeled, and the contour values are labeled.
. Provide an interpretation for each constant, in language appropriate for the general public.
. Discuss your results. Intuitively, how did you expect the variables to be related to each other? In what ways (if any) did your observations align with your expectations? In what ways (if any) did your observations disagree with your expectations?
. Relate your results back to your pod’s research question.
3. Conclusion. Briely summarize your results and discussion. Relate your results back to your research question. Suggest possible areas for future work on this topic. Be succinct and clear enough that the reader doesn’t need to read the rest of your report to understand your summary. Use language that is accessible to another student in MAT133.
3.3 Part II: Understanding Mathematics through Metaphors and Art
“Metaphor is not a mere embellishment; it is the basic means by which abstract thought is made possible. ”
– George Lakof and Rafael Nu?nez
In the second part of the pod project, you will create a mathematical art piece! The goals of this project are:
. Reinforce your understanding of the concepts you’ve learned in MAT133.
. Help your classmates reinforce their understanding of the concepts.
. Have fun!
3.3.1 Introduction: How do we make sense of abstract mathematical ideas?
In your journey through mathematics, from elementary school to MAT133, you have come into contact with a variety of abstract mathematical objects and ideas: numbers, matrices, vectors, functions, derivatives, integrals, and so on. You have drawn connections between these ideas and used them to investigate real-world problems. A person might look at the abstract symbols written across the pages in your notebook and wonder, what goes on in your head when you write this? How do you make sense of all this? This is a valid question!
As a mathematician, how do abstract ideas live in your head?
Where does your mathematical intuition come from?
According to some scholars in linguistics and cognitive science, our understanding of abstract ideas, in mathematics and elsewhere, is grounded in metaphor. In your hive meetings on March 21-22, you’ll discuss examples of metaphors you use in everyday language, as well as examples of metaphors for numbers and arithmetic. This will help you to understand what conceptual metaphors are and why they are important for understanding mathematical concepts.
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