Data Containers Worksheet
Data Containers
Linear Algebra // Intro to Vectors
Student Name:
Date:
1. The Hook: Spreadsheets as Space
Consider the following sample from a customer database:
| Cust_ID | Age | Income_k | Spending_Score | Visits_Per_Mo |
|---|
| 001 | 19 | 15 | 39 | 4 |
| 002 | 21 | 15 | 81 | 12 |
| 003 | 20 | 16 | 6 | 1 |
Discussion Prompt:
If a 2D vector represents a point in a plane $(x, y)$, how does a single row in this spreadsheet relate to our concept of a vector? What dimension would these vectors inhabit?
2. Abstraction: From Arrows to Lists
Watch the introductory segment of the video (0:00 - 4:15). Focus on the "Mia's Catering" example.
Key Vocabulary
n-dimensional Vector:
Dot Product (Interpreted):
The Catering Vectors
Define the vectors $q$ (quantity) and $p$ (price) based on the video example:
$q = $
----
$p = $
----
3. Project: The Nutrition Ledger
You are a data scientist for a nutrition app. You need to calculate the total nutritional intake for a specific meal plan using vector operations.
Dataset: Nutrition per Serving ($\vec{v}$)
| Food Item | Calories ($c$) | Protein ($p$) [g] | Fat ($f$) [g] | Carbs ($k$) [g] |
|---|
| Salmon ($\vec{s}$) | 200 | 20 | 13 | 0 |
| Quinoa ($\vec{q}$) | 120 | 4 | 2 | 21 |
| Broccoli ($\vec{b}$) | 30 | 2 | 0 | 6 |
Step A: Setup
Represent a meal consisting of 2 servings of Salmon, 1.5 servings of Quinoa, and 3 servings of Broccoli as a linear combination of the nutrition vectors.
Total Vector $\vec{T} = \_\_\_ \vec{s} + \_\_\_ \vec{q} + \_\_\_ \vec{b}$
Step B: Calculation
Show your work for calculating the final resultant vector $\vec{T}$. Round to the nearest tenth.
Step C: Dimensional Analysis
Explain why we consider this a 4-dimensional problem. If we added "Price per Serving" as a new column, how would the geometry of our vector space change?
4. Critical Reflection
In the video, the narrator mentions that vectors can represent "any number of quantities." In the context of modern data science (think of a recommendation algorithm for movies or a medical diagnostic tool), why is the ability to work in high-dimensional space ($n > 3$) more useful than simple 2D or 3D visualization?
Data Vectors Rubric
Assessment Rubric: Data Vectors
Linear Algebra // Visualizing Data Containers
| Criterion | Emerging (1-2) | Proficient (3) | Exemplary (4) |
|---|
| Conceptual Abstraction | | | |
| Understanding $n$-dimensional vectors as data containers. | | | |
| Confuses geometric arrows with abstract data; struggles to define dimensions in the context of the spreadsheet or video. | Accurately describes how rows/columns represent vectors and correctly identifies dimensionality. | Articulates a clear shift from spatial to abstract reasoning; explains how vector spaces scale with added data features. |
| Vector Modeling & Setup
Setting up linear combinations for the nutrition project.
| Incorrectly maps data to vector components; coefficients in the linear combination are misplaced or omitted. | Correctly represents the meal plan as a linear combination of food vectors with appropriate scalars. | Setup is flawless and demonstrates precise mathematical notation; handles fractional servings (1.5) accurately. |
| Computational Precision
Scalar multiplication and vector addition.
| Significant errors in component-wise addition or scalar multiplication; incorrect final resultant vector. | Minor calculation errors (e.g., rounding) but overall methodology is correct. Final vector is largely accurate. | Perfect execution of vector arithmetic; all components of the resultant vector are mathematically sound and checked. |
| Reflection & Application
Connecting linear algebra to modern data science.
| Reflection is superficial; fails to connect high-dimensional space to real-world utility like algorithms or medicine. | Connects $n$-dimensional math to data science; provides at least one specific example of why high dimensions matter. | Deep analytical response; discusses the limitations of human perception vs. the efficiency of linear algebra for machines. |
Total Assessment Score
Target: Mastery (14-16 points)
/ 16
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