Covariation & Rates of Change — Study Notes (Time-Distance and Vase Problems)
Covariation & Rates of Change
Imagining two quantities changing in tandem while attending to how they change in relation to each other.
Phrase from slides: "Imagining two quantities changing in tandem while attending to how they change in relation to each other" (Thompson, 1994; Carlson et al., 2002).
Core ideas: covariation (two quantities changing together) and rates of change (how one quantity changes with respect to the other).
Covariation with Time & Distance (core relation examined in this module).
Time and distance are a pair used to explore covariation.
Usain Bolt 100 m race as an implicit example: distance vs. time relation during a sprint.
Quantity and Attribute concepts (through a concrete example).
Quantity: a measurable attribute of an object.
Questions to define a quantity:
What is the object I’m referring to?
Can I imagine measuring this attribute?
Example: Kacie is 5 feet tall.
Object: Kacie; Attribute: height (measurable).
What is an attribute of Kacie I cannot measure? Eye color (an example attribute that isn’t a straightforward measurable quantity in this context).
Key terms to understand (from the video walk-through): Covariation and Quantity.
Covariation: how two quantities change together.
Quantity: what is being measured.
Covariation with Time & Distance: aim of the section.
Explore how time and distance co-vary and how to describe their relationship.
Walking graphs: graphing distance from a reference point vs. time elapsed.
Reference point: choose a fixed point in the room (base of a wall, a desk, etc.). Distances are measured from this point.
You imagine moving in a straight line toward/away from the reference point.
For each graph:
A) Describe your rate of change of distance with respect to time as time increases.
B) Describe how the amounts of change in distance and time co-vary as time increases by equal amounts.
Graph 1, Graph 2, Graph 3: co-variation models (described in subsequent slides).
Graph 1: Instructions for describing rate of change and co-variation.
Graph 2: Similar instructions; practice before checking the video.
Graph 3: More detailed description; includes interpretation of direction (away from wall, toward wall) and rate changes.
Graph 3 details: interpreting rate of change and direction.
At time 0: you are away from the wall (not at the wall yet).
From time 0 to time A: you walk away from the wall at a decreasing rate (the magnitude of distance increase per second is getting smaller).
From time A to time B: you walk toward the wall at a decreasing rate (the magnitude of distance decrease per second is getting smaller in the negative direction).
Points A and B sit on the graph: A marks a maximum of distance change; B is the horizontal intercept.
Note on speed vs rate: speed is the absolute value of velocity, i.e., speed = . The rate of change can be negative (moving toward the reference point) or positive (moving away).
The slope between A and B may be more negative (distance decreasing more rapidly), yet the speed is still a positive quantity when discussed as magnitude.
Additional resources for deeper understanding.
If you want a more detailed explanation of the concepts in Graph 3, there is a video: https://youtu.be/TFbFNUiHajU?si=zXEpbl8zDyrcwagi
Summary: Amounts of Change vs. Rates of Change (language and interpretation).
There are two related but distinct ways to describe the relationship between two quantities:
Amounts of Change language: describes how much one quantity changes for given changes in another quantity (e.g., for equal time intervals, distance changes by equal amounts).
Rates of Change language: describes how fast the quantity changes with respect to another (e.g., distance changes at a constant rate with time).
The difference:
Amounts of Change focuses on the magnitude of change over a fixed interval:
For equal increments in time Δt, ΔD is constant → the distance increases by the same amount in each equal time interval.This description emphasizes the actual amount of change per interval.
Rates of Change focuses on the rate of change per unit time:
As time increases, the distance is increasing at a constant rate →
rate = .This description emphasizes the slope/speed of the change.
Notation and relationships:
If distance is D(t) and time is t, the instantaneous rate of change is the derivative: .
For equal time steps Δt, the average rate is ; when Δt becomes infinitesimally small, this approaches .
Speed is the non-negative magnitude of velocity: .
Covariation with Bottle Problems: extending the idea to other quantities.
Instead of distance and time, consider height of water in a vase (H) and volume of water (V).
The core idea: describe how height and volume co-vary using either amounts of change language or rates of change language.
Cylinder Vase (a simple case): height vs volume.
Part A: Sketch a possible graph of height H as a function of volume V for the cylinder vase.
Part B: Describe how H varies with equal amounts of water added (i.e., equal ΔV) using amounts-of-change language.
Part C: Describe how V and the rate of change of height dH/dV vary together.
Cylinder Vase – Solutions (summary):
A: The sketch is consistent with a cylinder: the height vs volume graph is a straight line if the cross-sectional area is constant.
B: For equal amounts of water added (equal ΔV), the height increases by equal amounts (ΔH is constant).
C: As the volume increases, the height increases at a constant rate with respect to volume: .
Round Vase Problem: a vase with a rounded shape (not constant cross-section). A more nuanced analysis is required.
Part A: Sketch a possible graph of height H as a function of volume V for the round vase.
Part B: Describe H’s variation with equal ΔV using amounts-of-change language.
Part C: Describe how V and the rate of change of height dH/dV vary together; note: you may not use words like speed, faster, slower, acceleration, or deceleration in your explanations.
Round Vase – Solutions (summary for Part B and C):
B (amounts of change language): three sections describe how height changes with volume:
Section 1 (bottom section): for equal ΔV, ΔH is positive but increasing at first or increasing by smaller amounts (the transcript states: "the height increases by smaller and smaller amounts" for the first section).
Section 2 (middle/narrowing part): for equal ΔV, ΔH increases by larger amounts (the changes in height are positive and increasing).
Section 3 (top section): for equal ΔV, ΔH increases by equal amounts (the changes in height are the same each time).
C (rates of change language): as V increases, dH/dV behaves piecewise:
In Section 1: height increases at a decreasing rate with respect to volume (positive but decreasing slope).
In Section 2: height increases at an increasing rate with respect to volume (positive and increasing slope).
In Section 3: height increases at a constant rate with respect to volume (constant slope).
Interpretive notes about the vase graphs:
The graph’s slope corresponds to the vase’s cross-sectional area at the corresponding heights/volumes.
The narrowest part of the vase corresponds to the region where the rate of change of height with volume is greatest (steepest slope).
The corners/sections where the slope changes reflect changes in the vase’s width (wider sections yield smaller rises in height for the same added volume; narrower sections yield larger rises).
The final section (neck) shows a constant rate because the vase’s cross-sectional area becomes effectively constant again.
Part B continued (qualitative descriptions):
Section 1: for equal ΔV, ΔH is positive and small (increasing with volume? per transcript: “height increases by smaller and smaller amounts” in Section 1).
Section 2: for equal ΔV, ΔH is larger (increasing rate).
Section 3: for equal ΔV, ΔH is constant (equal changes in height per equal ΔV).
Part C continued (relationship between volume and rate of change of height):
Early in filling (before widest part): height increases with volume at decreasing rate (dH/dV decreases).
As you fill toward the widest part: height increases with volume at increasing rate (dH/dV increases).
After the widest part (toward the neck): height increases with volume at a constant rate (dH/dV = constant).
Visualization and drawing guidance:
When asked to sketch the vase from the graph, mark landmarks on both the vase and the graph to indicate where changes in the graph correspond to changes in the vase’s shape.
Possible solution idea: the narrowest part is where the height’s rate of change is greatest; a cylinder-like bottom implies constant rate initially; at the corner (transition to a wider section) slope decreases; this reflects widening of the vase.
Final observation from the slides:
Bottles, vases, and containers with different shapes can still share the same three key parts (bottom, middle, neck) that determine the qualitative covariation between height and volume, as reflected in their height-volume graphs.
Cross-cutting ideas for exam prep:
Distinguish between Amounts of Change language and Rates of Change language when describing graphs.
Be able to describe covariation using both time-distance graphs and height-volume graphs.
Use derivative terminology to connect intuitive descriptions (slope, rate) with precise mathematical language (dD/dt, dH/dV).
When interpreting piecewise graphs (like the round vase), be prepared to identify where the cross-sectional area changes and how that affects the slope of the height-volume function.
Title
Notes on Covariation & Rates of Change in Time-Distance and Height-Volume Contexts