Understanding Quantities and Their Relationships

Lesson Overview and Learning Objectives

  • The primary focus of this lesson is to understand quantities and their relationships with each other.
  • Specific Learning Objectives:
    • Understand the concept of quantities and how they relate.
    • Identify the independent and dependent quantities within a given scenario.
    • Match specific scenarios with their appropriate graphical representations.
    • Utilize a reasonable scale for graphs when modeling scenarios.
    • Identify and define the key characteristics of graphs.
    • Describe and analyze similarities and differences between pairs of graphs and their corresponding scenarios.

Fundamental Key Terms

  • Dependent Quantity: When one quantity is determined by another in a problem situation, it is called the dependent quantity.
  • Independent Quantity: The quantity from which the dependent quantity is determined is called the independent quantity.

Analyzing Relationships and Task Dependency

In many real-world situations, certain tasks or quantities depend on another being completed or established first. For example, when planning a birthday party, you might purchase ice, go grocery shopping, select music, prepare food, or clean. These tasks are often sequential; for instance, you would not make food before going grocery shopping.

Identifying Independent and Dependent Quantities in Specific Relationships:
  • Relationship: The number of movie ticket purchased and the total cost
    • Independent Quantity: The number of movie tickets purchased.
    • Dependent Quantity: The total cost.
  • Relationship: The number of eggs used and the number of cakes baked
    • Independent Quantity: The number of cakes baked.
    • Dependent Quantity: The number of eggs used.
  • Relationship: The number of students in attendance at school and the number of lunches served
    • Independent Quantity: The number of students in attendance at school.
    • Dependent Quantity: The number of lunches served.
  • Relationship: The number of hours driven and the number of miles to a vacation destination
    • Independent Quantity: The number of hours driven.
    • Dependent Quantity: The number of miles to a vacation destination.
  • Relationship: The number of minutes a swimming pool is filled with water and the number of gallons of water in the swimming pool
    • Independent Quantity: The number of minutes the swimming pool is filled with water.
    • Dependent Quantity: The number of gallons of water in the swimming pool.

Connecting Scenarios and Their Graphs

Graphs on a coordinate plane serve as a visual tool to see and interpret data, such as the monthly operating costs of a business or a runner's marathon pace. Relationships between points are represented using lines or smooth curves.

  • Interpretation Note: In some problem situations, every point on a line will make sense (continuous data). In others, not all points will make sense (discrete data). It is necessary to consider the specific situation to interpret the meaning of data values shown.
  • Graphing Protocols: When labeling a graph, you must label the xx-axis and yy-axis with the appropriate quantity, provide a reasonable scale, and interpret the meaning of the origin (0,0)(0,0). Always include appropriate units of measure for each quantity.

Case Study Scenarios for Graphical Modeling

Music Club
  • Scenario: Natalia belongs to "Songs When I Want Them," an online music store. She can purchase any song she wants for a price of $1\$1 per song.
  • Independent Quantity: Number of songs purchased (units: songs).
  • Dependent Quantity: Total cost (units: dollars).
  • Ask Yourself: How can graphs be used to tell a story in everyday life?
Something’s Fishy
  • Scenario: Kaya is the building manager of an office building and is responsible for cleaning a 200 gallon200\,\text{gallon} aquarium. She removes the fish and drains the water at a constant rate of 10 gallons per minute10\,\text{gallons per minute}.
  • Independent Quantity: Time elapsed while draining (units: minutes).
  • Dependent Quantity: Amount of water in the aquarium (units: gallons).
  • Ask Yourself: What strategies will you use to match each graph with one of the six scenarios?
Smart Phone, but Is It a Smart Deal?
  • Scenario: You want an upgraded smart phone but lack the funds. Your cousin offers to lend you money if you pay him back with interest. The initial interest payment is $1\$1, and the interest amount doubles each week thereafter.
  • Independent Quantity: Time (units: weeks).
  • Dependent Quantity: Interest amount (units: dollars).
It’s Magic
  • Scenario: The Amazing Alejandro performs a trick where he cuts a 20 foot20\,\text{foot} rope in half, then takes one of those halves and cuts it in half again. He repeats this process until the piece is too small to cut.
  • Independent Quantity: Number of cuts made (units: cuts).
  • Dependent Quantity: Length of the rope piece (units: feet).
Baton Twirling
  • Scenario: Samantha, a drum major, tosses her baton into the air. It reaches a maximum height of 22 feet22\,\text{feet}. She has 2 seconds2\,\text{seconds} to twirl twice before catching the baton as it returns to the ground.
  • Independent Quantity: Time since the baton was tossed (units: seconds).
  • Dependent Quantity: Height of the baton (units: feet).
Skateboarding
  • Scenario: Andrew is skateboarding on a 4 foot-tall4\,\text{foot}\text{-tall} half-pipe. He tracks the distance of his skateboard wheels from the ground as he skates from the top of one side of the half-pipe to the top of the opposite side.
  • Independent Quantity: Time or horizontal distance along the half-pipe (units: seconds or feet).
  • Dependent Quantity: Height of the wheels from the ground (units: feet).

Analyzing Graphical Characteristics

When examining multiple graphs, it is essential to analyze them from left to right to identify specific graphical characteristics.

  • Similarities and Differences: Relationships can be linear (forming a straight line), non-linear (forming curves), increasing (moving upward from left to right), or decreasing (moving downward from left to right).
  • Specific Comparison Pairs:
    • Smart Phone vs. Music Club: Both involve money and increasing values, but one increases at a constant rate ($1\$1 per song) while the other increases exponentially (doubles each week).
    • Something’s Fishy vs. It’s Magic: Both involve decreasing values. The aquarium water level decreases at a constant rate (linear), whereas the rope length decreases by half each time (non-linear/asymptotic).
    • Baton Twirling vs. Skateboarding: Both involve height changes over time. The baton starts at an initial height, goes up, and returns to the ground (inverted parabola). The skateboarder starts at a height of 4 feet4\,\text{feet}, goes down to the bottom of the half-pipe, and then returns to a height of 4 feet4\,\text{feet} (upward-opening parabola).

Graph Cutouts Reference

  • Graph A: Represents a linear relationship starting at the origin (0,0)(0,0) and increasing at a constant rate (matches Music Club).
  • Graph B: Represents a linear relationship starting at a positive yy-intercept and decreasing at a constant rate until it hits the xx-axis (matches Something’s Fishy).
  • Graph C: Represents an exponential growth curve that starts low and increases rapidly (matches Smart Phone interest).
  • Graph D: Represents a parabolic curve opening downward, beginning and ending at the xx-axis (matches Baton Twirling).
  • Graph E: Represents a curve that starts high and decreases, approaching but not necessarily reaching zero (matches It's Magic).
  • Graph F: Represents a parabolic curve opening upward, beginning and ending at a high point with a minimum in the middle (matches Skateboarding).