Seventh Grade Unit Two Lesson Twelve Study Notes

Seventh Grade Unit Two Lesson Twelve: Using Graphs to Compare Relationships

Problem One: Matching Equations to Graphs

  • Each equation corresponds to its respective graph based on the characteristics of rise and run.

Graph Characteristics:

  1. Graph One

    • Rise: 4 units

    • Run: 5 units

    • Equation: y = \frac{4}{5} x

    • Explanation: The slope calculated as rise over run results in the equation for the line being equal to four-fifths of x.

  2. Graph Two

    • Rise: 2 units

    • Run: 3 units

    • Equation: y = \frac{2}{3} x

    • Explanation: The slope calculated from the rise and run gives two-thirds of x.

  3. Graph Three

    • Rise: 4/3 units

    • Run: 1 unit

    • Equation: y = \frac{4}{3} x

    • Explanation: The units are divided into thirds, yielding a slope of four-thirds.

  4. Graph Four

    • Rise: 3/4 units

    • Run: 3 units

    • Equation: y = \frac{1}{4} x

    • Explanation: The slope is obtained by simplifying three-fourths divided by three, resulting in one-fourth.

  5. Graph Five

    • Rise: 2 units

    • Run: 1 unit

    • Equation: y = 2x

    • Explanation: The direct simplification yields a slope of two, meaning for each unit of x, y increases by two units.

  6. Graph Six

    • Rise: 3 units

    • Run: 2 units

    • Equation: y = \frac{3}{2} x

    • Explanation: The slope is calculated as three divided by two.

Problem Two: Analyzing Coffee Shop Menu Data

Part A: Identifying Graphs

  • The two graphs represent:

    1. Calories versus Drink Volume (in ounces)

    2. Cost versus Drink Volume (in ounces)

  • Reasoning: The graph showing high costs (up to $350) is likely not plausible for coffee prices. Therefore, it is inferred that those higher values represent calories instead of cost.

Part B: Proportional Relationships

  • Identifying Proportional Relationships:

    • The calories versus volume graph shows a proportional relationship.

    • Explanation: All points are on a line that passes through the origin, demonstrating a constant ratio.

    • Additionally, the calories double as the volume doubles, a characteristic of proportionality.

Part C: Finding the Constant of Proportionality

  • Calculation of Constant:

    1. Rise: 150 calories

    2. Run: 10 ounces

    • Constant of Proportionality:

    • \text{Constant} = \frac{150 \text{ calories}}{10 \text{ ounces}} = \frac{15}{1} = 15 \text{ calories per ounce}

    • Interpretation: There are 15 calories for every ounce of drink. This constant helps in determining calorie intake based on drink volume.

Problem Three: Comparing Lynn and Andre's Biking Speeds

Part A: Graphing the Bike Rides

  • The x-axis represents time (in minutes), and the y-axis represents distance (in kilometers) for both Lynn and Andre.

  • Graph Details:

    • Andre's data graphed in purple, coordinates: (8 minutes, 2 kilometers).

    • Lynn's data graphed in blue, coordinates: (5 minutes, 1.5 kilometers).

Part B: Highlighting Points on the Graphs

  • For Andre:

    • Rise: 2 kilometers

    • Run: 8 minutes:

    • \frac{2}{8} = \frac{1}{4} = 0.25.

    • Thus, for k: k = 0.25 (25 hundredths or a quarter kilometer per minute).

  • For Lynn:

    • Rise: 1.5 kilometers

    • Run: 5 minutes:

    • \frac{1.5}{5} = 0.3.

    • Thus, for k: k = 0.3 (three tenths of a kilometer per minute).

Part C: Comparing Speeds

  • Speed Comparison:

    • Lynn's Speed: 0.3 kilometers per minute

    • Andre's Speed: 0.25 kilometers per minute

    • Conclusion: Lynn is biking faster than Andre, as her speed (30 hundredths of a kilometer per minute) exceeds Andre's speed (25 hundredths of a kilometer per minute).