Investigating Rate of Change and Slope through Similar Triangles

Warm-up Exercise: Analyzing Price via Graphical Slopes

  • Exercise Prompt: Students were asked to identify which item costs more per pound based on a provided graph comparing peas and cabbage.

  • Graphical Analysis of Costs:     - The x-axis represents the weight in pounds.     - The y-axis represents the cost in dollars.     - Observation: As the value on the x-axis (pounds) increases, the cost increases for both items.     - Comparing Slopes:         - One student, Parker, initially suggested cabbage might cost more because the line "sloped more towards the pounds."         - Another student, Tran, correctly identified that peas cost more. This is because, at any given point on the x-axis, the slope for the peas line is higher on the cost range.         - Conclusion: At the same poundage (weight), the cost for peas is higher than the cost for cabbage because the line for peas is "taller" or steeper on the vertical axis.

Review of Linear Equations and Graphing Techniques

  • Methods for Graphing Linear Equations:     - Function Tables: Using structured tables to organize inputs and outputs.     - Ordered Pairs: Using the notation (x,y)(x, y) to determine specific points on the x-axis and y-axis.

  • Slope as a Mathematical Concept:     - Slope can be expressed as a fraction.     - It is defined as a number over a number, representing the ratio of "rise over run."

Investigating Rate of Change and Slope through Geometry

  • Objective: To understand why the slope is consistent between any two distinct points on a non-vertical line.

  • Similar Triangles:     - Similarity: Triangles are considered similar if they are proportional.     - Proportionality: This means that if a shape is enlarged or shrunk, it remains the same as the original triangle in terms of its ratios and angles.

  • Linear Characteristics:     - Vertical Lines: Described as lines that are "straight up and down."     - Non-vertical Lines: On these lines, the distance and ratio between any two points remain constant, which will be demonstrated using similar triangles.

Derived Equations for Linear Functions

  • The Origin:     - Defined as the point where the x-axis and y-axis meet.     - The ordered pair for the origin is (0,0)(0, 0).

  • Equation Structures:     - Through the Origin: The equation is represented as y=mxy = mx.     - Not through the Origin: The equation is represented as y=mx+by = mx + b.

Practical Application: "Music is a Must" Investigation

  • Context: The class explored real-world examples of music (e.g., in the car, elevators, at home, at dances, and in stores) to frame an investigation into CD pricing.

  • Task Objective: Evaluate the relationship between the number of CDs purchased and the total cost.     - Graph Data: The graph reflects the average cost of music CDs.     - Specific Data Points Discussed:         - 22 CDs cost $30\$30.         - 44 CDs cost $60\$60.         - 55 CDs cost $75\$75.         - 66 CDs cost $90\$90 (determined by finding the midpoint between $80\$80 and $100\$100, or following the established interval).         - 88 CDs cost $120\$120.

  • Calculating the Unit Rate:     - Definition: The unit rate is the cost for exactly one item (e.g., "per pound" or "per item").     - Calculation Method: To find the unit rate of CDs, one can divide the total cost by the number of CDs.         - Example: 302=15\frac{30}{2} = 15.         - The findings suggest a unit rate of $15\$15 per CD.

  • Validation: This unit rate should remain consistent across different points on the graph (e.g., 1208=15\frac{120}{8} = 15; 604=15\frac{60}{4} = 15).

Questions & Discussion

  • Question on Interpretation: "Which costs more per pound according to the graph? Peas or cabbage?"     - Response: The teacher confirmed that peas cost more because at the same poundage, the cost for peas is higher.

  • Question on Slope: "What do you mean by the slope?"     - Response (Tran): "How diagonal the line is… the peas slope is tall higher than the cabbage's."

  • Question on Similarity: "What do I mean by similar?"     - Response (Tran): "They're proportional… if you enlarged it, it would be the same as the other triangle."

  • Question on Unit Rate: "What does it mean by unit rate?"     - Response: The teacher explained it means the cost for one single unit (e.g., one pound or one CD).

  • Question on Origin Location: "Who can tell me where the origin is?"     - Response (Taylor): "Right where the x and y axis meet, right in the middle… that ordered pair is gonna be 0 0."

Lesson Structure and Procedure

  1. Private Think Time (5 Minutes): Students work individually on questions A through D on the "Music is a Must" worksheet.

  2. Small Group Discussion: Students compare their findings for A-D to ensure they are on the right track and then complete the rest of the worksheet.

  3. Poster Preparation (10 Minutes): Groups record their findings on large posters.

  4. Whole Group Discussion (Final 10 Minutes): The class shares and discusses the findings from the posters.