2.3 Linear Modeling Notes

2.3 Notes

General Context

  • Announcement of the current date as October 6, with a reminder about due assignments.

  • Exit ticket due on October 7, ensuring ample working time in class. (EDIT for 3rd period: 5-10 minutes will be given at the beginning of class tomorrow since Coach C. took so long to cover the notes)

Previous Content Review

  • Review of pair assessments previously conducted.

  • Clarification that assessments covered from sections 2.1 to 2.3, instead of the previously mentioned 1.1 to 1.2.

  • Mention of working with graphing problems, pointing toward today's focus.

Problem Scenario: Hart Family's Pretzel Quantity

  • Case study involving the Hart family's pretzel usage when making lunches.

  • Important variables to define:

    • L: Represents the number of lunches made.

    • P: Represents the number of pretzels remaining in the tub.

Determining the Number of Pretzels per Lunch

  • Discussion of the number of pretzels included in each lunch:

    • Counted solutions include 24, 18, and a note dismissing 126.

  • Correct answer was established as 18 pretzels per lunch, confirmatory reiteration of this number.

Use of the Slope Formula

  • Slope formula is introduced and applied:

    • The slope formula, expressed as m=(y2-y1)/(x2-x1) allows us to determine the rate of change between two points on a graph. This can be particularly useful in analyzing how changes in pretzel production might affect overall lunches served, yielding insights into the efficiency of our catering operation.

      • Relation of variables established:

        • x: Representing lunches (L).

        • y: Representing pretzels (P).

  • Observation of changes in variables to compute the slope:

    • Difference calculated: 3 to 7 (giving 4), and 78 to 306 (yielding 72).

    • Hence, 724=18\frac{72}{4} = 18 pretzels per lunch, confirming the rate.

Working with Lunch Quantities and Remaining Pretzels

  • Question posed: After making 16 lunches, how many pretzels remain?

    • Calculation led to conclusion of 144 pretzels remaining.

    • Used difference from table to reach a theoretical calculation based on previous points.

    • Clarification of 3's significance in determination.

Further Confirmation Mechanisms

  • Additional methods to reinforce understanding:

    • Either calculate by systematic subtraction or multiplying the difference of lunches made by the per lunch pretzel count (18).

    • Example: For every lunch, if 18 pretzels are used, (3 imes 18 = 54).

    • Alternative validation checked through calculators by pairing numbers properly.

Initial Total Calculation

  • Initial calculation of total pretzels in tub:

    • Recognization that 0 lunches corresponded to starting with a total of:

    • Engagement encouraged for peer discussions on confusion or questions.

Further Analysis of Lunch Capacity

  • Question: How many servings can Mr. Hart pack?

    • Possible solutions shared with diverse approaches.

    • Using total pretzels remaining (198) divided by number of pretzels per lunch (18):

    • Conclusion drawn: 11 more lunches can be made.

    • Summed with previously made lunches (13) resulted in a total of 24 lunches.

    • Alternatively, direct calculation with initial total pretzels (432) divided straight by 18 gives 24 as well.

Developing the Algebraic Equation

  • Formulation of a linear equation for remaining pretzels:

    • P=432−18LP = 432 - 18L./

  • Breakdown:

    • 432432 referred to as initial amount (like the y-intercept).

    • −18-18 reflects the rate of decrease in pretzels per lunch (constant rate indication).

  • Variations discussed: Switching the presentation, which can also be expressed as:

    • P=−18L+432P = -18L + 432.

  • Importance of maintaining correct signs noted for meaningful interpretation of data.

Graphing and Intercepts

  • Identification and labeling of coordinates on a graph:

  • Y-Intercept:

    • Confirmed at (0, 432).

  • X-Intercept:

    • Anticipated value calculated at (24, 0).

  • Final graphical representation showed critical points plotted:

    • Acknowledge both intercepts for context of equations.

    • The significance of the equation based on plotted data continued to reinforce understanding.

Summary of Learning

  • Reflection encouraged on equations, values, and various mathematical approaches taken during the lesson.

  • Emphasis placed on the importance of showing work step-by-step to validate processes used in calculations.

Conclusion and Next Steps

  • Preparation for the exit ticket that will utilize concepts applied in lessons.

  • Class closure with reminders to consolidate learning and prepare for further assessments.