Vital Vitals Project Booklet
VITAL VITALS
Clinical Mathematics Residency Project
Resident:
Date:
Residency Assignment
Welcome to your rotation at St. Jude’s Medical Center. As a senior math resident, you are responsible for pharmacy calculations, hospital staffing models, and cardiac monitoring. Precise mathematical modeling is the foundation of patient safety.
Part 1: The Pharmacy Lab (Exponents & Radicals)
A new antibiotic concentration \( C(t) \) (in mg/L) after \( t \) hours is modeled by: \[ C(t) = 80 \cdot (0.5)^{\frac{t}{4}} \]
1.1. Determine the initial concentration of the drug when first administered (\( t = 0 \)). Show your step.
1.2. Calculate the concentration after 8 hours. Is the medication still effective if the minimum required level is 15 mg/L?
1.3. A second drug's concentration is measured by \( R(t) = \sqrt{144 - 6t} \). After how many hours will the concentration reach 0 mg/L?
Part 2: Hospital Logistics (Linear Systems)
Total daily budget: $14,500. General NPs cost $500/day and Specialized Trauma NPs cost $800/day. The hospital board mandates a total hire of exactly 20 employees.
2.1. Define your variables and write the system of equations.
Variable 1
Let \( x = \)
Variable 2
Let \( y = \)
2.2. Solve the system to find the number of each NP type. Show all algebraic steps.
Part 3: The Recovery Curve (Quadratics)
A patient’s White Blood Cell count (in thousands/\(\mu L\)) during an infection is modeled by: \[ W(d) = -d^2 + 8d + 5 \]
3.1. Calculate the vertex \((d, W)\). What does this specific point mean in the context of the patient's recovery?
3.2. After how many days will the count return to the initial admission level of 5? Show your work.
Proceed to Cardiac Monitoring Suite
Part 4: Cardiac Monitoring (Trigonometry)
Blood pressure fluctuation:
P(t) = 20 sin(πt) + 100
Amplitude
Midline BP
Midline Eq
4.2. Sketch Cycle (Label Axis)
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Part 5: Stress Test (Transformations)
Case A: Tachycardia
Shift rises by 20; Amplitude doubles.
P(t) =
Case B: Beta-Blocker
Shift drops to 90; Amplitude halved.
P(t) =
Final Clinical Summary
How do changes in midline and amplitude help you diagnose a patient's cardiac health? Use Case A or Case B as an example.
St. Jude Medical Center Final Residency Assessment - Precalculus
Vital Vitals Project Slides
VITAL VITALS
Clinical Mathematics Residency
Your Medical Rotation
You are now a Senior Math Resident at St. Jude's Medical Center.
Analyze Patient Recovery
Manage Staffing Budgets
Monitor Cardiac Waves
Rotation Schedule
- Day 1: Pharmacy & Logistics 65m
- Day 2: Recovery & EKG 65m
- Day 3: Stress Test Review 45m
Day 1 Lab & Logistics
Pharmacy Calcs
Tracking drug decay using exponential models.
C(t) = 80 * (0.5)^(t/4)
Staffing Systems
Finding the balance point for personnel and budgets.
- People Equations
- Dollar Systems
The Vital Signs
Quadratics
Tracking the infection curve. Identifying the peak count.
Trigonometry
Modeling the rhythmic cycle of the heart monitor.
Clinical Vocabulary
Midline = Avg BP
Amplitude = Intensity
Vertex = Max WBC
Visualizing the Heart Beat
Midline (D)
The resting baseline. The middle of the fluctuation.
Amplitude (A)
The strength of the pulse. Distance from the baseline.
P(t) = 20 sin(πt) + 100
Board Certification
Apply your residency findings to the Stress Test Cases. Diagnostic accuracy determines graduation.
Algebra Complete
Units Labeled
Graphs Drawn
Reasoning Valid
Vital Vitals Teacher Guide
Teacher Guide: Vital Vitals
Residency Implementation & Answer Key
Pacing Guide
- Day 1 (65m): Intro Slides (10m), Parts 1 & 2 (55m).
- Day 2 (65m): Warm-up (10m), Parts 3 & 4 (55m).
- Day 3 (45m): Part 5 & Summary (35m), Exit Interview (10m).
Scaffolding
Key Support Strategies:
Standard Notation: Display \( y = a(x-h)^2 + k \) and \( y = A \sin(Bx) + D \) throughout the project days.
Solution Key
Part 1: Pharmacy
1.1. \( C(0) = 80 \text{ mg/L} \).
1.2. \( C(8) = 20 \text{ mg/L} \) (Effective).
1.3. \( t = 24 \text{ hours} \).
Part 2: Logistics
2.1. \( x + y = 20 \); \( 500x + 800y = 14500 \).
2.2. \( x = 5 \) (Gen), \( y = 15 \) (Trauma).
Part 3: Recovery
3.1. Vertex: \( (4, 21) \). Peak WBC count.
3.2. \( d = 8 \text{ days} \).
Part 4: Cardiac
4.1. Amp=20; Mid=100; Mid Eq: \( y=100 \).
4.2. Max 120, Min 80.
Part 5: Stress Test
5.1. \( P_{new}(t) = 40 \sin(\pi t) + 120 \).
5.2. \( P_{med}(t) = 10 \sin(\pi t) + 90 \).
Residency Grading Rubric
| Criteria | Exceptional (4) | Proficient (3) | Developing (2) | Initial (1) |
|---|
| Modeling | Perfect eq's for all 5 parts. | Minor sign errors in 1 part. | Structural errors in eqs. | Missing/invalid models. |
| Solving | All solutions verified correct. | 1-2 arithmetic slips. | Conceptual error in solving. | No work shown. |
| Graphing | Correct shape, labels, scale. | Correct shape, minor label. | Wrong scale or midline. | Non-trig graph. |
| Analysis | Deep link between math & clinic. | Clear but simple clinical link. | Partial understanding. | Inaccurate diagnosis. |
