Calculation Methods

This learning unit consists of two parts. 

The first aims to reinforce the high school skills that are essential for pursuing higher education in the sciences: understanding proportionality and linearity, performing calculations involving powers, working with fractions, and solving simple equations. 

The second part will be devoted to the study of real-valued functions: the focus will be on common functions, the graphical representation of functions, and the mathematical concept of the derivative (or instantaneous rate of change). 

Most of the concepts covered will be illustrated with concrete examples from biology. 

To provide the basic computational tools necessary for further study in the life sciences.  

10th-grade math

Recommended prerequisites*: Math track in 11th grade

A continuous assessment (CC) grade that will take into account: 

    - Participation and engagement in tutorials. 

    - the results of two midterm assessments (for each of the two parts)

A CT terminal check for the entire program. 

Max Rule: The score is calculated using the formula MAX(CT, AVERAGE(CT, CC))

• Basic Mathematical Techniques

1.a) Proportionality, linearity, and their various representations: 

• table of values, cross-product, coefficient of proportionality
• graphical representation (x-coordinate, y-coordinate, slope, and equation of a vector line)
• Solving the equation ax = b

Examples of illustrations: conversions between units of measurement (e.g., joules vs. kilocalories), the relationship between voltage and current, etc.

• Linear transformations: affine concepts, origin-aligned, equation y = ax + b

Examples of applications: determining bacterial concentration based on calibration data; converting degrees Celsius to degrees Fahrenheit. 

• Linear Regression

1.b) Fractions

• What is a fraction? (Ratio between integers, rules for simplifying fractions, concept of the GCD)
• Arithmetic rules (addition and multiplication, the concept of the least common multiple)
• inequalities (operations that preserve or reverse inequalities)

Examples of illustrations: calculating concentration after mixing, parallel resistance combinations, diagnostic tests (sensitivity, specificity, PPV, NPV—to be compared with prevalence)

1.c) Powers and orders of magnitude

• Integers raised to a power (rules for calculation, domain of definition for negative powers, scientific notation)
• Fractional powers and nth roots (domain of definition, equation x^n = c)
• orders of magnitude
• geometric growth

Examples of illustrations: dilution calculations, conversions (%, ‰, ppm, liters to cubic meters, etc.), "Fermi-style" order-of-magnitude estimates, and the reproduction number of an infectious disease. 

    2) Functions of a real variable

2.a) Vocabulary of Functions Through Examples

• Basic concepts (function, domain, graph, range, domain of definition). Examples from Part 1: linear functions, powers, polynomial functions. 
• The concept of a bijection. A detailed study of the logarithmic and exponential functions (logarithmic scale). 

Examples of illustrations: half-life, epidemiological models, allometry.

• The properties of functions and their representation on graphs (parity, monotonicity, periodicity: trigonometric functions). 

2.b) Limitations and Application

• the concept of a limit (examples using common functions already studied). 
• General results: the "policemen's theorem," comparative growth rates, limits of rational fractions. 
• continuity of normal operations.

Examples of applications: predictive use of a functional model, the Verhulst model, and load capacity.

2.c) Growth rate and derived number

• The concept of the derivative as the instantaneous rate of change. Graphical representation and the equation of the tangent line. 
• instantaneous speed

Examples of illustrations: all kinds of speeds. 

Note: The calculation of derivatives is part of the curriculum for the optional course in the second semester.

Hourly volumes*:

            CM: 12:00 p.m.

            TD: 9:00 p.m.

            Practical Work:

            Lot: