Mathematics
Grade 11
15 min
A.M. or P.M.
A.M. or P.M.
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1
Introduction & Learning Objectives
Learning Objectives
Define a piecewise function to represent the A.M./P.M. cycle.
Use the floor function to create a binary A.M./P.M. indicator function.
Apply modulo arithmetic to convert between 24-hour and 12-hour time formats within a function.
Analyze the domain and range of functions that model time.
Graph periodic functions that represent cyclical daily phenomena.
Compose functions to create a multi-step time conversion model.
Ever wondered how your digital clock knows whether to display 'A.M.' or 'P.M.'? ⏰ It's not magic, it's a function!
This lesson will explore how to mathematically model the daily A.M./P.M. cycle using advanced function concepts from your curriculum, including piecewise, floor, and periodic functions. You will learn...
2
Key Concepts & Vocabulary
TermDefinitionExample
Piecewise FunctionA function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain.A function for data charges: f(x) = { 20 if 0 < x ≤ 5; 20 + 2(x-5) if x > 5 }. The price is $20 for the first 5 GB, and $2 more for each GB after.
Floor FunctionThe function f(x) = ⌊x⌋, which returns the greatest integer less than or equal to x.⌊8.9⌋ = 8, ⌊7⌋ = 7, and ⌊-4.2⌋ = -5.
Modulo OperationThe operation a mod n, which finds the remainder after the division of a by n.17 mod 5 = 2, because 17 divided by 5 is 3 with a remainder of 2.
Periodic FunctionA function that repeats its values at regular intervals, called the period. Formally, f(x + P) = f(x) for some constant period P.The function f(x) = sin(x) is periodic with a per...
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Core Formulas
A.M./P.M. Indicator Function
P(h) = \lfloor h / 12 \rfloor
Use this function to determine if a given hour 'h' (in 24-hour format, h ∈ [0, 24)) is A.M. or P.M. The function outputs 0 for A.M. (when 0 ≤ h < 12) and 1 for P.M. (when 12 ≤ h < 24).
12-Hour Conversion Function
C(h) = ((h - 1) \pmod{12}) + 1
Use this function to convert an hour 'h' (in 24-hour format, h ∈ [1, 24]) to its 12-hour clock equivalent. It correctly maps 12 to 12 and 24 to 12.
Piecewise A.M./P.M. Definition
T(h) = \begin{cases} \text{'A.M.'} & \text{if } 0 \le h < 12 \\ \text{'P.M.'} & \text{if } 12 \le h < 24 \end{cases}
This piecewise function directly maps an hour 'h' from a 24-hour clock to its corresponding 'A.M.'...
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Challenging
A new indicator function I(h) is required to output -1 for any hour in the A.M. period ([0, 12)) and +1 for any hour in the P.M. period ([12, 24)). Which of the following functions correctly models I(h)?
A.2 * ⌊h / 12⌋ - 1
B.1 - 2 * ⌊h / 12⌋
C.⌊h / 12⌋ - 1
D.(-1)^⌊h / 12⌋
Challenging
Define a function f(h) that gives the fractional hour past the beginning of the current A.M. or P.M. block. For example, f(8.5) = 8.5 and f(14.5) = 2.5. Which of the following correctly defines f(h)?
A.h - 12 * ⌊h / 12⌋
B.((h - 1) mod 12) + 1
C.h / 12 - ⌊h / 12⌋
D.h mod 12
Challenging
A function is defined as g(h) = 1 - 2 * P(h), where P(h) = ⌊h/12⌋, on the domain [0, 24). What does this function represent and what is its range?
A.It's an A.M./P.M. indicator with range {0, 1}.
B.It's an A.M./P.M. indicator that maps A.M. to 1 and P.M. to -1; its range is {1, -1}.
C.It converts 24-hour time to 12-hour time; its range is [1, 12].
D.It's a periodic function with a period of 12; its range is all integers.
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