Central distinction. A function's domain tells us which inputs are allowed. Its outputs tell us what the function can produce. Whether a function is continuous concerns how its outputs behave as its input changes. These are different questions.
1. Three different questions about a function
A function assigns one output to each allowed input. Before graphing a real-world relationship, decide what each input and output means.
Don't confuse these: “the output must be a whole number,” “the input must be a whole number,” and “the graph has no jumps” describe three different properties. None automatically implies the others.
2. The same piggy bank can have two valid models
MVP 2.1, main task Problem 1, describes five pennies initially and three more added once each day. For these graphs, assume each deposit happens at the end of a completed day (times 1, 2, 3, …). The graphs describe the same deposits but answer different questions.
A. Count only at the end of each day
f(n) = 5 + 3n, n = 0, 1, 2, …
This model answers: “How many pennies after n completed days?” Only those day-number inputs belong to the domain.
B. Count at every moment
g(t) = pennies present at time t
This model answers: “How many pennies are there right now?” Every moment is an allowed input; at each deposit, the number jumps by three.
Check the actual quantities
At t = 0.5 days, there are still 5 pennies. At t = 1.5 days, there are 8 pennies. At t = 2.5 days, there are 11 pennies. The second model accepts fractional-day inputs, even though its outputs are always whole numbers.
Important: the blue points in A are not “wrong” or incomplete; they represent a deliberate choice to record only completed days. The orange steps in B are also valid under the stated deposit schedule.
3. Why this distinction matters in the packet
Situation
What a student must decide
Possible valid models
Pennies added daily, MVP 2.1 #1
Do inputs mean day numbers, or elapsed time at any moment?
Discrete-day sequence, or continuous-time step function.
Bank earns interest at the end of each month, MVP 2.1 #3
Are we recording balances after each month, or tracking the balance at every instant?
The packet’s natural answer is a discrete monthly sequence. A real-time step model is a possible alternative question, not a required correction.
Pool fills, MVP 2.1 #2
Are we modeling water volume continuously, or just measuring it once per minute?
Continuous-volume function, or a discrete table of samples from it.
Books shelved, MVP 2.2 #3
Are we counting books at each minute, or at every moment?
Whole-minute sequence, or an integer-valued step function of continuous time.
Electronic books downloaded, MVP 2.2 #2
Are we measuring completed books, or amount of data downloaded?
Completed books jump by whole numbers; data transferred can change continuously.
Student rule: First state what the input measures and which inputs the model allows. Then decide how to draw the graph. Don't decide solely from whether the output is a count.
Teacher / parent analysis · Source excerpts and critique
4. Where MVP’s terminology and models need clearer distinctions
The main concern is not that MVP chooses discrete day numbers or smooth approximations: those are often reasonable choices for ninth-grade modeling. The concern is that the notes sometimes use discrete to describe separated inputs and elsewhere to describe a quantity changing all at once, without clearly distinguishing these properties. Some tasks also leave the input and output variables unspecified while asking for a single classification.
A. Piggy bank: a valid discrete model, but an overbroad explanation
Student packet · §2.1, printed p. 1, Problem 1
“the number of pennies in the piggy bank on day n”
Teacher notes · §2.1, PDF p. 8
“Since money is only put in once a day, then it doesn’t make sense to have inputs like ½ or 3.5. That makes the domain the set of whole numbers.”
Analysis: Because the student problem explicitly asks for the count on “day n,” using whole-number day indices is appropriate; the resulting discrete sequence is not an error. The teacher’s reasoning is less precise when read as a general claim that a count cannot have fractional-time inputs merely because deposits occur once a day. A different question—how many pennies are present at any instant—allows a real-time input and yields a step function. Students need to learn that this is a change in the model and its input, not a contradiction in the mathematics.
B. Shelving books: distinguish continuous time from a continuous function
Student packet · §2.2, printed p. 7, Problems 3–5
“Would it make sense in any of these situations for there to be a time when 32.5 books had been shipped, downloaded into the computer or placed on the shelf?”
Teacher notes · §2.2, PDF p. 19
“the domain in this case is continuous, but the outputs are not because the librarians can’t shelve half of a book.”
Teacher notes · §2.2, PDF p. 19
“This is the type of situation that is often modeled as a continuous function for the sake of simplicity.”
Analysis: The manual correctly recognizes that elapsed time can be measured at any instant even though a count of completed books must be an integer. An exact running count rises in jumps and is not a continuous function on the whole time interval. By contrast, a smooth line of the form “books shelved ≈ 3 × hours elapsed” is a reasonable approximation for classroom modeling. The teacher notes even acknowledge using a continuous model “for the sake of simplicity.” The instructional improvement is to say explicitly when the graph is an approximation and not to equate a continuous domain with a mathematically continuous function.
C. Downloads: specify whether the output counts completed books or downloaded data
Student packet · §2.2, printed p. 6, Problem 2
“about 13 e-books can be downloaded onto the computer each hour”
Teacher notes · §2.2, PDF p. 19
“the graph should be continuous because parts of the books are being downloaded as the time continues”
Analysis: Treating downloads as a gradual process is a reasonable simplification if the quantity being modeled is the amount of data transferred or the equivalent number of e-books worth of data. However, “how many e-books have been downloaded” can also mean completed files, an integer-valued count that changes in steps. The difficulty is not choosing the smooth model; it is failing to specify which quantity that model represents.
D. Teacher guidance switches between input-domain and change-behavior criteria
Teacher notes · §2.2, PDF p. 18
“the change is happening gradually over the interval (continuous) or all at once (discrete)”
Teacher notes · §2.2, PDF p. 19
“Consider the domains for the two functions #1 and #2. How are the domains of these two functions different?”
Analysis: These are two distinct questions: which inputs belong to the domain, and whether the outputs change gradually or jump. For instance, an exact running count can accept every instant of time as an input yet jump whenever an event occurs. Using “discrete/continuous” for both distinctions without explicitly separating them can lead students to apply different classification rules to similar examples. This is the strongest conceptual problem identified in the teacher guidance.
E. Some homework classifications lack an identified input variable
Student packet · §2.1, Ready/Set/Go, printed p. 4, Set introduction
“Identify whether the following statements represent a discrete or a continuous relationship.”
Student packet · §2.1, Ready/Set/Go, printed p. 4, Problem 13
“The average person laughs 15 times per day.”
Analysis: This is a reasonable prompt for informal discussion: students can infer a likely count-based interpretation. But as a question about functions, it does not identify the input variable or domain. A sequence of end-of-day counts and a real-time running total are both legitimate models. If the exercise expects one answer, it should state the intended model, or allow students to explain their assumptions.
5. Recommended correction to the curriculum
State the variables and domain. When different models are plausible, identify the input and its allowed values rather than treating one sampling choice as mandatory.
Use distinct terms. Say “discrete-domain model” for isolated input values, “continuous-time model” for an interval of time inputs, and “continuous function” only when the values also change without jumps.
Identify the output. “Completed books,” “downloaded bytes,” and “fraction of a file transferred” are different measurable quantities.
Label approximations. If a smooth line stands in for stepwise inventory or book counts, explicitly state that it is an approximation.
Accept justified alternatives when the prompt permits them. Whole-number snapshots and exact real-time counts can answer different questions about the same situation; when the prompt explicitly asks for “day n,” a sequence is a natural default.
Suggested revised student prompt for §2.2, Problem 3: “Let n be the number of completed minutes. Model how many books have been shelved after n minutes and graph the allowed input values. Then suppose t is time measured continuously. Sketch how the actual count of completed books might change over time. Explain how the two models differ.”
6. Assessment and limits of the criticism
Mathematical nuance: The formal definition of continuity is made relative to the domain. A function with isolated domain points can be continuous at all its domain points; therefore “discrete-domain” and “discontinuous” are not strict logical opposites. The key classroom distinction here is between a function recorded only at separated inputs and a real-time function that jumps.
Supported conclusion: The materials contain a significant instructional consistency problem: they sometimes distinguish models by their domains and sometimes by how quantities change, without explicitly saying which property is being classified. The examples themselves are usually defensible once their inputs, outputs, and approximations are made explicit.
Important qualification: Choosing whole-number day indices for the piggy bank and monthly checkpoints for interest is reasonable. Smooth models of book handling can also be useful approximations. The criticism concerns the lack of consistent definitions and labeled modeling assumptions, not those ordinary choices or the authors’ knowledge.
§2.1 notes, PDF pp. 6–9; §2.2 notes, PDF pp. 17–19. Teacher-note PDF page numbers are viewer pages, not student-packet printed pages.
Quotations are short excerpts from the supplied packet and teacher notes; unquoted explanations, example models, and mathematical analysis are independent commentary. MVP source materials are licensed under Creative Commons Attribution 4.0. This file is independently created and not endorsed by MVP.