Last updated October 13, 2026.

Secondary Math I · Module 2 · Linear & Exponential Functions

2.2 — Shh! Please Be Discreet (Discrete)!

Supplement for Mathematics Vision Project (MVP) curriculum · Secondary Math I, Module 2, Section 2.2
Independently created to support students using MVP Math. Not official publisher material and not affiliated with or endorsed by the Mathematics Vision Project.

Today's goal. Choose and explain domains for models of books, paper, and medicine; distinguish discrete inputs from counting in jumps; compare linear and exponential growth; and practice unit rates and one-step equations.

Important mathematical clarification — Discrete does not mean discontinuous

The MVP student packet and teacher's notes sometimes use discrete to describe separated input values, and at other times focus on quantities that change in jumps. Those are not the same property. This lesson states the input, output, and allowed inputs for each model, and uses red notes where a statement in the original materials could lead to a misleading conclusion.

A separate companion explanation examines the mathematical distinction and quotes the MVP student and teacher materials.

1. In class: A Solidify Understanding Task

What makes a model discrete or continuous?

IN YOUR MVP PACKETSection 2.2 · Main task · pp. 6–10 · Problems 1–11

Every question in the main activity depends on identifying what the input is and which values are allowed.

A function matches each allowed input with exactly one output. Its domain is the set of allowed inputs. Before drawing a graph, decide what you are recording and at which inputs.

CLEAR DEFINITIONS — THREE DIFFERENT IDEAS

Discrete-domain function: a function defined at separated input values, such as . Its graph has individual dots at those input values.

Continuous function: a function defined throughout an interval whose output changes without jumps. Its graph is an unbroken curve over that interval.

Step function: a function whose value stays constant for stretches and jumps at particular inputs. A step function can be defined for every moment of time, but it is not a continuous function at its jumps.

Domain is not the same as range. A function may be defined at every instant (continuous-time domain) and still output only whole numbers.

In the next examples, a book count and the amount of data transferred will use different outputs. That choice matters more than the word “books” in the story.

Model the books carefully

IN YOUR MVP PACKETSection 2.2 · Main task · pp. 6–7 · Problems 1–5

Use tables, graphs, equations, and domains to compare shipments, downloads, and shelving books.

Problem 1: Boxes containing books

IN YOUR MVP PACKETMain task · p. 6 · Problem 1

The input is the number of boxes. Only whole numbers of boxes are permitted.

If a box contains about eight books, a simple average model is . The input is a count of boxes: . You cannot receive complete boxes in this particular box-counting model.

Worked example — Count boxes, not elapsed time

For six boxes, books. For ten boxes, books.

Boxes	Estimated books
0	0
1	8
2	16
3	24
4	32

Each increase of one box adds about eight books. This is an arithmetic pattern, and the graph uses isolated points because only whole-number box counts belong to the domain.

Only integer numbers of boxes are allowed. The straight alignment of the dots does not mean we should connect them.

Problem 2: Electronic books downloaded over time

IN YOUR MVP PACKETMain task · p. 6 · Problem 2

Decide whether your output measures transfer progress or a count of completed downloads.

The packet says an average of about 13 e-books can be downloaded per hour. To draw a smooth graph, we must choose an output that can change by fractional amounts, such as total data transferred measured in book-equivalents.

Worked example — Transfer progress

Suppose the transfer progresses uniformly. Let be elapsed time in hours and the total data transferred, in book-equivalents. Then

At , book-equivalents; at , . At , book-equivalents of data. This last value does not mean 19.5 finished e-books exist.

A straight-line approximation for transfer progress, which can include fractional book-equivalents of data.

MVP clarification — Main task, Problem 2; Teacher notes for Section 2.2

The teacher's manual says to connect the dots because “parts of the books are being downloaded.” That is reasonable if the output measures download progress. But the student question asks how many e-books can be added. A count of completed e-books can only take whole-number values.

The two outputs are different. A smooth graph for transferred data must not be presented as an exact graph of completed books. If the output is completed downloads counted at every time, the count changes in steps instead.

Problem 3: Shelving books one by one

IN YOUR MVP PACKETMain task · p. 7 · Problems 3–5

Use an elapsed-time input but distinguish a smooth average from the actual completed-book count.

A librarian shelves an average of three books per minute. A useful average-work model is , where can be any nonnegative number of minutes and measures progress in book-equivalents. For three hours, minutes, giving book-equivalents.

But if means the number of completely shelved books, its outputs must be whole numbers. For a simple illustrative model with one completed book every 20 seconds, the count makes a jump each time one book is finished.

One possible exact completed-book count if a book is finished every 20 seconds. Filled dots include the start of a step; open dots exclude its end. The input allows every time, but the count jumps.

Worked example — Two measurements of the same work

At minute, the smooth model gives book-equivalents of work. The simplified completed-book model gives finished book. Both models allow ; they just define different outputs.

The packet asks whether 32.5 books can be shelved. Not if the output is completed books. However, 32.5 book-equivalents can be a reasonable measure of partial work.

MVP clarification — Problems 3–5; Teacher notes for Section 2.2

The teacher's manual correctly observes that an input of time can include fractions even when the output is a whole-number count. But it also describes this as a “continuous function.” Those are different claims. The input can range over every time value while the function still jumps, so the completed-book count is not continuous at those jumps.

Correct language: “a function defined for continuous time” does not necessarily mean “a continuous function.”

How to answer Problem 5

First say what input and output you chose. Problem 1 is naturally a discrete-box model; Problem 2 supports a smooth model for download progress but a step model for completed downloads; Problem 3 has similar alternatives for shelving work and completed books. The question does not fix a unique interpretation for every case.

Discrete and continuous exponential models

IN YOUR MVP PACKETSection 2.2 · Main task · pp. 8–10 · Problems 6–8

Graph repeated paper-cutting rounds and the gradual decline in active medicine, identifying the domain of each.

Problem 6: Paper cut into thirds

At the start there is one piece. After one full round of cuts there are three pieces; after two rounds, nine. Each new round multiplies the previous count by three.

Worked example — An exponential sequence

Let be the number of completed cutting rounds and the number of paper pieces. The rule is

Completed rounds	Pieces
0	1
1	3
2	9
3	27
4	81

For 10 rounds, . For 20 rounds, . A completed-round count of 5.2 does not belong to this model's domain. A count of 53.6 paper pieces is likewise not a possible output.

Only completed rounds are included; the plotted points are not connected.

A geometric sequence is an exponential pattern recorded at successive integer inputs. The input is discrete because it counts completed rounds, not because exponentials must always be discrete.

Problem 7: Medicine decays by 20% per hour

If 20% becomes inactive each hour, 80% remains. So we multiply by for each hour of elapsed time. The initial dose is 60 mg.

Worked example — Exponential decay

Using a smooth percentage-decay model, the active amount in milligrams after hours is

After one hour mg; after two hours mg; after three hours mg. The model also accepts fractional times, such as or , because time is not restricted to whole hours.

For , mg. (The calculation uses a fractional exponent, but you do not need a new algebraic method to interpret the graph.)

Smooth exponential-decay model for active medicine. The input can be fractional hours, and the amount changes without jumps.

Could 35 mg remain? Yes, in this smooth model: mg and mg, so the continuous decreasing curve reaches 35 mg at some time between two and three hours. You do not need to solve for that exact time in today's lesson.

Compare Problems 6 and 7

Both are exponential because their values change by a constant multiplication factor (3 or 0.8) over each step of the same length. The paper model uses completed rounds ; the smooth medicine model allows all times .

Choose a model and explain it

IN YOUR MVP PACKETSection 2.2 · Main task · p. 10 · Problems 9–11

Summarize the modeling choices and distinguish arithmetic/linear from geometric/exponential behavior.

When a problem asks whether a model is discrete or continuous, begin with the defined function, not whether the objects in the story can be counted.

Question to answer	Why it matters
What is the input?	Number of boxes, completed cutting rounds, or elapsed time?
What is the output?	Completed books, transfer progress, paper pieces, or active medicine?
Which inputs are allowed?	Only separated values, or every time in an interval?
Can the output jump?	A full interval of inputs is not enough to guarantee a continuous function.
How does it change?	Add the same amount = linear pattern; multiply by the same factor = exponential pattern.

Worked comparison — Similarities can hide different domains

for box counts and for smooth shelving progress both have a linear form: a fixed rate times the input. But the first is defined only at whole-number box counts, while the second has a full interval of time inputs.

Likewise and both involve repeated multiplication, but the first uses discrete rounds and the second models continuous time.

For tables, list only allowed inputs. A continuous function may be shown in a table with a few samples, but a table containing only whole-number inputs does not prove the function's domain is discrete. For graphs, mark discrete inputs as separate dots. Draw smooth curves for continuous functions. Draw horizontal steps with correctly specified endpoints when representing a continuous-time integer count.

Packet clarification — Section 2.2, Main task, Problems 9–10Modeling assumptions must be named

These questions invite broad generalizations. A table, an equation, or a verbal description alone may not settle whether a model is discrete or continuous. For example, the formula can be used with whole-number inputs or with every real input. You must specify its domain.

2. READY — Compare rates with consistent units

Study the worked examples first, then try the problems yourself. An answer key at the bottom of this section is provided to check your work.

IN YOUR MVP PACKETSection 2.2 · Ready · p. 11 · Problems 1–4

Compare rates for stretch, growth, filling, and travel. Convert both rates to the same units first.

A unit rate tells how much output changes for one unit of input. To compare different situations, use matching units.

Worked example — Compare stretch per pound

A cord stretches 9 inches under 3 pounds, while a spring stretches 2 feet under 2 pounds. Compare:

Cord: inches per pound.
Spring: 2 feet = 24 inches, so inches per pound.
Since , the spring has the greater stretch rate in this simplified comparison.

Worked example — Compare rates with days and weeks

Plant A grows 3 inches every 2 days. Plant B grows 10 inches each week (7 days).

Plant A: inches per day. Plant B: inches per day. Plant A has the greater daily growth rate.

Worked example — Compare gallons per minute and miles per hour

A tank fills with 42 gallons in 6 minutes: gallons per minute. A second fills with 30 gallons in 5 minutes: gallons per minute. The first fills faster.

A cyclist riding 12 miles in 1.5 hours averages miles per hour. A runner covering 3 miles in 30 minutes takes 0.5 hour, giving miles per hour. The cyclist's rate is greater.

Watch the units: A rate of 3 miles in 24 minutes is not “3 miles per hour.” Convert 24 minutes into hours, or convert both comparisons into minutes.

Check your work — Ready 1–4 answer key

Slinky: 36 inches per pound, versus 2 inches per pound.
Amaryllis: \(\frac{18}{7}\) inches per day, versus 2.
Gas pump: \(\frac{25}{3}\) gallons per minute, versus 8.
Bike: 10 mph, versus \(\frac{15}{2}\) mph.

3. SET — Tell apart domain and growth pattern

Study the worked examples first, then try the problems yourself. An answer key at the bottom of this section is provided to check your work.

IN YOUR MVP PACKETSection 2.2 · Set · p. 11 · Problems 5–11

For each description, decide whether enough information identifies a model, then distinguish linear/arithmetic from exponential/geometric growth.

MVP clarification — Set, Problems 5–10

Many descriptions give an average rate but do not state the input, output, or permitted inputs. They cannot always be classified uniquely as discrete or continuous. A rate stated “per day” or “per hour” does not force the input to be a whole number. State your proposed model and domain before assigning a label.

Likewise, one rate does not always establish that a quantity will follow that same linear or exponential rule indefinitely; the question may be inviting a simplifying constant-rate model. Make that assumption clear.

Worked example — Fixed amount or fixed percentage?

Suppose a city adds 2,000 people each year. A whole-year checkpoint model is , an arithmetic/linear pattern.

Suppose instead a population grows by 5% each year. Then is a geometric/exponential pattern. Neither the words “each year” nor the formula by themselves require an integer-only domain; the model must state its domain.

Set problem	Likely intended growth pattern	Important limitation or domain choice
5 · Concrete	Linear model if 300 feet is added each day at a steady average rate	Footage could be measured throughout a day or recorded at daily checkpoints; output and domain are unspecified.
6 · Bacteria	Exponential (doubles each hour)	Could model hourly checkpoints or continuous-time spread.
7 · Bricks	Exponential/geometric (5% increase per day)	Daily counts are whole numbers, so an exact percent increase may require rounding; a smooth average model is also possible.
8 · Steps	Linear cumulative average if interpreted as about 10,000 additional steps each day	An actual running step count jumps; daily check-ins use separated inputs.
9 · Population	Exponential model (8% growth per year)	Specify yearly observations or continuous-time approximation; real population is integer-valued.
10 · Flow speed	Constant-speed distance model is linear	Speed alone does not define the requested function. If distance is output and elapsed time input, a continuous-time model is natural.

Problem 11: Read the representation, not a stereotype

IN YOUR MVP PACKETSet · p. 11 · Problem 11a–c

Distinguish a recurrence, isolated plotted points, and an explicit exponential equation.

11a. The packet gives a recursive rule with . That makes an arithmetic sequence on successive integer indices: . It is a discrete-domain linear pattern.

11b. The packet's graph shows isolated dots, decreasing by approximately the same multiplication factor (about 0.6). It depicts a discrete-domain exponential/geometric pattern over the plotted inputs. The dots themselves establish that the plotted model includes those separate inputs; they do not specify values in between.

11c. The packet writes , an exponential growth rule. The equation alone does not specify the domain. It can be evaluated for all real , but a model could also restrict to integer stages. In the absence of a stated restriction, the usual algebraic convention is all real inputs.

Packet clarification — Section 2.2, Set, Problem 11cDo not infer domain from the variable's letter alone

Using often signals integer indices in a sequence, while often signals a real input in algebra. These are useful conventions, not definitions. The same formula can be used on different domains.

Check your work — Set 5–11 answer key

The classifications refer to the modeled variables and domains stated below. A daily rate does not by itself force the time domain to be discrete.

Continuous linear model for concrete length versus elapsed working time.
Continuous exponential model for infected area versus elapsed time.
Discrete exponential model for each day’s brick count (factor 1.05).
Discrete linear model for cumulative steps at daily checkpoints; individual step counts are discrete.
Discrete exponential model for annual population counts (factor 1.08).
Continuous linear model for distance traveled by the surface water versus time.
(a) Discrete arithmetic/linear sequence, adding 3. (b) Discrete geometric/exponential sequence, multiplying by \(\frac35\). (c) Continuous exponential model, with real x inputs.

4. GO — Solve one-step equations

Study the worked examples first, then try the problems yourself. An answer key at the bottom of this section is provided to check your work.

IN YOUR MVP PACKETSection 2.2 · Go · p. 12 · Problems 11–19

Solve equations using inverse operations: undo addition/subtraction, multiplication, negative coefficients, and fractional coefficients.

An equation stays balanced when you perform the same operation on both sides. Undo the operation acting on the variable, then substitute your answer to check it.

Worked example — Addition and subtraction

Solve . Subtract 19 from both sides:

Check: .

Solve . Add 12 to both sides, giving . Check: .

Worked example — Multiplication and negative coefficients

Solve . Divide both sides by :

Check: .

Likewise, gives .

Worked example — Fraction coefficient

Solve . Multiply both sides by the reciprocal :

Check: . The same reciprocal technique handles negative fractional coefficients.

New fully worked GO problems

Problem A — Straightforward

Problem: Solve .

Divide both sides by 7: .
Simplify: .
Check: . ✓

Problem B — More challenging

Problem: Solve .

Multiply both sides by the reciprocal : .
Simplify: .
Check: . ✓

Because the coefficient was negative, the solution is negative; keep the sign when taking the reciprocal.

Check your work — Go 11–19 answer key

The packet repeats number 11: Set 11 is the classification exercise above; Go 11 is the first equation here.

\(x=20\).
\(x=23\).
\(x=5\).
\(x=7\).
\(x=-8\).
\(x=2\).
\(x=60\).
\(x=\frac74\).
\(x=-12\).

Additional Learning Resources

These resources support particular parts of this lesson. The scope guidance indicates where a useful resource goes beyond today's material.

Resource and direct link	What it covers / why use it	Scope / What to skip
Khan Academy — Function domain word problems (practice)	Choose an appropriate domain for quantities such as steps or units counted. Especially useful for main task Problems 1, 5, 6, and Set.	Close match: Focus on identifying meaningful inputs and whole-number versus real-number domains.
Khan Academy — What is the domain of a function? (video)	Clear introductory explanation of a function's permitted inputs, independent of the output.	Close match: The meaning of domain; examples involving denominator restrictions are extra and are not assigned today.
Paul's Online Math Notes — The Definition of a Function	Written definition, ordered pairs, function notation, and domain/range. Suitable for a more detailed alternative explanation.	Significantly broader: Read the working definition and introductory domain discussion. Skip piecewise-function algebra, radicals, and advanced domain examples.
OpenStax — 3.2 Domain and Range	Optional textbook treatment of domain choices, including meaningful inputs in real-world contexts.	Significantly broader: Use the introductory domain discussion; skip interval-notation techniques, algebraic domain restrictions, and piecewise formulas.
Companion analysis — Discrete Inputs, Jumping Counts, and Continuous Functions	Carefully distinguishes discrete domains, continuous-time inputs, and jumping counts, with analysis of the MVP manual.	Optional extended reading: Student explanation is useful now; teacher/parent critique is not required for today's problems.

Teacher's review of the original packet

This review identifies substantive mathematical problems, not ordinary simplifying assumptions.

Main task, Problems 2–5 — What exactly is being downloaded or shelved?

The student packet asks students to model numbers of e-books downloaded and books shelved. The teacher's notes justify a connected graph of downloads on the grounds that fractions of a book are transferred during download, while acknowledging separately that completely shelved books cannot have fractional output values. Those descriptions switch between work/data progress and completed-item count without explicitly changing the output variable.

Why it matters: the definition of the output changes the graph: a smooth progress model and a running integer count are not the same function. Correction: specify whether each model counts completed books or measures transferred data/work, and state the domain.

Teacher notes, shelving-books discussion — Continuous time versus a continuous function

The notes correctly recognize that elapsed time can include fractional values while the number of shelved books remains whole. However, the same passage then treats this as a continuous function for simplicity. A whole-number step count defined at all times has a continuous interval of inputs but is not continuous at the jumps. If the intended model is the smooth rate , it should be labeled an approximation to book-equivalent work, not an exact completed-book count.

Correction: explain the modeling choice and do not use “continuous domain” and “continuous function” interchangeably.

Set, Problems 5–10 — Classifications that need missing modeling information

Descriptions such as a quantity “per day” often do not identify the independent and dependent variables or their domains. The form of change (fixed addition or fixed percentage) can sometimes be inferred, but the discrete/continuous classification is not determined uniquely. Problem 10 specifies a surface flow speed without defining a distance or position function.

Correction: state the intended input, output, domain, and assumption of steady rate. Alternatively, require students to propose and justify a model rather than grade one implicit answer.

Set, Problem 11 — Domains must accompany equations

A recurrence with index naturally defines a discrete sequence and a dot graph displays separated plotted inputs, but an explicit rule does not establish a domain by itself. A letter such as or is a convention, not a mathematical domain definition. Correction: give the intended domain for each representation or explicitly ask students to choose one.

Coverage check — Section 2.2 only

Packet component	Original problems / skills	Where taught
Main, p. 6	1–2: boxes and download models, equations, tables, graphs, domains	In class, corresponding graph examples
Main, p. 7	3–5: shelving; fractional outputs and model interpretation	In class, step graph, clarified modeling choice
Main, pp. 8–9	6–8: paper-cut growth and medicine decay	In class with fully worked models and graphs
Main, p. 10	9–11: choice of domain; table/graph/equation; linear versus exponential	In class
Ready, p. 11	1–4: compare unit rates with matching units	Section 2
Set, p. 11	5–10: rate contexts and growth patterns; 11a–c: recursive, plotted, explicit	Section 3, including specific packet clarifications
Go, p. 12	11–19: inverse operations, negative/fractional coefficients	Section 4 and original new Problems A and B

Last updated October 13, 2026.