Last updated October 13, 2026.

Secondary Math I · Module 2 · Linear & Exponential Functions

2.1 — Connecting the Dots: Piggies and Pools

Supplement for Mathematics Vision Project (MVP) curriculum · Secondary Math I, Module 2, Section 2.1
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. Model quantities that change by a fixed addition or fixed percentage, decide which input values a model permits, and connect tables, equations, and graphs. Review arithmetic/geometric sequences, then practice identifying discrete versus continuous models and solving one-step equations.

Important mathematical clarification — Discrete and continuous functions

The MVP student packet and teacher’s manual do not always distinguish consistently between which inputs a model permits and whether a quantity changes in jumps. These are different mathematical questions. This lesson uses explicit definitions and states its modeling assumptions. The smaller red notes identify specific places where MVP’s explanation needs qualification.

A separate companion explanation examines the issue in greater depth, with exact passages from both MVP books.

1. In class: A Develop Understanding Task

Four ways quantities change

IN YOUR MVP PACKETSection 2.1 · Main task · pp. 1–2 · Problems 1–4

Build a table, equation, and graph for each situation. The first two involve adding a fixed amount; the next two involve multiplying by a fixed factor.

To describe a changing quantity, identify what you start with, how it changes, and which input values make sense. The last choice determines which points belong on its graph.

CLARIFYING DEFINITION — DomainThe domain of a function is the set of input values it is defined for. For a model of time, ask: are we studying only the end of each whole day, or the amount at every moment? The formula alone may not answer this question.

Problem 1: Pennies added once a day

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

Model the piggy bank using the number of completed days as input.

Savannah starts with 5 pennies and receives 3 more once each day. At the end of each new day, we add 3 to the previous total:

After completed days, the total is the starting number plus :

Meaning: counts completed days. The output is the number of pennies at those daily checkpoints.

Completed days	0	1	2	3	4
Pennies	5	8	11	14	17

Worked example — Find the amount after 6 days.

pennies. The starting five are counted only once.

Daily count: separate points, one for each whole-number day

CLARIFYING DEFINITION — Discrete-domain functionA discrete-domain function has inputs at separate values rather than every value in between. This particular model accepts whole-number days only; the graph therefore uses separate dots. We do not join them to assert that this formula gives the penny count at every instant.

MVP clarification — Section 2.1, Main Task, Problem 1

The MVP teacher’s notes say fractional-day inputs do not make sense because the deposit happens once per day. That is true of the chosen daily-checkpoint model, not a restriction on all possible models of the piggy bank. A count recorded at any time, including halfway through a day, is also meaningful; it stays constant between deposits and jumps when pennies are added. In this lesson the dots show the balance after whole numbers of days.

Problem 2: Filling a pool steadily

IN YOUR MVP PACKETMain task · p. 1 · Problem 2

Model gallons as a function of elapsed time while water flows into the pool.

The pool starts with 5 gallons. The hose adds 2 gallons each minute. The amount therefore increases by 2 for every additional minute, or by 1 for an additional half-minute.

Here is elapsed time in minutes; any nonnegative real time is allowed until the pool is full.

Time (minutes)	0	1	2.5	4
Gallons	5	7	10	13

Worked example — When will the pool be full?

It holds 1,500 gallons:

So the physical model has the domain minutes. It does not keep rising beyond capacity.

Water added steadily: all intermediate times are allowed

CLARIFYING DEFINITION — Continuous modelA continuous model allows the input to move through an entire interval and gives outputs without jumps. For the steady filling model, a time such as 2.5 minutes is meaningful. Its graph is an unbroken line over the relevant time interval.

Key comparison: Both the pennies and water increase by a fixed addition, so both use linear rules. But the daily-checkpoint pennies model uses separate input values, while the water-filling model uses any time in an interval.

Problem 3: Bank balance grows by 3% each month

IN YOUR MVP PACKETMain task · p. 2 · Problem 3

Write the monthly balance as repeated multiplication, not repeated addition.

Start with $50. The bank adds 3% of the current balance at the end of each month. Keeping 100% of the money and adding another 3% means each month-end balance is 103% of the previous one:

The exponent tells you how many monthly interest payments have been added.

Completed months	0	1	2	3
Ideal balance before rounding	$50.00	$51.50	$53.045	$54.63635

Worked example — Why not add $1.50 every month?

The first month’s interest is . The second month’s interest is , which is slightly more. The bank pays interest on the earlier interest too. For the idealized model, .

Balance recorded immediately after each monthly interest payment

MVP clarification — Section 2.1, Main Task, Problem 3

The teacher’s notes call for separate graph points. That is appropriate when the input means the number of completed monthly interest payments. However, the account still has a balance between payment dates. With no other transactions, that balance stays fixed until the next payment; it does not follow a smooth growth curve. The geometric sequence in this lesson describes month-end snapshots, not every moment in time. The unrounded table illustrates the pattern; a bank would round to cents.

Problem 4: Draining by 3% of what remains

IN YOUR MVP PACKETMain task · p. 2 · Problem 4

Write an exponential decay model for the remaining water.

The pool starts with 1,500 gallons. If 3% of the current amount drains in each minute, then 97% remains after each one-minute interval:

We use a smooth exponential curve to model water draining continuously from the pool.

Minutes	0	1	2	3
Gallons (approximately)	1,500	1,455	1,411.35	1,369.01

Worked example — Amount after 10 minutes.

Approximately 1,106 gallons remain. Subtracting 45 gallons every minute would be a different, linear model; here the gallons lost per minute become smaller as the pool empties.

Smooth exponential model of the draining pool

Compare the four models

IN YOUR MVP PACKETSection 2.1 · Main task · p. 3 · Problems 5–7

Explain how the models are alike or different in their rules, graphs, input domains, and real-world meanings.

Model	Rule	Change over one full unit	Inputs used in this lesson
Pennies	Linear: add 3	+3 pennies per day	Whole-number days
Filling pool	Linear: add 2 per minute	+2 gallons per minute	All times until full
Bank balance	Exponential: multiply by 1.03	+3% each month-end	Whole-number months
Draining pool	Exponential: multiply by 0.97	−3% each minute (relative to amount present)	All nonnegative times in the smooth model

Worked comparison — Why Problems 1 and 2 are both linear.

Pennies increase by 3 for each complete day; pool water increases by 2 for each minute. Neither multiplies its current value by a fixed percentage. They have the same type of change but different allowed inputs and different physical units.

Worked comparison — Why Problems 3 and 4 are both exponential.

A bank balance is multiplied by 1.03 each month and grows. Draining water is multiplied by 0.97 each minute and shrinks. In both cases the percentage applies to the current amount, not the original amount.

CLARIFYING DEFINITIONS — Linear, exponential, arithmetic, geometric

Linear: equal additions over equal input steps (a constant rate of change). Exponential: equal multiplication factors over equal input steps. An arithmetic sequence is a linear-type rule read at separate numbered terms; a geometric sequence is an exponential-type rule read at separate numbered terms. The two distinctions—kind of change and allowed inputs—are independent.

2. READY — Recognize arithmetic and geometric sequences

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 2.1 · Ready · p. 4 · Problems 1–9

Identify what stays constant between successive terms; extend a sequence and justify your answer.

Arithmetic: subtract each term from the next. If every difference is the same, it is arithmetic.

Geometric: divide each term by the previous nonzero term. If every ratio is the same, it is geometric.

Worked example — A constant difference.

Consider . Each time subtract 4:

This is arithmetic. The next two terms are .

Worked example — A negative multiplication factor.

Consider . Each term is the previous term multiplied by −3:

This is geometric. The next two terms are . Alternating signs do not mean a pattern is irregular.

Worked example — Neither pattern.

Consider . Its differences are 3, 5, 7, so the difference is not constant. Its ratios are also not constant. It is neither arithmetic nor geometric even though the pattern is predictable: it lists square numbers.

Common mistakes: Check every available step, not just the first two; for geometric sequences, the ratio may be a fraction or a negative number. A sequence can have an obvious pattern yet belong to neither category.

Check your work — Ready 1–9 answer key

$2500,-12500$; geometric, ratio $-5$.
$17,23$; neither. The successive increases are $2,3,4,5,6$.
$20.25,15.1875$; geometric, ratio $\frac34$.
$-1.5,-2.25$; arithmetic, difference $-0.75$.
$\frac5{32},\frac5{128}$; geometric, ratio $\frac14$.
$11111,111111$; neither. Append a 1 to each term; differences and ratios are not constant.
$-18.225,-27.3375$; geometric, ratio $1.5$.
$4,21$; arithmetic, difference 17.
Sample: $1,4,9,16$, the square numbers; neither constant differences nor constant ratios.

3. SET — When a problem has not defined the function

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 2.1 · Set · p. 4 · Problems 10–15

The packet asks you to label each statement discrete or continuous. Several descriptions do not define a function well enough for a unique answer. The examples below show what additional information is needed.

Important mathematical problem with the original packet
MVP clarification — Section 2.1, Set, Problems 10–15

These questions ask whether a relationship is discrete or continuous without clearly defining the function to be classified. To answer mathematically, we need to know what the input represents, what the output represents, and which input values are allowed (the domain).

Several statements can support either a discrete-domain model or a model using every time value in an interval. Problem 15 does not specify a function at all. As written, these questions do not have uniquely determined mathematical answers. There is no single correct classification without adding assumptions that the packet does not state.

For the homework, state a reasonable input, output, and domain before describing your model. Another student can legitimately choose a different model. A rate such as “per day” does not by itself establish a whole-number-day domain.

CLEAR DEFINITIONS — What must be specified?

Input: the quantity you choose, such as elapsed time or day number.

Output: the quantity the function gives for each input, such as total gallons or number of laughs.

Domain: all the input values the model allows. A discrete-domain model uses separated values, such as day numbers 0, 1, 2, … . A continuous-time model allows fractional times as well. Allowing every time value does not guarantee that the output changes smoothly; counts can jump.

Worked example — MVP Set Problem 12: Water flowing over Niagara Falls

The statement gives a volume of about 3.24 billion gallons per day, but does not say what the function's input or output should be. Consider two possible models, each assuming a steady average flow.

Model A — Record cumulative gallons only at the end of each day. Input: the completed day number n = 0, 1, 2, 3, … . Output: total gallons since day 0. The model is V(n) = (3.24 × 10⁹)n. Its domain is discrete, because we have chosen whole-day checkpoints.
Model B — Track cumulative gallons at any elapsed time. Input: t days, including fractions such as 0.5. Output: total gallons since the start. The model is V(t) = (3.24 × 10⁹)t for any real t ≥ 0. Its domain includes every nonnegative real time, and the idealized volume changes continuously.

Conclusion: the same daily rate supports both choices. The statement alone does not determine which model the exercise wants.

Worked example — MVP Set Problem 13: Laughs per day

The statement says an average person laughs about 15 times per day. It does not specify whether we are recording daily totals or tracking laughter moment by moment.

Model A — Daily record. Input: day number n = 1, 2, 3, … . Output: number of laughs during that day. We record one whole-number count per day, so the model has a discrete domain. The reported average of 15 does not mean every day has exactly 15 laughs.
Model B — Running count. Input: elapsed time t, including fractions of a day. Output: total laughs so far. The domain includes every moment of time, but the count stays unchanged between laughs and jumps by one when a laugh occurs. That is a step function, not a smoothly changing continuous function.

Conclusion: the output being a whole-number count does not tell us which inputs are allowed. And a function defined at every moment is not automatically a continuous function.

MVP problem	What's missing, and why a unique answer isn't justified
10 · Hair growth	Is the output hair length at any elapsed time or only the lengths measured at monthly check-ins? Continuous growth is a natural model, but monthly measurements give a discrete set of recorded values.
11 · Paper and trees	Is the input tons of paper (which may be fractional) or a count of full-ton recycling batches? Is “trees saved” an estimate or a count of particular trees? The relationship has not been specified.
12 · Niagara Falls	Does the input mean elapsed time, or the completed day number? Does the output mean cumulative volume or a single day's volume? See the worked example above.
13 · Laughter	Are we recording one count for each calendar day, or the cumulative number of laughs at every instant? These lead to different models. See the worked example above.
14 · Trash deposited	Is the input elapsed time or the number of completed days? Is the output cumulative deposited mass or a daily total? A daily rate alone does not choose between these models.
15 · Stock prices	A statement that prices fell 75% reports one change between two values. No independent variable, output rule, or domain is specified. There is no defined function here to classify as discrete or continuous.

CLARIFYING DISTINCTION — Discrete inputs are not the same as jumping outputs

A function defined only for day numbers 1, 2, 3, … has a discrete domain. A running count, defined at every instant but changing in jumps, has a continuous interval as its domain but is not a continuous function. These are different mathematical properties.

Check your work — Set 10–15 answer key

These statements do not specify both variables or their domains. The answers below state reasonable models; other clearly defined models may differ.

Hair length versus elapsed time: continuous.
Trees saved as a count: discrete; paper mass itself can vary continuously.
Accumulated water volume versus elapsed time: continuous.
A count of laughs: discrete.
Accumulated trash mass can be modeled continuously; daily recorded totals have separate day inputs.
A percentage fall alone does not specify a domain. Daily closing prices give separate inputs; the statement does not define a unique discrete/continuous function.

4. GO — Unit rates and 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 2.1 · Go · p. 5 · Problems 16–31

Find prices per item or per unit, solve simple equations, then match story problems to equations that represent them.

Calculate a unit rate

Unit rate = amount for one unit.

To find the price of one item when identical items cost the same, divide the total price by the number of items.

Worked example — Price per pound.

A 5-pound bag of apples costs $4.25. One pound costs:

Check the unit: dollars per pound. To express the result in cents, multiply dollars by 100.

Worked example — A fraction of a dozen.

If 12 muffins cost $9.00, then one muffin costs:

If six identical muffins are purchased, the model gives .

Undo a multiplication

To solve an equation such as , divide both sides by , provided .

Worked example — Solve a decimal-coefficient equation.

Check by multiplying: .

Worked example — A fraction multiplying x.

Dividing by is the same as multiplying by its reciprocal .

Connect an equation back to its story

The unknown is not always the same kind of quantity. In a grocery story, it might mean the number of dozens or the price of one item. Name the unknown, write an equation with units, and only then solve it. This is the skill used in Problem 31.

Worked example — Reverse a purchase.

If one notebook costs $3.60 and the total is $25.20, let be the number of notebooks:

Check: seven notebooks cost $25.20.

Two additional fully worked GO problems

Problem A — Straightforward.

Six identical pencils cost $2.70. Find the cost of one pencil in cents.

Solution: Divide by 6 pencils:

Each pencil costs 45 cents. Verify: .

Problem B — More challenging.

Three-quarters of a bag of trail mix weighs 1.8 pounds. How much does the entire bag weigh?

Solution: Let be the weight of a full bag. Because three-quarters of the bag is 1.8 pounds,

The full bag weighs 2.4 pounds. Check: pounds.

Check your work — Go 16–31 answer key

50 cents per pound.
About 17 cents.
About 17 cents.
6 dozen.
$17.31 per pair.
$17.31.
$x=12$.
$x=50$.
$x=\frac{50}{3}$.
$x=2.15$.
$x=34.62$.
$x=17.31$.
$x=16.5$.
$x=6$.
$x=8$.
Matching equations: story 16 → 23 (x is cents per pound); 17 → 24 (x is cents per apple); 18 → 28 (x is cents per egg); 19 → 29; 20 → 27. Story 21 uses the unit price from 20 and has no matching equation in 22–30.

Additional Learning Resources

Resource and direct link	What it covers / why use it	Scope / What to skip
Companion — Discrete Inputs, Jumping Counts, and Continuous Functions	Optional, ninth-grade explanation of different valid models of the same situation, followed by a teacher/parent analysis quoting the MVP student and teacher books.	Extra context. Start with the student explanation; the longer curriculum critique is optional. Keep this HTML file alongside the companion file so the link works offline.
Khan Academy — Intro to geometric sequences	Simple explanation of repeated multiplication, first term, and common ratio. Useful for Ready and the savings example.	Close match. Watch this introductory video; do not feel obliged to follow the entire sequences course.
Khan Academy — Graphing exponential growth and decay (practice)	Practice identifying the shape of smooth exponential graphs; useful for the draining-pool model.	Some extra material. Work with increasing/decreasing exponential curves; skip more advanced parameter transformations.
Professor Leonard — Geometric Sequences (College Algebra 71)	Optional full-length lecture on common ratios and formulas for geometric sequences.	Significantly broader. Use the opening explanation and examples of common ratios; skip formal generalized sequence notation or longer derivations not required here. Exact timestamps were not verified.
Paul’s Online Math Notes — Exponential Functions	Written reference explaining bases greater than 1 and bases between 0 and 1, with graphs and evaluated examples.	Some extra material. Focus on the basic definition and growth/decay graphs; skip the sections introducing the constant e and logarithms. The practice page has worked solutions, but some problems go beyond this lesson.

Teacher’s review of the original packet

Fundamental concern — Inconsistent criteria for “discrete” and “continuous”

The Section 2.1 teacher’s notes treat fractional-day inputs for the piggy bank as nonsensical because deposits occur once a day. Yet the Section 2.2 teacher’s notes recognize that shelving books can have a continuous time domain even though completed-book counts take only whole-number values. These are compatible modeling choices only if the input and sampling rule are stated. The notes instead alternate among the input domain, the possible outputs, and whether a process changes gradually. A student applying one explanation to another exercise is not given a consistent decision rule. See the separate companion analysis for precise passages and their locations.

Main task, Problem 1 — Distinguish daily checkpoints from an elapsed-time function

The teacher notes correctly call for separated points at whole-number days. But the physical number of pennies exists between deposits, so “the number of pennies as a function of time” is not intrinsically restricted to integer times. A whole-day snapshot model is valid, while an instantaneous count is a step function. Explicitly say which one the graph represents.

Main task, Problem 3 — Bank balance is not continuously compounded

The packet specifies interest payments at the end of each month. The ideal monthly-checkpoint model is geometric, but treating the account as continuously increasing between payments would misrepresent the stated terms. Its real-time balance is constant between postings, absent withdrawals or deposits. Amounts should also be rounded to cents in an actual bank, even if an unrounded model is useful for instruction.

Serious mathematical deficiency — Set Problems 10–15 do not specify functions

The student packet asks students to identify each statement as a discrete or continuous relationship, but generally does not provide an independent variable, dependent variable, or domain. These are not optional details: the classification depends on the mathematical model being defined. For instance, “17 trees saved per ton” leaves the meaning and permitted values of the input unresolved; “15 laughs per day” could mean a daily count or a running count defined at every instant. The description of stock prices falling 75% in Problem 15 does not define a function at all.

Consequently, these exercises do not have uniquely defensible classifications as written. This is not just awkward wording. The packet asks students to apply a mathematical classification before specifying the mathematical object to which it applies. A student can make reasonable assumptions and still produce a different answer from the one expected.

Suggested correction: For each context, state the independent and dependent variables, specify the domain, and distinguish the exact modeled quantity from daily measurements or approximations. Alternatively, explicitly ask students to propose two possible models for the same description and explain how their domains differ. Do not grade an unstated modeling assumption as though it were a mathematical error.

Coverage check — Section 2.1 only

MVP packet section	Skills	Supplement location
Main task, pp. 1–2, 1–4	Four equations, tables, graphs; linear addition vs exponential factor; selecting meaningful domains	In class: Four ways quantities change
Main task, p. 3, 5–7	Compare linear/exponential, increasing/decreasing, discrete/continuous models	In class: Compare the four models
Ready, p. 4, 1–9	Extend arithmetic, geometric and other sequences; negative/fractional ratios; justify	2. Ready
Set, p. 4, 10–15	Choose and explain appropriate model/domain; identify ambiguity	3. Set and packet clarification
Go, p. 5, 16–21	Unit prices, unit conversions, dozen/quantity interpretation	4. Go — unit rates and story equations
Go, p. 5, 22–31	One-step equations including fractional multipliers; match an equation to a context	4. Go — equations and original A/B examples

Last updated October 13, 2026.