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HSF.LE.A.1Common CoreMathFunctionsGrades 9-12

HSF.LE.A.1: Distinguishing Linear and Exponential Models

In plain English: HSF.LE.A.1 is the Common Core functions standard that asks students to tell apart situations modeled by linear functions and situations modeled by exponential functions. Students prove that linear functions change by equal differences over equal intervals while exponential functions change by equal factors, and they recognize constant-rate and constant-percent-rate situations. It is usually taught in Algebra I.

Distinguish between situations that can be modeled with linear functions and with exponential functions.

  1. a.Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
  2. b.Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
  3. c.Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
Common Core State Standards for Mathematics · Domain: Linear, Quadratic, and Exponential Models (LE) · Cluster: Construct and compare linear, quadratic, and exponential models and solve problems
Also written as HSF-LE.A.1 or F-LE.1 · Official standard
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01

Lesson Plan

65-70 min

Overview

Students learn to tell whether a situation calls for a linear or an exponential model. The lesson starts from the defining patterns: over equal intervals, a linear function always changes by the same difference and an exponential function always changes by the same factor. Students prove both facts algebraically for any interval, not only for the rows of one table.

Students then apply the patterns to words and tables. A constant rate per unit interval, such as gallons per minute, points to a linear function; a constant percent rate per unit interval, such as 2.5% per year, points to an exponential function. Tables with unequal input steps and phrases such as "percentage points" give practice at reading situations carefully.

Learning Objectives

By the end of this lesson, students will be able to:

  • Prove that a linear function f(x) = mx + b changes by the same difference mh over every interval of length h
  • Prove that an exponential function g(x) = a · bˣ changes by the same factor bʰ over every interval of length h
  • Recognize situations in which a quantity changes at a constant rate per unit interval and model them with linear functions
  • Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval and model them with exponential functions
  • Decide from a table, including one with unequal input steps, whether the data are linear, exponential or neither

Prior Knowledge Required

Students should already be comfortable with:

  • Linear functions, slope as a rate of change and the form y = mx + b 8.F.A.3
  • Constructing a linear function from a description or a table 8.F.B.4
  • Properties of integer exponents, such as bˣ⁺ʰ = bˣ · bʰ 8.EE.A.1
  • Converting percents to decimals and finding a percent of a quantity 7.RP.A.3

Lesson Procedure

65-70 minutes of class time across 5 phases.

  1. Warm-Up10 minutes

    Post the two situations below and give students three minutes to build both tables before any discussion.

    Warm-Up Prompt

    "A plant is 4 cm tall and grows 2 cm each week. A patch of mold covers 4 cm² and its area doubles each week. Make a table of each quantity for weeks 0 to 4. How does each one change from one week to the next?"

    Record the tables: the plant is 4, 6, 8, 10, 12 cm and the mold is 4, 8, 16, 32, 64 cm². Ask students to write the change under each pair of columns. The plant always adds 2; the mold adds 4, 8, 16 and 32, so its additions are not equal, but it is always multiplied by 2. Name the two patterns, "equal differences" and "equal factors," and tell students that the lesson will prove these patterns always hold for linear and exponential functions.

  2. Direct Instruction20 minutes

    Part 1: Prove the two growth patterns (standard a). A proof has to work for every interval of a given length, not only for the intervals in one table. Write the two general functions on the board, f(x) = mx + b and g(x) = a · bx with a ≠ 0, b > 0 and b ≠ 1, and walk through these steps:

    1. Pick an arbitrary interval: from x to x + h, where h > 0 is the interval length and x can be any input.
    2. Linear case, subtract: f(x + h) - f(x) = m(x + h) + b - (mx + b) = mh. The x cancels, so every interval of length h has the same difference mh.
    3. Exponential case, divide: g(x + h) ÷ g(x) = a · bx + h ÷ (a · bx) = bh. The x cancels, so every interval of length h has the same factor bh.
    4. Read the result: for h = 1 the difference is the slope m and the factor is the base b. Equal differences mean a linear function; equal factors mean an exponential function.

    Work Examples 1 and 2 as numerical versions of the proof. Then use Diagram 1: the linear staircase has equal rises, and the exponential staircase has rises that grow while the ratio stays 2.

    Part 2: Recognize the situation (standards b and c). Ask of every situation: "Does the same amount get added or subtracted in each unit of time, or does the quantity get multiplied by the same factor?" A constant rate per unit (gallons per minute, dollars per month) signals a linear model. A constant percent rate per unit (2.5% per year, 15% per year) signals an exponential model with factor 1 + r for growth or 1 - r for decay.

    • Proof: equal differences

      For f(x) = 4x + 7, compare the outputs at the ends of any interval of length 3, from x to x + 3.

      Equation: f(x + 3) - f(x) = (4x + 19) - (4x + 7) = 12 for every x

    • Proof: equal factors

      For g(x) = 5 · 2ˣ, compare the outputs at the ends of any interval of length 3.

      Equation: g(x + 3) ÷ g(x) = (5 · 2ˣ · 2³) ÷ (5 · 2ˣ) = 8 for every x

    • Constant rate (linear)

      A 500-gallon tank drains at 12 gallons per minute. The same amount leaves in every minute.

      Equation: V = 500 - 12t, a change of -12 gallons per minute; empty after 41⅔ minutes

    • Constant percent rate, growth (exponential)

      A town of 8,000 people grows by 2.5% per year. Each year the population is multiplied by 1.025.

      Equation: P = 8000(1.025)ᵗ, so P = 8,200 after 1 year

    • Constant percent rate, decay (exponential)

      A car bought for $24,000 loses 15% of its value each year, so 85% remains after every year.

      Equation: V = 24000(0.85)ᵗ, so V = $20,400 after 1 year

    After Example 5, use Diagram 2 to compare two decreasing quantities that start at 200: one loses 25 each hour, the other loses 20% each hour. The linear one reaches 0 after 8 hours; the exponential one loses less and less each hour and never reaches 0.

  3. Guided Practice15 minutes

    Pairs classify five tables. For each one they write the differences and the ratios of consecutive outputs, then decide: linear, exponential or neither.

    • Table A: x = 0, 1, 2, 3, 4 and y = 3, 7, 11, 15, 19 (linear: difference 4 per unit)
    • Table B: x = 0, 1, 2, 3, 4 and y = 2, 6, 18, 54, 162 (exponential: factor 3 per unit)
    • Table C: x = 0, 1, 2, 3, 4 and y = 1, 2, 4, 7, 11 (neither: differences 1, 2, 3, 4 and ratios 2, 2, 1.75, about 1.57)
    • Table D: x = 0, 2, 4, 6 and y = 80, 40, 20, 10 (exponential: factor 1/2 over every 2 units)
    • Table E: x = 0, 1, 3, 4 and y = 5, 8, 14, 17 (linear: the inputs skip 2, but the change is 3 per unit on every interval)

    Table E is the trap: the raw differences are 3, 6 and 3, but the intervals are not equal. Divide each difference by the length of its interval before comparing.

  4. Independent Practice15 minutes

    Students decide whether each situation is linear or exponential, name the constant rate or the constant percent rate, and write a function.

    • A gym charges a $25 sign-up fee plus $30 per month (linear: C = 25 + 30m)
    • A bacteria culture of 500 cells doubles every 20 minutes (exponential: factor 2 per 20 minutes)
    • A 400 mg dose of a medicine leaves the body at 18% per hour (exponential decay: A = 400(0.82)t)
    • A 24 cm candle burns down 1.5 cm per hour (linear: h = 24 - 1.5t)
    • $1,500 in a savings account earns 4% interest compounded yearly (exponential: B = 1500(1.04)t)
    • A phone battery at 100% drops 8 percentage points per hour of video (linear: P = 100 - 8t, because the same 8 points leave every hour)

    The last item is a deliberate check: "percentage points" of a fixed full charge is a constant amount, not a constant percent of what is left.

  5. Closure5-10 minutes

    Exit ticket: (1) A quantity increases by 6 units every 2 hours. Linear or exponential? (Linear: equal differences of 6 over equal 2-hour intervals, a rate of 3 per hour.) (2) A quantity increases by 6% every 2 hours. Linear or exponential? (Exponential: factor 1.06 over every 2-hour interval.) (3) Show that for g(x) = a · bx, the ratio g(x + 1) ÷ g(x) does not depend on x. (It equals b.)

Differentiation Strategies

For Struggling Students

  • Give a two-row template under each table: one row for "subtract" (differences) and one for "divide" (ratios), so students always test both patterns
  • Start with situations that name the change explicitly ("adds 30 dollars each month", "is multiplied by 1.04 each year") before phrases like "4% interest"
  • Have students compute two or three values of each situation before classifying it, so the pattern is visible in numbers first

For Advanced Students

  • Ask students to prove the converse for whole-number inputs: if a sequence has equal ratios between consecutive terms, then it has the form a · bⁿ
  • Ask why a constant percent rate of 10% per year is not the same as 20% every 2 years, and find the exact 2-year factor (1.21)
  • Ask students to explain why an exponential model with 0 < b < 1 never reaches 0, while a decreasing linear model always crosses 0

Assessment Guidance

What to Look For

Listen for proofs that start from an arbitrary interval from x to x + h, not from two specific rows of a table: standard a asks students to prove, not only to observe. When students classify a table, check that they compare changes over intervals of the same length. In word problems, look for a clear reason tied to the wording: "the same number of gallons leaves every minute" or "the value is multiplied by 0.85 every year." A student who says "it goes up fast, so it is exponential" needs to be pushed back to differences and factors.

02

Classroom Activities

3 Activities

1

Stack It or Fold It

15 minPairs

Students measure two ways a pile of paper can get thicker: by adding one sheet at a time or by folding the pile in half. The two tables show equal differences and equal factors side by side.

Procedure

  • Each pair gets several sheets of printer paper. Take 0.1 mm as the thickness of one sheet (a typical value)
  • Stack: start with one sheet (step 0) and add one sheet at each step. Record the thickness for steps 0 to 6: 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7 mm
  • Fold: start with one sheet and fold the pile in half at each step. Record the thickness for steps 0 to 6: 0.1, 0.2, 0.4, 0.8, 1.6, 3.2, 6.4 mm
  • Under each table, write the differences and the ratios between consecutive steps, then label each table linear or exponential

Discussion Questions

  • Which table has a constant rate per step, and what is it? (Stack: 0.1 mm per step.)
  • Which table has a constant percent rate per step? (Fold: the thickness grows 100% per step, a factor of 2.)
  • Most pairs cannot fold a sheet more than 6 or 7 times. Use the table to explain why.

Modification for Distance Learning

Students fold a sheet at home and report how many folds they managed. The class builds both tables in a shared spreadsheet, with one column for differences and one for ratios.

2

Situation Sort

20 minGroups of 3-4

Groups sort 12 situation cards into four piles: linear increasing, linear decreasing, exponential growth and exponential decay. For each card they must underline the words that decide the pile.

The 12 Cards

  1. A hiker descending a trail loses 300 meters of elevation per hour (linear decreasing)
  2. A concert venue sells 250 tickets each day (linear increasing)
  3. A new social media account gains 9% more followers each week (exponential growth)
  4. The temperature difference between hot cocoa and the room shrinks by 10% each minute (exponential decay)
  5. A rental car costs a $30 fee plus $45 per day (linear increasing)
  6. A radioactive isotope loses half its mass every 8 days (exponential decay)
  7. A $2,000 deposit earns 4.5% interest compounded yearly (exponential growth)
  8. A printer tray holds 500 sheets and the printer uses 20 sheets per minute (linear decreasing)
  9. A virus sample in a lab triples every 6 hours (exponential growth)
  10. A basketball bounces back to 70% of its previous height on each bounce (exponential decay)
  11. A worker earns $17.50 per hour (linear increasing)
  12. Ice on a pond thickens by 0.4 cm per day during a cold spell (linear increasing)

Procedure

  • Groups cut out the cards and place each one in a pile, underlining the deciding words
  • Check against the key: 4 linear increasing (2, 5, 11, 12), 2 linear decreasing (1, 8), 3 exponential growth (3, 7, 9), 3 exponential decay (4, 6, 10)
  • For two cards of their choice, groups write the function and a three-row table

Challenge Variation

Groups rewrite three cards so that each one moves to a different pile, changing as few words as possible. For example, "sells 250 tickets each day" becomes "sells 5% more tickets each day."

3

Prove It Two Ways

20 minPairs

Students write the two proofs from standard a in their own words, then test them on specific functions over intervals of length 2. The goal is to see why a table can suggest a pattern but only algebra proves it for every interval.

Procedure

  • Partner A proves that f(x) = mx + b changes by mh over any interval from x to x + h. Partner B proves that g(x) = a · bx changes by the factor bh. Partners swap and check each other's steps
  • Test the proofs with h = 2: for f(x) = -3x + 10, the outputs at x = 0, 2, 4, 6 are 10, 4, -2, -8, a difference of -6 = -3 · 2 each time
  • For g(x) = 96(1/2)x, the outputs at x = 0, 2, 4, 6 are 96, 24, 6, 1.5, a factor of 1/4 = (1/2)2 each time
  • Pairs write one sentence explaining why the x cancels in each proof and what that means

Discussion Questions

  • Why is checking three rows of a table not a proof?
  • In the linear proof you subtract; in the exponential proof you divide. Why does each operation fit its function?
  • For g, what happens to the differences over intervals of length 2? (They shrink: -72, -18, -4.5.) Does that contradict the proof?

03

Diagrams & Visual Aids

2 diagrams

Diagram 1: Equal Differences Versus Equal Factors

1 2 3 4 4 8 12 16 +3 +3 +3 +3 1 2 3 4 8 16 24 32 +2, ×2 +4, ×2 +8, ×2 +16, ×2 Linear: y = 3x + 2 Exponential: y = 2 · 2ˣ Differences: 3, 3, 3, 3 (equal) Ratios: 5/2, 8/5, 11/8, 14/11 (not equal) Ratios: 2, 2, 2, 2 (equal) Differences: 2, 4, 8, 16 (not equal)
Both graphs are drawn to scale for x = 0 to 4. On the line y = 3x + 2, every step of 1 unit to the right rises by the same amount, 3. On y = 2 · 2x, the rises grow (2, 4, 8, 16), but every step multiplies the output by the same factor, 2. Subtracting tests for linear; dividing tests for exponential.

Diagram 2: Losing a Constant Amount Versus a Constant Percent

1 2 3 4 5 6 7 8 50 100 150 200 Hours t A: 200 - 25t (dashed) loses 25 each hour B: 200(0.8)ᵗ (solid) loses 20% each hour t A B 0 200 200 1 175 160 2 150 128 3 125 102.4 4 100 81.92
Two quantities start at 200. Quantity A decreases at a constant rate of 25 per hour and reaches 0 at t = 8. Quantity B decreases by a constant percent rate of 20% per hour, so it is multiplied by 0.8 each hour: it loses 40 in the first hour but only about 10.5 between hours 6 and 7, and it never reaches 0. Points are plotted to scale at whole hours.

04

Homework Assignment

~30 min

HSF.LE.A.1 Homework: Linear or Exponential?

Directions: Show all work. When you classify a table, write the differences or the ratios you used, and adjust for intervals of different lengths. When you classify a situation, quote the words that decide it.

Part 1: Proving the Patterns (Problems 1-2)

  1. Let f(x) = -2.5x + 40. (a) Find f(0), f(4), f(10) and f(14), and show that f changes by the same amount on the intervals from 0 to 4 and from 10 to 14. (b) Prove that f changes by this same amount over every interval of length 4, from x to x + 4.
  2. Let g(x) = 81(1/3)x. (a) Prove that over every interval of length 2, from x to x + 2, the output is multiplied by the same factor, and find that factor. (b) Compute g(0), g(1), g(2) and g(3), and show that the differences g(2) - g(0) and g(3) - g(1) are not equal. Explain why this does not contradict part (a).

Part 2: Reading Tables (Problems 3-4)

  1. Classify each table as linear, exponential or neither, and give the constant difference or factor per unit when there is one. Table P: x = 0, 1, 2, 3, 4 and y = 12, 18, 27, 40.5, 60.75. Table Q: x = 1, 2, 3, 4, 5 and y = 50, 43, 36, 29, 22. Table R: x = 0, 1, 2, 3, 4 and y = 2, 3, 6, 11, 18.
  2. A table shows x = 0, 2, 5, 6, 10 and y = 7, 13, 22, 25, 37. A classmate says, "The differences are 6, 9, 3 and 12, so the data are not linear." Explain the error, decide whether the data are linear, and if they are, write the function.

Part 3: Situations (Problems 5-6)

  1. Decide whether each situation is linear or exponential, state the constant rate or the constant percent rate, and write a function. (a) A streaming service has 1,200 subscribers and gains 150 new subscribers each month. (b) Algae cover 30 m² of a lake, and the covered area grows by 12% each day. (c) A snowbank 90 cm deep melts 4 cm each day. (d) A population of 640 rare birds falls by 5% each year.
  2. A city bike-share program had 2,000 rides in its first month. Plan 1 predicts 300 more rides each month; Plan 2 predicts 12% more rides each month. (a) Make a table of rides for months 0 to 4 under each plan, rounding Plan 2 to whole rides. (b) Show that Plan 1 has equal differences and Plan 2 has equal factors. (c) Write a function for each plan.

Rubric

CriterionFull Credit (2 pts)Partial Credit (1 pt)No Credit (0 pts)
ProofsWorks from an arbitrary interval x to x + h and simplifies correctlyCorrect idea shown only with specific numbersNo proof or incorrect algebra
TablesDifferences or ratios computed and adjusted for interval lengthCorrect classification with incomplete evidenceIncorrect classification
SituationsDeciding words quoted, rate or percent rate named, function correctCorrect type with an error in the functionIncorrect type
AccuracyAll values correct and rounded sensiblyOne or two arithmetic errorsSeveral errors

05

Quiz: 20 Questions

Interactive, with answers

Instructions

Answer each question, then open the explanation to check your reasoning. Your score updates as you go, and Reset quiz clears every answer so the quiz can be used again.

Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.

0 of 20 answered · 0 correct

  1. Question 1 of 20 · Multiple Choice

    Which situation is best modeled by a linear function?

  2. Question 2 of 20 · Multiple Choice

    Which situation shows exponential decay?

  3. Question 3 of 20 · Multiple Choice

    A table shows x = 0, 1, 2, 3 and y = 5, 15, 45, 135. Which description fits?

  4. Question 4 of 20 · Multiple Choice

    A table shows x = 0, 1, 2, 3 and y = 20, 14, 8, 2. Which description fits?

  5. Question 5 of 20 · Multiple Choice

    For f(x) = 6x - 1, what is f(x + 5) - f(x) for every x?

  6. Question 6 of 20 · Multiple Choice

    For g(x) = 3 · 4x, what is g(x + 2) ÷ g(x) for every x?

  7. Question 7 of 20 · Multiple Choice

    A quantity grows at a constant percent rate of 6% per year. Each year, the quantity is multiplied by:

  8. Question 8 of 20 · Multiple Choice

    A quantity decreases by 35% each week. What is its weekly growth factor?

  9. Question 9 of 20 · Multiple Choice

    Which table shows an exponential function? Each table uses x = 0, 1, 2, 3.

  10. Question 10 of 20 · Multiple Choice

    Which statement about h(x) = 2x + 3 is true?

  11. Question 11 of 20 · Multiple Choice

    A table shows x = 0, 1, 3, 6 and y = 4, 8, 32, 256. What model fits, and what is the factor per unit?

  12. Question 12 of 20 · Multiple Choice

    A quantity starts at 500. Which additional fact shows that an exponential model fits, rather than a linear one?

  13. Question 13 of 20 · Multiple Choice

    Which of these is modeled by a linear function?

  14. Question 14 of 20 · Multiple Choice

    To prove that f(x) = mx + b grows by equal differences over equal intervals, which expression should you simplify?

  15. Question 15 of 20 · Short Answer

    Let p(x) = 1.5x + 4. Show that p changes by the same amount on the interval from 2 to 6 and on the interval from 9 to 13, and explain why.

  16. Question 16 of 20 · Short Answer

    Prove that q(x) = 250(0.9)x is multiplied by the same factor over every interval of length 2. What is the factor, and what percent of the quantity is lost over each 2-unit interval?

  17. Question 17 of 20 · Short Answer

    A club's membership grew from 400 to 460 in the first year and from 460 to 529 in the second year. Is a linear or an exponential model a better fit? Explain.

  18. Question 18 of 20 · Short Answer

    Two coffee shops charge $3.00 for a coffee now. Shop 1 raises its price by 10 cents each year. Shop 2 raises its price by 4% each year. Which price is linear and which is exponential? Write a function for each.

  19. Question 19 of 20 · Short Answer

    A table shows x = 0, 1, 2, 3, 4 and y = 1000, 900, 810, 729, 656.1. Describe the change as a percent rate and write a function.

  20. Question 20 of 20 · Short Answer

    Quantity M starts at 50 and increases by 6 over every 2-unit interval. Quantity N starts at 50 and is multiplied by 1.2 over every 2-unit interval. Find each quantity at x = 2, 4 and 6, and name each model type.

0 of 20 answered · 0 correct

06

Frequently Asked Questions

10 Questions

What does HSF.LE.A.1 mean?

HSF.LE.A.1 means students can tell whether a situation is linear or exponential and can justify the choice. Part a asks them to prove that linear functions change by equal differences and exponential functions by equal factors over equal intervals. Parts b and c ask them to recognize those patterns in words: a constant rate per unit (linear) or a constant percent rate per unit (exponential).

Is HSF.LE.A.1 taught in Algebra 1 or Algebra 2?

It is usually taught in Algebra I, when exponential functions are first introduced next to linear ones. Algebra II returns to the same ideas with continuous growth, logarithms and more complex models, so students who master the difference-versus-factor test here use it again later.

How can you tell from a table if a function is linear or exponential?

Check the differences and the ratios of outputs over equal input intervals. If the differences are all equal, the function is linear. If the ratios are all equal, it is exponential. If neither is constant, the data are neither, for example a quadratic pattern such as 3, 4, 7, 12, 19.

What if the x-values in the table are not evenly spaced?

Then compare changes per unit, not raw changes. For a linear function, divide each difference by the length of its interval. For an exponential function, the ratio over an interval of length h should equal bh, so a ratio of 9 over 2 units means a factor of 3 per unit. A common mistake is to compare raw differences across intervals of different lengths.

What does it mean to "prove" that linear functions grow by equal differences?

It means showing the result for every interval of a given length, not just for the rows of a table. Start with f(x) = mx + b and an arbitrary interval from x to x + h. Then f(x + h) - f(x) = mh, and because x has cancelled, every interval of length h has the same difference. The exponential proof divides instead: a · bx + h ÷ (a · bx) = bh.

What is the difference between a constant rate and a constant percent rate?

A constant rate adds or subtracts the same amount each unit, such as 12 gallons per minute; that gives a linear function. A constant percent rate changes the quantity by the same percent of its current value each unit, such as 2.5% per year; that multiplies by the same factor (1.025) each unit and gives an exponential function.

Why is simple interest linear but compound interest exponential?

Simple interest is always a percent of the original deposit, so the same dollar amount is added each year. Compound interest is a percent of the current balance, which keeps growing, so the balance is multiplied by the same factor each year. For $2,000 at 3%, simple interest adds $60 every year, while compound interest adds $60, then $61.80, then about $63.65.

Is "drops 8 percentage points per hour" exponential?

No, it is linear. Percentage points of a fixed whole, such as a phone battery's full charge, are a constant amount: the battery goes 100%, 92%, 84% and so on. It would be exponential only if the battery lost 8% of its remaining charge each hour. Wording like this is worth practicing because students often react to the % sign alone.

What are common mistakes with HSF.LE.A.1?

Common mistakes include calling anything that grows quickly exponential, using the percent itself as the factor (0.04 instead of 1.04, or 0.25 instead of 0.75 for a 25% decrease), comparing changes over intervals of different lengths, and treating a few checked rows as a proof. Asking "subtract or divide?" for every table addresses most of them.

How does this standard connect to later topics?

It is the starting point for the rest of the cluster: students next construct linear and exponential functions from graphs, descriptions and tables (HSF.LE.A.2), compare their long-run growth (HSF.LE.A.3) and interpret their parameters in context (HSF.LE.B.5). On the digital SAT, linear models belong to the Algebra domain and exponential models to Advanced Math.