Math 1 Quiz: Comparing Linear Vs Exponential Models
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Comparing Linear Vs Exponential ModelsQuestion 1 of 18

A technology company's user base grows from 10,000 to 40,000 users over 8 months. The marketing team assumes this represents constant monthly growth and projects 70,000 users after 16 months total. Which analysis of this projection method is most comprehensive?

The projection is reasonable assuming linear growth of 3,750 users per month continues at the same rate consistently
The projection uses linear growth correctly, but exponential growth (≈18.9% monthly) would predict about 73,000 users instead
The projection assumes linear growth yielding the stated result, but the actual growth pattern needs analysis to validate this approach
The projection method depends on whether 'constant growth' means constant absolute increases or constant percentage increases, significantly affecting the result
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Math 1 Quiz

Math 1 Quiz: Comparing Linear Vs Exponential Models

Practice Comparing Linear Vs Exponential Models in Math 1 with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Comparing Linear Vs Exponential Models, giving you a quick way to practice the rules, question types, and explanations that matter most for Math 1.

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Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

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Question 1

A technology company's user base grows from 10,000 to 40,000 users over 8 months. The marketing team assumes this represents constant monthly growth and projects 70,000 users after 16 months total. Which analysis of this projection method is most comprehensive?

  1. The projection is reasonable assuming linear growth of 3,750 users per month continues at the same rate consistently
  2. The projection uses linear growth correctly, but exponential growth (≈18.9% monthly) would predict about 73,000 users instead
  3. The projection assumes linear growth yielding the stated result, but the actual growth pattern needs analysis to validate this approach
  4. The projection method depends on whether 'constant growth' means constant absolute increases or constant percentage increases, significantly affecting the result (correct answer)
Explanation: The correct answer is D. The phrase 'constant monthly growth' is ambiguous - it could mean constant absolute increases (linear: +3,750 users/month, reaching 10,000 + 16(3,750) = 70,000 users) or constant percentage increases (exponential: ≈18.9%/month, reaching 10,000(1.189)^16 ≈ 73,000 users). The ambiguous terminology leads to different valid interpretations and significantly different results. Choice A assumes linear without acknowledging the ambiguity, choice B accepts linear as correct without questioning the assumption, and choice C doesn't address the fundamental terminology issue.

Question 2

A population starts at 1,000 individuals. After 10 time periods, it reaches 2,000 individuals. A student claims this must be exponential growth because "the population doubled." Which analysis of this claim is most appropriate?

  1. The claim is correct because doubling always indicates exponential growth regardless of the time frame involved
  2. The claim is incorrect because exponential doubling would require exactly 2^{10} = 1,024 individuals, not 2,000 individuals
  3. The claim cannot be evaluated without knowing the population values at intermediate time periods between start and end (correct answer)
  4. The claim is incorrect because doubling could result from either linear growth of 100 per period or exponential growth
Explanation: The correct answer is C. Knowing only the starting and ending values is insufficient to determine the growth pattern. A population could reach 2,000 from 1,000 through constant linear growth (+100+100 each period), exponential growth (≈7.2%\approx 7.2\% per period), or various other patterns. The intermediate values reveal whether growth is additive (linear) or multiplicative (exponential). Choice A incorrectly assumes all doubling is exponential, choice B misunderstands exponential growth calculations, and choice D correctly identifies that both models could work but doesn't emphasize the need for intermediate data.

Question 3

An economist analyzes income data where the relationship between years of experience (xx) and salary (yy) shows R2=0.78R^2 = 0.78 for y=35,000+2,500xy = 35,000 + 2,500x and R2=0.81R^2 = 0.81 for y=35,000(1.06)xy = 35,000(1.06)^x. Given that salary negotiations typically involve annual percentage increases, which model selection approach is most justified?

  1. Choose the exponential model because the higher R2R^2 value and theoretical basis both support percentage-based salary growth (correct answer)
  2. Choose the linear model because the R2R^2 difference is small and linear models are easier to interpret for salary planning
  3. Choose the exponential model based solely on the higher R2R^2 value since statistical fit should determine model selection
  4. Choose the linear model because it predicts more conservative salary growth and reduces the risk of overestimating future earnings
Explanation: The correct answer is A. The exponential model has both better statistical fit (higher R2R^2) and theoretical justification (salary increases typically occur as percentage raises, not fixed dollar amounts). This combination of statistical and theoretical support makes it the most appropriate choice. Choice B undervalues the theoretical justification, choice C relies solely on statistical fit while ignoring context, and choice D introduces risk considerations that aren't relevant to model selection based on data patterns.

Question 4

A company's quarterly revenue follows the pattern: Q1: $100,000, Q2: $140,000, Q3: $196,000, Q4: $274,400. The CEO claims this represents $40%40\% $ quarterly growth and should be modeled exponentially. Which evaluation of this claim is most thorough?

  1. The claim is correct because 140,000/100,000=1.4140,000/100,000 = 1.4, confirming 40%40\% growth, so exponential modeling is appropriate throughout
  2. The claim is correct about 40%40\% growth and exponential modeling; ratios of 1.4, 1.4, 1.4 show constant multiplicative growth (correct answer)
  3. The claim is partially correct about the growth rate but wrong about consistency; ratios are 1.4, 1.4, 1.4, confirming exponential
  4. The claim appears correct with consistent 40%40\% growth, but additional quarters of data would strengthen the exponential model choice
Explanation: When evaluating claims about exponential growth, you need to check two things: whether the growth rate is consistent and whether that rate matches what's claimed. Exponential growth means each term is found by multiplying the previous term by the same constant factor. Let's verify the CEO's claim by calculating the growth ratios between consecutive quarters. From Q1 to Q2: 140,000÷100,000=1.4140,000 ÷ 100,000 = 1.4. From Q2 to Q3: 196,000÷140,000=1.4196,000 ÷ 140,000 = 1.4. From Q3 to Q4: 274,400÷196,000=1.4274,400 ÷ 196,000 = 1.4. Each quarter shows exactly 40% growth (since a ratio of 1.4 means the new value is 140% of the previous, representing a 40% increase). The consistent multiplicative factor of 1.4 confirms this is indeed exponential growth. Choice A stops after checking only the first ratio, which is insufficient to establish a pattern. Choice C contradicts itself by saying the growth rate is wrong about consistency while simultaneously confirming the ratios are consistently 1.4. Choice D suggests the data is insufficient when we actually have perfect consistency across three growth periods, which is quite strong evidence for exponential behavior. Choice B correctly identifies that the CEO's claim is accurate on both counts: the growth rate is consistently 40% and the pattern is exponential, as demonstrated by the constant ratios. Study tip: For exponential growth problems, always calculate ratios between consecutive terms. If they're constant, it's exponential; if they vary, it's not. Don't just check the first ratio—patterns require multiple data points.

Question 5

Two students model the same dataset. Student A uses y=15x+120y = 15x + 120 and reports R2=0.89R^2 = 0.89. Student B uses y=120(1.08)xy = 120(1.08)^x and reports R2=0.91R^2 = 0.91. The data represents monthly sales over 24 months. Which factor should be most important in choosing between these models?

  1. Choose Student B's model because the higher R2R^2 value indicates better statistical fit to the observed data
  2. Choose Student A's model because linear models are generally simpler and easier to interpret than exponential models
  3. Choose based on which model makes more sense theoretically for sales growth and shows better residual patterns (correct answer)
  4. Choose Student B's model because exponential growth is more realistic for business applications than linear growth
Explanation: The correct answer is C. While R2R^2 values provide useful information, model selection should prioritize theoretical appropriateness (does the growth mechanism make sense for sales?) and residual analysis (are there patterns in prediction errors?). A slightly higher R2R^2 doesn't automatically make a model better if it lacks theoretical justification or shows problematic residuals. Choice A overemphasizes R2R^2, choice B inappropriately prioritizes simplicity over fit, and choice D makes an unjustified generalization about business growth patterns.

Question 6

A biologist studies two bacterial colonies. Colony A increases by 500 bacteria per hour. Colony B doubles every 4 hours. After 24 hours, both colonies have grown substantially. Which statement about long-term modeling and growth comparison is most accurate?

  1. Colony A follows linear growth and will eventually be overtaken by Colony B's exponential growth, regardless of starting populations
  2. Colony B follows exponential growth and will always grow faster than Colony A's linear growth after sufficient time passes
  3. The relative growth depends on starting populations; Colony A's linear growth could outpace Colony B's exponential growth indefinitely
  4. Colony B's exponential growth will eventually exceed Colony A's linear growth, but the timeline depends on initial population sizes (correct answer)
Explanation: The correct answer is D. Colony A shows linear growth (+500+500 per hour), while Colony B shows exponential growth (doubling every 4 hours, or 21/4≈1.1892^{1/4} ≈ 1.189 per hour). Eventually, exponential growth will exceed linear growth, but the crossover point depends on starting populations. If Colony A starts much larger, it could lead for an extended period before Colony B's exponential nature dominates. Choice A ignores starting populations, choice B incorrectly claims Colony B always grows faster initially, and choice C incorrectly suggests linear growth could outpace exponential growth indefinitely.

Question 7

A researcher studying plant height over time notices that heights increase by 3 cm each week for the first month, then begin increasing by 5%5\% each week afterward due to a fertilizer treatment. What modeling strategy addresses both the mathematical and practical considerations?

  1. Use a linear model for the entire period and accept some error in the later weeks for simplicity
  2. Use a piecewise function with linear then exponential segments, ensuring the transition point aligns with treatment timing (correct answer)
  3. Use an exponential model for the entire period since the later growth pattern will dominate long-term
  4. Use separate linear and exponential models independently without considering how they connect at the transition point
Explanation: When modeling real-world phenomena that change behavior over time, you need to consider both mathematical accuracy and practical constraints. This plant growth scenario involves two distinct phases: initial linear growth, then exponential growth after fertilizer treatment. Option B correctly addresses both considerations by using a piecewise function. The linear segment (3 cm per week) models the first month accurately, while the exponential segment captures the 5% weekly growth rate afterward. Crucially, ensuring the transition point aligns with the treatment timing reflects the actual biological change and maintains mathematical continuity between the segments. Option A oversimplifies by using only linear modeling throughout. While this might be computationally easier, it completely fails to capture the exponential growth pattern after fertilization, leading to significant errors in later predictions when the 5% growth rate compounds. Option C makes the opposite error by using exponential modeling for the entire period. This misrepresents the initial month's growth pattern and creates inaccuracy in the early data, even though it might capture long-term trends better. Option D suggests using separate models without considering their connection. This approach ignores a fundamental requirement: the plant's height must be continuous at the transition point. Disconnected models could predict different heights at week 4, creating an impossible mathematical discontinuity. When you encounter problems with distinct behavioral phases, look for piecewise functions that respect both the mathematical properties of each phase and the practical constraints of the real-world scenario, especially continuity at transition points.

Question 8

A researcher collects data on two different phenomena and calculates correlation coefficients. Dataset X has r=0.95r = 0.95 with a linear model and r=0.87r = 0.87 with an exponential model. Dataset Y has r=0.82r = 0.82 with a linear model and r=0.96r = 0.96 with an exponential model. What conclusion should the researcher draw?

  1. Dataset X is better modeled linearly and Dataset Y is better modeled exponentially based on correlation strength alone
  2. Both datasets should use linear models since Dataset X shows stronger overall correlation with the linear approach
  3. Dataset X is better modeled linearly and Dataset Y exponentially, but residual analysis and theoretical justification are also needed (correct answer)
  4. The correlation differences are too small to determine model preference; additional data collection is required before choosing
Explanation: The correct answer is C. While correlation coefficients suggest Dataset X fits better with a linear model (r=0.95r = 0.95) and Dataset Y with an exponential model (r=0.96r = 0.96), correlation alone is insufficient for model selection. Good statistical practice requires examining residual plots for patterns, considering theoretical justification for the model choice, and potentially using other fit statistics. Choice A relies solely on correlation, choice B incorrectly generalizes from one dataset to both, and choice D unnecessarily delays decision-making when clear preferences exist.

Question 9

Two datasets both show strong positive trends. Dataset M has evenly spaced yy-values when xx increases by 1. Dataset N has yy-values where each is 1.3 times the previous when xx increases by 1. A student claims both should use exponential models because "they both grow rapidly." Which analysis is most appropriate?

  1. The student is correct; rapid growth indicates exponential behavior regardless of the specific numerical patterns in the data
  2. The student is incorrect; Dataset M shows linear growth (constant differences) while Dataset N shows exponential growth (constant ratios) (correct answer)
  3. The student is partially correct; both datasets grow rapidly, but Dataset M needs exponential modeling while Dataset N needs linear
  4. The student's reasoning is flawed; growth rate alone doesn't determine model type, and the mathematical patterns suggest different models
Explanation: The correct answer is B. Dataset M has "evenly spaced yy-values" which indicates constant differences (additive growth), characteristic of linear models. Dataset N has each yy-value being 1.3 times the previous, indicating constant ratios (multiplicative growth), characteristic of exponential models. The student incorrectly equates "rapid growth" with exponential growth, when the mathematical pattern of change determines the appropriate model type. Choice A accepts the flawed reasoning, choice C reverses the correct model assignments, and choice D is less direct about identifying the specific growth patterns.

Question 10

A city's population grows from 50,000 to 75,000 over 10 years. A demographer needs to project population for the next 20 years. If the growth continues at the same rate, which statement about model choice and long-term projections is most accurate?

  1. Use linear growth (+2,500+2,500 per year) for reliable long-term projections since the rate has been consistent historically
  2. Use exponential growth (≈4.14%≈4.14\% per year) for more realistic long-term projections accounting for compound population effects
  3. The choice between linear and exponential models significantly affects long-term projections and requires additional demographic analysis (correct answer)
  4. Both models will give similar results for 20-year projections since the historical growth rate was relatively modest
Explanation: The correct answer is C. Over 20 years, linear growth would project 75,000+20(2,500)=125,00075,000 + 20(2,500) = 125,000 people, while exponential growth would project 75,000(1.0414)20≈170,00075,000(1.0414)^{20} ≈ 170,000 people - a substantial difference. Model choice requires understanding whether population growth follows additive or multiplicative patterns, which depends on factors like available land, resources, and migration patterns. Choice A assumes linear growth without justification, choice B assumes exponential growth without justification, and choice D incorrectly suggests the models yield similar long-term results.

Question 11

A company tracks its monthly profit over two years. During the first year, profits increased by $2,000 each month. During the second year, profits increased by 8%8\% each month.

Based on the passage above, which modeling approach would best capture the company's profit pattern, and what challenge might arise in the analysis?

  1. Use separate linear and exponential models for each year; ensure continuity between models (correct answer)
  2. Use a single exponential model for both years; account for the varying growth rates appropriately
  3. Use a single linear model for both years; average the different growth mechanisms appropriately
  4. Use separate quadratic models for each year; determine the appropriate curvature parameters carefully
Explanation: The correct answer is A. The first year shows constant additive growth ($2,000 monthly), requiring a linear model. The second year shows constant multiplicative growth (8% monthly), requiring an exponential model. The main challenge is ensuring the models connect properly at the transition point. Choice B incorrectly applies exponential modeling to linear growth, choice C incorrectly applies linear modeling to exponential growth, and choice D suggests quadratic models which don't match either growth pattern.

Question 12

A researcher collected data on two different bacterial cultures over 6 hours. Culture A started with 100 bacteria, and the population increased by 50 bacteria every hour. Culture B started with 100 bacteria, and the population doubled every 2 hours. After 6 hours, which statement correctly compares the growth patterns and final populations?

  1. Culture A follows linear growth reaching 400 bacteria, while Culture B follows exponential growth reaching 800 bacteria. (correct answer)
  2. Culture A follows linear growth reaching 350 bacteria, while Culture B follows exponential growth reaching 600 bacteria.
  3. Culture A follows exponential growth reaching 400 bacteria, while Culture B follows linear growth reaching 800 bacteria.
  4. Both cultures follow exponential growth, with Culture A reaching 400 bacteria and Culture B reaching 800 bacteria.
Explanation: Culture A increases by a constant 50 bacteria per hour (linear): 100 + 6(50) = 400 bacteria. Culture B doubles every 2 hours (exponential): after 2 hrs = 200, after 4 hrs = 400, after 6 hrs = 800 bacteria. Choice B incorrectly calculates Culture A as 100 + 5(50) = 350. Choice C reverses the growth types. Choice D incorrectly identifies Culture A as exponential when it has constant additive growth.

Question 13

A student analyzes data showing a population that starts at 200 and follows the pattern: 200, 240, 288, 345.6, 414.72. The student claims this is linear growth because 'the population keeps going up by more each time, which means it's adding larger amounts.' Which response best addresses the student's reasoning?

  1. The student is correct because linear growth means the rate of change increases over time, resulting in larger incremental additions.
  2. The student is incorrect about the growth type, but the reasoning about increasing additions is valid since exponential functions show decreasing rates of change.
  3. The student is partially correct because while the amounts added increase, this represents accelerated linear growth rather than exponential growth.
  4. The student is incorrect because linear growth requires constant differences, but this data shows constant ratios of 1.2, indicating exponential growth. (correct answer)
Explanation: When analyzing population growth patterns, you need to distinguish between linear and exponential growth by examining how the data changes. Linear growth shows constant differences between consecutive terms, while exponential growth shows constant ratios. Let's check the differences in this data: 240-200=40, 288-240=48, 345.6-288=57.6, 414.72-345.6=69.12. These differences are increasing, not constant, so this isn't linear growth. Now let's check the ratios: 240÷200=1.2, 288÷240=1.2, 345.6÷288=1.2, 414.72÷345.6=1.2. The constant ratio of 1.2 confirms this is exponential growth with a 20% increase each period. Choice A incorrectly defines linear growth as having increasing rates of change. Linear growth actually requires a constant rate of change. Choice B wrongly states that exponential functions show decreasing rates of change - they actually show increasing rates when the base is greater than 1. Choice C invents the term "accelerated linear growth," which isn't a standard mathematical concept, and misidentifies the growth pattern. Choice D correctly identifies that linear growth requires constant differences, recognizes that this data lacks constant differences, and properly identifies the constant ratio of 1.2 as evidence of exponential growth. Study tip: Always check both differences and ratios when analyzing growth patterns. Constant differences = linear growth; constant ratios = exponential growth. Don't be fooled by increasing amounts - exponential growth naturally produces larger and larger increases over time.

Question 14

Two investment accounts start with $1000 each. Account A earns $50 per year in simple interest. Account B earns 4.5% annual compound interest. An investor claims that 'Account A is better for short-term goals because it grows linearly, while Account B is better for long-term goals because it grows exponentially.' Which evaluation of this claim is most accurate?

  1. The claim is correct about the growth types, and Account A will have more money than Account B for approximately the first 10 years of investment. (correct answer)
  2. The claim is correct about the growth types, and Account B will always have more money than Account A due to the power of compound interest.
  3. The claim incorrectly identifies the growth types; both accounts actually follow exponential growth patterns at different rates.
  4. The claim is correct about growth types, but Account A only outperforms Account B for approximately the first 2-3 years.
Explanation: Account A grows linearly: $1000 + $50t. Account B grows exponentially: $1000(1.045)^t. The growth type identification is correct. To find when Account B overtakes Account A: At t=10, Account A = $1500 and Account B ≈ $1553. So Account A performs better for approximately the first 10 years. Choice B ignores the initial period where linear growth outperforms. Choice C incorrectly identifies simple interest as exponential. Choice D significantly underestimates the crossover time.

Question 15

A water tank is being filled. The depth increases by 2 inches every 3 minutes for the first 15 minutes. Then, due to increased water pressure, the filling rate doubles every 6 minutes thereafter. Which statement best describes the mathematical models needed to represent this situation?

  1. The entire process requires an exponential model because the rate changes over time, making it non-linear throughout.
  2. A quadratic model best represents the entire situation because the rate changes create a curved relationship between time and depth.
  3. The entire process follows a linear model because the initial rate of 2 inches per 3 minutes remains the base rate throughout.
  4. A linear model for the first 15 minutes, then an exponential model for the remaining time, since the growth patterns fundamentally change. (correct answer)
Explanation: When analyzing real-world scenarios involving changing rates, you need to identify whether the growth pattern is constant (linear) or proportional to the current amount (exponential). The key is recognizing when the mathematical relationship fundamentally shifts. Let's break down this water tank problem. For the first 15 minutes, the depth increases by 2 inches every 3 minutes - that's a constant rate of 23\frac{2}{3} inches per minute. Since the rate doesn't depend on how much water is already in the tank, this creates a linear relationship: depth = (initial depth) + 23t\frac{2}{3}t. After 15 minutes, the filling rate doubles every 6 minutes. This means if the rate was rr at minute 15, it becomes 2r2r at minute 21, then 4r4r at minute 27, and so on. This doubling pattern creates exponential growth because the rate depends on the previous rate, following the form y=abty = ab^t. Answer A incorrectly claims the entire process is exponential - but the first 15 minutes follow a constant rate, not exponential growth. Answer B suggests a quadratic model, but quadratic functions have rates that change linearly, not through doubling. Answer C misses that the rate fundamentally changes after 15 minutes - the doubling pattern isn't just an extension of the original linear rate. Answer D correctly identifies that you need two different models: linear for the constant-rate phase, then exponential for the doubling phase. Study tip: When rates are constant, think linear. When rates change proportionally (doubling, tripling, etc.), think exponential. Watch for transition points that require switching models.

Question 16

A city's population was 50,000 in 2010 and 65,000 in 2020. A demographer needs to choose between two models for projecting 2030 population: Model A assumes the same 15,000-person increase per decade, while Model B assumes the same 30% growth rate per decade. Which factors most strongly support choosing one model over the other?

  1. Model A is better because linear population growth is more realistic for established cities due to resource constraints and space limitations.
  2. Model B is better because the 30% growth rate accounts for compound effects like increased birth rates in larger populations.
  3. The choice depends on additional context such as economic conditions, migration patterns, and infrastructure development rather than mathematical preferences alone. (correct answer)
  4. Model A is better because it provides more conservative estimates, while Model B overestimates growth by not accounting for population saturation effects.
Explanation: Both models fit the given data equally well (Model A: 50,000 + 15,000 = 65,000; Model B: 50,000 × 1.3 = 65,000). Choosing between them requires understanding the underlying factors driving population change, such as economic growth, urban planning, migration trends, and resource availability. Choice A makes unsupported generalizations about city growth. Choice B assumes compound effects without evidence. Choice D makes assumptions about conservatism and saturation without contextual support.

Question 17

A company tracks its revenue growth over time. In Year 1, revenue was $50,000. In Year 2, revenue was $65,000. In Year 3, revenue was $84,500. The company wants to project revenue for Year 5. Which statement best describes the most appropriate model and reasoning?

  1. A linear model is more appropriate because the year-to-year differences are approximately constant at $15,000 per year.
  2. An exponential model is more appropriate because the revenue increases by approximately 30% each year, indicating constant multiplicative growth. (correct answer)
  3. A linear model is more appropriate because revenue cannot grow exponentially in real business contexts due to market limitations.
  4. An exponential model is more appropriate because the revenue amounts are increasing and any increasing pattern suggests exponential growth.
Explanation: To determine the appropriate model, we need to examine the growth pattern. Year 1 to 2: 65,000/65,000/50,000 = 1.30 (30% increase). Year 2 to 3: 84,500/84,500/65,000 = 1.30 (30% increase). The constant multiplicative factor of 1.30 indicates exponential growth. Choice A is incorrect because the differences aren't constant ($15,000 vs $19,500). Choice C makes an incorrect generalization about business contexts. Choice D incorrectly assumes any increase indicates exponential growth.

Question 18

Two investment accounts are being compared. Account A grows by $50 each month. Account B grows by $4%4\% $ each month. After analyzing both accounts over a 5-year period, which statement about model selection is most accurate?

  1. A linear model fits Account A better, while an exponential model fits Account B better throughout the entire period (correct answer)
  2. Both accounts should use exponential models since they both show growth over time periods
  3. Account A needs an exponential model for compound interest, while Account B needs a linear model for percentage growth
  4. The model choice depends on the initial balance amounts, with larger initial balances favoring exponential models
Explanation: The correct answer is A. Account A adds a fixed amount (50)eachmonth,whichrepresentsconstantadditivegrowth−thedefiningcharacteristicoflineargrowth.AccountBgrowsbyafixedpercentage(50) each month, which represents constant additive growth - the defining characteristic of linear growth. Account B grows by a fixed percentage ( 4%4\% $) each month, which represents constant multiplicative growth - the defining characteristic of exponential growth. Choice B incorrectly assumes all growth is exponential, choice C reverses the appropriate models, and choice D incorrectly suggests that initial balance determines the growth pattern rather than the growth mechanism.