Modelling skills AI HL

IB Mathematics: Applications and Interpretation HL · Unit 2: Functions

Lesson 2.5 · Modelling Skills

Choose, build, test and use a mathematical model.

Faith connection

Barnabas noticed that others needed encouragement and took action. When he visited the believers in Antioch, he “exhorted them all, that with purpose of heart they would cleave unto the Lord” (Acts 11:23, KJV). Loving others involves observing carefully and responding wisely. In modelling, we also examine evidence before deciding what to do. As you learn to use mathematics responsibly, come to Jesus, our Savior and Lord, and ask him to guide both your thinking and your care for others.

Learning objectives

By the end of this lesson, you should be able to:

recognise an appropriate model from the situation, data and graph;
define variables and establish a reasonable domain;
find parameters using initial conditions or simultaneous equations;
test whether a model fits the information and explain its limitations;
read, interpret and make predictions using a model;
distinguish between interpolation and extrapolation.

1. The modelling process

A mathematical model is a simplified description of a real situation. A useful modelling process has five stages:

Understand: Identify the question, variables, units and available information.
Choose: Select a function family whose behaviour fits the context.
Build: Find the model’s unknown parameters.
Test: Compare its predictions with the data and examine its assumptions.
Use and reflect: Answer the original question and decide whether the answer is reasonable.

A more complicated formula is not automatically better. A simple model is often easier to interpret, provided it captures the important behaviour.

Worked example · Choosing a model.

A machine produces approximately $120$ , $135$ , $150$ and $165$ items during four consecutive hours. The increase is $15$ items from one hour to the next, so a linear model is reasonable over this short interval. If $t=0$ represents the first hour, a model is $P(t)=120+15t$ for $0\leq t\leq3$ .

The model describes the recorded hours. Predicting production several years later would require new evidence.

2. Choosing a function family

Evidence or context	Possible model	Question to ask
Approximately constant amount added each period	Linear: $y=mx+c$	Are first differences approximately constant?
Arch, bowl or one turning point	Quadratic: $y=ax^2+bx+c$	Does a vertex have meaning?
Approximately constant percentage change	Exponential: $y=ab^x+c$	Is a constant multiplier plausible?
One quantity varies as a power of another	Power: $y=kx^n$	Is the stated proportional relationship justified?
Repeated peaks and troughs	Sinusoidal	Are the amplitude and period reasonably stable?

Worked example · Amount or percentage?

A value changes from €1000 to €1100 to €1200 in equal time intervals. The constant increase of €100 suggests a linear model. Another value changes from €1000 to €1100 to €1210. Its constant multiplier of $1.10$ suggests an exponential model. The first two values alone would not distinguish these models.

3. State the domain and interpret parameters

The mathematical domain is where a formula can be evaluated. The contextual domain contains the inputs that make sense for the situation. State which one the problem requires.

Worked example · Straight-line depreciation.

A building is valued at €50 000 when placed in service and depreciates by €2000 per year for $25$ years. Let $t$ be the number of years since it entered service. A model is:

$V(t)=50000-2000t,\qquad 0\leq t\leq25$

The intercept $50000$ is its starting value. The gradient $-2000$ means its modelled value falls by €2000 each year. After $10$ years:

$V(10)=50000-2000(10)=30000$

To find when the value is €39 000:

$50000-2000t=39000\quad\Longrightarrow\quad t=5.5$

Under a continuous-time interpretation of the model, it reaches €39 000 after $5.5$ years. The specified $25$ -year domain matters: continuing this formula indefinitely would eventually predict a negative value.

4. Find parameters from conditions

Parameters are the fixed numbers in a model. We can determine them by substituting known points or using conditions such as an initial value, gradient, maximum or asymptote. When several parameters are unknown, form simultaneous equations.

A linear model from two points

Worked example.

A tank contains $94$ L after $2$ minutes and $70$ L after $8$ minutes. Assume its water level falls at a constant rate. Write $W(t)=mt+c$ .

$2m+c=94,\qquad 8m+c=70$

Subtract the equations: $6m=-24$ , so $m=-4$ . Substituting gives $c=102$ .

$W(t)=102-4t,\qquad 0\leq t\leq25.5$

The model predicts an initial volume of $102$ L and an empty tank after $25.5$ minutes.

A quadratic model from three points

Worked example.

A projectile has heights $2$ m, $10$ m and $10$ m at times $t=0$ , $1$ and $3$ seconds. Assume $h(t)=at^2+bt+c$ .

At $t=0$ , $c=2$ . The other two points give:

$a+b+2=10,\qquad 9a+3b+2=10$

$a+b=8,\qquad 9a+3b=8$

Solving simultaneously gives $a=-\frac83$ and $b=\frac{32}{3}$ .

$h(t)=-\frac83t^2+\frac{32}{3}t+2$

Check: $h(0)=2$ , $h(1)=10$ and $h(3)=10$ , as required. This model is relevant only for times during the projectile’s flight.

5. Fit a model to imperfect data

Measured values rarely lie exactly on a simple curve. One approach is to choose a model, estimate its parameters and compare its predictions with the observations. A residual is:

$\text{residual}=\text{observed value}-\text{model prediction}$

Small residuals are useful, but also check whether they follow a pattern. A model that consistently underestimates early observations and overestimates later ones may have the wrong shape.

Worked example · Estimating from endpoints.

The following counts were recorded on consecutive days:

Day $d$	1	2	3	4	5	6
New cases $n$	15	30	50	90	140	200

Because the counts rise rapidly, try $n(d)=15b^{d-1}$ , which passes through the first observation. Requiring it also to pass through $(6,200)$ gives:

$15b^5=200\quad\Longrightarrow\quad b=\left(\frac{200}{15}\right)^{1/5}\approx1.679$

$n(d)=15(1.679)^{d-1}$

Using unrounded $b$ , the day 4 prediction is about $71$ cases, compared with $90$ observed: the residual is about $19$ cases. The model is an approximation, and passing through two observations does not make it an exact fit to the others.

Technology note: A calculator’s regression command can use all the data points to fit a curve. Regression is studied more fully in the statistics unit; here the main skill is to explain your model choice, check its fit and interpret its predictions.

6. New cases and total cases are different quantities

If $n(d)$ is the number of new cases on day $d$ , it does not directly give the total cases by day $d$ . For whole-number days, add the daily counts:

$T(d)=n(1)+n(2)+\cdots+n(d)$

Worked example · Interpreting a threshold.

For the observed figures above, the total through day 6 is:

$15+30+50+90+140+200=525\text{ cases}$

The endpoint-based exponential model instead gives an estimated cumulative total of about $473$ through day 6. The difference shows why model estimates should not replace available observations.

Using the fitted model to predict future daily counts gives approximately $336$ on day 7, $564$ on day 8 and $946$ on day 9. Using the six observed days as the starting total, the projected cumulative total passes $1000$ on day 8:

$525+336=861,\qquad 525+336+564=1425$

The modelled new cases per day do not exceed $1000$ until day 10. These are different questions and different answers.

7. Test and reflect on a model

Ask whether the model describes the process as well as the numbers. Check the shape of its graph, the units, the domain, its residuals and whether its long-term behaviour is realistic.

Worked example · Limits of exponential growth.

A population may initially grow by $8\%$ per year, giving $P(t)=240(1.08)^t$ . This might be useful for a short-term prediction. Over many years, food, space and other factors can change the growth rate. The model predicts unlimited growth, so it should not be treated as a permanent law of the population.

Interpolation estimates a value between observed inputs. Extrapolation predicts outside the observed interval. Extrapolation usually requires more caution because the conditions underlying the model may change.

Worked example · Interpolation and extrapolation.

A model is based on sales data from 2021 through 2025. Estimating 2023 sales is interpolation; predicting 2035 sales is extrapolation. The 2035 prediction depends on the assumption that the model’s pattern continues for another decade.

Calculator workflow for modelling questions

Enter the observed inputs and outputs in two lists, if data are supplied.
Plot the data and inspect its shape before choosing a model.
Determine parameters using the stated conditions; use a simultaneous-equation solver if necessary.
Graph the model with the data, using a window that includes the relevant domain.
Use the intersection, zero, maximum or minimum command when a numerical answer is needed.
Check the answer in the original situation and report suitable units and precision.

IB-style Paper 1 · Skill practice

Question 1 · Choose and justify

Three sets of equally spaced observations are shown below.

Set	Outputs
A	$12,\ 19,\ 26,\ 33$
B	$12,\ 18,\ 27,\ 40.5$
C	$4,\ 9,\ 16,\ 25$

Suggest a linear, exponential or quadratic model for each set. Justify each choice using differences or ratios.

Show solution

A is linear because its first differences are all $7$ . B is exponential because each output is multiplied by $1.5$ . C is quadratic because its first differences are $5,7,9$ , giving constant second differences of $2$ .

Question 2 · Two conditions

A linear model $f(x)=mx+c$ passes through $(4,23)$ and $(10,5)$ . Find $m$ and $c$ , then find the zero of $f$ .

Show solution

$m=(5-23)/(10-4)=-3$ . Using $23=-3(4)+c$ gives $c=35$ . Thus $f(x)=35-3x$ , with zero at $x=35/3\approx11.7$ .

Question 3 · Interpret a domain

A candle is $18$ cm tall when lit and burns at $1.5$ cm per hour. Write a height model, give its contextual domain and find when its height is $6$ cm.

Show solution

$H(t)=18-1.5t$ for $0\leq t\leq12$ . Solving $18-1.5t=6$ gives $t=8$ hours.

Question 4 · Residual

A model predicts $48.2$ units at $t=5$ , while the observed value is $53.1$ units. Find the residual and interpret its sign.

Show solution

The residual is $53.1-48.2=4.9$ units. It is positive because the observation is $4.9$ units above the model prediction; the model underestimated this observation.

Question 5 · Exponential parameters

A model has the form $P(t)=ab^t$ . Given $P(0)=80$ and $P(3)=135$ , find $a$ and $b$ . Give $b$ to three significant figures.

Show solution

$a=80$ . From $80b^3=135$ , we obtain $b=(135/80)^{1/3}\approx1.19$ . An appropriate model is $P(t)=80(1.19)^t$ , provided the unrounded multiplier is retained in calculations.

Question 6 · Evaluate a claim

A student’s model of a water tank is $W(t)=120-4t$ , where $W$ is in litres and $t$ is in minutes. The student predicts that after $40$ minutes the tank contains $-40$ L. Explain what is wrong and give a suitable contextual domain.

Show solution

The tank becomes empty when $120-4t=0$ , at $t=30$ minutes. Negative water volume has no physical meaning here. The suitable domain is $0\leq t\leq30$ ; after that, a different model would be needed if the tank remains empty.

Applications · Extended modelling practice

Application 1 · Depreciating a building

A building is purchased for €64 000 and is depreciated by a fixed €2400 per year. A manager intends to use this model for $20$ years. Let $t$ be the number of years since purchase.

Write a model $V(t)$ and give its domain.
Find the value after $12$ years.
When does the model predict a value of €40 000?
Explain why the same straight-line rule should not automatically be used forever.

Show solution

$V(t)=64000-2400t$ for $0\leq t\leq20$ . After $12$ years, $V(12)=€35\,200$ . Solving $64000-2400t=40000$ gives $t=10$ years. Beyond its intended interval, the model may cease to reflect the building’s actual value; it would eventually predict a negative value.

Application 2 · Two possible models

A wildlife survey reports $300$ animals in year $0$ and $432$ in year $4$ .

Construct a linear model through both observations.
Construct an exponential model through both observations.
Predict the year 8 population using each model and explain why two observations alone cannot establish which model is better.

Show solution

The linear rate is $(432-300)/4=33$ animals per year, giving $L(t)=300+33t$ . The exponential model is $E(t)=300b^t$ , where $b^4=432/300=1.44$ , so $b=1.44^{1/4}$ . In year 8, $L(8)=564$ while $E(8)=300(1.44)^2=622.08$ , approximately $622$ animals. Both models pass through the supplied observations; intermediate data and knowledge of the habitat would help decide which is more suitable.

Application 3 · New infections and cumulative infections

During a six-day period, the observed numbers of new cases are $15,\ 30,\ 50,\ 90,\ 140,\ 200$ in day order. To make a short-term estimate, a researcher uses $n(d)=15b^{d-1}$ and chooses $b$ so the model matches day 6.

Find $b$ to three significant figures.
Predict new cases on day 7.
Using the observed six-day total and the model’s predictions for later days, determine the first day on which the cumulative total exceeds $1000$ .
State one reason for treating this prediction cautiously.

Show solution

$15b^5=200$ , so $b=(200/15)^{1/5}\approx1.68$ . Retaining the unrounded value gives $n(7)\approx336$ and $n(8)\approx564$ new cases. The observed total through day 6 is $525$ . Adding day 7 gives approximately $861$ , still below $1000$ ; adding day 8 gives approximately $1425$ . Thus the cumulative total first exceeds $1000$ on day 8 according to this method. Infection patterns and reporting practices can change, so constant exponential growth is a short-term assumption.

Application 4 · A quadratic model from observations

A fountain’s water jet is modelled by $h(x)=ax^2+bx+c$ , where $x$ is horizontal distance in metres and $h$ is height in metres. Measurements give $h(0)=1$ , $h(2)=5$ and $h(6)=1$ .

Find $a$ , $b$ and $c$ .
Find the greatest modelled height and where it occurs.
Give a reasonable domain for the visible arc from the first to the last measurement.

Show solution

From $h(0)=1$ , $c=1$ . The other conditions give $4a+2b=4$ and $36a+6b=0$ . Solving gives $a=-\frac12$ and $b=3$ . Therefore:

$h(x)=-\frac12x^2+3x+1$

The vertex is at $x=-3/[2(-1/2)]=3$ , where $h(3)=5.5$ m. For the stated visible arc between measurement locations, a reasonable domain is $0\leq x\leq6$ .

Application 5 · Testing an extrapolation

A town has measured its water use for five consecutive years and fits $U(t)=1.8(1.06)^t$ million litres per day, where $t=0$ is the first measured year.

Interpret $1.8$ and $1.06$ .
Predict water use at $t=4$ and $t=20$ .
Identify which prediction is within the observation period and explain a limitation of the other.

Show solution

$1.8$ is the modelled initial daily use in millions of litres. The multiplier $1.06$ represents $6\%$ growth per year. $U(4)=1.8(1.06)^4\approx2.27$ million litres per day, while $U(20)=1.8(1.06)^{20}\approx5.77$ million litres per day. The $t=4$ estimate lies within the five-year observation period; $t=20$ is an extrapolation. Population, conservation measures and supply constraints could change the growth rate.

Paper 1 modelling checklist

Define your variables. Give units and a sensible domain. Explain why the function’s shape fits the situation. Use the given conditions to find parameters. Compare predictions with data where possible. Distinguish observed values from estimates, new amounts from cumulative totals, and interpolation from extrapolation. End with an answer in context.

Lesson 2.5 · Modelling Skills

Faith connection

Learning objectives

1. The modelling process

2. Choosing a function family

3. State the domain and interpret parameters

4. Find parameters from conditions

A linear model from two points

A quadratic model from three points

5. Fit a model to imperfect data

6. New cases and total cases are different quantities

7. Test and reflect on a model

Calculator workflow for modelling questions

IB-style Paper 1 · Skill practice

Question 1 · Choose and justify

Question 2 · Two conditions

Question 3 · Interpret a domain

Question 4 · Residual

Question 5 · Exponential parameters

Question 6 · Evaluate a claim

Applications · Extended modelling practice

Application 1 · Depreciating a building

Application 2 · Two possible models

Application 3 · New infections and cumulative infections

Application 4 · A quadratic model from observations

Application 5 · Testing an extrapolation

Paper 1 modelling checklist

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IB Mathematics: Applications and Interpretation HL 6 Sections80 LessonsLifetime

Curriculum

Modelling skills AI HL

Faith connection

Learning objectives

1. The modelling process

2. Choosing a function family

3. State the domain and interpret parameters

4. Find parameters from conditions

A linear model from two points

A quadratic model from three points

5. Fit a model to imperfect data

6. New cases and total cases are different quantities

7. Test and reflect on a model

Calculator workflow for modelling questions

IB-style Paper 1 · Skill practice

Question 1 · Choose and justify

Question 2 · Two conditions

Question 3 · Interpret a domain

Question 4 · Residual

Question 5 · Exponential parameters

Question 6 · Evaluate a claim

Applications · Extended modelling practice

Application 1 · Depreciating a building

Application 2 · Two possible models

Application 3 · New infections and cumulative infections

Application 4 · A quadratic model from observations

Application 5 · Testing an extrapolation

Paper 1 modelling checklist

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