Teacher Guide: Half-Life Problems

Learn how to use half-life to find how much of a substance remains, how much time has passed, and how long a half-life is.

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Learning Objectives

Use the half-life formula $N(t) = N_0 \left(\frac{1}{2}\right)^{t/T}$ to find the amount remaining
Find the fraction and the percent of a substance left after a number of half-lives
Find the elapsed time when the fraction left is a power of $\frac{1}{2}$
Find the half-life of a substance from two measurements
Apply half-life to carbon-14 dating, medicine and iodine-131

Prerequisites

• Introduction to Exponential Functions: the form $f(x) = a \cdot b^x$
• Exponential Growth: multiplying by a constant factor in equal time periods
• Powers of 2 and powers of fractions, such as $\left(\frac{1}{2}\right)^{3} = \frac{1}{8}$
• Converting fractions to decimals and percents

Discussion Starters

1. If a medicine has a half-life of 6 hours, why do doctors not tell you to take it every 12 hours at double the dose?
2. Will a radioactive sample ever reach exactly zero? Why or why not?
3. Why can carbon-14 date a 10000-year-old bone but not a 10-million-year-old dinosaur fossil?
4. How is halving every half-life similar to doubling every doubling time?

Common Misconceptions

After two half-lives, the whole sample is gone

Remediation: Build a halving table together: $100 \to 50 \to 25 \to 12.5$ . Ask what is left after the second half-life. Use the coin toss simulation to show that some coins always remain.

The exponent in the formula is the time $t$

Remediation: Ask: how many times did the amount get halved in 12 hours if the half-life is 4 hours? Students count 3 halvings, which shows the exponent is $\frac{t}{T}$ .

A fraction left of $\frac{1}{16}$ means 16 half-lives

Remediation: List $\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}$ and number them 1 to 4. The position in the list, not the denominator, is the number of half-lives.

Differentiation Ideas

For Struggling Students:

• Use halving tables (time, number of half-lives, amount) before the formula
• Keep to whole numbers of half-lives and starting amounts that are powers of 2
• Use the sequence visualizer to see each halving as a bar

For On-Level Students:

• Solve for the elapsed time from the fraction left
• Find the half-life from two measurements
• Convert between fraction left, percent left and percent decayed

For Advanced Students:

• Work with non-whole numbers of half-lives using a calculator
• Write the decay factor per time unit, such as $\left(\frac{1}{2}\right)^{1/8}$ per day for iodine-131
• Model repeated medicine doses and find the amount just before the third dose

Standards Alignment

HSF-LE.A.1c (CCSS.MATH.CONTENT.HSF.LE.A.1.C)
Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another
HSF-LE.A.2 (CCSS.MATH.CONTENT.HSF.LE.A.2)
Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs
HSF-LE.B.5 (CCSS.MATH.CONTENT.HSF.LE.B.5)
Interpret the parameters in a linear or exponential function in terms of a context
HSF-IF.C.8b (CCSS.MATH.CONTENT.HSF.IF.C.8.B)
Use the properties of exponents to interpret expressions for exponential functions

Lesson Resources

visualHalving Sequence
Geometric sequence starting at 64 with ratio 0.5, showing how the bars shrink by half each half-life
activityCoin Decay Simulation
Students toss 100 coins, remove the heads each round and record how many remain, then compare with 100 times one half per round
worksheetHalf-Life Detective
Carbon-14 dating, medicine doses and iodine-131 problems with whole numbers of half-lives

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

The half-life

T

of a substance is the time it takes for half of it to disappear (decay, break down or leave the body). After every half-life, the amount is multiplied by

\frac{1}{2}

N_0

is the starting amount, the amount left after time

t

is:

N(t) = N_0 \left(\frac{1}{2}\right)^{t/T}

where:

$N_0$ = initial amount (at $t = 0$ )
$T$ = half-life (same time unit as $t$ )
$\frac{t}{T}$ = the number of half-lives that have passed

After

n

whole half-lives, the amount left is

\frac{N_0}{2^n}

Half-lives	0	1	2	3	4	5
Fraction left	$1$	$\frac{1}{2}$	$\frac{1}{4}$	$\frac{1}{8}$	$\frac{1}{16}$	$\frac{1}{32}$
Percent left	$100\%$	$50\%$	$25\%$	$12.5\%$	$6.25\%$	$3.125\%$

This is exponential decay with decay factor

\frac{1}{2}

per half-life: the amount gets smaller and smaller, but it never reaches exactly zero.

Worked Examples

A lab sample contains $80$ mg of a substance with a half-life of $4$ hours. How much is left after $12$ hours?

Find the number of half-lives

$\frac{t}{T} = \frac{12}{4} = 3$ → 3 half-lives

Write the formula

$N(12) = 80 \left(\frac{1}{2}\right)^{3}$ → Formula set up

Compute the power

$\left(\frac{1}{2}\right)^{3} = \frac{1}{8}$ → $\frac{1}{8}$ of the sample is left

Multiply

$80 \times \frac{1}{8} = 10$ (or halve three times: $80 \to 40 \to 20 \to 10$ ) → $10$ mg

Answer: After 12 hours, 10 mg of the substance is left.

Common Mistakes

Halving once for every hour (or day, or year) instead of once per half-life

Why it's wrong: With a half-life of 4 hours, after 12 hours some students compute $80 \left(\frac{1}{2}\right)^{12}$ . But the amount only halves every 4 hours, not every hour.

Correct: The exponent is the number of half-lives $\frac{t}{T} = \frac{12}{4} = 3$ , so the answer is $80 \left(\frac{1}{2}\right)^{3} = 10$ .

Subtracting half of the ORIGINAL amount each half-life

Why it's wrong: Thinking $100 \to 50 \to 0$ treats the decay as linear. Each half-life removes half of what is left NOW, not half of the starting amount.

Correct: Halve the current amount each time: $100 \to 50 \to 25 \to 12.5$ . A sample never reaches exactly zero.

Using the denominator as the number of half-lives

Why it's wrong: If $\frac{1}{8}$ is left, some students say 8 half-lives passed. But 8 half-lives would leave $\frac{1}{256}$ .

Correct: Write the denominator as a power of 2: $8 = 2^3$ , so 3 half-lives have passed.

Mixing up the fraction left and the fraction that decayed

Why it's wrong: After 3 half-lives, $\frac{1}{8}$ is left, so $\frac{7}{8}$ has decayed. Questions can ask for either one.

Correct: Read the question carefully: fraction decayed $= 1 -$ fraction left. After 3 half-lives: $12.5\%$ left, $87.5\%$ decayed.

Why It Matters

Half-life is how scientists, doctors and engineers measure how fast something disappears:

Archaeology: Carbon-14 has a half-life of $5730$ years, so the carbon left in bones and wood tells us how old they are
Medicine: The half-life of a medicine tells doctors how often a patient should take the next dose
Hospitals: Iodine-131, used to treat thyroid illness, has a half-life of $8$ days, which tells staff when a patient is safe to go home
Energy and safety: Nuclear waste must be stored until enough half-lives have passed

The good news: when the time is a whole number of half-lives, you can solve these problems by halving, without any advanced tools.

Real World Applications

Carbon-14 Dating

Living things take in carbon-14. After they die, it decays with a half-life of 5730 years, so the fraction left shows the age of bones, wood and charcoal.

Example:

A piece of wood with $\frac{1}{4}$ of its original carbon-14 is $2 \times 5730 = 11460$ years old.

1Try It Yourself

Charcoal from an ancient fireplace contains $\frac{1}{16}$ of its original carbon-14. The half-life of carbon-14 is $5730$ years.

How old is the charcoal?

Step 1: Write the mathematical expression

Find the number of half-lives, then multiply by $5730$ :

Medicine in the Bloodstream

Doctors use the half-life of a medicine to decide how often to give the next dose, so the amount in the body stays safe and effective.

Example:

A dose of $400$ mg with a half-life of $6$ hours leaves $400 \left(\frac{1}{2}\right)^{12/6} = 100$ mg after $12$ hours.

2Try It Yourself

A patient takes $600$ mg of a medicine with a half-life of $4$ hours.

How many mg are left after $16$ hours?

Step 1: Write the mathematical expression

Calculate: $600 \left(\frac{1}{2}\right)^{16/4}$

Iodine-131 Treatment

Iodine-131 is used to treat thyroid illness. Its half-life of 8 days tells the hospital how fast the radiation drops and when it is safe to store or dispose of leftovers.

Example:

$100$ mg of iodine-131 becomes $100 \left(\frac{1}{2}\right)^{16/8} = 25$ mg after $16$ days.

3Try It Yourself

A hospital stores $160$ mg of iodine-131 (half-life $8$ days).

How many mg are left after $24$ days?

Step 1: Write the mathematical expression

Calculate: $160 \left(\frac{1}{2}\right)^{3}$

Key Takeaways

1The half-life $T$ is the time it takes for half of a substance to disappear
2The amount left is $N(t) = N_0 \left(\frac{1}{2}\right)^{t/T}$ , where $\frac{t}{T}$ is the number of half-lives
3After $n$ half-lives, the fraction left is $\frac{1}{2^n}$ : $\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \ldots$
4To find the elapsed time, write the fraction left as $\frac{1}{2^n}$ and multiply $n$ by the half-life
5To find the half-life, count how many halvings fit in the data and divide the time by that number
6Each half-life removes half of what is left now, so the amount never reaches exactly zero

Frequently Asked Questions

Does a substance ever completely disappear?

Mathematically, no:

\left(\frac{1}{2}\right)^{n}

is never zero. In practice, after about 10 half-lives less than

0.1\%

is left (

\frac{1}{1024}

), which is usually treated as gone.

Why is the exponent $\frac{t}{T}$ and not just $t$ ?

The factor

\frac{1}{2}

applies once per half-life, not once per time unit. Dividing the time

t

by the half-life

T

counts how many half-lives have passed.

What if the time is not a whole number of half-lives?

The formula still works, for example

400 \left(\frac{1}{2}\right)^{1.5}

. Use a calculator (type 400 × 0.5^1.5) or a given value like

\left(\frac{1}{2}\right)^{1.5} \approx 0.3536

. Finding an unknown time in that case needs logarithms, which come in a later lesson.

How is half-life related to the decay rate in exponential decay?

In the Exponential Decay lesson, an amount is multiplied by

b = 1 - r

every time unit. Half-life uses the same idea but chooses the time unit so that the factor is exactly

\frac{1}{2}

, which makes mental calculations much easier.

Glossary

Half-life: The time $T$ it takes for half of a substance to decay or disappear
Initial amount: The amount $N_0$ at the start, when $t = 0$
Number of half-lives: The ratio $\frac{t}{T}$ : how many times the amount has been halved
Radioactive decay: The process in which unstable atoms change into other atoms, losing a fixed fraction in every equal time period
Carbon-14 dating: A method that finds the age of once-living material from the fraction of carbon-14 left (half-life $5730$ years)

Formula Card

Half-Life Formula

N(t) = N_0 \left(\frac{1}{2}\right)^{t/T}

$N_0$ = initial amount, $T$ = half-life, $t$ = time passed (same unit as $T$).

After n Half-Lives

N = \frac{N_0}{2^n}, \quad n = \frac{t}{T}

Divide by 2 once for every half-life. The elapsed time is $t = n \cdot T$.

Teacher Guide: Half-Life Problems

Use this lesson with your class

For Teachers

For Struggling Students:

For On-Level Students:

For Advanced Students:

Lesson Content

Definition

Worked Examples

Common Mistakes

Why It Matters

Real World Applications

Carbon-14 Dating

Medicine in the Bloodstream

Iodine-131 Treatment

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Half-Life Formula

After n Half-Lives

More in This Topic

Introduction to Exponential Functions

Exponential Growth

Graphing Exponential Functions

Exponential Decay