AP Environmental Science Science Practice 6 - Mathematical Routines is the skill of using quantitative methods to solve environmental problems. You pick a method that fits the problem, apply the right math relationship while showing your work, and report a numeric answer with correct units. This is the "do the calculation" practice, and it shows up most heavily in FRQ 3.

This practice covers three connected steps: choosing an approach (6.A), applying the math correctly with work shown (6.B), and landing on an accurate answer with units (6.C). You will use it for things like the 10% rule, half-life decay, percent change, dimensional analysis, and energy or population calculations.

more resources to help you study

cheatsheets score calculator key terms

What Science Practice 6 - Mathematical Routines Means

Mathematical routines means turning an environmental scenario into numbers you can work with. The grouping description from the course is short and direct: apply quantitative methods to address environmental concepts.

Three subskills make up this practice:

6.A Determine an approach or method aligned with the problem to be solved.
6.B Apply appropriate mathematical relationships to solve a problem, with work shown (for example, dimensional analysis).
6.C Calculate an accurate numeric answer with appropriate units.

Think of these as a sequence. First decide what to do, then do it on paper, then check that your final number and units make sense.

What This Practice Requires

Each subskill asks for something specific.

6.A: Pick the method. Read the problem and choose the relationship that matches. Are you finding percent change? Applying the 10% rule across trophic levels? Using half-lives? Converting units? Picking the right setup is half the work.

6.B: Show the math. Write out the equation, plug in values, and keep your steps visible. Dimensional analysis is named directly in the course, so practice writing units in your setup and canceling them. Work shown also helps you earn partial credit when the final number is off.

6.C: Land the answer with units. A number alone is not enough. Report it with the correct unit, like kg, kBq/m², %, or kilowatt-hours. Match the unit the problem expects.

Skills You Need for This Practice

Set up and solve percent change problems.
Apply the 10% rule to estimate biomass or energy across trophic levels.
Work through half-life decay over multiple half-life periods.
Use dimensional analysis to convert units and cancel them cleanly.
Keep track of significant figures and rounding so your final value is reasonable.
Write units at every step, not just at the end.

A four-function, scientific, or graphing calculator is allowed on both exam sections, so focus your energy on setup and units rather than arithmetic alone.

How It Shows Up on the AP Exam

Based on the course framework:

Multiple choice: Practice 6 is 6 to 9% of the MCQ section.
Free response: Practice 6 is 20% of the FRQ section and is primarily assessed in FRQ 3.

FRQ 3 is described as analyzing an environmental problem and proposing a solution while doing calculations, and it is worth 10 points. That is where careful setup and units matter most.

On multiple choice, some Practice 6 questions ask you to identify the correct setup rather than compute. For example, one sample question asks which expression estimates the biomass of mice in a pyramid using the laws of thermodynamics, and the correct choice multiplies the starting biomass by 0.10 twice to move down two trophic levels.

Examples Across the Course

Practice 6 appears across many units because almost every topic has a quantitative side.

Unit 1, Energy Flow and the 10% Rule: Estimate biomass at a higher or lower trophic level. A producer biomass of 18,705 kg becomes roughly 187 kg two levels up after multiplying by 0.10 twice (6.A and 6.B).
Unit 6 and Unit 9, half-life decay: A soil sample near Chernobyl with 187 kBq/m² of cesium-137 and a 30 year half-life drops to about 23.38 kBq/m² after 90 years, since 90 years is three half-lives (187 to 93.5 to 46.75 to 23.38). This pairs 6.B and 6.C.
Unit 9, percent change in sea ice: Reading a graph and computing the approximate percent decrease from 1980 to 2005, which works out to about 30% (6.C).
Unit 8, pollution loads: Use dimensional analysis to convert pollutant concentrations or waste volumes into consistent units before comparing them (6.B). The course lists 6.B as a skill emphasized in this unit.
Unit 3, populations: Calculate values like total fertility rate effects or population change over time, then report with appropriate units (6.A and 6.C).

How to Practice Science Practice 6 - Mathematical Routines

These are practical study suggestions, not official rules.

Build a formula sheet. List the relationships you use most: percent change, the 10% rule, half-life, and common unit conversions. Know when each applies.
Always write units in your setup. If units do not cancel to what the question asks, your method is probably off.
Show every step. On FRQ 3, partial credit often comes from a correct setup even if the final number is wrong.
Box your final answer with its unit. Make it easy for a scorer to find.
Practice half-life as repeated halving. Count how many half-life periods fit in the total time, then halve that many times.
Redo a missed problem from scratch. Do not just read the solution. Reset and solve it again.

Common Mistakes

Forgetting units on the final answer. This is the fastest way to lose a 6.C point.
Multiplying by 10 instead of 0.10. Moving down a trophic level means keeping 10% of energy, so multiply by 0.10.
Mishandling half-lives. Dividing by 3 instead of halving three times for 90 years over a 30 year half-life gives the wrong answer.
Skipping work. A bare number with no setup limits partial credit on FRQs.
Mixing units. Comparing values in different units without converting first leads to wrong conclusions.
Rounding too early. Carry digits through the steps and round at the end.

Quick Review

Practice 6 is the calculation practice: choose a method (6.A), apply it with work shown (6.B), and report an accurate answer with units (6.C).
It is 6 to 9% of MCQ and 20% of FRQ, mainly in FRQ 3, which is the 10 point calculation question.
Core tools: percent change, the 10% rule, half-life decay, and dimensional analysis.
Show your work, write units at every step, and box the final answer with its unit.
Calculators are allowed on both sections, so spend your effort on correct setup and clear units.