Science Practice 2: Mathematical Routines

Overview

AP Physics 1 Science Practice 2: Mathematical Routines is the skill of doing the math of physics with purpose. You derive symbolic expressions, calculate or estimate unknown values with units, compare quantities across scenarios, and predict how one quantity changes when another changes. In short, you pick a logical mathematical pathway from what you know to what you want, then follow it cleanly.

This practice shows up everywhere in the course because almost every physics problem ends with some form of math. The four subskills (2.A, 2.B, 2.C, 2.D) all appear on both the multiple-choice and free-response sections, so getting comfortable with them pays off across all eight units.

What Science Practice 2: Mathematical Routines Means

The grouping description says it directly: conduct analyses to derive, calculate, estimate, or predict physical phenomena. That breaks into four moves.

• 2.A Derive: Start from a known equation or principle and produce a symbolic expression for the target quantity.
• 2.B Calculate or estimate: Plug in numbers and get an answer with correct units.
• 2.C Compare: Look at two scenarios, or two moments in one scenario, and rank or relate quantities.
• 2.D Predict: Use the relationship between variables to find new values or factors of change.

The common thread is choosing a pathway. You are not just memorizing one formula. You decide which principle applies, then chain steps together to reach the answer.

What This Practice Requires

Each subskill has a specific job. Here is what graders and questions expect.

A few details that matter:

• 2.A often asks you to "begin your derivation by writing a fundamental physics principle or an equation from the reference information." Show the starting equation, then the algebra.
• 2.B answers need units. A number without units is incomplete.
• 2.C answers are about relationships (greater, less, equal, or a ranking), not always a single number.
• 2.D is about proportional reasoning. If a quantity depends on another squared, doubling the input multiplies the output by four.

Skills You Need for This Practice

You can build this practice with a small set of habits.

• Know your reference equations. You get an equation sheet, so practice recognizing which equation connects your knowns to your unknown.
• Do algebra symbolically first. Solve for the variable in letters, then substitute numbers. This reduces arithmetic mistakes and earns derivation credit.
• Track units through every step. Units act as a built-in error check.
• Recognize functional dependence. Spot whether a relationship is linear, inverse, squared, or square root.
• Set up ratios for comparisons. Writing quantity 2 divided by quantity 1 cancels constants fast.

How It Shows Up on the AP Exam

The exam has 40 multiple-choice questions and 4 free-response questions, and a four-function scientific or graphing calculator is allowed on both sections. Science Practice 2 is assessed in both sections.

What this looks like in practice:

• MCQ: You might select the correct symbolic expression (2.A), pick the closest numerical answer (2.B), rank positions or energies (2.C), or choose how a value changes when conditions change (2.D).
• FRQ: FRQ 1 is the Mathematical Routines question type, which shares a name with this science practice and leans heavily on 2.A and 2.B. Other free-response question types also use Practice 2 skills, including deriving expressions, comparing values, and predicting changes.

Practical tip, not an official rule: on derivation problems, write the starting principle even if it feels obvious. It signals your pathway and protects partial credit.

Examples Across the Course

These come from sample questions and span several units, so you can see how one practice repeats in different content.

Unit 2, Forces (2.A derive): Blocks A, B, and C are connected with a hanging mass C and a fixed block A. You reason through the force balance to find the force Block A exerts on Block B equals $m_C g$. The pathway is a free-body analysis turned into a symbolic result.

Unit 1, Kinematics (2.C compare): A velocity-versus-time graph gives car positions at 1 s, 2 s, and 3 s. Since the car keeps moving in the positive direction, position keeps increasing, so the ranking is $x_3 > x_2 > x_1$. You compare values at different times in one scenario using area under the curve.

Unit 4, Momentum (2.B calculate): A position-versus-time graph for Block A gives its speed before colliding with Block B. Using momentum relationships, the momentum of Block B after the collision is most nearly $6.0 \ \text{kg} \cdot \text{m/s}$. You read data, compute, and report with units.

Unit 8, Fluids (2.D predict): A rock sits under water with buoyant force $F_0$. Replace the water with oil that is $\frac{3}{4}$ as dense. Since buoyant force scales with fluid density, the new force is $\frac{3}{4} F_0$. This is functional dependence: density down by a factor, force down by the same factor.

Unit 5/6, Rotation (2.D predict): Two disks have angular momentum graphs over 5 seconds, and Disk B has twice the rotational inertia of Disk A. Comparing average net torques from the slopes gives $\tau_B = 2\tau_A$. You connect the graph to the relationship and scale the result.

Notice the variety: a derivation, a graph ranking, a data calculation, and two scaling predictions, all under one practice.

How to Practice Science Practice 2: Mathematical Routines

Build the skill with focused reps.

• Solve symbolically, then plug in. For every numeric problem, find the algebraic answer first.
• Drill ratio reasoning for 2.D. Take any equation and ask: if I double this variable, what happens to the output? Write the factor of change.
• Practice graph reading for 2.B and 2.C. Convert slopes and areas into physics quantities like velocity, displacement, momentum, and torque.
• Write the starting principle every time on derivations. Make it automatic so you do it under time pressure.
• Check units on the final line. If the units do not match the quantity, find the error before moving on.
• Compare without numbers. Try 2.C problems by setting two scenarios side by side and canceling shared constants.

Common Mistakes

• Skipping the starting equation on a derivation. This can cost setup credit on free-response.
• Dropping units on 2.B answers. A bare number is not a complete physics answer.
• Treating every comparison as a calculation. For 2.C, sometimes you only need to know which is bigger, not the exact value.
• Misreading functional dependence. If a quantity depends on the square of another, do not just scale linearly.
• Plugging in numbers too early. Early substitution makes algebra messier and hides which variables cancel.
• Ignoring what changes between scenarios. In 2.C and 2.D, identify the one thing that is different and hold everything else fixed.

Quick Review

• Science Practice 2 covers four moves: derive (2.A), calculate or estimate with units (2.B), compare (2.C), and predict using functional dependence (2.D).
• All four subskills appear on both multiple-choice and free-response sections.
• Start derivations from a reference equation, solve symbolically, then substitute.
• Always attach units to calculated answers.
• For comparisons, use ratios and cancel shared constants.
• For predictions, read the relationship carefully: linear, inverse, squared, or root, then apply the matching factor of change.