Smarter Balanced Assessment (SBAC) Mathematics: Problem Solving and Modeling Syllabus

What is a mathematical model?

A representation (equation, graph, diagram, table, or function) that captures the essential relationships of a real-world situation to analyze it or make predictions.

What is the SBAC/CCSS practice 'model with mathematics' (Practice 4)?

Applying math to solve real-world problems by representing situations with equations, diagrams, tables, or graphs, interpreting results, and revising the model as needed.

List the main steps of the mathematical modeling cycle.

1) Identify variables and assumptions, 2) formulate a model, 3) compute/analyze the model, 4) interpret the results, 5) validate against the situation, 6) report or revise.

What is an 'assumption' in modeling, and why state it?

A simplifying condition taken as true to make the situation manageable (e.g., constant speed); stating it clarifies the model's limits and the validity of conclusions.

What does it mean to 'define a variable' when representing a real-world situation?

Assigning a letter to an unknown quantity and clearly stating what it represents, including its units (e.g., let t = time in hours).

How do you translate the phrase 'a number increased by 5 is 12' into an equation?

x + 5 = 12, where x represents the unknown number.

What real-world situation is best represented by a linear function, and what does its slope mean?

A situation with a constant rate of change (e.g., constant speed or fixed hourly pay); the slope represents that constant rate (units of output per unit of input).

What does the y-intercept of a linear model typically represent in a real-world context?

The initial value or starting amount—the output when the input is zero (e.g., a flat fee before any usage).

When would an exponential model be more appropriate than a linear model?

When a quantity changes by a constant percent or factor per period (constant ratio), such as population growth, compound interest, or radioactive decay.

What is the difference between a discrete and a continuous model?

A discrete model represents quantities that take separate, countable values (e.g., number of people); a continuous model represents quantities that vary smoothly over an interval (e.g., time, distance).

What does it mean to 'interpret results in context'?

Translating a mathematical answer back into the real-world situation, stating what it means in words with proper units, and answering the original question.

After computing a numerical answer, what does 'checking for reasonableness' involve?

Asking whether the result makes sense given the situation—correct magnitude, sign, and units—and comparing it to an estimate or known benchmarks.

In context, what does a negative value often indicate when the variable represents a physical quantity like length or count?

It usually signals an error or an answer that must be rejected, since lengths and counts cannot be negative—prompting a review of the work.

Why might a calculated answer need to be rounded based on context rather than mathematically?

Because real-world meaning constrains it—e.g., needing whole buses means rounding up, and money is rounded to the nearest cent regardless of the raw decimal.

What is the difference between 'rounding up' and 'rounding down' in a context like 'how many buses are needed'?

Such problems require rounding up (ceiling) to the next whole number because a partial bus still requires a full bus to carry the remaining people.

What does it mean for a solution to be 'viable' in a modeling problem?

The solution is realistic and satisfies all constraints of the situation (e.g., non-negative, within budget, a whole number when required), not just mathematically valid.

What is a multi-step problem?

A problem requiring two or more connected operations or sub-problems, where the result of one step is used as input for the next to reach the final answer.

What is a key strategy for organizing a multi-step problem?

Break it into smaller sub-problems, solve them in logical order, keep track of intermediate results with labels/units, and combine them to answer the original question.

What is a 'performance task' in the SBAC assessment?

An extended, real-world scenario requiring students to apply multiple skills, integrate information, and produce a reasoned, multi-step solution often with explanation and justification.

Why is it important to label intermediate results in a multi-step problem?

Labeling prevents confusion about what each number represents, reduces errors, and makes the reasoning easy to follow and check.

In a multi-step problem, why should you avoid premature rounding of intermediate values?

Rounding too early accumulates error; keeping full precision until the final step yields a more accurate result, then round only the final answer.

What is a mathematical argument?

A logical chain of reasoning, built from definitions, given facts, and valid steps, that justifies why a mathematical claim or conclusion is true.

What is the SBAC/CCSS practice 'construct viable arguments and critique the reasoning of others' (Practice 3)?

Building logical justifications using assumptions, definitions, and prior results, and analyzing others' arguments to judge their validity and identify flaws.

What is the difference between a conjecture and a proof?

A conjecture is an unproven claim based on observation or pattern; a proof is a complete logical argument establishing the claim as definitely true.

What is a counterexample, and what does it accomplish?

A single specific case that satisfies the hypothesis but violates the conclusion; it disproves a general statement by showing it is not always true.

What is the difference between inductive and deductive reasoning?

Inductive reasoning generalizes from specific observations/patterns (probable conclusions); deductive reasoning derives certain conclusions from accepted premises using logical rules.

What makes a mathematical argument 'valid' versus 'invalid'?

A valid argument has each step logically following from prior true statements with no gaps or errors; an invalid one contains a flawed step, unjustified leap, or false premise.

When critiquing another student's reasoning, what should you look for?

Whether each step is justified, the logic is sound, assumptions are valid, calculations are correct, and the conclusion actually follows—identifying any errors or gaps.

What is a productive way to respond to a flawed argument when critiquing it?

Identify the specific error or unjustified step, explain why it is wrong (e.g., with a counterexample), and, if possible, suggest a correction.

What does it mean to 'explain your method and results' in a math task?

Communicating, clearly and in order, the strategy you chose, the steps you took, why they are valid, and what your final answer means in context.

What are the qualities of a good written mathematical explanation?

It is clear, logically ordered, uses correct terminology and units, justifies each step, and connects the result back to the original question so a reader can follow the reasoning.