Math I
Build connected understanding across quantities, equations, functions, coordinate geometry, congruence, and data.
- Problem types
- 659
- Practice variants
- 2,636
Page 5 of 19
Each problem type has four distinct practice variants. Open a preview to move among all four.
Rewrite a percent-change description as a multiplier
Interpret terms, factors, and coefficients in linear and exponential expressions.
A percent increase combines the entire original amount with an added fraction; the change alone is not the new total. We’ll convert the stated percent to decimal form, add it …
Preview problemInterpret a sub-expression inside parentheses
Interpret complex expressions by treating meaningful sub-expressions as single units.
Parentheses define the exact scope of the requested subexpression. We’ll interpret each amount inside as part of one person’s cost, combine only those inside terms, and attach a per-person dollar …
Preview problemInterpret an outside coefficient as the number of repeated groups
Interpret complex expressions by treating meaningful sub-expressions as single units.
An outside coefficient acts on the entire parenthesized group. We’ll first read the inside as the complete contents of one bag, then interpret the outside factor as the number of …
Preview problemInterpret a nested expression from the inside out
Interpret complex expressions by treating meaningful sub-expressions as single units.
A nested expression is most reliable when read from the inside out. We’ll treat the parenthesized terms as one complete pre-tax subtotal, confirm their common dollar unit, and then interpret …
Preview problemChoose the expression that matches a verbal description with grouping
Interpret complex expressions by treating meaningful sub-expressions as single units.
The verbal structure determines the grouping before any arithmetic begins. We’ll combine the fixed charge and per-item contribution into one subtotal, convert the tax rate into a multiplier for the …
Preview problemInterpret an exponential sub-expression as an accumulated factor
Interpret complex expressions by treating meaningful sub-expressions as single units.
An exponential subexpression can be treated as one accumulated scale factor. We’ll interpret the base as a one-period multiplier, connect the exponent to the number of equal periods, and expand …
Preview problemInterpret a difference of two expressions
Interpret complex expressions by treating meaningful sub-expressions as single units.
Subtraction compares two complete quantities in a fixed order, so reversing them reverses the meaning of every sign. We’ll name the first and second dollar expressions, preserve minuend-minus-subtrahend order, and …
Preview problemInterpret a quotient as a unit rate or average
Interpret complex expressions by treating meaningful sub-expressions as single units.
A quotient’s meaning follows both its order and its units. We’ll identify the complete quantity in the numerator, the count in the denominator, and divide their units in the same …
Preview problemChoose the unit of a sub-expression
Interpret complex expressions by treating meaningful sub-expressions as single units.
Units multiply along with numerical factors. We’ll write the rate as output units per input unit, attach the count’s input unit, and cancel matching units in the product. The unit …
Preview problemRewrite an expression to reveal a meaningful grouped sub-expression
Interpret complex expressions by treating meaningful sub-expressions as single units.
Equivalent rewriting can expose which costs repeat and which occur only once. We’ll separate the terms that share the item-count factor from the standalone fee, factor the common count from …
Preview problemIdentify a shared sub-expression across two formulas
Interpret complex expressions by treating meaningful sub-expressions as single units.
An identical algebraic chunk represents the same intermediate quantity wherever it appears. We’ll name the shared parenthesized expression once, interpret its component dollar terms before any adjustment, and then examine …
Preview problemSelect the expression part that answers a specific context question
Interpret complex expressions by treating meaningful sub-expressions as single units.
To locate a requested quantity inside a larger expression, follow the operations in their actual order. We’ll identify the complete amount built from the fixed and per-item costs, then separate …
Preview problemBuild an explicit linear function from a starting value and constant rate
Build a function from context by determining an explicit formula, recursive rule, or calculation process.
A constant-rate model combines what is already present with the change accumulated over the input interval. We’ll define the time input and measured output, place the time-zero value as the …
Preview problemBuild an explicit proportional function
Build a function from context by determining an explicit formula, recursive rule, or calculation process.
A proportional model has no separate starting amount: the total comes entirely from equal contributions per input unit. We’ll define ticket count as the input and total cost as the …
Preview problemBuild a recursive arithmetic sequence model
Build a function from context by determining an explicit formula, recursive rule, or calculation process.
A recursive arithmetic model needs two pieces: an initial term and a rule linking each later term to its predecessor. We’ll translate the first stated amount into the initial condition, …
Preview problemBuild an explicit arithmetic sequence model
Build a function from context by determining an explicit formula, recursive rule, or calculation process.
An explicit arithmetic rule starts with the first term and adds one common difference for every transition after it. We’ll define the indexed output, identify the starting count and signed …
Preview problemBuild an explicit exponential function from percent growth
Build a function from context by determining an explicit formula, recursive rule, or calculation process.
Percent growth is multiplicative because each period’s change is based on the current amount. We’ll define elapsed time and the changing population, use the time-zero amount as the leading factor, …
Preview problemBuild a recursive geometric sequence model
Build a function from context by determining an explicit formula, recursive rule, or calculation process.
A recursive geometric model describes each new term by scaling the entire previous term. We’ll record the starting cell count as the initial condition, translate the word describing the hourly …
Preview problemBuild a function rule from a table
Build a function from context by determining an explicit formula, recursive rule, or calculation process.
A table’s pattern is identified by how outputs change across equal input steps. We’ll compute consecutive differences and ratios, use the consistent comparison to decide between additive and multiplicative structure, …
Preview problemWrite a linear or exponential function from graph features using the initial value and the per-unit change
Build a function from context by determining an explicit formula, recursive rule, or calculation process.
Graph features must be read from the labeled axis scales, not from raw grid-square counts. We’ll use the marked points, treat the point with zero input as the initial value, …
Preview problemBuild a piecewise function from a tiered context
Build a function from context by determining an explicit formula, recursive rule, or calculation process.
A tiered price needs one rule for each interval where the calculation changes. We’ll locate the time boundary, assign it to exactly one piece, and distinguish total hours from only …
Preview problemDefine a function from an ordered calculation process
Build a function from context by determining an explicit formula, recursive rule, or calculation process.
An ordered calculation process becomes a function by tracking the intermediate result after each instruction. We’ll begin with the input, apply the multiplication first, and then add to that completed …
Preview problemChoose whether an explicit or recursive model is more useful
Build a function from context by determining an explicit formula, recursive rule, or calculation process.
The most useful sequence representation depends on the retrieval task. We’ll compare a rule that accepts the term number directly with one that depends on preceding terms, then trace the …
Preview problemBuild a function with a restricted domain
Build a function from context by determining an explicit formula, recursive rule, or calculation process.
A contextual domain records every restriction on the input, not just numerical bounds. We’ll use the fact that tickets are counted objects to determine the allowed number type, set the …
Preview problemBuild a linear function rule from a verbal comparison of input and output
Build a function from context by determining an explicit formula, recursive rule, or calculation process.
A verbal comparison becomes a function only after the input and output roles are fixed. We’ll start from the input, translate the comparison phrase into the operation performed on it, …
Preview problemBuild a context function and evaluate it at a given input
Build a function from context by determining an explicit formula, recursive rule, or calculation process.
Build the general cost model before evaluating the requested trip. We’ll define mileage as the input, keep the one-time fee as a constant, and multiply only the per-mile rate by …
Preview problemCritique and correct a proposed function model
Build a function from context by determining an explicit formula, recursive rule, or calculation process.
A proposed model should be tested against structural facts in the context, not accepted from its appearance. We’ll evaluate it at zero to test the one-time fee, compare outputs one …
Preview problemAdd two functions and interpret the combined output
Combine standard functions using arithmetic operations to build models.
Function addition combines two complete outputs evaluated at the same input. We’ll place each supplier rule in parentheses, add corresponding variable and constant terms, and keep the common dollar unit …
Preview problemSubtract two functions and interpret the result
Combine standard functions using arithmetic operations to build models.
A function difference keeps the written order meaningful, especially when one output is revenue and the other is cost. We’ll subtract the entire second rule with parentheses, distribute the negative …
Preview problemScale a function by a constant and interpret the scaled model
Combine standard functions using arithmetic operations to build models.
Scaling a function means taking multiple copies of its entire output, not changing only one term or replacing the input. We’ll substitute the full rule inside parentheses, distribute the scale …
Preview problemForm a product function and interpret it in context
Combine standard functions using arithmetic operations to build models.
A product function multiplies two outputs produced at the same input. We’ll identify one factor as price per item and the other as item quantity, form their product before expanding, …
Preview problemForm a quotient function and state restrictions
Combine standard functions using arithmetic operations to build models.
Average cost per item is a quotient of the complete total cost by the item count. We’ll substitute the full cost rule into the numerator, divide each term by the …
Preview problemForm, evaluate, and interpret combined functions such as (f+g)(x), (f-g)(x), and (fg)(x) from given function rules
Combine standard functions using arithmetic operations to build models.
Combined-function notation applies the named outside operation to separate outputs at the same input. We’ll expand the notation first, substitute the input independently into both rules, complete each function’s internal …
Preview problemEvaluate and interpret combined functions such as (f+g)(a), (f-g)(a), and (fg)(a) from given function values
Combine standard functions using arithmetic operations to build models.
When both function values are already supplied for the same input, no underlying formulas need to be reconstructed. We’ll expand the combined-function notation, match each named output to its given …
Preview problemFind the domain of a combined function
Combine standard functions using arithmetic operations to build models.
A combined function accepts only inputs that every required component accepts. We’ll write both domain conditions, take their overlap rather than their union, and preserve endpoint inclusion from the original …
Preview problemBuild a net-change model from inflow and outflow functions
Combine standard functions using arithmetic operations to build models.
Net change depends on direction: entering contributes positively, while leaving must be removed. We’ll align the inflow and outflow over the same time input, write their difference in meaning-first order, …
Preview problem