Math I
Build connected understanding across quantities, equations, functions, coordinate geometry, congruence, and data.
- Problem types
- 659
- Practice variants
- 2,636
Page 10 of 19
Each problem type has four distinct practice variants. Open a preview to move among all four.
Connect sequence type to model type
Distinguish linear from exponential situations by equal differences versus equal factors over equal intervals.
Sequences connect to function models through the kind of change between consecutive terms. We’ll test subtraction for a repeated amount and division for a repeated factor, then link the surviving …
Preview problemShow data is neither linear nor exponential
Distinguish linear from exponential situations by equal differences versus equal factors over equal intervals.
Failing one pattern test does not finish the classification, because the data may belong to another family. With equal input steps confirmed, we’ll compute first differences and ratios, then examine …
Preview problemChoose a linear or exponential model from data evidence
Distinguish linear from exponential situations by equal differences versus equal factors over equal intervals.
Equal time steps let us put additive and multiplicative evidence side by side. We’ll calculate every consecutive difference and ratio, identify which pattern remains constant, and combine that pattern with …
Preview problemTranslate growth language into additive or multiplicative form
Distinguish linear from exponential situations by equal differences versus equal factors over equal intervals.
Growth language becomes clearer when we ask what happens in one time step. We’ll identify whether the repeated quantity is a fixed dollar amount, a percent, or a unitless factor, …
Preview problemFind constant rate of change from a table
Recognize constant rate of change as evidence for a linear model.
A rate is not just an output change; it is output change per unit of input change. We’ll take two consecutive table points, subtract coordinates in a consistent order, and …
Preview problemVerify constant rate of change with unequal input gaps
Recognize constant rate of change as evidence for a linear model.
Unequal input gaps can create unequal raw output changes even when the underlying rate is constant. We’ll normalize each interval separately by dividing its signed output change by its own …
Preview problemFind constant rate of change from graph points
Recognize constant rate of change as evidence for a linear model.
Two plotted points turn the line’s visual steepness into an exact rate. We’ll read their coordinates, move from the first point to the second in one consistent direction, and calculate …
Preview problemIdentify constant rate of change in context
Recognize constant rate of change as evidence for a linear model.
A contextual rate is a quotient whose order is determined by the requested units. We’ll place distance in the numerator and elapsed time in the denominator to create miles per …
Preview problemBuild a linear model from rate and initial value
Recognize constant rate of change as evidence for a linear model.
A linear model combines a starting amount with repeated constant change, and those ideas occupy different positions in the formula. We’ll place the rate on the input and keep the …
Preview problemInterpret constant rate of change with units
Recognize constant rate of change as evidence for a linear model.
Interpreting a rate means naming its input, output, units, and direction before doing arithmetic. We’ll read the sign to determine whether the output rises or falls, explain the effect of …
Preview problemUse constant rate to find a missing table value
Recognize constant rate of change as evidence for a linear model.
A constant rate can extend a table across any input gap, not just one row at a time. We’ll use the complete rows to find output change per input unit, …
Preview problemJudge whether a context should be modeled linearly
Recognize constant rate of change as evidence for a linear model.
Whether a context is linear depends on constant change, not on whether the formula contains multiplication or a bonus. We’ll identify the amount added for each equal input step and …
Preview problemCompare constant rates from different representations
Recognize constant rate of change as evidence for a linear model.
Rates from different representations must be converted to the same output-per-input form before they can be compared. We’ll read one rate from the equation’s input coefficient and calculate the other …
Preview problemDiagnose a nonlinear table
Recognize constant rate of change as evidence for a linear model.
One average slope cannot prove that a table is linear; every interval rate must agree. With equal input gaps confirmed, we’ll calculate all consecutive slopes, then inspect how those slopes …
Preview problemFind the input value when a linear model reaches a given output
Recognize constant rate of change as evidence for a linear model.
The target number here is an output, so the task runs the linear model backward to recover its input. We’ll set the function expression equal to that target, undo the …
Preview problemCorrect a linear model with swapped rate and initial value
Recognize constant rate of change as evidence for a linear model.
A quick zero-input test exposes which number is the initial value in a linear model. We’ll pair the standalone constant with the starting amount and the input coefficient with the …
Preview problemConvert percent increase to growth factor
Recognize constant percent growth or decay as evidence for an exponential model.
A growth factor must represent the whole new amount, not just the added portion. We’ll convert the percent increase to a decimal and combine it with the original whole, represented …
Preview problemConvert percent decrease to decay factor
Recognize constant percent growth or decay as evidence for an exponential model.
A decay factor describes the portion that remains after a loss, not the portion removed. We’ll convert the percent decrease to a decimal and subtract it from the original whole. …
Preview problemVerify constant percent change from a table
Recognize constant percent growth or decay as evidence for an exponential model.
Constant percent change appears as a constant output ratio over equal input intervals, even when the raw output differences grow. We’ll divide every later output by its predecessor and compare …
Preview problemWrite an exponential model from initial value and percent increase
Recognize constant percent growth or decay as evidence for an exponential model.
An exponential model separates the value at zero intervals from the factor repeated during each interval. We’ll place the initial value as the coefficient, convert the percent increase into a …
Preview problemInterpret the base of an exponential model
Recognize constant percent growth or decay as evidence for an exponential model.
The base of an exponential model carries both a retained percentage and a one-step percent change. We’ll separate the whole from the excess above one, express the model as a …
Preview problemIdentify percent growth versus fixed-amount growth
Recognize constant percent growth or decay as evidence for an exponential model.
A fixed percent and a fixed amount can both repeat, but they update values differently. We’ll write one rule that adds the same quantity and another that adds a fraction …
Preview problemEvaluate an exponential model after several intervals
Recognize constant percent growth or decay as evidence for an exponential model.
The exponent counts how many times the interval factor is applied, so several intervals mean repeated multiplication rather than multiplying once by the interval count. We’ll substitute the requested count, …
Preview problemReason backward in an exponential model
Recognize constant percent growth or decay as evidence for an exponential model.
Moving backward through an exponential model requires undoing the forward multiplier, not subtracting it. We’ll express one forward interval as multiplication by the base, reverse that operation with division, and …
Preview problemCompare exponential models by percent rate
Recognize constant percent growth or decay as evidence for an exponential model.
Initial value and percent rate are separate exponential parameters, so matching coefficients do not imply matching growth. We’ll subtract the retained whole from each base, convert the excess decimals to …
Preview problemIdentify an invalid exponential model for percent growth
Recognize constant percent growth or decay as evidence for an exponential model.
A percent-growth base must contain both the original whole and the decimal increase. We’ll simplify each candidate factor, identify which expressions are equivalent, and flag any base that uses only …
Preview problemChoose an exponential model from approximate ratio evidence
Recognize constant percent growth or decay as evidence for an exponential model.
Real data can support an exponential model even when its ratios are close rather than identical. We’ll compare consecutive outputs over equal input steps, use a representative factor to build …
Preview problemConstruct a linear function from graph points
Construct linear and exponential functions, including arithmetic/geometric sequences, from graphs, descriptions, or input-output pairs.
Two plotted points determine a linear rule through a constant rate and a starting value. We’ll divide the vertical change by the matching horizontal change to find the slope, then …
Preview problemConstruct a linear function from two input-output pairs
Construct linear and exponential functions, including arithmetic/geometric sequences, from graphs, descriptions, or input-output pairs.
A pair of input-output records gives the change information for a linear function, even when neither input is zero. We’ll compute output change per unit of input, place that slope …
Preview problemConstruct a linear function from a context
Construct linear and exponential functions, including arithmetic/geometric sequences, from graphs, descriptions, or input-output pairs.
A contextual linear model separates the charge that repeats with the input from the charge paid only once. We’ll identify hours as the input, attach the hourly units to the …
Preview problemConstruct an exponential function from graph points
Construct linear and exponential functions, including arithmetic/geometric sequences, from graphs, descriptions, or input-output pairs.
Plotted exponential points reveal one parameter at input zero and the other through repeated multiplication. We’ll use the zero-input output as the coefficient, then divide consecutive outputs whose inputs are …
Preview problemConstruct an exponential function from input-output pairs
Construct linear and exponential functions, including arithmetic/geometric sequences, from graphs, descriptions, or input-output pairs.
An exponential table is organized by an initial output and a multiplier repeated across equal input intervals. We’ll read the output at zero for the coefficient and compare every consecutive …
Preview problemConstruct an exponential model from percent growth
Construct linear and exponential functions, including arithmetic/geometric sequences, from graphs, descriptions, or input-output pairs.
Percent growth repeatedly multiplies the entire current amount, so the exponential base must include the retained whole as well as the increase. We’ll use the starting investment as the coefficient, …
Preview problemConstruct an explicit arithmetic sequence rule
Construct linear and exponential functions, including arithmetic/geometric sequences, from graphs, descriptions, or input-output pairs.
An explicit arithmetic rule starts from the first term and counts how many equal additions are needed to reach term n. We’ll identify the starting value and signed common difference, …
Preview problemConstruct a recursive arithmetic sequence rule
Construct linear and exponential functions, including arithmetic/geometric sequences, from graphs, descriptions, or input-output pairs.
A recursive sequence rule needs both an anchor and an update; either part alone leaves the sequence incomplete. We’ll translate the first week into the initial condition and the fixed …
Preview problemConstruct an explicit geometric sequence rule
Construct linear and exponential functions, including arithmetic/geometric sequences, from graphs, descriptions, or input-output pairs.
An explicit geometric rule records a first term and the number of times a common multiplier has been applied. We’ll translate doubling into a ratio and use n minus one …
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