Math I
Build connected understanding across quantities, equations, functions, coordinate geometry, congruence, and data.
- Problem types
- 659
- Practice variants
- 2,636
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Each problem type has four distinct practice variants. Open a preview to move among all four.
Rearrange a formula by dividing to isolate the target
Rearrange formulas to isolate a chosen quantity and interpret the result.
The target variable is attached to another quantity by multiplication. We’ll treat the other symbols as known, undo that multiplication by dividing both sides by the complete factor, and simplify …
Preview problemRearrange a formula by undoing an outer factor and then isolating the variable inside a group
Rearrange formulas to isolate a chosen quantity and interpret the result.
The parentheses reveal the operation layers around the target variable. We’ll work from the outside inward: first undo the factor multiplying the entire group, then undo the addition still inside …
Preview problemRearrange a quotient formula to solve for the denominator variable and state restrictions
Rearrange formulas to isolate a chosen quantity and interpret the result.
The target begins in a denominator, so we’ll clear that fraction before trying to isolate it. Multiplying by the original denominator turns the relationship into a product; then dividing by …
Preview problemUse a square root and choose the meaningful branch
Rearrange formulas to isolate a chosen quantity and interpret the result.
Undoing a square produces two algebraic branches, so the algebra and the context both matter. We’ll take square roots to expose the positive and negative possibilities, then use the meaning …
Preview problemRearrange a linear formula to isolate a chosen variable, especially when that variable appears in more than one term
Rearrange formulas to isolate a chosen quantity and interpret the result.
The target variable appears in two separate terms, so trying to divide immediately would leave part of it behind. We’ll first combine those terms by factoring out the shared target, …
Preview problemRearrange a formula with inverse operations when the target variable appears in one term
Rearrange formulas to isolate a chosen quantity and interpret the result.
We’ll read the operation layers around the requested length and undo them in reverse order. First remove the term that does not contain the target; then divide the complete remaining …
Preview problemRearrange a formula to isolate a chosen variable, then substitute values to find its value
Rearrange formulas to isolate a chosen quantity and interpret the result.
This task has a symbolic stage and a numerical stage, and keeping them in that order makes the variable roles clear. We’ll isolate time by undoing its multiplication by the …
Preview problemRearrange a formula by clearing a fraction or denominator to isolate a variable
Rearrange formulas to isolate a chosen quantity and interpret the result.
The fraction here is a coefficient multiplying the target, not a denominator containing the target. We’ll first remove the added constant, then multiply the entire remaining side by the reciprocal …
Preview problemRearrange a linear formula with subtraction or a negative coefficient to solve for a chosen variable
Rearrange formulas to isolate a chosen quantity and interpret the result.
The negative coefficient makes sign tracking the central issue. We’ll move the constant away from the target term, divide by the full signed coefficient, and then rewrite the negative quotient …
Preview problemRearrange a formula and state the variable restrictions that make the new equation valid
Rearrange formulas to isolate a chosen quantity and interpret the result.
The original equation and the rearranged equation can impose restrictions at different moments. We’ll record the variable forbidden by the starting denominator before clearing it, then note the variable used …
Preview problemDecide whether two algebraic forms are equivalent rearrangements of a formula
Rearrange formulas to isolate a chosen quantity and interpret the result.
Equivalence should be tested by transforming one form into the other, not by judging how different they look. We’ll start with the grouped quotient, apply its common divisor to every …
Preview problemName the equality property used in a solving step
Explain equation-solving steps as logical consequences of equality and justify solution methods.
An equality property is named for the operation performed between lines, not merely for a symbol visible in the starting equation. We’ll compare the before-and-after forms, reconstruct the same hidden …
Preview problemJustify an algebraic transformation
Explain equation-solving steps as logical consequences of equality and justify solution methods.
Only the left expression changes while the right side stays fixed, so this is an equivalent rewrite rather than an operation on both sides. We’ll track the outside factor to …
Preview problemFind the first invalid solving step
Explain equation-solving steps as logical consequences of equality and justify solution methods.
An error audit must move in chronological order, because later work may be consistent with an already broken equation. We’ll compare each line with the one immediately before it, beginning …
Preview problemVerify a solution by substitution
Explain equation-solving steps as logical consequences of equality and justify solution methods.
Verification means testing the proposed value in the original equation, not simply trusting the solving work that produced it. We’ll replace the variable while preserving the equation’s operation order, evaluate …
Preview problemClassify an equation as one solution, no solution, or infinitely many solutions
Explain equation-solving steps as logical consequences of equality and justify solution methods.
The solution count is determined by what remains after both sides are simplified, not by the equation’s original appearance. We’ll expand the grouped side, combine like terms, and compare the …
Preview problemDecide whether a transformation creates an equivalent equation
Explain equation-solving steps as logical consequences of equality and justify solution methods.
Two equations are equivalent only when their complete solution sets match. We’ll identify the balanced operation connecting these forms, check that applying it forward preserves equality, and then reverse it …
Preview problemClear denominators in an equation
Explain equation-solving steps as logical consequences of equality and justify solution methods.
Clearing a denominator is a whole-equation operation. We’ll use a common-denominator multiplier on every term on both sides, so the fractional term cancels while equality is preserved. After simplifying the …
Preview problemTest whether one equation step is equivalent
Explain equation-solving steps as logical consequences of equality and justify solution methods.
This audit concerns a local rewrite on one side, so the unchanged right side is not evidence of an imbalance. We’ll multiply the existing outside factor by every term in …
Preview problemComplete an equality-preserving solution chain
Explain equation-solving steps as logical consequences of equality and justify solution methods.
A correct endpoint is not enough; every arrow in the solution chain must preserve equality. We’ll first rewrite the grouped expression and combine its constants, then use the same inverse …
Preview problemTest whether an ordered pair satisfies an equation
Understand that a two-variable equation's graph is the set of all ordered-pair solutions.
An ordered pair supplies values for both variables, with the input first and the output second. We’ll substitute each coordinate into its matching position, evaluate the expression that predicts the …
Preview problemGenerate ordered-pair solutions from input values
Understand that a two-variable equation's graph is the set of all ordered-pair solutions.
The equation acts as one input-output rule for all three supplied inputs. We’ll substitute each input separately, use parentheses around the negative value to protect its sign, and keep the …
Preview problemIdentify listed points that are not solutions
Understand that a two-variable equation's graph is the set of all ordered-pair solutions.
Every listed point needs its own test; one successful point says nothing about the others. For each pair, we’ll use the x-coordinate to compute the output required by the equation …
Preview problemChoose the table that matches an equation
Understand that a two-variable equation's graph is the set of all ordered-pair solutions.
A table represents the equation only if every row is an ordered-pair solution. We’ll read each row from the supplied tables, calculate the output predicted by the rule for its …
Preview problemMatch graph features to a linear equation
Understand that a two-variable equation's graph is the set of all ordered-pair solutions.
Slope-intercept form gives each graph feature a different symbolic role. We’ll place the line’s rise-over-run in the coefficient position and its vertical-axis crossing in the constant position, keeping their signs …
Preview problemInterpret an ordered-pair solution with units
Understand that a two-variable equation's graph is the set of all ordered-pair solutions.
The ordered pair has to be interpreted in the equation’s variable order, not by the sizes of its numbers. We’ll identify which variable is the input and which is the …
Preview problemRepresent a vertical or horizontal solution set
Understand that a two-variable equation's graph is the set of all ordered-pair solutions.
An equation naming only one coordinate fixes that coordinate and leaves the missing one free. We’ll describe every solution as an ordered pair with the stated first coordinate and an …
Preview problemFind an equation that fits plotted solution points
Understand that a two-variable equation's graph is the set of all ordered-pair solutions.
The plotted points reveal both pieces of a linear rule. We’ll calculate the constant change in output per unit change in input to get the slope, then use the point …
Preview problemTest every row in a table and locate violations
Understand that a two-variable equation's graph is the set of all ordered-pair solutions.
The word “every” makes this a row-by-row test, so early successes cannot settle the question. We’ll use each x-entry to compute the output required by the equation and compare it …
Preview problemVerify consistency across equation, table, and graph features
Understand that a two-variable equation's graph is the set of all ordered-pair solutions.
Consistency requires agreement across all three representations, not just a few matching numbers. We’ll generate the equation’s outputs at the table inputs and compare them row by row, then decode …
Preview problemFind an exact intersection by solving f(x) = g(x)
Solve equations of the form f(x)=g(x) approximately by finding graph/table intersections.
At an intersection, the two rules produce the same output from the same input. We’ll set their output expressions equal and solve that equation for the shared input, then substitute …
Preview problemEstimate an intersection point from a graph description
Solve equations of the form f(x)=g(x) approximately by finding graph/table intersections.
The graph gives an estimate, so the goal is to enclose the crossing rather than claim more precision than the scale supports. We’ll read the horizontal and vertical coordinates separately, …
Preview problemUse a table to locate where two functions are equal
Solve equations of the form f(x)=g(x) approximately by finding graph/table intersections.
We’ll compare the two output columns at each common input, first looking for a row where the values are exactly equal. If no row matches, adjacent inputs where the functions …
Preview problemInterpret an intersection point in context
Solve equations of the form f(x)=g(x) approximately by finding graph/table intersections.
The coordinates must be translated through the axes before the intersection can be interpreted. We’ll attach the input unit to the first coordinate and the output unit to the second, …
Preview problemFind intersection points of a line and a quadratic
Solve equations of the form f(x)=g(x) approximately by finding graph/table intersections.
A line and a quadratic can meet more than once, so we have to account for every shared input. We’ll set their output expressions equal, move everything into a zero-product …
Preview problemApproximate a crossing point from a function-comparison table
Solve equations of the form f(x)=g(x) approximately by finding graph/table intersections.
A table can show an intersection exactly, so we should look for equality before settling for an interval estimate. We’ll compare both output entries in every common-input row. If a …
Preview problem