Math I
Build connected understanding across quantities, equations, functions, coordinate geometry, congruence, and data.
- Problem types
- 659
- Practice variants
- 2,636
Page 15 of 19
Each problem type has four distinct practice variants. Open a preview to move among all four.
Disprove a geometric claim with coordinate evidence
Use coordinates, distance, and algebra to prove or disprove simple geometric statements.
A disproof needs only one exact violation of a necessary condition, so look for the shortest decisive measurement instead of checking every property. For a parallelogram claim, compare an opposite-side …
Preview problemComplete a missing coordinate calculation in a proof step
Use coordinates, distance, and algebra to prove or disprove simple geometric statements.
First identify what kind of quantity the proof blank requests, because slope, displacement, and distance use the same coordinate changes differently. For distance, treat the horizontal and vertical changes as …
Preview problemDecide whether two lines are parallel by comparing their slopes
Prove and use slope criteria for parallel and perpendicular lines.
Slope records direction, so parallelism is about equality of slopes rather than whether they are positive or negative. Confirm that the lines are distinct, compare the two slope values directly, …
Preview problemDetermine whether two lines are perpendicular from their slopes
Prove and use slope criteria for parallel and perpendicular lines.
For two nonvertical lines, test perpendicularity by asking whether one slope is the negative reciprocal of the other. Multiplying the slopes is often the cleanest check: a product of negative …
Preview problemClassify horizontal and vertical line relationships
Prove and use slope criteria for parallel and perpendicular lines.
Read special-form equations geometrically: fixing x creates a vertical line, while fixing y creates a horizontal line. After identifying each direction, compare the constants to decide whether the equations describe …
Preview problemFind the slope of a line parallel to a given line
Prove and use slope criteria for parallel and perpendicular lines.
Put the equation in slope-intercept form and separate the coefficient of x from the constant term; they describe direction and vertical position, respectively. A parallel line keeps the same direction, …
Preview problemFind the slope of a line perpendicular to a given line
Prove and use slope criteria for parallel and perpendicular lines.
Read the original slope from the coefficient of x, then perform both parts of the perpendicular transformation: invert the slope ratio and change its sign. A quick product check should …
Preview problemWrite an equation of a line parallel to a given line through a point
Prove and use slope criteria for parallel and perpendicular lines.
A parallel-through-a-point problem separates direction from position: keep the given line’s slope, but do not copy its intercept. Insert the new point into point-slope form or into y equals mx …
Preview problemWrite an equation of a line perpendicular to a given line through a point
Prove and use slope criteria for parallel and perpendicular lines.
Build the line in two stages: first replace the given slope with its negative reciprocal, then use the required point to determine position. After simplifying the equation, check that the …
Preview problemUse slope to show two segments are parallel
Prove and use slope criteria for parallel and perpendicular lines.
Treat each segment as a direction vector and compute signed rise over signed run with one subtraction order per segment. Equal slopes show equal direction only after the supporting lines …
Preview problemUse slope criteria to prove two segments are perpendicular from coordinates
Prove and use slope criteria for parallel and perpendicular lines.
Compute both signed slopes before judging the drawing, especially preserving a negative rise when a segment falls from left to right. For nonvertical segments, multiply the slopes: a product of …
Preview problemFind the distance between two points on the coordinate plane
Use coordinate methods to compute polygon perimeters and areas of triangles/rectangles.
Straight-line distance is the hypotenuse formed by the horizontal and vertical coordinate changes, not the total of walking along those two legs. Square the changes, add them, and take the …
Preview problemFind the perimeter of a triangle from its vertices on the coordinate plane
Use coordinate methods to compute polygon perimeters and areas of triangles/rectangles.
Perimeter means walking the entire boundary, so list all three side lengths before adding anything. Horizontal or vertical edges come from one coordinate difference, while a diagonal needs the distance …
Preview problemFind the perimeter of a rectangle from its coordinates
Use coordinate methods to compute polygon perimeters and areas of triangles/rectangles.
Measure one horizontal side and one vertical side from absolute coordinate differences, then use the rectangle’s opposite-side equality. Perimeter counts the boundary, so double the sum of those adjacent lengths; …
Preview problemFind the area of an axis-aligned rectangle from its coordinates
Use coordinate methods to compute polygon perimeters and areas of triangles/rectangles.
For an axis-aligned rectangle, the coordinate spans are already perpendicular dimensions: use the absolute x-difference for one side and the absolute y-difference for the other. Multiply those dimensions to count …
Preview problemFind the area of a triangle on the coordinate plane when one base is horizontal or vertical
Use coordinate methods to compute polygon perimeters and areas of triangles/rectangles.
Use the horizontal side as the base, then measure height perpendicular to its line rather than along a slanted side. Because the base lies on a constant y-value, the height …
Preview problemFind the area of a right triangle from its coordinates
Use coordinate methods to compute polygon perimeters and areas of triangles/rectangles.
Start by spotting the two sides that meet at a right angle; those are already a base and its perpendicular height. Read each length from the coordinate span, then use …
Preview problemFind the area of a polygon by decomposing it into rectangles and triangles
Use coordinate methods to compute polygon perimeters and areas of triangles/rectangles.
Treat the L-shape as a complete outer rectangle with one rectangular corner missing. Find those two areas from their side lengths and subtract the missing piece, checking that the removed …
Preview problemUse the coordinate area formula on ordered vertices
Use coordinate methods to compute polygon perimeters and areas of triangles/rectangles.
List the vertices in one continuous trip around the boundary and repeat the first point at the end. Form the two diagonal-product sums, take the absolute difference so orientation cannot …
Preview problemFind a missing x-coordinate from the length of a horizontal side
Use coordinate methods to compute polygon perimeters and areas of triangles/rectangles.
First use the perimeter relationship to recover the rectangle’s positive missing width. Then translate that width into an absolute horizontal distance from the fixed coordinate; because distance does not encode …
Preview problemFind a missing coordinate from rectangle area
Use coordinate methods to compute polygon perimeters and areas of triangles/rectangles.
Use the area and known height to determine the rectangle’s positive missing width. That width is an absolute coordinate difference from the fixed endpoint, so solve the resulting distance relationship …
Preview problemCompare area and perimeter of coordinate figures
Use coordinate methods to compute polygon perimeters and areas of triangles/rectangles.
Compare the two measurements independently: area comes from multiplying side lengths, while perimeter comes from doubling their sum. Equal products do not force equal sums, so verify each rectangle separately …
Preview problemConvert units using a conversion factor
Use units to guide multi-step problem solving and choose/interpret units, scales, and graph origins.
Write the conversion relationship as a fraction equal to one, placing the starting unit where it will cancel and the target unit where it will remain. Then multiply and check …
Preview problemChoose the correct unit for a measured quantity
Use units to guide multi-step problem solving and choose/interpret units, scales, and graph origins.
Let the kind of quantity determine the unit, not just the object’s dimensions. Area multiplies one length by another, so the length unit is multiplied by itself and must be …
Preview problemInterpret a compound unit rate
Use units to guide multi-step problem solving and choose/interpret units, scales, and graph origins.
Read the word “per” as division: the first quantity belongs in the numerator and the second in the denominator. A unit rate then normalizes that denominator to one unit, so …
Preview problemIdentify invalid operations with incompatible units
Use units to guide multi-step problem solving and choose/interpret units, scales, and graph origins.
Classify the physical dimension of each term before doing arithmetic. Addition requires quantities that measure the same kind of thing, possibly after a conversion within that dimension; a conversion can …
Preview problemInterpret a point on a graph using axis labels and units
Use units to guide multi-step problem solving and choose/interpret units, scales, and graph origins.
Read the graph’s labels and scale before interpreting the point. The horizontal coordinate pairs with the horizontal quantity and its unit, while the vertical coordinate pairs with the vertical quantity …
Preview problemFind a unit rate from a total amount and a total time
Use units to guide multi-step problem solving and choose/interpret units, scales, and graph origins.
Let the requested unit tell you the order of division: the amount named first goes above the elapsed time named after “per.” Dividing the total amount by the total time …
Preview problemConvert a rate to new numerator and denominator units
Use units to guide multi-step problem solving and choose/interpret units, scales, and graph origins.
Convert the numerator and denominator as separate unit-cancellation tasks, using factors equal to one. Arrange each factor so the old distance and time units disappear while the requested units remain …
Preview problemJudge whether a unit and magnitude are reasonable
Use units to guide multi-step problem solving and choose/interpret units, scales, and graph origins.
Make two separate judgments: first ask whether the unit measures the right physical quantity, and only then compare the numerical value with the stated benchmark interval. Quantify any mismatch with …
Preview problemChoose a simpler scale for modeling large quantities
Use units to guide multi-step problem solving and choose/interpret units, scales, and graph origins.
Define exactly how many original units make one scaled unit, then divide the original numerical value by that group size. This counts scaled-unit groups without changing the underlying quantity; multiply …
Preview problemIdentify independent and dependent quantities in a context
Define appropriate quantities for descriptive modeling.
Use the sentence’s dependency language to draw a one-way arrow: the quantity recorded or set first points toward the quantity that changes in response. The first is the independent input …
Preview problemChoose clear variables with units for a model
Define appropriate quantities for descriptive modeling.
Define each symbol with four pieces: its input-or-output role, the exact quantity it represents, its unit, and its context-appropriate domain. Pay special attention to how the quantity is measured: counts …
Preview problemChoose relevant quantities for a model
Define appropriate quantities for descriptive modeling.
Start with the relationship that produces the requested outcome, then identify which contextual quantities fill its inputs. A useful relevance test is to imagine changing one detail while holding the …
Preview problemDefine measurements needed for a geometric model
Define appropriate quantities for descriptive modeling.
Identify both the shape being modeled and the geometric measure the model must produce, because together they determine the governing formula. The measurements you need are exactly the independent dimensions …
Preview problemIdentify variable, group, and units in a statistical question
Define appropriate quantities for descriptive modeling.
Separate three ideas that are easy to blur together: the variable is the characteristic measured, the observational units are the individuals measured, and the data source is the sampled or …
Preview problem