Math Foundations
Strengthen number sense, fractions, ratios, expressions, equations, geometry, and data—the bridge from middle-school math to high-school success.
- Problem types
- 634
- Practice variants
- 2,536
Page 15 of 18
Each problem type has four distinct practice variants. Open a preview to move among all four.
Read proportionality from a rule
Distinguish proportional relationships from linear but non-proportional relationships.
Read the rule by structure before computing: a proportional equation has only a constant multiplier times the input, with no added nonzero term. Substitute zero for the input to expose …
Preview problemCompare two linear forms for proportionality
Distinguish proportional relationships from linear but non-proportional relationships.
The two rules have the same rate, but equal slopes alone do not settle proportionality. Evaluate each rule at zero or inspect its constant term, because a proportional linear relationship …
Preview problemDecide whether a story is proportional
Distinguish proportional relationships from linear but non-proportional relationships.
Translate the story into total cost equals hourly rate times hours plus any starting fee. Evaluate that model at zero hours, then check whether every nonzero cost-to-hours ratio stays equal …
Preview problemModel a story, then test proportionality
Distinguish proportional relationships from linear but non-proportional relationships.
Build the table from a model, not from a guessed pattern: height equals the initial height plus the signed hourly change times elapsed time. Evaluate that rule at every requested …
Preview problemMatch a story to the right kind of linear form
Distinguish proportional relationships from linear but non-proportional relationships.
Separate the delivery cost into the amount charged before any miles and the amount repeated for each mile. The first becomes the equation’s constant term, while the per-mile rate multiplies …
Preview problemConvert without changing proportionality
Distinguish proportional relationships from linear but non-proportional relationships.
The table’s raw output change is not automatically the coefficient when the input jumps by more than one. Compute the unit rate as output change divided by input change, read …
Preview problemRewrite to compare proportional structure
Distinguish proportional relationships from linear but non-proportional relationships.
Rewrite both relationships as rate times hours plus a starting amount so the same features are visible. For the table, find the cost change per one-hour step, then use any …
Preview problemSort mixed representations by proportional structure
Distinguish proportional relationships from linear but non-proportional relationships.
Apply one structural test across equations, tables, contexts, and point sets: proportional output is a constant multiple of input with a zero start. Look for no added term, a constant …
Preview problemConvert to a smaller unit
Convert measurements within a system and use unit reasoning to choose the correct scale.
Use an equality between the units as a conversion factor equal to one, placing the original unit in the denominator so it cancels. Because the target unit is smaller, multiply …
Preview problemConvert to a larger unit
Convert measurements within a system and use unit reasoning to choose the correct scale.
Think of the conversion as grouping smaller units into larger ones. Arrange the unit ratio so the original unit cancels and the requested larger unit remains, which is equivalent to …
Preview problemChoose a sensible measurement unit
Convert measurements within a system and use unit reasoning to choose the correct scale.
A sensible unit matches the scale of the object and the needed precision, not simply the smallest unit available. Compare how many of each candidate unit would span the classroom-sized …
Preview problemCompare measurements in different units
Convert measurements within a system and use unit reasoning to choose the correct scale.
Convert before comparing or reporting, because numbers attached to different units are not directly interchangeable. Multiply the given yards by a feet-per-yard factor arranged so yards cancel, then keep the …
Preview problemConvert a measurement for a story problem
Convert measurements within a system and use unit reasoning to choose the correct scale.
The map’s requested unit determines the conversion direction. Multiply the trail length by a feet-per-mile ratio so miles cancel, account for every mile rather than using the one-mile factor alone, …
Preview problemPick the best scale for the job
Convert measurements within a system and use unit reasoning to choose the correct scale.
Judge reporting scale by expressing the same bottle capacity in both candidate units. Use the liter–milliliter relationship to compare whether each form is a familiar whole-number amount or an awkward …
Preview problemCheck whether a converted answer makes sense
Convert measurements within a system and use unit reasoning to choose the correct scale.
Check the direction before doing arithmetic: changing from hours to the smaller unit of minutes must increase the numerical measure. Then apply the actual units-per-hour factor and compare the converted …
Preview problemRewrite a measurement in a new unit form
Convert measurements within a system and use unit reasoning to choose the correct scale.
Build an equivalent-measure equation by multiplying by a conversion ratio equal to one. Place liters opposite liters so they cancel and milliliters remain, then scale the numerical value by the …
Preview problemPut both measurements into the same unit
Convert measurements within a system and use unit reasoning to choose the correct scale.
Measurements with different units cannot be compared by their visible numbers alone. Pick one common unit, convert the other measurement with a conversion factor equal to one, and only then …
Preview problemFind every equivalent or sensible measurement choice
Convert measurements within a system and use unit reasoning to choose the correct scale.
Use one common time unit as a benchmark for every representation. Convert the target duration first, then rewrite mixed hours and minutes, seconds, days, and plain minutes into that same …
Preview problemAdd the side lengths for perimeter
Find and interpret perimeter and circumference in numeric and contextual problems.
Perimeter measures one complete trip around the outside, so every boundary segment must be counted exactly once. A rectangle has two sides of each given dimension; add those four lengths …
Preview problemUse the circle boundary formula
Find and interpret perimeter and circumference in numeric and contextual problems.
Start by matching the requested quantity to a boundary formula: circumference is the distance around a circle. Because the given measurement is the diameter, use the form that multiplies diameter …
Preview problemDecide which boundary measure to use
Find and interpret perimeter and circumference in numeric and contextual problems.
Translate the context into both a shape and a type of measurement. The fountain is circular, and “all the way around the edge” asks for its boundary length, not the …
Preview problemCompare boundary distances
Find and interpret perimeter and circumference in numeric and contextual problems.
Compare like quantities by finding the full boundary length of each figure. Add both pairs of rectangle sides, use pi times diameter for the circle with the stated approximation, and …
Preview problemUse perimeter in a real-world situation
Find and interpret perimeter and circumference in numeric and contextual problems.
Words such as fencing and “all the way around” signal a boundary-length problem. Model the rectangular garden with two copies of its length and two copies of its width, then …
Preview problemUse circumference in a real-world situation
Find and interpret perimeter and circumference in numeric and contextual problems.
A single wrap of ribbon follows the circular boundary, so the needed length is a circumference. Identify the supplied measure as a diameter and use the diameter form of the …
Preview problemDecide what the context is asking you to measure
Find and interpret perimeter and circumference in numeric and contextual problems.
First classify what the situation measures: around a boundary, across a shape, or over the region inside. Here the rope traces one complete outside edge, so the needed quantity is …
Preview problemWrite the right boundary expression from a description
Find and interpret perimeter and circumference in numeric and contextual problems.
The requested expression should represent one complete trip around the rectangle without needing to evaluate it. Account for both copies of the length and both copies of the width, then …
Preview problemPut both boundary measures into a comparable form
Find and interpret perimeter and circumference in numeric and contextual problems.
A fair comparison begins by measuring the same feature of both gardens: their complete boundary lengths. Use the radius form of circumference for the circle and the doubled length-plus-width structure …
Preview problemFind every matching boundary representation
Find and interpret perimeter and circumference in numeric and contextual problems.
Treat the target boundary value as a test that every representation must pass independently. Add all outside sides for polygons and contextual figures, evaluate any given expression with its grouping …
Preview problemMultiply length by width
Use area formulas for rectangles, triangles, parallelograms, trapezoids, and composite figures.
Area counts how many unit squares cover the rectangle, which organizes naturally as rows by columns. Multiply the two perpendicular dimensions so both the length and width contribute to the …
Preview problemUse one-half base times height
Use area formulas for rectangles, triangles, parallelograms, trapezoids, and composite figures.
A triangle occupies half of a parallelogram with the same base and perpendicular height. Multiply those two measurements to form the corresponding full area, then apply the one-half factor exactly …
Preview problemMatch the shape to its area formula
Use area formulas for rectangles, triangles, parallelograms, trapezoids, and composite figures.
Imagine sliding the slanted end of the parallelogram to make a rectangle without changing the covered region. That rearrangement shows why its area is base times perpendicular height, with no …
Preview problemBreak a composite figure into easier pieces
Use area formulas for rectangles, triangles, parallelograms, trapezoids, and composite figures.
Use the visible horizontal segment to view the L-shape as two familiar rectangles that meet without overlapping. Find each piece’s area from its own labeled dimensions, then add those areas …
Preview problemUse area for covering problems
Use area formulas for rectangles, triangles, parallelograms, trapezoids, and composite figures.
Carpet covers the whole floor surface, so the relevant measure is area rather than the distance around the room. Model the floor as a rectangle and multiply its length by …
Preview problemModel a less-regular region with area formulas
Use area formulas for rectangles, triangles, parallelograms, trapezoids, and composite figures.
A trapezoid’s width changes steadily from one parallel base to the other, so use the average of those base lengths. Multiply that average by the perpendicular height, which is equivalent …
Preview problemTurn a shape description into an area setup
Use area formulas for rectangles, triangles, parallelograms, trapezoids, and composite figures.
Translate the shape description into the trapezoid area structure before inserting any measurements. Keep both parallel bases grouped as a sum, apply the one-half factor to that sum, and multiply …
Preview problemPut two area problems into a comparable form
Use area formulas for rectangles, triangles, parallelograms, trapezoids, and composite figures.
Put both figures into the same comparable form by calculating complete areas in square centimeters. The triangle needs one-half of base times perpendicular height, while the rectangle uses length times …
Preview problem