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
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Each problem type has four distinct practice variants. Open a preview to move among all four.
Judge proportionality in context
Determine whether a relationship is proportional from a table, graph, equation, or context.
The determining evidence is whether every ticket-cost pair shares one cost per ticket and whether the relationship starts at zero. We’ll divide each cost by its matching ticket count, write …
Preview problemCompare contexts for proportionality
Determine whether a relationship is proportional from a table, graph, equation, or context.
Both situations contain a repeated per-unit rate, so the decisive difference is what happens before any units are purchased or used. We’ll translate each context into a cost rule, substitute …
Preview problemDecide whether a context can continue proportionally
Determine whether a relationship is proportional from a table, graph, equation, or context.
The total includes both a one-time ride fee and a repeated hourly charge, so we must test the whole rule rather than the hourly portion alone. We’ll write total cost …
Preview problemName the feature that determines proportionality
Determine whether a relationship is proportional from a table, graph, equation, or context.
Straightness gives a constant slope, but a proportional graph must also pass through the origin. We’ll identify the point where the line crosses the y-axis, compare that point with zero …
Preview problemScreen a mixed set for proportional relationships
Determine whether a relationship is proportional from a table, graph, equation, or context.
We need the complete set, so each representation must be tested with the shortcut suited to its form. We’ll compare output-to-input ratios in both tables, inspect the equation and context …
Preview problemChoose a smart proportionality test
Determine whether a relationship is proportional from a table, graph, equation, or context.
Because the representation is a table, the decisive question is whether one multiplier carries every x-value to its paired y-value. We’ll evaluate the proposed tests by calculating y divided by …
Preview problemFind k from a table
Find and interpret the constant of proportionality in multiple representations.
In y equals kx, the constant tells how many output units correspond to one input unit, so its calculation is y divided by x. We’ll use one complete table row …
Preview problemFind k from an equation
Find and interpret the constant of proportionality in multiple representations.
The equation is already written in the proportional form y equals k times x. We’ll match each part of the given equation to that template and identify the numerical coefficient …
Preview problemFind k from a graph
Find and interpret the constant of proportionality in multiple representations.
For a proportional line through the origin, k is the vertical output per horizontal input. We’ll use a nonzero plotted point, divide its y-coordinate by its x-coordinate, and then repeat …
Preview problemUse k to fill a missing value
Find and interpret the constant of proportionality in multiple representations.
The known pair determines the one multiplier used by every row of this proportional table. We’ll divide its y-value by its x-value to find k, substitute the target input into …
Preview problemInterpret k in context
Find and interpret the constant of proportionality in multiple representations.
Here y measures dollars and x measures pounds, so k must carry output-per-input units: dollars per pound. We’ll divide the total dollar amount by the matching weight to find the …
Preview problemCompare constants across contexts
Find and interpret the constant of proportionality in multiple representations.
The total payments cover different amounts of time, so comparing thirty-six with fifty would not compare the same quantity. We’ll find each job’s constant by dividing dollars earned by hours …
Preview problemCheck a proposed contextual constant
Find and interpret the constant of proportionality in multiple representations.
Since x is gallons and y is miles, the proposed k must be checked as miles divided by gallons. We’ll calculate that output-to-input quotient from the trip, compare both its …
Preview problemWrite an equation that shows k
Find and interpret the constant of proportionality in multiple representations.
Dollars earned is the output and hours worked is the input, so the requested equation must place d alone and multiply h by k. We’ll divide dollars by hours in …
Preview problemMatch representations by constant of proportionality
Find and interpret the constant of proportionality in multiple representations.
Each card must be both proportional and have output divided by input equal to the target constant. We’ll read the coefficient in proportional equations, calculate y over x from every …
Preview problemChoose a smart first step for finding k
Find and interpret the constant of proportionality in multiple representations.
The constant on a proportional graph describes y-units for each one x-unit, so a valid method must calculate output divided by input. We’ll evaluate the proposed operations against that meaning, …
Preview problemSolve a basic proportion
Solve proportion equations and explain why the setup matches the context.
The two fractions are equal ratios, so their diagonal products must match. We’ll multiply x by the opposite denominator and seven by the opposite numerator, then divide the resulting one-step …
Preview problemSolve a proportion with the unknown in any position
Solve proportion equations and explain why the setup matches the context.
The unknown appears in a denominator, but cross multiplication still pairs the same diagonals. We’ll multiply nine by twenty and fifteen by x, set those products equal, and divide by …
Preview problemSolve an equivalent-ratio equation in varied forms
Solve proportion equations and explain why the setup matches the context.
The numerators correspond to each other, as do the denominators, so the equal ratios can be solved with diagonal products. We’ll multiply x by one hundred and forty by fifteen, …
Preview problemComplete a proportion-solving step
Solve proportion equations and explain why the setup matches the context.
The displayed cross multiplication is already complete, so the task is to finish the ordinary equation ninety equals nine x. We’ll isolate x by dividing both sides by the coefficient …
Preview problemWrite and solve a proportion from context
Solve proportion equations and explain why the setup matches the context.
Let x represent cups of flour, and keep flour over cookies in both ratios so corresponding units stay aligned. We’ll pair three cups with eighteen cookies and x cups with …
Preview problemPick the proportion that matches the context
Solve proportion equations and explain why the setup matches the context.
Each fraction must keep an actual distance paired with its travel time and use miles over hours on both sides. We’ll test the candidate setups against those pairings, then find …
Preview problemValidate a proportion setup
Solve proportion equations and explain why the setup matches the context.
The setup keeps map centimeters over actual kilometers on both sides, so corresponding quantities are aligned before solving. We’ll set the diagonal products equal, divide by the coefficient on x, …
Preview problemTurn a ratio representation into a proportion
Solve proportion equations and explain why the setup matches the context.
Each table row is a paired cups-and-servings relationship, so we’ll write cups over servings for both fractions. Keeping that order places three with five and nine with s, after which …
Preview problemTranslate words into a proportion
Solve proportion equations and explain why the setup matches the context.
This is a translation problem before it is an equation-solving problem. Treat each “is to” phrase as an ordered pair, place the first member of each pair in the same …
Preview problemIdentify all valid setups for one proportion problem
Solve proportion equations and explain why the setup matches the context.
Start by locking in the two matched pairs: each map distance belongs with its real distance. A valid equation may compare map to real, real to map, original to new, …
Preview problemChoose a smart way to solve a proportion
Solve proportion equations and explain why the setup matches the context.
The useful clue is the simple factor connecting the two denominators. Because equivalent ratios scale both positions together, apply that same factor to the matching numerator instead of building a …
Preview problemFind a part from a percent and whole
Use proportional reasoning to solve percent problems without relying on memorized templates alone.
Here the whole is known and the missing quantity is the part. Place that unknown part over the whole, and match it to the given percent written over one hundred. …
Preview problemFind a whole from a part and percent
Use proportional reasoning to solve percent problems without relying on memorized templates alone.
Eighteen is only the known part here, so the unknown whole belongs beneath it in the part-over-whole ratio. Set that equal to thirty over one hundred and solve for the …
Preview problemFind the percent from a part and whole
Use proportional reasoning to solve percent problems without relying on memorized templates alone.
The question gives both the part and the whole, so begin with their ratio in that order. A percent is the equivalent ratio whose denominator is one hundred; here, scaling …
Preview problemSolve a percent proportion with a rotating unknown
Use proportional reasoning to solve percent problems without relying on memorized templates alone.
Use the position of each term to identify what is missing: the known part sits above an unknown whole, while the other ratio records the percent out of one hundred. …
Preview problemModel a percent context with a proportion
Use proportional reasoning to solve percent problems without relying on memorized templates alone.
First decide which group is contained inside the other: the music-club students are the part, and all surveyed students form the whole. Preserve that order in part over whole, then …
Preview problemPick the percent proportion that fits
Use proportional reasoning to solve percent problems without relying on memorized templates alone.
Label the students who walk as the part and the entire club membership as the whole before comparing the two setups. The fitting proportion must preserve part over whole on …
Preview problemValidate a percent proportion in context
Use proportional reasoning to solve percent problems without relying on memorized templates alone.
Read the given equation as two equivalent ratios: part over whole and a matching amount out of one hundred. Solve for the number above one hundred, then attach the percent …
Preview problemTurn a representation into a percent proportion
Use proportional reasoning to solve percent problems without relying on memorized templates alone.
Read the table by meaning before writing any equation: cleaned stations are the part, while all stations are the whole. Put those quantities into part over whole and match that …
Preview problemIdentify all valid percent setups for one problem
Use proportional reasoning to solve percent problems without relying on memorized templates alone.
Anchor the meaning first: twenty-seven is the part, forty-five is the whole, and x names the percent out of one hundred. Different-looking proportions can still be equivalent if their cross …
Preview problem