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
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Each problem type has four distinct practice variants. Open a preview to move among all four.
Write and solve a one-step equation from a representation
Solve one-step equations and interpret the solution in context.
A level beam means the total weight on one pan equals the total on the other. Translate the mystery box and three known units as one side of an equation, …
Preview problemTranslate equation words into symbols
Solve one-step equations and interpret the solution in context.
Translate the sentence one relationship at a time: “increased by” describes addition to the unknown, while “is” marks equality with twenty. Once the symbolic equation matches those words, undo the …
Preview problemMatch solutions across several one-step equations
Solve one-step equations and interpret the solution in context.
Use substitution as a uniform test rather than judging equations by appearance. Replace x with four in every labeled equation, evaluate each left side carefully, and compare it with the …
Preview problemChoose the inverse operation and solve
Solve one-step equations and interpret the solution in context.
Identify exactly what has been done to x: eight has been subtracted. The useful first move must reverse that operation on both sides, while repeating subtraction or introducing division cannot …
Preview problemSolve a two-step equation with a constant and coefficient
Solve two-step equations and justify each inverse operation.
Undo the operations in reverse order from how the expression is built. First remove the constant added after the multiplication, using the same subtraction on both sides. Once the variable …
Preview problemSolve a two-step equation with a divided variable
Solve two-step equations and justify each inverse operation.
Work backward through the two operations: the addition occurred after x was divided by four, so undo that addition first. With the quotient isolated, multiplication by four reverses the division. …
Preview problemSolve two-step equations with negatives
Solve two-step equations and justify each inverse operation.
Keep the negative coefficient attached to x throughout the work. Remove the added seven first, then divide both sides by negative two to isolate the variable. The sign rule matters …
Preview problemSolve a two-step context equation
Solve two-step equations and justify each inverse operation.
Separate the total into three equal ticket costs and one fixed service fee. Model the repeated ticket amount with multiplication and add the fee only once. To find one ticket's …
Preview problemSolve a fixed two-step context model
Solve two-step equations and justify each inverse operation.
The six-dollar charge is fixed, while four dollars repeats for every class, so those quantities play different roles in the model. Reverse the cost-building process by removing the one-time charge …
Preview problemMatch a context to the right two-step equation
Solve two-step equations and justify each inverse operation.
Match each story quantity to its role before solving: five dollars repeats per hour, eight dollars is charged once, and thirty-three dollars is the total. A valid model keeps the …
Preview problemWrite and solve a two-step equation from a representation
Solve two-step equations and justify each inverse operation.
Read the level hanger as an equality: three identical unknown boxes and four known units balance nineteen units. Translate the three boxes as three copies of the same variable, not …
Preview problemTranslate a verbal two-step equation
Solve two-step equations and justify each inverse operation.
Build the expression in the order the quantities are related. Start with three times the unknown; “seven less than” means subtract seven from that whole quantity, rather than reversing the …
Preview problemMatch solutions across several two-step equations
Solve two-step equations and justify each inverse operation.
Use the given value as a test input for every equation. Substitute three for each x, honor grouping and operation order, and simplify each entire left side before comparing it …
Preview problemChoose a direct isolation step and solve
Solve two-step equations and justify each inverse operation.
Treat the left side as a product with an outside subtraction applied afterward. The direct way to isolate the product is to undo that outer subtraction on both sides; dividing …
Preview problemSimplify first, then solve
Solve multi-step linear equations, including equations requiring simplification first.
Simplify the equation before applying inverse operations. The two x-terms are like terms, so combine their coefficients while leaving the constant separate. With one variable term remaining, remove the constant …
Preview problemDistribute, simplify, then solve
Solve multi-step linear equations, including equations requiring simplification first.
First remove the parentheses by multiplying two by both terms inside, then combine the resulting constants. That produces a simpler two-step equation with one variable term and one constant. Undo …
Preview problemSolve a signed multi-step equation
Solve multi-step linear equations, including equations requiring simplification first.
Combine the signed variable terms before solving, remembering that the lone x has an understood positive-one coefficient. Keep the resulting negative coefficient attached as you remove the constant and divide. …
Preview problemSolve and check a multi-step equation
Solve multi-step linear equations, including equations requiring simplification first.
Work inward by reversing the equation's construction: undo the outside subtraction, then the multiplication, and finally the addition inside the group. For the required check, return to the untouched original …
Preview problemSolve a multi-step context equation
Solve multi-step linear equations, including equations requiring simplification first.
Model one pack as a grouped quantity containing the unknown granola bars and two juice boxes. Four identical packs repeat that entire group to make twenty-eight items. Divide first to …
Preview problemSolve a fixed multi-step context model
Solve multi-step linear equations, including equations requiring simplification first.
Keep the repeated and fixed seat groups distinct. Each of the three sections repeats both its unknown regular seats and two aisle seats, while the six back seats are added …
Preview problemMatch a context to the right multi-step equation
Solve multi-step linear equations, including equations requiring simplification first.
Build the cost of one bike before accounting for two bikes. One rental combines a two-dollar hourly charge for x hours with one three-dollar equipment fee, and the outer factor …
Preview problemWrite and solve a multi-step equation from a representation
Solve multi-step linear equations, including equations requiring simplification first.
Each dashed outline marks one complete group containing an x-tile and one unit tile. Because the hanger shows two identical groups, the entire grouped quantity repeats twice and balances fourteen …
Preview problemTranslate a verbal multi-step equation
Solve multi-step linear equations, including equations requiring simplification first.
Preserve the named sum before applying “twice.” The unknown and four belong inside one group, and the factor two multiplies that whole group rather than only the variable. Translate the …
Preview problemCheck which transformed equations stay equivalent
Solve multi-step linear equations, including equations requiring simplification first.
Establish a trustworthy chain by distributing through the original group and combining like terms. From there, reordering addends or applying the same inverse operation to both sides preserves the solution …
Preview problemChoose a specified simplification and solve
Solve multi-step linear equations, including equations requiring simplification first.
The specified first move is to remove the parentheses by distribution. Multiply three by both the variable and the constant inside the group; combining x with a constant is not …
Preview problemCollect variable terms on one side
Solve linear equations with variables on both sides and interpret special cases.
Because x appears on both sides, first collect the variable terms without breaking the balance. Subtract the smaller variable term from both sides so one side becomes constant-only. Then move …
Preview problemSimplify before collecting variables
Solve linear equations with variables on both sides and interpret special cases.
Simplify each side before moving anything across the equality. Combine the two like x-terms on the left while keeping the constant separate, then collect all variable terms on one side. …
Preview problemRecognize special solution cases
Solve linear equations with variables on both sides and interpret special cases.
Subtract the identical variable expression from both sides and inspect what remains. If the variables disappear, the resulting constant statement—not an assumed value of zero—determines the solution case. A true …
Preview problemSolve a comparison context
Solve linear equations with variables on both sides and interpret special cases.
Equal plan costs occur where the two cost expressions have the same value for the same number of months. Set those expressions equal, then collect the month terms on one …
Preview problemSolve a fixed both-sides comparison model
Solve linear equations with variables on both sides and interpret special cases.
The comparison needs an equals sign because both expressions describe costs at the same month. Keep each plan's fixed amount and monthly rate together, then move variable terms to one …
Preview problemWrite and solve a both-sides equation from a representation
Solve linear equations with variables on both sides and interpret special cases.
The balanced scale turns each tray into a complete expression: two mystery boxes plus three units on one side, and one box plus eight units on the other. Set those …
Preview problemTranslate a comparison statement into an equation
Solve linear equations with variables on both sides and interpret special cases.
The word “equals” connects two complete expressions built from the same unknown. Keep each verbal group intact, join the expressions with an equals sign, and collect the variable terms on …
Preview problemMatch equations to solutions or cases
Solve linear equations with variables on both sides and interpret special cases.
Simplify every equation far enough to see what remains after like variable terms are collected. A remaining variable equation gives one value, a true constant statement works for every value, …
Preview problemChoose a specified both-sides simplification
Solve linear equations with variables on both sides and interpret special cases.
The requested first move has a specific job: reduce the variable terms while keeping the equation balanced. Look for an operation applied to both sides that cancels one of the …
Preview problemWrite and solve an additive word-problem equation
Write and solve a linear equation from a short verbal or contextual situation.
This is a start-plus-change-equals-total story. Let the unknown represent Mia’s starting number, add the seven stickers she received, and set that sum equal to the final total of eighteen. Undo …
Preview problemWrite and solve a multiplicative word-problem equation
Write and solve a linear equation from a short verbal or contextual situation.
Four equal boxes mean four copies of the same unknown per-box amount, so the relationship is multiplicative rather than additive. Represent one box by a variable and make four of …
Preview problem