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 14 of 18
Each problem type has four distinct practice variants. Open a preview to move among all four.
Read the type of change from a rule
Distinguish additive change from multiplicative change in tables, graphs, and contexts.
Read the rule by asking what operation transforms an input into its output. A fixed amount added to the input and a fixed factor multiplying the input describe different structures, …
Preview problemCompare two growth patterns
Distinguish additive change from multiplicative change in tables, graphs, and contexts.
Analyze the tables independently even though they share the same inputs. For each output column, compute all adjacent differences; if those are not constant, compare adjacent ratios instead. One table’s …
Preview problemRead change type from a story
Distinguish additive change from multiplicative change in tables, graphs, and contexts.
Story language identifies the operation before any starting value is known. Ask whether the same amount is added each day or the current total is multiplied by a factor, and …
Preview problemModel a story with a table and classify it
Distinguish additive change from multiplicative change in tables, graphs, and contexts.
Build the table from the story in chronological order: enter the stated total after round one, then apply the same gain to each newest total for the next round. Once …
Preview problemMatch a story to the right structure
Distinguish additive change from multiplicative change in tables, graphs, and contexts.
Treat three inches per week as accumulated growth, so zero elapsed weeks must correspond to no accumulated increase. Multiply the weekly rate by each week number to generate an independent …
Preview problemTranslate words into a structure you can test
Distinguish additive change from multiplicative change in tables, graphs, and contexts.
The word triples specifies what happens to the current tile count at each new stage. Start with the stated stage-one count, apply that factor successively to the newest total, and …
Preview problemTrack the same structure across different forms
Distinguish additive change from multiplicative change in tables, graphs, and contexts.
Different forms can hide the same kind of change, so apply one invariant test to each representation. For equal one-unit input steps, look for a constant ratio between consecutive outputs; …
Preview problemChoose a smart test for the structure
Distinguish additive change from multiplicative change in tables, graphs, and contexts.
Since the figure numbers advance by equal steps, consecutive output differences are the direct test for a repeated additive change. Compute every difference and require them to agree; if they …
Preview problemRead rate of change from a simple table
Interpret rate of change informally as how much one quantity changes for each change in another.
Rate of change compares matching movements in the two columns, not the raw y-values themselves. Measure how x changes from one row to the next, measure the corresponding y-change over …
Preview problemCompare two points to describe the change
Interpret rate of change informally as how much one quantity changes for each change in another.
The two coordinates must be compared in the same direction, from the first point to the second. Subtract first x from second x and first y from second y separately; …
Preview problemTurn graph movement into a rate statement
Interpret rate of change informally as how much one quantity changes for each change in another.
Translate the graph’s movement into signed changes before forming a rate: right is positive x and up is positive y. Divide the vertical change by the horizontal change so the …
Preview problemMatch a representation to its rate statement
Interpret rate of change informally as how much one quantity changes for each change in another.
Read consecutive rows in one consistent direction and keep each x-change paired with its matching y-change. After finding the signed changes, express vertical change per horizontal change and translate the …
Preview problemInterpret rate of change in a story
Interpret rate of change informally as how much one quantity changes for each change in another.
The words in the story determine all three parts of the rate: magnitude, sign, and units. Use liters as the changing output and hours as the input, attach a negative …
Preview problemModel a context with a table and read the rate
Interpret rate of change informally as how much one quantity changes for each change in another.
Treat the hourly wage as a constant rate multiplying the number of hours. Evaluate that relationship for each requested hour, keep hours first and earnings second in every pair, and …
Preview problemMatch a story to the same rate in another form
Interpret rate of change informally as how much one quantity changes for each change in another.
Raw distance changes cannot be compared until their matching time intervals are included. For each table, subtract two mile values and the corresponding hour values in the same order, divide …
Preview problemMatch the same rate across different forms
Interpret rate of change informally as how much one quantity changes for each change in another.
Representations with different-looking changes can still describe the same rate, so reduce each one to output change per one input unit. Use matching intervals and consistent subtraction order for contexts, …
Preview problemChoose a smart way to compare rates of change
Interpret rate of change informally as how much one quantity changes for each change in another.
Both tables show the same y-change, but that alone does not make their rates equal because the x-intervals differ. Divide each y-change by its own x-change to put both relationships …
Preview problemFind the starting value
Interpret intercepts and starting values in contextual linear situations.
A starting value is tied to the zero input, not to the first nonzero row or the rate of change. Locate the table row where x is zero and read …
Preview problemMatch a representation to the right intercept meaning
Interpret intercepts and starting values in contextual linear situations.
Interpret the point through its coordinate labels before attaching meaning to either number. The first coordinate measures elapsed weeks, so zero identifies the beginning; the second coordinate gives the account …
Preview problemRead the start of a story
Interpret intercepts and starting values in contextual linear situations.
Starting value means the amount already present when the time input is zero. Identify what charge has occurred at month zero and keep it separate from any recurring monthly rate; …
Preview problemFind when a contextual quantity reaches zero
Interpret intercepts and starting values in contextual linear situations.
Model the remaining battery level as the initial percentage minus the hourly loss repeated for each elapsed hour. To find when it is empty, set that changing output equal to …
Preview problemUse a contextual table to read special values
Interpret intercepts and starting values in contextual linear situations.
The missing entry occurs at hour zero, so it is both a starting height and the y-intercept's second coordinate. First use the known rows to find the constant height change …
Preview problemBuild a representation that shows the special value
Interpret intercepts and starting values in contextual linear situations.
A coordinate point can display a starting value only when its first entry represents the input and is set at the beginning. Translate the requested coordinate order into a sentence, …
Preview problemMatch the same special value across forms
Interpret intercepts and starting values in contextual linear situations.
Use one invariant test across every form: a starting value of ten means the output is ten when the input is zero. Check zero-input rows or intercepts directly, substitute zero …
Preview problemChoose a smart way to compare special values
Interpret intercepts and starting values in contextual linear situations.
Month zero isolates each plan’s starting cost, so record each y-intercept from that shared input before comparing anything. Determine which start is larger, then subtract the smaller starting amount from …
Preview problemMatch a table to its rule
Match and compare words, tables, graphs, and simple equations that represent the same relationship.
An equation must capture two independent features of the table: how much y changes for each one-unit change in x, and the y-value when x is zero. Find those pieces …
Preview problemMatch words to numeric evidence
Match and compare words, tables, graphs, and simple equations that represent the same relationship.
Turn the verbal pattern into two requirements before examining the candidate tables: the stated value belongs at step zero, and each move to the next step applies the same signed …
Preview problemJudge whether two forms match
Match and compare words, tables, graphs, and simple equations that represent the same relationship.
Different forms can represent the same relationship, but the check must use shared input values. Substitute each table input into the equation and compare its computed output with the paired …
Preview problemCompare two near-miss relationships
Match and compare words, tables, graphs, and simple equations that represent the same relationship.
Compare like features rather than the most noticeable numbers. Read the equation’s coefficient as its rate and its constant as its zero-input value; calculate the table’s rate from consecutive rows …
Preview problemMatch a story to another form
Match and compare words, tables, graphs, and simple equations that represent the same relationship.
Translate the story into a fixed part plus a repeating part. The signup fee is present at zero visits, while the per-visit charge multiplies the visit count; any matching table, …
Preview problemBuild one form to compare with another
Match and compare words, tables, graphs, and simple equations that represent the same relationship.
Build the context equation from meaning: total cost equals the fixed initial charge plus the hourly rate times hours. Then compare that structure with the proposed equation, remembering that reversing …
Preview problemFind the story that fits the math
Match and compare words, tables, graphs, and simple equations that represent the same relationship.
Read the equation as a story template before considering the situations. The constant gives the amount present when the input is zero, while the coefficient gives the signed amount added …
Preview problemConvert without changing the relationship
Match and compare words, tables, graphs, and simple equations that represent the same relationship.
To preserve the relationship while changing formats, recover the table’s constant rate and its zero-input output as separate features. Divide matched output and input changes for the coefficient, use the …
Preview problemRewrite both forms before comparing
Match and compare words, tables, graphs, and simple equations that represent the same relationship.
Put both representations into the same start-and-rate summary before judging equivalence. In the story, separate the one-time signup fee from the repeating per-visit charge; in the equation, read the constant …
Preview problemFind every equivalent representation in a mixed set
Match and compare words, tables, graphs, and simple equations that represent the same relationship.
Use the target equation as a two-feature fingerprint: its zero-input value and its output change per one input unit. Translate every table, point set, rule, and verbal statement into that …
Preview problemDecide whether a linear pattern is proportional
Distinguish proportional relationships from linear but non-proportional relationships.
Proportionality requires more than a constant additive change. Pair the aligned values, check that y divided by x is constant for every nonzero input, and separately inspect what happens at …
Preview problem