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.
Use commutative structure to reorder
Use commutative, associative, and distributive structure to rewrite and simplify numerical expressions efficiently.
The goal is to make the addition friendlier without changing any addend. We’ll use the commutative property to move the two numbers that combine to a round hundred next to …
Preview problemUse associative structure to regroup
Use commutative, associative, and distributive structure to rewrite and simplify numerical expressions efficiently.
This product becomes easier when two of its factors are combined first. We’ll use the associative property to place parentheses around the pair that makes a friendly hundred, while keeping …
Preview problemDistribute across a sum or difference
Use commutative, associative, and distributive structure to rewrite and simplify numerical expressions efficiently.
The outside factor applies to the entire sum, so distribution must create one product for each term inside the parentheses. We’ll multiply six by both addends, keep the plus sign …
Preview problemChoose the best property-based rewrite
Use commutative, associative, and distributive structure to rewrite and simplify numerical expressions efficiently.
Ninety-nine is close to a friendlier factor, one hundred, but using that factor adds one extra group of seven. We’ll rewrite ninety-nine as one hundred minus one, distribute the multiplication …
Preview problemRewrite a contextual calculation efficiently
Use commutative, associative, and distributive structure to rewrite and simplify numerical expressions efficiently.
The story is about equal groups, and the product becomes easier if we pair the packs. We’ll rewrite eight as four times two, regroup so two multiplies twenty-five first, and …
Preview problemUse a common factor in context
Use commutative, associative, and distributive structure to rewrite and simplify numerical expressions efficiently.
Both clubs pay the same price for every sketchbook, so that unit price is a common factor. We’ll write each club’s cost as quantity times price, factor the shared price …
Preview problemFactor a common contextual rate or group count
Use commutative, associative, and distributive structure to rewrite and simplify numerical expressions efficiently.
The two ticket groups share the same price per ticket. We’ll write their revenues as two products, factor the common ticket price, add the ticket counts inside parentheses, and then …
Preview problemRecognize an equivalent structural rewrite
Use commutative, associative, and distributive structure to rewrite and simplify numerical expressions efficiently.
The outside factor multiplies the entire sum, so an equivalent distributed form needs two product terms. We’ll multiply seven by each addend, keep the addition between those products, and stop …
Preview problemTranslate a shortcut into symbols
Use commutative, associative, and distributive structure to rewrite and simplify numerical expressions efficiently.
The shortcut works by breaking one factor into smaller factors and regrouping them. We’ll replace sixteen with four times four, use the associative property to pair one four with twenty-five, …
Preview problemCheck rewrite validity and use a specified target structure
Use commutative, associative, and distributive structure to rewrite and simplify numerical expressions efficiently.
A rewrite is valid only if it preserves the original product. We’ll check whether each proposed form follows a true factor split, reject any form that changes a factor’s value, …
Preview problemSelect the best structural model
Use commutative, associative, and distributive structure to rewrite and simplify numerical expressions efficiently.
Nineteen is one less than the friendly multiple twenty, so using twenty creates one extra group of six. We’ll calculate the easier twenty-group product and subtract that extra group. This …
Preview problemList all factors of a number
Find factors, multiples, common factors, and common multiples to solve numerical and contextual problems.
To list every factor without omissions, we’ll search for whole-number pairs whose product is thirty-six. We’ll test possible small factors in order, record both members of each successful pair, and …
Preview problemWork with multiples of a number
Find factors, multiples, common factors, and common multiples to solve numerical and contextual problems.
A multiple of nine must equal nine times a whole number. We’ll divide fifty-four by nine and check whether the quotient is whole with no remainder, then multiply back to …
Preview problemFind common factors or the GCF
Find factors, multiples, common factors, and common multiples to solve numerical and contextual problems.
A common factor must divide both numbers evenly, and the greatest one is the largest shared divisor. We’ll build the factor sets for twenty-four and thirty-six, identify their overlap, and …
Preview problemFind common multiples or the LCM
Find factors, multiples, common factors, and common multiples to solve numerical and contextual problems.
The least common multiple is the first positive value reached by both counting patterns. We’ll list multiples of six and eight in increasing order, compare the lists as soon as …
Preview problemUse factors for equal-group arrangements
Find factors, multiples, common factors, and common multiples to solve numerical and contextual problems.
Every rectangular desk arrangement corresponds to a factor pair whose product is twenty-four. We’ll test row counts in order, pair each divisor with the number of desks per row, and …
Preview problemUse common factors in a grouping context
Find factors, multiples, common factors, and common multiples to solve numerical and contextual problems.
Making identical bags with nothing left over means the bag count must divide both supply totals. Because the teacher wants as many bags as possible, we’ll compare the factors of …
Preview problemUse common multiples in repeating schedules
Find factors, multiples, common factors, and common multiples to solve numerical and contextual problems.
Each light’s future flash times are multiples of its own interval. Since they start together and we need the first new match, we’ll list the six-second and eight-second schedules in …
Preview problemCheck a common-factor or common-multiple claim
Find factors, multiples, common factors, and common multiples to solve numerical and contextual problems.
A number can be a common multiple without being the least common multiple. We’ll test the student’s claim by listing multiples of four and six in increasing order, locating the …
Preview problemTranslate a representation into factors
Find factors, multiples, common factors, and common multiples to solve numerical and contextual problems.
Each tile array turns a picture into a multiplication fact: rows times columns equals twenty-four. We’ll read both dimensions from every rectangle, collect both members of each factor pair, remove …
Preview problemTranslate shared-number language into arithmetic
Find factors, multiples, common factors, and common multiples to solve numerical and contextual problems.
The phrase “divides both evenly” tells us to look for common factors, and “greatest” tells us where to stop. We’ll list the factors of eighteen and thirty, compare the overlap, …
Preview problemTest candidates against factor and multiple conditions
Find factors, multiples, common factors, and common multiples to solve numerical and contextual problems.
The two phrases describe the same test from opposite directions: the number must be divisible by both six and nine. We’ll translate each requirement into a whole-number multiplication or division …
Preview problemChoose between factor and multiple reasoning in context
Find factors, multiples, common factors, and common multiples to solve numerical and contextual problems.
This story splits two fixed totals into the greatest number of identical groups, so the group count must divide both totals rather than mark a repeated meeting time. We’ll compare …
Preview problemCheck divisibility with one rule
Use divisibility rules, prime factorization, and factor structure to analyze and solve whole-number problems.
The rule for divisibility by three turns a large division problem into a digit-sum check. We’ll add the digits of three hundred forty-two, test whether that sum is divisible by …
Preview problemClassify a number as prime or composite
Use divisibility rules, prime factorization, and factor structure to analyze and solve whole-number problems.
Prime or composite classification depends on factor count, so we only need to test possible divisors up to the square root. We’ll bound the search for twenty-nine, check the relevant …
Preview problemFind a prime factorization
Use divisibility rules, prime factorization, and factor structure to analyze and solve whole-number problems.
A prime factorization is complete only when every branch ends in a prime. We’ll split eighty-four into a convenient factor pair, keep breaking apart any composite branch, collect the prime …
Preview problemCombine divisibility clues and factor structure
Use divisibility rules, prime factorization, and factor structure to analyze and solve whole-number problems.
Testing nine possible divisors one by one is clearer if we first expose the prime structure of three hundred sixty. We’ll factor the target number, compare each listed number’s prime …
Preview problemCheck a no-leftovers context
Use divisibility rules, prime factorization, and factor structure to analyze and solve whole-number problems.
No leftovers means the total must be divisible by the box size. We’ll apply the digit-sum rule for nine to three hundred twenty-four, then use direct division to find whether …
Preview problemUse prime factor structure for specified numeric outputs
Use divisibility rules, prime factorization, and factor structure to analyze and solve whole-number problems.
The two requested results come from the same factor pair. We’ll split seventy-two into eight times the unknown row length, decompose both factors completely into primes, and count repeated primes …
Preview problemFilter contextual choices by factor structure
Use divisibility rules, prime factorization, and factor structure to analyze and solve whole-number problems.
There are three conditions, so we’ll filter the shipment sizes in a deliberate order. First keep even values, then apply the digit-sum rule for divisibility by three, and finally confirm …
Preview problemCheck a prime-structure claim
Use divisibility rules, prime factorization, and factor structure to analyze and solve whole-number problems.
A correct product is not automatically a complete prime factorization. We’ll verify that eight times nine gives seventy-two, check whether each displayed factor is prime, and continue decomposing any composite …
Preview problemTranslate a structure representation into standard form
Use divisibility rules, prime factorization, and factor structure to analyze and solve whole-number problems.
The information we need sits at the ends of the factor tree, not at its first split. We’ll follow both composite branches until the leaves are prime, collect every leaf, …
Preview problemApply a named whole-number structure conclusion
Use divisibility rules, prime factorization, and factor structure to analyze and solve whole-number problems.
The phrase “digits add to twenty-seven” points directly to the divisibility rule for nine. We’ll test whether the given digit sum is a multiple of nine and transfer that remainder …
Preview problemFilter candidates by number-structure conditions
Use divisibility rules, prime factorization, and factor structure to analyze and solve whole-number problems.
We can handle the three conditions as filters instead of testing everything at once. We’ll first keep numbers divisible by six, use digit sums to remove those divisible by nine, …
Preview problemChoose a specified whole-number structure tool
Use divisibility rules, prime factorization, and factor structure to analyze and solve whole-number problems.
The requested divisors all have quick tests, so full prime factorization would add work without helping. We’ll use the last digit of eight thousand one hundred ninety to test two, …
Preview problemGenerate equivalent fractions
Generate, compare, and simplify equivalent fractions in numeric and contextual settings.
Equivalent fractions come from multiplying by a form of one. We’ll apply each given scale factor to both the numerator and denominator of three fifths, keeping the requested order, and …
Preview problem