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.
Simplify to lowest terms
Generate, compare, and simplify equivalent fractions in numeric and contextual settings.
Lowest terms means the numerator and denominator share no factor greater than one. We’ll find the greatest common factor of eighteen and twenty-four, divide both parts by that same number, …
Preview problemFind the missing value in an equivalent fraction
Generate, compare, and simplify equivalent fractions in numeric and contextual settings.
Equivalent fractions scale both positions by the same multiplier. We’ll compare the denominators seven and twenty-one to find that multiplier, apply it to the numerator four, and check the completed …
Preview problemTest whether two fractions are equivalent
Generate, compare, and simplify equivalent fractions in numeric and contextual settings.
Different numerators and denominators can still name the same value. We’ll simplify six eighths by a common divisor and compare the result with three fourths, then use equal cross-products as …
Preview problemMatch the same part-whole amount in context
Generate, compare, and simplify equivalent fractions in numeric and contextual settings.
This is a part-to-whole comparison in a larger class. We’ll place the fifteen students over the total of twenty-five, then reduce that fraction and six tenths to the same simplest …
Preview problemSimplify a contextual fraction
Generate, compare, and simplify equivalent fractions in numeric and contextual settings.
First translate the story into a part-to-whole fraction, with twelve selected students above eighteen total students. Then we’ll divide both parts by their greatest common factor and check that the …
Preview problemMatch an equivalent contextual fraction
Generate, compare, and simplify equivalent fractions in numeric and contextual settings.
Each situation can be translated into a part-over-whole fraction. We’ll compare those values with two thirds by simplifying when a common factor is obvious and using cross-products otherwise. Matching just …
Preview problemCheck an equivalence claim
Generate, compare, and simplify equivalent fractions in numeric and contextual settings.
Fraction equivalence depends on value, not whether the written numerals match. We’ll simplify nine twelfths by dividing both parts by their greatest common factor and compare the result with three …
Preview problemTranslate a fraction model into equivalent symbols
Generate, compare, and simplify equivalent fractions in numeric and contextual settings.
Each strip must be read on its own: shaded parts over total equal parts. We’ll count those two quantities for the six-part strip and again for the twelve-part strip, then …
Preview problemTranslate verbal fraction descriptions into equivalent symbols
Generate, compare, and simplify equivalent fractions in numeric and contextual settings.
The words “eight of twelve” tell us part over whole, so the selected slices belong in the numerator and all equal slices in the denominator. We’ll write that fraction, divide …
Preview problemFilter a set for equivalent fractions
Generate, compare, and simplify equivalent fractions in numeric and contextual settings.
To find every match, we need a common reference value. We’ll reduce four sixths to lowest terms, simplify each fraction in the list or test it by cross-products, and keep …
Preview problemChoose the right equivalence-check strategy
Generate, compare, and simplify equivalent fractions in numeric and contextual settings.
To judge the two proposals, we’ll test each one against the definition of equivalent fractions. For each strategy, we’ll determine what evidence it actually provides about value, then confirm the …
Preview problemAdd or subtract with a shared denominator
Add and subtract fractions and mixed numbers with appropriate common denominators.
Both addends are already measured in eighths, so the pieces have the same size. We’ll keep that unit, add only the counts in the numerators, and then check whether the …
Preview problemUse a common denominator first
Add and subtract fractions and mixed numbers with appropriate common denominators.
Thirds and sixths are different-sized pieces, so we must rename them before adding. We’ll use six as the common denominator, scale one third to an equivalent number of sixths, combine …
Preview problemAdd mixed numbers
Add and subtract fractions and mixed numbers with appropriate common denominators.
Mixed-number addition is easier when whole and fractional parts are organized separately. We’ll add the whole numbers, combine the fourths, and check whether the fractional sum makes another whole. If …
Preview problemSubtract fractions or mixed numbers with regrouping
Add and subtract fractions and mixed numbers with appropriate common denominators.
The fractional parts must first use the same unit, so we’ll rename the fourths as eighths. Because the top fractional part is then too small to subtract five eighths, we’ll …
Preview problemAdd contextual fractional quantities
Add and subtract fractions and mixed numbers with appropriate common denominators.
The story asks for a combined cup measure, so we’ll add the two ingredient amounts and keep the unit attached. We’ll rename the third in sixths, combine like-sized sixths, and …
Preview problemSubtract contextual fractional amounts
Add and subtract fractions and mixed numbers with appropriate common denominators.
The ribbon left is the starting length minus the cut length. We’ll express both fractional parts in fourths, trade one whole foot for four fourths because two fourths is too …
Preview problemWork with mixed numbers in context
Add and subtract fractions and mixed numbers with appropriate common denominators.
We’ll combine the morning and afternoon distances by separating whole miles from fractional miles. The fourth and third parts need a common denominator, so we’ll rename both in twelfths, add …
Preview problemCheck mixed-number regrouping
Add and subtract fractions and mixed numbers with appropriate common denominators.
We’ll inspect the student’s work one move at a time rather than judging only the final line. After confirming the conversion to sixths, we’ll compare the two fractional parts, regroup …
Preview problemTranslate a fraction model into an operation
Add and subtract fractions and mixed numbers with appropriate common denominators.
Each bar shows a count of shaded parts out of eight equal parts. We’ll read the two fractions separately, use the plus sign to translate the model as combining rather …
Preview problemTranslate verbal fraction situations into equations
Add and subtract fractions and mixed numbers with appropriate common denominators.
The recipe uses both broth amounts, so the word total signals addition. We’ll write the equation with cups, rename fourths and thirds as twelfths, and combine those equal-sized units. If …
Preview problemMatch fraction operations by their final value
Add and subtract fractions and mixed numbers with appropriate common denominators.
The target’s simplified value will serve as the reference for the entire list. We’ll evaluate it first, then calculate and reduce each candidate expression independently before comparing values. Checking all …
Preview problemChoose the right fraction-operation strategy
Add and subtract fractions and mixed numbers with appropriate common denominators.
A common denominator must be divisible by both six and four, and the least one is the smallest value that passes both tests. We’ll test the three proposed denominators, scale …
Preview problemMultiply two fractions
Multiply and divide fractions and mixed numbers and interpret the result in context.
Fraction multiplication uses numerator times numerator and denominator times denominator, so no common denominator is needed. We’ll multiply straight across, reduce the resulting fraction by its greatest common factor, and …
Preview problemDivide by a fraction using the reciprocal
Multiply and divide fractions and mixed numbers and interpret the result in context.
To divide by a fraction, we keep the dividend and multiply by the reciprocal of the divisor. We’ll invert only four fifths, multiply the resulting fractions, and reduce the quotient …
Preview problemMultiply with mixed numbers
Multiply and divide fractions and mixed numbers and interpret the result in context.
A mixed number must be converted to one fraction before multiplication. We’ll rewrite one and one half as an improper fraction, multiply it by two thirds, and simplify matching factors …
Preview problemDivide mixed numbers
Multiply and divide fractions and mixed numbers and interpret the result in context.
We’ll first convert the mixed-number dividend into an improper fraction. Then we’ll keep that fraction unchanged, replace division by three eighths with multiplication by its reciprocal, and cancel common factors …
Preview problemFind a fraction of an amount
Multiply and divide fractions and mixed numbers and interpret the result in context.
Here, “three fourths of” the full yogurt amount signals multiplication. We’ll multiply three fourths by two thirds cup, interpret the numerator and denominator as an overlap in a fourths-by-thirds area …
Preview problemScale a quantity by a fraction
Multiply and divide fractions and mixed numbers and interpret the result in context.
Making two thirds of a batch means scaling the entire full-batch flour amount, not adding to it or scaling only one part. We’ll convert one and one half cups to …
Preview problemInterpret fraction division in context
Multiply and divide fractions and mixed numbers and interpret the result in context.
The question asks how many three-fourths-cup groups fit inside the total broth, so the structure is total divided by serving size. We’ll convert two and one fourth to fourths, multiply …
Preview problemCheck fraction-division reasoning
Multiply and divide fractions and mixed numbers and interpret the result in context.
We’ll check first which fraction the student inverted, because fraction division keeps the dividend and takes the reciprocal of only the divisor. Then we’ll rewrite the division correctly, multiply, simplify, …
Preview problemTranslate a fraction model into multiplication or division
Multiply and divide fractions and mixed numbers and interpret the result in context.
The two shadings divide one rectangle into fourths by columns and thirds by rows. We’ll identify the crosshatched overlap as a fraction of a fraction, translate that relationship into multiplication, …
Preview problemTranslate verbal fraction situations into equations
Multiply and divide fractions and mixed numbers and interpret the result in context.
This is a how-many-groups question: the total four cups is divided by the two-thirds-cup size of each scoop. We’ll keep that order, multiply by the reciprocal of two thirds, and …
Preview problemMatch fraction operations by their final value
Multiply and divide fractions and mixed numbers and interpret the result in context.
We’ll simplify the target product first so it becomes the reference value for every expression. Then we’ll evaluate each multiplication or division independently, using reciprocals for the division cases, reduce …
Preview problemChoose the right fraction multiplication or division strategy
Multiply and divide fractions and mixed numbers and interpret the result in context.
Because one and one half cups makes only three fifths of a batch, reaching one full batch requires scaling up by division. We’ll convert the mixed amount to an improper …
Preview problemCompare two rational numbers in different forms
Compare and order rational numbers written as fractions, decimals, and percents.
Fractions and decimals become comparable once they use the same form. We’ll convert three fourths to a decimal, align it with zero point seven two by place value, and compare …
Preview problem