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Multiplying Fractions
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Foundational skills
Recognise that fractions can always be multiplied
\[ \frac{2}{3} \times \frac{4}{5} \text{ directly?} \]
Understand that multiplying fractions does not require a common denominator.
Convert a mixed number to an improper fraction
\[ 3\frac{2}{5} \rightarrow \frac{17}{5} \]
Convert a mixed number to an improper fraction in preparation for multiplication.
Convert an improper fraction to a mixed number
\[ \frac{17}{5} \rightarrow 3\frac{2}{5} \]
Convert an improper fraction to a mixed number.
Write an integer as a fraction
\[ 5 = \frac{5}{1} \]
Express a whole number as a fraction with denominator 1.
Identify if cross-cancelling is possible
\[ \frac{2}{3} \times \frac{9}{4} \text{ cancel?} \]
Decide whether two fractions share a common factor that allows simplifying before multiplying.
Integer × fraction
Integer × unit fraction
\[ 4 \times \frac{1}{5} = \text{?} \]
Multiply a whole number by a unit fraction.
Integer × proper fraction — proper result
\[ 2 \times \frac{1}{5} = \text{?} \]
Multiply a whole number by a proper fraction where the answer is less than one.
Integer × proper fraction — improper result
\[ 4 \times \frac{2}{5} = \text{?} \]
Multiply a whole number by a proper fraction where the answer is greater than one.
Integer × proper fraction — with cross-cancelling
\[ 6 \times \frac{2}{3} = \text{?} \]
Multiply a whole number by a proper fraction where the integer and denominator share a common factor.
Integer × improper fraction
\[ 3 \times \frac{7}{4} = \text{?} \]
Multiply a whole number by an improper fraction.
Integer × mixed number
\[ 3 \times 2\frac{1}{4} = \text{?} \]
Multiply a whole number by a mixed number.
Proper × proper fraction
Unit fraction × unit fraction
\[ \frac{1}{3} \times \frac{1}{4} = \text{?} \]
Multiply two unit fractions together.
Proper × proper — no simplifying needed
\[ \frac{2}{3} \times \frac{4}{5} = \text{?} \]
Multiply two proper fractions where the answer does not simplify.
Proper × proper — answer simplifies
\[ \frac{2}{5} \times \frac{5}{6} = \text{?} \]
Multiply two proper fractions where cross-cancelling is possible and simplifies the answer.
Proper × proper — cross-cancelling with one pair
\[ \frac{3}{4} \times \frac{2}{9} = \text{?} \]
Multiply two proper fractions where one numerator-denominator pair can be cancelled.
Proper × proper — cross-cancelling with two pairs
\[ \frac{3}{4} \times \frac{8}{9} = \text{?} \]
Multiply two proper fractions where two numerator-denominator pairs can be cancelled.
Involving improper fractions
Proper × improper — no simplifying
\[ \frac{2}{3} \times \frac{7}{4} = \text{?} \]
Multiply a proper fraction by an improper fraction where no simplifying is needed.
Proper × improper — with cross-cancelling
\[ \frac{2}{3} \times \frac{9}{4} = \text{?} \]
Multiply a proper fraction by an improper fraction where cross-cancelling is possible.
Improper × improper — no simplifying
\[ \frac{5}{3} \times \frac{7}{4} = \text{?} \]
Multiply two improper fractions where no simplifying is needed.
Improper × improper — with cross-cancelling
\[ \frac{5}{4} \times \frac{8}{3} = \text{?} \]
Multiply two improper fractions where cross-cancelling is possible.
Involving mixed numbers
Mixed number × unit fraction
\[ 2\frac{1}{3} \times \frac{1}{4} = \text{?} \]
Multiply a mixed number by a unit fraction.
Mixed number × proper fraction — no simplifying
\[ 2\frac{1}{3} \times \frac{2}{5} = \text{?} \]
Multiply a mixed number by a proper fraction where no cross-cancelling is possible.
Mixed number × proper fraction — with cross-cancelling
\[ 2\frac{1}{4} \times \frac{2}{3} = \text{?} \]
Multiply a mixed number by a proper fraction where cross-cancelling is possible.
Mixed number × improper fraction
\[ 2\frac{1}{3} \times \frac{5}{4} = \text{?} \]
Multiply a mixed number by an improper fraction.
Mixed number × mixed number — no simplifying
\[ 2\frac{1}{3} \times 3\frac{1}{4} = \text{?} \]
Multiply two mixed numbers where no cross-cancelling is possible after conversion.
Mixed number × mixed number — with cross-cancelling
\[ 2\frac{1}{4} \times 2\frac{2}{3} = \text{?} \]
Multiply two mixed numbers where cross-cancelling is possible after conversion.
Special cases
Fraction × 1
\[ \frac{3}{5} \times 1 = \text{?} \]
Multiply a fraction by one to show the identity property.
Squaring a unit fraction
\[ \left( \frac{1}{4} \right)^2 = \text{?} \]
Square a unit fraction.
Squaring a proper fraction
\[ \left( \frac{2}{3} \right)^2 = \text{?} \]
Square a proper fraction.
Squaring an improper fraction
\[ \left( \frac{3}{2} \right)^2 = \text{?} \]
Square an improper fraction.
Squaring a mixed number
\[ \left( 1\frac{1}{2} \right)^2 = \text{?} \]
Square a mixed number.
Fraction × its reciprocal
\[ \frac{3}{5} \times \frac{5}{3} = \text{?} \]
Multiply a fraction by its reciprocal to get one.
Multiplying three fractions
\[ \frac{1}{2} \times \frac{2}{3} \times \frac{3}{4} = \text{?} \]
Multiply three fractions together.
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