6.2.e 6.3.a MA.6.2.2 6.3.b

Working with Fractions 6th Grade

Texas Essential Knowledge and Skills (TEKS): 6.2.e
extend representations for division to include fraction notation such as a/b represents the same number as a ÷ b where b ≠ 0.
Texas Essential Knowledge and Skills (TEKS): 6.3.a
recognize that dividing by a rational number and multiplying by its reciprocal result in equivalent values;
Texas Essential Knowledge and Skills (TEKS): 6.3.b
determine, with and without computation, whether a quantity is increased or decreased when multiplied by a fraction, including values greater than or less than one;
Florida - Benchmarks for Excellent Student Thinking: MA.6.2.2
Extend previous understanding of multiplication and division to compute products and quotients of positive fractions by positive fractions,including mixed numbers,with procedural fluency.

A fraction can be interpreted as a division problem
where the numerator is divided by the denominator.

In this case, the dividend is 1, and the divisor is 7.
So, 1 ÷ 7 is equal to

\frac{1}{7}

In this case, the numerator is 3, and the denominator is 4.
So,

\frac{3}{4}

is equal to 3 ÷ 4.

Dividing by a fraction is the same as multiplying by its reciprocal.

The reciprocal of a fraction can be found by
switching the numerator and the denominator.

The reciprocal of

\frac{3}{4}

\frac{4}{3}

, so dividing by

\frac{3}{4}

is the same as multiplying by

\frac{4}{3}

12 \div \frac{3}{4} = 12 \times \frac{4}{3}

EFFECTS OF MULTIPLICATION

Does the operation of multiplication on a number result in a larger number?
The answer is sometimes. Let's look at some cases.

When a number is multiplied by a fraction equivalent to 1,
the product is equal to the number.

Example:	$3$ × $\frac{2}{2}$ = $\frac{6}{2}$ = $3$
Example:	$\frac{1}{3}$ × $\frac{4}{4}$ = $\frac{4}{12}$ = $\frac{1}{3}$

When a positive whole number is multiplied by a fraction greater than 1,
the product is greater than the whole number.

Example:

3

1 \frac{1}{4}

3

\frac{5}{4}

\frac{15}{4}

3 \frac{3}{4}

When a positive whole number is multiplied by a fraction between 0 and 1,
the product is less than the whole number.

Example:	$3$ × $\frac{1}{3}$ = $\frac{3}{3}$ = $1$
Example:	$3$ × $\frac{1}{6}$ = $\frac{3}{6}$ = $\frac{1}{2}$

When a number greater than 1 is multiplied by a number greater than 1,
the product is greater than both numbers.

Example:	$3$ × $2$ = $6$
Example:	$3$ × $2 \frac{1}{3}$ = $3$ × $\frac{7}{3}$ = $\frac{21}{3}$ = $7$
Example:	$1 \frac{1}{2}$ × $1 \frac{1}{3}$ = $\frac{3}{2}$ × $\frac{4}{3}$ = $\frac{12}{6}$ = $2$

Working with Fractions

Texas Essential Knowledge and Skills (TEKS): 6.2.e

extend representations for division to include fraction notation such as a/b represents the same number as a ÷ b where b ≠ 0.

Texas Essential Knowledge and Skills (TEKS): 6.3.a

recognize that dividing by a rational number and multiplying by its reciprocal result in equivalent values;

Texas Essential Knowledge and Skills (TEKS): 6.3.b

determine, with and without computation, whether a quantity is increased or decreased when multiplied by a fraction, including values greater than or less than one;

Florida - Benchmarks for Excellent Student Thinking: MA.6.2.2

Extend previous understanding of multiplication and division to compute products and quotients of positive fractions by positive fractions,including mixed numbers,with procedural fluency.

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