Question

Ms. White's students are working on generating equivalent fractions like the ones shown. She asks her students to write a set of instructions for how to generate equivalent fractions. One student writes, "You have to multiply the bottom and the top of the fraction by a number."

Which of the following revisions most improves the student statement in terms of validity and generalizability?

A. You have to multiply both denominator and numerator by the same number.
B. You have to multiply both denominator and numerator by the same nonzero number.
C. You have to multiply both denominator and numerator by the same whole number.
D. You have to multiply both denominator and numerator by the same positive whole number.

Answer 1

To generate equivalent fractions, one must multiply both the numerator and denominator by the same nonzero integer, keeping the fraction's value the same. The operation must be identical for both to maintain the equivalence of the fraction.

Step-by-step explanation:

To generate equivalent fractions, one must multiply both the numerator (the top number) and the denominator (the bottom number) of a fraction by the same nonzero integer. This process does not change the value of the fraction, as you are effectively multiplying by 1 (any number divided by itself is 1). For instance, if we take the fraction 1/2 and multiply both the numerator and denominator by 2, we get 2/4, which is an equivalent fraction to 1/2. Similarly, multiplying both by 3 gives us 3/6, another equivalent fraction.

Multiplying fractions themselves is done by multiplying their numerators together and their denominators together. It is crucial to remember that when multiplying fractions, we do not add denominators. Simplifying the result by reducing common factors may be necessary to reach the simplest form of the new fraction. Following these rules will help maintain the equality of fractions.

It's important to note that the operation must be the same for both numerator and denominator to ensure we maintain an equivalent fraction. This principle ties back to the concept that an operation performed on both sides of an equals sign maintains balance and equality. This is why multiplying by the same positive whole number for numerator and denominator is essential. For example, multiplying the numerator and denominator of 1/3 by 4 would yield 4/12, which simplifies to 1/3, demonstrating the fraction's equivalency.