Once a person is comfortable with multiplying fractions, dividing one fraction by another becomes fairly straightforward.
An alternative to division by any number (not just a fraction) is “multiplying by the reciprocal”. Dividing by two has the same effect as multiplying by one half. Multiplying by the reciprocal of a number will always produce the exact same result as dividing by the original number.
Using this approach, any division problem can be rewritten as a multiplication problem. This applies to fractions as well, which provides us with a useful approach for dividing one fraction by another: When you need to divide one fraction by another, transform the problem into a multiplication problem. Multiply the fraction in the numerator by the reciprocal of the fraction in the denominator. This is often referred to as “invert and multiply”:
Note that as many divisions as you wish from this problem can be rewritten as multiplication by the reciprocal, and the answer will remain the same:
To divide one fraction by another, invert the fraction in the denominator, then multiply.
Now that multiplication and division of fractions are familiar processes, what happens when a quantity is multiplied by the number 1? Nothing. Since one is the multiplicative “identity”, the original quantity will remain unchanged.
This is a very useful property when dealing with fractions. Consider the following:
Two expressions are equivalent if they can be converted into one another. They may look very different initially, but if they are equivalent and the rules of algebra are followed, one can be made to look exactly like the other.
Two quantities are equivalent if they represent the same number.
Both of the above definitions of equivalence work with fractions. Two fractions are equivalent if they represent the same decimal number. Two fractions are also equivalent if they simplify to the same fraction, which means they will also represent the same decimal number.
To create an equivalent fraction, multiply the original fraction by one… in the form of a fraction whose numerator and denominator are equal:
By using different versions of “one” (changing the number used in both the numerator and denominator), this process can create an infinite number of equivalent fractions:
All of the above fractions will simplify to a value of 4, so they are all equivalent fractions.
The concept of equivalent fractions is important to master at this stage, because it is what allows you to transform fractions with different denominators into fractions with the same denominator… something that is necessary before two fractions may be added or subtracted.
Earlier posts in this series:
Algebra Intro 1: Numbers and Variables
Algebra Intro 2: Addition
Algebra Intro 3: Subtraction
Algebra Intro 4: Negative Numbers, Zero, Absolute Values, and Opposites
Algebra Intro 5: Addition, Subtraction, and Terms
Algebra Intro 6: Multiplication
Algebra Intro 7: Properties of Multiplication
Algebra Intro 8: Division
Algebra Intro 9: Fractions, Reciprocals, and Properties of Division
Algebra Intro 10: Fractions and Multiplication
Posts that continue this series:
Algebra Intro 12: Adding and Subtracting Fractions
Great post, Whit. I especially liked the Dividing Fractions section.
When I’ve worked with young children, they’ve called creating equivalent fractions “renaming” fractions (as do some text and activity books for elementary-age students). They also term 2/2, 3/3, etc. “other names for 1”, which comes after students have worked with materials to construct representations of 1 whole using equal parts.
Do you find the language many younger students learn (renaming, other names for 1, and so forth) helps or hinders their developing understanding of equivalencies as older learners?
The idea of “renaming” seems fine to me as long as everyone is clear about *what* is being renamed. Logical candidates would be either the simplest form of the fraction, or the decimal form of the fraction. From this perspective, 4/6 is another name for 2/3, and 8/12 is yet another name for 2/3. This then would presumably lead to the excellent question: how do you know when a fraction is a “renaming” versus a simplest form?
I like the phrase “equivalent fractions” because of its parallelism with “equivalent equations”. In both cases, expressions that look different represent the same quantitative relationship. “Other names for one” is subtly different than “equivalent to one” to my way of thinking. “Phone” is an “other name” for a “telephone”, yet both words refer to an object whose appearance remains unchanged. 2/2 and 3/3 are two abstract expressions which look very different, yet both simplify to the same abstract expression: the number 1. Since there is no physical object involved, I perceive the “renaming” approach to be potentially confusing at some point.
With young children, they are most likely learning all this by rote, and only later connecting some of the dots. So I guess I would be inclined to focus more on the question: what activities and experiences that follow this lesson will help them relate fractions to concrete objects or experiences in their life? If a young student can visualize 3/3, 2/2 or any fraction with equal numerator and denominator as representing a whole (pizza, block box, class of chairs, box of cookies, etc.), and understands from personal experience that each of these is equivalent to the number 1, then it probably does not matter what terminology was used. However, if young students are relying solely on phrases learned by rote, and never gain the personal experience I mentioned, then I do think the “renaming” terminology could be a source of later (mild) confusion.
Upon reflection, I think I would prefer to introduce fractions to young children as vocabulary words that represent scaling quantities: a half, a quarter, a tenth. Once students understand what each of these represents in life (would you rather have half a cookie or a third of a cookie?), I would then introduce more complex fractions in a way that mirrors our language: 3/5 is read as “three fifths”, so I would use multiplication and write (3)(1/5)… and leave the notation 3/5 for much, much later. This would all eventually lead to the question: what happens when I have three thirds, or five fifths? Any time you have a number times its reciprocal, the result will be “a whole”, or one.
Having established all of the above as a conceptual foundation, and given the students much experience with what these words represent in their world, we could eventually work our way around to the notion of equivalent fractions… but I am not certain whether this is a “crucial” concept to master in elementary school. I think of it as more of an enrichment activity (measure a series of objects to the nearest 1/2 of a inch, then measure them to the nearest 1/4 of an inch… what do you notice?). I think a strong “number sense” for what each reciprocal (1/2, 1/3, etc.) represents, along with complete comfort with “three fifths” being three of the thing called a “fifth” is probably more important. In short, I do not see “fractions” as a concept to be mastered… I see “fractions” as problems that involve multiplication by the reciprocal of an integer, which will turn out later on to be a number between 0 and 1. I think the effects of multiplication by whole numbers or their reciprocals are the most important concepts to be able to relate to one’s own real-life experiences for later comfort in math and science courses.