Alfinio Flores who wrote a chapter in the NCTM 2002 Yearbook *Making Sense of Fractions, Ratios, and Proportions*^{1} that provided many intriguing ways fraction division can be interpreted. The most common are measurement and partitive, but there are more. No wonder it’s so challenging to teach!

While there are many interpretations and connections, not all have to be made immediately since fraction concepts and proportional reasoning is developed over many grades (*not just 5*^{th}* grade)*. However, all are important concepts, not just to prepare children for learning algebra but to also build their foundation in proportional reasoning.

Flores wrote, “Some of the connections needed in division of fractions are fractions and quotients, fractions and ratios, division as multiplicative comparisons, reciprocals (inverse elements), and inverse operations.”

### Connections Needed in Division of Fractions

**Fractions and quotients.** That makes sense since all fractions are little division problems. We know that 2/8 is 2 divided by 8.

**Fractions and ratios**. What if 2 out of 8 bananas are brown and too ripe to eat? Ratios can easily be modeled by arrays.

**Division as multiplicative comparisons.** Now we get a bit more complicated. Recall the ‘as much as’ or ‘as many times as’ word problem. (I can’t think of any time in my life, outside a math classroom, that I’ve used those phrases). But, they are needed because they start our children down the path of thinking proportionally. I prefer to think of this in the context of rates. For example, a gold snail moved 2 inches, while the green snail moved 3 inches. The gold snail at moved 2/3^{rd} the speed of the green snail.

**Reciprocals.** I love anything related to identities!! Who doesn’t love the multiplicative identity, one! When you multiply by a number’s reciprocal, you get 1. That’s the best part of simplifying fractions, finding what numbers might “cancel out” because they are one. This is such a powerful tool! And fun!! My second favorite identity: the additive identity, 0. So cool! Then there’s the trig identities…wait I’m going off topic.

**Inverse Operations.** This harkens back to whole number connections where division and multiplication are inverse operations. Same is true with fractions.

### Example Problems Based on Different Interpretations

Each concept can be found in interpretations of fraction division and connected example problems mentioned in a variety of research articles.

They all have different conceptual bases.

*THINK: If each problem was given to children, what algorithms would they build? What pictures would they draw? *

**Measurement:** This asks how many ½’s are in ¾? A concrete problem might be: “You have ¾ cup of flour and for each batch of brownies you need ½ a cup. How many batches can you make?”^{1,2}

**Partitive (equal sharing): **The easiest examples are when a fraction is divided by a whole number, “If you have ½ a candy bar to share among 3 people, how much of a candy bar does each person get?”^{1,2}

**Finding a Whole Number:** Conceptually similar to the partitive model, examples would look like, “If ¾ of a gallon of water fills 1/2 a bucket, how much fills the whole bucket?” ^{3} Or, as a unit rate: “If a printer can print 10 pages in 1/2 a minute, how many can it print in 1 minute?” ^{2}

**Missing Factor:** Going back to the relationship between multiplication and division, “If ½ times a number is equal to ¾, what is the number?”^{1}

**Inverse of an Area Model:** A fancier way to say it is “Inverse of a Cartesian Product,” and example would be, “If the area of a rectangle is 1 ¾
feet long, and the width is ½ foot, find the height of the rectangle.”^{2}

If I had the time, I’d draw a picture for each. Or perhaps you can add them in the comments? For now, I’ll close with a quote from Flores (2002), where he aptly explains the importance of the concept of fraction division and how in elementary grades we are forming the foundation for years of mathematical learning.

“Division of fractions also provides a setting to develop for proportional thinking, which is at the core of mathematics in middle school. “ p. 238

Sources:

1. Flores, Alfinio. (2002). Profound understanding of division of fractions. In *Making Sense of Fractions, Ratios & Proportions* (Eds. Litwiller, B. & Bright, G.) p. 237-246. NCTM, Reston, VA.

2. Sinicrope, R., Mick, H., & Kolb, J. (2002). Fraction division interpretations. In Making Sense of Fractions, Ratios & Proportions (Eds. Litwiller, B. & Bright, G.) p. 153-161. NCTM, Reston, VA.

3. Van de Walle, J. & Lovin, L. (2006). *Teaching Student-Centered Mathematics: Grade 5-8.* Pearson, Boston.

“If the area of a rectangle is 1 ¾ feet long” really? Also, what is “Inverse of an Area”? or “Inverse of a Cartesian Product”, assuming we have the same definition of Cartesian Product. Your rectangle problem, when correctly formulated, becomes an example of Missing factor.

There is a fine line between numerator and denominator. Only a fraction of people will find this statement funny.

Love it!

You might want to try Wolfram Alpha for quick calculations. But as a hint, one of the easiest ways to make sense of this problem is to find a common denominator for both.