Most of us were taught dividing with fractions using algorithms – i) change the division to a multiplication and ii) invert the numerator and denominator (invert) of the divisor. This works for both dividing with fraction and with whole numbers. However this sole reliance on algorithm causes two potential problems.
Firstly, kids may become confused about the algorithm itself. Most often, they invert the dividend instead of the divisor, e.g.
Secondly, not been exposed to any conceptual modeling of the algorithm, many kids have difficulty with word problems that requires dividing with fractions. They usually do not have difficulty writing the fraction division equation from the word problem (CCSS 5.NF.7c). However, they may have trouble coming up with examples of word problems or scenario for which a particular fraction division equation is applicable.
Conceptual Modeling of Dividing with Fractions
Before fractions, division with whole numbers involves two different interpretation that are converse of each other, e.g. 12 ÷ 3 can be interpreted as “the number of units in each group when 12 units are split into 3 groups” (scenario 1) and “the number of groups when 12 units are split into groups of 3” (scenario 2). In both scenarios, the answer is the same (2), hence, many kids treat them as equivalent, or at best two equally correct way of modeling division.
Division ConceptsHowever, when it comes to fractions, these two scenarios correspond to different models.
Scenario 1 corresponds to cases when the divisor is a whole number. For example ½ ÷ 8 can be modeled by “half a pizza shared by 8 persons (1/16 of a pizza each)” or “half a pound of chocolate melted into 8 cups (1/16 pound in each cup)”.
Scenario 2 corresponds to cases when the divisor is a fraction. For example 2 ÷ ¼ can be modeled by “two whole pizzas cut into quarters (8 pieces)” and ½ ÷ ¼ can be modeled by “½ gallon of milk poured into ¼ gallon bottles (2 bottles)”
When kids are shown examples of each scenarios systematically and are able to categorize them internally, they would be able to come up with examples without any confusion, and are less likely to make mistakes when they interpret word problems which requires dividing with fractions.
Progression of Teaching
1) Whole number divide by whole number
Example 3 ÷ 4 = 3/4
Each girl will have 3/4 of a pizza.
and 4 ÷ 3 = 4/3
Each girl will have 1 and 1/3 of a pizza.
2) Fraction divide by whole number (CCSS 5.NF.7a)
Example, ½ ÷ 8 = 16
Half a pizza divided among 8 girls. Each girl will have 1/16 of the pizza.
3) Whole number divide by fraction (CCSS 5.NF.7b)
Example, 4 ÷ ⅛.
There are 4 pizzas. Each slice is 1/8 of a pizza. There are 32 slices.
4) Fraction divide by fraction (CCSS 6.NS.1)
Example, 3/4 ÷ ⅛
There is only 3/4 of a pizza left. Each slice is 1/8 of a pizza. There are 6 slices.
The most easily confused cases are 2) and 4), i.e. students have the mostly difficulty drawing model to explain, and coming up with scenarios for these two cases.
Number Sense
To develop number sense for fractions, we can also include the following thinking questions:
Without working out the answer,
- Is ½ ÷ ⅛ greater or smaller than 1?
- Is ½ ÷ 8 greater or smaller than 1?
In the first case, it is asking how many ⅛ are in ½? So, the answer is greater than 1. In the second case, since we are dividing ½ further into 8 equal parts, so the answer has to been less than 1, in fact, it is less than ½.
Such questions help students build a strong number sense for fractions by questioning the logic behind dividing with fractions. They are also interesting and help break up the monotony of learning,
Develop Rules by Induction
Dividing with Fractions is also a good chance for students to develop their inductive muscles. Let the students figure out the division rules by their own reasoning and then verify their rules with examples, before formally teaching the algorithm, for example “figure rules for calculating 4 ÷ ⅛ and ½ ÷ ⅛”.
Conclusion
It is time we teach the conceptual modeling of the algorithm for dividing with fractions! Not only does this make learning more interesting and meaningful, students are able at solving word problems and relate what they learn to the practical world they live in.
This article is part of a series of blog posts on Fractions:
- >> Read the next post on Fractions: Fractions and Decimals
- << Read the previous post on Fractions: Fraction Multiplication
- Or start from the beginning: Understanding Fractions as Equal Parts
Do you always use circular fractions? Or rectangular ones (which I find translate well into a number line and bar model) .., or?