Tuesday, July 2, 2024

Division of Fractions

I. Standard Algorithms
A. Division of Fractions Algorithm
Turn the division problem into a multiplication problem (Flip Method)
1. Turn the second fraction (the one you want to divide by) upside down
(this is now a reciprocal).
2. Multiply the first fraction by that reciprocal
3. Simplify the fraction (if needed)

Example:

1/2 ÷ 1/6 = 1/2 x 6/1 = 6/2 = 3

B. Division of Mixed Numbers
To divide mixed numbers:
1. Convert the mixed numbers to improper fractions.
2. Follow the general algorithm for dividing fractions.


II. Conceptual Understanding

1. Whole number divided by whole number with fractional answer
Example

3 ÷ 4

Using the partitive model of division we can translate to "After dividing 3 into 4 groups, how much is in each group?" 


How do we partition a number line 0 to 3 into 4 parts? By partitioning each unit into fourths we now have a number line with 12 segments. 12 divided by 3 provides us with the answer as to where our partitions should be made. After doing so, we can see that each group is comprised of 3/4 of a total unit. 

Here's another representation of the same problem.




2. Whole number divided by fraction
Example

4 ÷ 2/3

Using the quotative model of division we can translate this to "How many 2/3 are contained in 4?"












We have a number line 0 to 4.
We then subdivide each whole number into three parts (the denominator of the divisor).
Now we answer the question "How many 2/3 are in 4?" We count the number of 2/3 jumps and determine the answer is 6.

4 ÷ 2/3 = 6

Note: The partitive definition of division doesn't really work here. If the problem was 4 ÷ 4 we could imagine cutting the number line into four groups and with each group having 1 unit. If the problem was 4 ÷ 2 we could imagine cutting the number line in half at 2 forming two groups of 2. And with 4 ÷ 1 there wouldn't be any split since there would be a single group of 4. But when you try to mentally divide 4 by a fraction using the partitive model, it doesn't seem to make sense. How do you split something into 2/3 groups? This caused me a great deal of mental anguish till I discovered the quotative model. 

Update: I asked Chatgpt (now that AI is a thing) about this and it agreed that "the partitive model doesn't work well for this problem" but explained that it could be reframed to work by asking "If 2/3 of a group equals 4, how much is in one whole group?" Now it's a partitive problem which can be solved using algebra. 
(2/3)x  = 4 
(3/2)(2/3)x = 4(3/2)
x = 12/2 = 6


3. Fraction divided by a whole number
Example

3/4 ÷ 3

Using the partitive model of division we can translate this to "After dividing 3/4 into 3 groups, how much is in each group?"

With this simple example we divide 3/4 into three groups with each group containing 1/4

3/4 ÷ 3 = 1/4

Here is another representation of the same problem.




4. Fraction divided by a fraction
Example (Common Denominator)

6/8 ÷ 2/8

Using the quotative model of division we can translate this as "How many 2/8 are in 6/8?"


This number line shows the whole number 1 divided into eight partitions with each partition being 1/8. We want to know how many 2/8 are in 6/8 so we mark are end point as 6/8. Then we count the number of 2/8 jumps, and determine the answer is 3. 

6/8 ÷ 2/8 = 3 

Example (Uncommon & Noncompatible Denominators)

2/3 ÷ 1/2

Using the quotative model of division we can translate this into "How many 1/2 are in 2/3?"


Here is a number line with 2/3 marked as the end point. It might be tempting to simply split 2/3 in half and say the answer is 1/3. That is wrong. It would be correct to do so if the problem was 2/3 ÷ 2. Then you could partitively cut the number line in half and see that the answer is 1/3. But here we are trying to see how many 1/2 are in 2/3. The number line above doesn't allow us to answer this question. To find the answer using a number line, we have to reformat the patricians. We can do this by finding common denominators. 

2/3 ÷ 1/2 = 4/6 ÷ 3/6


Now the reformatted question asks how many 3/6 is in 4/6. We can see there is one full jump of 3/6, then another 1/3 jump of 3/6. Therefore, the answer is 1 1/3.

 2/3 ÷ 1/2 = 4/6 ÷ 3/6 = 1 1/3



III. Proof of the Flip Method of Division Algorithm
Here is a proof of the flip method where the problem is presented as a complex fraction.









Reference
Room to Discover: Teaching Fraction Division: A Visual and Conceptual Approach
https://mathbitsnotebook.com/JuniorMath/FractionsDecimals/FDdivide.html
https://www.mathsisfun.com/fractions_division.html
https://teachablemath.com/dividing-with-fractions/#google_vignette
https://pressbooks-dev.oer.hawaii.edu/math111/chapter/dividing-fractions-meaning/
https://www.scaffoldedmath.com/2019/11/dividing-fractions-by-fractions-using-models.html
https://mathmonks.com/number-line/division-on-a-number-line
https://mathgeekmama.com/how-to-teach-dividing-fractions/
https://teachablemath.com/dividing-with-fractions/#google_vignette
https://www.geogebra.org/m/SPnu62Tp
number line maker
https://mathsbot.com/tools/numberLine
https://apps.mathlearningcenter.org/number-line/

Practice


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