Using Pictures to Solve Compound Fractions

Today I find that Bridging the Gap stole my thunder (sort of) by posting on a method I have been using in class recently.  We have been looking at compound fractions when writing equivalent forms of equations.  Changing a standard form equation such as 2x + 1/2y = 4 to slope-intercept has really thrown a curve to my students, and it’s all about one concept- fractions.  My students admit they struggle with fractions and then to create a compound fraction?  Most just put the pencil down, worry about other things and try to nap in class.  I have tackled the problem by using pictures of fractions to help students visualize what is going on.

I first start with an improper fraction example:

normal fraction

We work on dialogue on what 4/2 means, and students came up with the wording you now see on the board.  They then draw 4 circles and split them into groups of 2.

We then look at an example when there is a fraction in the denominator:

fraction in denominator

Students once again write their wording to the fraction and attempt to complete it as instructed.  I have had students struggle on how to make groups of 1/2, but when you ask them to draw 1/2 they easily give you a semi-circle.  They they have that aha! moment and quickly draw lines bisecting the four wholes.  They created eight 1/2 pictures.

Students really grasp the first two examples quickly, and we then throw them the curve, a fraction in the numerator:

fraction in numerator

Many students automatically say they can’t split a half of a circle into a group of four, so I will go back and change the fraction to a whole number, 4.  they quickly draw four lines between the wholes and say that this problem is nothing like the one I asked them.  I have them explain their process to find 4/4, and then write it as 1/4.  Some students will see that you can cut the circle into four pieces, but others will still struggle- and this is where as an instructor I have to realize that inadvertently I gave them a specific context in which to look at creating groups.  I gave them a circle, so I ask the students if we need to start with a circle, and many say no.  They draw a rectangle on the board and are able to make it into a group of four:

s1na020Once they see this, they quickly come to the idea that they can cut the half into four pieces with either a rectangle or circular representation.  They identify the new pieces as eighths.

We do a few other problems where students practice with both cases, and then I ask students to find a pattern.  Although they do not tell me to multiply by the reciprocal, students understand what numbers to combine and why they are doing so.

The next step, of course is to extend this concept beyond the unit fraction…

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