and the fractions continue!

We’re still working on fraction in 5-8th grade resource.  Surprisingly (or maybe not!) we haven’t even started dealing with computations yet!  We’ve been working on understanding what fractions are truly all about.

My students have a pretty decent grasp of what a fraction is when it comes to an area model, or even a set model, however, a number line model seemingly blew their mind.  I made an activity a while back called Fraction Approximation.   In this activity, students first sort the fractions (from halves to twelfths) based on whether they are closer to zero, one-half, or one whole.  Then, they start placing them on a number line.

I decided to have them do the sort as I called out the fractions.  Basically, it was a pre-test to see how they would do.   Closer to Zero wasn’t a problem for them.  Most put any unit fraction in the closer to half, then the rest as closer to one whole.  Some students were able to break a number, such as ninths into 9 pieces and spread them close to evenly over the sections.  Others were all over the place.  YIKES!  Okay… time to regroup.

So that night I thought about how to get them to understand the number line better.  I decided even though we haven’t worked with fraction operations, that I would pose a kind of division problem the next day without saying it was division.   I have 8 kiddos in my 2nd period.  I placed 12 mini candy bars on the table and said that we needed to make sure that everyone had the same amount. We couldn’t purchase more, or hope that more would appear.  How could we make sure that each person had the same amount?  My 8th graders started drawing pictures and tearing paper (it was actually quite a sight to see as most of these kids groan whenever tasked to do something – though food was involved so they were ‘all in’)

Pretty quickly they came up with 3 halves for each person.  We had two or three different ways to get the answer and they were actually rather surprised that their wasn’t only one way.  Then, I used a new pair of scissors and cut each piece in half and asked them to come up and grab their three pieces.

Okay, so we figured out how to deal with a set of items again…. still, needed to relate it to a number line.  Enter a ball of yarn!  I gave each student a pre-measured piece of yarn.  I used yarn because it’s very difficult to measure it with a ruler precisely.  I wanted students to think outside the rhombus when it came to making even slices.  The task:  using whatever method they wished, and without throwing any pieces away, cut the yarn into 8 pieces of the same (or as close to the same as possible) slices.

Using a pencil to “eye” the measurements.

Trying hard to keep the yarn taught so they can measure.

Cutting little pieces of paper and trying to evenly space them before cutting.

Using their hands to visually cut the yarn into pieces.

This brave soul just started cutting!

A “make-shift” ruler out of a pencil end.

This student kept drawing a line and tried cutting it. She was the first to have a major “ah-ha” moment and was able to create the most evenly cut  yarn!  Her method? folding!

fold the string in half.

fold the string in half again
(fourths now!)

fold the string in half again
(eighths now!)

carefully cut the loops! VOILA!!

Needless to say, this student was super impressed with herself (not to mention that her classmates were in awe and voted her as most precise)!  We then took some discussion time to talk about how precise you need to be when using a number line to show distance with fractions.  The next day they came to class and we started working with the number line again.

After having a better grasp of how to cut a string into equal pieces, we took our thoughts to the number line.  Students were not able to use calculators to convert fractions to decimals.  We measured a length of adding tape to be 130 inches (120 for the number line and 5 inches past the beginning/end.  Students took turns marking inches on the number line and making tiny numbers on it to help with quick measurements.

We started by thinking about 0/2, then 1/2, then 2/2.  Students pretty quickly made the connection that 0/2 = 0 inches on the number line, 1/2 of 120 inches was 60 inches, and 2/2 = 120 inches on the number line.  They were really excited to be able to make the connection between measurement and where the fractions were located.  

The 7th grade number line wouldn’t stay up in the hall so we moved it to the board.
A view from the left…

A view from the right…

Close-up of the first 1/3

Close-up on the middle 1/3

Close-up of the final 1/3

After creating the number line we talked about how accurate we were and what pieces human error could have (and probably did) play into our line.  They got a bit frustrated by 7ths, 9ths, and 11ths but didn’t give up.  You can see how they reconciled the 11ths (notice them on the bottom of the number line as they were “hard to squeeze in”)

Now to keep referring to this amazing resource so that kiddos get more used to the distance concept.  We’ll be returning here tons in the next few weeks!

p.s. if you know me at all, you will know how hard it was for me to let the kiddos take complete control over the number line.  I wanted them to shift it downward so badly, and to use a ruler to drawn their lines, and to line things up neater….. ahhhhhhhh OCD at it’s finest!

Categories: conceptual math, fractional reasoning, mathematical reasoning, and representational math.

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