
The thing I've learned, in my years of using number lines,
is that kids are more apt to count the lines than the spaces. They need to learn right away that the only
right way to use a fraction number line is to count the spaces, and they need lots of opportunities to practice doing so. We draw a line under to form boxes and they
see it. We label blank number lines
starting with zero and they see it. Yet when they are asked to find 1/2 on a number line that
is divided into eighths, suddenly they forget all about equivalent
fractions. They draw their point on the
second eighth instead of the 4/8 mark.

After spending a while using number lines to relate improper fractions to
mixed numbers, it was time to find an algorithm. Once again, I had them write two
equivalent fractions (a mixed number that equals an improper fraction) on two
separate Post Its. They had their
labeled number lines to refer to so that they knew what to write. Then they stuck them randomly on the board
for me to sort, and they had to guess how I figured it out.
I did give them a hint in red on each Post It this time,
because I was afraid too many kids just wouldn't see any rhyme or reason to the
matches, or worse, see coincidences like "there are lots of even
numbers." You might notice that on 1 1/4 I drew a little 4 on the left and
a little 5 on the right. This really
helped a lot more students get a dialog going! "Method 1" was how the
majority of kids explained the process (different partnerships chose different
examples when they "turned and talked" but most explained it the same
way). Method 2 was the same steps, but
in a more visual, less wordy form for those kids who get bogged down with language. And finally, I added a picture so they could
go back and forth in their minds' eye that the 3 2/4 really does look just like
14/4. Circle fractions are used in our Every Day Counts program so it's a familiar model for
them.

Although Math in Focus is more about "decomposing numbers," and I focused my reteaching efforts on Method 2 (really just shorthand for method 1) a lot of the kids
overall liked the traditional "Method 3" the best because long division actually stuck with this group of kids especially well this year (could it be all the fun task card practice?) :D Nothing wrong
with it as long as they can see WHY it works.

Overall, their progress from the beginning to now has been amazing! I saw so much growth from my lowest students as well as my highest ones! I had a high student at the beginning ask on the pretest, "What even IS fraction of a set?! Is my answer going to be a fraction or a whole number," and get 8/11 wrong. He got a perfect score on the final.





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