I asked students to refer back to the equivalent fractions they found when using the plastic fraction bar manipulatives. I explained that when two different fractions, such as one yellow bar and 4 purple bars equal the same amount, they are called equal, or equivalent. Next, I had students write down a fraction that was equal to 1/3 and a fraction that was equal to 1/4 on two separate sticky notes. I asked them to stick them anywhere on the board, and try to trick me. "Pretend I'M the one who has to take this quiz! I need to sort them into the correct boxes."
Once this was done, I had them turn and talk with a partner about how they thought I figured it out so quickly. The results went onto an anchor chart, "How Can You Tell if a Fraction is Equivalent to 1/3 or 1/4?"
Once they had a few different ways to explain the algorithm, they were ready for lots of practice using it. And after another week of practice finding equivalent fractions and simplest form, (whether they practiced it in the context of adding, subtracting, comparing or ordering fractions with unlike denominators, finding equivalents and simplest form was the pervasive skill) I gave them their "third grade test" (Math in Focus, grade 3, chapter 14 test). And I am proud to say NOT ONE STUDENT thought that 1/4 + 2/8 = 3/12. They were so comfortable with the concept of denominators being a different fraction altogether that the thought did not even cross their minds. Not every single kid found the equivalent correctly, and many of them didn't pay attention to the direction to "write your answer in simplest form," but those were learning lessons for another day (I sat that half of the class down and showed them "the most popular wrong answers" and lots of them said "hey, that's what I wrote......ohhhhhhh!"). It was time to move on to the fourth grade unit.
Next, it was time to introduce the idea of "bigger" fractions: improper fractions and mixed numbers. However, using those plastic fraction pieces was going to get cumbersome really fast. My group this year are not really pros at sharing or taking turns, so combining sets of fractional pieces would have been a headache for all of us. Instead, I turned to a model that is very, very common on our MCAS tests: the fraction number line.
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.
What they needed was practice labeling number lines with confidence. That means that before they do anything else, they must remember to find the denominator of all the fractions on the number line. I gave them several blank number line worksheets (a future TPT product for my store!) that ask them to draw a line of varying lengths, but the first step is always to find the denominator and label the lines. The last step is to find an equivalent eighth for the given 1/2.
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.
Again, I wrote hints, however a few of my students predicted, before we even started, "it's just going to be the opposite!" "What do you mean?" "Instead of multiply then add, you divide and subtract!" Isn't it awesome when they can apply their thinking to new contexts? :)
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.
After a final (multi-day) lesson on problem solving, which I forgot to photograph, I felt like they were ready for their test. I made them SWEAR to me that they would check the directions on each questions to tell if their answer needed to be in simplest form, a mixed number, or an improper fraction. In the end, half the class missed, on average, only one question due to their answer being technically correct, but in the wrong form. And if you teach Math in Focus, Chapter 6, I bet you can guess which completely, and I have to think purposely, misleading question I'm talking about.
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. Another student, who is a low performer, who got 1 right on the pretest got 8/11 correct on the final. I wrote right on his test how proud I am of him. And although he can be a difficult kid, he was SO EXCITED. "I actually get fractions!" He has never looked so proud of himself.
P.S., Since Teacher Appreciate Week is next week (to coincide with our MCAS state test, BOOOO!) I'm going to have a couple extra very quick, time-sensitive posts in addition to my typical Wednesday post. Be sure to check back a few times before Wednesday! :)