Fourth Grade Fractions for CCSS

Last year I was very proud of my fractions unit.  This year our district cut back on the amount of time dedicated to fractions in fourth grade.  As a result, I had to make some tough decisions on which lessons to cut, and which were non-negotiable.  And I believe if my students only learn one thing, it has to be a true understanding of equivalent fractions.  So after introducing the idea of equal parts and equivalent fractions with 1/2 with paper folding just as we did last year, we continued on to finding other equivalents.

Another priority for me was to stick with the instructional strategy of starting with manipulatives, moving on to visuals, and culminating with computation.  The ultimate goal being for my students to use the models to come up with their OWN algorithms.  Using fraction bars, circle models, (affiliate links) and even pattern blocks (this was a voluntary challenge that every single student insisted on trying, yay!) here are some of the observations they made:

The next day, they brought their recorded observations with them to quiz ME.  I repeated the "how did she do it" sessions as I modeled how to sort equivalent fractions.  As some students were able to articulate the algorithm and demonstrated that they could apply it themselves, I gave them extension questions such as comparing and ordering fractions (which our curriculum cut this year).  Those kids who were able to find equivalents were, of course, able to add and subtract fractions with ease.  Those who did not have the concept yet continued to use manipulatives with me until they were able to draw their own conclusions about the algorithm.  There were several ways I helped my remedial group:

• We used fraction number lines as a tool for locating equivalents in order to add and subtract fractions.  This way they could retain the understanding that equivalents were needed, as well as reinforce the idea of adding numerators for like fractions (not denominators).
• We explored online tools here and here to help us find equivalents at home.
• We played games online to practice visualizing, and then calculating equivalents.

Then we moved on to changing from improper fractions to mixed numbers and back again.  We started with fraction number lines, since this is a common visual fourth graders are expected to be able to interpret on MCAS.  I felt like Math in Focus didn't give quite enough direct practice on using them, so I created several number lines pages of my own (link to product) to break down the process. And that's when something amazing happened.

After using the number lines to count up using mixed numbers and later improper fractions, most of my students found it simple to use fraction circles to tell 11/3 equals 3 2/3.  With a few hints, some were even able to find the algorithm of dividing and finding a remainder.  But one of my students said, "I have a different way to change a mixed number to an improper fraction.  Just draw 11 circles, box them into groups of 3, and you have 2 as a remainder.  The 3 groups is your whole number, the 2 is your numerator, and the denominator stays the same so it's 3."

She showed me a model she had drawn to prove her point (I tried to make it darker and in color so it would show up on camera).

I asked the kids, "Does this method always work?  Try it for one of the problems we've already answered on the numberline."  Of course, they came to the same conclusion:  this was a new strategy.  One their teacher had not used before!  From that point on I referred to this as "The Sally Method," and it became a favorite!

But even better was the fact that many of the other kids from that point on had become determined to find a method for the skills we were learning because they wanted a method named after them too!

Even those students who were struggling with equivalent fractions had a respite when we started fraction of a set, since we were all back to manipulatives and then visuals.  I explained to the kids how previously we divided a single circle into 4 parts.  However, we can also divide groups, not just shapes.  I said, "I can divide this class in half easily..." and many replied there would be 8 in each group, but one of my students (the sort of student you worry about, who struggles with math, normally lacks the confidence to participate, and has some difficulty expressing himself verbally) said, "Not...like a magician's box."  I laughed before the other kids caught on, and then explained for him.  "If I divide the class in half, I wouldn't need to do the magician's trick, like, 'I will now divide my lovely assistant in half!  Just step into this box...'"  They all said, "Ohhhhhh!"  That brought home the difference between dividing one thing and dividing a group for them! 

They really got into using the fraction of a set cards (link to product).  Once again they were all comfortable using the visuals to make observations, and several told me, "I think I've found a method!"  (They much preferred "method" to "algorithm," haha).  After using the level 1 set, "Junie" put it best, "You just divide the whole number by the denominator."  We now had a "Junie Method" instead of lifting a formula from the book.  Once we started the level 2 set, another student discovered, "To find 3/5 of 15, just do the Junie method and then multiply by the numerator."  They were developing their own language for concepts that they needed.  By the time we played 40 frogs during our computer time, they were telling me that fraction of a set is easy.

So despite a few disrupted weeks thanks to weather-induced days off, and outside pressures to speed things along, fractions was once again one of the highlights of our year.  We have today (Wednesday) off again for snow, but I think they are ready for their test on Friday (in record time).    

P.S., I know I missed a post last week.  The reason is I've been trying to get better about using Facebook.  I really had to immerse myself to get into it because after 2 years it wasn't clicking for me just going on once per week.  I think I've finally started to get into the spirit of it, so if you'd like to join me my Facebook is here.  On the other hand, if you enjoy reading about my lessons/units then not to worry; I do plan to keep up my Wednesday posts here at Shut the Door and Teach.

How to Teach Mixed Numbers and Improper Fractions

Once my fourth graders were finally comfortable with equivalent fractions, 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 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. I've made these these fraction number line worksheets available for your students as well!

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.      

Finally, we did the inverse, which was to change an improper fraction to a mixed number.  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.

Fraction Unit Wrap-Up: Grade 4 Math in Focus Chapter 6

As I mentioned before, I needed to teach my fourth graders the third grade chapter on fractions this year.  Since it wasn't in the third grade standards, my district decided to move it up to fourth.  However, it worked out just fine, because there was a lot of overlap.  And I actually really enjoyed laying the groundwork for improper fractions and mixed numbers.  I could be sure that the concept of what fractions really are was fresh in their mind, as was finding equivalent fractions.  So I spent a week just on ways to represent a half before we moved on.

Next 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.  You can read about this in my equivalent fractions post or check out my fraction activities bundle preview.

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 time I'll write about improper fractions and mixed numbers.