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, 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).
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.