A Critical Thought Approach
I opened the lesson today with a quick review of a concept that was confused by the majority of my students on Quiz 1. I wanted to be sure they understood the difference between using a fractional model to compare vs. adding. This is a conceptual misunderstanding that is part of the transition process of students learning to really think about the meaning of mathematics.
I explained that there would be a test tomorrow and that I noticed there was some confusion about fractional models and how to use them. I told them that they needed to remember why a fractional model is used and that there is meaning behind them.
I went up to the white board and drew a circle that was divided into thirds. I shaded two thirds blue and one third red. I also drew two bars (fractional models) and divided those into thirds. I shaded one third on one bar and two thirds on the other.
I wrote 1/3/+ 2/3 = n and 1/3 O 2/3 (< > =). I asked my students to choose which equation would be used to explain the drawings. Rather than just telling them about the error, and reteaching it, I thought this was a good way for them to think about the drawings.
A student raised her hand and said she thought the circle best showed the addition problem.I asked what the difference was between the drawings? She told me that the other drawing was about thirds too, and she thought that it would also show addition somehow. But she thought that the circle "looked more like" addition. I asked the class if anyone had any other thoughts about the problem.It was quiet.
I proceeded by clarifying the difference between showing how the whole needs to be considered with the addition problem because 1/3 + 2/3 would create 3/3. I reminded them of the prior mini unit where we learned to create equivalent fractions by drawing. I explained that if they had used the bar fractional for addition, it would be divided into thirds and there would be one bar. I asked if they could tell which fraction was larger by looking at the bars?
By the time we were done, students could see the difference. In order to satisfy the part of the standards about adding and subtracting fractions or comparing fractions with fractional models, I knew it was important to address the whole class.
With the whole group seated at their desks with notes out from lessons past, I reviewed the concepts of decomposition with addition and subtraction problems. I emphasized that it was important to be able to decide when decomposition was necessary to solve the problem.
I referred to this SB file as I taught. RTI Decomposition. As we went through and examined the problems, I heard a lot of "oh's". They worked on each problem on the board. I gave an example of two ways of solving another problem: 2 1/4 + 3 3/4. I decomposed each addend first. Then decomposed the answer. Then, I added without decomposing, creating the improper fraction and then reducing it. I asked which they thought was the correct way.
One student raised his hand and said "The first way." Another student interjected and told us that both of them were right. I asked which took less time and was apt to be more accurate because there were less steps.
I split students into three groups. I had four students who had not passed their quiz and invited them back to my work table. I also invited anyone who had any concerns that they might need some more instruction and help. Four more students raised their hands to join us.I used explicit direction, step by step and worked until all of them could independently solve the problems I presented. They worked in their notebook. I had them draw out each decomposition first, using circular fractional models,then create the improper fractions by counting the total pieces of the wholes and portion of the whole. This very concrete instruction was very important for them. Their difficulty was with the process. Then, we worked on different examples of when we needed to decompose and when it wasn't necessary. I used extreme opposite problems: 1 1/3 + 2 1/3 : no decomposition necessary to 2 4/6 + 3 5/6...decomposing after adding and getting the improper fraction. One student still had difficulty dividing up the circles into the denominator. She wanted to draw them as fourths regardless of the denominator.
The second group worked on practicing subtraction of mixed numbers using dice because subtraction remains a weaker skill. They created their problems by rolling first for the denominator. Then they rolled for the whole numbers. Finally, they rolled two more times for their numerators. I told them they needed to create their numbers and arrange them to properly subtract. This reinforced their thinking about the numbers and how they are placed. As they worked they purposfully set up their problems to decompose. Explaining Decomposition demonstrates how one student worked on her decomposition problem she had created from rolling the dice. I like how she chose to arrange the subtraction problem forcing her to decompose the numbers. They all were working hard at making sure that was happening in each problem they created from rolling the dice.
The third group worked on IXL.com math at different levels. I had assigned some students challenging problems in level G in various sections. I wanted them to extend their practice and challenge their thinking. I let them decide and choose what was challenging for them and then checked on their work, making sure they were on task. One student had chosen some algebraic function table work while another chose word problems with unlike denominators. IXL isn't completely Common Core based, so I have to be careful that what they are working on matches standards. I have to be familiar with fifth and six grade math standards in order to be sure that my advanced students are on track. It's a great site, but I know that I need to watch it!
I stopped everyone as soon as I thought they had practiced enough. I had been roving the classroom as they independently worked looking at how accurately problems were being solved. I could tell that they were stronger at subtracting using decomposition. Several had used the drawing and then the number conversion before they subtracted. They told me they like drawing the fractional models. I wrote these problems, Quick RTI Quiz Decomposition, on the white board for them to prove they had mastered the skill. The other students who were at mastery level, kept working on their individual tasks on their iPads. They solved on a loose leaf paper. When they were done, we corrected them together as they looked at their own sheet. I did this so they could see any mistakes they had made. I think that self correction is one of the best learning tools as they self assessed their progress.