To open today's lesson, I placed a comparison (3/4 > 1/2) on the board and ask the students to turn and describe how they know it is true. The strategies are varied. One of my students remarks that he understood 2/4 equals 1/2, so an extra 1/4 makes 3/4 larger. Many students draw models on their white boards to grapple with and prove.
Following the partner discussions, I ask the students why it seems they are struggling with this more than when we compared fractions with same denominator. They tell me exactly what I thought I would hear…everything is different, so they couldn't visualize it.
Now I remind them that we can only compare fractions when we have two of the same whole. If your students need this type of reminder as well, draw the standard small and large pizza and ask if half of each is the same, or why they are not equal. This is usually all it takes.
It is critical that students have various ways of comparing fractions, so we make a list on the board of models they can use. We came up with number lines, Cuisenaire rods, sketches, and fraction circles.
I put several comparison sentences on the board and ask students to use their choice of model to help them solve. The sentences for my class to model are:
2/4_____1/3 (on this one I was hoping someone would see the 2/4 and think about 1/2)!
You may use any fraction pairs for this lesson. The real task is in the students creating and using models to support their responses.
This partnership still struggles with how to begin. At the end, they stop "guessing" and start drawing!
In this video, my student is using number lines and hoping to find the solution. When I approach him, he has the lines drawn and labeled at the beginning, but has not partitioned the line. I ask him to go ahead and do that, and I was very pleased with his strategy!
My closing starts out as a review of using models to represent comparisons, remembering to use the same size whole. However, we take a sharp turn when the students start asking questions about fractions that take my breath away.
One student asks if there is a way to use fourths to create fifths. I did not have my camera in my hand to video the discussion, but it was fantastic because everyone just starts sketching, wondering, debating. Finally, the class decides you cannot use fourths to create fifths, because fourths are not multiples of fifths.
The next wondering comes from a student who receives extra support; fractions were a bit difficult at the beginning. He starts out asking a question and goes to the board to explain what he is wondering. Another student literally jumps up and adds his thinking.
The following two videos are what came from a little wondering.