I worked in the school's Learning Commons and made use of the Lego board that was placed on the wall.
In a similar way, I asked students to show me what one-fourth (or one-quarter) might look like. As I listened to the students think there way through this, it became clear that even though they might know the fraction words 'half' or 'fourth', that they did not necessarily know how this connected to the number of parts needed to make a whole. Indeed, when we tried to do some skip counting by fourths as a class, I could tell that there was some uncertainty. To clarify this, I gave them the following challenge:
If a 2-by-4 block is one whole, build seven-halves and tell me another way to say this.
As they did this, I moved from group to group and modelled some skip counting aloud with them:
"One half, two halves, three halves, four halves, five halves, six halves, seven halves."
Now we could use a 2-by-4 block to show that this was equivalent to 3 and one-half:
The students seemed to like this visual 'proof'.
The next challenge was to build twelve-halves and to find what this was equivalent to. The students were able to do this more quickly now and tell me the correct answer. If some students put all their blocks together like this:
we suggested that that their work might be more clear if they leave a gap between each whole:
Finally, we asked them to build six-fourths and to find out what this is equivalent to. Again, they were able to build this quickly:
Some students argued that this was one-and-one-half. Others argued that it was the same as one-and-two-quarters. Then one student suggested that it this meant that one-half was the same as two-quarters.
I seized on this idea: "Who agrees with this suggestion?"
Everyone did. Unfortunately, the bell rang for recess but it gave me plenty to think about what happened in the lesson and what the students are now ready for. I liked working with the Lego mainly because it was easy for the students to organise their thinking: once the blocks were placed, they stayed put and didn't get knocked all over the place. In fact, I could pick up one student's work and easily show the whole group.
One drawback of using Lego is that the fractions that you can use are somewhat limited, so I would have to think more carefully about what models and blocks I could use to show thirds, fifths, sixths, tenths etc.
I actually did a similar activity at a Parent Council meeting for one our elementary schools last week but with pattern blocks. I showed the parents a hexagon and asked them to show me what one-third of this was. After they confirmed it was the blue rhombus, I challenged them to build eight-thirds and to then tell me another way of saying this. Once they had lined up eight of these rhombii, they carefully arranged six of these into two hexagons and were able to tell me that eight-thirds is equivalent to 2 and two-thirds. A number of parents actually said "That's why it works!" It was a perfect moment to show them the power of the Concrete-Diagrammatic-Symbolic continuum and how we can use this to develop students' understanding of fractions:
My overall learning of this is then as follows:
1) Equivalence is a key concept in fractional understanding. Without it, fractional computations are built on shaky foundations. Students need to develop this knowledge of equivalence initially through concrete activities before moving on to diagrammatic and thence symbolic activities.
2) Skip counting with fractions is an important prerequisite for developing an understanding of equivalence. Again, this should be developed concretely first before moving on to diagrammatically (number lines) and then symbolically.