In my previous blog, I talked about a composite unit, what it is and how it plays an important role in many different aspects of students’ construction of mathematics. One of these areas is fractions. So how does the student’s ability to take a number as something that is countable affect their understanding of fractions? It begins with the simple understanding of sharing equally, and progresses into number sequences with a unit of 1/n.
Think about a candy bar. What would a piece look like if five people were sharing the candy bar equally. The way students do this will differ depending on whether or not they have constructed a composite unit. Some kids will mark off a piece, and then make four more pieces the same size to see if they take up the whole bar, and start over if they do not. A student with a composite unit will be able to “project” five equal parts onto the bar simultaneously, and will often be fairly accurate with their first attempt. The composite unit makes these sharing tasks much more accessible for these students.
Eventually, we want students to generalize their number sequence. What I mean by this is we want them to see any number as a possible unit from which to build a number sequence. One isn’t the only number by which we can count, but neither are 2, 3, 5 and 10. We see the importance of this skill and teach kids to count by two’s, and especially by tens. Think how powerful it would be for a student to realize that they can construct a number sequence of fourths. If you are an elementary teacher, you have probably come across children struggling to make sense of improper fractions. When students come to understand 1/4 as a unit that is iterated 5 times to make 5/4, it opens new possibilities for conceptual understanding of operations with fractions. This understanding is very similar to seeing the number 5 as the iteration of one, five times on the candy bar. Can you see the connection?
The development of a composite unit contributes to their ability to come to know fractions. The concepts developed with whole numbers lend themselves to making sense of improper fractions, as well as operating with these numbers. We already saw how the composite unit is important for additive as well as multiplicative thinking in previous blog posts. With the added importance it plays in fractional knowledge, I would say that the construction of a composite unit is a very important goal that should be a focus of mathematics instruction for all children.