Recently, I was invited into three Grade 3/4 classrooms to introduce fractions.

Cuisenaire rods give children hands-on ways to explore the meaning of fractions. After students built their towers, flowers, and robots, I asked, “If the orange rod is the whole, which rod is one half?” Students explained their thinking: “two yellows make an orange.” I emphasized, or rather, *students* emphasized that the two parts must be equal.

I asked students to find as many pairs as they could that showed one half. I let ’em go and they built and recorded the following:

Once more, with one third:

As children shared their pairs, we discussed the big ideas:

- the denominator tells how many equal parts make the whole (e.g., two purple rods make one brown rod, three light green rods make one blue rod)
- the same fraction can describe different pairs of quantities (e.g., one half can be represented using five different pairs, one third can be represented using three different pairs)
- the same quantity can be used to represent different fractions (e.g., white is one half of red and one third of light green, red is one half of purple and one third of dark green, etc.)

Something interesting and outside the lesson plan happened in each of these three classrooms.

Some students described each pair of rods using equivalent fractions (e.g., 1/2, 2/4, 4/8):

I asked the “we’re done” students to represent their own fractions using pairs of rods and determine each other’s mystery fraction. Many students chose fractions like 2/5 or 3/4, not simply unit fractions:

After students shared the three pairs of rods for one third, I asked if anyone found any more. “I did,” said one student, unexpectedly. Check this out:

I asked her why she chose to combine an orange rod and a red rod to make the whole. She explained that twelve can be divided into three equal parts. Without prompting, the rest of the class starting building these:

*adapted from The Super Source*