"Why does the math language have to be so difficult," remarked one of my students. "Why can't it just be about the numbers," she added.

We were studying rational and irrational numbers. I liked the visuals I had seen before on number categories but I felt like to connect the language and concepts, my students needed more. Rather than start with the language, we started with a set of visual and kinesthetic activities for exploring differences between rational and irrational numbers. (For those who read here regularly, I used these during practice/review/assessment, but you could just as easily set up at math centers for whole class mini-project.)

I posted a set of numbers at the center with the labels rational and irrational above. Students were asked to brainstorm about the difference. What do they notice?

Students then were asked to create one rational and one irrational number. They had a choice of three ways to represent the two types of numbers: cityscapes, beads, paper chains.

In a subsequent lesson students made a rational/irrational venn diagram. The purpose of this is to reinforce their understanding of the concept as well as to see that rational and irrational numbers never overlap.

Now, I have students go back and create a definition for rational numbers including examples and one for irrational numbers. These are shared and by this time most of my students have it.  If they are close, I might help them refine their definition through questions or examples. If they are not close, they might do another kinesthetic activity or watch a video on rational/irrational numbers.

Finally, I ask students to think about what irrational numbers look like in terms of rational numbers- in other words what is their value. Students have a scavenger hunt (individually or in small groups) to find number line representations of irrational numbers. Between these three different activities, students have rational/irrational numbers mastered.

As a final point, although this post was specifically about teaching rationality, in looking at the big picture, I am reminded of the multiple ways that students access information. Through the use of different types of activities, students understand the concepts and even use the "math language."