One thing students need to understand is that they can take any two quantities and make a ratio out of them.
They can.
It may not mean anything useful, but they can do it. For example, if I have 3 potted plants in my classroom and 28 students, I could say the ratio of potted plants to students is 3:28. Does that mean anything? Do you care?
I can’t say I care a whole lot about this ratio.
However, I can still talk about the relationship between these two quantities. With this ratio, I see that there are nearly 9 times as many students as there are potted plants. Another way to put it would be for every 1 potted plant there are nearly 9 students. Again, is there much meaning to that? Do you care that this relationship exists?
Probably not. I can’t say I really care.
Although, that gets me thinking. I might ponder with my students what our classroom would look like if that relationship were reversed. What if we had nearly 9 times as many potted plants as students? What would that look like? Going back to the original ratio, what if we looked in other classrooms? Would we see a similar ratio? Why or why not?
In the end though, we should realize that just because we can make a ratio, doesn’t mean that we should. Often we analyze ratios because the relationship between the quantities matters to us in some way. This is why teachers wrack their brains coming up with “real-life” examples of ratios. And we should do that!
But let the kids play, especially early on in their learning of ratios.
Let them come up with ratios based on objects in the classroom. If a student sees the ratio of girls to boys in the class is 9:16, ask the students to find another ratio of 9:16 in the class. If they can’t, ask them to find a relationship that’s close to the same. The ratio 9:16 tells me there are nearly ½ as many girls as boys, so look for another pair of objects in the room where there is nearly ½ as many of one as the other.
Give students small objects like race cars and other small toys. Ask them to model a 2:1 ratio using those objects. Challenge them to do it with more than 3 objects. Be sure to ask them to model relationships using different language. Instead of giving a ratio like 3:2, say something like, “For every 3 pencils, put out 2 paperclips.” If a student puts out 3 pencils and 2 paperclips, challenge them to keep that relationship but use more pencils and paperclips.
The more students can understand how this is all about relating quantities, the more they can appreciate why we talk about the real-world examples that we use. Teachers choose these examples because the relationships in them matter. And it’s fun to discuss what may happen if that relationship isn’t maintained. Let’s say some kind of dough requires a 1:5 ratio of wet to dry ingredients. What’s going to happen if we goof up and make a 1:5 ratio of dry to wet ingredients? Even better, bring in some flour and water and model what happens!
Teaching to the Beat of a Different Drummer 