Date of publication: 2017-08-28 02:02
The universal set is actually a good place to start: If you 8767 re talking about animals, draw that big box at the beginning of the discussion and label it clearly. That will keep the conversation on track and better organize your ideas.
Is a rectangle always a rhombus? No, because all four sides of a rectangle don't have to be equal. However, the sets of rectangles and rhombuses do intersect, and their intersection is the set of squares 656 all squares are both a rectangle and a rhombus.
The truth is, this is not a Venn diagram. Mathematically speaking, a Venn diagram represents sets and elements. The circles are sets, and the items in the sets are elements of those sets. The space in the middle, where the circles overlap, is the intersection of the sets. You’ve probably heard these terms before and they’re fairly intuitive, but we don’t use the formal language very often. And while you don’t have to use the formal language with your students, it will be easier for the moment if we use those terms. Now, don’t panic we 8767 ve just done sets and elements! In table 7, above, each column header is a set, and the headings along the left are elements.
You’re probably familiar with the Venn diagram, or 8775 double bubble chart. 8776 Research has shown that identifying similarities and differences is perhaps the single most powerful strategy for student learning.
There are still problems with this, though. It’s hard to see connections among the animals we’ve classified them, but we’ve lost the sense of similarities and differences, haven’t we? You can read it across the characteristics of one animal or down the animals that share one characteristics but it’s hard to move around visually within the boxes.
We know many quadrilaterals by their special shapes and properties, like squares. Remember, if you see the word quadrilateral, it does not necessarily mean a figure with special properties like a square or rectangle! In word problems, be careful not to assume that a quadrilateral has parallel sides or equal sides unless that is stated.
Of course, there are plenty of animals that breathe air and can’t swim, such as chimpanzees, spiders, and chickadees. And humans, dogs, and ducks, like sea turtles, breathe air and can swim but don’t have fins. We 8767 re moving beyond water-dwelling animals now, but that 8767 s all right we 8767 re thinking outside the box circle.
Inspired by their curiosity about animals, students work together to research an animal of their choice and present the information they gather to an authentic audience.
When you see the finished diagram, it looks easy and obvious, but it took a lot of thinking to come up with these answers. I brainstormed and bugged my colleagues, and still couldn 8767 t come up with anything for that last space until a teacher in one of my conference presentations suggested water moccasins. It 8767 s harder than it looks, and it 8767 s a good collaborative project for students working in groups, because they 8767 ll each have different ways of thinking about animals all of which will be helpful.
Well, sure. I’ve added shrimp to the animal comparison diagram. As we can see, shrimp are very different from both whales and fish, and they don 8767 t seem to have much in common at all with whales. A student might look at this and think, That 8767 s interesting three very different kinds of animals that live in the water. It looks like quite a lot of different animals live in water. So our graphic organizer is helping us to understand similarities and differences among animals that have one key trait in common.
Here, I 8767 ve simply drawn a box around my diagram and labeled it with the thing we 8767 re classifying animals. In mathematical terms, that box is the universal set , the set of all elements in the 8775 universe 8776 of our conversation. The universal set contains everything we might be trying to classify.
At a basic level, you could do the classification work yourself, make a blank diagram, give it to students with a list of elements, and have them figure out where the elements go. As a demonstration, you could use a diagram on a whiteboard, then let students work on their own diagrams in groups.
On the surface not much has changed, but I want you to stop and think about this a moment because it’s actually very powerful! In the first example (figure 6), we were comparing two animals and recognizing how they’re alike and different. That’s fine, but if we want to compare different animals, we have to start all over. And maybe the words we used to describe those two animals won’t work to describe the new animals. Or, to take my serious literary example, when you read two more novels, you 8767 ll have to compare the heroes separately. There’s no connection to what you’ve done before. We 8767 re not activating students 8767 prior knowledge.
Moreover, while both the comparison diagram and those classification charts we made are closed, either conceptually or visually, this diagram is open. A classification diagram invites us, practically demands us, to consider new ideas.