Who's a Widget? Making Sense of Definitions

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Objective

Students will be able to recognize and apply the components of a "good" definition (classification and differentiation). Students will be able to test definitions of terms by looking for counterexample.

Big Idea

Students will write precise definitions based on examples and non-examples and test these definitions by looking for counterexamples.

Launch: Who's a Widget? (How to Write Good Definitions)

30 minutes

The Common Core explicitly asks students to attend to precision (MP6), which students must do as they grapple with the preciseness of definitions and apply them correctly.  For this reason, I introduce the features of definitions (classification and differentiation) by showing the class Widget Examples and Non Examples. For this task I give silent, individual, think time (1-2 minutes) for students to identify and be able to explain which figure(s) are widgets.

After students have identified Widgets and non-Widgets themselves, I have them share out in their group and try to come to an agreement as to which figure(s) are widgets.  When circulating the room, I ask students how they determined who the widgets are and ask them about the characteristics widgets have in common. I am asking questions to help students focus closely on classification and differentiation.

After groups seem to have come to an agreement on the widgets, I ask each group to write a good definition for "the widget".  I tell groups that we will test the quality of the definition by searching for counterexamples.  

During the whole class discussion, I call on each group to read their definition.  I have found that typing the definition and projecting it helps the entire class can see the definition and be able to test it.  I then give 1-2 minutes for the other groups to search for a counterexample, which they can draw on the whiteboard. (See my reflection Unpacking the Qualities of a Good Definition for more information about this discussion.)

I give groups a chance to refine their definitions based on the previous discussion and have them tested by their peers.  Ultimately, after this round of refining and testing another definition, the class should agree on a good definition for a widget, which might be something like, "a creature (classification) with colorful bodies and with nothing else inside and two tails: one tail is a crescent moon and the other is like an eyeball (differentiation)."

 

Whole Class Discussion and Practice

10 minutes

This is a great chance to link the "Who's a Widget?" discussion to yesterday's Definitely, Maybe activity.  In Definitely, Maybe, the last image students worked with appeared to be a square but was actually a rectangle.  The main point of having this discussion is to impress upon students that observations are different from inferences, and that we often need more evidence to actually prove that our inferences are true!

I now ask students to test a definition for a rectangle, "a shape that has four right angles," by trying to find counterexamples.  After students share out counterexamples, focus again on the idea of definitions needing classification as well as differentiation, e.g., a better definition for a rectangle would be "a quadrilateral with four right angles."

If it seems like students might want another example on which to practice, they can test this definition for parallel lines, "two lines that never meet/intersect/cross."  Ideally, after testing this definition, students will see that they to mention that the lines must be in the same plane.  Using pencils is often a convincing way to show students the difference between parallel lines and skew lines; additionally, holding up a cube and showing students edges that will never intersect (but are definitely not parallel) can help.

Investigation

20 minutes

Investigation: Defining Angles

In the Defining Angles Investigation, I pass out an envelope containing examples and non-examples of different types of angles (right, acute, obtuse, vertical, linear, complementary, and supplementary).  Using the ideas of classification and differentiation, students will write and test definitions for different kinds of angles.

Directions:

  1. Ask students to sit as a group of four, but with the group divided into two pairs.
  2. Pass out an investigation envelope to each pair (the directions should be glued to the outside of the envelope and examples and non-examples of each type of angle should be inside).
  3. Pairs should come to an agreement on the definition for each type of angle and write the definition on their copy of the Defining Angles Workspace.
  4. Pairs may not trade definitions for testing until both pairs are done writing.

While students work, I circulate the room and look at students' definitions, making sure they start writing their definitions with the correct classification ("an angle...") and that they focus on the characteristics that differentiate the angles from one another (the angle's measure if it is acute, right, or obtuse, or, the angle's relative location to another if the pair of angles is vertical or linear.

Debrief

10 minutes

By this point, all students have worked on defining all the types of angles as well as tested others' working definitions of these angles.  Now is a great time to clean up and formalize these definitions for the whole class as students take notes.

I call on students to share out their definitions orally.  I write out their definitions and ask for feedback from the class.  We revise the Angles Definitions, if necessary, by taking notes in our Angles Note Taker.