A B C

Solutions for Session 10, Part C

See solutions for Problems: C1 | C2 | C3

Problem C1

a.

Answers will vary. Some of the goals:

 • To visualize and describe smaller shapes that will be created by cutting larger shapes • To identify the number of sides and vertices of simple two-dimensional shapes • To identify figures that are congruent (the same shape and size)

b.

Prerequisite knowledge includes identifying and naming squares, triangles, and non-square rectangles, as well as identifying the sides and vertices of figures.

c.

In this activity, students work across levels. They create and identify particular shapes. But they also visualize, make predictions, and think about properties and how to create them (especially when students try to create six different outcomes). This means they are working at level 0 and level 1.

d.

The activity prepares students for other dissection-related activities (like the one in Session 5) and for thinking about extreme cases (which creates the most sides, cutting vertex to vertex or side to side; and does it matter which sides?). It also prepares students for thinking about congruence and similarity.

e.

Possible extension: Begin with an equilateral triangle or a trapezoid instead of a rectangle. Have the students follow the same procedure. (1) Identify the shapes that can be made with one straight cut from one side or vertex of the original shape to another side or vertex, and (2) compare the sizes and shapes of the cut figures to identify those that are congruent.

Problem C2

 a. This activity encourages students to think about different properties that figures can have, including sorting by straight sides and curved sides, concave and convex ("dented" and not), right angles and no right angles, and so on. b. This activity has no prerequisite skills, since students are not asked to name the shapes or to sort them with regard to any particular attribute. Allowing students to think creatively about sorting and properties can lead to more structured activities like the ones on Venn diagrams in Session 3. c. This activity is primarily level 0, as students are focused on particular shapes and not families. But categorizing by properties helps move them towards level 1 thinking. d. To extend students' thinking, it is important to ask them to verbalize what the shapes in each of their sorted categories have in common. You can then ask them to draw new shapes that belong to either of their categories. e. Ideas for lessons will vary. One way to do this may be to begin with sorting drawings of everyday objects as a whole class. For example, hold up several drawings of different types of shoes, different types of clothing, and different types of food. Tell students you want to make three groups, putting together all of the things that are alike. When that is done, tell students they will receive an envelope of shapes. Their goal is to put the shapes into two categories, so that everything in the same category has something in common. When students have completed the task, have different students share how they sorted the shapes.

Problem C3

 a. The goal of this activity is for students to identify shapes by their properties. b. To work through this activity, students need to know the particular vocabulary you choose to focus on. Possible vocabulary words include, "right angle, parallel, 90°." This will help them develop the skill of moving between the familiar name for an object (e.g., "square") and the properties of that object that might be currently relevant (e.g., "four sides that are the same length.) c. This activity is much like Problem C2, but moving students more towards level 1. When students name the shapes that remain after a sorting and sort by the same information in a different order, it helps them shift their attention from the particular shapes to the classes of shapes they represent. They may also use some "if-then" thinking: For example, if a shape is a rectangle, it will have four 90° angles. d. To extend students' thinking, you can ask them to choose a property to sort by. Ask a student to look at one of the shapes in front of him and to think quietly about something that is true of that shape. Then have the student announce just the property. Everyone must sort by that student's property. e. A lesson could be very much like the one in Problem C2. For example, start with everyday objects (clothing, animals, furniture, etc.) and ask students to sort by properties. For example, if you asked for things with legs, you could choose many of the animals, and perhaps some of the furniture (chairs, tables, etc.) After a couple of examples, give students their shapes and a few minutes to explore them and become familiar with them. Start with easier properties, like the number of sides or angles, all straight sides or curves, etc. Then move into the more challenging properties.