## Learning Math: Geometry

# Classroom Case Studies, K-2 Part C: Activities That Illustrate Geometric Reasoning (55 minutes)

## Session 10: K-2, Part C

### In This Part

- Cutting Corners Activity
- Shape-Sorting Activities

In this part, you’ll look at some activities designed for students in grades K-2. As you read each activity, answer these questions:

- What is the geometry content in this activity?
- What skills do students need to work through this activity? What skills will this activity help them develop for later work?
- What level of geometric thinking is expected of students in the activity? Does it ask students to bridge levels?
- What other questions might extend students’ thinking about the activity?
- Describe a lesson that you could develop based on the content of this activity.

**Note 6**

**Activity Summary**

Students explore ways to cut rectangles to make other shapes, including smaller rectangles, triangles, and shapes of the same size.

**Materials Needed: one pair of scissors for each student
**

- six sheets of 8 1/2″ x 11″ paper
- paste
- six sheets of 12″ x 15″ colored construction paper for each student
- a straightedge for each group of students
- a pencil for each student

**Begin Activity**

Hold up a rectangular piece of paper and ask students to identify the shape. Also ask questions such as, “How many sides does a rectangle have? How many vertices does a rectangle have? What is special about the angles of a rectangle?” As students answer, point out the sides and vertices to reinforce the vocabulary and review right angles.

Ask, “How can I cut this rectangle to get two smaller rectangles whose shape and size are the same?” Take several suggestions from different students and follow their methods to cut several sheets of paper. Have the class verify that the resulting pieces are rectangles, and have them compare the rectangles’ sizes and shapes. You can introduce the word “congruent” to describe two figures that fit perfectly on top of each other.

Using a new sheet of paper, call on students to tell where the paper should be cut to make two triangles. Have students compare a couple of different methods, as well as the sizes and shapes of the triangles they create. Also ask students to identify sides and vertices of the triangles to reinforce the vocabulary.

**Main Activity**

Hold up a fresh sheet of paper and say: “I will make one straight cut. I will start here (point to one side) and stop here (point to an adjacent side). Tell me about the shapes you think I will get.” After the students describe the shapes, cut off a corner of the rectangular piece of paper, leaving a small triangle and a large pentagon. Have the students identify the number of sides and vertices of each shape, and repeat the word “pentagon” with students as that shape is discussed.

Review the names, numbers of sides, and number of vertices for several figures, including circle, triangle, square, rectangle, trapezoid, and pentagon. Give students scissors, paste, sheets of paper, and large construction paper. Tell them that their task is to create shapes by drawing a line from one side or vertex of the white paper to another side or vertex with a straightedge and then cut along the line. The students can then paste the resulting two pieces on the construction paper and record the names of the shapes. Have them repeat the activity six times, each time trying to create two new shapes.

**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 C1**

Answer the following questions about the Cutting Corners activity:

- What is the geometry content in this activity?
- What skills do students need to work through this activity? What skills will this activity help them develop for later work?
- What level of geometric thinking is expected of students in the activity? Does it ask students to bridge levels?
- What other questions might extend students’ thinking about the activity?
- Describe a lesson that you could develop based on the content of this activity.

Cutting Corners activity adapted from Findell, Carol R.; Small, Marian; Cavanagh, Mary; Dacey, Linda; Greenes, Carole E.; and Sheffield, Linda Jensen. *Navigating through Geometry in Prekindergarten-Grade 2.* pp. 22-25. Copyright © 2001 by the National Council of Teachers of Mathematics.

Used with permission. All rights reserved.

### Shape-Sorting Activities

### Problem C2

- What is the geometry content in this activity?
- What skills do students need to work through this activity? What skills will this activity help them develop for later work?
- What level of geometric thinking is expected of students in the activity? Does it ask students to bridge levels?
- What other questions might extend students’ thinking about the activity?
- Describe a lesson that you could develop based on the content of this activity.

**Shape Sort #2**

Ask students to sort shapes by naming properties, not by naming the shapes. When two or more properties are combined, have them sort by one property at a time. Example: “Find all of the shapes that have four sides.” (Find these.) “Now find those that also have all right angles.” (This group should include squares as well as non-square rectangles.) After sorting, discuss what the name of the shapes is. Also try sorting by the same combination of properties but in a different order.

### Problem C3

- What is the geometry content in this activity?
- What other questions might extend students’ thinking about the activity?
- Describe a lesson that you could develop based on the content of this activity.

Shape-Sorting activities adapted from Van de Walle, John A. Geometric Thinking and Geometric Concepts. In Elementary and Middle School Mathematics: Teaching Developmentally, 4th ed. pp. 342-349. Copyright © 2001 by Pearson Education. Used with permission. All rights reserved.

### Notes

**Note 6
**

It’s difficult to identify the important content and how students might approach an activity without actually doing the mathematics yourself. Allow yourself time to work through the mathematics, even briefly, before going on to answering the other questions.

### Solutions

**Problem C1**

- 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)

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

- 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.
- 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.
- 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.
- 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.
- 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

- The goal of this activity is for students to identify shapes by their properties.
- 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.)
- 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.
- 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.
- 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.