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# Classroom Case Studies, K-2 Part A: The Concept of Area (25 minutes)

## Session 10: K-2, Part A

To begin exploring what the teaching of measurement might look like in the classroom, participants in the Measurement course first reexamined the big ideas around one topic: area. They considered how children make sense of these ideas and discussed ways to present these concepts to young students.

 Video Segment In this video segment, Dr. Chapin reviews four big ideas of area. To plan lessons that will help students understand area concepts, it is essential that teachers consider these and other big mathematical ideas when designing instructional sequences. You can find this segment on the session video approximately 1 minute and 15 seconds after the Annenberg Media logo.

### Problem A1

Answer the questions based on what you saw in the video:

1. What is one of the fundamental concepts of area? Why?
2. What vocabulary must students understand to make sense of area?
3. Are there related concepts or skills that will affect whether or not students can make sense of area?
4. Thinking back to the big ideas of this course, what are some other ideas that young students should encounter to help extend and deepen their understanding of the topic?

Problem A2

Choose one of the concepts that you listed for Problem A1 and describe an instructional activity that you might use to help students grasp that concept.

 Video Segment In this video segment, kindergarten, first-, and second-grade teachers discuss some of the important ideas about area. One teacher mentions that the conservation of area has to be considered when teaching young children. You can find this segment on the session video approximately 3 minutes and 30 seconds after the Annenberg Media logo.

Problem A3

What types of experiences engage students in thinking about the conservation of area?

### Solutions

Problem A1

1. A fundamental concept of area is that it is the measure of how much surface is covered. For students of this age, understanding what it means to cover a surface completely with a particular unit is central. Some other concepts of area are as follows: a) some shapes cover a surface more completely than other shapes; b) the units associated with area measurement are square units; and c) the smaller the square unit, the more square units are needed to determine the area.
2. To make sense of area, students need to be familiar with vocabulary such as surface, covering, and squares. Vocabulary that students will acquire as they explore area include unit and square unit.
3. Students who can recognize a square and the two-dimensionality of a square and who also have a solid understanding of rectangles are in good shape to tackle the concept of area. Students will also need prior experience covering surfaces with different objects to know that some shapes fit together with no holes or gaps (e.g., rectangles and triangles), while other shapes leave holes (e.g., circles).
4. Some other ideas, in addition to the ones already mentioned, include conservation and transitivity. Students need to learn that the area of a shape will not change if it is moved to a different position, or if it is cut and transformed in a certain way. They also need to understand that when you can’t compare two objects directly, you can compare them by means of a third object.

Problem A2

Answers will vary. The following activity can be used to help students start thinking about some of the concepts of area: On a large piece of paper, draw several shapes of different sizes. Then ask students to cover one of these shapes with different pattern blocks. (Pattern blocks are a commercial product found in most primary classrooms that consist of blocks in six shapes — triangles, squares, hexagons, trapezoids, and two rhombuses.) Have students cover one shape at a time and count the number of blocks needed to cover each shape. Students will find that they need more smaller blocks than larger blocks to cover the shape on the paper. This helps students start to internalize the idea that the size of the unit (in this case, pattern blocks) affects the number of units needed to cover a surface. It also adds to children’s development of the idea of area as a covering with no holes or gaps.

Problem A3

During their study of area, many students will be challenged by the idea of conservation. Conservation is the principle that an object maintains the same size and shape even if it is repositioned or divided in certain ways. But at the heart of teaching this principle, as one teacher said, is the notion of taking an abstract idea like area and making it concrete or tactile for young children. Students should have many experiences with determining area by tiling shapes or figuring out how much surface a shape takes up. Understanding that a shape will preserve its area regardless of its orientation is also an important first step.