Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

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Learning Math Home
Measurement Session 10: Classroom Case Studies
Session 10 Session 10 k-2 Part A Part B Part C Homework
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Session 10 Materials:

A B C 


Solutions for Session 10, Part A

See solutions for Problems: A1 | A2 | A3

Problem A1


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.


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.


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).


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.

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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.

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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.

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