Private: Learning Math: Measurement
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.
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.
Answer the questions based on what you saw in the video:
- What is one of the fundamental concepts of area? Why?
- What vocabulary must students understand to make sense of area?
- Are there related concepts or skills that will affect whether or not students can make sense of area?
- 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?
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.
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.
What types of experiences engage students in thinking about the conservation of area?
- 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.
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.
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.
Session 1 What Does It Mean To Measure?
Explore what can be measured and what it means to measure. Identify measurable properties such as weight, surface area, and volume, and discuss which metric units are more appropriate for measuring these properties. Refine your use of precision instruments, and learn about alternate methods such as displacement. Explore approximation techniques, and reason about how to make better approximations.
Session 2 Fundamentals of Measurement
Investigate the difference between a count and a measure, and examine essential ideas such as unit iteration, partitioning, and the compensatory principle. Learn about the many uses of ratio in measurement and how scale models help us understand relative sizes. Investigate the constant of proportionality in isosceles right triangles, and learn about precision and accuracy in measurement.
Session 3 The Metric System
Learn about the relationships between units in the metric system and how to represent quantities using different units. Estimate and measure quantities of length, mass, and capacity, and solve measurement problems.
Session 4 Angle Measurement
Review appropriate notation for angle measurement, and describe angles in terms of the amount of turn. Use reasoning to determine the measures of angles in polygons based on the idea that there are 360 degrees in a complete turn. Learn about the relationships among angles within shapes, and generalize a formula for finding the sum of the angles in any n-gon. Use activities based on GeoLogo to explore the differences among interior, exterior, and central angles.
Session 5 Indirect Measurement and Trigonometry
Learn how to use the concept of similarity to measure distance indirectly, using methods involving similar triangles, shadows, and transits. Apply basic right-angle trigonometry to learn about the relationships among steepness, angle of elevation, and height-to-distance ratio. Use trigonometric ratios to solve problems involving right triangles.
Session 6 Area
Learn that area is a measure of how much surface is covered. Explore the relationship between the size of the unit used and the resulting measurement. Find the area of irregular shapes by counting squares or subdividing the figure into sections. Learn how to approximate the area more accurately by using smaller and smaller units. Relate this counting approach to the standard area formulas for triangles, trapezoids, and parallelograms.
Session 7 Circles and Pi (π)
Investigate the circumference and area of a circle. Examine what underlies the formulas for these measures, and learn how the features of the irrational number pi (π) affect both of these measures.
Session 8 Volume
Explore several methods for finding the volume of objects, using both standard cubic units and non-standard measures. Explore how volume formulas for solid objects such as spheres, cylinders, and cones are derived and related.
Session 9 Measurement Relationships
Examine the relationships between area and perimeter when one measure is fixed. Determine which shapes maximize area while minimizing perimeter, and vice versa. Explore the proportional relationship between surface area and volume. Construct open-box containers, and use graphs to approximate the dimensions of the resulting rectangular prism that holds the maximum volume.
Session 10 Classroom Case Studies, K-2
Watch this program in the 10th session for K-2 teachers. Explore how the concepts developed in this course can be applied through case studies of K-2 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for K-2 students.
Session 11 Classroom Case Studies, 3-5
Watch this program in the 10th session for grade 3-5 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 3-5 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for grade 3-5 students.
Session 12 Classroom Case Studies, 6-8
Watch this program in the 10th session for grade 6-8 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 6-8 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for grade 6-8 students.