Learning Math: Measurement
Classroom Case Studies, 3-5 Part C: Problems That Illustrate Measurement Reasoning (55 minutes)
Session 10: 3-5, Part C
In this part, you’ll look at several problems that are appropriate for students in grades 3-5. For each problem, answer the questions below. If time allows, obtain the necessary materials and solve the problems.
|Questions to Answer:
Take 24 square tiles. Make all the rectangles that have an area of 24 square units. Record the dimensions of each rectangle. What do you notice about the relationship of the length and width of a rectangle to its area? How are the dimensions of the rectangles related to their areas? Write a rule for finding the area of any rectangle given its length and width.
Examine the following measurements collected by students:
Length of Math Book
|Carlos and Pam||58 cm||29 cm|
|Linda and Jen||56 cm||29.5 cm|
|Yoji and Pete||57.5 cm||28.8 cm|
Why aren’t the measurements the same? What affects precision?
Cut out the net of a box with dimensions of 4 by 6 by 2 from centimeter grid paper. Without folding the net into a box, predict how many centimeter cubes will fit into the box when folded. Then connect two centimeter cubes together to form a 1-by-1-by-2 package. How many 1-by-1-by-2 packages fit into the box? What strategies did you use to predict how many centimeter cubes and how many 1-by-1-by-2 packages fit into the box? Check your answers by filling the box with cubes. Finally, generalize an approach to determining the number of cubes in a box.
Use geometry software such as Geometer’s Sketchpad for this problem. If geometry software is not available, collect pictures or models of the different types of triangles and measure the angles with a protractor.
Use the software to measure the angles in each of the triangles in the table. Find the sum of the angles of the triangle. Record your findings.
What patterns do you notice about the sum of the angles in a triangle? Create a few more triangles and find the angle sum for each of them. Do you think these patterns will hold for all triangles? Why or why not?
A net is a two-dimensional representation of a three-dimensional object, like a cube. It shows all the faces of the three-dimensional object and is connected in such a way that it lies flat, but can be folded up to form the three-dimensional object. For more information about nets, go to Learning Math: Geometry, Session 9, Part B.
Here is a sample net:
Using 24 square tiles, you can make four distinct rectangles: 1 by 24; 2 by 12; 3 by 8; and 4 by 6. Even though the lengths and widths differ from rectangle to rectangle, their product is always the same, and equal to the area of the rectangles — 24. Each of these rectangles represents an array of tiles with particular dimensions. If we count the tiles in each array, we get the area of the rectangle. The rule for finding the area of any rectangle, given its length and width, is length multiplied by width.
|Answers to Questions:
The measurements aren’t the same because the students may have used different measuring tools and techniques. Also, physically measuring an object is likely to produce some degree of measurement error. The measurement process is, by its nature, never exact. Precision is affected by the measuring tool. The smaller the unit on a measuring tool, the more precise it is.
| Answers to Questions:
The number of centimeter cubes that will fit into a 4-by-6-by-2 box is 48. The number of 1-by-2-by-2 cubes that will fit into this box is 24. Answers will vary for the type of strategy that can be used to predict how many packages of a particular size will fit into the box. One strategy is to first see how many centimeter cubes are needed to cover the base of the box (4-by-6 rectangle), which is 24. Then, because the height of the box will be 2, multiply the number of cubes in the base by 2 to get 48. Similarly, you can find out how many 1-by-1-by-2 cubes will cover the base of the box. Since the height of the box is the same as one dimension of the 1-by-1-by-2 cube, you can place the cubes vertically to determine how many will cover the base of the larger box, and in the process, entirely fill the space that will be encompassed by the folded box. Answers will vary on what kinds of approaches help determine the number of cubes in a box. One method is to multiply all three dimensions of a box to determine the number of centimeter cubes that will fit in that box
|Answers to Questions:
Depending on the triangles drawn, the measurements of the particular angles will vary (with the exception of the equilateral triangle, whose angles will all measure 60 degrees). The sum of the angles for every triangle is 180 degrees, and this will remain true for any triangle. Testing more triangles will demonstrate this further, but it is not a proof.
|Answers to Questions:
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.