Learning Math: Measurement
Classroom Case Studies, 35 Part C: Problems That Illustrate Measurement Reasoning (55 minutes)
Session 10: 35, Part C
In this part, you’ll look at several problems that are appropriate for students in grades 35. For each problem, answer the questions below. If time allows, obtain the necessary materials and solve the problems.
Questions to Answer:

Problem C1
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.
Problem C2
Examine the following measurements collected by students:
Names 
Circumference 
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?
Problem C3
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 1by1by2 package. How many 1by1by2 packages fit into the box? What strategies did you use to predict how many centimeter cubes and how many 1by1by2 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. Note 4
Problem C4
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?
Notes
Note 4
A net is a twodimensional representation of a threedimensional object, like a cube. It shows all the faces of the threedimensional object and is connected in such a way that it lies flat, but can be folded up to form the threedimensional object. For more information about nets, go to Learning Math: Geometry, Session 9, Part B.
Here is a sample net:
Solutions
Problem C1
Solution:
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:

Problem C2
Solution:
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:

Problem C3
Solution:
The number of centimeter cubes that will fit into a 4by6by2 box is 48. The number of 1by2by2 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 (4by6 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 1by1by2 cubes will cover the base of the box. Since the height of the box is the same as one dimension of the 1by1by2 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:

Problem C4
Solution:
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:
