Private: Learning Math: Measurement
About the Learning Math Courses
When teachers have a deep conceptual understanding of mathematics, it can help their students develop strong mathematical skills and knowledge. Learning Math is a series of five multimedia, college-level courses designed to teach mathematics content to elementary and middle school teachers. Organized around the content standards developed by the National Council of Teachers of Mathematics (NCTM), the courses cover Number and Operations; Patterns, Functions, and Algebra; Geometry; Measurement; and Data Analysis, Statistics, and Probability.
Synopses of sessions, video lists, production credits and copyright information.
- To help teachers better understand mathematics content
- To provide engaging explorations of mathematics using video, interactive activities, and problem solving
- To encourage teachers to view mathematics as more than rote sets of rules and procedures
How to Use These Courses
Each Learning Math course includes sequentially organized problems, video viewing, interactive activities, readings, and homework. The multimedia elements of the course create an exciting environment for probing mathematics content. The course can be taken entirely on the Web, followed in a print guide, or completed using a combination of Web and print.
The following sequence of activities will give you a sense of what you will do as a student using Learning Math:
1. Watch the session video in its entirety. You can watch the video before you begin the session to become comfortable with the material, or you can view the video after completing the session (so as not to view answers to problems).
2. Do problems sequentially. Each session is divided into several parts indicated in the tabs along the top of the page. The problems are numbered within each part. (e.g., Problem A1)
3. If you are having difficulty with a problem, look for a Tip icon.
4. Find the solutions to problems in the Solutions section below.
5. If you want a more rigorous challenge, try out a Take It Further problem.
6. Read the Notes as you go along, to establish a context for the content and for suggestions on doing the activities in groups.
7. Watch video segments strategically placed throughout the session by fast-forwarding or skipping to the approximate time code indicated. Use the time codes provided to locate the segment.
8. Do homework problems and readings (available as PDF files online) at the end of each session to reinforce your learning.
Ways To Take Learning Math
Learning Math was flexibly designed for a variety of users and situations. You may choose to work through the sessions on your own, in a study group, or as part of a facilitated, face-to-face, professional development workshop.
Facilitating the Course
You can prepare for facilitating the course by reading through each session and its “Notes” section prior to meeting with your group. Reading through the material will help you become clear about the activities, plan how much time you need to spend on each one, and pull together necessary materials. The course is designed for use by an individual, but the Notes may suggest ways for groups to work through the sessions.
Each Learning Math course is comprised of 10 two-and-a-half-hour sessions. The first nine sessions are devoted to mathematics content; the 10th session covers classroom applications. Concepts are developed within and across the sessions, and the sessions increase in difficulty as they progress. Each session includes reading, problem solving, and group or individual activities that are available on the web and in print, and a half-hour of video viewing, available on the web. There are additional problems and readings to complete for homework.
The following components are in each course:
Key mathematical terms relevant to each session are listed at the beginning of that session. These terms are divided into two parts: terms that are new in that session and terms that were introduced in a previous session. Definitions for key terms may be found by clicking on the Glossary link.
Notes can be used by facilitators, study groups, or individuals working alone. They provide extended information about the topics presented in the course, including help for dealing with stumbling blocks that may come up and recommendations for different ways to approach the content.
Each session contains mathematical problems to be solved individually or by groups. Problems build upon previous concepts and increase in difficulty as the course progresses.
Take It Further
The problems marked “Take It Further” are optional and are not counted as part of the two-and-a-half hour time-frame for each session. These problems are designed for individuals who would like to explore a topic in greater depth. They are often more difficult than the other problems in the session, and they may introduce new information or concepts not previously discussed.
Interactive Activities (ALL FLASH ACTIVITIES ARE NOW DISABLED)
Each session in the course includes at least one interactive activity. These activities help you learn new mathematics content or reinforce existing knowledge through hands-on exploration directly on the Web. The interactive activities require the Flash plug-in, which you can download for free from Macromedia’s Web site. There are also non-Flash versions of each activity that don’t require the Flash plug-in.
A Tip is provided at the end of many problems in the sessions. The tips are hidden so those who prefer to try to solve the problem without any help can do so. To view the tip for a selected problem, click on the “Tip” button. To hide it, click “Close Tip” at the end of the tip text.
A solution is provided for every problem in Learning Math, with the exception of a few open-ended questions. When solving a problem with multiple parts, consider writing down your answers to all of the parts on paper first before checking the solution, because the answers to each part of the problem will be visible at once on the solution page.
Write and Reflect
Open-ended problems that encourage you to think about a particular concept include a Write and Reflect button rather than a Solution button.
Each Learning Math session includes viewing a video that is available on the course website. The videos feature K-8 teachers working on the Learning Math course materials in a workshop with a facilitator. The videos for the nine content sessions show onscreen participants as they are introduced to the concepts, work through the problems, sometimes struggle to reach an understanding, and then reflect on what they have learned. At the end of most videos there is an example of how the content from the session is applied in a “real world” situation. The videos for the 10th session show participants from the videotaped workshops as they apply the content that they have learned back in their own classrooms. You may choose to watch each of these videos before or after you work on the associated course session.
Each session includes short excerpts from the associated video, which you watch (or review) and reflect on to see how the onscreen participants grapple with the same or similar problems and concepts you encounter in the course. The segments are available on the course website, if you are watching the videos online.
Each session includes approximately 45 minutes of homework problems and/or reflective writing assignments that reinforce the session’s content.
Readings from journals and books are cited at the end of some sessions. They are available on the course website as downloadable PDF files.
To use the Learning Math courses online, we recommend the following:
Minimum 56K modem connection, but ISDN or high-speed connection is recommended. The slower your connection, the longer it will take to load larger features, such as the Flash activities. To view the video programs and video segments online, a DSL connection, cable modem, or LAN connection to a T1 line or greater is required.
Here are links to sites where you can download (for free) the plug-ins you’ll need to get the most out of the courses:
Notes on printing:
If you are having trouble printing some of the course content pages, you may try doing one or more of the following (from the “Print Preview”, “Print…”, or “Page Setup…” menu):
• Turn on “Shrink to Fit” mode (IE 5 only)
• Print the page in “Landscape” mode
• Reduce the scale of the printer output
• In the Measurement course, certain PDF documents are drawn to scale. When printing them, be sure that the Acrobat print dialog box settings for page scaling or shrinking oversized pages are turned off.
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.