Session 10, Part B:
Reasoning About Number and Operations (40 minutes)

In This Part: Exploring Standards | Examining Students' Reasoning

The National Council of Teachers of Mathematics (NCTM, 2000) has identified number and operations as a strand in its Principles and Standards for School Mathematics. In grades pre-K through 12, instructional programs should enable all students to do the following:

 • Understand numbers, ways of representing numbers, relationships among numbers, and number systems • Understand the meaning of operations and how they relate to one another • Compute fluently and make reasonable estimates

In grades 6-8 classrooms, students are expected to do the following:

 • Develop, analyze, and explain methods for solving problems involving proportions • Work flexibly with fractions, decimals, and percents to solve problems • Compare and order fractions, decimals, and percents efficiently, and find their approximate locations on a number line • Develop an understanding of large numbers, and recognize and appropriately use exponential, scientific, and calculator notation • Develop meaning for integers, and use them to represent and compare quantities • Develop and use strategies to estimate the results of rational-number computations and judge the reasonableness of the results

"In the middle-grades mathematics classrooms, young adolescents should regularly engage in thoughtful activity tied to their emerging capabilities of finding and imposing structure, conjecturing and verifying, thinking hypothetically, comprehending cause and effect, and abstracting and generalizing" (NCTM, 2000, p. 211).

Watch another segment from Ms. Miles's class, and think about how the students are developing this understanding of number and operations.

 Video Segment In this video segment, Ms. Miles leads students through the process of using the division algorithm instead of manipulatives to convert from base five to base ten and from base ten to base five. The students then work in groups to convert base ten numbers to base five. If you are using a VCR, you can find this segment on the session video approximately 17 minutes and 39 seconds after the Annenberg Media logo.

 Problem B1 What reasoning processes are the students using to solve the problems?

 Problem B2 How does working in a different base develop students' sense of number and operations? What does working with different bases tell us about place-value systems?

 Join the discussion! Post your answer to Problem B2 on Channel Talk; then read and respond to answers posted by others.

 Problem B3 How did the manipulatives help students understand how to use long division to solve the problem?

 Problem B4 What are some ways that you see the NCTM Standards being incorporated into Ms. Miles's lesson?

 Principles and Standards for School Mathematics Copyright © 2000 by the National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or redistributed electronically or in other formats without written permission from NCTM. standards.nctm.org Standards are listed with the permission of the National Council of Teachers of Mathematics (NCTM). NCTM does not endorse the content or validity of these alignments.
 Session 10, Grades 6-8: Index | Notes | Solutions | Video

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