# Classroom Case Studies, 6-8 Part B: Reasoning About Number and Operations (40 minutes) – Exploring Standards

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.

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?

### Solution B1

Students are using logical reasoning and their understanding of place value in base ten to help them interpret base five numbers and convert them to base ten. Students are also using mental mathematics and the order of operations to solve the problems. The students realize that they must choose the appropriate place value first, meaning using the exponents to find the greatest power of 5 that is less than the number they are converting.

### 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?

### Solution B2

When students transfer what they know about base ten to another base, their understanding of place value is extended and deepened. Also, when students have an understanding of the base ten place-value system, their ability to do complex computations increases. They are more able to use mental mathematics to solve problems.

### Problem B3

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

### Solution B3

Because students can picture what the manipulatives look like, they can understand that they first have to figure out what is the greatest power of 5 that is less than or equal to the number. This translates into dividing by the same power of 5 in the algorithm.

### Problem B4

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

### Solution B4

Ms. Miles’s lesson focuses on understanding numbers and ways of understanding numbers and number systems. For example, by converting from one base to another and vice versa, the students become more aware of the ways we represent numbers and the meanings underlying those representations, such as place value. They also make connections between the representations and are able to extend them when working with other bases.