Private: Learning Math: Number and Operations
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.
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.
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.
What reasoning processes are the students using to solve the problems?
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.
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?
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.
How did the manipulatives help students understand how to use long division to solve the problem?
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.
What are some ways that you see the NCTM Standards being incorporated into Ms. Miles’s lesson?
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.
Session 1 What Is a Number System?
Understand the nature of the real number system, the elements and operations that make up the system, and some of the rules that govern the operations. Examine a finite number system that follows some (but not all) of the same rules, and then compare this system to the real number system. Use a number line to classify the numbers we use, and examine how the numbers and operations relate to one another.
Session 2 Number Sets, Infinity, and Zero
Continue examining the number line and the relationships among sets of numbers that make up the real number system. Explore which operations and properties hold true for each of the sets. Consider the magnitude of these infinite sets and discover that infinity comes in more than one size. Examine place value and the significance of zero in a place value system.
Session 3 Place Value
Look at place value systems based on numbers other than 10. Examine the base two numbers and learn uses for base two numbers in computers. Explore exponents and relate them to logarithms. Examine the use of scientific notation to represent numbers with very large or very small magnitude. Interpret whole numbers, common fractions, and decimals in base four.
Session 4 Meanings and Models for Operations
Examine the operations of addition, subtraction, multiplication, and division and their relationships to whole numbers. Work with area models for multiplication and division. Explore the use of two-color chips to model operations with positive and negative numbers.
Session 5 Divisibility Tests and Factors
Explore number theory topics. Analyze Alpha math problems and discuss how they help with the conceptual understanding of operations. Examine various divisibility tests to see how and why they work. Begin examining factors and multiples.
Session 6 Number Theory
Examine visual methods for finding least common multiples and greatest common factors, including Venn diagram models and area models. Explore prime numbers. Learn to locate prime numbers on a number grid and to determine whether very large numbers are prime.
Session 7 Fractions and Decimals
Extend your understanding of fractions and decimals. Examine terminating and non-terminating decimals. Explore ways to predict the number of decimal places in a terminating decimal and the period of a non-terminating decimal. Examine which fractions terminate and which repeat as decimals, and why all rational numbers must fall into one of these categories. Explore methods to convert decimals to fractions and vice versa. Use benchmarks and intuitive methods to order fractions.
Session 8 Rational Numbers and Proportional Reasoning
Begin examining rational numbers. Explore a model for computations with fractions. Analyze proportional reasoning and the difference between absolute and relative thinking. Explore ways to represent proportional relationships and the resulting operations with ratios. Examine how ratios can represent either part-part or part-whole comparisons, depending on how you define the unit, and explore how this affects their behavior in computations.
Session 9 Fractions, Percents, and Ratios
Continue exploring rational numbers, working with an area model for multiplication and division with fractions, and examining operations with decimals. Explore percents and the relationships among representations using fractions, decimals, and percents. Examine benchmarks for understanding percents, especially percents less than 10 and greater than 100. Consider ways to use an elastic model, an area model, and other models to discuss percents. Explore some ratios that occur in nature.
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.
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.