Private: Learning Math: Number and Operations

Professional Development > Private: Learning Math: Number and Operations > 12. Classroom Case Studies, 6-8 > 12.5 Homework – Grades 6-8

Mathematics

K-2, 3-5, 6-8

Classroom Case Studies, 6-8 Homework – Grades 6-8

Solutions are not provided for these homework problems, since answers will vary depending on individual experiences.

Problem H1

Assume that you need to report back to your grade-level team or to the entire school staff at a faculty meeting on your experiences and learning in this course. What are the main messages about the teaching of number and operations that you would share with your colleagues? Prepare a one-page handout or an overhead or slide that could be distributed or shown at the meeting.

Problem H2

Look at a lesson or activity in your own mathematics program for your grade level that you think has potential for developing students’ reasoning about number and operations. If you were to use this lesson or activity now, after taking this course, how might you modify or extend it to bring out more of the important ideas about number and operations?

Series Directory

Private: Learning Math: Number and Operations

Credits

Produced by WGBH Educational Foundation. 2003.

ISBN: 1-57680-678-2

Sections

12.1 Part A: Observing a Case Study (25 minutes)

12.2 Part B: Reasoning About Number and Operations (40 minutes) – Exploring Standards

12.3 Part B (continued): Reasoning About Number and Operations – Examining Students’ Reasoning

12.4 Part C: Problems That Illustrate Reasoning About Number and Operations (55 minutes)

12.5 Homework – Grades 6-8

Programs

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.

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.

Private: Learning Math: Number and Operations

Classroom Case Studies, 6-8 Homework – Grades 6-8

Problem H1

Problem H2

Series Directory

Credits

Sections

12.1 Part A: Observing a Case Study (25 minutes)

12.2 Part B: Reasoning About Number and Operations (40 minutes) – Exploring Standards

12.3 Part B (continued): Reasoning About Number and Operations – Examining Students’ Reasoning

12.4 Part C: Problems That Illustrate Reasoning About Number and Operations (55 minutes)

12.5 Homework – Grades 6-8

Programs

Session 1 What Is a Number System?

Session 2 Number Sets, Infinity, and Zero

Session 3 Place Value

Session 4 Meanings and Models for Operations

Session 5 Divisibility Tests and Factors

Session 6 Number Theory

Session 7 Fractions and Decimals

Session 8 Rational Numbers and Proportional Reasoning

Session 9 Fractions, Percents, and Ratios

Session 10 Classroom Case Studies, K-2

Session 11 Classroom Case Studies, 3-5

Session 12 Classroom Case Studies, 6-8

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