NOTE: This is taken from the Teacher's Edition of Bits and Pieces II

Bits and Pieces Mathematical and Problem-Solving Goals

Bits and Pieces II was created to help students:

• Continue to build understanding of fractions, decimals, and percents and the relationships among these concepts and their representations
• Explore situations that involve operations with rational numbers
• Use strategies to quickly estimate sums and products
• Use 0, ½, 1, 1½, and 2 as benchmarks to make sense of how large a sum is
• Develop strategies for adding and subtracting fractions and decimals
• Understand when addition or subtraction is the appropriate operation
• Develop ways to model sums and differences
• Become facile at changing a fraction to a decimal and at estimating what fraction a given decimal is near
• Develop an understanding of the multiplication of fractions and of decimals
• Use an area model to represent the product of two fractions
• Explore the relationship between two numbers and their product to generalize the conditions under which the product is larger than both factors, between the factors, or smaller than both factors
• Understand how to use percent as an expression of frequency when a data set contains more than or fewer than 100 pieces of data
• Represent $1.00 as 100 pennies so that a special application of the hundredths grid can be used to visualize percents of a dollar
• Use percents to estimate or compute taxes, tips, and discounts
• Draw pictorial models to represent a situation; for example, showing ½ of ½ is ¼ by drawing an area model
• Look for and generalize patterns
• Use estimation to help make decisions
• Use a problem’s context to help reason about the problem

The overall goal of Connected Mathematics is to help students develop sound mathematical habits. Through their work in this and other number units, students learn important questions to ask themselves about any situation that is represented and modeled mathematically such as: In what kinds of situations is it appropriate to use percents? What kinds of graphs can be created from information in percent form? How are percents like fractions and decimals? How are they different? What kinds of models can be developed to show computation with fractions and decimals? What algorithms can be developed from these models? Will these algorithms apply to all fractional quantities? How does the concept of multiplication of whole numbers extend to multiplication offractions? To multiplication of numbers in decimalform? Do results always match those found by using models? How can estimation skills and algorithm skills reinforce one another?”

The Mathematics in Bits and Pieces II

This unit does not teach specific algorithms for working with rational numbers. Instead, it helps the teacher create a classroom environment where students consider interesting problems in which ideas of fractions, decimals, and percents are embedded. Students bump into these important ideas as they struggle to make sense of problem situations. As they work individually, in groups, and as a whole class on the problems, they will find ways of thinking about and operating with rational numbers.

The teacher’s role is to help students make explicit their growing ideas about the world of rational numbers and, when students are ready to inject ideas and strategies into the conversation along with the ideas and strategies generated by the students. Simply giving students algorithms for moving symbols for rational numbers around on paper would be a mistake—and the temptation to do so is often great. All teachers want their students to succeed, and showing them how to do something—such as how to cross multiply to compare two fractions—gives the impression of immediate success. Students can do the algorithm by memorizing. However, evidence from student assessments shows that students do not understand algorithms that are given to them in this way and therefore cannot remember or figure out what to do in a given situation.

This unit provides a rich set of experiences that focus on developing meaning for computations with rational numbers. We expect students to finish this unit knowing algorithms for computation that they understand and can use with facility.