The Number strand develops far-reaching ideas about number: how to represent numerical quantities in various ways, and how to reason with those representations. A particular unit may focus more on developing algorithms and fluent skills, but in fact developing an algorithm for division of fractions, for example, calls upon students' ability to use models and to reason proportionally. Thus, computing with algorithms, and reasoning about quantities are complementary ideas.

The first unit in CMP, Prime Time, focuses on factors and multiples of whole numbers. The 6th grade sequence of Bits and Pieces units applies these understandings to fractions and decimals. In the 7th grade unit Comparing and Scaling students apply their understanding of the multiplicative structure of numbers to develop an understanding of rate and proportion. Accentuate the Negative introduces Integers, and negative fractions. This developmental sequence expands students' ideas about number, and strategies for working with numbers, from the familiar set of Whole Numbers, to the set of Rational Numbers. The set of Irrational Numbers is first introduced in Looking for Pythagoras. Meanwhile, ideas developed in the Number strand are continually used in intervening units from other strands, giving students many opportunities to review, practice and refine their understandings about Number.

Overall Goals for CMP Number Strand

Connected Mathematics develops four mathematical strands: Number and Operation, Geometry and Measurement, Data Analysis and Probability and Algebra. The mathematical learning goals below signify what students should be able to do in Number by the end of eighth grade.

Number Sense

Use numbers in various forms to solve problems (6, 7, 8)
Understand and use large numbers, including in exponential and scientific notation (6, 7, 8)
Reason proportionally in a variety of contexts using geometric and numerical reasoning, including scaling and solving proportions (6, 7, 8)
Compare numbers in a variety of ways, including differences, rates, ratios, and percents and choose when each comparison is appropriate (6, 7, 8)
Order positive and/or negative rational numbers (6, 7, 8)
Express rational numbers in equivalent forms (6)
Make estimates and use benchmarks (6, 7, 8)

Operations and Algorithms

Develop understanding and skill with all four arithmetic operations on fractions and decimals (6)
Develop understanding and skill in solving a variety of percent problems (6)
Use the order of operations to write, evaluate, and simplify numerical expressions (7, 8)
Develop fluency with paper and pencil computation, calculator use, mental calculation, and estimation; and choose among these when solving problems (6, 7)

Properties

Understand the multiplicative structure of numbers, including the concepts of prime and composite numbers, evens, odds, and prime factorizations (6)
Use the commutative and distributive properties to write equivalent numerical expressions (7, 8)

Number Units

There are 6 Number Units in CMP.

Prime Time (Grade 6)
Bits and Pieces I (Grade 6)
Bits and Pieces II (Grade 6)
Bits and Pieces III (Grade 6)
Comparing and Scaling (Grade 7)
Accentuate the Negative (Grade 7)

Every important idea addressed in the CMP Number Strand is carefully placed to make initial development appropriate to student developmental level, and also to connect productively to other units already studied. For example, the idea of "Equivalent Fractions" comes in formally with Bits and Pieces I, but prior to this Unit students have been finding common factors and common multiples in Prime Time, tools which they need to rewrite fractions in equivalent forms. The concept of equivalence of fractions is then revisited in Bits and Pieces II and III, and becomes a useful strategy for solving proportions in Comparing and Scaling.

It is important to note that many goals are revisited in later units, in the same grade level or later, either within classroom problems or in the Connections problems in the ACE homework assignments. For example, the goal under Number Sense of "Express rational numbers in equivalent forms" is labeled as a Grade 6 goal, because the unit Bits and Pieces I includes this goal as a "big idea." However, practice with this goal occurs throughout the curriculum.

In order to have a clearer idea of the particular goals for each unit, the Mathematical Help section lists the goals for each unit.

Relating Number in CMP to High School

It is unusual for high schools to offer a separate course in Number, in the same way that Geometry and Algebra are offered as separate courses. High School teachers assume that students come in with an understanding of number concepts as they relate to the set of Real Numbers, that is, Rational Numbers (Bits and Pieces I, II and III, and Accentuate the Negative) and Irrational Numbers (Looking for Pythagoras) and expect students to apply these understandings in high school classes. For example, the ability to apply the properties of Real Numbers to algebraic expressions, to reason proportionally, and to compute with Real Numbers are skills that are frequently called upon in high school Algebra classes. CMP students are well prepared to meet these expectations. In high school CMP students' understanding of Number will expand to include the set of complex numbers.

The approach in CMP focuses on making sense of numerical expressions with estimates, benchmarks and models, on using equivalent forms of the same quantity (fraction, decimal, percent, exponential), and on reasoning about relationships (order and equivalence). While computation skills are valued and developed in CMP, and practiced throughout, the focus on reasoning and making sense means that CMP students are encouraged to decide when and how to use strategies from their repertoire of representational and computation and estimation skills. Knowing when to apply a skill is as important as knowing how. This approach fosters a sense of confidence in students as they apply and extend their middle school understanding of Number in unfamiliar contexts.