Connected Math Project

Authors• Glenda Lappan, Michigan

State UniversityElizabeth Difanis Phillips, Michigan State UniversitySusan N. Friel, University of North CarolinaWilliam M. Fitzgerald,

• Michigan State University(Deceased)James T. Fey, University of Maryland

http://connectedmath.msu.edu/index.shtml

Developing summary

Features and benefits of CMP2• NSF funded. CMP2 has been classroom tested for five years as part of a

new NSF grant prior to publication to ensure student success with the materials.

• Problem-centered, research-based approach. The same problem-centered, research-based approach proven successful with students as the original; content is developmentally appropriate for middle-school students.

• Embeds important mathematical concepts in interesting problems. Students learn important mathematical ideas in the context of interesting, interconnected problems. This exploration leads to understanding and the development of higher-order thinking skills and problem-solving strategies.

• Accessible to all levels of students. CMP2 is an effective combination of content and methodology designed to foster more "a-ha!" moments, regardless of a student's skill level or learning style.

• New technology to support learning! CMP2 now comes with updated technology to support teachers and students. Support is provided for digital presentations and StudentEXPRESS™ provides an interactive version of the textbook, with built-in homework help!

Features and benefits of CMP3• Engage students in active, personalized learning • CMP3 takes inquiry-based learning to the next level. New digital tools

engage students while driving conceptual understanding, procedural skill, and real-world applications.

• Teach the Common Core, teach with greater ease • CMP3 aligns to the Common Core State Standards and prepares

students for college and careers. Technology applications help you manage your classroom with fidelity, maximize instructional time, and capture needed student data.

• Apply a research-proven instructional approach • CMP3 offers the most comprehensive research base of any middle

school mathematics program. Download Connected Mathematics Project research and evaluation reports.

CMP Instructional Model

• Launch• Explore • Summarize

Practice With Concepts, Related Skills, and Algorithms

• Immediate practice should be related to the situations in which the ideas have been developed and learned.

• Continued practice should use skills and procedures in situations that connect to ideas that students have already encountered.

• Students need opportunities to use the ideas and skills in situations that extend beyond familiar situations. These opportunities allow students to use skills and concepts in new combinations to solve new kinds of problems.

• Students need practice distributed over time to allow ideas, concepts and procedures to reach a high level of fluency of use in familiar and unfamiliar situations and to connect to other concepts and procedures.

• Students need guidance in reflecting on what they are learning, how the ideas fit together, and how to make judgments about what is helpful in which kinds of situations.

• Throughout the Number and Algebra Strands development, students need to learn how to make judgments about what operation or combination of operations or representations is useful in a given situation, as well as, how to become skilful at carrying out the needed computation(s). Knowing how to, but not when to, is insufficient.

Students’ materials

• Unit opener • Mathematical Highlighting

Investigations

• Problems • Getting ready?

Did you know?

Applications- Connections- Extensions• Applications Connections Extensions

• Mathematical Reflections

• Unit project

Looking Back and Looking Ahead

Glossary

Technology and other resources

• Calculators• Applets (provided on the Students Activities CD-ROM)• Parent Materials• Special Needs Booklet• TeacherExpress CD-ROM, ExamView CD-ROM• Additional online resources (NCTM Illuminations; Project

Interactive; National Library of Virtual Manipulative; SELECT Math Website; MISTM MATH Portal; NEIRTEC Mathematics Website; Texas Instruments online course in the TI-83, 84; TI and CMP Activities; Geometer’s Sketchpad and CMP Activities; Tinkerplots)

Developing Depth of Understanding and Use

• Grade 6 Bits and Pieces I and II introduce students to fractions and their various meanings and uses. Models for making sense of fraction meanings and of operating with fractions are introduced and used. These early experiences include fractions as ratios. The extensive work with equivalent forms of fractions builds the skills needed to work with ratio and proportion problems. These ideas are developed further in the probability unit How Likely Is It? in which ratio comparisons are informally used to compare probabilities. For example, is the probability of drawing a green block from a bag the same if we have 10 green and 15 red or 20 green and 30 red?

• Grade 7 Stretching and Shrinking introduces proportionality concepts in the context of geometric problems involving similarity. Students connect visual ideas of enlarging and reducing figures, numerical ideas of scale factors and ratios, and applications of similarity through work with problems focused around the question: "What would it mean to say two figures are similar?"

The next unit in grade seven is the core proportional reasoning unit, Comparing and Scaling, which connects fractions, percents, and ratios through investigation of various situations in which the central question is: "What strategies make sense in describing how much greater one quantity is than another?" Through a series of problem-based investigations, students explore the meaning of ratio comparison and develop, in a progression from intuition to articulate procedures, a variety of techniques for dealing with such questions.

A seventh grade unit that follows, Moving Straight Ahead, is a unit on linear relationships and equations. Proportional thinking is connected and extended to the core ideas of linearity- constant rate of change and slope. Then in the probability unit What Do You Expect?, students again use ratios to make comparisons of probabilities.

• Grade 8 Thinking With Mathematical Models; Looking For Pythagoras; Growing, Growing, Growing, and Frogs, Fleas, and Painted Cubes extend the understanding of proportional relationships by investigating the contrast between linear relationships and inverse, exponential, and quadratic relationships. Also in Grade Eight, Samples and Populations uses proportional reasoning in comparing data situations and in choosing samples from populations.

These unit descriptions show two things about Connected Mathematics-the in-depth development of fundamental ideas and the connected use of these important ideas throughout the rest of the units.

Student Learning: Rationale for a Problem-Centered Curriculum

• A problem-centered curriculum not only helps students to make sense of the mathematics, it also helps them to process the mathematics in a retrievable way.

• In CMP, important mathematical ideas are embedded in the context of interesting problems. As students explore a series of connected problems, they develop understanding of the embedded ideas and, with the aid of the teacher, abstract powerful mathematical ideas, problem- solving strategies, and ways of thinking. They learn mathematics and learn how to learn mathematics

Alignment with NCTM Principles and Standards 2000• Content Standards• Number and Operations• Prime Time (Grade 6)• Bits and Pieces I (Grade 6)• Bits and Pieces II (Grade 6)• Comparing and Scaling (Grade 7)• Numbers Around Us (Grade 7)• Accentuate the Negative (Grade 7)• Looking for Pythagoras (Grade 8)• Clever Counting (Grade 8) @ 2004• Algebra• Variables and Patterns (Grade 7)• Moving Straight Ahead (Grade 7)• Thinking With Mathematical Models (Grade 8)• Looking for Pythagoras (Grade 8)• Growing, Growing, Growing (Grade 8)• Frogs, Fleas, and Painted Cubes (Grade 8)• Say It With Symbols (Grade 8)• Shapes of Algebra (Grade 8)

• Geometry• Shapes and Designs (Grade 6)• Ruins of Montarek (Grade 6)• Stretching and Shrinking (Grade 7)• Filling and Wrapping (Grade 7)• Looking for Pythagoras (Grade 8)• Kaleidoscopes, Hubcaps, and Mirrors (Grade 8)• Measurement• Shapes and Designs (Grade 6)• Covering and Surrounding (Grade 6)• Stretching and Shrinking (Grade 7)• Filling and Wrapping (Grade 7)• Data Around Us (Grade 7)• Looking for Pythagoras (Grade 8)• Data Analysis and Probability• Data About Us (Grade 6)• How Likely Is It? (Grade 6)• What Do You Expect? (Grade 7)• Data Around Us (Grade 7) @ 2004• Distributions (Grade 8)• Samples and Populations (Grade 8)• Clever Counting (Grade 8) @ 2004

• Progress Standard• Problem Solving All units

Because Connected Mathematics is a problem- centered curriculum, problem solving is an important part of every unit.Reasoning and Proof All unitsThroughout the curriculum, students are encouraged to look for patterns, make conjectures, provide evidence for their conjectures, refine their conjectures and strategies, connect their knowledge, and extend their findings. Informal reasoning evolves into more deductive arguments as students proceed from Grade 6 through Grade 8.Communication All unitsAs students work on the problems, they must communicate ideas with others. Emphasis is placed on students' discussing problems in class, talking through their solutions, formalizing their conjectures and strategies, and learning to communicate their ideas to a more general audience. Students learn to express their ideas, solutions, and strategies using written explanations, graphs, tables, and equations.Connections All unitsIn all units, the mathematical content is connected to other units, to other areas of mathematics, to other school subjects, and to applications in the real world. Connecting and building on prior knowledge is important for building and retaining new knowledge.Representation All unitsThroughout the units, students organize, record, and communicate information and ideas using words, pictures, graphs, tables, and symbols. They learn to choose appropriate representations for given situations and to translate among representations. Students also learn to interpret information presented in various forms.

Brief introduction of units• 6th Grade• Prime Time• Factors and Multiples• number theory, including factors, multiples,

primes, composites, prime factorization• Bits and Pieces I• Understanding Rational Numbers• move among fractions, decimals, and percents;

compare and order rational numbers; equivalence• Shapes and Designs• Two-Dimensional Geometry• regular and non-regular polygons, special

properties of triangles and quadrilaterals, angle measure, angle sums, tiling, thetriangle inequality

• Bits and Pieces II• Understanding Fraction Operations• understanding and skill with addition,

subtraction, multiplication, and division of fractions

Covering and SurroundingTwo-Dimensional Measurementarea and perimeter relationships, including minima and maxima; area and perimeter of polygons and circles, including formulas

Bits and Pieces IIIComputing With Decimals and Percentsunderstanding and skill with addition, subtraction, multiplication, and division of decimals, solving percent problemsHow Likely Is It?Probabilityreason about uncertainty, calculate experimental and theoretical probabilities, equally-likely and non-equally-likely outcomesData About UsStatisticsformulate questions; gather, organize, represent, and analyze data; interpret results from data; measures of center and range

• 7th GradeVariables and Patterns

• Introducing Algebra• variables; representations of

relationships, including tables, graphs, words, and symbols

• Stretching and Shrinking• Similarity• similar figures; scale factors; side length

ratios; basic similarity transformations and their algebraic rules

• Comparing and Scaling• Ratio, Proportion, and Percent• rates and ratios; making comparisons;

proportional reasoning; solving proportions

• Accentuate the Negative• Positive and Negative Numbers• understanding and modeling positive and

negative integers and rational numbers; operations; order of operations; distributive property; four-quadrant graphing

Moving Straight AheadLinear Relationshipsrecognize and represent linear relationships in tables, graphs, words, and symbols; solve linear equations; slopeFilling and WrappingThree-Dimensional Measurementspatial visualization, volume and surface area of various solids, volume and surface area relationshipWhat Do You Expect?Probability and Expected Valueexpected value, probabilities of two-stage outcomesData DistributionsDescribing Variability and Comparing Groupsmeasures of center, variability in data, comparing distributions of equal and unequal sizes

• 8th Grade•

Thinking With Mathematical Models• Linear and Inverse Variation• introduction to functions and modeling;

finding the equation of a line; inverse functions; inequalities

• Looking for Pythagoras• The Pythagorean Theorem• square roots; the Pythagorean Theorem;

connections among coordinates, slope, distance, and area; distances in the plane

• Growing, Growing, Growing• Exponential Relationships• recognize and represent exponential growth

and decay in tables, graphs, words, and symbols; rules of exponents; scientific notation

• Frogs, Fleas and Painted Cubes• Quadratic Relationships• recognize and represent quadratic functions

in tables, graphs, words and symbols; factor simple quadratic expressions

• Kaleidoscopes, Hubcaps and Mirrors• Symmetry and Transformations• symmetries of designs, symmetry

transformations, congruence, congruence rules for triangles

• Say It With Symbols• Making Sense of Symbols• equivalent expressions, substitute and

combine expressions, solve quadratic equations, the quadratic formula

• Shapes of Algebra• Linear Systems and Inequalities• coordinate geometry, solve inequalities,

standard form of linear equations, solve systems of linear equations and linear equalities.

• Samples and Populations• Data and Statistics• use samples to reason about populations

and make predictions, compare samples and sample distributions, relationships among attributes in data sets

Contents of CMP2• Grade 6• Prime Time• Bits and Pieces I• Shapes and Designs• Bits and Pieces II• Covering and Surrounding• Bits and Pieces III• How Likely Is It?• Data About Us• Common Core Additional Investigations

for Grade 6• Grade 7• Variables and Patterns• Stretching and Shrinking• Comparing and Scaling• Accentuate the Negative• Moving Straight Ahead• Filling and Wrapping

• What Do You Expect?• Data Distributions• Common Core Additional

Investigations for Grade 7• Grade 8• Thinking With Mathematical Models• Looking for Pythagoras• Growing, Growing, Growing• Frogs, Fleas, and Painted Cubes• Kaleidoscopes, Hubcaps, and

Mirrors• Say It With Symbols• The Shapes of Algebra• Samples and Populations• Common Core Additional

Investigations for Grade

Content of CMP3

2061 Benchmarks-Under the Instructional Categories rows, there are only two, that do not completely meet the high potential for learning to take place.

- Promoting Student Thinking about Mathematics (Number Concepts)-Enhancing the Mathematics Learning Environment (Number Skills, Algebra

Graph Concepts, Algebra Equation Concepts)- Under the Content row, there are two concepts that have been evaluate to only have partial content

- Number Concepts- Geometry Concepts

-There are no instructional categories seen as having little potential for learning to take place, or that are not present.-All content is seen as having partial or most content.

Note: The edition used for this study was the 1998 version (the Original CMP), there have been two revisions to the Curriculum since then. If the Benchmarks were redone, the results may be drastically different.

Activities

• Grade 6

• Grade 7

• Grade 8