How well does Connected Mathematics address the content in the selected benchmarks?

This In Brief chart provides profiles showing how this textbook scored on content and instructional quality. For the content profile, the coverage of each specific mathematical idea in the selected benchmark was rated on a 0 to 3 scale (no coverage to substantive coverage). These ratings were then averaged to obtain an overall rating for each benchmark (Most content 2.6-3.0, Partial content 1.6-2.5, Minimal content 0-1.5). For the instruction profile, the score for each instructional category was computed by averaging the criterion ratings for the category. This was repeated for each benchmark, to produce ratings of instructional quality on a 0 to 3 scale (High potential for learning to take place 2.6-3.0, Some potential for learning to take place 1.6-2.5, Little potential for learning to take place 0.1-1.5, Not present 0).

### Connected Mathematics in Brief

 Benchmarks Number Concepts Number Skills Geometry Concepts Geometry Skills Algebra Graph Concepts Algebra Equation Concepts Content Instructional Categories Identifying a Sense of Purpose Building on Student Ideas about Mathematics Engaging Students in Mathematics Developing Mathematical Ideas Promoting Student Thinking about Mathematics Assessing Student Progress in Mathematics Enhancing the Mathematics Learning Environment
 Content Scale Instructional Categories Scale Most content Partial content Minimal content High potential for learning to take place Some potential for learning to take place Little potential for learning to take place Not present

The content ratings are estimates of what the textbook series attempts to present on only these benchmarks and are not an indication of overall content coverage or accuracy. The ratings also do not indicate whether or not the content will be learned. The instructional analysis provides information on the potential the series has for helping students actually learn the concepts and skills it attempts to present.

The following indicates how well Connected Mathematics attempts to address the substance, breadth, and sophistication of the ideas contained in each of the six mathematics benchmarks that were selected for the analysis.

Number Concepts — Partial Content

 The expression a/b can mean different things: a parts of size 1/b each, a divided by b, or a compared to b. (Chapter 9A, grades 6-8, benchmark 5, pg. 213.)

The material only addresses two parts of the benchmark. The grade 6 unit Bits and Pieces I focuses on the meaning of fractions, decimals, and percents. The lessons explore the part-whole conception of fractions treating the idea "a parts of size 1/b." Benchmark idea "a divided by b" is addressed only implicitly in converting fractions to decimals and fractions to percents. The grade 7 unit Comparing and Scaling addresses the "a compared to b" idea of the benchmark through a study of ratios and proportional reasoning. One investigation focuses on the idea that a ratio can be written using fraction notation, presenting the concept of rates. The unit also contains activities that convey this benchmark idea when finding the percentage equivalent for ratio comparisons. Across the units, the lessons become more complex, and the variety of applications increases, with the more complex idea "a compared to b" discussed mainly at the grade 7 level.

Number Skills — Most Content

 Use, interpret, and compare numbers in several equivalent forms such as integers, fractions, decimals, and percents. (Chapter 12B, grades 6-8, benchmark 2, pg. 291.)

All parts of the benchmark are addressed in the material. The activities that target the benchmark ideas are focused more on rational numbers than on integers. The activities throughout the materials allow the students to use, interpret, and compare numbers. Interpreting fractions includes comparing denominators and numerators, finding equivalent fractions, and locating fractions on number lines. The grade 6 unit Bits and Pieces I emphasizes the equivalence of fractions, percents, and decimals. In the grade 7 unit Comparing and Scaling, the focus is on comparisons among ratios, fractions, decimals, and percents. Work with integers and with using and comparing large numbers as statistics is treated in two grade 7 units—Accentuate the Negative and Data Around Us.

Geometry Concepts — Partial Content

 Some shapes have special properties: Triangular shapes tend to make structures rigid, and round shapes give the least possible boundary for a given amount of interior area. Shapes can match exactly or have the same shape in different sizes. (Chapter 9C, grades 6-8, benchmark 1, pg. 224.)

All ideas in the benchmark are addressed except the idea about circles having the least boundary. The grade 6 unit Shapes and Design focuses on regular polygons and develops the special properties of quadrilaterals and parallelograms. The text explains the concept of the rigid structure of a triangle in a building polygons investigation. The grade 7 unit Stretching and Shrinking presents the concept of similarity and special properties of shapes including enlarging figures, patterns of similar figures, and similar triangles. In the lessons, students learn about the different properties of figures and how to determine the different properties that may cause them to be similar or not. A lesson on tessellations introduces and treats the concept of congruence. An entire grade 8 unit, Looking for Pythagoras, presents and applies the Pythagorean Theorem to determine the properties of special triangles such as isosceles, equilateral, scalene, and right triangles.

Geometry Skills — Most Content

 Calculate the circumferences and areas of rectangles, triangles, and circles, and the volumes of rectangular solids. (Chapter 12B, grades 6-8, benchmark 3, pg. 291.)

Connected Mathematics addresses all of the benchmark skills in different units of the material. In the grade 6 unit Covering and Surrounding, circumference and perimeter of figures and areas of rectangles, triangles, and circles are calculated by counting square units. Later, the usual formulas for perimeter, circumference, and area are discussed and practiced. The grade 7 unit Stretching and Shrinking focuses on determining the area of circles and triangles and on using scale factors. The grade 7 unit Filling and Wrapping presents the ideas of volume of the three dimensional figures including rectangular prisms, leading to the use of the formula v=lwh. The grade 8 unit Looking for Pythagoras presents the area of figures by applying the concept of the unit square to right triangles to find areas of triangles and half-circles.

Algebra Graph Concepts — Most Content

 Graphs can show a variety of possible relationships between two variables. As one variable increases uniformly, the other may do one of the following: increase or decrease steadily, increase or decrease faster and faster, get closer and closer to some limiting value, reach some intermediate maximum or minimum, alternately increase and decrease indefinitely, increase or decrease in steps, or do something different from any of these. (Chapter 9B, grades 6-8, benchmark 3, pg. 219.)

The grade 7 unit Variables and Patterns addresses various types of graphs including scatterplots and line graphs to illustrate changes in variables. The grade 8 unit Thinking with Mathematical Models introduces patterns of change by representing the relationship between the variables. The unit uses equations to introduce linear and nonlinear relationships between variables. Most equations are accompanied by graphs to help illustrate a more in-depth understanding of two changing variables. For some of the more complex problems or linear equations, a graphing calculator is introduced for graphing purposes. The grade 8 unit Frogs, Fleas, and Painted Cubes presents quadratic relationships and discusses graphing functions with minimum and maximum values. Growing, Growing, Growing, also for grade 8, treats exponential patterns of change and graphing functions with limiting values.

Algebra Equation Concepts — Most Content

 Symbolic equations can be used to summarize how the quantity of something changes over time or in response to other changes. (Chapter 11C, grades 6-8, benchmark 4, pg. 274.)

The grade 8 unit Thinking with Mathematical Models addresses both of the benchmark ideas. A variety of equations represent several different types of relationships between two variables, including linear, non-linear, inverse relationships, and exponential expressions of growth and decay. Data collection is used as a setting for deriving equations that model relationships between two variables. Both graphs and symbolic expressions are used to present equation concepts. Across grade levels, the complexity of the applications and relationships progressively increases. The grade 8 units Frogs, Fleas, and Painted Cubes, which presents quadratic relationships, and Growing, Growing, Growing, which treats exponential relationships, also address symbolic expressions to summarize change.