How well does Mathematics: Applications and Connections 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).

### Mathematics: Applications and Connections 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 Mathematics: Applications and Connections 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 focuses attention on the idea of a/b as meaning "a parts of size 1/b" only one time—in grade 6. There are only three instances in which the material refers to a/b as meaning "a divided by b." In grades 7 and 8, students are told that a fraction indicates division, and they use this knowledge to convert a fraction to a decimal. The concept of a/b meaning "a compared to b" is introduced to students in a Cooperative Learning Lab on ratios, where the idea of comparison is emphasized. This idea is repeated numerous times throughout all three grades.

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.)

This benchmark receives ample treatment, and students are given many opportunities to use, interpret, and compare equivalent forms of many numbers throughout the material beginning with number grids in grade 6. Equivalent forms of fractions, decimals, and percents are given the most substantive treatment across the three grades. There are a few explicit references to integers and to equivalent forms of exponents and scientific notation.

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.)

This benchmark receives only partial coverage. The material gives ample treatment to the idea that shapes have special properties, most substantially in grades 7 and 8. There are only two implicit references to the idea that triangular shapes tend to make a structure rigid. There is no reference to round shapes and boundary related to interior area. Students are introduced to the concepts of congruence and similarity in grade 6. These concepts are not addressed again until grade 8.

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.)

The ideas in the benchmark are covered extensively. A great deal of time is devoted at all grade levels to finding perimeters and areas of rectangles, triangles, and circles and to finding volume. Calculating the perimeter of triangles receives the least amount of attention. Students are introduced to the formulas for each of the other calculations and have ample opportunity to use these, although many of the lessons in grade 8 and much of the practice merely reiterates the work that was done at earlier levels.

Algebra Graph Concepts — Minimal 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.)

While the meaning and purpose of line graphs are defined beginning in grade 6, there is little opportunity for students to work with the various possible relationships. For the most part, the relationship is simply increasing or decreasing, with no work on the benchmark ideas related to maximums, minimums, step graphs, or other relationships. Most lessons on graphs in grades 7 and 8 are repetitions of those found in grade 6.

Algebra Equation Concepts — Minimal 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.)

Early in grade 6, students are introduced to the concept of variables as symbols that represent a number that changes. The concept of symbolic equations summarizing change is implicit in the opening story, but there is no explicit discussion throughout the lesson. In grade 7, a similarly implicit explanation is given. Further work with variables in grades 7 and 8 follows this same pattern with little concept development.