### How well does Middle Grades Math 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).

### Middle Grades Math 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 Middle Grades Math 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 — Most 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.)

Middle Grades Math addresses all parts of the benchmark. The idea "a parts of size 1/b" is explored mainly in grades 6 and 7 while the idea "a divided by b" is addressed more in grades 6 and 8. The last benchmark idea, "a compared to b," is addressed across all three grades. The text uses fraction bars with the shaded parts of a whole to illustrate the first benchmark idea, explaining that the fraction bar represents the numerator and denominator. The grade 6 material specifically states, "You can write a fraction as a decimal by dividing the numerator by the denominator. The fraction symbol itself means division." The idea "a compared to b" is explored in lessons that define ratio as a comparison of two numbers (or quantities) by division written as a fraction.

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.)

Middle Grades Math addresses all the benchmark ideas across grades 6, 7, and 8. The text places an emphasis on comparing and interpreting numbers. Integers are covered mainly in grade 6, and negative integers and exponents are explored in grade 8. The material defines and gives examples of the various forms of numbers and their equivalents. Lessons focus on converting fractions to decimals or fractions to percents, or vice versa, and improper fractions to mixed numbers. Numbers and their equivalent forms are compared using the symbols greater than (>), less than (<), or equal to (=).

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.)

Middle Grades Math addresses two of the benchmark ideas, "Some shapes have special properties" and "Shapes can match exactly or have the same shape in different sizes." The rigid property of triangles and the boundary and area of circles are not mentioned. Other special properties of circles are covered as well as the classifications of triangles, quadrilaterals, and other polygons. The text defines congruence and similarity. Specific corresponding parts of figures and the rules for congruent and similar triangles are covered in grades 6 and 7.

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.)

Middle Grades Math addresses all of the ideas in the benchmark. In calculating the perimeter of rectangles and triangles, the grade 6 material explores adding the lengths of the sides of figures and using the formula P = 2(l + w) for rectangles. The text also presents the formulas for the circumference and area of a circle, the areas of rectangles and triangles, and the volume of rectangular prisms.

Algebra Graph Concepts — Partial 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.)

Most of the graphing lessons are directed towards reading and interpreting graphs or making graphs that correspond to a described situation or data table, rather than towards the wide variety of possible relationships between two variables. The grade 8 text covers matching a graph with situations that reflect a change over time, e.g., representations of a ball reaching a maximum height, step-wise changes, alternating increase and decrease, and others. There is one lesson on direct and inverse variation that explains the relationships between two variables and includes examples from physical science.

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.)

All ideas in this benchmark are presented in grades 7 and 8 in lessons that describe relationships between distance, rate, and time, slope of a line, and formulas that show relationship between two quantities. Spreadsheet applications are employed to describe how various quantities change as one variable increases. In activities involving variation, the text examines data, states whether the data vary directly or inversely, and presents equations and corresponding graphs.