Methods for Seventh Grade Program Reviews

Seventh grade is problematic in most math programs. Unlike the early grades, where all students are approximately equally prepared to master the appropriate material, and unlike the named courses of high school (Algebra, Geometry, Algebra II, Trig), where students should enter the courses with appropriate preparation regardless of their school grade, seventh graders may show a wide degree of variation in the absence of clear guidelines as to how to deal with these differences. A critical decision must be made, prior to grade 7 or prior to grade 8, as to when a student will take Algebra. In many curricula, the program is geared, from kindergarten, to preparing children for Algebra in ninth grade. In these programs, Algebra in eighth grade is an accident for a lucky few, usually those with educational advantages outside the classroom which allow them to succeed above the level at which they are taught. Thus, most programs, whether they decide who enters the "eighth grade Algebra track" after grade 6 or after grade 7, inherently choose to make reaching Algebra by grade 8 a less likely goal for those with fewer resources at home.

This system is being challenged in the development of various sets of standards. For example, the California Academic Standards Commission deliberately chose to design K-7 mathematics standards that put students on track to study Algebra and Geometry by grade 8. Plans such as this one start at kindergarten and present all students, and their teachers, with the expectation that students will keep up and be ready for Algebra in grade 8. Whether all students succeed at this level, all will have had the opportunity to succeed at a level that is now available only to the lucky few. This plan requires that the seventh grade math course be a Pre-Algebra course. This is the philosophy we have used in developing the criteria this review.

Topic Area Evaluations

Our first area for review, and the most important, was Mathematical Depth. A set of key topics, representing an array of algebra readiness concepts, knowledge, skills and problem-solving applications, was chosen for examination. Key benchmarks within these areas were drawn from various high level standards documents and used as the content basis for review of each mathematical topic.

The reviews for seventh grade differ somewhat in form for those compiled for grades 2 and 5. In the earlier grades, a smaller set of topics was selected and the reviews summarize the presentations of the programs in rather extensive detail. In seventh grade, on the other hand, the set of topics selected for review is extensive and covers most of the material that is important at this grade level. Consequently, there are more topic areas with less detail about the presentation for each one than found in the reviews for earlier grades.

The topics selections for review are

Properties, Order of Operations
Exponents, squares, roots
Fractions
Decimals
Percents
Proportions
Expressions and Equations - Simplifying and Solving
Expressions and Equations - Writing
Graphing
Shapes, Objects, Angles, Similarity, Congruence
Area, Volume, Perimeter, Distance

Topic Area Criteria

The expectations within each of the selected topic areas is presented below.

Properties, Order of Operations

At this level students should master the rules of order of operations and the properties of the real number system (commutative, associative, distributive, identity and inverse) and be able to apply them, with justification, with all four operations in calculations involving fractions, decimals, percents, positive and negative numbers, and in the simplification of simple roots and powers. This should include the simplification of numerical expressions, the evaluation of expressions with substitution of numerical values and the solution of simple one and two step algebraic equations.

Exponents, squares, roots

Books should support mastery of the basics of the manipulation of exponents including knowing how to raise rational numbers to whole number powers, recognizing that raising to a whole number power is repeated multiplication, multiplying and dividing expressions involving exponents with a common base, including fractions with exponents, and evaluating monomials with whole number exponents. More advanced students should learn to view negative exponents as the inverse of a number raised to a power and evaluate expressions involving negative exponents. Students should learn the relationship between squares and square roots and should be able to determine without calculators the two integers between which the square root of a number lies. They will recognize and simplify perfect square monomials, including those involving squared numerical and variable terms and use this to evaluate expressions involving simple roots. Finally, students should convert to and from scientific notation for numbers both greater than 1 and less than 1 and use scientific notation in calculations.

Fractions

This is a key topic because of its usefulness in its own right and because these are essential skills that need to be developed for success in algebra and all algebra-based future mathematics. Mastery of fractions and the conversion of fractions to other forms is critical at this level. Books should support students learning to add and subtract any fractions, finding common denominators by factoring or prime factorization if necessary. Students should know that every fraction is equivalent to a terminating or repeated decimal and should be able to express any fraction as a decimal or a percentage.

Decimals

This topic goes hand in hand with the topics on fractions and percents. All need to be mastered at this level. It is hoped that students will have mastered addition, subtraction, multiplication and division with decimals (including negatives), as well as multiplying and dividing by powers of ten before this level, but books reasonably contain some review. At this level, students should in addition be able to convert terminating decimals to reduced fractions (potentially review), and to percentages. It might also be hoped that they would learn to convert repeating decimals to reduced fractions.

Percents

Along with fractions and decimals, this is the third topic at this level that completes elementary arithmetic. In previous years it is hoped that students learned how to calculate percent, and memorized the percent equivalents of commonly occurring fractions such as 1/2, 1/4, 1/5 and 1/10. They might also have mastered finding a given percent of a specific quantity and converting simple percents to decimals in the tenths or hundredths. Review of these topics might be expected, but initial instruction in these topics is too late. At this grade, students should be able to convert any given percent (including percents greater than 100 or less than 1) to a fraction or decimal. In addition they should be able to find percent increase or percent decrease for the change in a quantity and should be able to solve problems involving discounts, markups, commissions, profits and simple interest.

Proportions

This is one of the algebraic topics that comes to the fore in this year, with major applications in many real world situations including probability, and changes of scale. In a previous grade students may have been introduced to ratios and proportions as probabilities or in forms such as batting averages or miles per hour. They may also have learned to recognize rates and proportions presented in problem situations and have learned how to solve proportion problems for a missing term. If not, these are a key goal for this year. In addition to the above, students at this year should learn to use dimensional analysis to check that the symbolic representation of a problem, and the answer, contain the appropriate units, and should be able to solve one and two step problems involving direct variation. They should be able to perform changes of scale (maps, models, recipes), express the relationship between parts of a figure and find unknown lengths in similar figures. They should convert between and within unit systems (e.g. monetary systems, measurement systems) and should know the difference between rates (e.g. miles/hour) and product measures such as person-days.

Expressions and Equations - Simplifying and Solving

After mastery of signed numbers and manipulations of fractions, decimals and percents, the skills of writing and solving linear equations are the key to preparing students for mastery of algebra and true algebraic problem solving. Some of the skills in this area will have been introduced in earlier grades, including some use of variables to represent unknown quantities and solution of simple one step linear equations. Some of the goals related to this topic (e.g. direct variation) are mentioned under "Proportions". At this level students should be able evaluate numerical expressions and symbolic expressions with substitution for the variables. Top students should simplify expressions involving multiple variables. Students should solve and check one and two step linear equalities and inequalities in one variable, including those involving rational solutions or decimals.

Expressions and Equations - Writing

This topic is a key to mastery of algebraic problem solving in later years. If one cannot write an equation or expression to fit a verbal description, the strongest of problem solving techniques is lost. This skill should begin development well before grade 7, but by the end of grade 7 students should aim to be able to write, in response to problem situations, linear equalities and inequalities in one variable, expressions with up to three variables, expressions with a non-linear term and expressions or equations involving proportions and rates (e.g. as in "Proportions").

Graphing

This topic links algebra, analytic geometry, classic geometry and data analysis. At earlier grades students should have been exposed to various forms of data graphs and their interpretation in terms of problem situations. Students should also have learned to graph ordered pairs in the coordinate plane and to use this to plot values of expressions for given replacement values and to graph simple linear equations. They should also have learned to plot inequalities on the number line. At grade 7 students should be able to graph a linear equation in two variables when given the equation, the slope and y intercept, or the x and y intercepts and they should be able relate the equation, graph and table of values for a linear function. Students should recognize situations involving direct variation, represent such situations on a graph and recognize such situations from their graphs. They should be exposed to the concept of slope, know that it is a constant in linear equations, and know the definition of slope. Top students may be able to plot single or multiple linear inequalities on the coordinate plane and to plot equations of the form y=nx² or y=nx³.

Shapes, Objects, Angles, Similarity, Congruence

At this level, students will have mastered a substantial amount of practical geometry and the underpinnings for success in high school geometry. In prior years they should have learned to recognize the simple polygons and know and be able to apply the definition of regular polygons. They should know that a circle has 360 degrees, the sums of the angles of triangles and quadrilaterals, and how to use this information to solve problems involving missing angle or arc measures. They should already know how to use a compass, protractor, straight edge, ruler and so on. They may already have learned and applied the concepts of supplementary, complementary, adjacent and vertical angles, and parallel and perpendicular lines. They may also know the concept of similar and congruent figures, be able to determine if two figures are similar or congruent and apply these concepts to find the measure of individual sides or angles in a figure. In addition to mastery of any topics not yet mastered at previous grades, students at grade 7 should identify and sketch central and inscribed angles, arcs, radii, diameters and chords of circles. They should be able to determine the number of diagonals of polygons, and the measures of central, interior and exterior angles of regular polygons. They should be able to classify quadrilaterals based on their properties (e.g. quadrilateral, rectangle, rhombus, square, etc.). They should be able to identify and construct (using compass and straight edge) line segments, altitudes of triangles, medians, angle bisectors and perpendicular bisectors.

Area, Volume, Perimeter, Distance

This topic is developed over a number of years. Students should already know how to calculate area and perimeter for circles, triangles and rectangles. In doing so, they should already be familiar with approximate values of pi and application of pi to estimate the radius, diameter or circumference of a circle. They should know how to calculate the volume and surface areas of rectangular solids. By the end of grade 7 it is hoped that a student will know how to find the formulas for area and perimeter, and use the formulas to find the area and perimeter of basic two dimensional figures (e.g. rectangles, parallelograms, trapezoids, squares, triangles, circles) as well as to combine these skills to find the area and perimeter of polygons or irregular figures composed of parts of circles, quadrilaterals and triangles. The student will also know and use the formulas for volume and surface area of prisms, cones, cylinders and pyramids and combine these rules and the knowledge of their derivation to determine the surface area or volume of irregularly shaped three dimensional figures. Finally, as special triangles and distance become more important, a student should be able to identify Pythagorean triples and should apply the Pythagorean theorem to find the approximate length of the missing side of a right triangle or the diagonal of a square or rectangle. A top level book might expose students to one or more proofs of the Pythagorean theorem.

Program Quality Evaluations

In addition to addressing the Mathematical Depth for each topic, Quality of Presentation was judged as well. We looked for clarity of objectives within each lesson and within a topic as a whole, clarity of presentation, a sufficient number and appropriateness of examples, reasonable guided practice or scaffolding of the early stages of implementation of new knowledge, and a general sense that students would have a high probability of mastering the material at the level at which it was taught.

In conjunction with judging Presentation, we also judged Quality of Student Work in terms of both amount and level. Is there enough chance to practice what is taught, is the work well designed, and is the work at an appropriately high level? Importantly, a range of depth and scope of student work should be provided, appropriately building from the simpler, less abstract cases to the more difficult ones.

After completely analyzing a mathematical topic in a given book, a numeric rating from 1 (low) to 5 (high) combining aspects of depth, presentation and student work was given for that topic, such that a review of each book contains topic-specific number grades and reviews. In addition, at the end of the review, overall ratings for each subset of the criteria (Depth, Presentation and Student Work) are given.

Finally, the ratings of each mathematical topic were taken into account to generate an overall rating for the entire program.