September 9, 2009

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This resource guide provides links to exemplary resources and insight on how to teach mathematics and science concepts at the middle school level. Concepts supported by this particular guide include decimals, fractions, division of whole numbers, and geometry.

The guides provide information on the needed content knowledge, science and mathematical pedagogical knowledge, exemplary lessons and activities, career information, and correlations to national mathematics and science standards.

- Mathematics > General

- Grade 6
- Grade 7
- Grade 8

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This resource was reviewed using the Curriki Review rubric and received an overall Curriki Review System rating of 3, as of -0001-11-30.

Curriki Review System

September 21, 2009

This resource received a 3* rating because it is part of the larger resource, Math Focal Points: Grade 8, which received a rating of 3-Exemplary in the Curriki Review System. You can learn more about this larger resource by reading its review and comments.

With the goal of highlighting “the mathematical content that a student needs to understand deeply and thoroughly for future mathematics learning,” the National Council of Teachers of Mathematics has developed Curriculum Focal Points for Prekindergarten Through Grade 8 Mathematics. A “focal point” is an area of emphasis within a complete curriculum, a “cluster of related knowledge, skills, and concepts.”

This is the fourth and last in a Middle School Portal series of publications that highlight the focal points by grade level. Others in the series are Math Focal Points: Grade 5, Math Focal Points: Grade 6, and Math Focal Points: Grade 7. This publication offers resources that directly support the teaching of the three areas highlighted for eighth grade: (For a complete statement of the NCTM Curriculum Focal Points for grade 8, please see below.)

NCTM recommends that students in grade 8 analyze linear functions, translating among their verbal, tabular, graphical, and algebraic representations. They should also solve linear equations and systems of linear equations in two variables as they apply them to analyze mathematical situations and solve problems. In our section titled Linear Functions and Equations, we offer tutorials, games, carefully crafted lessons, and online simulations that provide varied approaches to these algebraic concepts. You will also find opportunity for the practice needed for understanding.

Eighth-graders are expected to use fundamental facts of distance and angle to analyze two- and three-dimensional space and figures. NCTM recommends that they develop their reasoning about such concepts as parallel lines, similar triangles, and the Pythagorean theorem, both explaining the concepts and applying them to solve problems. In the section titled Geometry: Plane Figures and Solids, we feature visual, interactive experiences in which your students can work with concepts of angle, parallel lines, similar triangles, the Pythagorean theorem, and solids. You will find games as well as lessons and challenging problems.

In grade 8, the emphasis is on understanding descriptive statistics; in particular, mean, median, and range. Students organize, compare, and display data as a way to answer significant questions. In the Analyzing Data Sets section, you will find tutorials, lesson ideas, problems, and applets for teaching these topics, and even full projects that involve worldwide data collection and analysis.

In Background Information for Teachers, we identify professional resources to support you in teaching the materials targeted in the focal points for grade 8. In NCTM Standards, we relate the curriculum focal points to Principles and Standards for School Mathematics.

**NCTM Curriculum Focal Points for Grade 8**

**Algebra: Analyzing and representing linear functions and solving linear equations and systems of linear equations.** Students use linear functions, linear equations, and systems of linear equations to represent, analyze, and solve a variety of problems. They recognize a proportion (y/x = k, or y = kx) as a special case of a linear equation of the form y = mx + b, understanding that the constant of proportionality (k) is the slope and the resulting graph is a line through the origin. Students understand that the slope (m) of a line is a constant rate of change, so if the input, or x-coordinate, changes by a specific amount, a, the output, or y-coordinate, changes by the amount ma. Students translate among verbal, tabular, graphical, and algebraic representations of functions (recognizing that tabular and graphical representations are usually only partial representations), and they describe how such aspects of a function as slope and y-intercept appear in different representations. Students solve systems of two linear equations in two variables and relate the systems to pairs of lines that intersect, are parallel, or are the same line, in the plane. Students use linear equations, systems of linear equations, linear functions, and their understanding of the slope of a line to analyze situations and solve problems.

**Geometry and Measurement: Analyzing two- and three-dimensional space and figures by using distance and angle.** Students use fundamental facts about distance and angles to describe and analyze figures and situations in two- and three-dimensional space and to solve problems, including those with multiple steps. They prove that particular configurations of lines give rise to similar triangles because of the congruent angles created when a transversal cuts parallel lines. Students apply this reasoning about similar triangles to solve a variety of problems, including those that ask them to find heights and distances. They use facts about the angles that are created when a transversal cuts parallel lines to explain why the sum of the measures of the angles in a triangle is 180 degrees, and they apply this fact about triangles to find unknown measures of angles. Students explain why the Pythagorean theorem is valid by using a variety of methods—for example, by decomposing a square in two different ways. They apply the Pythagorean theorem to find distances between points in the Cartesian coordinate plane to measure lengths and analyze polygons and polyhedra.

**Data Analysis and Number and Operations and Algebra: Analyzing and summarizing data sets.** Students use descriptive statistics, including mean, median, and range, to summarize and compare data sets, and they organize and display data to pose and answer questions. They compare the information provided by the mean and the median and investigate the different effects that changes in data values have on these measures of center. They understand that a measure of center alone does not thoroughly describe a data set because very different data sets can share the same measure of center. Students select the mean or the median as the appropriate measure of center for a given purpose.

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