# MiddleSchoolPortal/Math Focal Points: Grade 8

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<metadescription content="A free, standards-based, online publication developed for teachers offering digital resources that support teaching of the NCTM Curriculum Focal Points for grade 8: linear functions, equations, similar triangles, Pythagorean theorem, mean and median." />

## Math Focal Points - Grade 8 - Introduction

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.

## Contents

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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.

## Background Information for Teachers

If looking to refresh your mathematical content knowledge, or simply to find a new approach to teaching the material targeted in the Grade 8 Focal Points, you will find these professional resources valuable.

Learning Math: Patterns, functions, and algebra In this online course designed for elementary and middle school teachers, each of ten sessions centers on a topic, such as understanding linearity and proportional reasoning or exploring algebraic structure. The teacher-friendly design includes video, problem-solving activities, and case studies that show you how to apply what you have learned in your own classroom.

Linear functions and slope In one session from the online workshop described above, teachers gather to explore linear relationships--as expressed in patterns, tables, equations, and graphs. Video segments, interactive practice, problem sets, and discussion questions guide participants they consider such concepts as slope and function.

Similarity Explore scale drawing, similar triangles, and trigonometry in terms of ratios and proportion in this series of lessons developed for teachers. Besides explanations and real-world problems, the unit includes video segments that show teachers investigating problems of similarity. To understand the ratios that underlie trigonometry, you can use an interactive activity provided online.

Indirect Measurement and Trigonometry For practical experience in the use of trigonometry, look at these examples of measuring impossible distances and inaccessible heights. These lessons show proportional reasoning in action!

Pythagorean Theorem A collection of 76 proofs of the theorem! From the diverse approaches used by Euclid, Da Vinci, President Garfield, and many others, these proofs are clearly and colorfully illustrated, often accompanied by an interactive Java illustration to further clarify the brief explanations. Incredible as it sounds, this page is far from boring.

Variation about the mean Just what do we mean by “the mean”? This workshop session, developed for K-8 teachers, explores this statistic in depth. Participants work together to investigate the mean as the "balancing point" of a data set and come to understand how to measure variation from the mean.

Gallery of Data Visualization: The Best and Worst of Statistical Graphics This site offers graphical images that represent data from a range of sources (historical events, spread of disease, distribution of resources). The author contrasts the differences between the best and worst graphics by showing how some images communicate data clearly and truthfully, while others misrepresent, lie, or totally fail to "say something." If you are looking for innovative representations of data or examples of misrepresentation, you will find this resource helpful.

## Linear Functions and 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 (NCTM, 2006, p. 20).

These resources offer a variety of ways to learn the material targeted in this Focal Point: tutorials, games, carefully crafted lessons, and online simulations. Your middle school students will also find plenty of opportunity for practice in real-world as well as imaginative scenarios.

Lines and Slope At this site, students learn to draw a line and find its slope. Joan, a cartoon chameleon, is used throughout the tutorial to demonstrate the idea of slope visually. Background information on solving equations and graphing points is laid out clearly, followed by a step-by-step explanation of how to calculate slope using the formula. Finally, the slope-intercept form (y = mx + b) is carefully set out.

Walk the Plank Students place one end of a wooden board on a bathroom scale and the other end on a textbook, then "walk the plank." They record the weight measurement as their distance from the scale changes and encounter unexpected results: a linear relationship between the weight and distance. Possibly most important, the investigation leads to a real-world example of negative slope. An activity sheet, discussion questions, and extensions of the lesson are included.

Writing Equations of Lines This lesson uses interactive graphs to help students deepen their understanding of slope and extend the definition of slope to writing the equation of lines. Online worksheets with immediate feedback are provided to help students learn to read, graph, and write equations using the slope intercept formula.

Linear Function Machine The functions produced by this machine are special because they all graph as straight lines and can be expressed in the form y = mx + b. In this activity, students input numbers into the machine and try to determine the slope and y-intercept of the line.

Algebra: Linear Relationships Seven activities focus on generalizing from patterns to linear functions. Designed for use by mentors in after-school programs or other informal settings, these instructional materials have students work with number patterns, the function machine, graphs, and variables in realistic situations. Excellent handouts included.

Explorelearning.com The following three resources come from this subscription site; a free 30-day trial is available. Experiment with the online simulations, particularly selected for their use in teaching equations of a line. Subscriptions include inquiry-based lessons, assessment, and reporting tools.

Slope calculation
Examine the graph of two points in the plane. Find the slope of the line that passes through the two points. Drag the points and investigate the changes to the slope and to the coordinates of the points.
Point-slope form of a line
Compare the point-slope form of a linear equation to its graph. Vary the coefficients and explore how the graph changes in response.
Slope-intercept form
Compare the slope-intercept form of a linear equation to its graph. Find the slope of the line using a right triangle on the graph. Vary the coefficients and explore how the graph changes in response.

Slope slider What difference does it make to the graph of a function if you change the slope or the y-intercept? Students can see the changes in the equation itself and in its graph as they vary both slope and y-intercept. Excellent visual! The activity could be used for class or small group work, depending on computer access.

Grapher : algebra (grades 6-8) Using this online manipulative, students can graph one to three functions on the same window, trace the function paths to see coordinates, and zoom in on a region of the graph. Function parameters can be varied as can the domain and range of the display. Tabs allow the student to incorporate fractions, powers, and roots into their functions.

Planet hop In this interactive game, students find the coordinates of four planets shown on the grid or locate the planets when given the coordinates. Finally, they must find the slope and y-intercept of the line connecting the planets in order to write its equation. Players select one of three levels of difficulty. Tips for students are available as well as a full explanation of the key instructional ideas underlying the game.

Constant dimensions This complete lesson plan requires students to measure the length and width of a rectangle using both standard and nonstandard units of measure, such as pennies and beads. As students graph the ordered pairs, they discover that the ratio of length to width of a rectangle is constant, in spite of the units. This hands-on experience leads to the definition of a linear function and to the rule that relates the dimensions of this rectangle.

Barbie bungee Looking for a real-world example of a linear function? In this lesson, students model a bungee jump using a doll and rubber bands. They measure the distance the doll falls and find that it is directly proportional to the number of rubber bands. Since the mathematical scenario describes a direct proportion, it can be used to examine linear functions.

Exploring linear data This lesson connects statistics and linear functions. Students construct scatterplots, examine trends, and consider a line of best fit as they graph real-world data. They also investigate the concept of slope as they model linear data in a variety of settings that range from car repair costs to sports to medicine. Handouts for four activities, spread out over three class periods, are provided.

Supply and Demand Your company wants to sell a cartoon-character doll. At what price should you sell the doll in order to satisfy demand and maintain your supply? The lesson builds from graphing data to writing linear equations to creating and solving a system of equations in a real-world setting. Discussion points, handouts, and solutions are given.

Printing Books Presented with the pricing schedules from three printing companies, students must determine the least expensive way to have their algebra books printed. They compile data in tables, spreadsheets, and a graph showing three equations. Throughout the lesson, students explore the relationships among lines, slopes, and y-intercepts in a real-world setting.

Purplemath - Your Algebra Resource Algebra modules provide free tutorials in every topic of algebra, from beginning to advanced. Lessons concentrate on "practicalities rather than the technicalities" and include worked examples as well as explanations. Of particular interest are the modules on Systems of Linear Equations and Systems-of-Equations Word Problems. A site worth visiting!

## Geometry: Plane Figures and Solids

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 (NCTM, 2006, p. 20).

These activities offer your eighth graders visual, interactive experiences with geometry. Through games as well as lessons and problems, they work with concepts of angle, parallel lines, the Pythagorean theorem, and solids.

Angles This Java applet enables students to investigate acute, obtuse, and right angles. The student decides to work with one or two transversals and a pair of parallel lines. Angle measure is given for one angle. The student answers a short series of questions about the size of other angles, identifying relationships such as vertical and adjacent angles and alternate interior and alternate exterior angles.

Triangle Geometry: Angles This site directly addresses students as it leads them to explore angles and their measurement. Most important, it offers applets to introduce the Pythagorean theorem by collecting data from right triangles online and provides an animated picture proof of the theorem.

Manipula math with Java : the sum of outer angles of a polygon This interactive applet allows users to see a visual demonstration of how the sum of exterior angles of any polygon sums to 360 degrees. Students can draw a polygon of any number of sides and have the applet show the exterior angles. They then decrease the scale of the image, gradually shrinking the polygon, while the display of the exterior angles remains and shows how the angles merge together to cover the whole 360 degrees surrounding the polygon.

Parallel Lines and Ratio Three parallel lines are intersected by two straight lines. The classic problem is: If we know the ratio of the segments created by one of the straight lines, what can we know about the ratio of the segments along the other line? An applet allows students to clearly see the geometric reasoning involved.

Area triangles This applet shows triangle ABC, with a line through B parallel to base AC. Students can change the shape of the triangle by moving B along the parallel line or by changing the length of base AC. What happens to the length of the base, the height, and the area of the triangle as students make these moves? Why?

Understanding the Pythagorean Relationship Using Interactive Figures The activity in this example presents a visual and dynamic demonstration of this relationship. The interactive figure gives students experience with transformations that preserve area but not shape. The final goal is to determine how the interactive figure demonstrates the Pythagorean theorem.

Distance Formula Explore the distance formula as an application of the Pythagorean theorem. Learn to see any two points as the endpoints of the hypotenuse of a right triangle. Drag those points and examine changes to the triangle and the distance calculation.

Measuring by Shadows A student asks: How can I measure a tree using its shadow and mine? This letter from Dr. Math carefully explains the mathematics underlying this standard classroom exercise.

Finding the Height of a Lamp Pole Without using trigonometry, how can you find the height of a lamp pole or other tall object? Two methods, both depending on similar triangles, are outlined and illustrated. A rich problem.

Polygon Capture In this lesson, students classify polygons according to more than one property at a time. In the context of a game, students move from a simple description of shapes to an analysis of how properties are related.

Sorting Polygons In this companion to the above game, students identify and classify polygons according to various attributes. They then sort the polygons in Venn diagrams according to these attributes.

Fire hydrant : what shape is at the very top of a fire hydrant? This activity begins an exploration of geometric shapes by asking students why the five-sided (pentagonal) water control valve of a fire hydrant cannot be opened by a common household wrench. The activity explains how geometric shape contributes to the usefulness of many objects. A hint calls students' attention to the shape of a normal household wrench, which has two parallel sides. Answers to questions and links to resources are included.

Diagonals to Quadrilaterals Instead of considering the diagonals within a quadrilateral, this lesson provides a unique opportunity: Students start with the diagonals and deduce the type of quadrilateral that surrounds them. Using an applet, students explore characteristics of diagonals and the quadrilaterals that are associated with them.

Image:Geometric Solids and their Properties A five-part lesson plan has students investigate several polyhedra through an applet. Students can revolve each shape, color each face, and mark each edge or vertex. They can even see the figure without the faces colored in — a skeletal view of the "bones" forming the shape. The lesson leads to Euler’s formula connecting the number of edges, vertices, and faces, and ends with creating nets to form polyhedra. An excellent introduction to three-dimensional figures!

Slicing solids (grades 6-8) So what happens when a plane intersects a Platonic solid? This virtual manipulative opens two windows on the same screen: one showing exactly where the intersection occurred and the other showing the cross-section of the solid created in the collision. Students decide which solid to view, and where the plane will slice it.

Studying Polyhedra What is a polyhedron? This lesson defines the word. Students explore online the five regular polyhedra, called the Platonic solids, to find how many faces and vertices each has, and what polygons make up the faces. An excellent applet! From this page, click on Polyhedra in the Classroom. Here you have classroom activities to pursue with a computer. Developed by a teacher; the lessons use interactive applets and other activities to investigate polyhedra.

## Analyzing 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 (NCTM, 2006, p.20).

As reflected in this set of resources, the emphasis here is on understanding descriptive statistics; in particular, measures of center. You will find tutorials, lesson ideas, problems, and applets for teaching these topics, and even full projects that can involve your middle school students in worldwide data collection.

Describing Data Using Statistics Investigate the mean, median, mode, and range of a data set through its graph. Manipulate the data and watch how these statistics change (or, in some cases, how they don't change).

Understanding Averages Written for the student, this tutorial on mean, median, and mode includes fact sheets on the most basic concepts, plus practice sheets and a quiz. Key ideas are clearly defined at the student level through graphics as well as text.

Plop It! Users click to easily and quickly build dot plots of data and view how the mean, median, and mode change as numbers are added to the plot. An efficient tool for viewing these statistics visually.

Working hours : how much time do teens spend on the job? This activity challenges students to interpret a bar graph, showing only percentages, to determine the mean number of hours teenagers work per week. A more complicated and interesting problem than it may seem at first glance! A hint suggests that students assume that 100 students participated in the survey; a full solution sets out the math in detail. Related questions ask students to calculate averages for additional data sets.

Stem-and-Leaf Plotter Can your students find the mean, median, and mode from a stem-and-leaf plot? They can use this applet to explore the measures of center in relation to the stem-and-leaf presentation of data. Students use the online plotter to enter as much data as they choose; then they determine measures of center and have the program check and correct their values. Ideas for class practice and discussion are provided in a lesson outline.

Train race In this interactive game, students compute the mean, median, and range of the running times of four trains, then select the one train that will get to the destination on time. Players extend their basic understanding of these statistics as they try to find the most reliable train for the trip. Students can select one of three levels of difficulty. There are tips for students as well as a full explanation of the key instructional ideas underlying the game.

Comparing Properties of the Mean and the Median Through the Use of Technology This interactive tool allows students to compare measures of central tendency. As students change one or more of the seven data points, the effects on the mean and median are immediately displayed. Questions challenge students to explore further the use of these measures of center; for example, What happens if you pull some of the data values way off to one extreme or the other extreme?

The Global Sun Temperature Project This web site allows students from around the world to work together to determine how average daily temperatures and hours of sunlight change with distance from the equator. Students can participate in the project each spring, April-June. Students learn to collect, organize, and interpret data. You will find project information, lesson plans, and implementation assistance at the site.

Down the Drain: How Much Water Do You Use? This Internet-based collaborative project will allow students to share information about water usage with other students from around the country and the world. Based on data collected by their household members and their classmates, students will determine the average amount of water used by one person in a day. Students must develop a hypothesis, conduct an experiment, and present their results.

Data analysis : as real world as it gets Resources that firmly place data analysis in context! The lessons and interdisciplinary projects were selected to promote student interest by focusing on real-world situations and developing skills for using the power of mathematics to form important conclusions relevant to life. Students learn that working with data offers insights into society’s problems and issues.

## SMARTR: Virtual Learning Experiences for Students

Visit our student site SMARTR to find related math-focused virtual learning experiences for your students! The SMARTR learning experiences were designed both for and by middle school aged students. Students from around the country participated in every stage of SMARTR’s development and each of the learning experiences includes multimedia content including videos, simulations, games and virtual activities.

## NCTM Standards

You may be asking yourself, "What do the curriculum focal points have to do with the Principles and Standards for School Mathematics (PSSM)?" NCTM answers that identifying areas of emphasis at each grade level is the next step in implementing those principles and standards. Curriculum Focal Points for Prekindergarten Through Grade 8: A Quest for Coherence "provides one possible response to the question of how to organize curriculum standards within a coherent, focused curriculum, by showing how to build on important mathematical content and connections identified for each grade level, pre-K–8" (NCTM, 2006, p. 12).

The curriculum focal points draw on the content standards described in PSSM, at times clustering several topics in one focal point. Also, the process standards are pivotal to well-grounded instruction, for "it is essential that these focal points be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations" (p.20).

This Middle School Portal publication offers resources intended to support you in teaching the key mathematical areas identified for grade 8: linear functions and simple systems of linear equations, parallel lines and angles in polygons, the Pythagorean theorem, polyhedra, and descriptive statistics. The selected resources are grounded in the Algebra, Geometry, and Data Analysis standards, and particularly in the process standards of Problem Solving and Representation.

A variety of formats (tutorials, lesson plans, games, problems, and projects) are provided for your use in teaching these focal points. We believe you will find here resources that engage your eighth graders in probing the deeper and increasingly abstract concepts of middle school mathematics.

A full description of the Curriculum Focal Points for Grade 8 is available at http://www.nctm.org/standards/focalpoints.aspx?id=340&ekmensel=c580fa7b_10_52_340_10

Curriculum Focal Points for Prekindergarten Through Grade 8: A Quest for Coherence may be viewed in its entirety at http://www.nctm.org/standards/content.aspx?id=270

Reprinted with permission from Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence, copyright 2006 by the National Council of Teachers of Mathematics. All rights reserved.