# MiddleSchoolPortal/Teaching Strategies for Middle School Math

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### Introduction - Teaching Strategies for Middle School Math

In explaining its Teaching Principle, one of six principles from Principles and Standards for School Mathematics, the National Council of Teachers of Mathematics took care to emphasize that “teachers have different styles and strategies for helping students learn particular mathematical ideas, and there is no one ‘right way’ to teach” (NCTM, p. 18). Our aim in this publication is to provide resources that support your personal instructional style while, perhaps, introducing materials that encourage you to experiment with a wider range of teaching techniques.

In the section titled Assessment as Instruction, we offer resources that connect these seemingly opposing activities. In another section, Games That Teach, we add to your collection of math games. Each game selection deals with middle school content, such as fractions, linear equations, factors, and geometry.

We all want to teach mathematics that is relevant and interdisciplinary, but it can be difficult to find supporting resources. Connecting to the Wider World offers lesson ideas that integrate math across the school curriculum and beyond the classroom. Taking Advantage of Technology offers activities that use the Internet as a teaching tool, both to explore and to visualize math concepts.

If you are looking for problems that encourage your students to think outside the box, try Challenging with “Rich” Problems. Finally, Launching Through Literature recommends books that will involve your students in mathematics scenarios.

For professional resources, you will find interesting online books in Background Information if you like to dig into theory of teaching and learning. And in the section on The Teaching Principle, we discuss how the aim of this publication aligns with the NCTM Principles and Standards.

We hope these resources support your teaching strategies and add to your repertoire of effective instructional materials.

### Background Information for Teachers

For those of you who are interested in exploring mathematics teaching and learning in depth, the books Measuring What Counts, Adding It Up, How Students Learn, and Beating a Path to the Brain provide the theory and research findings underlying current recommendations for reform in mathematics education. For practical, well-explained ideas for classroom teaching, Measuring Up offers exercises that challenge the usual types of assessment. All three books are available free online. The final resource here is a unique site that presents strategies for teaching math to visually impaired students.

**Measuring What Counts: A Conceptual Guide for Mathematics Achievement**
Arguing for a better balance between educational and measurement concerns in the development and use of mathematics assessment, this book sets forth three principles — related to content, learning, and equity — that can form the basis for new assessments that support national standards in mathematics education.

**Adding It Up: Helping Children Learn Mathematics**
Adding It Up explores how students in pre-K through grade 8 learn mathematics and recommends ways of changing teaching, curricula, and teacher education to improve learning during these critical years. Based on research findings, the book details the processes by which students acquire proficiency with whole numbers, rational numbers, and integers, as well as beginning algebra, geometry, measurement, and probability and statistics.

**How Students Learn: History, Mathematics, and Science in the Classroom**
In this book, questions that confront every classroom teacher are addressed using the latest research on cognition, teaching, and learning. Leading educators explain in detail how they developed successful interdisciplinary curricula and teaching approaches. Their recounting of personal teaching experiences lends strength and warmth to this volume.

**Beating a Path to the Brain**
Written from one teacher to another, this short article outlines several instructional strategies based on recent research on the human brain. Practical ideas on "getting the brain's attention" and helping students retain the lesson over time.

**Measuring Up: Prototypes for Mathematics Assessment**
This book features 13 classroom exercises that demonstrate the meaning of inquiry, performance, communication, and problem solving as standards for mathematics education. Even though the examples are at the fourth-grade level, all middle grades teachers can learn from the use of these genuine exercises to challenge and prepare students.

**Teaching Math to Visually Impaired Students**
This site presents strategies for teaching visually impaired students as well as information about math tools, adaptive tools and technology, and the Nemeth code.

### Assessment as Instruction

Assessment and instruction can seem opposed to each other, with assessment serving only as a measure of success of instruction. These resources show that assessment can be used as an essential part of the instructional process itself.

**Balanced Assessment**
A set of more than 300 assessment tasks actually designed to inform teaching practice! Most tasks, indexed for grades K-12, incorporate a story problem and include hands-on activities. Some intriguing titles include Confetti Crush, Walkway, and Hockey Pucks. Rubrics for each task are provided. MSP full record

**Mathematics Assessment Instructional Support Modules**
Developed by the Washington education agency, these modules embed the content of the state assessment within the process of problem solving. Each module consists of problems that engage students in practicing/learning mathematical concepts. Following each module is a set of assessment items that test the content covered in that module. Rubrics are included. Although intended for high school students, most of the material is appropriate for the upper middle grades.

**Math Partners: Mathematics Mentoring for America’s Youth**
The materials here were designed for use by mentors for K-9 students in after-school programs, but are useful in any teaching situation. In each grade band (K-2, 3-5, 6-8, and 8-9 algebra), you will find four units focused on number and operation, geometry and measurement, statistics and probability, and patterns and functions. Each unit has pre-assessment activities to help determine prior knowledge as well as student needs.

### Games That Teach

You probably already incorporate games in your teaching. Games focus students’ attention as few other teaching strategies can. The ones selected here deal directly with the math content covered in the middle grades. Each has a learning objective; each could be embedded in a lesson plan. We believe that they will add to your store of games that teach.

**Fraction Game**
For work on fractions, this applet is a winner! It allows students to individually practice working with relationships among fractions and ways of combining fractions. It helps them visualize what is meant by equivalence of fractions. A link to an applet for two-person play is also given here.

**Polygon Capture**
This excellent lesson uses a game to review and stimulate conversation about properties of polygons. A player draws two cards, one about the sides of a polygon, such as "All sides are equal," and one about the angles, such as "Two angles are acute." The player then captures all the polygons on the table that fit both of the properties. Provided here are handouts of the game cards, the polygons, and the rules of the game.

**The Factor Game**
A two-player game that immerses students in factors! To play, one person circles a number from 1 to 30 on a gameboard. The second person circles (in a different color) all the proper factors of that number. The roles are switched and play continues until there are no numbers remaining with uncircled factors. The person with the largest total wins. A lesson plan outlines how to help students analyze the best first move in the game, which leads to class discussion of primes and squares as well as abundant and deficient numbers.

**Planet Hop**
In this online one-person computer game, four planets are shown on a coordinate grid. A player must pass through each on a journey through space. The player must find the coordinates of the four planets and, finally, the equation of the line connecting them. Three levels of difficulty are available.

**Towers of Hanoi: Algebra (Grades 6-8)**
This online version of the Towers of Hanoi puzzle features three spindles and a graduated stack of two to eight discs, a number decided by the player, with the largest disc on the bottom. The player must move all discs from the original spindle to a new spindle in the smallest number of moves possible, while never placing a larger disc on a smaller one. The algebra learning occurs as the player observes the pattern of number of discs to number of moves needed. Generalizing from this pattern, students can answer the question: What if you had 100 discs? The final step is expressing the pattern as a
function.

### Connecting to the Wider World

It is through an integrated unit of study that students can see measurement and data analysis in the context of science, or improve their sense of shape, symmetry, and similarity through the study of art. Applying mathematics to other subject areas helps students see where mathematics fits into the world at large.

**It’s a Math World! A WebQuest for Writing Across the Curriculum**
A well-developed WebQuest that integrates language arts and mathematics! Reading the book Math Curse sets the stage for creating a children's book that shows the role mathematics plays in life. Working in groups, students investigate suggested web sites to learn about publishing and book development and mathematics applications. Students from each group become the measurement, statistics, and computation experts who complete a series of mathematics activities that are incorporated into a story. Finally, desktop publishing completes the unit.

**The National Math Trail**
Ideas on this site lead students to explore the mathematics in their own communities; for example, the geometric shapes in buildings or the data available on monuments and in cemeteries. As students explore, they develop math problems related to their findings. Teachers can submit the problems to the web site, along with photos, drawings, sound recordings, and even videos, which are then made available to other educators and students.

**Living the Story Problem**
An eighth-grade math teacher leads her students out of the classroom and into the community. As the article states, her students "have honed measurement skills while helping to build houses for the local chapter of Habitat for Humanity. They have also mastered concepts of algebra by taking an inventory of forest snags and woody debris alongside staffers from the Oregon Department of Forestry, and applied their understanding of area, perimeter, and volume to helping Oregon Fish and Wildlife workers reseed a thinned tract of forestland." An inspiring read!

**Culturally Situated Design Tools: Teaching Math Through Culture**
These materials are designed to show the mathematical principles embedded in the traditional designs created by various cultures; in particular, African, African American, Native American, and Latino cultures. Using applets on the site, students explore mathematics as they re-create existing patterns and structures, or make their own designs. Standards-based lesson plans, evaluations, and other teaching materials support the software.

**Measuring the Circumference of the Earth**
Through this online project, students learn about Eratosthenes and actually do a similar measurement that yields a close estimate of the earth’s circumference. Even with access to only one computer, students can obtain data from other schools that lie approximately on their own longitude. Careful instructions guide the students in carrying out the experiment and analyzing the data collected. The project also provides activities, reference materials, online help, and a teacher area.

**Connections: Linking Mathematics to Social Studies, Art, and Science**
Here you will find online resources that connect mathematics to social studies, art, and science. Each section contains lesson plans, problems to solve, and examples of mathematics at work in an interdisciplinary setting.

### Taking Advantage of Technology

The computer can be a distraction and a frustration, but it can also be a teaching tool. Such commonplace but abstract concepts as fractional equivalence and the "size" of large numbers can be made visual through technology. And students can interact with virtual manipulatives to change algebraic variables on a balance scale, or rotate a 12-sided solid to see its regularity and symmetry. These resources are examples of the potential of the Internet as a teaching strategy.

**The MegaPenny Project**
This site shows arrangements of large quantities of U.S. pennies. It begins with only 16 pennies, which measure one inch when stacked and one foot when laid in a row. The visuals build to a thousand pennies and in progressive steps to a million and even a quintillion pennies! All pages have tables at the bottom listing the value of the pennies on the page, size of the pile, weight, and area (if laid flat). The site can be used to launch lessons on large numbers, volume versus area, or multiplication by a factor of 10.

**Cynthia Lanius' Fractal Unit**
In this unit developed for middle school students, the lessons begin with a discussion of why we study fractals and then provide step-by-step explanations of how to make fractals, first by hand and then using Java applets—an excellent strategy! But the unit goes further; it actually explains the properties of fractals in terms that make sense to students and teachers alike.

**Project Interactivate Activities**
This site offers numerous opportunities for online exploration of middle school mathematics. The following two resources are examples. MSP full record

**Fraction Sorter**
Using this applet, the student represents two to four fractions by dividing and shading areas of squares or circles and then ordering the fractions from smallest to largest on a number line. The applet even checks if a fraction is correctly modeled and keeps score. A visual support to understanding the magnitude of fractions!

**Transmorgrapher 2**
Another way to "explain" geometric transformations! Using this applet, students can explore the world of translations, reflections, and rotations in the Cartesian coordinate system by transforming polygons on a coordinate plane.

**National Library of Virtual Manipulatives**
In this impressive collection of applets, each applet presents a problem and prompts the student for a solution. The ease of use and clear purpose of each applet make this a truly exceptional site. Below is an example of an activity that fits well in the middle school curriculum.

**Algebra Balance Scales — Negatives**
This virtual balance scale offers students an experimental way to learn about solving linear equations involving negative numbers. The applet presents an equation for the student to illustrate by balancing the scale using blue blocks for positives and red balloons for negatives. The student then solves the equation while a record of the steps taken, written in algebraic terms, is shown on the screen. The exercise reinforces the idea that what is done to one side of an equation must be done to the other side to maintain balance.

**Illuminations, National Council of Teachers of Mathematics Vision for School Mathematics**
The site was developed to illuminate the vision for school mathematics in NCTM’s Principles and Standards for School Mathematics. The activities, lesson plans, and other resources are designed to improve the teaching and learning of mathematics for all students. Below are two examples of material for the middle grades level.

**Exploring Angle Sums**
Students explore the sum of the interior angles of triangles, quadrilaterals and other polygons. To do this, they mark a midpoint on any side, then rotate the figure 180 degrees about that midpoint. They eventually get all interior angles together at one vertex and consider what the figure suggests about the angle sum.

**Geometric Solids**
This tool allows learners to investigate various geometric solids and their properties. They can manipulate and color each shape to explore the number of faces, edges, and vertices, and to answer the following question: For any polyhedron, what is the relationship between the number of faces, vertices, and edges?

### Challenging with Rich Problems

What makes a problem "rich?" In my opinion, rich problems have multiple entry points, force students to think outside the box, may have more than one solution, and open the way to new territory for further exploration. The problems in these resources can challenge your students and enliven their study of mathematics.

**Problems with a Point**
A collection of problems designed to help students in grades 6-12 learn new mathematical ideas by building on old ones, this resource is exceptional in the quality of the problems. Varying in difficulty and approaches, the problems can be searched not only by topic but also by problem-solving strategy, class time, technology needed, and students’ mathematical background. Each problem offers ideas for exploration, classroom discussion, and extensions.

**Ohio Resource Center for Mathematics, Science, and Reading**
Among this site’s resources is a collection of rich math problems intended generally for the high school level, but the following three could appropriately challenge middle school students.

**What Is the Average Birth Month?**
What is the average month for births? The class may start out with assigning a number to each month — January = 1, February = 2, and so forth — and then find "the average". Examining just what "average" means in this case leads to selecting and graphing the best way to find an "average" with categorical data. Next, students examine class data and are asked if this information is representative of the entire population. In this way, students explore a question that engages them even as it leads to deeper understanding of basic statistical concepts. Questions for class discussion and teaching tips are included.

**The Mouse**
City Hall has a rectangular lobby with a floor of black and white tiles. The tiles are square, in a checkerboard pattern, lined up with the walls: 93 tiles in one direction and 231 in the other. There are two mouse holes, at diagonally opposite corners of the floor. One night a mouse comes out of one mouse hole and runs straight across the floor, and into the other mouse hole. How many tiles does the mouse run across? A complete solution and handouts are provided.

### 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.

### Careers

**The FunWorks**
Visit the FunWorks STEM career website to learn more about a variety of math-related careers (click on the Math link at the bottom of the home page).

### NCTM Standards

*Students learn mathematics through the experiences that teachers provide.* — Principles and Standards for School Mathematics, p. 16

The Teaching Principle is one of six principles describing the National Council of Teachers of Mathematics’s vision of high-quality mathematics education. As noted in the discussion of this principle, teaching mathematics well requires several types of knowledge, including pedagogical knowledge, which "helps teachers understand how students learn mathematics, become facile with a range of different teaching techniques and instructional materials, and organize and manage the classroom" (NCTM, 2000, p. 17). The resources featured in Teaching Strategies in Middle School Math present a wide range of techniques and supporting materials in each type. They may give you opportunity to explore a teaching strategy new to you, or simply build up your store of activities in a strategy already familiar to you.

In a discussion of professional standards for mathematics teachers, Mathematics Teaching Today, NCTM emphasizes that "the tasks and activities that teachers select are mechanisms for drawing students into the important mathematics that composes the curriculum. Worthwhile mathematical tasks are those that do not separate mathematical thinking from mathematical concepts or skills, that capture students’ curiosity, and that invite students to speculate and to pursue their hunches" (NCTM, 2007, p. 33).

The activities in the resources highlighted here, both online and offline, directly address middle school curriculum while challenging students to make sense of the mathematical concepts through their own reasoning. We hope these resources will add to your own list of "worthwhile mathematical tasks".

## Author and Copyright

Terese Herrera taught math several years at middle and high school levels, then earned a Ph.D. in mathematics education. She is a resource specialist for the Middle School Portal 2: Math & Science Pathways project.

Please email any comments to msp@msteacher.org.

Connect with colleagues at our social network for middle school math and science teachers at http://msteacher2.org.

Copyright December 2007 - The Ohio State University. This material is based upon work supported by the National Science Foundation under Grant No. 0424671 and since September 1, 2009 Grant No. 0840824. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.