# MiddleSchoolPortal/Algebraic Thinking: A Basic Skill

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<keywords content="middle school mathematics generalization abstraction argument algebraic expressions

solving equations understanding graphs moving from patterns to rules to functions" />

<metadescription content="This free, standards-based, online publication, developed for middle school mathematics teachers, supports teaching algebraic thinking by linking to and describing inquiry-based lessons and activities." />

### Numbers at Work! - Introduction

Algebraic thinking should be considered a basic skill for middle school students because it is compatible with their changing, maturing intellectual capabilities. Glenda Lappan, a teacher educator and author of textbooks for the middle grades, noted consequences of this intellectual growth for school mathematics programs:

## Contents[hide] |

Students in middle grades are growing in their ability to reason abstractly. They become capable of generalization, abstraction, and argument in mathematics. This signals the need for programs that give students the opportunity to expand their experiences with "doing mathematics," with controlling variables and examining the consequences, with experimenting, making conjectures, and developing convincing arguments to support or disconfirm a conjecture (Lappan, 2000, p. 23).

The resources highlighted here aim to reflect students' growing mathematical capacity over the span of the middle school years. The activities and lessons, intended as supplementary materials, range from introduction to the fundamentals of algebra to work on linear functions. Uniformly, they take into consideration the preference of the middle school student for concrete models, visual representations, and interactive tasks. You will find resources on:

- Working with algebraic expressions
- Solving equations
- Understanding graphs
- Moving from patterns to rules to functions

Some are games, others are online simulations that can complement a lesson, and yet others are full-blown lesson plans. We believe you will find tasks here that motivate your students to expand their basic skills in algebra.

### Background Information for Teachers

Being expected to teach algebra—at any level of middle school—can arouse anxiety. Questions may range from the actual teaching of the material to the content knowledge needed to teach such an abstract subject. The first three resources here speak directly to these questions. Each is a professional course offered free and online. The next two are large sets of activities, virtual treasure troves!

**Insights into Algebra 1: Teaching for Learning**
In this online professional development workshop for middle and high school teachers, participants explore strategies for teaching 16 topics found in most Algebra 1 programs. In each session, teachers view two half-hour videos that feature effective strategies for teaching specific algebra topics. Then, using a workshop guide, teachers participate in activities designed to help them incorporate these strategies in their own practice. From Annenberg/CPB: Teacher Professional Development.

**Learning Math: Patterns, Functions, and Algebra**
An online course designed for K-8 teachers who want to explore or deepen their understanding of the "big ideas" in algebraic thinking. Each of 10 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. From Annenberg/CPB: Teacher Professional Development.

**Algebra in Simplest Terms**
This video series could be valuable for teachers who want to review algebra. Intended for high school classrooms and adult learners, the course offers 26 half-hour video programs and coordinated books. From Annenberg/CPB: Teacher Professional Development.

**Algebra Problem of the Week**
You will find excellent algebra problems from a variety of sources here, at varying levels of difficulty. The problems challenge students to think through their solutions and put them in writing. A small subscription fee is charged for entry to the problems section.

**King's List of On-line Math Activities**
This comprehensive site contains links to numerous online math activities. What makes it invaluable to the teacher is its listing of specific topics. For example, under Pre-algebra/Algebra, you will find links to activities on functions machines, order of operations, and solving equations.

### Working with Algebraic Expressions

These activities can provide a first encounter with positive and negative numbers, order of operations, and factoring polynomials. Ranging from games to full lesson plans, these resources could supplement your classwork in these areas of algebraic manipulation.

**Space Coupe to the Rescue**
In this online activity, students key in a positive or negative number to raise or lower their space coupe. The challenge is to line up the coupe with a virus pod on the screen; this destroys the pod, one of eight to be destroyed within a given time. The distance the ship needs to travel to destroy each pod is counted using a scale on the left side of the screen. An excellent first experience with integers!

**Late Delivery**
In this game, the student helps the mail carrier deliver five letters to houses with numbers such as 3(a + 2). The value of a is held by the dog. This is a good exercise in substituting for variables. Three levels of difficulty are available. The game is part of the Maths File Game Show.

**Amby's Math Resources: Order of Operations**
This resource is a tutorial and practice on a topic that often frustrates the younger middle school student. Immediate feedback is given when an incorrect answer is chosen, plus a full explanation of the correct solution.

**Explaining Order of Operations**
How would you explain the order of operations to a fifth or sixth grader who has not yet studied the subject? This is a question from a teacher. An answer from Dr. Math sets out how he would develop the accepted order of operations in much the same way that students might develop the rules for a game.

**Balance Pans-Expressions**
This interactive pan balance allows students to enter numeric or algebraic expressions. They can "weigh" the expressions by entering them on either side of the balance and seeing if they are equivalent.

**Understanding Algebraic Factoring**
An excellent set of lesson plans that introduce factoring through finding areas of rectangles! Each step in the procedure is well explained and illustrated. Questions for the class are included. This unit is meant to be worked with algebra tiles, either the usual plastic ones or cut-out paper shapes.

**Difference of Squares**
Designed for two class periods, this lesson begins with computing the squares of any two consecutive integers and finding the difference of the squares. Students build on further arithmetic experiences as they are led to generalize to the algebraic rule on factoring the difference of two squares. Steps are carefully set out. As an extension of the lesson, a geometric interpretation of the rule is explained and illustrated.

### Solving Equations

Each activity here involves solving equations or investigating the rationale underlying the basic rules for solving equations. All challenge students in unique ways.

**Gone Fishing: My, My, Little Fish-How You've Grown!**
The activity opens with a cartoon showing the weights of three combinations of fish. The challenge is to determine the weight of each fish. Three solutions are set out graphically and in terms that can be easily translated to algebraic symbols. An excellent introduction to the manipulation of equations and the reasoning that underlies it!

**Equation Match**
Students must solve equations, from the most simple to more complex, and in this way find pairs of equations that "match"; that is, both equations in the pair have the same value of x. When a match is found, part of a picture is revealed. At each return to the game, a new set of equations is given. From the Maths File Game Show.

**Algebra-Fun with Calendars**
Clever tricks-sleight of mind, if you will-involve calendars and the patterns of numbers on them. Each "trick" is found to be an equation that simplifies to an algebraic expression. Students end by creating their own calendar problems and simplifying their own equations. From Mathematics Lessons That Are Fun! Fun! Fun!.

**Balance Beam Activity**
This simulation of a balance beam allows students to explore the meaning of balance, a key concept in developing mathematical understanding of solving equations. Working with two to four shapes of differing weights, the students must experiment to balance the virtual scale by adding shapes of unknown weights. Finally, although the weight of one shape is known, students must use basic equation principles to find the weights for the other shape or shapes.

**Algebra Balance Scales-Negatives**
Like the Balance Beam Activity above, this online manipulative also features a virtual balance scale, but the challenge is very different. The activity offers students an experimental way to learn about solving linear equations involving negative numbers. The applet presents an equation for students to illustrate by balancing the scale, using blue blocks for positive units and variables and red balloons for negative units and variables. Students then work with the arithmetic operations to solve the equation. A record of the steps taken by the student is shown on the screen and on the scale. The applet reinforces the idea that what is done to one side of an equation must be done to the other side to maintain balance.

### Understanding Graphs

Graphing holds a central place in algebra. It connects the algebraic statement to a geometric representation. These resources begin at the beginning—plotting points—then move on to examining slope and the graphs of linear and nonlinear functions. Lots of opportunity for experimenting with graphs!

**General Coordinates Game**
Here is a game for those just learning the Cartesian coordinate system. An applet allows players to name the coordinates of a house placed on the grid by the computer, or input their own coordinates for the house. A brief history of Descartes' invention and a good discussion on plotting points are provided, but, unfortunately, no game rules are included. Your students will enjoy inventing their own.

**Maze Game**
The game is to move a robot from the bottom corner of an x-y grid to the top right corner, but mines block the way. To move the robot, players have to give the coordinates of the next point; one mistake and BAM! Students can request up to 30 mines and, after a little experience with the game, will probably want that many. Good practice on locating and naming x-y coordinates.

**Understanding Distance, Speed, and Time Relationships**
In these two lessons, students use an online simulation of one or two runners along a track. Students control the speed and starting point of the runner, watch the race, and examine a graph showing time versus distance. Students can use the activity to come to conclusions on the distance, speed, and time relationship. They can also use it to consider the graphical representation and the concept of slope.

**Stressed Out-Slope as Rate of Change**
A story and a graph begin this activity. The graph shows how performance is related to stress; questions ask students to analyze the graph in terms of the story situation. Students then consider a graph of speed over time, and are asked to create their own graph showing "a direct relation between the rate of change of a function and the slope of its lines." This activity is part of Mathematics Lessons That Are Fun! Fun! Fun!

**Planet Hop**
You are traveling through space. You must find the coordinates of four planets and then write the equation of the line you have created. This game is part of the Maths File Game Show.

**Lines and Slope**
A chameleon named Joan helps illustrate how to graph linear equations, first using only two points that are solutions to an equation, then gradually introducing the idea of slope, and building to the slope-intercept form of the equation. The goal is to visualize the concept of slope and understand the reasoning behind the standard formulas.

**Slope Slider**
Using this simulation, students can manipulate a linear function of the form f(x)=mx+b and explore the relationship between slope and intercept in the coordinate system. They can see the slope of the line change as they change the value of m.

**Grapher: Algebra (Grades 6-8)**
An interactive tool for practicing graphing or for visualizing functions! Students can graph one to three functions, of different colors, in the same window. Functions can be written with absolute value, square roots, exponents, or algebraic fractions, and the domain can be restricted.

**Function Flyer**
This simulation tool allows students to graph not only linear but also exponential and polynomial functions. Students create the function, see it graphed, and can then change the constants, noting the effects visually and immediately on the graph.

### From Patterns to Relations to Functions

Since elementary school, your students have worked with patterns. The lessons and activities featured here move students mathematically forward to consideration of the rules underlying the patterns and then to formulation of those rules in algebraic terms.

**Barbie Bungee**
Looking for a "real-world" example of a linear function? In this lesson, students model a bungee jump using a Barbie 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.

**Function Machine**
Operating under a secret rule, the function machine uses numbers input by the students to generate output. Students compare the input (domain) to the output (range) to find the function rule. The analogy of a function machine is a basic, strong visual that holds up even in advanced study of functions.

**Walk the Plank**
One end of a wooden board is placed on a bathroom scale and the other end is suspended on a textbook; students can literally "walk the plank" and record the weight shown on the scale as their distance from the scale changes. It turns out that the relationship between the weight and distance is linear, and this investigation leads to a real-world occurrence of negative slope. An activity sheet, its solutions, and questions for class discussion are included in this one-period lesson.

**The Handshake Problem**
This two-lesson unit allows students to discover patterns in a fictional but real-world scenario: How many handshakes occur when the nine Supreme Court justices shake hands with each other? Students explore—through a table, a graph, and finally an algebraic formula—the number of handshakes in any size group. A second pattern is explored, that of triangular numbers; again, students generalize the pattern with variables. The lessons are well illustrated and include background information for the teacher.

**Building Bridges**
Designed expressly for middle school classes, this lesson is built on the premise that "teachers need help in building a bridge between their current instructional goals and new goals that emphasize an earlier introduction to algebraic thinking." As students work through tasks, they organize values into tables and graphs as they move toward symbolic representations of the functions involved. The problem situations, carefully explained, employ linear, quadratic, and exponential models.

**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.

### SMARTR: Virtual Learning Experiences for Students

Visit our student site **SMARTR** to find related 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. Visit the virtual learning experience on **Equations** and **Graphing.**

### 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

In presenting its "ambitious, focused mathematics program for all students in the middle grades," the National Council of Teachers of Mathematics identifies "ambitious expectations . . . in algebra and geometry that would stretch the middle-grades program beyond a preoccupation with number" (NCTM, 2000, p. 211). Algebra is not proposed as a one-year course for a select few but as an integrated component of a program that extends throughout the middle school math curriculum.

The NCTM Principles and Standards for School Mathematics document presents algebra as more than an exercise in manipulating symbols or even solving equations; it is a way of thinking that permeates a broad range of math content. The document states that "students in the middle grades should learn algebra both as a set of concepts and competencies tied to the representation of quantitative relationships and as a style of mathematical thinking for formalizing patterns, functions, and generalizations" (p. 223). In particular, the Standards recommend that students see "algebra and geometry as interconnected," experiencing "both the geometric representation of algebraic ideas, such as visual models of algebraic identities, and the algebraic representation of geometric ideas, such as equations for lines represented on coordinate grids" (p. 212).

You will find the resources featured here to be appropriate for a wide range of students. Moreover, they focus not only on facility with algebraic manipulation but also on algebra as a way to represent a mathematical situation. The connection between geometry and algebra is highlighted through activities that focus on students' understanding of graphs, from simply plotting points to actually graphing linear functions.

For more information on the NCTM Standards, check out the nine specific expectations for algebra at the middle school level. For more insight into how to reach those expectations, Navigating through Algebra in Grades 6-8 offers problems and activities that show how students can model mathematical situations through algebraic thinking.

## 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 June 2006 - 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.