Solving Equations: Lesson Ideas

Your own textbook has numerous pages on equations and their solutions. What I’ve pulled together here are scenarios that involve students in solving equations in unusual contexts.  A change of pace, a different approach. If you have found lesson ideas on working with equations that you’d like to share with colleagues, please share the wealth!

Algebra—Fun with Calendars
Using any calendar, tell a friend to choose four days that form a square, then tell you only the sum of the four days. You tell her the four numbers! The trick lies in setting up an equation and simplifying it to an algebraic expression.

The Yo-Yo Problem
In these activities, students explore patterns, write a pattern in algebraic language, and solve equations using algebra tiles, symbolic manipulation, and the graphing calculator.

Balance Beam Activity
Students explore the meaning of balance, a key concept in developing mathematical understanding of solving equations. Working with shapes of differing weights, the students must experiment to balance the virtual scale by adding shapes of unknown weights. They use basic equation-solving principles throughout the activity. 

Algebra Balance Scales-Negatives
This 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.

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.

Planet Hop
Here students concentrate on writing an equation. In an interactive online game, they find the coordinates of four planets shown on a 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. Tips for students are available as well as a full explanation of the key instructional ideas underlying the game.


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We want and need your ideas, suggestions, and observations. What would you like to know more about? What questions have your students asked? We invite you to share with us and other readers by posting your comments. Please check back often for our newest posts or download the RSS feed for this blog. Let us know what you think and tell us how we can serve you better. We appreciate your feedback on all of our Middle School Portal 2 publications. You can also email us at msp@msteacher.org. Post updated 11/10/2011.

Making Math Visual

The abstract concepts of mathematics, usually expressed through symbols and un-common vocabulary, can frustrate the visual learners in your middle school classroom. Here is where the computer can become a powerful teaching tool. Such commonplace but abstract concepts as fractional equivalence and the “size” of large numbers can be made visual through technology. 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.

Below are a few examples of what I mean. If you have found other sites that make math visual for your students, please use our comment box below to share them with other teachers!

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 covered (if laid flat). The site can be used to launch lessons on large numbers, volume versus area, or multiplication by a factor of 10.

The Pythagorean Theorem
This site invites learners to discover for themselves “an important relationship between the three sides of a right triangle.” Five interactive, visual exercises require students to delve deeper into the mystery; each exercise is a hint that motivates and entices. The tutorial ends with information on Pythagoras and problems that rely on the theorem for their solutions.

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!

Algebra Balance Scales — Negatives
This virtual balance scale offers students an experimental way to learn about solving linear equations. Blue blocks represent positives and red balloons represent negatives. The student solves an equation by adding or removing the blocks and balloons, while a record of the steps taken, written in algebraic terms, is shown on the screen.

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?

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

Cynthia Lanius’ Fractal Unit
This unit developed for middle school students begins with a discussion of why we study fractals at all. Lessons 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.


We Want Your Feedback
We want and need your ideas, suggestions, and observations. What would you like to know more about? What questions have your students asked? We invite you to share with us and other readers by posting your comments. Please check back often for our newest posts or download the RSS feed for this blog. Let us know what you think and tell us how we can serve you better. We appreciate your feedback on all of our Middle School Portal 2 publications. You can also email us at msp@msteacher.org. Post updated 11/10/2011.

History to Enrich Mathematics Learning!

Although the NCTM Standards do not have student expectations for learning mathematics history, exposure to this history can help students see real-world connections in mathematics.

Approximating Pi
Mathematics and science owe a great deal to Archimedes, including a way to approximate pi. Here is historical information along with an applet to approximate pi using the perimeter lengths of polygons inscribed within and circumscribed around a circle.

The Beginnings of Probability
Teachers can share some of this mathematics history as they work with students to compute probabilities for simple compound events, an NCTM expectation for students in grades 6-8.

The Golden Ratio
This rich site connects linear measurement, ratio and proportion, art, and mathematics history.

Measuring the Circumference of the Earth
This Internet project is hands-on, real-world, and historical. Students join with classes around the world to repeat the experiment of Eratosthenes — measuring the shadow of a meter stick and making calculations to approximate the circumference of Earth.

Pythagorean Puzzle
The Pythagorean theorem is at the intersection of algebra and geometry. At this site, learn about the life of Pythagoras and the development of the Pythagorean theorem. And use an applet to explore the meaning of the most famous equation in algebra.

Manipula Math with Java: Pythagorean Theorem
Here is another applet offering a more sophisticated approach to affirm in a visual way the validity of the Pythagorean theorem.

Tortoise and Hare Race
Uing an applet, students can vary parameters for the race. There is information about Zeno’s paradox along with exploration questions for students that can lead to a discussion about infinity.


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We want and need your ideas, suggestions, and observations. What would you like to know more about? What questions have your students asked? We invite you to share with us and other readers by posting your comments. Please check back often for our newest posts or download the RSS feed for this blog. Let us know what you think and tell us how we can serve you better. We appreciate your feedback on all of our Middle School Portal 2 publications. You can also email us at msp@msteacher.org. Post updated 11/21/2011.

Graphs: An Algebra Basic

Algebra can be hard to visualize. Learning to show relationships between symbolic expressions and their graphs opens an avenue to really seeing algebraic patterns. According to the NCTM Standards, at the middle school students move from plotting points to exploring the graphs of lines, including slope and intercept.

An Introduction to the Coordinate Plane
This is a fun but serious introduction to Cartesian coordinates. Students work interactively online to plot points of integer pairs, to deal with changing the scale of a graph, and to consider “tricky” graphs.

Coordinate Graphing
Another site on the basics of coordinate graphing with opportunities to practice plotting points. This is actually a complete lesson on the subject—introduction, objectives, procedures, and extensions—plus excellent interactive games for students to practice their point-plotting skills.

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. Students also observe the slope of horizontal, vertical, parallel, and perpendicular lines, not a usual topic at a beginning level. Finally, the slope-intercept form (y = mx + b) is carefully set out.

Grapher: Algebra (Grades 6-8)
Using this online manipulative, students can graph functions of their choice—including functions with square roots, exponents, and fractions. In fact, they can graph up to three functions in the same window, a useful capability when considering points of intersection. Also, they can zoom in on a region of the graph and trace the function paths to find particular coordinates. A powerful tool for exploring graphs!


We Want Your Feedback
We want and need your ideas, suggestions, and observations. What would you like to know more about? What questions have your students asked? We invite you to share with us and other readers by posting your comments. Please check back often for our newest posts or download the RSS feed for this blog. Let us know what you think and tell us how we can serve you better. We appreciate your feedback on all of our Middle School Portal 2 publications. You can also email us at msp@msteacher.org. Post updated 11/08/2011.

Order of Operations Rules

The order of operations rules are important now in evaluating numerical expressions and later in the study of algebra. The featured resources offer explanations, examples of what goes wrong when the rules are misapplied, and problems to solve.

Amby’s Math Resources: Order of Operations
Check out this online tutorial offering a pre-test, practice exercises, and a post-test. It is especially handy as a quick review for students working independently.

Explaining Order of Operations
This online article explains the how and why for the simplification of numerical expressions.

Expression Evaluation
Here is information about applying the order of operations rules, as well as an online quiz with a practical application.

Matho 2
Try to win a bingo game while practicing simplifying numerical expressions.

Number Card
Students must correctly combine four numbers on a number card to form a target number.

Operation Order
This online game is a good practice tool because the correct solution is given when a wrong answer is entered. The easiest level involves addition and subtraction while the hardest level requires the use of all operations.

Order of Operations
This site features a clear explanation of the rules for the order of operations with examples. It offers several problems for students to solve, including one practical application.

Please Excuse My Dear Aunt Sally
Here is an explanation of the popular mnemonic with problems and puzzles to solve.


We Want Your Feedback
We want and need your ideas, suggestions, and observations. What would you like to know more about? What questions have your students asked? We invite you to share with us and other readers by posting your comments. Please check back often for our newest posts or download the RSS feed for this blog. Let us know what you think and tell us how we can serve you better. We appreciate your feedback on all of our Middle School Portal 2 publications. You can also email us at msp@msteacher.org. Post updated 11/08/2011.