# Testing! Beyond One-Step Math Problems

Middle school students often do well on straight calculations but feel lost when faced with more complex problems. And many tests these days require critical thinking and ask for an extended response. As in all test preparation, students need practice, especially on problems requiring more than one-step computation. Here are sites that offer test items you might use in class reviews, challenges, quizzes, etc.  Please let us know of other treasure troves of practice problems or how you prepare your students for testing.

Searching for Solutions
Within this WebQuest is a set of individual lessons on several problem-solving techniques, such as finding patterns, making a table, working backward, and solving a simpler problem. Each strategy is explained simply, with students in mind, then activities requiring that strategy are presented. An excellent guide from an e-learning specialist on ways to attack math problems!

Balanced Assessment
A set of more than 300 assessment tasks, indexed for grades K-12. Each incorporates a story problem format and includes hands-on activities. Rubrics provided.

Figure This! Math Challenges for FamiliesCreated for students in grades 6 to 8, the site offers math challenges that focus on everyday life, such as how fast your heart beats, what shape container holds the most popcorn, and how much of you shows in a small wall mirror.

Word Problems for Kids
A wide range of carefully selected problems! Organized by grade level from 5 through 12, each problem links to a helpful hint and to the answer; the more difficult problems offer complete solutions.

Problems with a Point
Here you can search for word problems by topic, lesson time, required mathematical background, and problem-solving strategy. Take the time to do the short guided tours of the site, and then look at favorite problems selected by teachers — a good set of problems at the middle school level.

Fermi Questions
Fermi questions emphasize estimation, numerical reasoning, communicating in mathematics, and questioning skills. Students often believe that word problems have one exact answer and that the answer is derived in a unique manner. Fermi questions encourage multiple approaches, emphasize process rather than the answer, and promote non-traditional problem solving strategies.

NAEP Questions
Over 2000 questions archived.  Online tools allow you to search the collection by content area, grade level, and difficulty. The site also shows what students at each achievement level are likely to know and how NAEP questions are scored.

# 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.

# Let’s Play Math!

Math games open an arena for strategic planning and reasoning, one of the NCTM Process Standards. Many games also provide practice in basic operations or problem solving, and all motivate the active participation we hope for in the classroom.

The Maths File Game Show
A collection of 12 interactive games that deal with basic concepts of mathematics: data handling, numbers, algebra, and measurement. Students can use order of operations to ensure that letters reach the right mailbox, compete in a test of fractions and percentages, and even guide a spaceship across Cartesian coordinates.

Towers of Hanoi: Algebra (Grades 6-8)
An online version of the ancient Towers of Hanoi puzzle, featuring three spindles and a graduated stack of two to eight discs, as decided by the player, with the largest disc on the bottom. The challenge is to 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 disc. By observing the pattern of number of discs to number of moves, students can generalize the relation and answer the question “What if you had 100 discs?”

Traffic Jam Activity
Why the jam? There are seven stepping stones and six people. Three stand on the left-hand stones and three on the right-hand; all face center. Everyone must move so that the people on the right and the people on the left pass each other, eventually standing on the side opposite from where they started. But no two people may stand on the same stone at the same time! This problem requires reasoning, but its solution also reveals a pattern that leads to an algebraic expression. A lesson plan is provided.

Clever Games for Clever People
Here are 16 strategy games from the book On Numbers and Games by mathematician John Conway. Each develops those critical thinking skills so valued in math. As for materials, only paper and pencil or, occasionally, crayons or a checkerboard, are needed. Game rules and setup are clearly described and illustrated for each game.

# Word Problems

Problem solving, one of the NCTM Process Standards, is critical to learning mathematics. Students may feel confident with computation, measurement, and statistics, but they can feel completely lost when asked to apply those skills to a word problem. How to start? What to do next? This page offers an introduction to problem-solving strategies, followed by a treasure trove of word problems at the middle school level.

Figure This! Math Challenges for Families
Created for students in grades 6 to 8, the site offers math challenges that focus on everyday life, such as how fast your heart beats, what shape container holds the most popcorn, and how much of you shows in a small wall mirror.

Word Problems for Kids
A wide range of carefully selected problems! Organized by grade level from 5 through 12, each problem links to a helpful hint and to the answer; the more difficult problems offer complete solutions.

Problems with a Point
Here you can search for word problems by topic, lesson time, required mathematical background, and problem-solving strategy. Take the time to do the short guided tours of the site, and then look at favorite problems selected by teachers—a great set of problems at the middle school level.