Preparing Middle School Students for Algebra The National Council of Teachers of Mathematics' Principles and Standards for School Mathematics lists the following algebra standards for students in grades 6-12: Understand patterns, relations, and functions Represent and analyze mathematical situations and structures using algebraic symbols Use mathematical models to represent and understand quantitative relationships Analyze change in various contexts Being aware of these fundamental standards and incorporating elements of these standards into daily lessons is critical to preparing middle school students for algebra. One of the ways middle grades teachers can do this is by fostering their students' algebraic thinking. Algebraic thinking can be thought of as a way of thinking and reasoning about relationships. This includes recognizing and analyzing patterns, representing relationships, making generalizations, and analyzing change. A strong foundation in algebraic thinking is a necessary precursor to more traditional algebra activities such as using abstract symbolism. How to Jumpstart Algebraic Thinking in the Middle Grades Encourage abstract thinking—Success in algebra involves the ability to think abstractly and to form generalizations. Middle grades students should be encouraged and fostered to do this in their daily math activities. Give practice in symbolic representation—Success in algebra also entails the ability to represent and analyze relationships symbolically. Middle grades students should be encouraged to do this by setting up equations and expressions to represent numeric and quantitative relationships. Students should also be expected to represent relationships graphically, and should be comfortable creating and analyzing a variety of graphs. Foster reasoning—Middle grades teachers can prepare their students for more advanced algebra classes by fostering their reasoning abilities. Students should be encouraged to make conjectures and gather information to prove or disprove these conjectures. As students do this, they should be encouraged to look for patterns and to represent information graphically and symbolically. A Sample Activity The following sample activity demonstrates several important elements of middle grades math lessons that can help prepare students for algebra. The National Council of Teachers of Mathematics recommends that middle grades students experience a mathematics program that includes "significant amounts of geometry and algebra." It also recommends that these students should "see that these subjects are interconnected with each other." The following cooperative-group activity: connects algebra and geometry standards is project-based has students make conjectures emphasizes patterns and various ways to represent those patterns. These lesson elements can be incorporated into most any middle grades math lesson as a way to help prepare students for algebra and to foster their algebraic thinking. In addition to the algebra standards listed earlier, this activity addresses several geometry standards and expectations: Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships. Understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects. Create and critique inductive and deductive arguments concerning geometric ideas and relationships. Use two-dimensional representations of three-dimensional objects to visualize and solve problems such as those involving surface area and volume. Use geometric models to represent and explain numerical and algebraic relationships. Use coordinate geometry to represent and examine the properties of geometric shapes. A Lesson on Boxes Summary For this activity, cooperative groups will work as packaging research teams charged with coming up with a new box for packaging their company's product. Objectives The groups will: Investigate relationships between box dimensions and their surface area and volume. Make conjectures, gather information, and notice patterns. Represent relationships symbolically. Use variables to represent rules and formulas. Represent data in tables and graphs. Make and present a final product and share their findings with the class. Warm-Up Activity-Connect to Previous Learning For this activity, you may want to start with several area and perimeter examples as a warm-up activity for your students. This will serve to connect the lesson content to prior knowledge, and can be used as an informal assessment to monitor student understanding and identify any students who may need intervention. Ask students to explain the relationship between the side lengths of a rectangle and its area and perimeter. Ask students to provide several examples of how a change in the length of the sides of a rectangle will change the rectangle's perimeter and area. Ask students to discuss any patterns they notice in how a change in dimension changes the perimeter and area of a rectangle. Have students discuss various ways they can represent the formulas for area and perimeter of rectangles, and how they can use variables to represent the relationships between side lengths and perimeter and area. Project Introduction After the warm-up activity, describe the project that students will undertake. Explain to students that they will work in product research teams to investigate various boxes that could be used to package their team's product. Optional activities at this point include having the teams come up with names for their companies and for their products. You may either assign products to the teams, such as cookies, trail mix, or jigsaw puzzles, or have teams choose their products. Establishing the context for this investigation will help engage student interest and make real-life connections. Investigation Once students are in their teams and the basic groundwork has been covered, have students begin their investigation. Explain to students that they are to determine the "best" packaging option for their product. Explain that they are to investigate various boxes that could be used. Begin by having students measure and calculate the surface area and volume of several boxes. Have students begin to record their findings in tables. Once students have measured several boxes, have them investigate what would happen if they were to alter the dimensions of one of the boxes. Have students calculate and record the corresponding changes to the surface area and volume of the box. Have students continue to investigate the measurements of potential boxes to use as packaging for their product. They will need to record the corresponding surface area (materials required to make the boxes) and volume (amount boxes can contain) of potential packages for their product. As students conduct their investigation, they should: Make conjectures about the relationships among box dimensions and surface area and volume. Gather information to prove or disprove their conjectures. Record their findings in tables and in graphs. Look for and discuss patterns. Symbolically represent their findings (write expressions to represent the observed relationships). Share Results After students have completed their investigations, have them develop a presentation that they will give to the class. Teams should create a model of the packaging (box) that they decide to use, and they should prepare a chart that shows a table and graph of their findings. Teams will also need to explain their findings and why they chose the particular box for packaging their product. Beyond Boxes As you plan your lessons, keep this type of activity and the skills it encouraged in mind. Activities such as this one can help students build the skills and knowledge they will need to be successful in their future algebra classes. This article was contributed by Heidi Janzen, a former classroom teacher and mathematics specialist. She now works as an educational consultant in the areas of professional development, curriculum, standards, and assessment.