This algebra module has been designed to develop grade 6 students’ understanding of ten key algebraic concepts and to enhance their problem solving skills. Each section begins with a brief description of the problem set and of the concepts and skills that will be developed. This is followed by the solutions to the problems in the problem set. The first problem in each set is the “teaching problem.” It is completed following the five-step problem solving model and is designed to be used by teachers as the centerpiece of the instructional program. The teaching problem is followed by problems that are to be completed by students working on their own.
This module may be used as an algebra unit to complement the existing instructional program. It also may be used to show connections between algebra, strands of number, and measurement, to provide practice of number computational algorithms, and to reinforce problem solving skills.
The Extra for Experts problem sets provide additional opportunities for students to apply newly learned concepts and skills toward solving problems similar to those developed in this algebra module.
The Key Algebraic Concepts
Equality/Inequality: In this module, students gain greater understanding of equality as they learn about and apply the fundamental order of operations, which include exponentiations and the distributive property, to determine the values of expressions that name the same number. Understanding and using the distributive property is the key to working with polynomials in formal courses in algebra.
Variables as Unknowns: Variables may be letters, geometric shapes, or objects that stand for a number of things. When used to represent an unknown, the variable has only one value. For example, in the equation, , the variable is an unknown and it stands for the number 2. When is replaced with 2 in the equation, the expression to the left of the equal symbol names the same number as the expression to the right of the equal symbol; is another name for 7. Students solve for the values of unknowns in a variety of settings that model systems of equations, including letter grids and hanging scales.
Variables as Varying Quantities: In some equations, variables can take on more than one value. For example, in the equation , if the value of is 5, then the value of is 3. Or, if the value of is 9, the value of is 11. So, in this case can be any number. However, once a value for is chosen, then the value for is fixed. Likewise, once a value for is chosen, then the value of is fixed. Variables as varying quantities appear in functions and formulas. In this module, students identify and write rules for two-step (e.g., ) functions in words, and apply perimeter and area formula to solve problems.
Proportional Reasoning: A major method for solving algebraic problems is by reasoning proportionally. Proportional reasoning is sometimes called “multiplicative reasoning,” because it requires application of multiplication or its inverse, division. In this module, students reason proportionally when they determine weights of coins in grams.
Interpret Representations: Mathematical relationships can be displayed in a variety of ways including with text, tables, graphs, diagrams and symbols. Having students interpret these types of displays and use the data in the displays to solve problems is critical to success with the study of algebra. In this module, students interpret circle and arrow grid diagrams, weight scales, and line graphs.
Write Equations: Although an equation is a symbolic representation of a mathematical relationship, the writing of an equation is a key algebraic skill that requires separate attention and instruction. In this module, students write systems of equations using letters to represent weights of blocks pictured on scales, purchases of items, number of shapes in patterns, and distance/time relationships.
The Problem Solving Five-Step Model
The model that we recommend to help students move through the problem solving process has five steps:
Describe focuses students’ attention on the information in the problem display. In some cases, the display is a diagram. Other times it is a table, graph, equation, model, or a combination of any of these. Having students tell what they see will help them interpret the problem and identify key facts needed to proceed with the solution method.
My Job helps students focus on the task by having them explain what they have to do; that is, rephrase the problem in their own words.
Plan requires identification of the steps needed to solve the problem and helps students focus on the first step. Knowing where to start is often the most difficult part of the solution process.
Solve is putting the plan to work and showing the steps.
Check is used to verify the answer.
We recommend that you “model” these steps in your instruction and that you encourage your students to follow the steps when solving the problems in this module and when relating their solution processes to others.
Note: Although the instructional pages show only one solution plan, many of the problems have more than one correct solution path. These problems provide excellent opportunities for engaging your students in algebraic conversations about how their solutions are the same, how they differ, and perhaps, which solution method is “most elegant.”