Carole Greenes, Associate Vice Provost for STEM Education, Professor of Mathematics Education, and Director of the PRIME Center, ASU (firstname.lastname@example.org)
Mary Cavanagh, Executive Director of the PRIME Center (email@example.com)
Carol Findell, Professor of Mathematics Education, Boston University (firstname.lastname@example.org)
This algebra module has been designed to introduce grade 2 students to ten key concepts of algebra and to enhance their problem solving skills. Each section begins with a brief description of the problem set and the concepts and skills developed. This is followed by the solutions to 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, to be completed by students with the teacher’s guidance, is followed by problems to be completed by the students working on their own or in pairs.
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 and the strands of number and measurement, to provide practice of number computational algorithms, and to reinforce problem solving skills.
The Extra for Experts provides additional opportunities for students to apply newly learned concepts and skills to the solution of problems like those developed in this Algebra module.
The Key Algebraic Concepts
In this module, students explore equal and unequal relationships by interpreting and reasoning about pictures of pan balances. Their job is to figure out which boxes to place in an empty pan to balance the weight in the other pan. This is preparation for the study of variables as unknowns in equations, and reinforcing the concept that there are often multiple solutions to a problem.
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, t+5=7, the variable t is an unknown and it stands for the number 2. When t 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; 2+5 is another name for 7. In this module, students solve for the weights of blocks on scales, and for unknowns in two-by-two grids representing systems of equations.
Variables as varying quantities:
In some equations, variables can take on more than one value. For example, in the equation q=2+r, if the value of q is 5, then the value of r is 3. Or, if the value of r is 9, the value of q is 11. So, in this case q can be any number. However, once you choose a value for q, then the value for r is fixed. Likewise, once you choose a value for r, then the value of q is fixed. Variables as varying quantities appear in functions and formulas. In this module, students complete tables of values for two-step (e.g., Add 5 and then subtract 2) functions, and continue patterns in which the elements vary based on their position in the pattern.
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 as they trade “critters” as well as compute the unit costs of items in order to determine the better buy.
Mathematical relationships can be displayed in a variety of ways including with text, tables, graphs, diagrams and with 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 pan balances, circle and arrow grid diagrams, tables of values, weight scales and patterns of geometric shapes.
Although an equation is a symbolic representation of a mathematical relationship, the writing of an equation is a key algebraic skill and one that requires separate attention and instruction. In this module, students learn to write letter and number equations for the various collections of weights that can balance the pans.
The Problem Solving Five-Step Model
The model that we recommend to help students move through the solution problems 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, or 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 tell what they have to do, that is, rephrase the problem in their own words.
Plan requires identification of the steps to follow 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 with the first problem in each problem set and that you encourage your students to follow the steps when solving the problems 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.