AT1a. Explain how to solve equations such as 283 + 19 = x + 18 by "thinking relationally" (e.g., by recognizing that because 19 is 1 more than 18, x should be 1 more than 283 to make both sides equal) rather than by applying standard algebraic methods.

AT1b. Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols by applying the order of operations conventions.

AT1c. Write, read, and evaluate expressions that include letters for quantities, recognize and produce equivalent expressions, and understand the difference between an expression and an equation.

AT1d. Recognize a "run-on equation" as an invalid equation. For example, 12 - 4 = 8 + 5 = 13 - 6 = 7 is not a correct equation.

For many students, the equal sign can be interpreted as "the answer is coming" or an operator instead of a symbol that represents a relationship. This is because many equations are written in the form: 5 + 4 = 9 or 3 x 8 = 24 (in both of these instances "the answer" is on the other side of the equal sign). It is important for us to use the equal sign to show equivalent relationships (e.g., 5 + 4 = 3 + 6 or even 5 + 4 = 5 + 2 + 2; as well as arithmetic properties: 5 + 4 = 4 + 5). We must be very cognizant about the way we record the mathematical symbols because when students only see things written in one way, they think that is the only way to write them.

When recording students' thinking, many teachers create "run-on equations." We must develop PSTs' understanding of the equal sign as a relationship so that they do not promote these same misconceptions.

Additionally, because the equal sign is a button on the calculator (just like +, -, x, etc.), it furthers their (incorrect) thinking that the equal sign is an operator/operation.

Pattern tasks give students an opportunity to connect their verbal descriptions to mathematical expressions. As teachers use mathematical language to notate students' thinking it can help students see mathematical expressions as meaningful.

AT1 Article "What is Algebra in Elementary School?" by Jennifer Bay-Williams, Teaching Children Mathematics, Dec 2001 (Need NCTM membership to access.)

AT1.a Activity "Solving Equations by Reasoning about Expressions," Class Activity 9H in Beckmann, Mathematics for Elementary Teachers with Activities, 5th Ed

AT1.b Activity "Writing Expressions for Dot, Star, and Stick Designs," Class Activity 9A in Beckmann , Mathematics for Elementary Teachers with Activities, 5th Ed

AT1.c Article describing activity "Using Pattern Tasks to Develop Mathematical Understandings and Set Classroom Norms," Smith, et al., Mathematics Teaching in the Middle School, August 2007. (Need NCTM membership to access.)

AT1.c Source for additional pattern tasks visualpatterns.org

AT1.c Activity "Be a Mathemagician," NCTM Student Math Notes, January 2008

AT1.c Activity "Equivalent Expressions," Class Activity 9F in Beckmann, Mathematics for Elementary Teachers with Activities, 5th Ed

AT1.c Activity "Expressions for Quantities," Class Activity 9G in Beckmann, Mathematics for Elementary Teachers with Activities, 5th Ed

AT1.d Article "Children's Understanding of Equality: A Foundation for Algebra" in Teaching Children Mathematics, Dec 1999, 232-236

OP2a. Make sense of models for addition, subtraction, multiplication and division that progress from concrete through pictorial to abstract. Justify the choice of model for the context of a problem. Write and solve equations with the unknown in each location.

OP2b. State, recognize and apply: associative property (addition, multiplication), closure (addition, multiplication), commutative (addition, multiplication), identity (addition, multiplication, inverses (addition, multiplication), distributive property, zero property of multiplication.

These indicators are from earlier work on Number and Operations (the Operations indicators). They are reiterated here because it is essential that future teachers see the connections between operations with numerical values and operations within algebraic expressions. The resources listed at right are a subset of those listed above; these are some ideal opportunities to highlight these connections. For instance, in writing expressions for pattern tasks, students can be reminded that the expression 2x means either "2 groups of x items" or "x groups of 2 items." They should identify how they are connecting the expression to the groups of items they see in the pattern task.

In solving equations It is important that future teachers understand the inverse relationship between the inverse operations (addition & subtraction; multiplication & division) and how the operations relate to one another. We want students to be see a subtraction problem and be able to "think addition" to help them solve it; and likewise with division (e.g., see 72/8 and think "what times 8 equals 72?").

We also want to be sure to relate counting and skip counting to these operations. For example, addition can be seen as "counting up or counting on" where subtraction can be seen as "counting back." Multiplication can be seen as skip counting or repeated addition and division can be seen as skip counting backwards or repeated subtraction. Making these connections explicit is very important because the foundation for success with calculations starts with counting.

Be sure to discuss the properties with the different methods for each of the operations because that is where they are useful. It is helpful to understand that I can think of 9 + 6 as 9 + 1 + 5 = 10 + 5 using the associative property OR 4 x 17 as 4(10 + 7) = 4(10) + 4(7) = 40 + 28 = 68 using the distributive property. These properties are the reason we are able to decompose and recompose numbers to compute quickly and efficiently (especially mentally) and should be discussed when explaing WHY one is able to "do the calculations in that way."

OP2 Article "What is Algebra in Elementary School?" by Jennifer Bay-Williams, Teaching Children Mathematics, Dec 2001 (Need NCTM membership to access.)

OP2a Activity "Writing Expressions for Dot, Star, and Stick Designs," Class Activity 9A in Beckmann , Mathematics for Elementary Teachers with Activities, 5th Ed

OP2a Book Number Talks, Parrish

OP2a Book Making Number Talks Matter, Humphreys and Parker

OP2a Blog post "Numberless Word Problems"

OP2a, OP 2b Activity "Solving Equations Algebraically and with a Pan Balance," Class Activity 9I in Beckmann, Mathematics for Elementary Teachers with Activities, 5th Ed

OP2a, OP 2b Activity "Solving Word Problems with Strip Diagrams and with Equations," Class Activity 9K in Beckmann, Mathematics for Elementary Teachers with Activities, 5th Ed

OP2a, OP2b Article describing activity "Using Pattern Tasks to Develop Mathematical Understandings and Set Classroom Norms," Smith, et al., Mathematics Teaching in the Middle School, August 2007. (Need NCTM membership to access.)

OP2a, OP2b Source for additional pattern tasks visualpatterns.org