Weekly Overview

Weekly Topics

The focus of this week’s instruction is to deepen students’ understanding of:

  • Generating Equivalent Expressions (both lessons 1 and 2)
  • Understanding Equations
  • Using If-Then Moves in Solving Equations (both lessons 4 and 5)

Materials Needed

  • Manila Envelopes (enough for each student in the class to have one)
  • Expressions
  • Equations
  • Tape Diagram
  • Pencil and Paper

Standard(s) Covered

7.EE.B.3

Solve multi-step real-world and mathematical problems posed with positive and negative rational numbers presented in any form (whole numbers, fractions, and decimals).

  1. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate.
  2. Assess the reasonableness of answers using mental computation and estimation strategies.

Representations

  • Tape Diagram
  • Triangles and Quadrilaterals
  • Numerical Sentences

Additional Terms and Symbols

  • Variable:  A variable is a symbol (such as a letter) that represents a number (i.e., it is a placeholder for a number). A variable is actually quite a simple idea:  it is a placeholder—a blank—in an expression or an equation where a number can be inserted.  A variable holds a place for a single number throughout all calculations done with the variable—it does not vary.  It is the user of the variable who has the ultimate power to change or vary what number is inserted, as he/she desires.  The power to vary rests in the will of the student, not in the variable itself.
  • Numerical expression:  A numerical expression is a number, or it is any combination of sums, differences, products, or divisions of numbers that evaluates to a number.
  • Value of a numerical expression:  The value of a numerical expression is the number found by evaluating the expression. For example, \({1 \over 3} \) ∙ (2 + 4) - 7 is a numerical expression, and its value is -5.
  • Expression:  An expression is a numerical expression, or it is the result of replacing some (or all) of the numbers in a numerical expression with variables. There are two ways to build expressions:  We can start out with a numerical expression, such as \({1 \over 3} \) ∙ (2 + 4) - 7 and replace some of the numbers with letters to get \({1 \over 3} \) ∙ (x + y) - z. We can build such expressions from scratch, as in x + x(y-z) , and note that if numbers were placed in the expression for the variables x, y, and z, the result would be a numerical expression. The key is to strongly link expressions back to computations with numbers through building and evaluating them. Building an expression often occurs in the context of a word problem by thinking about examples of numerical expressions first and then replacing some of the numbers with letters in a numerical expression.  The act of evaluating an expression means to replace each of the variables with specific numbers to get a numerical expression, and then finding the value of that numerical expression.
  • Equivalent expressions:  Two expressions are equivalent if both expressions evaluate to the same number for every substitution of numbers into all the letters in both expressions.
  • An expression in standard form:  An expression that is in expanded form where all like terms have been collected is said to be in standard form.  Expanded form is where the like terms are not collected (3x + 2y + 5x – y, is an example of expanded form.  To write this in standard form would be 8x + y).
  • Term (description):  Each summand of an expression is called a term.
  • Coefficient of a term:  The coefficient of a term is the number multiplying a variable.  For example, 3x + 2y – c + 2.  3x, 2y, -c, and 2 are all terms.  3 is the coefficient of x, 2 is the coefficient of y, -1 is the coefficient of c, and 2 is the constant term and will therefore have no coefficient.
  • Equation:  An equation is a statement of equality between two expressions. If A and B are two expressions in the variable x, then A=B is an equation in the variable x.  Students sometimes have trouble keeping track of what is an expression and what is an equation.  An expression never includes an equal sign (=) and can be thought of as part of a sentence.  The expression 3+4 read aloud is, “Three plus four,” which is only a phrase in a possible sentence.  Equations, on the other hand, always have an equal sign, which is a symbol for the verb is.  The equation 3+4=7 read aloud is, “Three plus four is seven,” which expresses a complete thought (i.e., a sentence).  Number sentences—equations with numbers only—are special among all equations.
  • Number sentence:  A number sentence is a statement of equality (or inequality) between two numerical expressions.  A number sentence is by far the most concrete version of an equation.  It also has the very important property that it is always true or always false, and it is this property that distinguishes it from a generic equation.  Examples include 3+4=7 (true) and 3+3=7 (false).  This important property guarantees the ability to check whether or not a number is a solution to an equation with a variable:  just substitute a number into the variable.  The resulting number sentence is either true or it is false.  If the number sentence is true, the number is a solution to the equation.  For that reason, number sentences are the first and most important type of equation that students need to understand.
  • Solution:  A solution to an equation with one variable is a number that, when substituted for all instances of the variable in both expressions, makes the equation a true number sentence.

Materials List

The following materials list will be used for the entire four weeks: Materials List.