Unit 6: Expresions, Equations, and Inequalities

Unit 6: Expressions, Equations, and Inequalities

  • In this unit, students solve equations of the forms px+q=r and p(x+q)=r, and solve related inequalities, e.g., those of the form px+q>rand px+q≥r, where p, q, and r are rational numbers.

    In the first section of the unit, students represent relationships of two quantities with tape diagrams and with equations, and explain correspondences between the two types of representations (MP1). They begin by examining correspondences between descriptions of situations and tape diagrams, then draw tape diagrams to represent situations in which the variable representing the unknown is specified. Next, they examine correspondences between equations and tape diagrams, then draw tape diagrams to represent equations, noticing that one tape diagram can be described by different (but related) equations. At the end of the section, they draw tape diagrams to represent situations in which the variable representing the unknown is not specified, then match the diagrams with equations. The section concludes with an example of the two main types of situations examined, characterized in terms of whether or not they involve equal parts of an amount or equal and unequal parts of an amount, and as represented by equations of different forms, e.g., 6(x+8)=72 and 6x+8=72. This initiates a focus on seeing two types of structure in the situations, diagrams, and equations of the unit (MP7).

    In the second section of the unit, students solve equations of the forms px+q=r and p(x+q)=r, then solve problems that can be represented by such equations (MP2). They begin by considering balanced and unbalanced “hanger diagrams,” matching hanger diagrams with equations, and using the diagrams to understand two algebraic steps in solving equations of the form px+q=r: subtract the same number from both sides, then divide both sides by the same number. Like a tape diagram, the same balanced hanger diagram can be described by different (but related) equations, e.g., 3x+6=18 and 3(x+2)=18. The second form suggests using the same two algebraic steps to solve the equation, but in reverse order: divide both sides by the same number, then subtract the same number from both sides. Each of these algebraic steps and the associated structure of the equation is illustrated by hanger diagrams (MP1, MP7).

    So far, the situations in the section have been described by equations in which p is a whole number, and q and r are positive (and frequently whole numbers). In the remainder of the section, students use the algebraic methods that they have learned to solve equations of the forms px+q=rand p(x+q)=r in which p, q, and r are rational numbers. They use the distributive property to transform an equation of one form into the other (MP7) and note how such transformations can be used strategically in solving an equation (MP5). They write equations in order to solve problems involving percent increase and decrease (MP2).

    In the third section of the unit, students work with inequalities. They begin by examining values that make an inequality true or false, and using the number line to represent values that make an inequality true. They solve equations, examine values to the left and right of a solution, and use those values in considering the solution of a related inequality. In the last two lessons of the section, students solve inequalities that represent real-world situations (MP2).

    In the last section of the unit, students work with equivalent linear expressions, using properties of operations to explain equivalence (MP3). They represent expressions with area diagrams, and use the distributive property to justify factoring or expanding an expression.

Last Modified on June 8, 2020