 Unit 3

Expressions, Equations, and Inequalities

Unit Description:

Unit 3 consolidates and expands students’ previous work with generating equivalent expressions and solving equations and inequalities. By the end of this unit students should fluently solve multistep problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. This work is the culmination of many progressions of learning in arithmetic, problem solving and mathematical practices.  In solving word problems leading to one-variable equations of the form px + q = r and p(x + q) = r, as well as px + q < r and px + q > r, students should solve these equations fluently. This will require fluency with rational number arithmetic, as well as fluency to some extent with applying properties of operations to rewrite linear expressions with rational coefficients.  Students use the properties of operations and the relationships between addition and subtraction, and multiplication and division, as they formulate expressions, equations, and inequalities in one variable and use these to solve problems. They solve reallife and mathematical problems using numerical and algebraic expressions, equations, and inequalities. At the end of the unit students’ work with expressions and equations is applied to finding unknown angles in a figure.  By using facts about supplementary, complementary, vertical, and adjacent angles, students are able to write and solve simple equations to solve for the unknown angle.  The work and required fluency expectations in this unit are a major capstone leading to the mathematical development necessary to perform operations with linear equations in Grade 8.

Standards for Mathematical Practice

MP.1 Make sense of problems and persevere in solving them.

MP.2 Reason abstractly and quantitatively.

MP.3 Construct viable arguments and critique the reasoning of others.

MP.4 Model with mathematics.

MP.5 Use appropriate tools strategically.

MP.6 Attend to precision.

MP.7 Look for and make use of structure.

MP.8 Look for and express regularity in repeated reasoning.

Louisiana Student Standards for Mathematics (LSSM)

 EE:  Expressions and Equations A. Use properties of operations to generate equivalent expressions 7.EE.A.1 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients to include multiple grouping symbols (e.g., parentheses, brackets, and braces). *I can combine like terms with rational coefficients. *I can factor and expand linear expressions with rational coefficients using the distributive property. *I can apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. 7.EE.A.2 Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.  For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.” *I can write equivalent expressions with fractions, decimals, and integers. *I can rewrite an expression in an equivalent form in order to provide insight about how quantities are related in a problem context. B. Solve real-life and mathematical problems using numerical and algebraic expressions and equations. 7.EE.B.3 Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making \$25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or \$2.50, for a new salary of \$27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation *I can convert between numerical forms as appropriate. *I can solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. *I can apply properties of operations to calculate with numbers in any form. *I can assess the reasonableness of answers using mental computation and estimation strategies. 7.EE.B.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. a. Solve word problems leading to equations of the form and where , , and are specific rational numbers.  Solve equations of these forms fluently.  Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.  For example, the perimeter of a rectangle is 54 cm.  Its length is 6 cm.  What is its width? b. Solve word problems leading to inequalities of the form or , where , , and are specific rational numbers.  Graph the solution set of the inequality and interpret it in the context of the problem.  For example:  As a salesperson, you are paid \$50 per week plus \$3 per sale.  This week you want your pay to be at least \$100.  Write an inequality for the number of sales you need to make, and describe the solutions. *For an algebraic equation in the form and , I can fluently solve with speed and accuracy and identify the sequence of operations used to solve. *I can solve word problems leading to equations of the form and . *I can compare an algebraic solution to an arithmetic solution by identifying the sequence of the operations used in each approach. G: Geometry B. Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. 7.G.B.4 Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. *I can determine the parts of a circle including radius, diameter, area, circumference, center and chord. *I can identify . *I can recognize and use the formulas for area and circumference of a circle. *I can find the circumference of a circle, given the area of the circle. *I can justify that can be derived from the circumference and diameter of a circle. *I can apply the circumference or area formulas to solve mathematical and real-world problems. *I can justify the formulas for area and circumference of a circle and how they relate to . *I can informally derive the relationship between circumference and area of a circle. 7.G.B.5 Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. *I can identify and recognize types of angles: supplementary, complementary, vertical, adjacent. *I can determine complements and supplements of a given angle. *I can determine unknown angle measures by writing and solving algebraic equations based on relationships between angles. 7.G.B.6 Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. (Pyramids limited to surface area only.) *I can determine the formulas for area and volume of triangles, quadrilaterals, polygons, cubes and right prisms. *I can determine the procedure for finding surface area for two- and three- dimensional objects composed of triangles, quadrilaterals, polygons, cubes and right prisms. *I can determine when to use these formulas (area, volume, surface area) in real-world and math problems for two- and three- dimensional objects composed of triangles, quadrilaterals, polygons, cubes and right prisms.

Enduring Understandings:

*Variables can be used to represent numbers in any type of mathematical problem.

*Understand the difference between an expression and an equation.

*Expressions can be manipulated to suit a particular purpose to solve problems efficiently.

*Mathematical expressions, equations, inequalities, and graphs are used to represent and solve real-world and mathematical problems.

*Constructing simple equations and inequalities to solve real life word problems is a necessary concept.

*Writing and solving real-life and mathematical problems involving simple equations for an unknown angle in a figure helps students as they engage in higher geometry concepts.

*Reason about relationships among two-dimensional figures, which leads to gaining familiarity with the relationships between angles formed by intersecting lines.

*Geometry and spatial sense offer ways to interpret and reflect on our physical environment.

*Analyzing geometric relationships develops reasoning and justification skills.

Essential Questions:

*How can I apply the order of operations and the fundamentals of algebra to solve problems involving equations and inequalities?

*How can I justify that multiple representations in the context of a problem are equivalent expressions?

*How do I assess the reasonableness of my answer?

*How can I use and relate facts about special pairs of angles to write and solve simple equations involving unknown angles?

*What is the total number of degrees in supplementary and complementary angles?

*What is the relationship between vertical and adjacent angles?

*When and how are expressions, equations, and inequalities applied to real world situations?

*What are some possible real-life situations to which there may be more than one solution?

*How does the ongoing use of decimals apply to real-life situations?

*How can geometry be used to solve problems about real-world situations, spatial relationships, and logical reasoning?