Unit 3

Expressions, Equations, and Inequalities

 

Grade 7

Math 

Unit Length and Description:

 

36 days

 

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:

 

7.EE.A.1  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.”

o   Standards Clarification:  Students will apply their conceptual understanding and procedural skill in rearranging expressions to make them more meaningful within a particular context.

 

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 ½ 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.

o   Standards Clarification:  The equations and inequalities in this unit should provide the students an opportunity to work with all types of rational numbers, not just integers.

 

7.EE.B.4  Use variables to represent quantities in a realworld 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 px + q = r and p(x + q) = r, where p, q, and r, 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?

o   Standards Clarification:  It is not necessary nor is it encouraged to use the distributive property to solve the multi-step equations in this unit.  The equations in this unit should provide the students an opportunity to work with all types of rational numbers, not just integers.

b.   Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r 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.

o   Standards Clarification:  Although the mathematical examples in the standard only show the less than and greater than signs, it is appropriate to use the less than or equal to and the greater than or equal to signs as well.  The inequalities in this unit should provide the students an opportunity to work with all types of rational numbers, not just integers.

 

7.G.B.5  Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and use them to solve simple equations for an unknown angle in a figure.

o   Standards Clarification:  Students will be expected to write and solve equations for an unknown angle in a figure. This provides an opportunity to reinforce students’ fluency in solving equations; also, this unit provides an opportunity to model real-world situations involving angles with equations and inequalities. The standards do not explicitly address at any grade level the measure of a straight angle. It may need to be discovered by applying the students’ understanding of right angles and angle addition.

 

Focus Standards of 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.

 

Instructional Outcomes:

Full Development of the Major Clusters, Supporting Clusters, Additional Clusters and Mathematical Practices for this unit could include the following instructional outcomes:

 

7.EE.A.1

·        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

·        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.

 

 7.EE.B.3

·        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.4a

·        I can identify the sequence of operations used to solve an algebraic equation of the form px + q=r and p(x + q) = r.

·        I can fluently solve equations of the form px + q=r and p(x + q) = r with speed and accuracy.

·        I can solve word problems leading to equations of the form px + q=r and p(x + q) = r, where p, q, and r are specific rational numbers.

·        I can compare an algebraic solution to an arithmetic solution by 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 the width? This can be answered algebraically by using only the formula for perimeter (P=2l+2w) to isolate w or by finding an arithmetic solution by substituting values into the formula.

 

7.EE.B.4b

·        I can graph the solution set of the inequality of the form px + q>r or px + q<r, where p, q, and r are specific rational numbers.

·        I can use variables and construct equations to represent quantities of the form px + q=r and p(x + q) = r from real-world and mathematical problems.

·        I can solve word problems leading to inequalities of the form px + q>r or px + q<r, where p, q, and r are specific rational numbers.

·        I can interpret the solution set of an inequality in the context of the problem.

 

7.G.B.5

·        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.

 

Enduring Understandings:

 

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

·        Understand the difference between an expression and an equation.

·        Understand the difference between an equation and an inequality.

·        Properties of operations allow us to add, subtract, factor, and expand linear expressions.

·        There are precise terms and sequences to describe operations with rational numbers.

·        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.

·        Properties, order of operations, and inverse operations are used to simplify expressions and solve equations and inequalities efficiently.

·        Generating equivalent, linear expressions with rational coefficients using the properties of operations will lead to solving linear equation.

·        Discovering that rewriting expressions in different forms in a problem context leads to understanding that the values are equivalent.

·        The ability to solve and explain real life and mathematical problems involving rational numbers using numerical and algebraic expressions is important in preparation for algebra concepts in future math courses.

·        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.

 

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?