Algebra I, Unit 1 Relationships between Quantities and Reasoning with Equations

Linear and Exponential Expressions

Relationships in One Variable

Relationships in Two Variables

Relationships between Quantities and Reasoning with Equations

Overview (Big Ideas): Linear and Exponential Expressions Students are expected to model problem situations using expressions focusing on linear and exponential models during Unit 1. In order to do this, students must be able to identify and apply appropriate units and quantities when modeling real-world situations and reasoning through possible solutions within context. Units should be chosen carefully in order to achieve a suitable level of accuracy and precision in the context of the problem. Additionally, students should be able to identify and interpret parts of an expression, such as terms, coefficients and powers, using the vocabulary of mathematics and in the context of the given problem. Students will use the rules for the order of operations to describe expressions and will apply those rules to simplify expressions. In addition, students will apply properties of exponents to simplify and evaluate expressions. This includes working with products and quotients of powers; powers of powers, products, and quotients; and negative and zero exponents. Students have had some previous exposure to laws of exponents. All exponent properties will be applied to rational exponents. Students will learn the connection between radicals and rational exponents in order to rewrite radical expressions using rational exponents. They will work with small perfect squares and perfect cubes, using both square root and cube root symbols in addition to rational exponents.

Relationships in One Variable Students will analyze the relationships between quantities in real-world problems to write and solve one-variable equations and inequalities. In the 8th grade, students used mathematical properties to solve one-variable equations and inequalities requiring multiple steps. Emphasis should be placed on constructing arguments to justify their solutions and the process by which they arrived at their answers. The structure of equations and inequalities should be used in order to develop a plan for solving. Students will be able to apply the mathematical reasoning behind solving one-variable equations in order to rearrange formulas and literal equations in order to highlight variables or quantities of interest.

Relationships in Two Variables Functions are relationships defined with one output, or element of the range, for every input, or element of the domain. Students will be introduced to functional relationships and function notation. Students may use mapping to help visualize the relationship. The function is often defined as f where the inputs (elements of the domain) are called x. While students are comfortable with having the range defined as y, they now will use notation to show the range is f(x) = y. Function notation will be used to define a given relationship between two variables. Students will evaluate a function for a given input within the domain of the function. When given a real-world application, context will be used to interpret the meaning of a statement given in function notation. During Unit 1, emphasize the multiple representations of linear functions. Students will be required to model real-world situations symbolically in two variables and to create graphs from a table of values. Students should understand that the graph of a line represents the set of all real solutions to a linear function. They should apply appropriate labels, scales, and units. Students should properly interpret graphs in terms of the relationship between the quantities and in order to arrive at solutions to a real-world problem. Students will recognize the relationship between the contextual, graphical, numeric and symbolic representations of a problem situation. Students will learn to distinguish between equations and inequalities, particularly, when pertaining to real-life scenarios. Students will use inequality signs and equal signs appropriately for a given situation. They will explain why the solutions of an inequality may or may not include the solutions to the equation. Students will be able to combine graphing and modeling in order to represent constraints on a given problem situation using linear inequalities and equations, both symbolically and graphically. In addition, students will interpret solutions as viable or nonviable within the context of such constraints. It is important that students recognize the difference between mathematical solutions and contextual solutions. Students will be able to graph a linear boundary using a table of values and indicate the area of the coordinate plane that contains all viable solutions, both mathematically and contextually.

Enduring Understandings: 1. Expressions are powerful tools for exploring, reasoning about, and representing situations. 2. Two or more expressions may be equivalent, even when their symbolic forms differ. A relatively small number of symbolic transformations can be applied to expressions to yield equivalent expressions. 3. Variables have many different meanings, depending on context and purpose. 4. Using variables permits writing expressions whose values are not known or vary under different circumstances. 5. Using variables permits representing varying quantities. This use of variables is particularly important in studying relationships between varying quantities. 6. The equals sign can indicate that two expressions are equivalent. It is often important to find the value(s) of a variable for which two expressions represent the same quantity. Finding the value(s) of a variable for which two expressions represent the same quantity is know as solving an equation. 7. An inequality is another way to describe a relationship between expressions; instead of showing that the values of two expressions are equal, inequalities indicate that the value of one expression is greater than (or greater than or equal to) the value of the other expression. 8. In solving an inequality, multiplying or dividing both expressions by a negative number reverses the sign that indicates the relationships between the two expressions. 9. The equals sign can be used in defining or giving a name to an expression or function rule. 10. Functions provide a tool for describing how variables change together. Using a function in this way is called modeling, and the function is called a model. 11. Functions are single-valued mappings from one set – the domain of the function – to another – its range. 12. Functions can be represented in multiple ways—in algebraic symbols, graphs, verbal descriptions, tables, and so on—and these representations, and the links among them, are useful in analyzing patterns of change. 13. Some representations of a function may be more useful than others, depending on how they are used.

Sources: Lloyd, G., Herbel-Eisenmann, B., & Star, J.R. (2011). Developing essential understanding of expressions, equations, and functions for teaching mathematics in grades 6-8. Reston, VA: The National Council of Teachers of Mathematics, Inc. Cooney, T., Beckmann, S., & Lloyd, G. (2010). Developing Essential Understanding of Functions Grades 9-12. Reston, VA: The National Council of Teachers of Mathematics, Inc.

Essential Questions: Linear and Exponential Expressions

How can expressions be used to model real-world problems?

What units should be used to model a given problem in order to achieve an appropriate level of accuracy?

In what situations is the use of exponents practical?

Does it matter in what order properties of exponents are applied?

How can properties of exponents be applied to simplify expressions?

Relationships in One Variable

Can the structure of a linear equation or inequality be used to devise a plan for solving the equation or inequality in the context?

o How can mathematical properties be correctly applied when solving one-variable equations and inequalities?

o Can a solution or solution set and the method by which the solution was obtained be justified using the language of mathematics?

o What does a solution or solution set mean in the context of the problem situation?

o How can mathematical properties be correctly applied in order to solve literal equations and formulas for a given variable?

Relationships in Two Variables

Is the given relationship a function? Why or why not?

For what situations is function notation useful?

When is it appropriate to restrict the domain?

How can a linear function be created, using function notation, to represent the relationship between the two quantities given?

What representations can be created that will portray the solution set to a linear function? By what methods can these representations be created?

What units, scales and labels must be applied to accurately represent a linear function in the context of a problem situation?

What are the advantages of representing the relationship between quantities symbolically? Numerically? Graphically?

Which representation is best applied to the real-life scenario, and what constraints does this representation have?

Common Misconceptions: Linear and Exponential Expressions

Students generally struggle with units and unit conversions.

Representing problem situations symbolically is difficult for students to conceptualize. Many students become disconnected froom the real-world context once the problem is re-written symbolically. Additionally, some students have difficulty understanding how to interpret contextual information into symbols.

o There is a tendency to memorize rules rather than internalize the concepts behind the laws of exponents.

o Students struggle with the appropriate application of exponents to coefficients.

o Students often confuse power of a power and product of powers.

o As a result of weaknesses with fractions, students struggle with the simplification of negative exponents. Many students do not fully grasp that division is defined as multiplication by a reciprocal. As a result, students memorize rules for negative exponents without conceptualizing the reason behind the rules.

o Students sometimes confuse x0 = 1 with x0 = x1. In addition, they are often tempted to simplify an entire expression to 1 when presented with a zero exponent, ignoring the rules that govern the order of operations.

o When simplifying using the quotient of powers rule, subtraction mistakes are often made.

Relationships in One Variable

Students may have difficulty using inverse operations and reversing the order of operations in order to solve equations for a given variable.

o Students often struggle with the difference between combining like terms on one side of an equation and eliminating a variable on one side of an equation.

o Inequality symbols are often difficult for students to master with fluency.

Relationships in Two Variables

The notation f(x) does not mean “f multiplied by x.”

The notation f(x) represents the range, or y-values.

A vertical, not horizontal, line test can help to verify graphically that a relationship is a function.

Domain is all of the x-values and range is all of the y-values. Students often confuse the two.

Domain and range should be listed in intervals from least to greatest.

Students find it difficult to choose a scale that displays all of the data accurately.

Many students do not fully understand that a line is made up of infinite points and have difficulty understanding that solutions are not limited to integers.

Students often confuse the x and y axes when graphing.

It may be difficult for students to conceptualize that an equation, a table of values, and a graph can all represent the same problem situation.

Students confuse “at least” with “less than,” and “no more than” with “greater than.”

The comparison of mathematical versus contextual solutions can be difficult.

Relationships between Quantities and Reasoning with Equations

Relationships between Quantities and Reasoning with EquationsOverview (Big Ideas):Linear and Exponential ExpressionsStudents are expected to model problem situations using expressions focusing on linear and exponential models during Unit 1. In order to do this, students must be able to identify and apply appropriate units and quantities when modeling real-world situations and reasoning through possible solutions within context. Units should be chosen carefully in order to achieve a suitable level of accuracy and precision in the context of the problem.

Additionally, students should be able to identify and interpret parts of an expression, such as terms, coefficients and powers, using the vocabulary of mathematics and in the context of the given problem. Students will use the rules for the order of operations to describe expressions and will apply those rules to simplify expressions. In addition, students will apply properties of exponents to simplify and evaluate expressions. This includes working with products and quotients of powers; powers of powers, products, and quotients; and negative and zero exponents. Students have had some previous exposure to laws of exponents. All exponent properties will be applied to rational exponents. Students will learn the connection between radicals and rational exponents in order to rewrite radical expressions using rational exponents. They will work with small perfect squares and perfect cubes, using both square root and cube root symbols in addition to rational exponents.

Relationships in One VariableStudents will analyze the relationships between quantities in real-world problems to write and solve one-variable equations and inequalities. In the 8th grade, students used mathematical properties to solve one-variable equations and inequalities requiring multiple steps. Emphasis should be placed on constructing arguments to justify their solutions and the process by which they arrived at their answers. The structure of equations and inequalities should be used in order to develop a plan for solving. Students will be able to apply the mathematical reasoning behind solving one-variable equations in order to rearrange formulas and literal equations in order to highlight variables or quantities of interest.

Relationships in Two VariablesFunctions are relationships defined with one output, or element of the range, for every input, or element of the domain. Students will be introduced to functional relationships and function notation. Students may use mapping to help visualize the relationship. The function is often defined as f where the inputs (elements of the domain) are called x. While students are comfortable with having the range defined as y, they now will use notation to show the range is f(x) = y. Function notation will be used to define a given relationship between two variables. Students will evaluate a function for a given input within the domain of the function. When given a real-world application, context will be used to interpret the meaning of a statement given in function notation.

During Unit 1, emphasize the multiple representations of linear functions. Students will be required to model real-world situations symbolically in two variables and to create graphs from a table of values. Students should understand that the graph of a line represents the set of all real solutions to a linear function. They should apply appropriate labels, scales, and units. Students should properly interpret graphs in terms of the relationship between the quantities and in order to arrive at solutions to a real-world problem. Students will recognize the relationship between the contextual, graphical, numeric and symbolic representations of a problem situation.

Students will learn to distinguish between equations and inequalities, particularly, when pertaining to real-life scenarios. Students will use inequality signs and equal signs appropriately for a given situation. They will explain why the solutions of an inequality may or may not include the solutions to the equation. Students will be able to combine graphing and modeling in order to represent constraints on a given problem situation using linear inequalities and equations, both symbolically and graphically. In addition, students will interpret solutions as viable or nonviable within the context of such constraints. It is important that students recognize the difference between mathematical solutions and contextual solutions. Students will be able to graph a linear boundary using a table of values and indicate the area of the coordinate plane that contains all viable solutions, both mathematically and contextually.

Enduring Understandings:1. Expressions are powerful tools for exploring, reasoning about, and representing situations.

2. Two or more expressions may be equivalent, even when their symbolic forms differ. A relatively small number of symbolic transformations can be applied to expressions to yield equivalent expressions.

3. Variables have many different meanings, depending on context and purpose.

4. Using variables permits writing expressions whose values are not known or vary under different circumstances.

5. Using variables permits representing varying quantities. This use of variables is particularly important in studying relationships between varying quantities.

6. The equals sign can indicate that two expressions are equivalent. It is often important to find the value(s) of a variable for which two expressions represent the same quantity. Finding the value(s) of a variable for which two expressions represent the same quantity is know as solving an equation.

7. An inequality is another way to describe a relationship between expressions; instead of showing that the values of two expressions are equal, inequalities indicate that the value of one expression is greater than (or greater than or equal to) the value of the other expression.

8. In solving an inequality, multiplying or dividing both expressions by a negative number reverses the sign that indicates the relationships between the two expressions.

9. The equals sign can be used in defining or giving a name to an expression or function rule.

10. Functions provide a tool for describing how variables change together. Using a function in this way is called modeling, and the function is called a model.

11. Functions are single-valued mappings from one set – the domain of the function – to another – its range.

12. Functions can be represented in multiple ways—in algebraic symbols, graphs, verbal descriptions, tables, and so on—and these representations, and the links among them, are useful in analyzing patterns of change.

13. Some representations of a function may be more useful than others, depending on how they are used.

Sources:Lloyd, G., Herbel-Eisenmann, B., & Star, J.R. (2011). Developing essential understanding of expressions, equations, and functions for teaching mathematics in grades 6-8. Reston, VA: The National Council of Teachers of Mathematics, Inc.

Cooney, T., Beckmann, S., & Lloyd, G. (2010). Developing Essential Understanding of Functions Grades 9-12. Reston, VA: The National Council of Teachers of Mathematics, Inc.

Essential Questions:Linear and Exponential ExpressionsRelationships in One VariableRelationships in Two VariablesCommon Misconceptions:Linear and Exponential ExpressionsRelationships in One VariableRelationships in Two Variables