In module 4, students work with numerical and algebraic expressions and equations. First, they learn that exponents represent repeated multiplication, evaluate powers with whole number, fraction, and decimal bases, and use the order of operations to evaluate numerical expressions. Then, students learn why and how to use variables to represent unknown numbers and quantities. They write and evaluate algebraic expressions and use properties of operations to generate equivalent expressions. Students reason about and solve single-variable, one-step equations, and they understand the meaning of a solution to an equation or inequality. At the end of the module, they revisit ratio relationships and write and graph equations in two variables, identifying independent and dependent variables in real-world situations.
In topic E, students write and graph two-variable equations and identify independent and dependent variables. After representing ratio relationships with graphs and two-variable equations, students transition to representing relationships of the form y = x + b and y = x - b. Students interpret the meanings of points on graphs and also interpret the meanings of coefficients, variables, operators, and constants in equations that represent real-world situations. At the end of the topic, students analyze and model relationships between two quantities when they complete a modeling task about climbing the steps of the Statue of Liberty.
6.NS.10: Use reasoning involving rates and ratios to model real-world and other mathematical problems (e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations).
6.AF.10: Use variables to represent two quantities in a proportional relationship in a real-world problem; write an equation to express one quantity, the dependent variable, in terms of the other quantity, the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.
I can...
Represent a ratio relationship with a table and a two-variable equation.
Identify the independent and dependent variables in a real-world or mathematical situation.
Lesson at a Glance
In this lesson, students use a graph to determine unknown values in a ratio table and on a double number line. When given a real-world situation relating two quantities, students use a variable that represents one quantity to write an expression that represents the other quantity. They progress to writing equations in two variables that represent ratio relationships, and they identify independent and dependent variables. After the Take a Stand routine, students realize that it can be unclear which variable is independent and which is dependent in a situation. Students also write two equations that each represent the same situation. This lesson introduces the terms independent variable and dependent variable.
I can...
Analyze the relationship between the independent and dependent variables in the graph of a ratio relationship.
Represent a ratio relationship with a table, a graph, and a two-variable equation.
Lesson at a Glance
In this digital lesson, students define independent and dependent variables for a ratio relationship between the number of gallons of gasoline and the total cost of the gasoline in dollars. Students initially choose whether to label the x-axis or the y-axis as the independent variable. By comparing two graphs, students realize that the relationship is clearer when the independent variable is on the x-axis and the dependent variable is on the y-axis. Continuing with the same context, students complete a table, make a graph, and write an equation to represent the relationship. Students identify the value of the ratio in each of the representations of the relationship and use the value of the ratio to write an equation from a graph and create a graph from an equation. After students consider other real-world contexts, they engage in a discussion about the advantages and disadvantages of using each representation to model a ratio relationship.
Use the digital platform to prepare for and facilitate this lesson. Students will also interact with the lesson content and activities via the digital platform.
If student computers or devices are not available, use the alternate version of this lesson.
I can...
Represent a real-world situation with a table, a graph, and a two-variable equation.
Analyze the relationship between the variables in a real-world situation.
Lesson at a Glance
Students solve problems related to real-world relationships represented by equations in the form y = x + b or y = x - b. Students begin by using the Co-construction routine to write contexts for a set of unlabeled graphs. Then students work in pairs to represent real-world situations with tables, graphs, and equations. Students identify independent and dependent variables, interpret the meanings of points on graphs, and determine whether points on a graph should or should not be connected. With equations, students interpret the meanings of coefficients, variables, operators, and constants. Students make predictions by using the tool of their choice.
I can...Use tables, graphs, and equations to estimate the solution to a real-world problem.
Lesson at a Glance
This lesson is an open-ended modeling exploration. First, students watch a video showing a person climbing the stairs of the Statue of Liberty, and they create a class list of questions about the video. Then students work in groups to determine the elevation of the viewing platform at the top of the crown. As they solve the problem, students combine rate reasoning with their ability to create tables, graphs, and equations to model the relationships between quantities in this situation. At the end of the lesson, groups of students present their solution strategies to the class.