TITLE:  Family Portraits: Function Introduction and Linear Functions 

CONTENT  AREAS (What areas of mathematics does this lesson cover?):   
 Direct and Inverse Variation, Slope and Intercept

GRADE  LEVEL:   8 & 9

MATERIALS   NEEDED: graph paper, rulers, hand-outs  

KEY   CONCEPTS:  Simple functional relationships, equations of the form y=kx, and y=k/x, the slope and y-intercept of the equation y=mx+b 

EALR'S and GLE'S (Make the connections clear and specific)

Grade 8

Component 1.5:  Understand and apply concepts and procedures from algebraic sense.

Patterns, functions, and other relations

1.5.1 Apply understanding of linear and non-linear relationships to analyze patterns, sequences, and situations.  W

·   Extend, represent, or create linear and non-linear patterns and sequences using tables and graphs. [RL]

·   Explain the difference between linear and non-linear relationships. [CU]

·   Predict an outcome given a linear relationship (e.g., from a graph of profit projections, predict the profit). [RL]

·   Use technology to generate linear and non-linear relationship. [SP, RL]

1.5.6 Understand and apply a variety of strategies to solve multi-step equations and one-step inequalities with one variable.  W

·   Solve multi-step equations and one-step inequalities with one variable.

·   Solve single variable equations involving parentheses, like terms, or variables on both sides of the equal sign.

·   Solve one-step inequalities (e.g., 2x<6, x+4>10).

·   Solve real-world situations involving single variable equations and proportional relationships and verify that the solution is reasonable for the problem. [SP, RL, CU]

Component 2.2:  Apply strategies to construct solutions.

2.2.2 Apply mathematical tools to solve the problem.  W

·   Implement the plan devised to solve the problem or answer the question posed (e.g., in a table of values of lengths, widths, and areas find the one that shows the largest area; check smaller increments to see if this is the largest that works).

·   Identify when an approach is unproductive and modify or try a new approach (e.g., if an additive model didn’t work, try a multiplicative model).

·   Check the solution to see if it works (e.g., if the solution for a speed of 19 feet per second is 5 steps per second, perhaps the assumption of linearity was incorrect).

Grade 9/10

Component 1.1:  Understand and apply concepts and procedures from number sense.

1.1.4 Apply understanding of direct and inverse proportion to solve problems.  W

·   Explain a method for determining whether a real-world problem involves direct proportion or inverse proportion. [SP, CU, MC]

·   Explain a method for solving a real-world problem involving direct proportion. [CU, MC]

·   Explain a method for solving a real-world problem involving inverse proportion. [CU, MC]

·   Solve problems using direct or inverse models (e.g., similarity, age of car vs. worth). [SP, MC]

·   Explain, illustrate, or describe examples of direct proportion. [CU]

·   Explain, illustrate, or describe examples of inverse proportion. [CU]

·   Use direct or inverse proportion to determine a number of objects or a measurement in a given situation.

GOALS (Remember the difference between goals and objectives):   
 Part A- Function Introduction:  Students will understand and apply the definition of a function.  Students will determine whether or not a graph represents a function.  Students will write equations of direct and inverse variation functions.  Students will understand what it means for quantities to have a direct or inverse variation relationship.

Part B- Linear Functions:  Students will be able to understand and apply the definition of a slope as the ratio of rise to run.  Students will be able to calculate the slope of a line through two given points.  Students will be able to identify the slope and the y-intercept of an equation in the form y=mx+b.  Students will be able to compare slopes and y-intercepts of graphs by looking at their equations.  Students will be able to write equations that satisfy given conditions and y-intercepts.

OBJECTIVES: Part A:  Students will demonstrate their understanding of function equations by working out examples that take place in real life.  They will give examples of direct and inverse variation.

Part B:  Students will complete a homework assignment that will provide evidence of their ability to identify the slope of a line, compute a rise to run ratio, and solve and check linear equations.  

PROCEDURES:  (Label each step in the process:  Activating Prior Knowledge, Disequilibration, Elaboration, Crystallization)

  • Introduction/Preassessment
  • Part A:  Short introduction of unit.  Explain how students will be learning about various function equations and tell them they will compile a Function Family Album.
  • Part B:  Working in groups, students will be given graph paper, rulers, and a hand-out.  They will be asked to plot these points on a graph.  When they are finished, we will discuss the graphs mad and introduce them to the slope of a line, y-intercept, and methods for solving and checking their work.
  • Activity
  • Part A:  Working in pairs, students will complete worksheet #1.  Students will then discuss their answers as a class.  They will understand what makes an equation a function.  They will know how to tell if a graph is a function.  Students will then work to solve worksheet #2, direct variation.  As a class, students will discuss direct variation, they will give examples of these relationships in real life.  Students will then be given instructions for completing “How long is a meter?” activity.  After completing activity, students will discuss their findings and give real life examples of inverse variation.
  • Part B
  • Ask students to form five groups of four.  Distribute handouts, one to each student.  One handout is a starting points graph and the other is equations to be used.  Have students work with their groups to find the slope of their line, find the y-intercept, and write an equation for their line.  When everyone is done, ask for volunteers to share their graph and equation with the rest of the class on the overhead projector.  Have more discussions about slope, intercepts, and linear equations, if necessary.
  • Closure
  • Part A:  Students will make direct and inverse variation pages for their Function Family Album.
  • Part B:  Answer any questions about linear equations, slope, and intercepts. Instruct students to write about what they learned today in their Function Family Album. 

POST-ASSESSMENT :  Each student should have include a written definition of the functions covered, the corresponding equation, and a graph of the function.  They should also include how to solve and check equations of the form y=mx+b.