Linear? Yey or Nay

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Objective

SWBAT make comparisons and determine if a function is linear or non-linear.

Big Idea

Students analyze graphs, equations and table data, recognizing specific characteristics of linear and non-linear functions.

Launch

20 minutes

I begin today's lesson by pairing up students as they enter. I discretely pair them up by ability level, because I want to avoid any one student dominating too much over the other. I hand each student the FUNCTIONS handout. I also place a copy under the document camera to project on the board. 

Today's opening task involves making comparisons of graphs and equations to identify differences between linear and non-linear functions. I ask my students to begin by comparing two graphs, y = x and y = x^2. Then, we move onto other functions. I ask my students to answer the guiding questions below while I walk around monitoring their work.

Questions I may ask a group as they work are: 

  • Asking one of the partners: "Do you agree with him/her?" Why?
  • What do you mean by constant? 
  • Do you think all linear functions have a constant change like this?
  • Does having variables with an exponent of 1 determine a linear function? (I ask that they graph xy = 2, for example and analyze that function)
  • Is this true for all numbers?

As I oversee students work, I am expecting to see and hear students use words and expressions like "constant", "curved", "exponent", "steady", "interval". When I hear them using correct terminology, I encourage them to use the terms in their comparison statements. If a student is grasping for the correct word, I will revoice his/her statement or idea using mathematical language. I also motivate students to use their graphing calculators to test ideas and look at more examples. This is an efficient way for them to assess the ideas that they have about graphs. (See my Asking the Right Questions reflection for more about how this activity progresses in my course.)

Once students are done I call on volunteers to share their groups answers and ask that other groups write responses that they did not write into their sheets. 

 

New Info

10 minutes

As usual, I want to include new information in my lessons.  Today, the new information will be generated and shared through a whole class discussion. The ideal is that the new information will be provided by students as they report back on their comparisons of linear and non-linear functions. I record responses on the board in a way that formalizes their observations as our learning objectives for the lesson. 

These are some responses I've had and expect from the class, and which I write and keep on the whiteboard during the lesson. 

Linear

  • the rate at which y and x increases is constant
  • points of the equation lie on a straight line
  • the growth is steady 

Non linear

  • as x changes, the y value changes quicker or "as x increases, y increases faster"
  • points of the equation lie on a curve

I like to write the comparison statements in two columns.

I always get a student that says in linear functions, as x increases y increases, or as x decreases, y decreases. In this case I simply refer to the 3rd graph y = 2^x and ask..."well is that happening in this function?" Since students have been working with this graph, they are usually quick to make the necessary connection.

 

 

 

Application

25 minutes

The next activity in this lesson can be carried out with the same student pairs. Each student receives a Linear Functions Activity Sheet. The task on the sheet is to evaluate each graph, equation, table, or verbal expression to determine if it represents a linear or nonlinear function. Students must then explain their reasoning for their choice.

The table of linear/nonlinear characteristics from our earlier discussion will remain on the board for students to refer to as they work. As I walk around checking up on students, I ask question which may include: 

  • How did you determine your answer?
  • What particular criteria "nailed it" to help you decide that the function was linear or nonlinear?
  • Do you think observing the graph of a function is always enough to determine if it's linear or not? 

Once students complete the work, we will discuss the examples. I will ask volunteers to share their answers and explanations.

For reference, here are some notes about the functions (students' answers may vary): 

#1: Nonlinear because the rate of change varies. / there are intervals wher the graph is a curve 

#2: Linear; the rate of change is constant

#3: There are no exponents other than 1 in equation / The points of the equation lie on a straight line

#4: Nonlinear; the equation has no exponents other than 1. 

#5: Nonlinear; the rate of growth is not constant. 

#6: Linear; the graph has a constanr change. / points lie on a straight line

 

 

Closure

5 minutes

Once the discussion on the activity is over I ask each pair of students to use the back of their activity sheet to: 

  1. Create one representation of a linear function and one of a nonlinear function. They should be equations or a table of values for a particular situation. 
  2. Write what makes your equation or table linear or nonlinear. 

Example: y = 2x + 11

Linear function because solutions to the equation are points that lie on a straight line and as x increases by 1, y shows a constant change of 2. 

x:  1,    2,    3,    4,    5

y: 13,  15,  17,  19   21