##
* *Reflection: Discourse and Questioning
Linear? Yey or Nay - Section 1: Launch

Asking strategic questions to assess student thinking and guide their understanding is part of what classroom discourse involves and part of many teachers' instructional tactics. In my intention of encouraging conceptual explanation and asking appropriate questions I must be able to anticipate responses and help students make connections between what they already know and what there is to learn. So I definitely need to grasp, what they already know. This may entail questions that may be way too obvious for some students, to the point where they may feel their intelligence is being "insulted". On the other hand, if I assume too much, I may lose students who need the scaffolding questions in order to make connections. Also, some students have a difficult time explaining their reasoning, making it tough to get a good idea of what they know. Then using the "easy" questions is essential. I may disguise questions like in this example:

I can't assume that students know what the graph of a linear function looks like. For some it is quite obvious, but others may find that the function y = 5^x, for example, is linear because of its linear appearance at certain intervals or for any other reason. In this case I say:

*"A student in my other class said that this function is also linear because of the behavior of the graph at these intervals, do you agree or disagree? Explain why?" *Instead of asking* "what does the graph of a linear function look like? *

When I do ask a question with an obvious answer, when I find it necessary, I make sure not to follow with similar questions which they may find "mindless". It's this repetition that really turn kids off and "loosens" respect for the subject matter.

This careful scaffolding of questions give me opportunities to appropriately assess different levels of comprehension and also evens the ground for student participation.

*Asking the right questions. Don't "insult", but don't over-estimate.*

*Discourse and Questioning: Asking the right questions. Don't "insult", but don't over-estimate.*

# Linear? Yey or Nay

Lesson 6 of 11

## 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.

*60 minutes*

#### Launch

*20 min*

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.

#### Resources

*expand content*

#### New Info

*10 min*

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.

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#### Application

*25 min*

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

#### Resources

*expand content*

#### Closure

*5 min*

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

- 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.
- 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

*expand content*

*Responding to eboni eaton*

And thank your for your comment....they are so helpful.

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- UNIT 1: Number Sense
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- LESSON 1: Relations that Function (Part 1)
- LESSON 2: Relations that Function (Part 2)
- LESSON 3: Functions Rule (Part 1 of 2)
- LESSON 4: Functions Rule (Part 2 of 2)
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