##
* *Reflection: Trust and Respect
Rates of Change - Section 2: Activity

English-language learners (ELLs) need many opportunities to interact in classroom discussions. Lets not fall into a pattern of allowing ELL students to be less involved in oral interactions. In order to help these students develop the language skills and be more engaged we must take into account a few things.

First is that not all questions are clearly understood by these students, and if that's the case, we should rephrase or clarify things in order to facilitate understanding.

When asking questions, we must also wait long enough for students to consider the questions and formulate a response. Sometimes we move too quickly from one student to the other. When the ELL student perceives this, he/she tends to stay back and just listen.

Maintain the same expectations and don't just call on ELL students to answer the "easy" questions. Some students may be turned off to participation if they know that the teacher expects them to respond to the low level knowledge questions. So I don't lower my expectations because of the language differences.

When an ELL student answers a question and it is only partially responded to, or partially correct, what we should always do is encourage the student to ellaborate their response. ELLs usually know more than they might readily say, so even when correct, I ask them to explain and ellaborate. Of course, commenting positively about their effort and their english is always encouraging.

Some ways we can respond to ELL student interventions are:

"You're absolutely right about that. Can you tell me more?"

"Yes, that's good. What else do you know about that?"

"Correct, great answer. How do you know that?"

"That's a good answer. Can you tell me why that is important?"

"Well said. that's good thinking" (then I repeat what was said)

One more thing I'd like to say is that since my native spanish speaking students know that I speak Spanish, I sometimes get responses in Spanish. I've found that most of the time, their answers were correct, so I've learned to simply ask the student to repeat the response in English. I also ask them to repeat their response when the answer is given in a mixture of both languages, which is very common with lower level ELL students.

*ELL student participation in discussion*

*Trust and Respect: ELL student participation in discussion*

# Rates of Change

Lesson 9 of 11

## Objective: SWBAT interpret slope in real world situations and compare rates of change in various linear functions.

#### Launch

*15 min*

Today's entrance ticket, Bathtub Case, is designed to require my students to access prior knowledge and use it productively. Students must analyze the graph and relate it to the real life situation it depicts. We dealt with these graphs earlier in the course when analyzing the distance v. time functions.

The new wrinkle in this task is the curve between A and B, which is the only segment of the graph where the rate is not constant. Once students complete the task, I will call on a few students to share their stories. I make sure to ask the students the following questions if these are not covered within the stories:

- Does the story start off with an empty tub?
- What does the curved line mean?
- Is the graph at points near B and from E to F vertical? What might have caused it to be nearly vertical near B?
- Explain the difference in slopes between segments CD and GH with respect to rates of change.

#### Resources

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#### Activity

*30 min*

The main activity for today's lesson, Activity Worksheet, asks students to apply their understanding of slope in several different situations. As they work, students analyze different representations of functions and respond to questions about the rate of change described by a function.

It's often hard to get my students to use units of measure in their answers. Today, I will stress the need to use proper units and I tell students that including the units gives meaning to the slope of the graphed situation. I find that relating the units to slope (vertical change / horizontal change) helps students place the correct units in the numerator and denominator (**MP4**, **MP6**).

When students finish the worksheet, we will discuss the meaning of the slope calculation in each problem. And, we will review their answers to the rate of change questions. I expect that my students will be able to verbally express the connections between the graph, the changing slope, and the problem situation.

#### Resources

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#### Closure

*10 min*

The last page of the Activity Worksheet contains a graph that I will use as a closure activity for this lesson. I want each student to work individually to analyze the problem and answer the corresponding questions. Students need to figure which animal, a cat or a mouse is faster. Students should calculate the rates and determine that the cat is faster.

At this point, my students should be able to find both rates of change and answer this problem correctly. I don't expect students to make the mistake of thinking that the line further left, which reaches the top of the graph (20 feet) in less time, is the faster animal. But, I look out for this interpretation as something that I need to address if students are making this interpretation.

The y-intercept question provides an opportunity for us to discuss how the mouse had a head start, and how this shows up in graphs of functions.

#### Resources

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- UNIT 1: Number Sense
- UNIT 2: Solving Linear Equations
- UNIT 3: Relationships between Quantities/Reasoning with Equations
- UNIT 4: Powers and Exponents
- UNIT 5: Congruence and Similarity
- UNIT 6: Systems of Linear Equations
- UNIT 7: Functions
- UNIT 8: Advanced Equations and Functions
- UNIT 9: The Pythagorean Theorem
- UNIT 10: Volumes of Cylinders, Cones, and Spheres
- UNIT 11: Bivariate Data

- 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)
- LESSON 5: Crickets Tell Temperature
- LESSON 6: Linear? Yey or Nay
- LESSON 7: Comparing Linear and Exponential Functions
- LESSON 8: Time-Distance Graphs
- LESSON 9: Rates of Change
- LESSON 10: Sequences as Functions (Part 1 of 2)
- LESSON 11: Sequences as Functions (part 2 of 2)