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# Sketching Graphs for Real-World Situations

Lesson 7 of 13

## Objective: SWBAT sketch graphs to model real-world situations and to justify their sketches using key vocabulary.

## Big Idea: What will happen to the length of an object's shadow throughout the day? Students discuss this and other questions and attempt to represent these relationships using graph sketches.

*100 minutes*

#### Warm-Up

*30 min*

This warm-up allows students the opportunity to review all of the key skills they have learned so far in this unit. They can get started working directly and the most important coaching that I try to provide is to make sure that students fully understand any problems that they choose to skip. Many students tell me that they prefer not to do word problems, so I always encourage students to go back to this first problem. If another student has fully solved it, I ask them to put their work on a whiteboard so that other students can refer to this if they are struggling.

Some key questions to ask students are:

- How can you determine whether the rate in this situation is constant?

- Can you create a data table to fit this situation?

- What do the
*x*values and the*y*values represent in this data table or situation?

- If there is a constant rate, how can you use this to find other missing information about this problem?

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I like to allow students to finish tasks from the day before in class, especially when these tasks will prepare them to make sense of the new task. The follow-up on yesterday's matching task is to ask students to finish writing their justifications, using the same key terms as they will use in the new lesson. You can allot however much time you want to this follow-up, because students can transition individually or in small groups to the next task when they are ready. Or you can give all students a set amount of time to work on this before having the class transition as a whole. I prefer to allow students to transition when they are ready, but this can be a little bit chaotic at times, so it is up to you and how much structure you think your students need.

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For this task, I partner students homogeneously. The purpose of this is two-fold. First, I want to make sure that I am getting reasonably accurate formative assessment information as I circulate, and this is best done if students are figuring things out together, rather than having one partner do all the work and the other partner just tag along. Secondly, this enables me to differentiate slightly by giving different pairs of students different numbers of problems to complete. I can quickly circulate and tell one pair of struggling students that I want them to focus on the 4 graphs that make the most sense to them, for instance.

I tell them that they should be able to explain each of their sketches using key math vocabulary, which I write on the board:

--rate

--constant

--increasing/decreasing

--starting point/y-intercept

I also ask them to use sentences in the form, "As ______________ happens, __________ happens." For now I tell them to focus on speaking about their justifications with their partner, but they will need to write them eventually.

#### Resources

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#### Closing

*10 min*

For the closing I like to focus on the most challenging aspect of the lesson to ensure that all students are given some guidance. Today, the most challenging part is not necessarily sketching the graphs, but defending or justifying a sketch using mathematical vocabulary. I use the last 5-10 minutes of this class to show one "model justification" which I create and display on the document camera.

I modeling precise use of mathematical language, I emphasize the key words that I use, the transition words, and the "As _____ happens, ______ happens" sentence frames. I then ask the students to write one justification using these same key words and sentence frames. As much as possible during the last few minutes, I either circulate to read each student's justification or I stand at the door and give them some quick feedback on their justification.

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- UNIT 1: Linear and Nonlinear Functions
- UNIT 2: Piecewise Functions
- UNIT 3: Absolute Value Functions and More Piecewise Functions
- UNIT 4: Introduction to Quadratic Functions through Applications
- UNIT 5: More Abstract Work with Quadratic Functions
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- UNIT 10: Circles
- UNIT 11: Radical Functions
- UNIT 12: Cubic Functions

- LESSON 1: Patchwork Tile Patterns
- LESSON 2: Investigating Linear and Nonlinear Tile Patterns
- LESSON 3: More Tile Patterns
- LESSON 4: Constant Speeds and Linear Functions
- LESSON 5: Linear and Nonlinear Functions
- LESSON 6: Real World Relationships
- LESSON 7: Sketching Graphs for Real-World Situations
- LESSON 8: Slopes of Linear Functions
- LESSON 9: Different Forms of Linear Equations
- LESSON 10: Linear Function Designs
- LESSON 11: Verbal Descriptions of Linear and Nonlinear Functions
- LESSON 12: Linear and Nonlinear Function Review and Portfolio
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