##
* *Reflection: Perseverance
Graphs ----> Tables ----> Rules - Section 2: Investigation

My students definitely struggled to find a rate of change that was both negative and not a whole number. As I wrote about in the lesson plan, we looked at changes over set periods of time that were easy to find on the graph. For example, we drew a step to show that over 40 days, 10 gallons of water was consumed. Students did NOT struggle with the idea that the amount of water was changing at a constant rate. This worked in our favor! Once we drew in the step and asked the question, "If they drank 10 gallons over 40 days, how can we find how much they drank in 1 day?" students were able to figure out that they needed to divide in order to determine the rate of consumption.

Students also struggled to write an equation to the second problem. I think this was due to the negative rate of change. Once we determined that they drank .25 gallons of water a day, I kept asking them questions like, "Well how would you figure out how much water was left after 5 days?" After performing some calculations together I would ask for a few more days. For example, "How much water is left after 10 days?, 20 days?" each time queuing students to calculate, rather than refer to their graphs. Finally, I asked, "So how much water would be left after "x" days?" From there, students were quickly able to figure out the equation.

Once we had the equation we talked a lot about where things show up in the graph. We compared how the equation and graph in this problem were different than the cafeteria table problem that we worked on first.

*Perseverance: Writing an equation for a negative rate of change*

# Graphs ----> Tables ----> Rules

Lesson 6 of 17

## Objective: SWBAT generate tables from graphs. SWBAT connect rules to graphs. SWBAT connect the straight line of a graph to the idea of a linear rate of change.

#### Opening

*5 min*

Today's lesson asks students to make sense of linear graphs by extrapolating information from the graph, putting data in a table, and finding a rule. The activities in this lesson are derived from the IMP curriculum but modified to fit my students' needs.

I begin class today by telling students that they have previously spent a lot of time generating rules from tables and also making tables that match with rules. Today's activity will help them to understand how ** graphs are another way to use algebra to tell the story of a situation using algebra**.

A primary focus today will be the use of correct terminology for graphs. The Discussion section of the lesson will develop students' knowledge of this vocabulary.

To begin, I hand out the worksheet Graphs_Tables_Rules.

#### Resources

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#### Investigation

*30 min*

I let students get to work on the Graphs_Tables_Rules worksheet in small groups. If students have trouble getting started, I guide them through Question 1. I ask, "How many people can be seated at 5 cafeteria tables? to help them identify where this point is on the graph.

As students generate tables, I make sure they are labeling the In and Out values correctly. In order to push students to reflect on the relationship between the variables, I typically use language like, "The number of seats depends on the the number of tables, right? So, the Out depends on the In."

Once students have generated the first table, the rule should be fairly easy for them to see. I make sure they pay special attention to Question 6, which asks them to connect the rule to the number of seats each cafeteria table provides. I ask them to write *specifically* about this connection. I also like to ask students to predict the number of seats provided by a larger number of tables. For example, "Can your rule predict how many people could be seated if there were 200 tables?"

In the second problem, students may have a harder time finding the rate of change. If so, I will ask if the water is being consumed at a constant rate and how they know that. Once students recognize that there is a constant rate involved, I'll ask them to figure out how much water was consumed over a certain period of time. For example, "If 10 gallons of water was consumed over 40 days (clean points to identify on the graph), how can you figure out how much water was consumed each day?" I encourage students to use their tables to find the constant change, if there are not able to see it otherwise.

For this task students need to account for a starting value that is not zero. If scaffolding is necessary, I might ask them, "How much water would be left after 20 days?" Since they know how much water is consumed each day, and, they know the starting point this is an approachable question. This is a good opportunity for students to practice **SMP 8: Look for and express regularity in repeated reasoning**. In order to take advantage of this, I will ask students to do the same calculations for two other days (see Graphs, Tables, Rules_Video). I want to draw students' attention to the idea that each time they look at a different number of days elapsed, they multiply that number by 1/4 or 0.25 and then subtract that answer from the starting value (20). This work should help students make connections between the algebraic equation and the meaning behind the numbers. This is the same connection that Questions 5 and 6 are highlighting.

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#### Discussion + Closing

*25 min*

Some discussion points are included in the Investigation section as ways to guide students in their work. Many of the points are worth repeating as they allow students to revisit the problem once it's finished and may help solidify their understanding.

Other teaching moves that I will try to accomplish:

- Be sure students see the connection between the rules they created and the situations. Make sure students see that the rate of change is showing up before the x-value.

- Discuss the difference between the two rules and the graphs. Point out that one graph starts at zero (a good time to remind students of the word "origin") and the other starts at 20. Ask students, "Where does this show up in the equations?" It may be helpful to write y = 6x + 0 or Out = 6(In) + 0 so students can see how it's taken into account.

- Students may have written the second equation as y = 20 - .25x or -.25x + 20. If so, discuss equivalence and probe to see what students have learned about y = mx + b. It's a judgment call about how much you want to go into y = mx + b, but students should begin to notice the pattern of x being multiplied by a value and then something being added or subtracted. There is no need to use a formal definition here.

- Ask students, "Does it makes sense to talk about a negative amount of cafeteria tables or a negative amount of school days?" Connect this idea looking in Quadrant I of the coordinate system for relevant values. Make sure students understand that although negative numbers don't make sense for x values in the context of these two problems, the line of the equations continue on in both directions. You might zoom out on the graph here if possible and show the continuation of the lines.

**Make a big deal out of the idea that the set of all points that fits a rule is called the graph of the equation**. The process of plotting these points to get an overall picture is graphing an equation.

Before the end of class, I will give students the opportunity to reflect on their learning. I might do a verbal "popcorn" exercise where students can just jump in and share their thinking. I may ask a question like, "How are graphs and rules connected?" If so, I will record some of the student responses and leave them posted for the next lesson so students will remember the key ideas.

*expand content*

This material is adapted from the IMP Teacher’s Guide, © 2010 Interactive Mathematics Program. Some rights reserved.

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- UNIT 1: Introduction to Algebra: Focus on Problem Solving
- UNIT 2: Multiple Representations: Situations, Tables, Graphs, and Equations
- UNIT 3: Systems of Equations and Inequalities
- UNIT 4: Quadratics!
- UNIT 5: Data and Statistics
- UNIT 6: Arithmetic & Geometric Sequences
- UNIT 7: Functions

- LESSON 1: Border Tiles: Seeing Structure in Algebraic Expressions
- LESSON 2: A More Complicated Border: Day 1 of 2
- LESSON 3: A More Complicated Border: Day 2 of 2
- LESSON 4: Kitchen Tiles
- LESSON 5: Brainstorming Algebraic Expressions
- LESSON 6: Graphs ----> Tables ----> Rules
- LESSON 7: Rules ----> Tables ----> Graphs
- LESSON 8: Predicting Water Park Attendance
- LESSON 9: Battery Life
- LESSON 10: A Friendly Competition
- LESSON 11: Equations on a Graphing Calculator
- LESSON 12: Plotting Data on a Graphing Calculator
- LESSON 13: Multiple Representations - Linear Functions
- LESSON 14: Working at the Ice Cream Stand
- LESSON 15: Free Throw Shots
- LESSON 16: Unit Portfolio - Day 1 of 2
- LESSON 17: Unit Portfolio - Day 2 of 2