##
* *Reflection: Checks for Understanding
Emerging Pictures: Graphing Inequalities in Two Variables - Section 2: Investigation

This piece of student work show two typical responses that I see in my classes to this task. In the first graph, the student started out plotting five points in green that would satisfy the inequality and five points in red that would not. In all of these cases, the student tried equal numbers of cats and dogs and ended up with what would look like a line from the origin. I see students make this line quite a bit. Presumably, what happened next is that the student realized the key to figuring out where the green and red points would be is finding the dividing points that are right on the boundary line. I believe he added the purple points next, realized they were collinear and then shaded the red and green sections of the graph.

Students who struggle will generally not take this same approach. They usually have a lot more discrete red and green points and less purple points. In the second example, we can see that this student made a lot of green points first. He also made them very systematically, starting with 1 dog and figuring out how many cats all the way up the y-axis and so son. In general, this kind of student adds more points before s/he sees a pattern but usually reaches the same conclusion.

I think it's important not to show them a "math shortcut" here but let students develop their own understanding of where the dividing line is between points that satisfy the inequality and those that do not. It can be tricky to keep students from overhearing what others who are further along might have figured out!

*Checks for Understanding: Typical Student Graphs*

# Emerging Pictures: Graphing Inequalities in Two Variables

Lesson 2 of 17

## Objective: SWBAT graph linear inequalities in two variables. SWBAT understand the boundary line between points that work and points that do not work in an inequality.

#### Opening

*5 min*

This is one of my favorite lessons. I love this lesson because it allows students to try out different points that will either fit an inequality, fit the inequality exactly (use up all of the constraint), or do not satisfy the inequality. In my opinion, this approach to graphing linear inequalities in two variables helps students to develop a conceptual understanding of graphing inequalities and from there they can learn shortcuts to make the process faster (but the conceptual understanding will (hopefully) remain).

I start this lesson by telling students that today we will get a picture of all the points that will work for Carlos and Clarita. We then read through Too Big or Not Too Big.pdf together. The task asks them to find five points of each kind (fit the inequality, fit exactly, and do not fit) in different colors, but I encourage students to continue plotting points until a picture emerges for them. Students begin with the Start Up Constraint. They can work in small groups or pairs, but each student must generate his/her own graph.

#### Resources

*expand content*

#### Investigation

*25 min*

Next, I let students get to work. Students usually start to have a lot of interesting ideas about how to find points that are exactly on the line. Some students may use the x and y intercepts to find points that are on the line and then use a rough idea of where that line is to find other points. Other students may notice a relationship between the rate of change, and might start to note how many you have to go over and down in order to find another point. I like to look for these ideas so that I can have students share out their thinking when we get to the discussion part of class. This year I took some video of students explaining their "ah ha" moments. I often find that this task reaches a point where students suddenly have a new understanding.

Other students will have trouble seeing the "picture" that emerges and may even get frustrated. For these students, I continue to encourage them to keep adding more points until they have an "ah ha" moment. By the end of class, all students usually have at least one graph finished that shows the two half planes that make up an inequality.

Students who finish the first inequality graph with time to spare can move on to the Space graph.

#### Resources

*expand content*

#### Discussion

*20 min*

Depending on how far students get with their graphs, today's discussion may occur at the end of class or at the start of the next lesson. If some students finish early, I might have them draw their graphs on the Smartboard, or take pictures of their graphs and project them. I like to give students ample opportunity to discuss how they figured out what parts of the graph would be shaded in which color and share their strategy for finding points that fit right on the line. Then I use the Too Big or Not Too Big PowerPoint to talk about what this all means in the context of inequalities. I especially spend time focusing on the difference between the points that fix *exactly* and the points that satisfy the inequality.

#### Resources

*expand content*

#### Closing

*10 min*

I like to ask students to reflect on their problem solving skills in order to get a complete picture of what an inequality looks like. At the end of today's class, I ask them to reflect on the following prompts:

- As you worked on your inequality picture, what changes did you have to make to see the whole picture?
- What strategies did you use to find points that were on the line?

*expand content*

Too Big or Not Too Big is licensed by © 2012 Mathematics Vision Project | MVP In partnership with the Utah State Office of Education Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license.

http://www.mathematicsvisionproject.org/secondary-1-mathematics.html

*expand content*

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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
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- LESSON 1: Introduction to Inequalities: Working with Constraints
- LESSON 2: Emerging Pictures: Graphing Inequalities in Two Variables
- LESSON 3: Two Methods, One Equation: Finding the Boundary Line
- LESSON 4: Writing and Graphing Inequalities to Represent Constraints
- LESSON 5: Finding the Feasible Region
- LESSON 6: Where Do the Lines Cross?
- LESSON 7: Solving Systems Using Elimination: An Intuitive Approach -- Day 1 of 2
- LESSON 8: Solving Systems Using Elimination: An Intuitive Approach -- Day 2 of 2
- LESSON 9: Introducing Inconsistent & Dependent Systems
- LESSON 10: Finding the Feasible Region: From Start to Finish
- LESSON 11: Maximizing Profit: An Introduction to Linear Programming
- LESSON 12: REVIEW: Systems Review and Word Problem Practice
- LESSON 13: SUPPLEMENT: Linear Programming Application Day 1 of 2
- LESSON 14: SUPPLEMENT: Linear Programming Application Day 2 of 2
- LESSON 15: ASSESSMENT PROJECT: Writing Linear Programming Problems Day 1 of 3
- LESSON 16: ASSESSMENT PROJECT: Writing Linear Programming Problems Day 2 of 3
- LESSON 17: ASSESSMENT PROJECT: Writing Linear Programming Problems Day 3 of 3