The Angel and the Cowboy - Working with Inequalities

I came across the Angel and the Cowboy years ago. I did not create this, but I did add the bow tie. I love using this with inequalities. I have found it most useful when working with piecewise functions. The idea is that the hats and the face deal with graphing, the body is used for writing domain and range in interval notation form and the legs help with writing inequalities. All the pieces are related. Notice how all the Angel symbols are for NOT EQUAL and the Cowboy symbols are when things are EQUAL.

I found this slide from when I taught piecewise functions last year:

Let's break this idea down a little further. If you asked your students to give the domain of only the left arrow, then they would see an open circle which is the angels face, so I call this an angel problem. The student would notice that the domain goes from negative infinity to -1. If you want the students to write this in inequality form, they would use the angel symbol and say: x < -1 or I allow my students to write: -∞ < x < -1. If you asked them to write the domain in interval notation form, they would write (-∞,-1). All symbols came from the angel!

Now let's look at the right arrow. This is a cowboy problem (well not completely). The enclosed dot indicates that the problem is a cowboy problem, but since there is an arrow on the opposite end which represents going to infinity, this problem is also an angel problem. The domain in inequality form would be x > -1 or -1 < x < ∞.  In interval notation you would write [-1, ∞). The second inequality answer and the interval notation answer contain both cowboy and angel parts. (Since you can never reach infinity, you can never equal infinity. That's why I added the infinity symbol to the angel.)

Anytime graphing and writing inequalities is a part of a lesson the angel and cowboy can be useful. You may not need all the parts of the angel and cowboy. Interval notation is not taught until Algebra II in Texas, so when I'm using this in Algebra I, I usually just tell the kids that they will learn about the rest of the symbols later.

The solid and the dotted lines above the faces come in handy when graphing inequalities. If we are graphing y > 2x - 3, then the students will realize that the symbol came from the angel, so the line will need to be dotted. The one thing that the angel and cowboy do not help with is shading above or below the line. You could easily add the words above and below to the symbols on the legs if you wanted.

I'd love to hear from other teachers that have used this before. Let me know how you use it or how you have tweaked it. If you've never seen this before, I hope you will find this handy and you will be able to use it in your own classroom!

I've made a free poster of the angel and cowboy that is in my TpT store. If you are interested, please use this link and download this resource.     Cowboy and Angel Poster

In Algebra II, it's hard to decide how to start the year. I want to review, but I don't want to keep us from moving forward. I need to know how much the students remember from Algebra I. My students were in Geometry last year. How much Algebra was incorporated into their Geometry class? I feel like my best bet is to begin by solving equations and inequalities. The activity that I will use, starts easy and gets progressively harder. The students will not be able to use a calculator because I need to know who REALLY knows how to solve equations without tricks or help. This activity is sold in my TpT Store:

Solving Equations and Inequalities

I really like this activity because there are options. I can make it fun by using the answer banks. The answer banks have the answers with an activity. A few of the activities have the students draw a picture. Another activity has the students fill in a movie title. There are progress checks along the way as well. I think that I will use the easiest page as a bellringer on the first day of class. I'll have them work through as much as they can on the rest and send it home to be finished. I can use the progress checks as quizzes or as pages for their interactive notebooks. This resource will give me an idea of where the students are and will be a nice segway into solving absolute value equations which is what I plan on doing next.