Linear Inequalities in 2 variables

What was interesting when we first started working with the common core curriculum in our state workshops four summers ago is that linear inequalities in 2 variables was NOT in the topic list but systems of linear inequalities was included!  So we realized pretty quickly that we had to add this topic if the students were going to have a successful experience with systems.

Here are my ISN pages. I realize that I need to have one of my examples be a negative slope. So I'll change one of my examples for next year. Here is the document with both pages.

Here is a homework worksheet I used for graphing and a nice enrichment worksheet for those who catch on to this topic quickly and need a challenge (given graph, write linear inequality).

I'm a fan of having students compare & contrast things in math class and Venn Diagrams are a great way to do that. These make a good warm up activity for students to fill in with their groups and then we have a giant one on the board to have students come up and fill in. Here is my worksheet.

My advanced students are very good at learning procedures and working through problems. But I want them to be able to articulate what the solutions are for each of these. (we talk about solutions quite a bit all year - haven't mentioned it in the blog so far - but a lot of what we are doing in algebra one is working with solutions.....and a solution isn't necessary a single number). This is a nice critical thinking worksheet getting them to think about the solution sets for equations and inequalities. 

We also did applications of linear inequalities in two variables - but I didn't get to make an ISN foldable for these. At this point in our school year it was the end of semester one and we were wrapping up so we could have a unit test and start to prepare for our midterm exam. But here are some materials I used with my students.

Application Introduction.

Applications HW

Linear Inequalities

We now go back to inequalities. We touched upon them briefly at the start of the unit. But now we see how to solve & graph them. Headings link to documents for the foldables.

Basic Solving

Flip Flop Rules

Then we've added compound inequalities back into our curriculum (original common core Alg one did not have it, but it snuck back in an update we recently read through). I like to start by exploring the real meaning of AND vs OR with this worksheet. And for my ISN entries I have the following:

Compound Inequalities

Interval Notation

Solving Compound Inequalities

Solution Checking with Your Graphing Calculator

https://en.wikipedia.org/wiki/Graphing_calculator

Technology is a beautiful thing and I want my students to get more comfortable with their graphing calculators, so we pause a bit and see how we can check our solutions with the graphing calculator).

This involved these ISN pages:

and you can get the Calculator Graphic HERE.

Equation Solving

So now we get into equation solving. My students got a very good background in that last year in 8th grade. So we just have to review two-step and then get into 3-step. We really focus on the "undo" aspect of equation solving at first, looking at what was used to build and then go backwards to solve. I don't get into distributive property yet (even though they saw it last year). I really focus on the "undo" process.

Here are my ISN pages:

While we were going over the mechanics of equation solving, I saw some trouble spots and created a foldable to address those.

Common Problem Spots

and here is where you can access the ISN pages for your copying needs. (it's got the templates and also created examples from the internet that I used for reference when creating mine).

What are Solutions?

So before I get into all the mechanics of equation solving I like to talk about what a solution is. I used this really great resource to guide my lesson - Mathematics Assessment Project CLASSROOM CHALLENGES Sorting Equations and Identities.  And we followed up by completing this worksheet. 

Then we filled in our ISN pages...

and here is the document above.

Expressions, Equations & Inequalities

This is a really starting unit in Algebra One and often is what people think of when they think of Algebra. An internet search of “what is algebra” gave this as a response

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols.

This is pretty fundamental to the study of Algebra – it gives us a symbolic way to work with numbers and to generalize. So this unit gets us started in how to do that. Students really need to understand what they are doing and not just memorize.

Sadly the symbolic nature of Algebra gets a bad rep and people can get very frustrated and joke about their frustration.

We are just going to move on from this and try to shed some light on how to teach this unit using ISNs.

Just to give some context, where is what we are working with from our curriculum:

Essential Questions:

·         How does one create an equation/inequality using symbolic algebra, specifically how are operations used in building? 

·         How are inverse operations used to deconstruct an equation/inequality by creating a series of equivalent equations/inequalities ultimately resulting in a solution?

·         How and in what ways do you use symbolic algebra to represent and explain mathematical relationships and real world situations? 

·         What is a solution set for a linear equation or linear inequality?

·         How do we utilize technology to justify a solution method?

Corresponding Big Ideas:

·        Obtaining the solution to an equation or inequality, no matter how complex, always involves the process of undoing the operations.

This is a long unit, so here is my TOC listing - I'm going to have to break down my foldables into a few blog posts (thus this is part 1). just remember my handwriting disclaimer...

I started the unit with the students being given a variety of mathematical items - numerical expressions, algebraic expressions, equations and inequalities. Check out my sorting cards HERE.  They had to sort them into 4 groups any way they saw fit (they weren't given those category names) - then we discussed and played around with what they came up with (at one point I gave them the category names and asked them to adjust their sorting into those groups). This was to warm them up to the vocabulary of these symbolic representations. From their we did little frayer diagrams of each in our notebooks:

For a structured right hand side sample, check this out. 

equation solving "practice" on the right side (my students should know these without instruction from last year's class). You can also find more equation solving ISN samples here. 

I added the "backwards inequalities" from the foldables I found on line. Here is what I started with...

and corresponding practice on the right hand side opposite each frayer diagram. Some of the "practice" was somewhat informational as you can see (I included those that were).  What's a frayer diagram? Learn more HERE.