## Introduction to 2 Variable Inequalities

Previously, I shared how I use a Google Jamboard to introduce 1 variable linear equalities with a focus on conceptual understanding. I use the same approach for the 2 variable version (example problem below).

A conceptual gap that typically arises is the students do not understand what the shading represents. This is what I am addressing from the start using a Jamboard. First, the focus is on understanding the inequality and identifying a single point that works (below).

The next step is for students to determine more points that are solutions for the inequality, with no equal to part. (below).

The equal to part is addressed separately (below).

The equal to and the greater parts previously addressed are combined together.

The inequality is will be expanded to include an operation (+ 2) with a focus on the equal to part first.

The greater than with no equal to is addressed.

Then the equal to and greater than are addressed sequential. The equal to results in dots in a straight line and in lieu of plotting all the points, a line is drawn (building on the intro to 1 variable inequalities). This is followed by the greater than part and shading in lieu of plotting all of the dots above. THIS is where they gain an understanding of what the aforementioned shading is.

Finally, the dashed line is addressed by showing, as was done with the 1 variable inequalities, that there is a cutoff point that is not part of the solution set so in lieu of plotting a bunch of open circles, a dashed line is drawn.

## Planning and Preparing for Math in the Fall

If you are reading this post, it is likely that you have a student or you teach students who struggle with math. Here are suggestions to help your students prepare for the math they will encounter in the fall.

Many students are behind in their math education. This has long term implications. The sooner you can address the gaps, the better chance your student has for post-secondary success or competence with math.

## Life Skills Math – Not So Easy

As I wrote previously, shopping is dense with math tasks as are grocery stores. Here are some division situations that are sneaky challenging and require a student to know when and why to divide before even reaching for the calculator. I will use these to help illustrate the fact that life skills math is not simply counting money or using a calculator to add up prices. There is a great deal of problem solving and thinking skills that need to be developed.

For example, if a student has \$60 to spend on gifts for her 3 teachers the student needs to understand that she can spend up to \$20 per teacher (before even talking about taxes).

An entry point for division can involve a dividing situation the students intuitively understand, e.g., sharing food. Start with 2 friends sharing 8 Buffalo wings evenly (below).

This can lead into the 3 teachers sharing the \$60 evenly (below). In turn, this can be followed by the online shopping shown above.

This approach can be used to develop an understanding of unit cost (cited in the shopping is dense post). Start with a pack of items to allow the students to see the cost for a single item before getting into unit cost by ounces, for example.

I have had success with teaching these division related concepts using sheer repetition as much of our learning is experiential learning. Using a Google Jamboard as shown in the photos allows for the repetition.

## Introduction to Inequalities

For students with special needs, the teacher speaking “math” to students sounds like the teacher from the Peanuts cartoons.

This is apparently the case when students are learning about inequalities such as x < 4 because I have seen many high school and college students struggle with this topic. The challenge is that teachers are often focused on the math symbols and steps as opposed to the math concepts. In contrast, below are Google Jamboard slides I use (you can make a copy and edit) to introduce the concept of inequalities.

First, I start with a topic of interest and possibly prior knowledge for the students (age to get a drivers license – below). I present the idea of an inequality in context before I show any symbols. In this case, students identify ages that “work”.

Then I introduce the symbol (below). In this case, I include equal to for the inequality (x > 16 vs x > 16). The students plot the same points then we discuss that there are many other ages that work. These ages are called solutions. We put a closed circle on all of the solutions. Then discuss that ages are not exactly whole numbers so we can plot points on all the decimals. Then we discuss that the solutions keep going to the right so we keep drawing dots to the right. There are so many dots we draw a “line” instead of all the dots.

Then we do the same steps for a situation in which the number listed (52 in the case below) is NOT a solution. The students put dots closer and closer to the number but cannot put a closed circle on 52 as a solution (top photo below). Then we present the symbols and talk about the number as a cutoff point that we get really close to but cannot touch. Therefore we use an open circle to show the number is NOT a solution.

This introduction can be followed by problems on a handout, ideally with context then without.

## K-12 Math Education is a Line of Dominoes

Critical dominoes in math education start falling in 6th and 7th grade with the last ones falling in college. If you have a student who struggles with math and is entering or returning to middle school, now is the time to intervene to avoid more serious issues related to math education in the future. If your student is not going to college or is not accessing the general curriculum, I suggest you read this.)

Below is a chart showing the different categories of Common Core of State Standards (CCSS) math (called domains) at different grade levels. For the majority of students who will attend college, the traditional algebra based sequence (algebra 1, algebra 2, and maybe pre-calculus, calculus) is the path of math courses to be taken. Given this, for students who struggle in math but have a post-secondary education as a goal, the domains I emphasize in middle school are Expressions and Equations, Ratios and Proportional Relationships, and Functions. For high school, I emphasize Algebra and Functions.

Looking at the overviews for CCSS math standards (below) you can see the dominoes line up.

• In 6th grade, Ratios and Proportions are an entry point for Functions in 8th grade which leads to Functions in high school.
• In 6th grade, Expressions and Equations are the entry point for Expressions and Equations in 7th and 8th grade, which lead to Algebra in high school.

If your student is struggling with the middle school topics I cited and the gaps are not filled, the struggle will be carried with them into high school and into college.

I recommend the following:

• Focus IEP math objectives on the priority units of the math curriculum, as cited above.
• Ask for examples of mastery for the objectives to help you evaluate progress and mastery. Have this in place from day 1.
• Focus on study skills, not just content mastery.

## Shopping is Dense with Math Tasks

I recently worked with a student on an online grocery shopping activity – finding ingredients for mac and cheese. We had the ingredients listed in a column on a Google Doc (allows both of us to edit the doc simultaneously) and then he cropped and pasted a photo of each ingredient (see photo below). The goal was for him to identify the total he need and the total cost in planning for actual shopping or to continue with the online shopping. Note: he wasn’t actually buying anything at this point but this was a step in preparing him to do so.

This activity is dense with math tasks and shopping related tasks. The math tasks include the following:

• Identify the price (vs quantity of the item or unit price).
• Interpret the quantity for the ingredient.
• Identify the units (oz and cups)
• Convert units
• Compare amount in box with amount needed.
• Determine how much more is needed, if any.
• Compare choices before selecting the item, (Barilla Pasta vs another brand).

To convert units, the “mathy” approach can be used or the student may simply use an app. For this student we chose an online unit converter (see below). This is more complicated that it appears. The student must choose the units and the order (in this case convert cups to ounces or vise versa), distinguish between imperial and US cups, understand that you enter the quantity (the search results in 1 US ounce appearing by default), and then interpret the decimal (keep in mind the ingredient quantities are in fractions).

Life skills math is more complex and challenging that parents and educators may realize. As a result, the planning for developing these skills should begin much sooner rather than later – not to mention the actual logistical tasks of shopping, e.g. finding an item in the grocery store.

## Critical Thinking

Often we adults engage students with closed-ended questions and then consider this as having a conversation with the student. I witnessed this first hand in a high school consumer math course I co-taught. The adults sat with the students the first day after December break for a conversation about their break. The questions were consisted of and were similar to the following. “Did you have fun?” “Did you eat a lot?” For some, like my son, this is appropriate. For many others, we are offering low hanging fruit that does little to move them forward.

Ask open-ended questions that prompt the student to engage in critical thinking such as analyzing and evaluating – below, courtesy of Jessica Shabatura. Work this into IEPs and 504 to have teachers implement this. For example, I asked the students what they liked about break. Then I asked why they liked it. Here is an example of me questioning my son, who does not have a disability, when he was maybe 4.

## Curriculum Based Assessments

Most testing for IEPs involves standardized testing. As I wrote in a previous post, this is important testing but is not sufficient. A major focus of special education is to make the general education accessible as possible. Hence, curriculum based testing is an important complement to the standardized based testing. For example, the KeyMath3 assessment will speak to problem solving or geometry but those are broad categories. If I am working with a 3rd or 4th grade student, I would be interested in the student’s level of mastery in computing the perimeter of a rectangle.

Also, math is very different than reading because math has a variety of categories of math, aka domains. A student testing at a 4th grade level in math does not reveal much information, as I explain in this previous post.

When I conduct evaluations or assessments, I go to the Common Core Standards and assess each with curriculum based problems, see below. The photo shows my planning document and then I transfer the problems to a student handout for the student to complete.

## Secondary Characteristics: A Performance Factor

For students with a disability, performance does not align with ability.

In my view, there are 3 different categories of performance factors: the disability, gaps in achievement, and secondary characteristics. (Percents are contrived to provide a visual representation.)

To address these secondary characteristics, which manifest as a set of behaviors, I suggest a focus on shaping with a token board.

Here is a video explaining this.

## Using \$10 and \$1 Bills to Represent Regrouping in Addition or Subtraction

Money is intuitive for many students, even when the underlying math is not. For example, I often find that students who do not understand well the concept of Base 10 place value do understand \$10 and \$1 bills. With this in mind, I created a virtual scaffolded handout that builds on student intuitive understanding of the bills through the use of \$10 and \$1 bills to represent regrouping. Here is a video showing how I use it.

In the photo below, at the top, a \$10 bill was borrowed into the ones column. The reason is that \$7 needed to be paid (subtracted) but there were only five \$1 bills. In the photo below, bottom, the \$10 bill was converted into ten \$1 bills. On the left side of the handout, the writing on the numbers shows the “mathy” way to write out the borrowing.

Once the student begins work with only the numbers, the \$10s and \$1s can be referenced when discussing the TENS and ONES places of the numbers. This will allow the student to make a connection between the numbers and their intuitive, concrete representation of the concept.