Tag: inequality

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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.

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Intro to Graphing Inequalities

For years over many settings from middle school to college I have witnessed students struggle to make sense out of inequalities like x < 4. Not only is the concept of an inequality of an inequality challenging, merely reading the symbol is problematic. This post shows an concept based approach to making sense of inequalities.

The Symbol

In lieu of neumonic devices such the “alligator eats the bigger number” or “the symbol points to smaller numbers”, I recommend unpacking the shape of the symbol.

Jamboard

Below are Google Jamboard slides I use (see photo at bottom for access) to introduce the concept of inequalities. Here is a FB Reel that shows how it works, as well as a YouTube video.

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.

Jamboard Access

Make a copy of the Jamboard to have access.

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More Than: Concept and Symbol

The alligator eats the bigger number is the common approach for student to use inequality symbols (<  > <  > ). I find that students remember the sentence but many do not retain the concept or use the symbols correctly, even in high school. This post presents an alternative approach.

Alligator Method

The reason, I believe, is that we introduce additional extraneous information: the act of eating, the mouth which is supposed to translate into a symbol, the alligator itself. For a student with processing or working memory challenges this additional information can be counter productive.

alligator eats bigger number

Dot Method

I use the dot method. By way of example here is the dot method. I show the symbols and highlight the end points to show one side has 2 dots and the other, 1.

Compare-with-Dots

Then I show 2 numbers such as 3 and 5 and ask “which is bigger?” In most cases the student indicates 5. I explain that because 5 is bigger it gets the 2 dots and then the 3 gets the 1 dot.

3 dots 5

I then draw the lines to reveal the symbol. This method explicitly highlights the features of the symbol so the symbol can be more effectively interpreted.

3 less than 5.

Concept of Inequalities

That is the presentation of the symbol. To address the concept of more, especially for students more severely impacted by a disability, I use the following approach. I ask the parent for a favorite food item of the student, e.g. chicken nuggets. I then show two choices (pretend the nuggets look exactly the same) and prompt the student to make a selection. This brings in their intuitive understanding of more.

concept of more chicken nuggets just plates

I think use the term “more than” by pointing to the plate with more and explain “this plate has more than this other plate.” I go on to use the quantities.

concept of more chicken nuggets more than words

Finally, I introduce the symbol to represent this situation.

concept of more chicken nuggets more than symbol

Below is the example my 3rd grade son used to explain less than to a classmate with autism. This method worked for the classmate!

Lucas less than example
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Unpacking the Inequality Symbols

I have found that many students I have taught, including in college, struggle with the inequality symbols. Typically, they cannot remember which symbol is less than or greater than despite various efforts such as the “alligator eats the bigger number” method.

I’ve found traction by using the dot method. It is like a light bulb clicking on for many of the students. I believe the reason is works is that it explicitly addresses the shape. Here’s the approach.

  • Show them an inequality statement
  • Tell them to think of the end points as dots.
  • Draw the dots.
  • Ask, “which number is bigger?”
  • Point out that the bigger number has more dots.
  • Show them two numbers, say 4 and 6.
  • Ask which is bigger?
  • Draw two dots for the 6 because the bigger number gets two dots.
  • Draw one for the 4.
  • Finally, draw the symbol.

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