## Introduction to Equations – (Meaning Making)

This is a meaning making approach to introducing equations. I will walk through the parts shown in the photo in the space below this photo. (A revised edition of this handout will be used in a video on this topic.) First I explain the difference between an expression (no =) and an equation (has =). An equation is two expressions set equal to each other (21 is an expression). I then develop the idea of a balanced equation and will refer to both sides of the see saw as a prelude to both sides of the equation. I also focus on the same number of people on both sides as necessary for balance. At this point I am ready to talk about an unknown. Here is the explanation I use with the photo shown below.

• I start with the seesaw at the top. The box has some guys in it but we don’t know how many.
• We do notice the seesaw is balanced so both sides are equal.
• This means there must be 2 guys in the box.
• I follow by prompting the students to figure out how many guys are in the box(es) in the bottom two seesaws.
• Finally, I explain that the number of guys in the box is the solution because it makes the seesaw balanced. There are multiple instructional strategies in play.

• Connection to student prior knowledge – they intuitively understand a seesaw. This lays the foundation for the parts of an equation and the concept of equality.
• Visual representation that can be recalled while discussing the symbolic representation, e.g. x + 1 = 3
• Meaning making which allows for more effective storage and recall of information.

## Trick for subtracting integers

I have algebra students, in high school and college, who struggle with evaluating expressions like 2 – 5.  This is a ubiquitous problem. I have tried several strategies and the one that is easily the most effective is shown below. When a student is stuck on 2 – 5 the following routine plays out like this consistently.

• Me: “What is 5 – 2?”
• Student pauses for a moment, “3”
• Me: “So what is 2 – 5?”
• Student pauses, “-3?”
• Me: Yes! I implemented this approach because it ties into their prior knowledge of 5 – 2. It also prompts them to analyze the situation – do some thinking.

## Example of Using Support Class to Support Current Math Content

I have posted on how to effectively provide support for current math topics. Here is an example (below) of how support can focus on both the current topic and prerequisite skills.

For example, on the 22nd in this calendar the current topic is solving equations. The steps for solving will include simplifying expressions and may involve integers. The support class can address the concept of equations, simplifying and integers which are all prerequisite skills from prior work in math. This approach allows for alignment between support and the current curriculum and avoids a situation in which the support class presents as an entirely different math class. For example, I recently encountered a situation in which the support class covered fractions but the work in the general ed classroom involved equations. Yes, equations can have fractions but often they do not and the concepts and skills associated with the steps for solving do not inherently involve fractions.

## Basic Skills Older Students

A widespread problem at the secondary level is addressing basic skills deficiencies – gaps from elementary school. For example, I often encounter students in algebra 1 or even higher level math who cannot compute problems like 5÷2. Often the challenges arise from learned helplessness developed over time.

How do we address this in the time allotted to teach a full secondary level math course? We cannot devote class instruction time to teach division and decimals. If we simply allow calculator use we continue to reinforce the learned helplessness.

I offer a 2 part suggestion.

1. Periodically use chunks of class time allocated for differentiation. I provide a manilla folder to each student (below left) with an individualized agenda (below right, which shows 3 s agendas with names redacted at the top). Students identified through assessment as having deficits in basic skills can be provided related instruction, as scheduled in their agenda. Other students can work on identified gaps in the current course or work on SAT problems or other enrichment type of activities.
2. Provide instruction on basic skills that is meaningful and is also provided in a timely fashion. For example, I had an algebra 2 student who had to compute 5÷2 in a problem and immediately reached for his calculator. I stopped him and presented the following on the board (below). In a 30 second conversation he quickly computed 4 ÷ 2 and then 1 ÷ 2. He appeared to understand the answer and this was largely because it was in a context he intuitively understood. This also provided him immediate feedback on how to address his deficit (likely partially a learned behavior). The initial instruction in a differentiation setting would be similar. ## Graphing Linear Functions Graphing linear functions and the underlying concept are challenging for many students. The video below shows a scaffolded approach to teaching how to graph. This approach also addresses the concept of the graph as a visual representation of all possible solutions (see photo above). Students often do not realize that the line is actually comprised of an infinite set of points which represent all the solutions. Here is a link to the document used in the video.

## Function Notation for Algebra

Below is a video of a lesson I recorded on function notation using the Explain Everything app. The lesson starts by addressing the concept of function notation by connecting it to the use of the notation “Dr.” as in Dr. Nick of Simpson’s fame. The lesson builds on prior knowledge throughout with a focus on color coding and multiple representations.

This videos shows an instructional approach to teaching function notation and concepts in general and video lessons can be used for students who miss class or who need differentiation.

## Function Notation Function notation is challenging for many students yet we teachers overlook the reasons for the challenges. For example, students see the parentheses and rely on the rule they were taught previously, y(5) is “y times 5”. Because this is overlooked by teachers we often skim over the concept of notation and delve into the steps. This document is a means of presenting the concept and the use of function notation in a meaningful way. Feel free to use or revise and use the document as you wish.

## – symbol: negative vs subtraction

The use of the “-” symbol is challenging for many students. They don’t understand the difference between the use of the symbol in -3 vs 5 – 3. To address this I use a real life example of multiple uses of the same symbol (1st 2 photos below) then break down the “-” symbol (photo below at bottom). I suggest this be introduced immediately prior to the introduction of negative numbers.  ## Functions Introduction Video Lesson

This video provides instruction to introduce the definition of and conceptual understanding behind algebraic functions.

Functions video ## Relations Introduction Video Lesson

This video provides instruction to introduce the definition and conceptual meaning behind algebraic relations.

Relations Video 