There are several layers to solving equations that can be unpacked using a task analysis approach. This includes written and mental steps (such as what we teachers mean when we tell a student to do the same thing to “both sides of the equation”). Here is how I develop the concept of equations and solving.
To unpack the layers for students, I have had a lot of success with the scaffolded handouts below (see last photo of example of what to write.) Here is a link to a Dropbox folder with all 4 handouts, in WORD format. Feel free to use and revise as desired.
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.
I am rolling out a series of videos on math topics from middle school and algebra 1. Here is a link to the folder with the accompanying handouts. This post will be updated periodically.
Let me know if your school is covering a topic that is challenging and students would benefit from an alternative presentation. I will see what I can do. The links to the videos are listed below the photo.
In general math is taught by focusing on the steps. Conduct a Google search for solving equations and you will see the steps presented (below). You need a video to help your student understand solving and you typically get a presenter standing at the board talking through the examples. (I’ve posted on my approach to solving equations.)
When the math is taught through the skill approach the student may be able to follow the steps but often does not understand why the steps work (below). The brain wants information to be meaningful in order to process and store it effectively.
To help flesh this situation out consider the definitions of concept and skills (below). Concept: An idea of what something is or how it works – WHY. Skill: “Ability” to execute or perform “tasks” – DOING.
Here is how the concept first approach can play out. One consultation I provided involved an intelligent 10th grader who was perpetually stuck in the basic skills cycle of math (the notion that a student can’t move on without a foundation of basic skills). He was working on worksheet after worksheet on order of operations. I explained down and monthly payments then posed a situation shown at the top of the photo below. I prompted him to figure out the answer on his own. He originally forgot to pay the down-payment but then self-corrected. Then I showed him the “mathy” way of doing the problem. This allowed him to connect the steps in solving with the steps he understood intuitively, e.g. pay the $1,000 down payment first which is why the 1000 is subtracted first. Based on my evaluation the team immediately changed the focus of this math services to support algebra as they realized he was indeed capable of doing higher level math.
First I develop an understanding of a balanced equation vis-a-vis an unbalance equation using the seesaw representation.
I then explain that the same number of guys must be removed from both sides to keep the seesaw balanced.
I then apply the subtraction shown above to show how the box (containing an unknown number of guys) is isolated. I explain that the isolated box represents a solution and that getting the box by itself is called solving.
I use a scaffolded handout to flesh out the “mathy” steps. This would be followed by a regular worksheet.
I extend the solving method using division when there are multiple boxes. I introduce the division by explaining how dividing a Snickers bar results in 2 equal parts. When the boxes are divided I explain both boxes have the same number of guys.
The students are then provided a scaffolded handout followed by a regular worksheet.
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.
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.
Dragon Box (link to website here) presents solving equations, proportions (and fractions and expressions) using an alternative representation and in a highly engaging game format (different platforms!).
In the photo above, the goal is to get the treasure chest by itself. To do this, the fly looking thing and the snake have to be eliminated. First eliminate the snake with the black background (a night card) by placing the other snake card on top of it (a day card). The day card snake must be placed on the right side as well. The day and night card (representing positive and negative numbers) become a circulating hurricane looking card (which represents a zero).
I used it with my 5 year old son (video here) and he could solve 2 step equations (on the game) independently within a couple hours. This is a game changer in teaching kids algebra.
Solving equations with a variable on both sides proves to be exceedingly tricky for many students. My approach is to focus on the individual expressions taken from both sides of the equation and to present them in the context of a relevant real life situation. The photo shows a snippet of the handout I use. The table is scaffolded to help students compute costs based on number of toppings. The pizza places charge the same at 3 toppings. Domino’s charges more for 0-2 toppings and Pizza Hut charges more for 4 or more toppings. The color coding fleshes this out.
Overall the kids are actively engaged and the variable, expressions and the overall equation has meaning.