Solving 2-step Equations is an escalation in terms of task demand. Starting in elementary school, they are exposed to math sentences in which there is a single operation. Now they are asked to choose a number to eliminate. This post provides details about a scaffolded handout. It addresses the choice of number to eliminate and how one choice may provide the path of least resistance.
Similar to the handout for solving 1-step equations, this scaffolded handout engages the students with the mental steps in addition to the steps they write out when they show their work. A key concept is identifying that there are two numbers to eliminate and then eliminating them, one at a time.
Choosing the Number
In lieu of teaching the method of doing PEMDAS or order of operations in reverse, I focus on presenting both the respective steps for eliminating either number in the expression with the variable. I believe it strengthens their understanding of the steps for solving and it shows them a reason to choose one route over the other – but they are free to choose! Here is a FB Reel and Youtube video addressing this.
Here is an introduction to solving equations using a Jamboard (see photo at very bottom for how to make a copy). A seesaw is used to unpack the concept of an equation as two sides that are equivalent. The box is used to unpack the concept of a variable representing an unknown number (or oranges in this context). The form of a solution for an equation is established, with students revising to create other solutions. Here is a YouTube video and a FB Reel showing how it works.
Students are then provided a couple slides to match seesaw representations with actual equations. The matching provides a scaffold to support the connection between representations.
Then the seesaw and box representation is used to unpack the steps. Students are provided the steps as written directions on how to model the given solving steps with the seesaw.
The students are then provided the equation and seesaw representation, along with the solving steps provided as moveable pieces. Students slide pieces and move seesaw objects to make the connection between the two. Here is a link to a post with an updated version of this handout.
Finally, students are given the equation and tasked with completing all of the steps including the initial set up. This slide can be copied with new equations entered for additional practice (including having the variable on the right or writing the number before the variable (e.g., 5+m=8).
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.
All too often math topics are introduced first with the skills and steps. This is backwards. The photo above shows how I introduced solving equations a high school student with autism using the concept as an entry point.
We discussed what was involved in buying a car, including payments (no interest) then I posed the problem seen at the top. I asked him to figure out the monthly payment. He worked out the problem, overlooking the down payment. With a minimal prompt he self corrected. I followed this by “showing him the mathy way of doing the problem.” (Seen in the bottom half of the photo). He conceptually understood why the -1,000 was the first step and x had meaning.
This is a photo of a handout from a special ed conference shared by Dr. Shaunita Strozier (firstname.lastname@example.org) of Valdosta State University. It shows the use of algebra tiles to provide a concrete level of the concept of an equation and solving the equation. The photos on the left show the R or representational level of the concept. Her approach is called SUMLOWS which is an acronym explained in this handout.
Solving equations can be very abstract and inaccessible for students. The photo below shows a scaffolded handout that introduces students to solving two-step linear equations. The students are introduced to buying a car: down payment and monthly payment. This provides meaning for the variable (x is the monthly payment), for the constant (down payment) and coefficient (12 months – fabricated situation). This approach introduces the concept of having two steps and for which to choose. Take note of the little “PEMDAS” on the left side. I explain that solving involves using PEMDAS in reverse