Substitution for Systems – Scaffolded

Solving systems of equations using the Substitution Method may be the most task intensive algorithm the students learn. It is easy to overlook concepts as the students attempt the numerous steps. This post provides details about a scaffolded handout that guides students through the mental and conceptual steps, as well as the traditional written steps.


The focus is on systems with a variable that has a coefficient of 1. The variable is not isolated. The entry point is typically with the variable already isolated with the written step being substitution. This handout works for both situations. For the variable already isolated, step 2 can be skipped.

The Scaffolded Handout

The scaffolding addresses the skills used the method but also reinforces vocabulary and concepts.

  • The first step is a mental step but the students are asked to circle as a means of focusing attention on this step. It is useful for a think aloud as the teacher talks through the process.
  • Substituting an algebraic expression in for a variable allows for focus on the terms variable, expression, equation and the idea that a variable is a simple expression being replaced by another one.
  • The solution is an ordered pair. Often, students may stop once they find the value for the first variable.

Scaffolded Steps

Page 2 of the handout is scaffolded to guide the students through the steps I found to be problematic. Replacing a variable with an algebraic expression is the central component of this method and is new to students at this point. previously, they would have substituted in only numeric values for variables.

The Handout

Here is a link to the handout. The image below shows page 3, a blank template.

Systems Word Problems Scaffolded

Word problems are challenging for many students. Writing a system of equations to model a word problem has unique challenges. This post provides details about a scaffolded handout with color coding can unpack the process for generating the appropriate system of equations.

Unpacking the Word Problem

A mistake I have witnessed over the years is students mistakenly using given values for both equations. In the problem below, students are far more likely to generate the equation for the yellow part: 2x + 3y = 24. The challenge is that the blue part has only 1 number so students will often write 2x + 3y = 10, using the dollar quantities a second time.

By highlighting the two parts of the word problem with given values, the students can match parts of the word problem with respective equations. The scaffolding separates the parts, and the color allows for matching.


The rest of the problem is prior knowledge with the students using one of the methods for solving. The scaffolding continues to lower the task demand by reducing the need to remember all the steps. This allows them to focus more bandwidth on the new steps.

Access to Handout

Here is a link to the handout.

Math Videos for Middle School and Algebra 1

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.