## Students start work on solving equations in 6th grade. They work on it in 7th and 8th and into algebra classes in high school and college. Despite this, many struggle at all levels with solving. I have witnessed and heard about this at each level. This post addresses one possible reason, which is the students are not grasping the concepts. This may be due to the way equations are presented. This post unpacks the possible reason and presents a scaffolded handout to unpack the concepts and vocabulary.

### Add-on Steps We Teach

The image below shows a earlier scaffolded handout I had used for years. It includes two elements that in my experience are common: referring to the “sides” of the equation, and the vertical orientation of the step with inverses.

I think these are “add-ons” that we teachers incorporate into math. I assume there are advantages and disadvantages for each. In this case, the disadvantages as I see them are a follows.

- The term
*side*may be somewhat ambiguous. We use it in the contexts of which side are you on, stand off to the side, side of a building, there are two sides to every story etc. We know this because most of us probably have experienced something like what is shown in the image below left. - The vertical orientation (image above “-1”) does not produce an expression that they have seen before. We are changing the rules of the game and then wonder why they are confused -see another in image below right. (I have used my share of add-ons.)

### Explanation of the Handout

I am attempting to wean myself off of the use of add-ons. This handout The handout scaffolds both mental steps and written steps. In addition to the skill based steps, they focus on concepts and vocabulary. Here is a YouTube video showing how it works, shown on a Jamboard accessible using this link. (You have to make a copy to access it.)

- The first 5 steps focus on mental steps which are identifying components of the equations using vocabulary. The steps prompt them to write what their thinking is. The example can be presented in “I Do” fashion with think alouds.
- The solving steps guide a horizontal orientation. This allows students to focus on the expressions as a whole and not in split-level fashion.
- Because the expressions with steps included are horizontally oriented, it is easier for the students to see the expressions as a whole and then simplify them.
- The 0 is written and addressed with an additional step to highlight the identity.
- I considered adding another step for the students to indicate that the final equation shows the solution.

### Blank Templates

The second sheet has blank templates. One possibility is that you could assign students problems and they can complete the first 4 on here.

### Accessing the Handout

Here is a link to the handout.

Awesome, thank you!! In the Add-on Steps We Teach example m=2.

YW!!

Good and useful writeups and scaffolded hands out. Many thanks! You are right about the add ons and the challange of the vertical list. I do have a question about something I struggle to give a good explanation for.

Your example is x+3=8 and the operation is addition.

For the equation 3+x=8 how to you start?

Yes! That is an important situation that I need to include. There are two ways, I believe. First is to unpack the commutative property (without actually using the term) and have them switch. The other is to circle the terms and any operation in front. I did this in another scaffolded handout. I think I prefer the former.

https://ctspedmathdude.com/solve-equations-with-like-terms-scaffolded/

Consider 3+x=8. We have suppressed the positive signs, since we all know they are there. Moreover we use the + and – signs to mean two different things. Operations and part of the name of the number. So the mathamatician that wants to create clarity might write (+3)+(+x)=(+8) where the +3 (for example) means postive 3 and the )+( means doing the addition operation. This helps, I think, decode the expression x-2(x-1) which might be written x+(-2)(x+(-1)) so that you can see you are distributing the number negative 2.

However, I don’t know that it clarifies 3+x=8. There really is only one operation and to use it’s inverse you really do need to rewrite it as x+3=8.

Thanks again for provoking my thoughts on the subject!