I have a group of struggling algebra students. They were taught to solve equations last year in 8th grade. When we approached the Mystery Bags activity (which covers solving equations with variables on both sides) I gave them some simple one-step equations and realized that they needed some more practice - especially with equations involving integers. The first day I "retaught" solving equations using inverse operations it did not compute with the majority of the class. I discussed with Mrs. New whether or not I should try to find a creative way to teach equations. After teaching using our Meaningful Math books in which we have a context and discovery/inquiry based activities it was so hard to just teach by giving notes and examples. I had seen a cover-up method for solving equations but the worksheet I had was a little confusing. Mrs. New had learned about a Cover Up Math app when she was at ISTE last Summer. I decided to let the students work with the app and see their reactions. When they told me that they understood it better using this method I did a little more searching. I found this worksheet along with this video which teaches solving equations with a method very similar to the Cover Up Math app. I had some students really grasp hold of this method because it made more sense to them. I still would kind of go to the side a be a "real" math teacher and talk about inverse operations. I had one student who did great using the traditional inverse operations.
Now...fast forward a week or so and we are now working on solving inequalities. I have realized that since I have so many students in the class that did not solve by inverse operations it is difficult to explain to them about when they need to change the direction of the inequality. We did an exploratory activity in Cookies where they discovered that when you multiply or divide by a negative (when solving inequalities) that we have to flip the inequality for the statement to remain true. However, when we looked at solving inequalities and we "reached back" to the blob/cover up method we do not talk about multiplying or dividing by a negative. So...in order to modify for the students to get the answers correct I told them to do these steps:
1. Replace the inequality symbol with an equal sign and solve the equation. (It is amazing how happy some of my students were when they didn't have an inequality symbol anymore.)
2. Draw a number line and decide whether or not to use an open or closed circle based on the inequality symbol (we had already discussed this)
3. Test a value on either side of the circled number to see if it makes the original inequality true. If it does shade on that side and if it doesn't shade the opposite side.
4. Lastly, make sure your "solution" matches the graph. This helps them to write the inequality with the sign going in the correct direction.
I did not write these steps on the board. We just worked through several together. I can't help but stand there thinking that it would be so much easier to just use the inverse operations with the "flip if you divide or multiply by a negative" rule. I had a few students tell me today that it all made sense to them now. I just wanted to say, "Really??"
So, I write this entry with a conflict brewing in my head over whether or not I have done the right thing. My reasoning for using the other method was that the students had been taught last year and this year with the inverse operation approach and it just did not seem to work for them. The math teacher in me tells me that it is so important for them to understand how to solve using inverse operations. The common sense portion of my brain says if I can get these students to improve their ability to solve 1 and 2-step equations then I am doing good. These are students whose math confidence level is so low.
As we go through these problems I often show the inverse operations method beside the cover up/blob method in order to show them the connection between the two. I want them to get to where they can understand the formal mathematics of what they are doing.