Teaching equations using the cover-up or blob method

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.

IMP Cookies - Rock 'n' Rap, formative assessment and collaboration

Rock 'n' Rap is an activity that talks about producing rock and rap albums. The students will once again be led to find profit lines which should help them on solving the Cookies unit problem. There are 3 constraints this time. Also some of them shade below but one shades above so there is a little variety.

When graphing the inequalities Sonya New, my IMP teaching buddy, has stressed with the students that one way to graph the lines is to find values for x and y that make the equation true (they really do this by trial and error using number sense). I love that she has done that (even though I have talked mostly about using intercepts or slope-intercept form). She has taught SI form and graphing using intercepts also, but she says alot of her students still graph by finding ordered pairs that work.

When I did the problems with my class we said that there had to be more rock albums than rap albums. However, smarty pants Sonya corrected me (I hate when I'm wrong) and told me that it should have been rock is greater than or EQUAL TO rap. When you read the paragraph about this constraint it says that the company promises that it would not release more rap than rock. Then it goes on to say that the company is more closely related to rock music in the public mind. Even though I know she is right I could really make an argument about why I chose to publish more rock than rap! (But the mathematician in me knows she is right...dang it!)

Lastly, I had my students put this activity on chart paper. I have been in such a rush lately I haven't done this as much. It made me realize that having them put it on chart paper "makes it real" for the students and they invest more in the problem. Also, I get the opportunity to formatively assess at a glance by reviewing their work. The picture below is of a group of students explaining their findings from the Rock 'n' Rap activity.

IMP Cookies - You are What You Eat and Changing What You Eat

This is once again practice for the students to write and graph inequalities using the constraints given in the activity. The difference in these 2 activities if you have a less than or equal to and a greater than or equal to. This is the first time they have had the shading to go up on any of the graphs. Also, we are looking for a minimum instead of a maximum.

Systems of Inequalities is a difficult concept because of all the "math" involved. I love how this book just keeps giving the students multiple opportunities to write and work with constraints.

Graph for You Are What You Eat

P.S. - The answer to You Are What You Eat is 2.5 Fruit Nuts.

Graph for #1 on Changing What You Eat
Answer to #1 is 2.5 Fruit Nuts and 2.5 Crispies

Graph for #2 on Changing What You Eat

Answer to #2 is 1.5 Crispies and .75 Fruit Nuts for a total of 2.25 total ounces

IMP Cookies - Finding Linear Graphs

We kind of did a "fly-by" on this activity. It has the students create a table in order to graph a linear equation. Then they are asked to graph 5 more equations while looking for shortcuts. Lastly they are asked to discuss in details the steps they use to graph a linear equation and what special methods can be used. They are also prompted to tell how they decided on the numbers to use in the table.

I had to be out one day before this activity and we had already been discussing how to graph from slope-intercept form and how to graph using intercepts. I think that somewhere in this area of the curriculum it is time to explicitly teach these concepts. We have been working on them for a long while now in Overland Trail and in Cookies. Therefore, I gave them a "traditional" worksheet on graphing using the 2 methods. Therefore when my students did #2 in this activity we focused our discussions on which method would be easier to use in order to graph the line.

Graphing lines is such a fundamental part of algebra and it is so frustrating how the concepts seems to escape the majority of the students. If they go a day or 2 without graphing it is like they totally forget. One thing I like about IMP is that they graph over and over and over again. In Cookies we are using constraints to find the feasible region and answer questions about different situations (systems of inequalities). Sonya and I decided that while we are working through Cookies we are going to SIT on graphing until they get it...we hope.

IMP Cookies - Profitable Pictures...very important

This activity builds off of Picturing Pictures. Mrs. New and I decided to draw a graph for Picturing Pictures and make copies of it so the students would for sure have an accurate graph on which to do their "profit lines." It is so wonderful to "share a brain" with Sonya (Mrs. New) because she has already taught Cookies twice. We are needing to trim as much as possible because we started the curriculum late. Therefore she knows when there is an activity that you should not skip and I need to remember NOT TO SKIP PROFITABLE PICTURES.

This activity gives students the tools they need to answer the Cookies unit problem. The students should discover that the profit lines are parallel and that the higher the y-intercept the higher the profit. Therefore they need to extend their parallel lines until they find the highest line that clips a point in the feasible region. After trying to get my students to explore this activity over 2 different days I finally EXPLICITLY told them that the maximum profit is going to come from one of the corners (if you will) of the feasible region. Sonya and I  also EXPLICITLY told them that the methods we used to discover the feasible region for Profitable Pictures are the same as what they need to use to solve the Cookies problem.

Profit Lines...

When I first started the curriculum I was so diligent to read through the teacher's guide for almost every lesson. So far this semester I have not been able to do that! It is a long story... Anyway, I really value having someone (who just happens to share a planning period with me) to collaborate and discuss lessons with! It makes such a difference. Sonya is totally committed to our "new curriculum" and we have so many aha moments that we get to share with each other.

Cookies - A Simpler Cookie thru What's My Inequality

My personal life has been turned upside down. I enjoy blogging about our new curriculum. I look forward to being able to look back next year in order to help me when we go through a 2nd time. However, when I have to cut some things out because of a lack of time blogging is the first thing cut. The thing is...blogging just helps me to organize my thoughts and see where I have been and where I am going. I think I might could really be diagnosed with ADHD...I doubt that any of my colleagues would disagree. I am kind of joking but I really do think it is true. Forcing myself to focus and "verbalize" what I am doing with my classes helps me to reflect and think. So...here is a brief summary of what we have done so far in Cookies.

How Many of Each Kind? - The students brainstorm the number of dozens of iced and plain cookies that meet the 4 constraints that are mentioned.

A Simpler Cookie - This activity allows the students to only worry about prep time. They assume unlimited amounts of dough, icing and oven space. Then they are asked to try to find the most profit. We believed the max profit in this case was 150 plain and 0 iced.

Manipulating Inequalities - This activity allowed the students to "discover" the rules for solving inequalities. They also practice graphing inequalities in one variable. When going over #3 ask the students what would make the inequalities where you multiplied or divided by a negative "ok" again.

My Simplest Inequality - In Part I the students practice solving inequalities in one variable. We supplemented with an additional worksheet on solving and graphing one-variable inequalities. In Part II we investigated equivalent inequalities. Then in #3 we took inequalities and wrote them in simpler ways.

Simplifying Cookies - In this activity you take the Cookies Constraints and write them as simpler inequalities. We did not spend much time on this but I did write the simpler inequalities on chart paper.

Picturing Cookies - In this activity students take the constraints and use 2 colors to indicate ordered pairs that either work or do not work for the cookies constraints. They will hopefully recognize a "dividing line" that halves the graphs into correct solutions and incorrect solutions. I gave each group one constraint to test and had them put it on chart paper after they did it individually on graph paper.

Inequalities Stories - I gave this as a bonus opportunity...

A Hat of a Different Color - This POW was assigned when I had a sub. I think it would be a good idea to use hats or objects and act out the situation a few times in class before the students try to figure out the color of Carletta's hat. Her hat is blue. If Arturo couldn't tell then the 2 girls could NOT have both had red hats. One of them must have been blue. If Belicia had opened her eyes and seen a red hat on Carletta she would have known her hat was blue. However, since she passed Carletta must have been wearing a blue hat.

Healthy Animals - You create 3 inequalities - one for fat, one for protein and one for total amount. I did not have my class graph this activity.

**Picturing Cookies - Part II - We used this to "reteach" graphing using slope-intercept form. We used Desmos to show the graphs but I don't know if I should have waited on the technology... Anyway, it is extremely important to have each student to graph these correctly and identify the feasible region. They will need it to solve the unit problem. I did the graph on chart paper and left it hanging in the room.

What's My Inequality - I really liked this activity to help us tie the work in the unit to "traditional algebra problems." This is the first time the text mentions the differences in when to make the line solid and when to make the line dashed.