IMP Overland Trail - Water Conservation and stopping to find balance...

My 5th period class is Algebra IA year-round. They should be covering Overland Trail and Cookies before the end of the year. I just realized that they are at a point of Overland Trail where I had to skip some of the activities with my Algebra IA Fall Block classes. I was determined to finish Overland Trail by Christmas in order to start Cookies in January.

The Water Conservation activity is another good one to put on chart paper. Most students will generate a chart in order to answer questions 1 and 2. However, I had some students who graphed the lines first and then tried to go back and answer how many gallons each family had left after a given number of days. This did give us the opportunity to discuss how it is easier to be exact when you use a table instead of the graph due to the estimation that is required when looking at the graphs.

I loved that the group that generated the work shown above included the table and the graph along with their answers to the questions. One of the questions that this group did not address is how to find how much water each family has after x amount of days. This forces them to think about the starting point and rate of change. They know to muliply the amount of water used per day times the number of days but then they have to consider that they are trying to give an expression for the amount of water LEFT not the amount of water consumed.

After covering this lesson I pulled some practice problems where graphs were given and we were asked to find equations. I also talked about the slope formula and had them find the slope given 2 points. I know that I need to take time to stop and explicitly teach topics that we have covered but they may not know the "math vocabulary" so I am going to list them here:

• graphing from slope-intercept form
• writing the equation of a line when given a graph
• finding the slope of a line given 2 points
• finding the slope of a line given a graph

Our students have done so much more with this curriculum and creating graphs given real-world circumstances. However, when they take the end-of-course exam and the ACT they are going to need to be familiar with how to do these problems when there is NOT a context. After having taught through this unit (Overland Trail) the first time I realized that there are times I needed to stop and teach the students what types of "traditional algebra" problems they might see from the content we have been covering. The problem was that having never been through the curriculum before I did not want to "steal the thunder" of a lesson that we were going to cover in the future. I feel that to teach using the "exploration/inquiry/problem-based" method you have to also find a balance with showing students what the "traditional/worksheet/drill and practice" version of the algebra looks like. I KNOW that I can do a better job of finding a balance between the two. 

IMP Cookies - Solving Systems of Equations (Only One Variable and The Classic Way to Get the Point)

I love the way the Cookies unit starts with solving systems of inequalities and then works into solving systems of equations. In the Cookies and Inequalities portion of the unit the students are introduced to "real-world" problems where systems of inequalities can be used for solving them. The students practice writing constraints and then graphing the inequalities that go with them. In the real world problems often have more than one possible answer - much like is true of the feasible regions you get when graphing systems of inequalities.

Then, as you try to solve the Cookies unit problem or other problems in the unit involving minimums and maximums you discover that you need to find the "corners" where the lines intersect. We had our students just estimate using their graphs in order to find the points of intersection. However, it is neat to be able to introduce the topic of solving systems of equations by talking about how we really need to be more precise when we find the "corners" of the graph. For a couple of days this week our classes worked on an additional activity that helped them to explore the elimination method.

Today I first had my students do the Only One Variable activity. We had alot of fun with it because we integrated some technology too. Each group had an Ipad and they were projecting their screens using air server/airplay. I gave the group 3 minutes to work on the problem and then after the time was up they had to put something on their screen (they were using the Educreations app). It is so cool to be able to see several samples of student work on the board at the same time. We can critique the work without the class knowing whose work it is. We are able to celebrate different approaches for solving the problem. Also, we are able to talk about common mistakes. They love when they are working on a problem and I point out their work. I can say, "Hey, this person is really moving in the right direction."

Each group's problems projected simultaneously! Cool stuff!

Next we worked on The Classic Way to Get the Point which introduces the substitution method. Mrs. New and I had looked through the rest of this unit and we saw so many activities that give the students the opportunity to practice solving systems again and again. I used to teach these two methods of solving systems of equations separately. Then after covering each method I gave them some "mixed practice" where they had to choose the method to use.  However, I now like the idea of giving them some time to practice a few of each type and then let them decide which method to use as we go. The ACT and other tests are not going to tell them which method to use so they need to learn how to choose which method works better for each type of problem. There are several activities in this portion of the Cookies unit that give them a few more systems of equations to practice solving so hopefully by the time we revisit the topic several times my students will retain the information.

IMP Cookies unit problem with a "New" twist

The unit problem in IMP Cookies has to do with finding the maximum profit for a bakery when they meet constraints pertaining to oven space, preparation time, amount of dough and amount of icing. Throughout the unit the students do activities that give them all the information and tools they need in order to answer the unit problem. My often-mentioned teaching buddy Mrs. New has done a couple of really cool things with this unit problem. She is having the students to present their findings to a panel of administrators, our instructional partner and another math teacher. They are going to have to tell them what amount of iced and plain cookies will give them the highest profit and prove to them why.

The BEST part is the idea that Sonya had of telling the students that they could change ONE and ONLY ONE constraint. She has told the students to make a proposal to the panel of which item they should increase. Should they buy another stove, hire another person, or buy more dough or icing? There are some constraints that even if their limits are increased they will not increase the profit. I LOVED this idea. I am doing the "lazy version" of Sonya's activity and just having the groups present to the class. I have tried to encourage the groups to be creative as to why they chose the constraint to increase. I told them they could make up a story to go along with it in order to convince us that it is the best thing to do. However, they are also supposed to have a new graph that reflects the changes in the constraint. I had students to realize today as they were preparing that they chose to increase an item that did not improve their profit. Their feasible region was still the same.

I love teaching with creative people! Sonya is ALWAYS coming up with really cool ideas and activities. I will update this post after her groups have done their presentations...I'm sure she will have some awesome things to share.

UPDATE after presentations:
Things we like:

• We love the problem...especially having the students change one constraint and then explain if it improves the profit!
• Mrs. New loved having the students present in front of a panel
• Mrs. New's students loved that they went in a separate room to present...no other students were present during their presentations
• Mrs. New required them to have a digital presentation - most used Prezi or Powerpoint

Improvements:

• The students didn't seem to fully understand what we wanted from them. We need to somehow do a better job of explaining the purpose.
• To address the above bullet...Ms. Whitt mentioned she wished we could model a presentation for the students...maybe use the Picturing Pictures activity to model a presentation, then let them practice presentations in the classroom using Rock 'n' Rap
• Remind the students to explain the problem like nobody has a clue what they are talking about...give some background info...

The students presented in this small "conference" room

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.