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 - 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 - Feasible Diets and Picturing Pictures

My students did Feasible Diets on this past Friday. The constraints that are needed for feasible diets were actually found during the Healthy Animals activity.

Today I used Picturing Pictures as a quiz but I added a twist to it. I allowed each student to ask me 3 questions (or get 3 hints). I actually wrote down their names and kept up with it. I encouraged the students to listen when others came up just in case they might learn from the questions/answers that other students have/get. I made the students use their "privacy fences" and did not allow them to help each other. I found that students were still struggling with graphing lines. The majority of the graphing we have been doing so far in Cookies have been easier to graph using intercepts. Once again I have found that students seem to understand the concept better since we have the context to tie it to.

I had a workshop today so I left "traditional" algebra problems where they practiced graphing using intercepts and graphing using slope-intercept form. I intend to give them a quiz on these 2 concepts this week.                                        

IMP Day 33 - Sublette's Cutoff or Who Will Make It? (5th period - Laced Travelers) - They don't need me!!!

Today I took the suggestion from Mr. Webb and Ms. Whitt and divided Sublette's Cutoff and Who Will Make It between the groups in my classes. In other words some groups did Sublette's Cutoff and some did Who Will Make It. The 2 activities were similar enough that I was able to introduce the activities pretty easily. They are asked to plot data for 3 different families and then draw 3 different lines of best fit. Then they are asked to answer questions and make estimations with the lines. I think that even I have a better understanding of why we do lines of best fit after teaching these last few activities. I know that I "knew" what it is for but I do not think that I have applied it within a context very many times. Over the course of this week my students will have used lines of best fit within at least 3 or 4 different contexts. The discussions about what the x and y intercepts were so meaningful and productive! We are really talking about slope and the y-intercepts without formally introducing the equation. It is so cool to see how the authors develop each of the concepts. I think it is hard to appreciate until you teach an entire unit.

Today after we finished graphing and "sharing out" the answers I asked the students to share with me the meaning for the x-intercept and the y-intercept. I had one student say, "The y-intercept is the starting point and the x-intercept is the ending point," and the BELL RUNG!!! UGH! I was very excited that she said that and I wanted to discuss it with the class...there is always tomorrow.

In 5th period today I did something different. I wrote the assignment on the board and referred the students to the assignment as they walked in the door. All I did was tell them to read over and it and put the "entry" into their notebooks. Dude! They got on it! I had one student figure out the answer to #1 (which deals with multiple constraints) by the time I had answered roll! AMAZING! As a whole they worked harder today than I have ever seen them work. Once the one student got the answer he walked around and helped other groups. I would ask them to explain how they got their answer and wouldn't give them candy (the great motivator) unless their explanations made sense. They don't need me! Well...not as much as I think they do. HAHA!

IMP Day 32 - Previous Travelers (and Shoelaces with my 5th pd)

Today my 3rd block finished Previous Travelers. The last two class days my 2nd block class was cut for different reasons. Therefore they are behind! Anyway, in Previous Travelers the students plot some data points and then draw the line of best fit. Then there are instructed to find the equation for the line and lastly they have to predict the amount of supplies they need for their Overland Trail families. I only gave each group one of the 3 supply items to graph. That made it where each item had at least 2 groups graphing the data. The neat thing about that was that the two groups could compare information AND when I had that one group that was behind everyone else because they wouldn't get on task in a timely manner I just had them to sit down and we still had another group that finished. I had the students write the rule for their supply on their chart paper and pointed the information out to the class. Then I let them use the rules to predict the supplies for their families. I was proud of myself for "shortening" the assignment by not making each group do all three of the supplies. We "divided and conquered" but I still think that they got enough experience with the activity to get a feel for using the line of best fit. I had some time to get my 3rd block to reflect on what they had learned so far about graphing. A couple of them really impressed me by saying something about how we took discrete data and learned how to create an equation using the line of best fit. I didn't even mention the fact that the data was discrete so they made me proud!

Another "victory" today was that I completely finished Shoelaces (from beginning to end) in my 5th period class which is 55 minutes. I used the tips of some experienced IMP teachers and did not wait for each group to get every part of each question correct. I allowed time for "productive struggle" but then had a student to share their findings on the board. I think I am more relaxed about not waiting for each group to "get it" because I am more familiar with the way the book "circles around" on most of the important topics. We will have another activity (or sometimes several more) that will reinforce what we covered today. Shoelaces uses constraints again and introduces the students to using algebraic expressions in the place of more complicated verbal explanations.

IMP Day 25 - Categorical Data on the Wagon Trian

This was a good activity for apply percent questions and logical reasoning. It was like putting a puzzle together. The students were using number of people as the percentage of people. I had one student who did an awesome job of using the idea of percent as per 100 and he took the 17/25 and instead of dividing the fraction to get the percentage he multiplied top and bottom by 4. I was excited about his prior knowledge because that is something we have not discussed this year. RETENTION STILL EXISTS!!  I HAVE SEEN PROOF!

I think this assignment could be assigned as a homework activity although we did it in class.

IMP Day 17 - Shoelaces and Diagonals Illuminated and Creating Families (5th period)

Today I handed out my rubric for the Haybaler POW and then gave (through questioning and getting really excited when someone got close) them the hint that two of the bales are 39 and 41. HINT HINT HINT! Some of my students still didn't take the hint!

Then we read the intro to Setting Out with Variables and did the Shoelaces activity. I really liked this activity! Jim had warned us in training that many of the students were going to forget to calculate for 2 shoes in each pair and he was right! Reading the teacher's guide is VERY IMPORTANT for this activity because it has you lead the students through assigning variables and coming up with an algebraic expression for the amount of shoelaces each family needs to buy. *Family folders are needed for this activity because they figure how much they need for their own family first.* Then we talked about substitution...  Having not taught out of this book before I keep being anxious to see when the topic is going to be revisited. I know that the topics spiral but I wish I had time to investigate and see how they deepen as they go. Ain't nobody got time for that!!  (not right now anyway)

The next activity we did was Diagonals Illuminated. This activity revisits In-Out tables and creating a rule.

My 5th period started Creating Families today. I used Jim's method of grouping them randomly using playing cards and then assigning the type of family by suits. They relocated and worked on their families in the "new" groups so that they were working with the same constraints. I liked this much better than the way I did it with my 2nd and 3rd blocks.

I have also taken some advice from Theodora Psitos (a trainer with It's About Time) and started posting the definitions from the unit on chart paper. Today I would pause every time I came to one of the vocabulary words and point to the definition for them to read to me. I hope it helps!

Popcorn reading seems to work even though they groan a little when we start using it. I also find myself getting impatient and just reading it myself...ALOT. I noticed when we were in training and I was in "student mode" it took me several reads for the info to sink in. I figure that they don't pay great attention to detail the first time through anyway so I speed read to hurry up the process. #trueconfessions    I do read slower on the most important parts. Wow...today I really rambled!

O yea!! We received the teaching kits from It's About Time last week for the PD. In the kit there were 2 books about the Oregon Trail and I read excerpts to my classes today. I didn't read long but it was fun. The book I read from has a collection of journals from the perspective a 15-year-old boy. I plan to read a little every day.

Day 9 of IMP - Creating Families - working with constraints

When I first looked at this activity I kind of thought it was just to do something a little different and have some cross-curricular work too. However, after helping the students work through the process of creating their families I appreciate more how this is intended to give a real-life context to working with inequalities. Making sense of words and phrases like at least, at most, between, less than, etc...

Sonya and I had created the charts for them to keep up with their family members easier and we also printed off a picture of a covered wagon that the students were able to decorate after they completed their work. This worked very well since some students finished much faster than others. However, I wish I had done a better job of making each group help each other after they finished created their own family. The group members who were responsible for the large families were (of course) always the last ones finished. I need to do a better job of reminding the group to help each other once they finish with the family that they were assigned.

I have also noticed that doing these types of problems make it so easy to talk about a real-life situation. For instance, before they left today I asked the students if anyone was interested in architecture or carpentry. I asked them if they were aware of the building codes that one must follow when building a new structure. I told them that they are constraints similar to the ones they had to follow today.

Above are some of the wagons where the students wrote the names and ages of their family members. Tomorrow I will be introducing the haybaler problem and I am a little concerned because I personally have not figured out the best way to approach the problem. Good thing it is a POW so I have some time to decide the best way to solve the problem (in my opinion). The students will probably figure out a much easier way on their own anyway!