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.

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.