Solving Equations Scavenger Hunt - Thanks Mr. Webb

It is great to have fellow algebra teachers that share their fun activities! Mr. Webb is another algebra teacher at Etowah High School and he made a scavenger hunt using QR codes in which the students had to solve equations in order to find the missing digits in the next room number. He bought cupcakes for the winners and we also awarded bonus points to the top 2 teams (everyone got a grade for participating). So during 5th period today we combined our  classes and created random teams of students.  Each team got a paper with a QR code that gave them an equation to solve in order to get the next room number.  They ran all over the school!

They had to get the room numbers in the correct sequence to win so they often returned to my checkpoint a little frustrated...

They do eventually get the hang of it though and they seemed to have alot of fun too! This was a great way to spend the last class of the day on the Friday before Spring Break! (And I got to eat a cupcake too!!)

Thank you again Mr. Webb for creating this activity and sharing with us!

Winners eating their cupcakes:)

IMP Alice - Stranger Pieces of Cake and Confusion Reigns

Today I first had my students do Stranger Pieces of Cake. I started off by reading over the into and giving them 5 minutes to think about number 1. When they were asked to put their findings on the ipads and project them (using airplay) I realized that they were in need of help. I did have one girl say that 2 to the 3/5 power would be like Alice's height increasing by 60 percent. I was excited that she used that description but I needed to lead them to a point where we could explain the problem using ideas we had already learned in Alice...so that we could develop the rule for fractional exponents.

So...I then asked them if they could figure out a problem where they either raised a power to a power or multiplied powers with the same base and the answer was 2^(3/5). Luckily, at least one group in each class eventually came up with 2^(1/5) * 2^(1/5) * 2^(1/5). We related this back to the activity where we developed the rule for exponents that are unit fractions which eventually got us to the idea that the 5th root of 2^3. Anyway...I had to do ALOT of leading through this activity but I really believe that the students have a better chance of remembering the rule for fractional exponents after they have had their "hands on" trying to figure it out themselves.

Confusion Reigns is a good activity that makes students just rethink through a few of the rules they have learned. I LOVE the first problem because so many students didn't pay any attention to the fact that they were adding 2 powers with the same base instead of multiplying. I hope this helps them to pay closer attention to the operation signs in the future.

A "case" for a problem-based curriculum with group work

As we have gone through Alice we have seen the need to just pull alot of practice problems for each of the rules. I still have students that approach these problems in a variety of ways. At this point there are some that have memorized the rules. I still have some who use expansion . I have explained to them that expansion is not always feasible but hopefully being able to expand problems correctly will help them to remember a rule.

Yesterday I spent the majority of class answering questions and going over examples from a couple of worksheets that I had left for them to work on when I had a sub. We have developed all the rules they needed throughout our Alice lessons but these problems had more "moving parts" for them to work through. After changing my teaching methods this year yesterday was such a "drag." I asked the students if they liked working through the packet instead of the Alice activities and some of them said yes. Today (while we were working through an Alice activity) I asked them whether they would prefer me just tell them the rule and give them practice problems or allow them to explore their way through a problem and help me to develop the rule. Once again I had some that preferred both ways. HOWEVER, when I asked them which way do you think might help them to remember the material 2 weeks from now not one student chose the "worksheet method." I think we will always have students that battle with you over having to think for themselves. There are many times that my students don't actually get ALL THE WAY to the rule or formula or method that they need to do the algebra. However, once they have had some time to "productively struggle" with a concept our discussion is so much more MEANINGFUL than when I just told the students the rule and had them practice problems using it.

One of my first major AHA! moments came when I attended an ACT Quality Core workshop in which Roy Dean was one of the presenters. I actually had to look back for some email communication between me and Roy in order to get his name. This was the first time that I remember sitting through a workshop (I hadn't attended AMSTI at this point) where MATH teachers modeled how to use strategic teaching strategies. He was very patient with me because I was often picking his brain about how he did things in his classroom instead of doing the actual activity assigned. When I found the emails I just wanted to share them because I learned so much from his answer. Below is the email I sent to him:

Roy, 

Thanks for the information and your willingness to share.  So...when I
came back to school and started telling other teachers about the
movement to teach using the methods we discussed I often get the same
question.  Teachers wonder if the group work translates into higher
test scores since the students take the tests (standardized - like
EOC, ACT, etc...) by themselves.  Concerns are also expressed about
having students in our classrooms to learn primarily through the
methods we discussed when most college classrooms are going to be
lecture based.  I didn't want to ask these questions at the workshop
because I feel like they seem argumentative.  However, I wondered if
you have any direction or advice on how I might answer these
questions.  I certainly share their concerns and understand the
questions. 

I have pulled out the "On Course for Success" book and am trying to
skim it to get answers...but I wondered if you could give me some
direction. 

Thanks!
Teri Owens
Etowah High School
Attalla, Alabama

And here is his reply...

Hi Teri, 

Sorry it's taken a couple days to get back to you. The severe weather yesterday had me "enjoying" the Bham airport for most of yesterday.
In answer to your question, I know the "on Course for Success" book has some data that you can use.  I would also think if you Googled reform math scores, group work and test scores, and such if you would find some data on your questions. 

As far as my personal experience, at my school our scores rose 6 points (a statistically significant rise for Colorado testing) the first year and 2 to 3 points over the next 8 when we switched out curriculum to a group/problem solving approach to mathematics. 

I also had a summer school class (not exactly your star students) that I taught with the reform/group/prob solve math for 6 weeks in the summer instead of the usual fraction ws, decimal ws, etc. The students seemed to enjoy the class more.  Of the 24 students I had that actually attended thes ummer classes, 23 improved the test scores the next spring and 8 moved from unsatisfactory to proficient (we have unsatisfactory, partially proficient,proficient, and advanced) and all but one improved their scores a substantial amount. 

As far as students to college, students that returned from college to chat mentioned that the college classes were different (having more lecture) but they weren't hard.  It seemed to me that since they knew the concepts and not algorithms, they could adjust. 

Sorry I don't have any hard data for you.  I at least hope this helps some.Thank you for your hard work during the training. Have a great rest of the year. 

Cheers,Roy

Going from the "Context" to the "Abstract"

I have been mulling over the advice of a veteran IMP teacher (Michael Reitemeyer). He mentioned in an email how he tries to use the context of the lessons to introduce topics but then moves as quickly as possible to the abstract. Sonya and I were discussing today how we would like to develop around 8-10 "homework/classwork" problems that reiterate the concept of the day/lesson. This is not appropriate for every lesson but there are several that it would work well with. For instance, each time we develop/investigate a new exponent law in Alice we could assign 8-10 practice problems. There is a point in Overland Trail where I wish I had given them more problems in which they practiced graphing lines without a context.  The problems they will see on standardized tests won't always have a story to go with them.

Now...having said all of the above...I have been practicing solving equations with my 5th period.  We have been working through a packet together of different types of equations. Today I once again heard some grumblings. I told them that they didn't realize how much they liked the textbook we were using (IMP Meaningful Math Algebra). They were talking about LOVING the textbook. They enjoy the math so much more when there is a story/context to help make sense of it all. I have especially found this to be true with my students who struggle with the math.

IMP Alice Days 9-10 - Continuing the Pattern and A Half Ounce of Cake

The last 2 days we finished some discussion of Having Your Cake and Drinking Too and then moved forward. Continuing the Pattern develops the rule for negative exponents and A Half Ounce of Cake starts the development of fractional powers.

Today I handed back the worksheets I left for them to do on Tuesday.  The worksheets were an exploration on exponents.  I gave them feedback on the ones they missed and told them to make corrections. I started class with a warm up which reviewed the rules which will be quizzed tomorrow.  I had each group project their answers using the Ipads. (Airplay allows up to 10 to be projected simultaneously.) I went over each problem by having students to defend their answers. It was great. I have really been enjoying the Ipads!

Some students used expansion...some used the rules...and some used the Alice context... fun stuff!

IMP Alice Day 8 - Having Your Cake and Drinking Too

When I saw that the teacher's guide allowed 60 minutes for this activity I thought there was no way. I thought we would get through it quicker...haha! We didn't even finish and we had OVER an hour. This activity helps the students to explore and HOPEFULLY discover the rule for dividing powers with the same base. I love it! It is really revealing the lack of number sense that my students have. The questions are not really that difficult...that is why I thought we would breeze right through.

I have allowed myself to "sit" on this activity and not rush it because I really believe it might help to cement the concept. I am going to be absent tomorrow from my classes and I have some good exploration worksheets on exponents to leave for my students. However, it has reminded me how blessed we are to have found a book that combines the exploration with the context. Someone had a wonderful imagination!

**Update 3/7/15** Original post was made 3/16/15
This year when I taught this lesson I took the time to help the students to explore #2. I did not let anyone blurt out an answer when we first started looking at it. I have been using the random integer generator on the TI-83 (which is awesome but I just now figured out how to do it!!) to call on students. In my classes I just happened to call on students who were unsure of what to do so I just asked them to start by making a guess. I told them to tell me a number of ounces of cake and a number of ounces of beverage and then I showed them how to "test" their answer to see if this would give an answer where Alice's height was multiplied by 8. Then I randomly called on more students. The first 3 students guessed more beverage than cake (ugh!) so I asked the next student to make a "conjecture" on whether or not more beverage than cake would EVER allow Alice's height to be multiplied by 8. My 2nd class arrived at the correct number of ounces much faster than my 1st one did but I feel that modeling to the students how to "guess and check" was valuable. I am always telling them not to ever erase their guesses. I want them to learn to look back over the ideas that didn't work in order to help them to identify new theories that might work!! Anyway...the day I taught this lesson this year I felt really good about what we had accomplished:)

Show me how to teach it differently!

I remember walking into our Instructional Partner's office a few years ago because of my frustration with my algebra classes. My failure rate was climbing and I was teaching algebra just like I always had - and it had worked well in the past! I remember asking her if she knew of a different way to teach algebra. That year I taught the "repeater" Algebra IA class in the Spring and I chose to teach the majority of the class using the sample units we had received at the ACT Quality Core workshop. That was the first time that I had seen a math teacher model some strategic teaching strategies that I had seen in some ARI workshops. I was often frustrated thinking that the strategies worked for other content areas but the majority of them just didn't fit in the math classroom. That Summer I attended my first AMSTI training and decided (although I wasn't completely sold out on all the activities yet) to make the commitment to put my students in groups and keep them in groups for the entire school year. I hated it at first but once I adjusted I doubt that I ever go back!

That next school year I looked for and used several station activities and review activities. I also came across some inquiry/discovery-based activities that I loved. I was constantly scouring the Internet for resources and making copies for my classes. My student engagement improved in that school year. The following Summer I attended year 2 of AMSTI training and the light bulbs started going off. I realized that the "AMSTI activities" were the type of activities that I had been looking for on the Internet. I had always thought of them as activities to do AFTER I had taught the concepts and I didn't think I had time for that. A phrase that one of the trainers kept saying to me was , "Quit thinking like a math teacher. Your students wouldn't do that!"

Through these experiences I have come to realize that for the majority of my teaching career I have taught all my students as if they were all good math students who were going to pursue educational goals that involved upper level math classes. I do have a few students who fit that description...but the majority of them don't. However, I was teaching them all the way I preferred to be taught...but the majority of my students aren't like me. They don't love math and math does not come easy to them.

Fast forward a few months to my implementation of the IMP Meaningful Math Algebra curriculum...almost everything I teach is tied to a context. I am finding that even my students who are not great at math will interact more with the activities because the context makes the problems more accessible to them. They might not be able to just see a "naked math" problem and figure it out but when they have a storyline surrounding it they have a context that helps them to make sense of the math. So now I not only have a curriculum that is discovery/inquiry based- I have a curriculum that also presents the algebra concepts within a context. It may have taken me a few years to find the resources/ways to teach it differently, but I have definitely found it. Now...I just have to learn how to teach using them. I am working on that!

I appreciate all of the people who have helped me to transform my teaching. I don't want to try to name them for fear of leaving someone out.

IMP Alice Days 6-7 - Piece After Piece, Many Meals for Alice, and In Search of the Law

For Many Meals for Alice and In Search of the Law I really put my Ipads to use. I am learning better ways to implement using the Ipads. I gave each group a few minutes to brainstorm on each problem and then told them that at the end of the allotted time they had to put SOMETHING on the Ipad (using the Educreations app). The Airserver app allows you to project up to 10 Ipads at a time so I have an Ipad in each group (sometimes more). I had told my students that if you have an Ipad you HAVE to be mirroring the screen. I can see all the screens on my computer. This has been a tremendous help for keeping my students from using the Ipads for purposes other than the math! When I turn on my projector where the students can see everyone else's work it is interesting. I am still amazed how my students will approach problems so differently when I don't stand at the board and show them an example of what to do (which causes them to all try to do it my way). We have an easy way to compare and critique the work without it being totally obvious whose group it came from. Then they are able to edit their work when it is needed. Formative assessment on the fly!!

I had a student volunteer to "teach" - she was great!

The context given in All About Alice provides such a neat alternative to simply just giving the students the rules of exponents. Piece after Piece establishes the rule for multiplying powers with the same base. Some of my students wanted to ADD 2^3 and 2^5 so to help them I told them to remember that we are always looking at what we are MULTIPLYING Alice's height by. I am interested to see how other IMP teachers handle this...

Many Meals for Alice establishes the rule for raising a power to a power. The original power is for the number of ounces of cake she eats at each meal - #1 is 3 ounces of base 2 cake so it is actually 2^3. Then the students figure out what is happening to her height after certain numbers of meals. In the end they have hopefully developed the idea that (2^3)^4 means that Alice ate 4 meals where she had 3 ounces of base 2 cake. It takes a few times of looking at it to start making the connection. I LOVE #3.

In Search of the Law actually has the students to explore exponent rules that I have not usually put much thought into. It took me a minute or two of exploring myself - I actually had to tell my students to start working on #1 while I went to read the teacher's guide! When I glanced over the activity the day before I taught it I didn't realize it was going to throw me a curve. Anyway...this activity is actually not difficult AT ALL. It gives the students several scenarios to just work with exponents.

IMP Overland Trail - solving equations using mystery bags - 5th period

I really enjoyed using Mystery Bags and Scrambled Equations. However, when I gave my students some "naked math" one-step equations they have really struggled. I have one class that is taking all year to work through the 1st half of algebra. They were identified to possibly have a struggle in algebra. I have several incredibly bright students in there. I have often thought - why was this student put in this class...   I had an eye-opening experience today. There were a few of my students who excel when we work in our IMP books who really struggled with today's work. I mean...the ones who are always "piping up" to answer the questions and seem to just "get" the big picture. One student even started the age-old, "When am I ever going to use this? This is stupid!" It took hearing him say that for me to realize that I have not heard that near as often since we started using our IMP Meaningful Math Algebra I books.

Ok...I am rambling a bit in this post so I am going to resort to using bullet points to make sure I get the main ideas that I am trying to convey:

• If I am teaching students who are struggling math students I am going to have to supplement material to help them learn to solve 1 and 2 step equations ...for sure. When integers were thrown in (after mystery bags) it has totally blown their minds. I think I am going to go "old-school" tomorrow and just teach the process in steps.
• After today I have a greater appreciation for how the context and story lines in the books give the algebra more meaning. I think it helps all students but I think it makes an even bigger difference in struggling students who don't usually do well in math.

IMP Alice Day 5 - Here Goes Nothing

Today we spent the first part of class taking our 3rd version of the systems of equations quiz. I had 9 students in each class to improve so I think it was worth it. Overall I am disappointed because I still have many students who just still don't get it. UGH!

Here Goes Nothing is an activity that the students can do pretty quickly. It helps the students to further analyze the nature of 2 to the zero power. I am glad that we did the Graphing Alice activity on chart paper so we had them hanging in the room to help us. I also wish that I had at least done one graph on chart paper of A Wonderland Lost - just for discussion purposes. The teacher's guide really adds to this activity so make sure you read through it before you teach it.

Since we did Here Goes Nothing quicker than I expected I had my students to graph #5 from A New Kind of Cake on chart paper. This is a comparison of y=2^x and y=x^2 and I really like it. We will discuss it tomorrow.

IMP Alice Day 4 - A Wonderland Lost and Scrambling Equations (Overland Trail)

We started class today with another warmup on solving systems of equations. My students will have their 3rd and final opportunity to retake the quiz in order to improve their grade for this 9 weeks.

Once we went over the warmup I assigned A Wonderland Lost. We read it together and discussed it some first. I love the question the teacher's guide advises you to ask about how long it will take the rain forest to be completely gone.  Several students think it is 10 years. I did an example at the board using a beginning amount of 100 and worked through decreasing by 10 percent per year. I led them to the "shortcut" of taking 90% of the previous (or beginning) amount each time. I am learning to be more comfortable writing exponential equations within the contexts we've used so far. I don't know if my students are going to retain these lessons but I'm quite sure I will. This was a time when the teacher's guide gives you the rule.  I had to stop and think about it to be able to explain it. AND relating back to the walk thru we did for the Alice problem certainly made it easier for me to understand and explain! After today's lesson I took a few minutes to give notes on the differences between exponential, quadratic, and linear functions. I also made sure they heard me use the vocabulary terms of exponential growth and decay.

My 5th period finished the Scrambling Equations activity from Overland Trail. I invested a little more time in this activity this time through and I think it went better.  I found many mistakes in the students' work and tried to clarify some misconceptions. I had my students write their "complicated equations" on notecards. I also provided a problem I made up and wrote on the board for students to use if they didn't feel certain of their problem. After they worked each other's problems I asked them to write down a brief description of the process they used to solve the equation. They used phrases like "canceling out" and "did the opposite." We also emphasized that looking on the side with the variable tells you what to do. After the activity I passed out a worksheet for them to practice 1-step equations. We didn't have much time to work on them today. We will build up in difficulty and have a quiz by the end of the week.

Quiz retakes and Mystery Bags

Today my Algebra IB students retook their quiz on systems of equations. The majority of my students improved their grade. This semester I have taken two concepts so far and worked hard to try to ensure that my students have mastered them. (This statement actually has me shaking my head...I promise I hope that all my students master all of the topics...but in my experience with struggling math students that doesn't happen.) One of them was graphing lines. We gave them a quiz and then gave the students individual feedback on what they missed. Then we gave a retake for all students who did not ace the first quiz. We did the same thing with systems of equations (substitution and elimination). This topic seems to really get the better of my students every year.

I have been "transforming my teaching" for a couple of years now. For years I taught a topic and then tested it and moved on. There would be some times that I could tell that the students were struggling with a topic and I would reteach it or spend extra days practicing. However, once I taught it and tested I moved on. Well...I still think that you can't sit a topic until every student gets it. I wish I could. However, we would never cover the required objectives if we do that. Sometimes my students either don't care enough to get it or do not come to school enough to get it. So...I have tried to do a better job of giving smaller quizzes on a few topics and then giving the students individual feedback. After the feedback they are given at least one opportunity to retake the quiz (possibly more). I wish we had the time and opportunity to do this with every topic. I truly believe if my students would pay attention in class, do all of their work, and ask questions when they don't understand so that I can help them at their points of confusion they would ALL PASS. However, I do not teach in a perfect world. I do not have perfect students who come to class prepared and ready to learn each day. Also...my students do not have a perfect teacher. So...until these things change I am going to continue to take a few topics each 9 weeks and give students individual feedback with the opportunity to retake. If any of you reading this blog (all 2 of you - haha!) have any ideas that might help this process to be less painful please shoot me an email at towens@attalla.k12.al.us. I thought I would ask...just in case...

Today my 5th period continued working on the 2nd mystery bags activity in Overland Trail. This is definitely a place where we will need to pull extra practice on solving equations. I think that if your students come to your algebra class and they can already solve one and two-step equations successfully then you won't have to supplement as much. However, many of my students still struggle solving basic equations. Therefore, after the More Scrambled Equations and Mystery Bags activity I think I am going to do a quick quiz to assess their fluency on one and two-step equations. The Mystery Bags activities have actually taken them through working on equations with variables on both sides. However, students will need more practice on equations that involve subtraction (integer coefficients). I really believe that referring back to the Mystery Bags context will help. We will see!

IMP Alice Day 3 - Logic POW and Mystery Bags revisited

Today my Algebra IB students took a short quiz on systems of equations. The grades were not all that great but Ms. Whitt (my incredible coteacher) and I did our best to talk to each of the students about what they missed. The students will be given the opportunity to retake the quiz within the next week.

Ms. Whitt tutoring a student one-on-one on solving systems of equations.

 After the students turned in the quiz they were instructed to start working on the Logic POW. This is a very different POW than any that we have done before. It gives the students several pairs of statements and asks them to figure out if there is a logical conclusion that can be made. Some of the pairs do not have logical conclusions. I think this is a great way to introduce them to the type of thinking and reasoning that is required in geometry!

My 5th period Algebra IA class started Mystery Bags this week. Today we did the More Mystery Bags activity. I appreciated the context the first time I went through this last semester. I went back and read the blog I wrote after that day. The first time through most of my students seemed to have a pretty firm grasp of solving basic equations. However, my 5th period class needed more prompting. When the students had a problem like 5M +3 = 2M + 15 the context gives me such an easy way to explain why you would first want to take 2 Ms (or mystery bags) from each side of the equation (balance). In the mystery bag activity the constants are referred to as the weights that are on the balances. Therefore it once again makes logical sense that the next step is to take 3 ounces of weight from each side. We want to get mystery bags only on one side and weights only on the other side. Almost every student then understands to divide the amount of weight by the number of bags. (They are told that each mystery bag has the same amount of weight in it.)

Tomorrow they will be thrown some problems with some negative numbers. I explained to them today that the concept of getting all the mystery bags (variables) on one side and all the constants (weights) on the other will still work.

IMP Alice Day 2 - Graphing Alice - raising to the zero power

I remember Sonya New telling me how much she loved Alice because it gives a context for any number raised to the zero power equaling one. Now I have taught it with the Alice context and it is so exciting! For years I have just told my students to memorize it as a fact. Today I got to use the Alice context... Just in case you might be reading this and you don't know what I'm talking about let me share.

As I'm sure you know Alice (in Alice in Wonderland) shrinks when she drinks the beverage and grows when she eats the cake. The first activity in the unit tells the students to imagine that Alice's height doubles for every ounce of cake she eats and is cut in half for every ounce of beverage she drinks. In graphing Alice they are asked to graph the two situations for 1-6 ounces. For the cake situaion you are graphing y=2^x and the beverage situation is y=.5^x. The outputs (y values) are actually giving you what you will be multiplying Alice's original height by. The  x values represent the number of ounces eaten or drank. The teacher's guide advises you to talk about what happens when she eats zero ounces. This would mean you are raising 2 to the zero power. It was so awesome to have the context to explain that when she eats zero ounces of cake her multiplier (y value) is 1 because multiplying by 1 doesn't change Alice's height!! My students can't appreciate how exciting it is to be able to explain the why. I told them today that I have never been able to explain why anything to the zero power equals one and I just got blank stares. Haha! Some of you may be looking at the computer screen like...

It doesn't matter. I know I am a math nerd but today's lesson made me HAPPY! And I felt the need to write about it even though I am sure it is hard to follow.

IMP All About Alice Day 1 - Alice in Wonderland

We have had multiple delays and missed school days in the past few weeks. Therefore I have decided to skip the last portion of the Cookies unit and start Alice. I'm going to try to make up for it by using systems of equations problems as my warm-ups for the next several days.

Today we read the intro and then watched a film clip from the movie. Thank you to Lori White for providing the film clip this past Summer at AMSTI training.  After giving the students about 10 minutes to work on the 4 questions in the Alice in Wonderland activity I had them to write their answers on 2' by 2' marker boards. It has been a while since we have used them.  I like placing all of their boards at the front of the class and comparing/critiquing the answers with the class.  I rolled my 8-sided dice and called on a group to explain the answer to each question. After the explanation we compared the answers on all the other boards. It was interesting to see the different ways the students worded their answers. For the first part of #1 some students said Alice height doubled twice, some said multiplied by 4, and some said quadrupled. A common mistake made for the 2nd part of #1 was students thought they should multiply by 10 for 5 ounces. Some students did say that Alice's height was doubled 5 times. I did have one student to say her height was multiplied by 32 when she ate 5 ounces. He is such a good math thinker. It was sad to me that none of the groups created a table to answer the questions. I showed them how we could use a table to help us to recognize the patterns which helped us to write the rules.