Monday, March 18, 2013

Functions and Algebra I Series - Post 2

Working with the algeblocks again to solve equations hopefully facilitated an effective way of helping students articulate the mathematical procedures they are doing when  solving equations. Students in grade 5 on up should have these kinds of experiences. I think you agree that if you don’t understand the fact that the product of any two factors is a rectangle can cause angst in an algebra course when working with binomials. Obviously, given a trinomial in which like terms have already been calculated is a lot more difficult then when all the terms are nicely represented in the rectangle. 

Factoring a variety of trinomials is crucial in helping all students understand the differences the signs of the trinomials represent. For instance if all the signs are positive then the factors are all positive,  x2 + 4x + 4 or (x + 2)(x+2) for example. Or if the first operator is negative and the second positive then the factors have two negatives  x2 – 4x + 4 or (x – 2)(x - 2). But the tricky part is when you have a trinomial where you have two negatives such as x2 – x – 2 which indicates like terms have been computed from the binomial multiplication (x-2)(x+1) . Not easy to visualize unless you are using manipulatives.

I enjoyed the excitement you displayed with the CBRs when modeling graphs. I was impressed that many of you made up more and more challenging graphs to model. This engenders the spirit of inquiry that we are trying to foster with our students as well as helping to visualize what distance graphs actually represent.

Two hands on activities hopefully broke up the paper and pencil work we have to do as well. I realize how hard some of us found the toothpick problem and how important it is to hear from everyone as each time one of you presented the way you solved the problem it helped others who were stuck in their own thinking and could not change that vision in their head or the values in the tables.

The inclusion of functions and function notation is a critical component of algebra and algebraic reasoning. So much so it permeates the Common Core. The idea of function machines is so appropriate in the lower grades that the transition to the notation in algebra will be seamless. We did some work with functions but will solidify it next weekend. I felt that if I pushed up to compositions of functions Saturday,  it would have been overload so we will begin with functions at our next session.

I appreciate the emails some of you send asking for clarifications…always happy to oblige. Your homework assignment is very clear and Katie is sending the blogging info for those of you who still need it.

In terms of giving the homework on Sat or Sun, I need to make sure the homework reflects what we did in class so I do need the extra day to make it appropriate. On Sun, I am so exhausted and like you work all week and need some family time I have to give myself permission to get the homework to you on Monday. I hope you understand.

Another exhilarating weekend for me.


  1. I am very excited that I am teaching linear functions and slope intercept form to my advanced 7th graders right now. This course is exactly what I need since it is the first year I am teaching the content at this level. Thankfully, my own understanding of the math is growing steadily every week. Many of my students are making sense of it all, but quite a few are struggling with solving multi-step equations or manipulating equations into y=mx+b form. It is time for the cutting string activity as well as the round robin equation solving. We are asking a lot of these children, but at times they are working so hard to understand, you could hear a pin drop in the classroom. After looking at explicit and recursive patterns in several tables, I really want to get started on investigation activities; I can't wait to see what they do, and listen to what they say.

  2. Another exhausting weekend for sure, but also another weekend well spent. The algeblocks are becoming easier to work with the more time I spend manipulating them. Factoring with the blocks is still an area I need to work on, but I can see that the more time I spend with them the clearer the picture gets. We have just finished factoring polynomials and I would love to have the blocks to use in my class right now.
    The CRP activity was great. Having to think about all of the different ways we could mirror the graphs, and then putting or theories to the test was awesome. I can really see middle school students loving this activity.
    Then there was the toothpick activity. Talk about feeling dumb. I was so lost at one point I wanted to cry. My brain just couldn’t do it. I sat there feeling useless. It really gave me perspective on how my students feel at times. I know I am not dumb, and I know that I can solve many types of problems, but without the help of my table mates, I might have just given up. I am still struggling as to why I could not see the patterns in the more complex problems. I would love to say that I was too hungry to think at that point, but that would be an excuse. Which standard is that,…persevere in problem solving? Well I think I put that one to the test this weekend.
    Functions are the first real concept that we “math talk “ over in my classroom each year. I am glad to know that the language is changing in the lower grades and hopefully our early year discussions regarding functions will take on a whole new level in the coming years. With that said, I look forward to extending my knowledge about functions next time. Dawn Tomasini

  3. I was defintely challenged in class last weekend. Functions have always been confusing for me and I'm glad to get the chance to work with them in different ways. Just like some of my classmates, I also felt frustrated with the toothpick activity. I could not see the patterns and it was hard to stay motivated to continue working. I kept trying to start over with fresh eyes, but often got stuck in the same places. However, I felt downright triumphant when we finally figured out the equations. It reminded me to allow students to struggle, persevere, and ultimately feel accomplished in their own learning. I also really enjoyed the Round Robin and have already thought of so many applications for it.

    Laura Shea

  4. I absolutely loved working with the algeblocks again this past weekend. I have worked with the flat, magnetic ones in class before, and I have had the students draw the algebra tiles, but I have not worked with the three-dimensional ones. I am looking forward to being able to get a class set and show students the differences in the polynomials. I think that getting their hands on them and really seeing for themselves that the different monomials represent different shapes, that they will really grasp the inability to combine them when adding and subtracting polynomials so much better.
    I also thoroughly enjoyed using the motion sensors in class. I think more than the algebra students, the pre-algebra students will love using these. I also think that it will be most beneficial for the lower students, so that they create that memory of actually experiencing making those graphs. That re-creation of the graphs and situations will hopefully aid them to remember how to interpret graphs when it gets time to take the MCAS.

  5. This past weekend was another example of time well spent in the classroom. I can’t say I’m a natural when it comes to the Algeblocks. My “ah ha” moments seem to come few and far between. Learning new material can be frustrating but I have to say my knowledge level has improved so much within this last year. I am using the material I’ve learned and presenting it in the classroom with students. It is a great feeling to pass the information on and see the excitement on the students’ faces when they are as engaged in the material as I am.

    On a side note, I really like the conjecture boards. I’ve never seen anything like that in a classroom and I think it’s a great tool to encourage students to explain why they believe something to be true. I’m excited to try that in the classroom.

    I really liked the session we had on functions. It was interesting to be able to determine from a diagram whether or not it represented a function. It was nice to see the domain and ranges as visual diagrams and not just listed out. Understanding how to set up a domain and range is helpful when constructing a function.

    The motion detector activity was great. Figuring out how the data collected was directly related to the outcome of the graph was fun. I also thought it was different than what we usually work with because we were the main variable that was influencing the graph. It was nice to see our group as well as other groups altering their techniques to achieve the desired graph. It’s not too often in math class that you get to do that.

    Commenting on the toothpicks project will just give me another headache. I’m just kidding. It was a great team building exercise and also taught us to be specific when determining how an object grows from one stage to another. I still struggle with creating the equations from the established data tables. I suppose that’s why I’m taking this class.

  6. Last weekend was definitely time well spent and I was certainly challenged by the problems and activities presented. I realized that modeling is a way for me to access the problems, but that sometimes a 2D model was more helpful than the 3D (string problem).
    After last week's homework, I was more skilled in using the Algeblocks but back in class I was stretched to progress to using the blocks past modeling and simplifying expressions without additional exemplars.
    As others have mentioned, it is helpful to realize that when I struggle I can better comprehend how my students feel when they struggle. My students and I both benefit from multiple practice opportunities on important concepts; so I welcome varied and extensive problems presented!

  7. I've enjoyed reading the posts so far! I join with some of you with the frustration of trying to figure something out and if I was 11, how that frustration would manifest. Being a student again only makes me a better teacher.
    I liked working with the Algeblocks so much I took some home to practice! The weekend course is an intense amount of time so this extra practice with help me. I showed them to a six year old who immediately starting playing with them.

    I've been using conjecture boards with my math classes and it is a great indicator of where they are at and of their misconceptions. It's interesting to note that if a student remembers a topic from last year's math, he/she has a story about a tactile experience associated with it. Perhaps candy was introduced or they had some other experience. It does prove true that the more senses /experiences are linked to the topic, the more the student will retain. One of the differences between adults and children is that our prior knowledge is so much more vast. We can make generalizations that they can't because we have so many more experiences to draw upon.

    So, I do value the hands-on experiences, especially the motion sensor. I liked that it was immediately related to a symbolic representation.
    Looking forward to learning more!

  8. I really love the Algeblocks. I find myself wishing I had them add the beginning of the school year when I taught solving equations. Even now, at this point when I still have student struggling with solving I wish I had them as a resource. I do not teach factoring in my grade level yet so factoring with Algeblocks won’t be too useful for me but I still find it fascinating how you can use them to show students a wide variety of math concepts.

    The word function always created this feeling of dread in me whenever I heard it. I took this course to get over that anxiety. So when we started talking about function on their most basic level there was a lot of clarification for me. Using the words domain and range as a high school student was difficult. I don’t think anyone ever said to me that domain is the x-values and range is the y-values. No one ever said that this is what we already know about y-intercept lines but now we are just giving everything a different title. I look forward to gaining more clarification so that I can be a more confident teacher and more educated.

  9. I also agree with a lot of you, I love the Algeblocks, and I could not believe it when I found some (not a whole set) in a closet at school. I am just introducing them to my students and I explained to them this will be a learning experience for all of us. Although I figure out the problems before hand it is fun for us all to problem solve together.

    I loved the CBR activity and how we played around with the sensor to develop different types of graphs. I can clearly see how this activity can foster a better understanding of how graphs can tell a story. This was a powerful lesson. On the other hand, I found the tooth pick activity very difficult, the first two problems were fine but I could not wrap my head around the others, I am still playing around with them.

    Thank you for not pushing the composition of functions and looking forward to meeting them next class.

  10. I loved working with the algeblocks in class. I think it is a really great way to take something that seems to me to be very abstract and turn it into something very concrete and tactile. I think a lot of students that are of the age where they learn this material are not “grabbed” or intrigued by it because it just seems like a lot of confusing numbers on a paper with no real meaning. I think the algeblocks help students to see what these numbers really represent and what it really means or looks like when numbers “cancel each other out” (making legal trades) and when your results are all positive or all negative, for example. Anne mentioned in the blog that it can get tricky when you have a trinomial “where you have two negatives such as x2 – x – 2 which indicates like terms have been computed from the binomial multiplication (x-2)(x+1). Not easy to visualize unless you are using manipulatives,” and I couldn’t agree more. I teach a much lower grade (third) but I am always a fan of pulling out manipulatives or letting my students do so in order to help them understand something in a more concrete fashion.

    I also liked working with the graph making and the CBRs. As an adult, I appreciate and enjoy the opportunity to get up out of my seat to learn, so I am positive that our students feel the same way. I like that Anne sort of throws these things at us and lets us figure out the best way or the right way, without telling us how we should be learning the material or performing the task. Third grade students respond well to this, too, and I find that they sometimes come up with more clever ways to solve than I can! It is a learning experience for both, and we should certainly keep this “spirit of inquiry” alive for all of our students.

  11. I agree with my colleagues that the Algeblocks were a great hands on activity! They give a great concrete model of how to solve algebraic equations. I took a class this past summer with Professor Mahesh Sharma and he explained that students need that hands on, concrete activity in order to master the content. As a special educator, I am a firm believer in this. I teach a much younger grade (3rd), but even my students’ benefit from a concrete representation of the math. This year I taught the strategies for addition/subtraction using Cuisenaire rods (same company as the Algeblocks) and the progress my students have made is remarkable!
    As others have mentioned, I also like the conjecture boards. I use them all the time when introducing a new topic with my students. We always list what we think we know about the topic and them add new information or cross it off as we progress through our learning! It is really cool to see the same concept used in a graduate level course!
    As far as the toothpicks are concerned, I love those type of activities! I love puzzles and trying to come up with functions for a set a data. This problem reminded me of the past course, Number Theory, which I thoroughly enjoyed! I really like to see how others came up with their answers and compare it with my thinking! The math geek inside me would love more problems like this!!

  12. Okay, so here's how the scorecard would go: Algeblocks 1, me 0. I cannot believe how one little piece of plastic tile is unhinging me, and it is. I left Friday's quiz feeling defeated that I couldn't figure how to model the expressions. Those blocks are giving me a nervous tick...just kidding, sort of ...(lol). Having had some time to reflect on this, it really helps me appreciate and understand the value of multiple representations and processes discussed during problem-solving. I can appreciate that my modeling may not always be best for my students and if their exposure to other processes may provide better connections for them. Lesson learned.
    I will say there is some hope for me as I did find using the Algeblocks a bit easier to manage when solving the multi-step equations. Timing is everything and as it happens, I can currently covering this concept with my eight-graders. I had been using the vocabulary of "cancelling out" like-terms but really appreciated Anne's discussion about how we understand what that means but some or many of our students won't. I have, instead, been using the term "zeroing out" and have found that the kids are more responsive to this. I am appreciating how important language can be and need to really make this a goal of mine. Another lesson learned.
    I really enjoyed the graphing and CBRs. Teaching in the middle school, it so important to infuse hands-on, collaborative activities to motivate and engage the students. Again, the timing of this was perfect. We just completed the English MCAS, the kids were a bit tapped and I wanted to capture their interest. I put together a lesson on bringing different types of graphs to life, and then had the kids write and perform skits that modeled this. As the students performed, the other groups would graph the story. After each skit, the performing group would check with rest of the class to see if they had graphed the story right - it fostered lots of great discussions, debates. A formative assessment confirmed they can show mastery with this topic. Student-centered learning is so powerful and I want to incorporate more of this into my planning. Yet another lesson learned.

    Another great week-end. Looking forward to our next one.

  13. I am the first to admit that I was not a big fan of manipulatives. The type of learner that I am always found the use of toys (that’s what I usually call manipulatives) somewhat “messy”. Well, until I worked with the Algeblocks! The part I enjoyed the most was the multiplication of binomials. I teach middle school and polynomials are not part of the curriculum. For my own edification, it was great to see concepts that seem so abstract actually represented concretely.

    I had so much fun using the motion sensor gadget (CBR? I guess…) to construct graphs related to time and distance. It would be great to be able to expose my students to such activity. The concept of “distance from a particular location” over time usually causes an issue because students often mistake it with “distance traveled” over time. I would be great for them to see how to produce desired graphs in addition to interpreting given graphs.

    The toothpick activities were quite fun even with the challenge they posed. Solving the problems would have probably been easier as a group, but it seemed that all of us on our table prefer working individually. As a result, we used different approaches that we ended up sharing, creating a more enriching experience. I had promised myself to solve all the problems on my own, but it did not happen: life took over my time, as usual.

    I should also mention that I greatly enjoy the discussions that spontaneously arise from (not so) random comments. They usually push me (or us) up one more notch in my thinking. As a big fan of algebra, I look forward to the next session. Difficult week-end but definitely worth it!

  14. Hi Everyone, Sorry, I posted under the wrong class. Thank you Lisa and Evelyne for telling me so I could fix it!!
    It is interesting how after class on Saturday I was really excited, but still really frustrated with the toothpick problem. I am a very “black and white” learner. If the teacher tells me how to do something I will follow the steps and solve any problem like it. Have me experiment and figure something out on my own, forget about it…… Even with the Algeblocks I have a difficult time understanding how the pieces “fit together” but once I completed few correctly, I could then visualize how the other factors and products should look. I enjoyed them so much, I actually purchased a set for my classroom. I have an extra support math class for 3 days out of the 6 day school cycle. I have begun using the Algebolcks with this class. I told them that these were new to me and we were going to figure out how to use these together. We have only added and subtracted integers using the basic mat, but a few of my students have actually said “OHHHHHH, I really get this now.” This made me so happy!!! I also had my students create problems that they thought were “really hard” for their classmates to complete. I told them that they needed to be able to explain what the correct answer was and explain to the other students how to evaluate the expressions it using the Algeblocks if the other students were having trouble. I have also started having the students write the properties down that they used to evaluate to expressions. This made each student not only challenge the other students, because they wanted to stump them, but they were challenging themselves in that they had to explain their work.

    I am thrilled that we are starting linear relationships in my 7th grade class. I know the 8th grade teachers at my school are excited that we are starting them too :) !! I have been really trying to “mix up” the variables used in the equations to emphasize that x and y are not the only variables that we should use. I am excited (a little scared :) ) to see what we will learn next.

  15. There were a few activities from our last class that I found to be particularly valuable. The first was our continued work with Algeblocks. I have used Algebra Tiles limitedly in the past, but never knew how to use the tiles with negative terms. I really like this modeling approach and think that my students would appreciate this hands on approach – especially when factoring. I also just finished a unit on polynomials with my class and using these blocks would again be a great way to model multiplying polynomials. I feel that the Algeblock approach to multiply x•x and x^2•x would help students solidify their comprehension and give a more visual and memorable representation as to why the products are x^2 and x^3.

    I have to admit that I was frustrated with the toothpick activity at first. I couldn’t get past the second page and struggled to find the correct equation to model the data we created in the table. I definitely “preserved in problem solving” and stuck with the problem. I reached my “aha moment” when someone else in the room was explaining how they looked at the pattern in the table. It was through another student’s explanation that I was able to gain more insight into the problem and was able to finish it. Group work is such an essential part of the classes that I teach, and this was a perfect reminder of how beneficial it is for students to work with each other, and also how valuable hearing and reviewing a variety of different methods that can be used to solve the same problem can be.

    I also really enjoyed using the CBR’s. I definitely need to put in a grant through my district to try and purchase these for the department. My group had fun trying to model other relationships than the one’s we were given and were especially proud of our sinusoidal function. Time was definitely a factor when trying to create our graphs, but that would also help lead to a great classroom discussion (especially with sinusoidal functions) on period and frequency. I can picture myself taping the floor, having students measure distances, write equations from the data in the graph they’ve created, and so on...I’m excited!

  16. Perhaps it is a function of age (advanced!) and the teaching methods practiced of the times (prehistoric!!), but I too do not naturally gravitate toward manipulatives as a teaching method. In fact I often have to LOOK REALLY hard to SEE the relationships that are being demonstrated by manipulatives in general. As a student myself I learned mathematics by learning the rules and applying them in somewhat of a piecemeal approach by type of problem. I'm not sure if it was how I was being taught or if it just worked for me and it was a way that I gave me success. It wasn't until I started teaching mathematics that I realized many of the connections that exist between various concepts, sort of bringing it all together. Particularly through some work at BU's PROMYS program I've gained an appreciation for the use of manipulatives. I have used cuisinare (sp) rods (limited) and algebra tiles more extensively, but I always found it difficult to demonstrate negatives and subtraction. I am impressed with the relative ease of use of the Algeblocks and envision utilizing them in my classroom. I also liked the work we did with the CBR's because it felt like we were doing a scientific experiment, rather than a mathematics activity. Because of that, for me, it is a true STEM approach to mathematics. Finally, one of the best things that I will get out of this class is the document viewer. I have never before seen one utilized in a classroom and it is on my MUST HAVE list. I am so impressed that I'll either buy my own, or write a grant to get them for the department.

  17. Referring to the self-inflicted roadblock that many of us encountered during the toothpick activity as "getting stuck in our own heads" is a very accurate description. This idea of getting "stuck", for me, extends beyond this particular problem. At times as an educator I can get stuck on a certain teaching method or modality, just as I did as a student during this activity. As I step back and reflect on the activity I realize that I wasn't committing myself to using the manipulatives. Quite often it can be a stretch to connect a problem to a hands-on, but this problem started with a visual (the toothpick drawing) that we could use the actual item, toothpicks, as the manipulative. Instead, I was too stubborn and tried to prove that I could solve it without the manipulative. The result: input-outputs tables for my input-output table's input-output table ... still with no solution!

    Returning to my class this past week we started equations, so I thought it was a perfect time to dust of the Hands-On-Equations sets. I gave the worksheets as a packet and allowed students to work ahead as they successfully completed each lesson. I found that my stronger students jumped out quickly without relying on the manipulatives too heavily. These students eventually became "stuck" themselves as the lessons became too complicated to do with mental math and fell behind the progress of their peers who committed to the hands-on. As they struggled and asked for help, I would say "first show me your set-up using the Hands-On-Equations". Like me during the toothpick activity, they were reluctant and wanted to prove they could do it on their own. My first thought was, "I can sympathize with how they feel". My Next thought: "Wow! I acted like a 6th grader!". Either way, I need to be more open to multiple ways to solve a problem. Although having empathy for students can be good at times, just because I share in a struggle that they have does not excuse them from being open to new ways of solving problems. It may be a little frustrating for them know, but may provide them with the tools to get "unstuck" in the future.

  18. The first time we worked with the Algeblocks, I was not a fan. All I wanted to do was use a pencil and paper to solve. It is hard to learn a new method when you are so used to another! However, after a few examples I wondered why I was not taught algebra this way. Even though I thought I “got” algebra before, actually seeing what is happening really opened my eyes. I have a better understanding of algebra and believe all students should have the opportunity to practice math using manipulates.
    I loved solving the toothpick problems. While the third and fourth problem were particularly challenging, I enjoyed working through them. I had such a feeling of accomplishment when I finally figured out the answers. I think it is important as teachers to realize we must allow students to work through problems on their own, rather than just giving the solution. I also benefitted from talking to my classmates about strategies, although it is sometimes hard to look at a problem from a different point of view.
    I have incorporated function tables into the fourth grade over the past two weeks. I realize that students need to be introduced to these words and concepts in the younger grades. Even some of my stronger students had trouble finding relationships between basic Input and Output table at first. They have been making progress and we will continue to work on this.

  19. I thoroughly enjoyed the many algebra challenges of the second weekend. Learning how to help our students understand algebra begins with exposure to creative techniques and sound reasoning while availing ourselves of all the current manipulatives and technology. This class is certainly providing that opportunity for me!

    My favorite activity was using the CBRs to model graphs. It was exciting to see our group work together and experiment with the many different type of graphs that could be made. I can definitely see myself using this tool, especially in my Algebra 1 review class as it would be so helpful to create visual representations for and with my students. We all learn by doing and CBRs would be a new approach to graphing.

    I am really interested in the use of factoring with Algeblocks, as I know this would be a great addition to my Algebra 2 class. These would really come in handy during my after school late night when I can work in small groups or one on one. They would be another tool available to me to “show” algebra concepts as opposed to writing out the problems. I am also really looking forward to expanding my knowledge about functions and function notation during our next class.

    I felt most challenged by the toothpick activity. However, I found our class discussion about individual techniques so helpful. This always proves to be a great way to broaden one’s perspective and think outside the box. Our class mirrors what I hope to achieve in my own classes... enthusiasm for learning!