Monday, March 25, 2013

Ratio, Rate and Proportion Series - Post 2

I wrote an epistle reflecting on our past three sessions but somehow it did not publish. Now, I am trying to reconstruct what I wrote.

On Tuesday, we introduced fraction as number and fraction as ratio on the Cartesian coordinate plane. Both operate differently due to the continuous nature of fraction as number and the discrete nature of fraction as ratio. What they both have in common however is the visual representation that allows us to see what it means to have a common denominator. Too often that is an abstract concept that students can determine but which many just do not understand. The way in which we introduce division can set up our students for success or for reliance on cutesy sayings rather than helping them understand we are simply looking to see how many groups of something we have or how many items each member of a group will get.

Using the graphs to convert ratios to percents is also a valuable tool for students especially when thinking about slope as percent…think about a 40% grade on a hill. Students need to be able to move seamlessly within the ratio, decimal/percent representations and this is yet one more model to use.

On Thursday, we really worked hard solving problems on the Cartesian coordinate plane. It is really challenging to use a model for the first few times. None of us were taught this as students yet the CCSS progressions insist that our students use it as a problem solving strategy. This model really emphasizes the concept of ratio as being discrete and answering the question "How Many?"  But, what I found most interesting is how hard it is for some of us to move to a “relative” representation rather than an exact numerical one. This is really a stretch since we have not been trained to think that way. What an advantage your students will have from their exposure to thinking like this!

The highlight of the week was our field trip to the Science Museum. I was really impressed with the depth of conversation around the different exhibits especially the Mt Everest exhibit. I was listening to a news show this morning and they mentioned that the base camp is 17,000 ft above sea level. Imagine!!! Tom mentioned how cold it must be, but I was thinking about the thin air. I had trouble breathing atop Pike’s Peak which is a measly 14,000+ feet above sea level. Of course today hikers bring portable oxygen tanks, but imagine Sir Hillary doing it au natural.

The other interesting point for me was seeing the difference between the pencils made to a scale of 12:1 versus a scale of 10:1. The end products were amazing.

I look forward to reading your posts.


  1. I found the coordinate plane an interesting way to teach ratio to my students. i usually don't do it until we study slope, and then I go to great lengths to explain that slope is a ratio. By introducing the coordinate plane earlier with all kinds of rates and ratios, slope should be an easier sell.

    As increasing importance is placed on the students' ability to move smoothly back and forth between numerical expressions, graphs, and tables, the use of the coordinate plane for ratios should really help this translation process from one type of representation to the other. I think it's important to use this a a learning tool on problems that work easily with it, and not try to "force it" to the point where the tool becomes confusing. I know AI got confused using it on a few of the problems. But maybe as I get more familiar with it, my use of it in teaching will improve as well.

    I really liked the exercise we did with the circles on the floor. The intro do Radians as angle measure and the applications with gear ratios are an area I hope to pursue more in the future. The "frequencies" display at the museum also hit on the same area.

    Our museum trip was great. I found each of our exercises to be fun and they offer some good examples for the classroom. Having fun always makes for engaged students (well sometimes!)

  2. What a great way to spend a Saturday grad class! I've been to the Museum of Science more times than I can count, but I have never spent time thinking about the exhibits like we did, as a class, on Saturday. As we were solving the chair ratios, in the Ratio and Proportion area of the museum, I kept thinking about my middle school students. I can really see them getting a lot out of this activity. I'm working on an upcoming field trip for them to be able to experience and learn from the different mathematics the Museum of Science has to offer.
    The hands on problem solving activities were just what I needed to spark my creativity in bringing new lesson plans into my classroom. I teach in an IB school in which project based learning is a large part of our curriculum. We are always tweaking our IB units and are about to work on our Scale Model unit. I just love the idea of incorporating the passage from Gulliver's Travels into a math class on ratios, proportions, and ultimately building a scale model representation. You can't get a better cross curricular lesson plan than that! In our current Scale Model IB unit, the students in 7th grade have to make a scale model of an item by scaling the object larger than the original size. In years past my students have all made different scale models, rarely duplicating one another’s objects. I really like the idea of the entire class making the same scale model representation. I think this could lend itself to some great classroom discussions amongst my students. I can definitely see myself taking this lesson from Saturday's class and bringing it into my classroom.
    So far, I have really been enjoying our ratio and proportions class. I'm constantly bringing new ideas and lessons from our classes to my middle school students. Last week I taught my 7th graders how to graph fractions. I know they thought I was crazy when I told them what we'd be doing, but they really thought the concepts of adding and subtracting fractions on a graph were cool. I've been thinking about re-teaching/revisiting circles with my 6th graders and I think the circle activity on the tile floor would be a perfect lesson for them.
    I'm really hoping to be able to take more grad classes like this one in the near future. It has been very beneficial to my students and my teaching .

    1. Julie, I totallay agree about incorporating Gulliver's Travels into a math lesson. I love to read and try to find meaningful ways to bring literacy into the math classroom. We get so caught up in our content area that our lessons become isolated. I think students get so much more out of lessons that incorporate other subject areas. Especially for those that "don't like math".

  3. Reflection on our lessons of this week – What a great field trip to the Museum of Science. Not only was it fun and inspiring, but it gave me insight into my students’ experiences when learning something new. Working in groups has many plusses (sharing information, fun), but it also has some frustrations, such as when something is clear to others but not to you. The MOS staff did a great job of focusing us on gathering information and trying to make sense of it. They reminded us that it was the process of inquiry that was important even if we did not get the right answer or finish answering all the questions. This made it feel safer to put work out for review. I am mindful of cultivating the same environment in my classroom. In the last month, we have been doing lots of sharing of work among my students using the document camera. Today I asked a student who had started the second part of a question if she would share her work. She had not finished but had gotten further than other students. She said that she did not want to show her work because she did not think that it was right. I told her that it did not matter whether it was right or not but that it would give us a place to start our discussion. She shared her work and with some questions that I asked of her and the class, together we were able to understand the problem better. Her willingness to risk showing us she did not understand helped many in the class move forward with their thinking.

    The best part of the museum trip was making the pencils. There was so much math involved in the activity, which, of course, was the point (pardon the pun). But again I was struck by the other lessons learned. It is eye-opening to see the different ways others solve a problem. Most important was that we, as a group, made a decision on the scale (even if it was not the actual one used in the book because we had not read far enough), and then carried out a plan using that scale. We made decisions along the way that affected the math we had to use and how our model turned out in the end. It was an achievable challenge with a fun payoff.

  4. I truly enjoyed our day at the museum. I am only sad that my day was cut short and I did not get to see the final products for the scaled pencils. I have been to the museum several times for field trips and have often thought there was probably something more I could do from a math perspective but have never had the time to really look in to it. I am glad to have spent the time in a learning capapacity. I thought all of the activities were engaging and worthwhile. I can see my students really enjoying a similar experience. I also felt it was a nice reminder that hands-on tasks are generally the ones from which we grow the most. In addition to the math ideas that were shared and learned there are so many other life lessons that can be learned from actually working together on a task.

    I feel as though I have gained so much from this course. Each class I feel as though I have had something to takeaway and assist me in my classroom. Working with the cartesian plane for fractions was really eye opening to me. Since we are rolling out a new program this year I have not had the autonomy in the classroom that I have had in the past. However, this visual model was something I was able to show as an alternative model. In fact I have one student that has significant reading deficiencies. She has found the modeling very difficult because working in the numbers is where she has her successes. I sat with her and showed her the graphing exercise (which she immediately took to while her paraprofessional looked at me like I had two heads!) to help her divide fractions. She was able to use the graph to find her answer and better understand the models her classmates were using. In addition she is now able to share with her classmates her understanding of dividing fractions rather than feel defeated and frustrated with the other model.

  5. On Tuesday we ventured into the world of Cartesian planes. What a fantastic visual way for students to compare, add, subtract, or even multiply and divide fractions. I have never seen this method used before and was quite intrigued. The word problems got a little tricky as we all got caught up on what scale we should be using when graphing only to find out that it wasn’t that important. My favorite part was mapping out the 10 x 10 section and graphing a fraction in order to find its percent.

    Thursday brought more practice on the Cartesian plane as we shared the word problems we had created. Alfred had us all stumped as he presented what I would call a “humdinger” of a problem. It was great to see how each of us tackled that problem. Next we went into the hallway to create circles with dry erase markers on the floor as we tried to measure the radius and circumference using only string and our shoes. It was a great hands-on activity which lead into the post-it note activity. Trying to use the post-it notes to fill the quadrants of the circle without wasting any paper had me wishing that I had my glue gun or a glue stick in my bag with me. I will certainly be sharing that with my class so they can see where the area formula for a circle comes from.

    Our week ended with a fabulous trip to the Science Museum. After getting through the stress of the drive and going through the tunnel, I was ready to get down to business. As the class discussed their results of the 3 chairs problem, I found out that my chair heights were very close but my seat dimensions were way off. Off we went to explore Mt. Everest and a giant grasshopper. The discussions were great and each group’s approach was a little different. How does one exactly find the volume of this giant insect? Ken and I decided that we were putting it into a box and proceeded accordingly by using my footsteps as our unit of measure to find a length width and height. After lunch we traveled into the land of Gulliver’s Travels. Our team worked feverishly to create a model of a pencil that was ten times bigger than the original. The challenge of creating the hexagonal body of the pencil worked out better than expected. All of the pencils came out great and no two were alike. I can’t wait to show it off to my classes.

    Today in my class my students were going over their homework last night which showed a picture of a circle inscribed in a square. At first they were asked what information would be needed to find the ratio of the area of the circle to the area of the square. Then they were provided with a radius of 1 inch which allowed them to calculate both areas and to create the ratio. The final question asked if that radius was needed in order to calculate the ratio. Of course the answer from the students was “yes”. My student teacher quickly corrected them as he wrote the ratio of both areas just using the two formulas. With the help of a little algebra, he quickly simplified the ratio so that it looked exactly the same as the answer derived when given a radius of 1 inch. The students were amazed and instead of moaning about the work, asked for another problem requiring such thinking. Well it’s only first period and I could say that my work was done.

  6. Spending the day at the Museum of Science was a delightful way to apply the concepts we are investigating in this graduate class. I’ve been to the museum quite a few times with my children, but it was more to “play” with the exhibits, than to really dive deep into the mathematics that supports them. It was so much easier to find the dimensions of the chairs that we were supposed to figure out for homework, in person. Depth perception is very deceiving in pictures.
    I enjoyed connecting literature to math and art when we built scale models of pencils. We had to find evidence from the story to get the scale factor, and then use that scale factor to compute the size of the materials to make the pencil. Our team had a very difficult time making cones, it was a great activity to learn our strengths and weakness. I hope that I can find a way to fit it into my math lab curriculum.

  7. What a great week! We learned how to represent fractions and ratios in the Cartesian plane, on double number lines and ended the week with a fabulous trip to the museum of science to apply what we have learned in class. What so interesting about the week are the different strategies we learned in class. Since the students in our classrooms come with different learning styles, the more strategies and skills we have, the better we are equipped to teach them. Anne put it beautifully by saying,” we cannot change the way the students learn, but we can change how we teach.”

    It was my first time seeing the representation of fractions and ratios in the Cartesian plane and I was very impressed with it. I wish I had learned it in the beginning of the year when I was teaching fractions and ratios to my 6th graders. That would really offer them a new/different perspective. The trip to the museum of science did it for me. The different exhibits we saw show how ratios are used in real life. It would be nice to bring the students to the museum after working with fractions and ratios. They would have a great experience, understand how ratios are used in real life, and make meaning of it.

    The activity we did to enlarge the pencil with the ratio of 12:1 was interesting. This is an activity we can all do in our classrooms. Students would learn a lot from this activity. Not only they would have to deal with ratio, but also they would have to deal with surface areas of prism, hexagon, cylinder, and cone.

  8. Tuesday, March 19, 2013

    Graphing ratios and fractions was a new experience for me! I always thought that you had to have a dependent and independent variable. I never knew variables could be discrete. I can see this method being helpful for student who really struggle with ratios and fractions. These are Math concepts which continue to challenge even some of our more “Math savvy” students. We learned a new way to help explain ratio problems, as well as how to add, subtract, multiply, & divide fractions. I am anxious to share this with my MSN colleague.

    Thursday, March 21, 2013

    The string activity we did where we measured the length of the radius with our feet and counted the radians was such a concrete method to help the students understand the relationship between radius, diameter, circumference, and pi. This is a standard that I will be testing my students on soon. Even though I have approached this relationship through definition, proportions, and equations, many still don’t see it. I hope to do this activity with them.

    Saturday, March 23, 2013

    The Museum of Science – what a great opportunity to experience the museum as a “student”! It reinforced why “hands-on” activities are so important. For example, when we were given the chair problem with the pictures of Emma sitting on them with details of her measurements, it was difficult to determine the heights of each chair and the dimensions of each seat. However, when we saw the chairs and used the tools provided at the exhibit to make these calculations, it was easy to see the relationship between the three chairs and the scale model provided.

    The Mount Everest problem was a fun challenge. Again, the picture of the scale model gave rise to many questions. However, when we were at the exhibit, we could use some of the information provided along with our informal and formal calculations to determine that the base camp must be above sea level and that the model was therefore not built at sea level.

    Finally, constructing the scale model of the pencil was a great culminating activity. It was interesting to see how each group came up with the scale they used to construct their pencils. I shared this activity with my students and showed them my group’s pencil! It was a great visual of scale modeling which I will be working on with my students later this week.

  9. As I progress through this class I have both “ah ha” moments and “oh no” moments. I can’t say which moments outweigh the others. I continue to struggle with fraction as number and fraction as ratio in the context of word problems. As I look at my notes, I need to internalize that “how much” in relationship to capacity or length is fraction and “how many” is ratio. Of the various ways we have been asked to represent ratio, I find the bar model the most accessible for my thinking process. I know with practice using the Cartesian coordinate graph will become a more usable model. I will say the reference made about the 40% grade struck a cord with me. I saw many gradient signs as I traveled across country pulling a 35ft trailer. I had a relative understanding of gradient, but using the coordinate plane to describe gradient was definitely an “ah ha” moment. Finally, the trip to the Museum of Science energized my thinking as teacher. As I reflect on the trip, the activities we did are the type of activities “we” should be doing in class on a regular basis. It was an enlightening, enriching experience --- and we only visited four exhibits.

  10. I truly enjoyed going to the Science Museum for our last class. It reminded me of how beneficial a well-designed, hands-on activity can be for making concepts have meaning to a student. When presented the question whether the grasshopper that had been scaled up would be able to support the new volume, I thought “Of course, its legs would be bigger too!” After playing with the idea and thinking about the numbers that we looked at, I still had to think about what the volume of the legs would be before I realized why it would be a problem for this grasshopper. I am hoping to implement an activity similar to the pencil activity, when we are in that unit. I think it will help make a connection for the students as well, particularly when looking at how the scaling affects the volume.

  11. The graphing we did on Tuesday was a very different way of looking at fractions. I brought it back to my 6th grade Math lab for Thursday. Some of these students seemed to struggle with adding and subtracting fractions. And even though you cannot use it for every problem, it is a different way of seeing the relationship of the fractions. In a very short time they were able to add and subtract the two fractions. I plan on using this method to introduce adding and subtraction of fractions and then utilize the same graph latter when we multiply and divide.

    The double number lines were just another great way to work with ratios; I plan on introducing this method as a warm up. This is yet another way of visualizing the comparison of ratios. I feel I need to continue to work on a few more problems before introducing this method to the class. Most of my students are more of hands on – visual type of learners and most of what we do is not.

    The Museum of Science was a great hand on experience and we discussed how this might be beneficial to our students. I thought about the chairs which could be done very easily in the classroom - because they were just pictures. It will be interesting to see what strategies the students will use – compared to what I would use to solve the problem. The most fun was when we read the story and determined the scale factor. The making of the pencil was a very fun activity – that is what learning should be about – fun and learning. I actually had the students building rectangular prisms and triangular prisms to determine area- surface area and volume. We will use these latter on to scale them up and down. The students seem to be having a lot more fun and understanding in the process of building our models.

  12. 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.

  13. We’ve worked on doing ratios on the double number line and graphing for solutions, interesting strategies. The Science Museum began our work on scale. Using stated information we solved scaled ratios. Later we built a scale model from a reading selection, an activity students may find interesting. The museum can be a useful place for field trips and class activities, a reference center for teachers.

  14. I really like the double number line diagram. It is a great visual tool for thinking. I introduced the double line model (diagram) to my 6th graders, they were impressed with it. We solved about 9 problems using the double number line diagram and the students fell in love with it. It is a good tool to have to solve problems involving rates and ratios. I can see myself using next year. I can say I learn some great key mathematics skills on rate, ratio, and proportion and I am ready to pass on these skills to my students. It is my belief that my students will learn more from me and become more self-sufficient learners.

    I gave the assignment – Snails on the move- to my 6th graders, they did very well answering the first three questions using the double number line. Only a few students answered the 4th question correctly. They used the answers to questions 1, 2, and 3 to answer question #4 not realizing they did not have the same units. One unit was inch / 1 hour, one was inch / half hour, and one was inch / 2hours. I had to instruct them to keep all four rates in the same units (inch / hour or inch/half hour or inch / 2 hours) in order to compare them.