Many of us successfully solve ratio problems partly
because we know how to represent them algebraically and can solve for the
unknown, and some of us because we have innate or developed number sense and
can “see” the relationships between and among quantities that are being
compared. We are lucky again partly because as adults we have been working on
these types of relationships for a long time and have been successful
implementing the algorithms we were taught.
BUT, think back to when you were
learning about ratios and did not understand what it meant to be related to
some other value. I recall being told that the ratio of two things were in a 5
: 7 ratio and there were a total of 763 items. I sat mesmerized when my math
teacher proceeded to write 5x + 7x = 763. It was like magic. At no time was I
shown a visual representation of what was going on nor was I encouraged to
think about the fact that there was a multiplicative relationship happening.
When I finally realized that teaching as telling is totally ineffective I
became a deeper thinker, sought out ways in which to help students visualize,
touch and manipulate the mathematics and relate it to things they are
interested in I discovered my students, no matter how old, started to progress
in their own learning.
I hope that after today’s class the inclusion of the
bar model makes sense to you and it is something that you can bring into your
own battery of teaching strategies. Modeling how the original problem looks and
comparing it to the results after an activity has occurred allows students to
make sense of a given situation and adds sense making to doing the mathematics.
Anne
Please share your thoughts here!
ReplyDeleteAs an algebra teacher, I first approached our problems by forming equations, solving them, and relating the solutions to the questions. Then, with some prodding from Anne, I started using the bar models. I can see how they offer a nice visual insight to the problems and they might also be more easily received by struggling students than going directly from a word problem to an equation. I am having some trouble using them with messy fraction situations, and figuring out how to use them in non-discrete (continuous) problems involving fractions. Like I tell my students, I'll get better with practice (I hope).
ReplyDeleteSo, one of the big differences between being able to efficiently solve a problem to being able to unpack it in a way that makes the mathematics make sense to students is to recognize the importance of moving from the concrete to the pictorial to the abstract. You are functioning at the abstract level but your students are still learning this mathematics and need to build a deep understanding of the math before you push them to computation.
DeleteAnne
While it took me a while to get my head wrapped around using the blocks, I did manage to get it. I observed Jon in class manipulating the blocks and I asked him to explain how he was thinking. His explanation helped me out a lot and gave me the understanding of trying it myself. I found that by manipulating the blocks I was better able to visualize the ratio and the math took care of itself. All I had to do was to find the value of one of the blocks and then calculate the result.
ReplyDeleteI think that the most important thing that I learn from taking a Lesley course is that there are multiple ways of solving problems and that by being familiar with some of these ways, we, as educators, are better able to reach and teach different types of learners so that they are successful.
Tom, Have you used this method with students Yet? I am curious to see how they did with it. The other students in the class seem to have had good success with using it in their math classes which is great.
DeleteI haven't yet used the bar models in class. I have been playing with them and I like how the process of forming a ratio is staggered. I do plan to use the web site,
Deletehttp://thinkingblocks.com/
when I do an MCAS review in April.
By the end of the first class, I felt like I was back in my high school math class – nervous and scared about what was to come. One of the reasons I signed up for this course was to feel more confident about my own ratio and proportion mathematical skills. It’s always been one of those areas I have dreaded teaching – an area I’ve never felt 100% confident in my own ability, so how could I possibly feel like I was teaching and demonstrating the proper skills so that my students would feel successful!? Within minutes of class number 2 I couldn’t wait to get back to my classroom and show my students and colleagues what I had learned. I couldn’t believe ratios and proportions could be so easily solved by using the bar method.
ReplyDeleteSince last week’s class I have been working on ratio and proportion word problems, with my students, as part of our daily warm up exercises. I modeled the very visual bar model method to them and, as I expected, they picked up this strategy fairly quickly. Several of my students even saying, “that’s all you had to do!?” I just love seeing my students’ reactions when a scary concept becomes so comfortable for them to be successful.
Great, wait until you see the next method for representing ratios and converting them to decimals and percents. Hopefully it will be as amazing to you and your students as the bar models.
DeleteAnne
As someone who understands algebra easily, it is often hard for me to turn to manipulatives and pictoral representations in my high school classes. I demonstrated the bar method to one of my colleges as a way to help kids in a math lab class to help prepare for the MCAS. He was unsure if the students would laugh off the pictures. I then remembered a geometry student that was having trouble solving single step equations. I went back to “cups and beans” with him. This was a 10th grade boy who just didn’t understand that 4x +3=11 could be translated into x = 2 until I explained that the x’s could be cups that each held the exact same thing. I think the bar method will help my students get a grasp of ratios and fractions that they were never able to get before. While I do teach in a high school, it is an alternative school. We must find alternative methods to teach our kids. I look forward to learning more ways to think outside of my mathematical box.
ReplyDeleteHi Jean,
DeleteI am so glad you are willing to work with the manipulatives and pictorial representations. It infuriates me when teachers won't give them a try either because they are uncomfortable or because they have decided for their students that they won't work. Adult engineers have been fascinated by algeblocks as a model to represent many algebraic concepts and would never think to blow them off.
Maybe this teacher will like the graphical model. It is a bit more sophisticated so it might be acceptable to him as another means of representing ratios.
Anne
Hi Everyone,
ReplyDeleteI am not math minded (some will say otherwise... but this is how I feel.) I have worked very hard to become more math minded over the past two years. These classes are another step in that process. When I solve problems, it is like roulette or the lottery. I pick what I am comfortable with and pray it all works out. I suppose for most students learning any new math process creates a similar experience. We read the problem until we understand it – at least a little –, latch on to anything that makes sense, go to our bag of tricks (strategies) and hope to use them to get a reasonable / logical answer. If we are really lucky the answer is correct. As teachers, the bigger our bag of strategies the better. The modeling using bars was the first step in adding to our strategies as teachers and as learners. This along with the other strategies will allow me to solidify this idea of ratio, proportion, and rate, and in the end, solidify it for my students.
Maria,
DeleteHow did you find working with the bar model? Does it make sense to you? Will you be able to model it with your students? Could you use it to check your answers from another model? How uncomfortable were you using this model?
Anne
In the last class we learned about how to use the bar model and quisi-something rods to solve problems involving ratios. My first intention when solving a problem is to do it algebraically. The rods were in front of me so I decided to go outside of my comfort zone and use the rods. Once I determined an appropriate way to use them I found that I could make sense of a problem faster with the rods over any algebraic work I did. Because we were trying multiple strategies I found out it even strengthened my algebraic equivalent answers. The work had a diagram to match and it helped create a thorough understanding of the problem.
ReplyDeleteI tried one of the ratio problems out in my fundamental math course. The question was …
A classroom has a ratio, boys to girls of 9:8. If half of the girls leave then there are 15 more boys than girls. Determine the total number of students. Determine the number of girls that left.
Many of the students did not know how to even read a ratio in the form a:b. Those that did had a misconception that there were exactly 9 boys and 8 girls. A lot of students answered the first question to be 17. We then talked about equivalent fractions and that the ratio could read as “for every 9 boys there are 8 girls.” This was helpful for some but other students were stuck because they didn’t understand that two requirements needed to be met. The ratio of 9:8 must be kept and that the difference is 15 if half the girls leave. The students for the most part avoided using diagrams or models. They worked with the numbers and fractions using trial and method techniques.
Jon,
ReplyDeleteWhat if you asked them to use the Cuisenaire rods...or to draw a picture of the ratio? Would that have helped? You have your work cut out for you in terms of bringing your students up to the expectation in the new frameworks since ratio is such a big component of the expectations.
Keep pushing them to do the modeling...once they try it they too might find it useful.
Anne
Manipulating blocks was confusing in the beginning (in another course). Watching colleagues and listening to their explanations have given me many ways of solving questions. Ratios seemed much easier as I worked through them using blocks. I learned once we determine the value of a block, the remainder of the question seems to fall in place. For some questions, however, I had to return to solving through equations.
ReplyDeleteI work with SPED students who require repeated explanations and sometimes modified questions. Using blocks and other manipulatives can provide another approach for these students.
With enough practice most SPED students are able to determine the value of the rod, then build from there. Often they rely on their pictures which replicates the blocks and at least gives them entry into the problem. The good think is the use of the blocks never change so the method is extremely reliable.
DeleteSolving the ratio problems using the bar model method was very helpful last week. I know this method from the Number Sense course in the fall and other courses, but I had not used it extensively. I did a lot more with modeling in my fraction unit this year and students have made progress, but they still show significant weakness in their understanding of fractions and fraction operations. I was determined to try some of the methods we used in the Number Sense, including using Cuisenaire rods. I ordered a class set and have begun planning how to help students learn to use them. So in our class last week, I focused on how students might use them to solve the bar model problems.
ReplyDeleteI did not have enough time to get them to a point to use the Cuisenaire rods before our next lesson, but I have begun the lessons to build to that point. Today I gave students one of the problems on the bar model note sheet as a warm up. Most of my students had no idea what to do. Some of my pre-algebra students did get the right answer, but many of these were not sure they were right. After we discussed students’ work, I presented the bar model as a way to look at the problem and told them we would be learning the method next week. I am eager to help them learn a way to think about these types of problems and I am hopeful that the Cuisenaire rods will help with the bar model.
Whenever we have our next class meeting (if mother nature allows us to have a next meeting) I will show you another method that may, or may not, help students who struggled with the blocks. We will be solving problems by graphing them. It relates to the bar models but provides a different perspective. I am anxious to hear how the lesson went with your students this week.
DeleteI have only been a middle school math teacher for a few years and constantly looking for ways to reach the struggling students. I realize that all students are different and many of the students state immediately – I don’t get it and I am not good at math and show little or no effort. This gives a lot of the students another way of visualizing the problem. When I first presented the bar method with a warm up problem – some students responded – that was easy. Most all the students were asking for more problems. I on the other hand am still working on utilizing the bar method to solve problem – I know the more I work with the bar method the better I will become- the whole reason I took the class. So that I can improve my teaching skills for ratios, rates and proportions. I presented the bar method to my 6th grade lab class and they were very receptive, after about three problems the students were solving problems on their own and looking for more challenging problems.
ReplyDeleteGReat, I have a lot of additional problems to share...pretty complex ones that all can be solved using the bar model. To a mathematician this may not look very sophisticated but it actually is an algebraic solution with the bar representing the variable we use in Algebra.
DeleteTo solve problems that involve fractions, ratios, rates and proportions, I often use equations. My middle school students however, find equations and abstract calculations difficult to understand. To make sure that they are able to solve problems that involve fractions, rates, and proportions, I use tables and other types of visuals. Before I took the class, I have never heard of the bar model concept. After solving the first problem using the bar model, I was still not convinced that the (bar) model was that efficient. As I continued to use the bar model to solve problems, I discovered how powerful it was.
ReplyDeleteIt really helps to convert the data provided in a problem into concrete visual images. It can enable students to comprehend and convert problem situations into relevant mathematical expressions and to solve them. I can see students use the model to solve simple, difficult, and multistep problems and for any problems that involve comparisons, part-whole calculations, ratios, and proportions. The bar model not only can enable students to solve math problems, but to represent them symbolically. With that solid foundation, students can smoothly make the transition from arithmetic to algebra.
As we ventured into the world of bar models, I sat perplexed as all I saw was a bunch of blocks. My house is filled with Legos which my son and husband can’t get enough of but again, to me, it’s just a pile of blocks. I have an algebra brain and am having a hard time wrapping my head around this bar model thinking. When I presented my class with the problem of the children and adults at the birthday party, they were just as perplexed as I was. Unfortunately we have not done any ratios this year, but I left them alone to see what they could come up with. The answer: not much. Many of them went immediately to the basic proportion set up which they quickly found did not work. Some tried random multiplication and division, while others started to build a table. The part about 180 more children was turned into there were 180 children, mostly because that is the strategy that I started to use. I then got my student teacher on the band wagon. He is a math major so of course his brain went right into algebra mode, but he was successful enough to get an answer. As our snow days have kept us from class, I have tried to look and relook at these problems hoping to get a visual perspective on them. I think I have completed them all with some degree of success but I am not yet confident enough to relay these strategies to my students.
ReplyDelete