What a whirlwind class we had Fri evening and Sat. I hope
the clarification on how to set up a tape diagram and the manner in which one
is used to illustrate comparisons, and comparisons when values are added or
subtracted resulting in revised ratios is clear. It was satisfying to see how
well you, as a group, took to illustrating computation of fractions on the
Cartesian coordinate plane. How many of you have used this in your classrooms?
How successful has it been? Do you use the common denominator method …actually
anything is better than giving them a “cutesy saying” or a rule to follow. We
really need to be at the fore front of improving the way in which we teach mathematics.
I was listening to the news this morning and the newscaster was reporting on a
survey about how well mathematics is being taught to their children. 83% of the
respondents said the teaching is deplorable and their children do not understand why any of the procedures
work. That is evident in my own teaching and coaching when I visit classrooms
and children are doing rote procedures. When I ask why, most answer, “that’s
what my teacher told me to do!”
I keep thinking about the problem with the ratio of girls to
boys being 2:1 and if 30% of the girls leave how many boys must be added to
have a 1:1 ratio. Have any of you tried to graph it since class? I hope so and
challenge you to but suggest you don’t label the x-axis in percents…we did see
a great example of how this looks (no cheating and asking your fellow two
students who did it this way) how to model it. (no names to protect the
innocent) Also, try to solve some of this week’s homework using the Cartesian
plane…in fact I will insist upon it!
I am also curious about what you thought about TAU versus PI
as the value for the ratio for the circumference to either the diameter or the
radius and why TAU is so much more accurate.
Curious to hear about your experiences doing Gulliver.
I look forward to reading your blogs.
Anne
Wendy Lee
ReplyDeleteWhile I have not had the confidence (yet) to begin teaching my students to solve problems using the coordinate plane, it is one of my goals by the end of the school year. At the very least I will introduce them to the process of arithmetic with fractions. My ideas and intentions for next year are to definately include them. While my level of understanding is increasing with practice I expect that when I begin to guide my students through the process that an even deeper understanding will evolve. I am certain that my students will have insights that I have not even considered yet. This sure is different then how I learned.
I too, have been unable to incorporate the use of the coordinate plane to me students due the their current math apptitude. I did introduce the coordinate plane yesterday with some plotting skills, developing a pattern.
DeleteThey were very excited about this. I will incorporate this regularly until the end of the year.
This is new to me, I forgot to enter my name above - Brian Clancy
DeleteTom Canty
DeleteI agree with both of you guys! I teach 8th grade math, when we first started discussing the use of the Cartesian coordinate grid to do these operations with fractions I thought It would be a great opportunity to do incorporate skills on plotting points and applying slope. _ Once this idea was introduced, the majority of my students preferred this method when asked to find a common denominator or to compare fractions, ( they could very clearly see which fraction was greater by its location in reference to the other one ) BUT, like myself, they still preferred to do the pen and paper calculations to determine the answers to the operations.
Brianna Clancy
ReplyDeleteI am excited to continue to utilize the Cartesian coordinate plan to solve problems and bringing this knowledge into the classroom.
I did have my students do the calculations for Gulliver's eraser while I had extra time after administering a final to my night school students and they loved it!! I had limited supplies with me so we are going to continue with creating objects next week.
I am interested in researching the new concept (to me) of TAU and am intrigued by the accuracy compared to pi.
Victoria Ellis
ReplyDeleteI have used the Cartesian coordinate plane to solve problems with my students and they loved it. Many of my students come to me with an underdeveloped concept of fractions and this is one method that helps them develop it. I loved watching them gain confidence with adding and subtracting fractions as well. Many of them actually had the "a-ha" moment in connection to the common denominator method! Next, I will try multiplication and division using this method.
Also, I am very excited to bring the Gulliver's Travels Activity into class. It is the perfect end of the year project for grade seven since it incorporates so many of the standards for this grade level. I showed a few of my students the picture of what we did (my partner took the pencil) and they immediately bought into the idea! I am thinking that I may have some groups scale items up and others scale down.
Anyone else doing this?
Kari Courchesne
DeleteFortunately, I was able to keep the pencil and my students absolutely loved it! They also bought into the idea of doing this activity. One of my students even came up with the idea to make an eraser. I teach 6th grade, but I think this is something I could do with my students; with some scaffolding. Victoria, I really like the idea of having some groups scale down while some scale up.
Gisella Grimaldi
DeleteI'm doing the pencil activity with both the 6th and 7th LLD kids in my building and it's been going great. We tried reading the passage first, but the language was to hard, so we showed the movie. Now we're going back and reading, breaking down and chunking the material. The students are getting excited, I told them how we had 5 adults, relevantly intelligent, and it took us 2.5 hrs to figure out measurements and create the pencil. Students said we were slow, so bets on. Chinese food if they win or posters saying they have the best teachers in the building for the staff lounge. Think we will be published next week as the best teachers, games on.
I'm really liking the activities they're great to bring back to the students.
I have kept the pencil myself but found it rather difficult to complete the activity with my students since I do not have enough time to finish it in one class period nor have the place to complete the task so I have decided to make a shirt for Gulliver. It was almost as lengthy as the pencil to start with but we were able to keep the material safe for them to work on next day. I will keep you guys updated as to what the results are. So far we have measured our thumb and wrist and talked about the ratios. Two classes were able to make the outline of the shirt. They did move faster than I thought they would. Go Math Class!!!
DeleteSelda Seppala
Lindsay Adams
DeleteThe Cartesian coordinate plane has also helped my students will misconceptions, too! My lower groups really liked how visual it was for them to compare fractions. I also had them fill in where 1 would be in comparison to the fractions they were graphing and they made the connection that improper fractions would be greater than one on the graph, or on the other side of 1. They started graphing fractions for fun when they are done with their work!
I have a hard time transferring the information from class into my classroom. I teach grade 4 Special Education in a pullout setting. I have introduced fractions and adding fractions with like denominators. I also recall you mentioning tape diagrams to add and subtract. Are there any websites or lessons you suggest? I don't recall this being rolled out in the Common Core for the elementary grades.
ReplyDeleteAmneris Narvaez
Brian Clancy
ReplyDeleteGood Day,
I was challenged in regard to the "Similar Triangle" (pond distance), question one.
Angle B is shared by both triangles, but it would need to be assumed that line AC is parallel to line DE, or angle A equals angle D or angle C equals angle E to show that they are similar.
I accepted these assumptions, or at least one of them, to continue answering the remainder of the questions.
What am I missing?
Sheila Commisso
DeleteI made the same assumption that triangle BDE was similar to triangle BAC in order to do the problems, but I do have a few questions. In part 1, it asks to show how they are proportional, but we do not have enough given information. Am I missing something? I have stated the proportion that shows that the ratio of corresponding parts are the same, but until I actually find the measure of segment DE, I don't think I can show that they really are in proportion. I could go back and show that the proportion does work after I find DE. Is there another way?
Also, in part 4, I understand that I can draw any line parallel to sides DE or AC, but since there are only the given measurements, how do you show that the newly created triangle is in fact similar?
I also struggled with this problem. I made the same assumptions in order to go forward with solving for missing side lengths and similar triangles. When I tried to show this in a drawing, I wasn't sure how to make a line parallel to line AC or line DE. I finally saw a relationship using scale factor and ratios of side lengths.
DeleteI have struggled with this problem as well and kept thinking I was missing something because of the assumption that you guys were struggling with. I am still a bit hazy with my answer.
DeleteKathy Chamberlain
ReplyDeleteI am a fourth grade teacher. I am going to try the tape diagrams and the double number lines as a draw a picture strategy for my children with ordinary word problems. I think it could be a valuable new strategy for them. If they see it now, maybe it will help in middle school. Coordinate grids have been moved up to the fifth grade, so I will not be doing that yet. It saddens me to hear that parents like the weatherman do not think that the children are being taught math for understanding. We try so hard to teach for thorough understanding. Luckily we are departmentalized so that I have the advantage of only concentrating on math instruction which seems to make a difference.
I would welcome any comments from middle school teachers on what you think we could do better on the elementary level to better prepare them for you.
I agree with Kathy - it would be great to know what middle school teachers would want for incoming students. I always try to think of what skills to work on with my kids to help them be better prepared but if there were specific strategies that would help them, too, that would be great to know! I have briefly exposed my fifth graders to tape diagrams and done a few lessons on the coordinate grid. Anything else we can work on to better prepare students for grade 6?
DeleteWendy Lee
DeleteWhile I cannot speak for 6th grade, I can tell you that by 7th grade we would really love to see students have a good understanding of fractions and their operations. I can tell you that we battle the entire year encouraging students to move away from decimals and use fractions. The second thing that would be helpful would be to have students understand that they really can divide a small number by a larger number. Any work that you can provide in these two areas would be greatly appreciated by your colleagues. More important is the fact that you asked! This shows real reflection on your teaching practices which I'll bet your students already benefit from.
Kari Courchesne
ReplyDeleteThis past week I’ve been working with my students on solving problems using double number lines. It’s amazing to me how quickly the students picked up on this model. They are able to successfully solve problems involving percentages, ratios and rates, and conversions between units of capacity, mass/weight, and measurement. When I first taught these concepts, students used proportions to solve the problems. They really struggled with identifying the part and whole in each problem and setting up the proportion correctly. I struggled with teaching this as well and helping students to identify (independently) the part and whole. While students solved the problems with the double number lines, many of them were able to see that the whole or total in each problem matched up with the 100%. They were also able to articulate the part and whole on the number lines. In fact, a couple students instantly connected it to the proportion. I also found that there was a greater success rate in solving the problems opposed to students just using proportions. My plan is to wait a week or two and have students solve additional problems involving these concepts using any strategy they prefer. I’m interested to see how many students will continue to use the double number lines.
There were some students who saw vertical multiplicative relationships between the two corresponding numbers on the number lines opposed to the horizontal way we discussed in class. They tended to solve problems using this method instead of going back to the unit rate or a “benchmark” percent. They were able to come up with the correct solution using this method. I’m assuming this is ok for students to do. Has anyone else used double number lines with their students and observed this?
I haven't used the double number lines with students yet, but I think that drawing the double number lines vertically and not just horizontally would be fine. It seems similar to using a horizontal number line or a thermometer (vertical number line). I was able to show some teachers how the double number line works today and they really liked it. I will check in with them to see if they showed them to their students and if not, suggest it. They make things much more clear for me. I think that if students show a fraction of the enthusiasm and understanding that we did with double number lines, they will have great success.
DeleteThe last post was from me. Forgot to put my name. Sorry, Sheila
DeleteRecently we started working on the Gulliver's travels project in 7th grade. So far it is going well, we have read through the passage outloud and identified the scale factor for both locations. We have also watched the movie, and discussed the symbolism the author uses for the political things going on in that era (which I'm not extremely familiar with) but we had a history teacher come in and talk a little about it.
ReplyDeleteWe then had the students pick an object that will fit in their hand to have them reconstruct. Some of them picked really neat objects that will be a challenge, and others were more simple. We haven't started the construction of these items, but they have been brainstorming ideas of materials, etc. It is going to be really neat when they come to fruition. Thank you Anne for this task! really really neat!
I just created a blog account with google to possibly use in my classroom so I didn't publish this as anonymous, sorry :(
DeleteMy students started expreimenting with double lines and I am excited to see their version of vertical double number lines on their homework. One student of mine asked for extra home enjoyment packet so she can use the double number line to solve the problems. Colorful lines make it so pretty that I had to replace my student work wall to show it to the rest of the students including other grades.
ReplyDeleteSelda Seppala
Selda, it is great to hear the enthusiasm with your students doing the double number line. I have had similar experiences when introducing the concept to my students. There have been a lot of moments when they finally have the light bulb go off!
DeleteBrianna
Selda,
DeleteYour comment about posting the new colorful work resonated with me. I think posting student work with specific feedback is so helpful to far more than the student that it was intended for and I know it takes time to do it, but bravo to you for taking that time at the end of the year when many people may figure why bother, school's almost out.
Also, it's really cool that your student asked for an "extra home enjoyment packet" and even cooler that you called it that. The way we portray math has such an impact on our students.
Sheila
Nicolas Gendreau
ReplyDeleteAfter doing the Gulliver's problem it made me happy that I never went into engineering. I'm happy with my paper and whiteboard but it did turn out to be a really fun project. Working in a high school I would find it hard to see that a teacher would have time to properly complete such a project but it would be awesome to see what the geometry students could come up with given the time.
I was hoping someone could explain to me why TAU is "so much more accurate" than PI. I remember the TAU method obviously working but as to why it comparison it seems that much better didn't fully click with me. Thanks!
Lindsay Adams
ReplyDeleteI have been using the Cartesian plane to graph with my students and they absolutely love it! They have just gotten the hang of actually graphing the fractions (as this is their first year working with the coordinate grid) and comparing fractions, too. I've shown them how to use the graph to find common denominators and plan on showing them how to use the grid to compute operations with fractions, as well.
I've also briefly used the tape diagrams. In fifth grade, we don't work with ratios yet so they have only experienced them when multiplying with fractions. Again, they loved it. They said they really like these strategies because "they don't have to do any thinking"!
I love that they think they are not thinking and really are!! I have had similar comments made when introducing these different strategies with my students. I am also glad to hear that you are introducing these concepts in elementary school to build the foundation for the future!!
DeleteBrianna Clancy
Just give a shout if you need help. I'll do my best, but so far a few of the problems on the end of course hw are pretty challenging. Nonetheless, I'm all for supporting and collaborating.
DeleteCongratulations on your new sign!!
Sheila
Hey guys, just wanted to let you know that I love teaching these new strategies, even with my grade 4 ELLS! (just typed about an hour of reading and lost it without publishing - this will be much shorter, GRR)
DeleteI had used the coordinate plane with coordinate pairs (gathered from the internet) and the kids plotted the points then connected them with lines to form pictures - they enjoyed this so much and put so much effort into being precise that I also gave them large poster size grid paper and they numbered the x and y axes themselves then "magnified" or enlarged their pictures using the same ordered pairs. They loved it!! Begged for more, asked to take home to finish and brought back completed!
I really love getting kids to love and understand math!!!
mo callahan
As the end of the school year approaches I am very grateful for the activities you have shared with us, Anne! A few of my 7th graders have checked out "MCAS is over, why are we still doing work?" Time to get creative!! I love the Gullivers travels scale project. It aligns perfectly with the 7th grade standards and will be a fun review the wrap up the year. I shared this idea with the other 7th grade math teachers and we have decided to make a whole week ( maybe more) of it. We'll start with figuring out the scale factor together as the passage may be a bit challenging to them to be through themselves. Then on to construction! We are going to let the students choose what object they would like to enlarge. Some student will really get into it and challenge themselves! Then wrap it all up by showing the movie! I'm looking forward to seeing their finished products!
ReplyDeleteSince the beginning of class I have implemented a lot more of the tape diagram and double number lines. The kids were uncomfortable at first but since requiring to use at least one different strategy they have become more comfortable and even use it as their primary method of solving problems. MISAEL Ramos
ReplyDeleteJust finished the pencil activity with my students this week and they were so excited. The students were led to believe, to build confidence that it was a high school math problem and they took the bait. Unfortunately they lost the bet, it took them over 2.5 hrs to put them together, so the I have the best teachers sign going up in staff lounge on Tuesday.
ReplyDeleteWhen I start with the end of course assignment, I may (know) I will have questions on the Cartisian graphing. So if any volunteers are out there blog back, please so I can get some help. The one main issue, for me, are those graphs so an SOS will be going out.
Victoria Ellis
ReplyDeleteGisella, I would be happy to help you with Cartesian coordinate plane graphing. I have done this with many of my students that have trouble with fractions and have had some very good results. Students that were not understanding fraction operations had the opportunity to visualize the multiplication and division. Also, they had an "aha' moment with the idea of common denominators because they are stronger with graphing so seeing where the two fractions had the same x-coordinate made a lot of sense to them.
I have the Gulliver's activity scheduled for our "send out" project during the last week of school. I am excited because proportional relationships and surface area have been the two major critical focus areas for grade 7 math according the Common Core and this is the perfect way to tie up those concepts in a way to keep the students engaged at a time when they just don't want to do work and have their mind focused on summer!!
I am having trouble with question number 13. I see that it took him 5 hours to travel some distance @ an average speed of 60km. So wouldn't that mean he traveled 300km? If yes, then I cannot figure out the problem because; 90km = 60% of the trip which would be 270 miles. For the remaining 40%, he traveled 25% of it at 60km. which if I am doing my math right (LOL) would mean he has gone farther than 300 miles? Any thoughts
ReplyDeleteI'm also having problems with this one! I was thinking that we needed to know the total distance to solve. I'm totally lost!
DeleteNicolas Gendreau
DeleteIt does mean he traveled 300 km but for an average speed of 60 km/hr he would need a negative speed...
mo Callahan
ReplyDeleteHey all! I know it's late, but I did want to mention a couple of things. I agree w/ Dr. Collins, too many students do not understand the concepts they are trying to learn - this is why I chose to be a Mathematics teacher; I didn't understand the why and how of math, and ended up dropping my attempts in high school.
When I was attending my undergrad classes and teaching mathematics was a requirement, I found that learning HOW to teach the background of math was exciting to me - here I am!
This has been a more than horrendous year for me and my family, but getting back to the Lesley courses has been a tremendous distraction for me. Thank you all for all of your support and helping me to learn!!!