Dear Class,
I felt as though we competed in a marathon of mathematics
last weekend. For a large class I was extremely pleased to see that everyone
was extremely engaged and embraced the tape diagrams and double number lines. I
think as we work through using these models we need to keep aware of the fact
that the cells in the Tape Diagrams actually represent a variable used in
algebra but by working with the models before the variables, students
will/should be able to transition from the arithmetic to the algebra. The power
of these models is that they allow students to visualize a situation…not just
teach them to solve one problem.
Using the Tape Diagram in a part-to-whole measurement
problem is effectively portrayed using the Cuisenaire Rods. It doesn’t matter
which rod represents the whole as long as the shorter rods model the ratio
distribution. Students in the younger grades should be familiar with the Rods
if their teachers used Investigations or modeled the mathematics they were
teaching. So, although they may be new to some of us they are not so new to
students.
The comparative model which lists one ratio portion under
another is extremely powerful when there are more than two comparisons being
made. What is really nice about the models is that they work regardless of how
many comparisons are being made and each additional comparison does not need to
be taught separately. I was observing in one classroom when the students
completed comparing two ratios but when the next problem introduced three
ratios, the teacher stopped the lesson and said, “I will show you these
tomorrow.” My reaction was shock because there is no difference in how we
operate on two, three or more ratio comparisons and when students understand
the multiplicative underpinnings the number does not matter.
I personally like the double number lines for percent
problems and rate problems. They plus the graphical representation are
extremely powerful for introducing ratio as slope…an important concept in
algebra. But, the bottom line is all the models are algebraic in nature and
students who are comfortable representing the problems using the models
transition well to representing them algebraically as well.
Hope you all enjoyed the course as much as I did teaching
it.
Anne
I have enjoyed the class so far and have found the information to be very useful to my class. I have used much of it already! (Val B.)
ReplyDeleteHi Val,
DeleteI am happy to hear that. I love your enthusiasm in class...and what you add to the discussions.
Anne
Wendy and Charlene working on the chair problem. We are having a problem finding the dimensions of the large chair due to the angle of the photo. We found the height of Emma in the photo (from the seat to the top of her head) to be somewhere between 2.2 cm and 2.7 cm. Not really sure what to use. Any suggestions?
ReplyDeleteNick Gendreau -
ReplyDeleteThe first thing I wanted to do was find the height of the back of the big chair. Using a ruler I measured the height of Emma's head to the height of the back of the chair (in cm) and got 2.4:3.7. I compared this to the actual height of Emma's head in the big chair which was given as 54 cm. I then made a proportion of 2.4/3.7 = 54/x. This gave me the height of the back of the big chair as 83.25 cm.
Looking at the medium chair we can see that we can measure with a ruler the ratio of the height of the back of the chair to the height of the whole chair. I got a ratio (in cm) of 2.3:4.0. We can use this ratio to find the height of the big chair because we just found the height of the back of the big chair. I set a proportion of 2.3/4 = 83.25/x. I got x = 144.8 cm as the height of the big chair.
As for the length of the big chair's seat I found the length of the small chair to be 22.86 cm using an inch to cm calculator, it states it is 9 inches long. It also states that the length is 1/4 of the big chair so it's safe to assume 22.86 x 4 is the length of the seat of the big chair which I got to be 91.44 cm.
Tom Canty
Deletehey Nick, wow, what a clear explanation for your thought process and approach- ( I wish I would have read your post earlier!) - this problem set was a challenge for me- I was a having a tough time applying the physical measurements using a ruler based on a photographic image ) I also like how you used the proportion to determine the height on the back of the big chair- which you could then use to find other missing dimensions, great job, way to show and explain.
Sheila Commisso
ReplyDeleteThanks for both of the posts about the chair problem. Kari and I started it during class and when we went back to it we were blocked about what to do next. I don't have it in front of me right now, but I do believe we started the same way, but weren't sure how to use that information to go to the next step. It makes me think that I should also double check my measurements.
Also, during weekend, two we had quite a few conversations with Steve about how the wording of problems can be interpreted differently by people and that will possibly change the way that it is done and/or the answer. I mention it because I think that this applies to the problem with the taps in it. It says "after 2 minutes" and I think that this is very confusing. Does it mean that you should turn Tap B on at 3 minutes or at 2 minutes? My first instinct was that if it is turned on at 2 minutes, the tap has been from 0 -1 and 1 - 2, which means in order to include after 2 minutes, you would need to include from 2 - 3. I did the problem this way at first, but then decided that since I wasn't sure if this was the correct way to interpret that language, I went back and re-did the problem with Tap B turning on at 3 minutes. What do people think about this question and the reading of "after 2 minutes"? Did anyone interpret it a third way? I'm curious about the different approaches that were taken. Thanks!
after 2 minutes means just that...2 min 1 sec or any interval between 2 and 3 minutes
DeleteI didn't even think about the language in that problem. I read it that when 2 minutes had passed, the drain would be opened so at the 2 minute mark I had the water start draining. Reading it again, I can see the confusion in the language. It is very interesting how language can make such a big difference in how problems are solved.
DeleteAnne wrote that "anonymous comment...sorry my name was not listed.
ReplyDeleteAnne
Jillian Dart
ReplyDeleteI tried to follow the instructions, but I am not sure I am doing it right, so here goes!
I have been doing tape diagram problems with my 6th graders as a do it now all week. They are really getting good at them. Thanks for such great resources.
ReplyDeleteVictoria Ellis
DeleteTape diagrams have been one of the best strategies for my students to learn this year. Especially for my students that are far below grade level and working on basic skills such as addition and subtraction, they can still be using these tape diagrams! The visual model really helps them!
Daniel Christensen
DeleteVictoria, I couldn't agree more with that well written statement. I also have really enjoyed introducing the tape diagrams to my students because its a great visual for them. In a few cases, I had students come up to me and say why didn't we learn this before. To have students gain a better concepts of ratios using the tape diagrams has been extremely beneficial to both me and the students
Sheila Commisso
ReplyDeleteI really enjoyed the activities that we did last weekend. My favorite being Gulliver's pencil. I know that most groups used a ratio of 12:1. Kari and I used 10:1 and I believe that a few other groups might have also. Since one the Brobdingnags was 60 feet and we figured out that Gulliver was 6 feet tall from the Lilliput:Gulliver ratio, we calculated 60 feet:6 feet and got 10:1. I've been thinking about it since our discussion and I am wondering how people arrived at a ratio of 12:1 and the only thing that occurs to me is that there was mention of a measurement to the Brobdingnag's face. If you used that as the 60 feet, did you assume that from their face to the top of their head was 12 more feet? That makes 72 feet: 6 feet and therefore, 12:1. It is the only thing that comes to mind.
I thought that the comic strip enlargement was great and have reflected on the comment about it being an art project versus a math project. As a matter of fact, I was telling someone about it today and they had an aha moment when I said that.
We spent a lot of time going over our homework and I appreciate that greatly. It gives me a chance to see other approaches and points of view about a problem that I might not have thought of or come to on my own. Thanks.
I agree about reviewing the homework. I find that to be very helpful to me. I like seeing the other approaches and how others solved the problems. I find it useful to help me teach concepts to my students.
DeleteDon't laugh anyone I've never done this but here it goes. I'm having a great time in class, this class seems to be a more hands-on class and more debating going on even at our own tables. Some of the thinking outside the box is so cool, like Boubacars' double, triple line bar to solve one of the problems last week. For someone who enjoys math, but the practical side of it, this thinking outside of the box is weird to me. I grew up in an age of formulas, nuns, and this is the way we do it. Dr. Collins is great with her pushing but not embarrassing, way of teaching, only regret next weekend is the last.
ReplyDeleteAlso can't wait to do Gulliver's Travel with my kids next week, I figure if it took us 2 hrs it's a weeks worth of class for my kids, thanks Dr. Collins for the great activity.
I have really enjoyed our classes so far. I have never taught tape diagrams before (or even heard of them before this year) and was very excited to learn it. I also like the many uses for double number lines, also something I haven't used before. I feel more confident about teaching it to my students now.
ReplyDeleteMo
ReplyDeleteI have been really enjoying the classes. Coming from an upper-elementary grade (4) and working alongside people who teach both middle school and high school has been wonderful for me. I pay close attention to what others share about student misconceptions, and I try very mindfully to be aware of times that myself or others may be (not intentionally) planting those seeds of misconception at the lower grades.
I really like the tape diagrams - I think that if we begin to use them for multiplication, even addition and subtraction, at lower grades the students will just grab onto the idea for ratios.
My favorite new concept, that I had never seen before was using the Cartesan coordinate grid for the fraction problems. I think that students will be able to use this concept to get used to the ideas of common denominator, common numerator and what they show about fractions, as well as understanding comparing fractions and then moving onto using the operations with fractions.
Just a small note - I have attempted several times to comment here, but sadly never "copied" before I tried to post, and my posts have never shown up here. I'm copying this one and will not give up (trying from my work pc this time).
Hi!
ReplyDeleteI have included the double line in my class and saw that a few of my students simply loved the idea of visual learning. I have not expected as much but I am now seeing the double number line being used more than I have expected. Did any one of you had the same success? Of course not as a whole but at least with some of the students.
Selda
I found that the majority of my students preferred the double number line. They liked the visual and seemed to understand problems much better than just trying to use proportions. I found a much higher success rate when they used double number lines to solve problems.
DeleteKari
Tom Canty
DeleteI have also found that students prefer the double number, I think the structure is more concrete for them. They know there has to be 4 values and that each value corresponds to another. I think I will start to incorporate double number lines in a our geometry unit as well- hopefully students make the connection to the concept of a scale factor for similar figures.
Dan Christensen
DeleteMy students loved the double number lines, but when it comes to overall excitement I took the cake. I wish I had these great visual for my students when I first started teaching. A double number line is like an angel on your shoulders. I have seen a lot of students who struggled with rates, and to see them grow from using the double number line has been very rewarding. I love the double number lines.
Selda,
DeleteYou said you've used the double number line and it went great. Have you tried the tape diagram, and if so how did it go? I've been afraid to show them because most are at a 2.3 level in the 6th grade and I don't know how they'll work with it. I'm starting the double number line this week and am hoping it goes ok. If you have and electronic copies of anything for that level can you send it through sps..
Thanks, Gisella
I am definitely an algebraic thinker and I teach this way in class. Given the requirements of common core and various types of learners I am becoming more open to implementing strategies such as the use of tape diagrams and double number lines in my class. Misael RAMOS
ReplyDeleteTom Canty
DeleteI am the same way, my mind automatically goes to the algebra, but as I have practiced more with both the tape diagrams and double number lines, I have found that I can utilize these strategies as visual representations to help support the algebraic reasoning. My students don't like the monotonous steps that I make them to do show out step by step, so when given an opportunity to model something with a picture- they jump on it.
Tom hate to say I think algebra also, but I'm starting this week with double number lines and I think the kids will find it much easier to understand. I would like to try the tape diagrams, but I'm not sure the kids at their level would understand it, any ideas for those kids in front of me.
DeleteGisella Grimaldi
Daniel Christensen
ReplyDeleteI really enjoyed scaling up and creating a big pencil. That activity was awesome and it hit so many different standards at once. This is going to be a perfect thing to do with the kids towards the end of the year to review some concepts on scaling. In addition, I am going to make it a competition which will really engage the students and they will be hitting just about every mathematical practice. Peace out.
Dan,
DeleteI posted before but I can't find it. Glad to hear that your doing the pencil activity with your kids, they'll love it. I bet my kids a sign up in teachers lounge they have the best teacher vs Chinese food and teachers won. The bet was they had to do it in 2.5 hrs or less like we did it, and I gave them hits. Needless to say teachers get sign up next week.
Have fun with it.
Gisella
Daniel Christensen
ReplyDeleteI really enjoyed comparing shoe size to height. I am extremely excited to introduce this activity to my students because it allows them to move around the classroom and engage in accountable conversation. Also, having them create a big graph on the floor will motivate students to participate and be more active learners. Please reply!!!!