Problem Solving

I am currently Teaching two Extended Algebra 1 classes. Because we have extended time I covered the necessary material and we are using the next 4 weeks to review for the Algebra 1 Core 40 Exam. I have been revisiting old activities and using the "High Achiever" website on the doe.

The address is ..... http://dualmus.doe.state.in.us/hiach/login.asp?Valid=1

All the questions are multiple choice and students take the exercises on the computer. I am able to create any length of exercise and pick questions from a bank of questions that meet the Algebra 1 standards. We generally have a review day/activity and then complete an exercise on High Achiever. My hope is that this will help the students feel more comfortable when they take the Core 40 via the computer..... Ok enough about that.
Because we had been reviewing all week I decided to use Friday as a day for problem solving and at the end of class a we went over yesterdays High Achiever exercise.

I gave my students the following problems.
Half of the room received this problem the March Issue of the Mathematics Teacher.

The other half of the room worked on the following problems from Menu Collection Problems Book.

4.5 Nails and Pails
A pail containing 40 nails weighs 175 grams. The same pail with 20 nails weighs 95 grams. How much does the pail weigh? How much does each nail weigh?

4.2 Leading by a Head
A six foot tall person walks around the earth. How much further does the person's head travel than the person's feet?


If students finished early they were to work on the other problems. I really enjoyed working with the students on all 3 problems. When we finished the students posted their answers on the board and we discussed different solution methods to the problems. The students struggled the most with the Mowing the Lawn problem and found the Nail problem to be pretty easy. However, we had multiple solutions methods to the Nail problem. Between my two classes we saw at least 5 different solutions methods, including one that involved drawing pictures, one that involved systems of equation, and one that used system of equations thinking but wasn't presented in that manner. As a result of the variety of solution methods I knew that it was a good problem. Students were engaged during the solution process and as we discussed the different solution methods. I know the problem might seem elementary, but it was the thinking and problem solving process that I was evaluating. We were also able to reviewthe elimination method for solving systems of equations and about how we could or couldn't use a proportion to solve the problem.

The mowing problem also had varied solutions, but what held up the students was knowing where to start and what to find first. After some hints the students were able to arrive at an answer and present a solution. The various methods for this problem after arriving at the areas involved, proportions, the d=rt formula, and unit rates.

Just wanted to share my experiences.

Video Proofs

I watched several of the video proofs, but my favorite by far was the Burning Tent. The voices and the extra character was entertaining. As for the math, the thought process seemed logical. It would have been nice if they had provided a column proof on the page like some of the other students provided. I really like the idea of posting the proofs of your students. It is pretty cool to have a math proof floating on the Internet. If I am at the High School next year and have access to the GSP software I would like to record some of things my students complete and post them on the school website. Is the JING software you used free???

Caterpillar Project

I used the activity that Dennis shared two weeks ago with Pre-Algebra students this week. They really enjoyed the activity and hopefully learned something. I thought it was great the next day when we were reviewing that an absent student was able to follow what we had done by looking at the caterpillars and was able to derive the area formula.

We had completed an exploration activity for circumference as well, and I am hoping that the students will remember more about circumference and area in the future. Several of them swore that this is the first time they have ever seen Area and Circumference formulas. I nicely told them not possible. Its not possible for a 9th grader to miss these two things given that they first appear in the 5th grade standards. Some days I wonder what I am teaching?????

Presentations + Last Week = Awesome

Last week was awesome and I am excited to see what will be shared tonight. The ideas and lessons were great last week. I am set to use Denis' circle activity tomorrow. I just wanted to thank all of you that have shared activities and offered suggestions this semester. I have learned so many things this year and I am excited to use them next time I teach geometry.

Update on Wheel of Theo...
I was working on GSP assignments this week and in the Pythagoras section (Chp.8) It has a lesson on how to create the wheel using GSP. It also gives instructions for creating a tool. Denis asked about this in class. I had tried it, but I was missing the first part. The book says that you need to identify reference points before selecting the new parts.

See you tonight.
Is tonight pizza nigh??? or next week???

Wheel of Theo in the Gym

I just finished building the Wheel of Theodorus in the gym with my 4th period class. It went pretty well and the kids did a great job with their accuracy. They were a little out of control at times, but it was a nice activity for a Friday. Compared to the hand drawings we were right on target with the number of triangles that we could make without overlapping. The students were able to explain what we were doing to both the Principal and the Ast. Principal. The only thing I would do different is to tie the bottles together with something thicker so that you could see the string on the picture.

ARTICLES

Perimeter and Area:
This was a nice article and was similar to the activity that Bill shared in class last week. One part that really spoke to me was in the first paragraph. The following things were stated...
1. Students develop knowledge through interaction between the student and the knowledge
2. Students do not think like adults
3. Students learn well through social interaction
There is a lot of truth to these 3 statements.

Chp. 9 Exploring Connections between teaching, learning and assessing Mathematics for Understanding

I haven't finished this article, but if you are going to read on I would suggest this article. The teacher/researcher is focusing on opened-tasks. She had been a math consultant for 2 years and discusses the steps that it took to change her classroom.

Wheel Of Theodorus

This is a great activity that covers several objectives. Rational Numbers, Irrational Numbers, Geometry, and Pythagorean Theorem.
The activity starts with creating a unit of length. Once I have the unit I cut a unit by unit square out of the corner of a note card. This tool will act as a ruler for the students throughout the activity. Using the "ruler" start in the middle of a large piece of paper and make a right isosceles triangle that has legs one unit long and then connect the legs creating the hypotenuse. The next step is repeated as many times as you like. Make the previous hypotenuse the base of the new triangle and use the ruler to add a leg that is one unit long and is perpendicular to the base. Then connect the unit length leg back to the middle. Repeat, Repeat, Repeat.... I have my students use the Pythagorean Theorem to find the length of the hypotenuse for the first ten triangles. They see a pattern quickly and could give you the first ten hypotenuses without the work, but I make them show the work.
I only require that the students draw about 15 triangles, but many of them go much further than this. Several of them will make 50 to 60 triangles attaching extra papers everywhere to make it work. When the students finish drawing all of the triangles and the math, they spend some time decorating or adding on to the triangles.
See links below for examples
http://www.ldlewis.com/Teaching-Mathematics-with-Art/images/Wheels-of-Theo/Kyles-turkey.jpg
http://www.chatham-nj.org/156620131125737203/lib/156620131125737203/IMG_0198.jpg

This year I have thought of two different extensions.
a) Using GSP. I figured it would be pretty cool to make on GSP and it was :) I will post the file on the discussion board if you would like to see it.
b) The second activity has yet to be completed. I would like to take my class to the gym and use yarn and water bottles to build a large scale version of the Wheel of Theodorus. If I am successful, I will let you know.

I originally got this idea from an article in the Mathematics Teacher.

Bell Tower

I really enjoyed Monday night's class. It is nice to be able to go outside the classroom and see the concepts that we are discussing be applied to our surroundings. I especially liked the Bell Tower activity. I was actually quite surprised at how close our estimate was to the actual height of the bell tower. Using sketchpad to model the situation was also exciting because it allowed you to complete the problem without knowing anything about the tangent function. However, it did require me to think about scale (something that hadn't been part of the lesson).

Chp. 6 Linear Measurement

I was surprised at how poorly some of the students did on the activities. Especially the students were calling something 5 1/2 when it was only 4 1/2. I guess I understand the confusion, but it doesn't seem to make much sense. As I was reading this article I was reminded of my father-in-law who is always talking about teaching the grand kids "stuff." I think that response to him sometimes is "oh they are to young (1, 2, or under) they couldn't possibly learn something," however as I go through this article some to the issues are just misconceptions. My guess would be that not all of these students have a grandpa who is encouraging them to work with different toys or "equipment." I don't know if this is even making sense.......What I am trying to say is that this article really made me a believer of what my father-in-law does/says. He really is teaching our kids "stuff" that will be useful to them in the future.

GSP!!!!!!!!! IT WORKS

I was working with my students today on squares and square roots. We started the lesson with finding the area and perimeter of different rectangles and squares. Then I gave the students squares with a given area and asked for the side length and perimeter. They all did fine until the they got to the square that had area equal to 110 sq meters. Some said 55 and others said 10 point something....maybe 10.5 they said. We then checked a square that was 10.5 by 10.5 and the area was to big. So others suggested 10.4 and the conversation continued.
I had GSP pulled up on the SMARTboard to see if we could make a square that had area 110 and then measure the side lengths. I was going to have the square ready, but I decided to see if they could create a square as a class. The initial response from the students was to just draw a square. I said how and they said with a ruler and pencil. I told them the program didn't have a ruler that we would have to do something different. One person said, "use that line thing and just do one that looks right." This brought up some debate about whether or not our sides were the same or if we had right angles. We used the measure tool and found that we didn't. We decided to erase everything but the line segment. I gave them a hint and drew a circle using the segment on the screen. From there they were able to create the square. They knew that radius would help them and they knew it had to be perpendicular. The most surprising part was that the students were telling me to make another circle. They wanted another circle that had the same radius with the center at the other end of segment.
After creating the square we measured the side lengths and area. We manipulated the square to make it match the squares in the exploration of area and perimeter.
The most amazing part.........these were Freshman taking Pre-Algebra extended. If you don't know what that means....they are the lowest 30 kids of the freshman class. They were able to figure out GSP and some were saying...."do we get to do this." I was shocked. I usually hear "I'm not doing this, it is stupid."

Supplementary Angles / Colinear Angles

FYI I could not find a definition for supplementary angles that didn't involve the word "pair" or the phrase "two adjacent angles."
I also could not find collinear angles.
So I guess the proof needs revised.

I can see my blog now, I don't know what was wrong.

Testing Blog

My blog is not showing up. If you can see my blog, please let me know because I can't see it.

Algebra/Geometry and Frustrations and GO CUBS

Algebra and Geometry
This is my first year teaching algebra and I use geometry mainly by using area models as examples for distributive property. I used area and perimeter both last week when I was working with quadratics. I was giving students a fixed perimeter for a rectangle and asking them to come up with whole number sides that satisfied the perimeter. Then they were to find the area of each rectangle, make a table, and graph. This was one our our introductory graphs to quadratics. When using geometry ideas to teach algebra, I try to stop and ask students some questions about the vocabulary and explore any questions or misinterpretations.

Frustrations
So our corporation is cutting spots, just like every other corporation. We are thinking about combining our two middle schools which is creating a huge fight. As a result of this combining I may be moved back to the Middle School next year. In fact I may be moving back regardless. I said all summer that I didn't want to go to high school, but now I don't want to go back. I am excited about the prospect of teaching high school geometry next year and was planning to use my masters project. Now I might not even be teaching high school....and that's just the beginning. My wife is an art teacher in the same corporation and her job is on the line and she may be cut or only be half day.

GO CUBS
My CUBS are doing awesome 4-1 so far in spring training. I know spring training doesn't mean anything, but it is looking good.
GO CUBBIES

Slavit (Has he worked in a Middle School???)

I thought this was a well written article and enjoyed following the activity. The students did a nice job of using maniplulatives to prove or disprove ideas of congruence and similarity. However, he starts the article by saying "The development of intuitive understandings of geometry is highly important at the middle school level" and then moves on with the article. As I am reading, I try to picture this taking place in a middle school classroom that I have taught in. I am amazed by the conjectures that his students are making and can't believe how they are exploring the ideas presented. Then at the end the of the article he says "even though these activities were used in a college classroom, they are immediately adaptable for use in a middle school classroom with little or not change." Hmmm...... Really. You think so, have you taught middle school before....Ok enough with that.

As I said earlier, I think the activity is a great idea and if I ever get the chance to teach geometry again I will try to use parts of this activity to discover the ideas behind triangle congruence. I also agree with Slavit's approach to teaching. He believes that our students should be "doing math" not just "hearing or seeing mathematics." This a difficult law to live by though. It takes a lot of preparation and you have to be willing to face defeat. Not all activities work as well as this one. Despite the difficulty of this type of teaching, if we believe it is best for students, we must persevere.

SPRING BREAK
March 23-27.
I am pretty sure that we are planning on going.....that's right no where. Unless I get a big fat stimulus check in the mail......lots of sarcasm.

Learning and Masters Project

These two things seem to be connected. I would really like to learn more about the proof process and how I can use GSP and other activities to help students connect to the ideas. Some of my early thoughts have been about setting up a template for a digital dictionary for geometry definitions. I have thought about what format I would use and what things might be with each word. Ideas....
1.) Picture/Sketch
2.) Definition in own words
3.) Formal Definition
4.) Examples and even counter examples
I also would like to build activities that would allow students to discover some of the definitions themselves. I am a little concerned about how many things I can use because many of the activities are already in the GSP activity book. Another concern that I have is availability of a computer lab and the programs that I would be able to use.
Even if I can't make a digital dictionary I would definitely like to find a unique way for students to keep track of the things they have discovered/learned during the year. I would also like to compile and create activities that engage students in geometric thought.
I have been researching different geometry texts this week and I have been looking at the topics covered in geometry. I have never taught geometry so I have been talking with teachers in my building and in Muncie to see what is covered and how they are currently covering the material.

Memory is the residue of Thought!!!!

Response to the Jane Keiser article and an Article from the American Teacher by Daniel T. Willingham.

It seems that the main point from Jane's article is that students need to produce their own definitions first. The concise definitions that we have for terms may not and in most cases do not register with the students. It is very possible that there are words in the definition itself that they don't understand. However, I think the idea goes beyond just definitions. The key is that the students are working with and idea and making discoveries. They aren't being feed definitions and examples. My experience has been that if students discover something themselves they are much more likely to remember the definition, concept, or procedure in the future.

"What Will Improve a Student's Memory?" In this article by Willingham there is a statement the supports the thought of students struggling with an idea, making their own conjectures, and then revising. His statement has nothing to do with mathematics, but definitely applies.
"..., the first principle for students is that memories are formed as the residue of thought." If are students are just given a definition and are not asked to think about the concept or make their own definition there most likely won't be a memory. If a student creates a definition and then defends the definition he or she has definitely devoted thought to the definition. Then as the created definition receives praise or is revised the student is engaging in thought.

Therefore, (sounds like a proof) it make sense that we give students opportunities to think about mathematics through activities and investigations.

Technology

Technology:
I really enjoy using technology as a learning tool and so do my students (most of them). I have enjoyed working with Logo. Although it seems slow at times, it makes me think visually. I find myself picturing the turtle in my head and thinking about the direction I should make him turn. It also makes me think in my head about how far I should rotate the turtle to achieve the angle or direction needed. I also find that if you are precise when working with the drawings the activity encourages mental math. I find my self using multiple mathematical ideas such as the Pythagorean theorem to know how many turtle steps to take.
Logo:
I think this program would be useful in both middle school classes and in high school geometry. It would help students visualize the geometrical ideas and place them in control of creating the shapes. In order to correctly produce the shapes using logo students will have to understand the basic ideas and properties of the shapes.
GSP:
I have used GSP in the past and really enjoy working with the program. I used the program when I taught in middle school. We would use it to explore properties of shapes and their area and perimeter. I used the program to show students a proof for the Pythagorean theorem. We also used the program to look at transformations of objects. It was helpful to use the coordinate plane as a background and then we could look at rotations, translations, and reflections.
Vocabulary:
Vocabulary in Geometry is important just like it is any other class or setting. If we are a football game and you don't know what a first down is, then you are in trouble. If you don't know what a QB or a receiver is then you aren't going to understand the game. When you find yourself in a setting where you don't know the vocabulary it seems as though everyone else is speaking in a different language.
I find that some of my students are this way. On the surface it just appears that they don't understand or aren't trying. However, if I start asking questions, I find that the student has no idea what we are talking about. He or she doesn't understand the terms that we are using and therefore can't follow the discussion, activity, or examples. I also find that students may know a word or its definition, but they don't understand how to use it or what it really means. Example....I asked my Pre-Algebra students early in the year "what is area?", several of the responded by saying "length times width." I said, "no, that is the formula for finding the area of rectangle, what is area?" They again said "length times width." Many of students have partial understanding of concepts or ideas that we talk about on a regular basis in the classroom. This particular lesson wasn't even about area, it was just using area as an application of the idea. In conclusion, vocabulary is very important and it is important that we (teachers) meet students at their current level of understanding and work upward. As said in the Van Hiele article, if you speak in level 2 to someone who is currently in level 0 they won't understand what you are talking about regardless of how well you are explaining the idea.

Last Night

Good Morning,
Sorry I missed class last night. We had the first, which ended up being the last game of our GBAA tourney last night. We got WAXED. The other team had a 6'3'' 7th grader who could play. Ok, enough about basketball. I hope that a few of you can help me catch up from last night. To those of you who took the ISTEM class in the fall, do you know if our NCTM orders have arrived? Look forward to seeing all of you in class next Monday.
I'm out.....like my team in last nights game :(
Mark