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).