Tuesday, November 27

Teacher S and I just finished working with surface area. We were surprised at the conceptual difficulties many of the students experienced when trying to find the surface area for rectangular prisms - even when they were given a rectangular block of wood and a ruler. The problem was only compounded when the students were given triangular prisms to hold and measure or when they were asked to find the surface area from a drawing instead of from a hand held object. However, through the combination of hands-on activities and flow maps, it seems many students were able to finally experience success. After the initial activity of measuring blocks of wood, we proceeded as follows:

1. Students were given a net of a rectangular prism. Before cutting it out, they 1) measued the lengths of the sides 2) calculated the area of each surface 3) calculated the entire surface area and 4) calculated the volume. They had difficulty calculating the surface area and volume from the net before cutting it out and using glue stick to create it. However, it was a very worthwhile activity because they were engaged and they were able to "see" that surface area is the sum of the area of all of the surfaces. I was surprised that many, many of the studnets really needed this level of concrete experience to begin to understand surface area. Click the image to see a larger image of the net we used. When printed on 8.5" X 11", the dimensions are convenient.

2. Some students mastered the above activity quickly. For those who finished early, a new question was added: Construct a new net for a rectangular prism with volume equal to one-half of the volume of the original. Most students wanted to divide each dimension by 2, but they learned you only need to divide one dimension by 2 to get half of the volume of the original.

3. After the net activity described above and the time spent measuring blocks by hand, we thought the students were ready to calculate surface area and volume for figures drawn on the board using formulas. We tried to lecture about the general formula S.A. = 2lw + 2lh + 2wh. We also tried to include V = Bh. There was lots of confusion about the B for "area of the base" and the b in "the base of a triangle". In addition, the students really weren't engaged and were unsuccessfull in finding the surface area and volume of a basic rectangular or triangular prism drawn on the board.

4. To help with the confusion, we decided to use a flow map. Teacher S and I created a flow map with 9 boxes to describe how to find the surface area of a prism. I don't have an image, but it went like this: 1) locate the bases 2) find the area of the base 3) multiply the base area by 2 4) locate the sides 5) find the area of each of the sides 6) add together the areas of all of the sides 7) add together the total base area with the total side area 8) write down your sum 9) add the correct units.
It did not work well for several reasons. They couldn't locate the bases because some thought the base was always on the bottom. When asked to locate the sides, many students thought that meant the length of the sides, so when I asked "Where is the side?" they said "They are 4, 5, and 10." The phrases "total base area" and "total side area" were too confusing and not well defined.

5. To modify the flow map, we did not think it would alleviate confusion to add more steps or try to clarify edge, surface, face, dimension, etc. Instead, we constructed more prisms out of wood and asked each student to contruct their own flow map with their own words. This worked much better. We had several blocks available for the students to calculate, and they had to calculate the surface area and construct a flow map of the process they used. After they calucated a surface area from measuring, etc, they checked their answer with either myself, Teacher A or Teacher R(we pulled in reinforcements). If their calculation was correct and their flow map made sense, then they used their same flow map to do another block. If they got the wrong answer they would modify their flow map, etc. This worked well except many students had difficulty constructing their flow maps becuase they had forgotten what they had done.

6. Finally, we spent one more day on the activity of getting a block to hold in their hands (this time they were triangular prisms - both right triangles and equilateral triangles), calculating the surface area and contructing a flow map. However, the directions this time were to divide their paper in half and to record calculations on the top half of the page and to immediately record their process in a flow map on the bottom half as they were doing it.

This process worked much, much better. For the triangular prisms, with 5 sides, many students made big connections when they realized there were 2 triangular faces and 3 rectangular faces. So, in their flow maps, they wrote things like "multiply the triangular area by 2 and the rectangular area by 3 and add them." Click on the image of the Flow Map to see the student sample.

We are moving on now. The maps and activities were beneficial to many students for the purpose of constructing their own understanding about surface area. How these activities translate into the ability to apply formulas for solids drawn on paper is yet to be seen.

Tuesday, November 13

After I attended the first NUA offsite training, I worked with Teacher S to implement some of the strategies with his Geometry class.

Journals Each student has their own composition book. Each journal has a table of contents and is organized chronologically instead of by topic. For example, instead of a section for notes, a section for homework, and section for handouts, etc., the students always turn to the next page in the book and write the date, topic, and page number in the table of contents. Handouts are folded and attached with glue sticks. Teacher S is grading the jounals on whether or not they are complete and organized exactly as specified. There is a para in Period 4 who keeps a journal as well. Students can refer to it if they were absent, and it is the model for grading purposes.

Grading Teacher S is beginning a seating chart this quarter that is connected to student improvement. All students were randomly assignmed seats on day 1. After the first week, each students' percentage was calculated and recorded based on homework and assignments from the first week. This first score is the base score. All students' base scores were given to them on Monday. The following Monday, new scores will be calculated. If the student sitting in the second row IMPROVES more than the student in the first row, then those two students switch places. If the 3rd students improves more than the second, they switch, etc. It's like a bubble sort. It is only possible to move up one chair per week, but it is possible to move back more than one chair. At the end of the quarter, each student receives AT LEAST the letter grade they earned. However, they could earn a higher letter grade based on their seat. If a student with a C average ends up in the first seat, he earns an A for the quarter. A student with a C average in the second seat receives a B. However, you grade will not be lowered. If you have earned an A you will receive an A even if you are not in the first seat. Note: This grading strategy does not come from NUA.

To begin the 2nd quarter and begin the unit on volume and surface area, we used the following strategies:

1. A-Z Taxonomy for "Topics from First Quarter Geometry". Students put the date and topic "A-Z Taxonomy" in their journals and the page number. They turned to the new page and wrote the letters A-M in the first column and N-Z in the second column. They had 3 minutes to silently write a word beginning with the corresponding letter. They did not have to use each letter, and they could write multiple words on each letter. After 3 minutes they shared with a partner. They were directed to increase their taxonomy by getting at least 5 words from their partner. Next, in a whole class format, students shouted out their favorites and they were written on poster paper in the front of the room. All students were encouraged to share their favorite word. The final poster was taped on the whiteboard and the following day compared with the taxonomies from the other classes. It went really well because the students were very engaged. A photo of the Period 4 taxonomy is included above.

2. To introduce volume, they turned to a new page in their journal, noted it in the table of contents, and wrote the word volume with a circle around it. In 3 minutes, they silently wrote words that they associate with the word volume - math or not math related. This is a significant change from how Lizzy and I did it. We were way to focused. We wrote "direct variation" and had students write words from the definition. The circle map for defining needs to start with a broad concept - not a specific definition. We included a frame for which the question was, "How do you know these words?" A photograph of the period 4 circle map included above.

3. Next, I actually wrote the definition of volume on poster paper along with an association.
"Volume is the amount enclosed.
Volume is the amount occupied.
The volume of the metrodome is enormous.
The volume of an iceberg is enormous."

I said it three times like a chant (but within my comfort level!). The students repeated it back once while looking at it, and then again but with the poster covered. Hand movements were included. This activity went well, but it takes a little convincing that chanting a definition isn't just for little kids!

4. Next, we had several toy blocks and the students were asked to find the volume and surface area of each. Each student got two different blocks and a ruler. They wrote the date, topic and page number in their journal and were asked to sketch each block before finding its volume and surface area. This was much more challenging than expected. Some students thought l, w, and h had to be specific sides and they didn't know which sides were which. The real challenge was surface area. We prompted with suggestion like, "How many surfaces are there?" and "Find the area of each surface and add them together." We will finish the activity on Wednesday, and from their experience try to write a meaningful definition of surface area.