Wed, Dec 19 #6

Here is a teacher generated Tree Map about graphing rational functions. I used it to summarize three days of activities, discussions and examples about the complexities of rational functions. The students were asking lots of questions about particular procedures and parts of the examples, but I didn't think they had a strong grasp of the whole picture. Using the Tree Map was very helpful because it made obvious the differences between local and global behavior. It also helped to categorize all of the procedures and definitions and examples we had discussed.

Wed, Dec 19 #5

Here is a student generated Flow Map for graphing a rational function. We had just spent 3 days learning about essential and removeable discontinuities, vertical, horizontal and oblique asymptotes, local and global behavior, etc. The student came for extra help so I did one more example, and then I asked her to create a Flow Map in preparation for the exam. It seemed to work well for her. I know it was on her mind during the exam because when I asked a "stretch" question she let me know that it wasn't on her Flow Map!

Wed, Dec 19 #4

Here is a Double Bubble for comparing and contrasting procedures used to solve rational equations and simplify rational expressions. Students often confuse the two, so I used the Double Bubble to help show the similarities and differences.

Wed, Dec 19 #3

Here is a Dancing Definition for the purpose of the quadratic formula. I put it together quickly during one of the NUA workshop days. I haven't used it with students yet becuase I am not teaching it this year. However, I have proposed to my trig students that they create and play songs for all of the core trig identities. Extra credit is involved if they record their tune and post it on YouTube. I will mention that my 5 year old was having difficulty memorizing our phone number until my wife and I set it to Lullaby. With the tune, he learned it immediately. I will keep the blog posted about the YouTube Trig Identity project. We'll see how it goes.

Wed, Dec 19, # 2

After working with volumes of revolution for 3 days in my calculus class, I asked the students to create a Flow Map for the process used for these particular kinds of problems. The students took between 5 and 10 minutes to create their Flow Map. For me, it was a good way to quickly assess their understanding without having to look at the details of the calculus and algebra of how they set up and solved the problem. In other words, I got better and quicker information about their conceptual understanding from this Flow Map assessment than I do from looking at their alebra and calculus work from a solved problem. In particular, I learned that for most students, the first decision they tried to make was whether to use a disk, washer or shell despite the fact that I had emphasized repeatedly in class that the first decision should be whether or not to use a vertical or horizontal slice. That observation of mine generated some good discussion.

It was obvious the students were using higher order thinking skills, and it obviously pushed them a bit more than the standard requirement of just solving the problem. I didn't tell them to use any specific language or to use any specific number of boxes. I did require that they follow the left to right Flow Map procedure.

Wed, Dec 19 #1

The Multi-Flow map at the left for the Intermediate Value Theorem is a useful way to present any theorem because it emphasizes the hypotheses of the theorem as well as the conclusion. The students tend to focus on the conclusion. The Multi-Flow presentation helps emphasize both parts. This map was teacher generated not student generated.

Tuesday, Dec 11

In calculus class today I used the basic NUA strategy of movement because it was 15 minutes before the end of the school day and the 6th hour students were falling asleep. We were working on volumes of revolution. I asked the students to stand up and put their left hands in the air to represent the revolution axis. I really didn't know what to do next, so I asked them to use their right hand to create a slice perpendicular to the revolution axis. They automatically put their right arms out perpendicular to their bodies and rotatated around creating nice examples of the washer/disk method. Next, they put their right arms up and spun around to make examples of the shell method. After the example they quieted down quickly and the last 10 minutes were, I think, more productive then they would have been without the stretch.

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.

Mon, October 22

Teacher S and I have been working together in Geometry. We used Flow Maps last week for an algorithm for working with the formula for the area of a trapezoid. See image above.

We are going to develop the equations for the circumference and area of a circle using some hands on activities. For the circumference, I will bring in about 20 different cans and bottles. Each student will have a string and a ruler and they will measure the diameter and radius of their can or bottle. They will enter their can's diameter in List1 on Teacher S's overhead TI-83, and their can's circumference in List2. I will plot a scatterplot and line of best fit and we will discuss how to use the line to find the circumference for a can with a given diameter. I will tell them the slope of the line is 2(pi), hence C = 2(pi)r. For the area of a circle, each student will be given a circle with various sizes of pieces of pie marked. Their job will be to cut out the pieces of pie and change their circle into a rectangle. The height of their rectangle will be r, and their length will be (1/2) of the circumference. Hence Area = r*(1/2)(2)(pi)(r) = (pi)(r)^2. See the image above.

At the end of the chapter, the students will create a tree map of the areas of the shapes studied in the chapter. Instead of asking the students to create the tree map from scratch like we did in Lizzy's class, I have created a blank tree map with a list of labels. The students need to put the labels on the map in the correct places. See the two images above - one of the blank tree map and one of the scrambled labels.

Wed, Oct 10

The images above include the Telescop Lab with a student sample, a student sample of a Double Bubble used for comparing graphs, and Thinking Maps used for Direct and Inverse Variation.

Tues, Oct 2

Teacher J and I finished AA sections 2.1 and 2.2 using circle, tree and flow maps. If we were to do it again, I think we would stick with the tree map and flow map but not the circle map for problems involving direct and inverse variation. We were both very surprised at how many students found it very difficult to construct a tree map from the important information from a problem. I would definitely ask students to construct tree maps again because it puts them in the position of having to group and label before doing the algebra. If they cannot organize the information they will not be successful with the problem.

Time was an issue because fully asking students to create their own flow map, post them, and discuss them as a class would have taken 2 or 3 days longer than we wanted.

My conclusion is that there is a place for thinking maps in ch 2, but I would do less maps and spend a little time with the one I chose. They were a good way to organize a constructivist approach, but I don't think we wanted quite so much of a constructivist approach for lessons 2.1 and 2.2.

Teacher J had a great idea about using dubble bubbles for comparing and contrasting the graphs of y=kx, y=k/x, y=kx^2 and y=k/x^2. We will use these at the end of Ch 2.

Wed, Sept. 26

Teacher J and I finished creating a hands on activity based on a website I found for direct and indirect variation. The kids make telescopes and take measurements on three variables. On Thursday and Friday we'll do the activity and lesson 1.1 using some NUA strategies.

Friday, Sept 21

Today I subbed for a biology teacher during period 3 so she could meet with Teacher R to discuss NUA strategies for biology. I am available to do this for anybody during 3rd or 4th hour. And Teacher R can cover a math teacher's class during 3rd or 4th hour so they can meet with me.

Also, Teacher J and I are going to change Adv Alg lessons 2.1 and 2.2 to a lab with a cardboard telescope. It's a discovery lessons for direct and inverse variation. We'll use thinking maps to organize the ideas, and then we'll proceed with the lessons from the textbook. I'll keep you informed as to how it goes. I found the activity online and tagged it under AdvAlg on my del.icio.is account.

EPDM Logic Map

To review for the EPDM Ch 1 test today, I gave the students a list of about 20 words from chapter 1, including: statement, for all, negation, argument, valid, converse error, etc, etc. There job was to create a tree map that included all of the words. All of the students were able to create the tree map, and three students wrote theirs on the whiteboard. All students maps were different, and there was some good discussion and why the maps were different and why some were right, some wrong, and for some it didn't matter. It took about 10 minutes for the students to create the map and 10 minutes to discuss it. It allowed me to emphasize key points from the chapter, and it seemed to spark some engagement with the students.

When I asked these juniors what they knew about thinking maps, the class was able to list off most all of them. They were very familiar with them.

I will definitely use tree maps again for a review activity.

Adv Algebra 2.1

I spent last Friday, Monday and Tuesday looking at Ch 2 in Adv Algebra to develop lessons that integrate thinking maps and NUA thinking strategies to improve instruction.

Adv Algebra For Lesson 2.1, the students will start with a Possible Sentences activity. For each of the following words or phrases they will write one sentence that uses the word or phrase. It doesn't have to be a math sentence. 1)... is proportional to... 2) function 3) varies 4) directly proportional 5) constant. A few sentences will be shared and the words and phrases will be informally defined and discussed. The purpose of the activity is to engage the students with the words and phrases used in the lesson, and it should take about 10 minutes. Next, students will read p 72 and then use a circle map to define direct-variation function. The purpose of the circle map is to present the key word along with the ideas, symbols, and examples that are associated with it. It's a way to see the key word in addition to the orange box in the text. (5 minutes). Next, students will read example 2 and represent it first with a circle map. The key idea goes in the middle circle and all the associated parts in the outside circle. The purpose of using a circle map for a word problem is to sort out the important parts from the unimportant parts. (5 minutes). Next, students will represent example 2 with a tree map. The purpose of the tree map is to organize the important parts from the circle map. (10 minutes). Now, we'll solve the problem by finding k, etc. If time, we will do another example, and then tie it all together with a flow map.

That's the plan. I wonder how it will actually work?!

Thursday, Sept 13

I read the Thinking Strategies for Student Improvement book we received last year and found some good ideas that should work well in a math class. I would like to try them. I visited Teacher B's and Teacher RT's classroom.

I think the next step is for me to help plan and teach an entire chapter. The best candidate is Teacher J because she has Adv Alg Per 3 and then Prep during per 4. We can plan together and I can help her during period 3. Teacher S is the other possibility but I don't want to jump in with a Cog Tutor class because of the added complications of texts and computer work I am not familiar with. I think it's better if I start my "coaching" with a class with a regular book. After working with Teacher J, and blogging about it, I'll move to one of the basic classes.

I spoke with Teacher R about this strategy and she thinks it is a good idea. She is using the same model with the English Department. I will meet with her again tomorrow because she has some examples of using Thinking Maps to completely work through a science problem.

Mon., September 9

I visited Teacher JE's classroom today and she started with a comparison of mean and median. There was a place for a double bubble. After she presented about box and whisker plots I went down to Teacher T's art room to see if she ever uses Thinking Maps to describe artwork. She does but what she finds more useful than the thinking maps are the NUA strategies.

Friday, Sept 7

After being in Teacher B's 1/3 class, I consulted with Teacher R today to see how she would use Thinking Maps to solve one of the word problems from AA LM1.1A. It was really funny watching Teacher R try to solve a math problem! She misread it, then tried to do a Tree Map and screwed that up, then we tried replacing the variables with numbers and she still got it wrong. Of course, she was laughing the whole time! Anyway, what helped her the most was first doing a circle map to sort of the important words from the problem from the unimportant words. Next, she organized the important words with a Tree Map. She tried another problem that was somewhat similar and she was much, much more efficient. The Circle Map followed by a Tree Map may be a good strategy for a variety of word problems.