Wednesday, May 28, 2008
12.3 Quiz
I have a quiz covering 12.1 - 12.3. At first I worried that the quiz would be too long, but after my first hour I am much more confident that students will be able to finish the quiz in the time provided.
12.3 Law of Sines
This lesson was split over two days.
On a Friday, when most of the students were gone on a field trip, I went over the derivation of the SAS Triangle Area Conjecture, and the Law of Sines. After the derivation I gave students time to catch up on past assignments.
Many students were able to follow the proof, and I didn't expect all to understand the proof fully.
On the second day I quickly reviewed the Law of Sines and SAS-Area-Conjecture. After a couple examples students began working on homework in class.
On a Friday, when most of the students were gone on a field trip, I went over the derivation of the SAS Triangle Area Conjecture, and the Law of Sines. After the derivation I gave students time to catch up on past assignments.
Many students were able to follow the proof, and I didn't expect all to understand the proof fully.
On the second day I quickly reviewed the Law of Sines and SAS-Area-Conjecture. After a couple examples students began working on homework in class.
12.2 Problem Solving with Right Angles
This is mainly a word problem chapter. We spent one day and went over a couple examples in class. Students had time in class to work on homework and ask for help.
12.1 Trigonometric Ratios
This lesson was split into two parts.
Day 1: We looked at the basic trig functions: sin, cos, and tan.
To introduce the tangent we looked at the ratios between the legs of a right triangle that I drew. The drawing was by hand so the ratios weren't perfect, but they were close to a constant. Then I told them that this ratio is important so it has a special name "tan", I let students guess what "tan" was short for and most classes had a student who would guess "tangent".
Then we went over the names for other ratios, i.e. the sine and cosine.
Our angles are in degrees so we need to check our calculators are in degree mode, then we went over how to use the sin, cos, and tan buttons on the calculator.
Day 2: The arcsine, arccosine, and arctan.
I mainly refer to the arc-functions as "inverse sine/cosine/tangent". This is easier, and worked with my explanation of how the inverse functions undo the sine/cosine/tangent.
We went over how to enter these in the calculator and began working on some problems.
Day 1: We looked at the basic trig functions: sin, cos, and tan.
To introduce the tangent we looked at the ratios between the legs of a right triangle that I drew. The drawing was by hand so the ratios weren't perfect, but they were close to a constant. Then I told them that this ratio is important so it has a special name "tan", I let students guess what "tan" was short for and most classes had a student who would guess "tangent".
Then we went over the names for other ratios, i.e. the sine and cosine.
Our angles are in degrees so we need to check our calculators are in degree mode, then we went over how to use the sin, cos, and tan buttons on the calculator.
Day 2: The arcsine, arccosine, and arctan.
I mainly refer to the arc-functions as "inverse sine/cosine/tangent". This is easier, and worked with my explanation of how the inverse functions undo the sine/cosine/tangent.
We went over how to enter these in the calculator and began working on some problems.
Practice Test, Review, and Test
I did another practice test. Students didn't try as hard on this one, but I think it was still benificial. We also did some review from the book.
Many students have been late to take the test and procrastinating. It may be due to the end of the year, or extreme laziness.
Overall students are doing well on the tests.
Many students have been late to take the test and procrastinating. It may be due to the end of the year, or extreme laziness.
Overall students are doing well on the tests.
11.5 Proportions with area and volume
This is a lot to cover in a single lesson so I split it into two parts.
First we looked at proportions with area. This lead to the proportions of length squared (m/n)^2.
Then we looked at proportions of volume. (m/n)^3.
This lead to an interesting propoerty that the exponent is just the power of how many dimensions we have. I think that when volume and area came together with dimension the students understood this better.
First we looked at proportions with area. This lead to the proportions of length squared (m/n)^2.
Then we looked at proportions of volume. (m/n)^3.
This lead to an interesting propoerty that the exponent is just the power of how many dimensions we have. I think that when volume and area came together with dimension the students understood this better.
11.4 Corresponding Parts of Similar Triangles
This is true for any polygon (or any shape really), so we talked about triangles, then thought about how it is true for any shape in general.
11.3 Indirect Measurement with Similar Triangles
We used similar triangles to solve real world problems.
This chapter is mostly word problems. We worked on 3 in class then did 3 for homework.
This chapter is mostly word problems. We worked on 3 in class then did 3 for homework.
11.2 Similar Triangles
For Similar triangles we can develop some special properties to determine if they are similar.
AA similarity shortcut
SSS similarity shortcut
SAS similarity shortcut
AA similarity shortcut
SSS similarity shortcut
SAS similarity shortcut
11.1 Similar Polygons
We are looking at similar polygons this chapter. We start out by looking at what makes polygons similar. Then we make a distinction between how the word similar is used in English and math classes.
We looked at how making a shape bigger or smaller, but keeping the "shape" the same results in a similar shape.
The projector served as a good example for this.
We looked at how making a shape bigger or smaller, but keeping the "shape" the same results in a similar shape.
The projector served as a good example for this.
11.0 (pg 560) Proportions
Before starting chapter 11 we did a quick review of ratios and proportions.
With scaling and dealing with similar polygons it is important to refresh proportions and how to work with them.
With scaling and dealing with similar polygons it is important to refresh proportions and how to work with them.
Chapter 10 Test
The Chapter 10 test went well. The practice test helped a lot and increased understanding of concepts. I will continue to use this format for future tests.
Tuesday, May 13, 2008
Ch 10 Practice Test
For Chapter 10 I gave a practice test on the first review day. I let students work together on the practice test and help each other to understand the test. I collected the practice test and graded the tests looking for the common mistakes. Then the following day I went over the common mistakes with the class in the hopes that they wouldn't make the same mistakes on the final test. I let them use the practice test on the final test and this seams to have helped student scores improve.
10.7 Surface Area of a Sphere
I started out this lesson by getting the kids to think about infinity. This was crucial because I was going to use infinitely many polygonal pyramids to find the surface area of a sphere.
From the volume of pyramids: V = 1/3* B1 * h + 1/3 * B2 * h + ... + 1/3 * Bn * h
or V = h/3 * (B1 + B2 + ... + Bn)
Let n go to infinity and we can find the area of this and the sum of bases is the surface area. From this we come up with the formula for the surface area of a sphere. I just want the kids to know how to use the formula and to enjoy the presentation of the proof.
From the volume of pyramids: V = 1/3* B1 * h + 1/3 * B2 * h + ... + 1/3 * Bn * h
or V = h/3 * (B1 + B2 + ... + Bn)
Let n go to infinity and we can find the area of this and the sum of bases is the surface area. From this we come up with the formula for the surface area of a sphere. I just want the kids to know how to use the formula and to enjoy the presentation of the proof.
10.6 Volume of a Spheres
I did another experiment similar to the cone/cylinder experiment.
But this time I used a hemisphere. I asked students how many hemispheres would fill a cylinder with height 2*pi. Then from this we worked out that the volume of a sphere is (4/3)*pi*r^3.
But this time I used a hemisphere. I asked students how many hemispheres would fill a cylinder with height 2*pi. Then from this we worked out that the volume of a sphere is (4/3)*pi*r^3.
Orthographic Drawing (pg 539)
Students need to learn to think and draw in 3-D so I felt it was important to take a day to do this activity in the book. We practiced drawing 3-D shapes using isometric graph paper, then we looked at making an orthographic sketch from a 3-D shape.
Ch 10 Quiz
I gave a simple quiz. Students were quizzed on vocab, and some basic facts of volume learned so far. Mainly the volume of a prism and pyramid. So V = Bh and V = 1/3 * Bh
10.4 Volume Problems
This is a word problem section. We are pressed for time so I skipped this. The problems looked good and I would like more time to do some.
10.3 Volume of Pyramids and Cones
To introduce this topic I told the class that we would use democracy to try and do math. So I showed the students a cone and a cylinder with the same base and height. Then I took a poll to see how many cones are needed to fill up the cylinder: 1, 2, 3, 4, 5. The poll resulted in answers of 2 and 4, but never 3 so democracy lost to math today. From this experiment we found that the volume of a cone was 1/3 the volume of a cylinder. Pyramids follow the same rule, so students used the new formula to work on their homework.
I had fun because I got to play with water.
I had fun because I got to play with water.
10.2 Volume of Prisms and Cylinders
This was split into two days. One day for prisms and one day for cylinders.
We looked at prims, in two parts. First a box shape, that students were familiar with. Then we looked at skewing as cutting into a bunch of layers, infinitely many in fact. This helped students to understand why the volume doesn't change.
On the second day we did the same for cylinders. Extra time was used to go over questions from homework, and give the students a chance to get their work done in class.
We looked at prims, in two parts. First a box shape, that students were familiar with. Then we looked at skewing as cutting into a bunch of layers, infinitely many in fact. This helped students to understand why the volume doesn't change.
On the second day we did the same for cylinders. Extra time was used to go over questions from homework, and give the students a chance to get their work done in class.
10.1 The Geometry of Solids
I didn't like this chapter because it is mainly a huge vocab list. I think it is good to get the vocab, but split it over a few chapters and mix in some math.
Ch. 9 Test.
Students took the chapter 9 test a week and a half into the semester, they were not happy about this, but the scores were OK. The test focused on the new stuff we had learned, and the kids who tried did OK.
9.5 Distance in Coordinate Geometry
We looked at the coordinate plane, and refreshed our memories on what the (x,y) coordinates mean. Then we looked at the line between two points (a,b) and (c,d), then we used a right triangle to derive the distance formula. Once students had the distance formula and saw it in action a couple times they seamed to be able to use it.
One other aspect of this chapter was the equation of a circle. Students liked this, and it worked well because they got to see how shapes can be shifted and moved in the coordinate plane. Exciting!
One other aspect of this chapter was the equation of a circle. Students liked this, and it worked well because they got to see how shapes can be shifted and moved in the coordinate plane. Exciting!
9.4 Story Problems (Skipped)
I didn't do this section because we are pressed for time. Looks good, but we already see enough story problems and we didn't do this.