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.

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.

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.

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.

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.

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

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.

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.

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.

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.

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.

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.

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.

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!

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.

Falling Bridges (extra activity)

We had planned a special "Falling Bridges" activity based on some 3-letter acronym teaching style.

I began by talking about the importance of math in building structures, then showed a couple of video clips about the Tacoma Narrows Bridge collapse.



After the video we looked at a worksheet where an engineer determined a value for what the bridge can hold, and Sam made a sign with an improper value. We looked at how rounding error can drastically effect the end result of a problem.

After teaching a couple sections I changed my wording in the presentation of the problem and had the students pretend to be judges trying to figure out who is at fault for the bridge collapse. This activity engaged the students and I would like to rework the activity to increase how engaged students are.

9.3 part 2: 30-60-90 triangles

Here we used isometric dot paper to run an experiment similar to the one from 9.3 part 1. This experiment involved creating triangles that had the base 'a' and hypotenuse '2*a' with integer values. Using the Pythagorean theorem we notice that the second leg was 'a*sqrt(3)'.

After looking at the case were the base was a=1,2,3,4,... students started to piece together the pattern.

Now both of the "special" right triangles were covered and we took a little bit of time to look at how to find the other two sides given one of the other sides of this triangle.

9.3 part 1 (Isosceles Right Triangles)

I took chapter 9.3 and broke it into two parts. First we looked at isosceles right triangles. We used square graph paper to compute the hypotenuse of some isosceles right triangles, and we noticed a pattern to extend to any isosceles right triangle. So the students figured out that if both legs have length 'a' then the hypotenuse has length 'a*sqrt(2)'. Now to figure out how to get the 'sqrt()' symbol to work in web-pages and blogs. I'm surprised more people don't use math symbols all the time...

9.2 Converse of the Pythagorean Theorem

Here we looked at how we can use the Pythagorean theorem to determine if a triangle is a right triangle. So we use the Pythagorean theorem in reverse. We also learned about some common Pythagorean triples.

If a^2 + b^2 = c^2, then we have a right triangle.

There was time to do homework in class.

Chapter 9.1 The Theorem of Pythagoras

This chapter begins the Pythagorean theorem for the students.

We went over that if you have a right triangle you know that the three sides have the relationship a^2 + b^2 = c^2, where a and b are the legs, and c is the hypotenuse.

We learned to find the length of the hypotenuse given 'a' and 'b'. And we learned to find the length of a leg (a or b) given the other leg and they hypotenuse.

This was basic and repeated many times over the semester.

Chapter 9 (Tri Square Rug Game)

We used the tri-square rug game to begin the unit on the Pythagorean theorem. Unfortunately my class didn't take well to this. There was a lot that the students needed to cut out, and would probably go more smoothly if things were cut out and prepared for students ahead of time.

Also the game doesn't make sense to the kids. I would like to try and create a game that we could modify into a tri-square game. I think that if we had a game, and played it, then it would connect better with the students.

Chapter 8 Review and Test

For the Chapter 8 Review I spent 2 days.

I thought I was clever to use questions 1-10 from the review section as a warm up. They were a set of matching questions where you match the area formula with the shape it works for.

After that we went over questions from 8.7, that used elements from the whole chapter.

On the second review day I used another set of questions from the review section to warm up, then gave a review assignment. So half of the review was in warm-up form, and the other half was classwork. I guess that was my attempt at being tricky.

On Tuesday I gave the test. I still have to grade it and will post my general feelings about that when I have some results to share.

8.7 Surface Area

Surface area was introduced through the context of cutting up a cube, or box, and laying it out and measuring the area of each piece.

I think that this is a massive section to cover in a day or two. In this chapter we are looking at:

• Prisms
• Pyramids
• Cones
• Cylinders

I think that this would be a good chapter to break up into a couple parts and look at with a little bit more detail then I was able to. I think things would go more smoothly if we looked at prisms and pyramids one day, then cylinders and cones the next day. This would allow for students to make a stronger connection from prism/pyramid formula to the cone/cylinder formula.

I think it is good to develop an intuition for looking at objects in 3-D. But rushing through this section doesn't do much to help build this intuition.

The vocabulary and 3-D intuition comes back into play in chapter 10 when looking at volumes of 3D objects.

Exploration: Geometric Probability II

This Exploration looks really interesting, but I didn't have time to do this with my class. I think that geometry is a great way to look at probability and I would like to try this with a class in the future.

8.6 Any Way You Slice It

This chapter looks at parts of circles.

My students found this very challenging. In this section we are looking at the area of 3 types of sections of a circle: a sector, an anulus, and a segment.




In the book they explain how to find the areas of these and I think that these were confusing to my students. As I think about this more I believe that the concept of how to find areas is more important then the formulas.

For the area of a sector a complicated formula is presented, and I think that the best way to think about this is by comparing fractions.

partial degrees / 360 = partial area / total area

This will help understand the idea of comparing proportional parts. The degrees are proportional to the area. This concept is very useful, and could be used for a wide variety of problems, I think this alone might deserve a full day in the future.

8.5 Areas of Circles

The chapter looks at Areas of circles.

There is an activity which I didn't do but I would like to do if I had time. Cut a circle up into a bunch of sectors and lay them out to form a rectangular-ish shape. The rectangle will have a length of ( C / 2) and a width of r. I would like to do an activity where students cut up a circular object, like a paper plate or a pizza and find the area, then derive an area formula.

One thing I like about cutting up a circle into pieces is that it is a good way to introduce the concept of infinity, and thinking about cutting something up into infinitely many pieces to find the area.

8.4 Areas of Regular Polygons

8.4 has a nice investigation where students break a regular polygon up into triangles, and find the area of each triangle, then multiply this area by the number of sides. Interesting how the number of triangles and the number of sides are the same, eh?

I started this by asking the students to try the Investigation as homework. It is my hope that some of them will notice the pattern on their own and better understand the formula. Only a few students seam to have made an attempt at the homework, but the people who have tried seamed to do well on it.

In class we looked at a regular polygon inscribed in a circle and looked at how as we add more sides to the polygon it is becoming more like a circle. I am trying to prepare students for tomorrow's area formula.

8.3 Area Problems

8.3 Brings together all the area formulas we have found so far. And it is bringing them together in word problems. I am used this lesson to go over how to approach word problems.

1: Read the problem carefully
2: Draw a picture and label the apropriate parts
3: Think of what equations might help with solving this problem
4: Fill the equations with the parts from the picture and solve

I would like to come up with a nice concise way of explaining word problems for my Geometry students. Any help would be appreciated.

8.2 Areas of Triangles, Trapezoids, and Kites

Three more shapes come to join our area finding party.

As I worked through these problems it became apparent to me that students need to understand how to set up problems and solve for a variety of quantities. I like these problems because they require students to pull out their old algebra hats, dust them off, and get a little dirty working out these problems. It was a fun time for me, not sure if the students appreciated this as much as I did.

8.1 Areas of Rectangles and Parallelograms

In this chapter students are visited by an old friend, the area of a rectangle. Something students should know from early days of multiplication. But a new friend comes to join the party, the parallelogram hangs out to mix things up a bit.

The parallelogram and the rectangle have the same area formula: A = bh

I tried to look at two reasons why this is true.

The first uses translations, just like we did last unit with tessellations. We can translate part of a parallelogram to make a rectangle, by translating a triangle. This doesn't work as easily for some parallelograms, so another idea is important to look at.

I looked at skewing a parallelogram. I showed this by looking at a stack of calculators, when I skew the stack the area stays the same. The same thing would be true if I skewed a stack of paper. I like this idea because it starts a more calculus based way of thinking.

I'd like to hear more thoughts on this.

Tessellation Packet

This semester we have been using a homemade Tessellation packet instead of chapter 7 in the book. I am going to summarize everything from that packet into a single post, please comment if you would like to add comments to what I have.

I think that Tessellations are a great way to learn about math. There are a few things I wish I would have done differently.

First I wish I would have looked at this as more of an Art Unit. Great art is born under constraint, or at least that is what I heard some rock and roll group say on the radio one day. With tessellations we are constrained to the operations on tessellating shapes. The next time I teach tessellations I would like to challenge my students to think more creatively about the subject. I think I had too much of a cut and paste approach, until the final project the students were never asked to create their own tessellation.

Second, I would like to increase the math content I use in my discussion. While I did discuss vectors briefly I found that I didn't go in depth because little was asked of the students in the packet. Tessellations would be a good topic to lead into, or follow a discussion on transformations acting on points in a coordinate plane.

I'm still waiting for my final projects to come in, and when I have some pictures I will try and post them.

Cheers,
Mr. Sevre

What I'm Doing Here

Hello!

I am Mr. Sevre, a math teacher at Roosevelt High School in South Minneapolis. I am creating this blog to share my thoughts on what I am teaching with other math teachers. By posting my reflections here it is my hope that these ideas will be easily accessible in the future. I am hoping to use these in the future, and I hope that other teachers find this useful as well. If you happen to find your way to this blog and find it useful please let me know, this will help motivate me to be diligent about my blogging.

This Semester I am teaching geometry from the Discovering Geometry text. If you want to know more about the book check out the site: http://www.keypress.com/x5233.xml

I promise to do my best to post everyday and have an active blog of geometry topics!