"I flip for reflection, I slide for translation, I turn for rotation in geometry!"

Last week we were introduced to translations, reflections, and rotations. We completed three worksheets in which we had to translate, reflect, and rotate certain shapes. I felt the translations were pretty easy to accomplish. The reflections and rotations were a little harder.

The reflections in which the reflection line did not pass through the shape were not too difficult. However, in the first reflection, the reflection line goes right down the middle of the shape. This confused me at first. I had a hard time imagining how the shape was supposed to reflect and what it would look like. The the part that confused me most was the fact that there were points on both sides of the line. Most reflections I had seen before were simply all points on one side of the line reflecting to the other side. After messing around for a bit, while trying to picture the whole shape reflecting, I realized if I just focused on one original point at a time, the task became much simpler. I would just pick a point, count how many spaces to the reflection line, and then use that same number of spaces to count past the reflection line to plot my reflected point. I would then move on to the next point until I had reflected all points. Taking the reflection one point at a time made the process much more manageable, rather than trying to imagine the whole shape  
                                                                reflecting.


I had a similar experience with rotating the shapes. Initially I had a hard time imaging the rotation of the actual shape. Professor Klassen then showed us the trick of turning the paper. I used this for the first couple rotations. I soon realized, once again, that if I focused on each individual point, rather than the whole shape rotating, the rotation became much more manageable and I no longer need to rotate the paper. Focusing on one point at a time was especially helpful when the shape would cross over into multiple sections of the number graph.

I thought this game here would be great practice for kids to become familiar with translations, reflections, and rotations.

Also, the most helpful thing I encountered all semester was this video here. After watching it, everything just fell into place and made sense.

Procrastinating and problems..

EDIT 11/28/2012 6:00PM : WHEN WRITING THIS POST, FOR SOME REASON, I THOUGHT THE HOMEWORK WAS DUE THE NEXT DAY. FOR THE SAKE OF THE POST LET'S JUST PRETEND IT WAS.
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So I just got done with my math homework tonight. Although I am a horrible procrastinator, or awesome procrastinator depending on how you look at it, I usually spread my online math homework out well over the section. This is due to the fact that I have a 2 hour break between classes. However, even with a 2 hour break, my all too familiar procrastination crept in. The math homework for other chapters hasn't been too difficult, so I figured I could knock out most of it fairly quickly. Boy was I wrong. As luck has it, the section in which I decided to do the homework last minute has proved to be the most difficult for me yet.

One of the biggest problems I ran into was the fact that I did not know all the formulas by memory. There were many different formulas used for the many different shapes and calculations. I found myself having to work through the examples on many of the problems. This was very time consuming. I am glad we have access to these types resources when doing our homework. Otherwise, I probably would have just skipped many problems altogether.

The following is one of the problems in which I had to work through the example to solve:

A right cylindrical can is to hold 1 L of water. What should be the height of the can if the radius is 9cm? Use 3.14159 for π.

The first thing you do is convert your L into cm. 1 L = 1000cm³.

Next you set your 1000cm³ equal to the volume of a cylinder.

1000cm³ = πr²h

Then you plug in your numbers and solve for h.

1000cm³ ≈ 3.14159(9)²h

h ≈ 3.93cm

Pyramids and Cones

Last week in math class we did an activity called "Pyramids and Cones". It helped to show us the relationship between the volume of a pyramid and prism with the same base and height and also the volume of a cone and cylinder with the same base and height.

First, we started out with nets for the prism, pyramid, cone, and cylinder. We cut them out, folded, and taped them. I have to admit, I did have a little trouble with this part. By the time I completed the pyramid, my partner had practically finished all the rest. After assembling our shapes we observed that our pyramid and prism shared the same base and height. Our cone and cylinder shared the same base and height as well. We were then asked to estimate how many pyramids full of rice would it take to fill the prism. My estimation was that it would take 2 pyramids of rice to fill up the prism. We then filled up our pyramid with rice and began to pour it into the prism. Note that our shapes had no cover over the base so that we could fill them. To my surprise, it actually took 3 pyramids to fill up the prism. We then went on to estimate that it would take 3 cones to fill up the cylinder. After filling up the cylinder we verified our estimation. For some reason I could imagine it taking 3 cones to fill up a cylinder, but I was surprised that it took 3 pyramids to fill the prism. Because we were able to measure the volumes ourselves, this activity was very helpful to me. If I had just been told that a prism is 3 times the volume of a pyramid, I could have easily forgotten this. However, this activity provided me with a visual memory that I will be much more likely to remember.

Here is a video of a similar demonstration being done.

Mmm piii..

Last week we did an activity to discover pi. I knew pi was about 3.14, however, I never really understood it's relationship to the diameter and circumference of a circle. I don't think I was ever taught this or, if I once had been told, I forgot. Either way, I found the activity to be helpful.

For the activity we measured both the diameter and circumference of a circle using a string and ruler. We measured different sized circles found randomly in a home or classroom. Some of the things we measured were a white lid, an oatmeal container, a pumpkin tin, a large red button, and a cup. After getting our measurements we divided the diameters by the circumferences and recorded the results. We then proceeded to answer questions on the worksheet and discuss it with the class. The following were some of the questions asked: How do circumference and diameter seem to be related? How are radius and diameter related? How does this tell us that radius and circumference are related?
 In the end I realized that the circumference is  about 3.14 times the length of the diameter. I found the activity to be both engaging and substantial. I would definitely use this activity in the future in my own class.

I found the following video and transcript helpful in regards to the measurements of a circle: What To Know When Measuring Circles

That triangle is right, right?

Last week in class we learned a helpful way to tell whether a triangle was obtuse, acute, or right. The activity we did also helped give me a better idea of what the Pythagorean theorem was. The Pythagorean theorem states that in a right triangle A² + B² = C². A and B are the two shorter legs of the triangle. C represents the longest leg or the hypotenuse.

For Example: The sides of a triangle are 8, 6, and 10. You then plug the sides into the formula and see if the statement is correct.       
                                                 A² + B² = C²
                                                  6² + 8² = 10²
                                                 36 + 64 = 100 
                                                       100 = 100 
                                                   
                                                Since the above statement is true, the triangle would be a right triangle.

Now, if the sum of the areas of the two smaller sides are less than the area of the largest side, then the triangle would be an obtuse triangle.

                                              3² + 6² =8²
                                               9 + 36 = 64
                                                      45 = 64

                                                This would make the triangle obtuse. 

If the sum of the areas of the two smaller sides are more than the area of the largest side, then the triangle would be an acute triangle.

                                            4² + 5² = 6²
                                           16 + 25 = 36
                                                    41 = 36
                                                                           
                                               This would make the triangle acute.

This came in handy since it isn't always easy to tell what type of triangle you are looking at. Even with an "angle finder" it can be difficult to tell since the angle can be only slightly above or below 90º.

The activity we did also helped me to understand what the Pythagorean theorem really represents. I knew the formula, however, I did not realize that the sides were squared in order to represent the total area of the side. This is hard to explain without a visual.

Here is a great video that visually represents the Pythagorean theorem.

For a game/quiz involving the Pythagorean theorem click here.

 

Who's Theorem??

Last week we did the activity "Discovering Pick's Theorem".  Before this, I had never even heard of Pick's Theorem. We used Geoboards and rubber bands to perform our investigation.  We began by making a designated polygon on the board. We then counted both the pins on the perimeter (P) and the pins on the interior (I). After that, we determined the area of the figure in square units (A).

We then went on to make our own figures and record the perimeter pins, interior pins, and area. We compared the classes results to see if we could find any pattern between the P, I, and A. Through observation, discussion, and guided questions on our packet, we ultimately came out with the formula A= 1/2P + I - 1. We could easily verify the formula by trying out different polygons and seeing if the formula was still applicable. I enjoyed this activity because it was hands on. Rather than just giving us a formula, we were able to discover and test the formula for ourselves. Since I was not familiar with Pick's Theorem, this activity allowed for a deeper understanding I would not have received had I just been given the formula.

Click here for an online interactive Geoboard!