I've been working on creating math "labs" that allow students to explore math concepts in a hands-on way. My first lab was in geometry; my goal was to help students understand the relationship between the lengths of three line segments that is necessary for creating a triangle.
First I had students try to make a triangle with three line segments which could NOT create a triangle. They concluded that it was impossible because two of the line segments were not long enough. Then I had students make a triangle with three line segments that were sufficient for creating a triangle.
I then wanted students to record a hypothesis for evaluating any three line segments' ability to form a triangle. This was really tough for students. They were able to describe why three segments could or could not form a triangle, but they had trouble generalizing their findings.
I want to keep working at making my students think about math in a broader sense. They are used to solving problems (as long as the problems are set up in a familiar format), but are not able to apply what they know to scenarios beyond what they are familiar with. My hope is that if I continue to challenge them to think beyond the box, they will begin to learn to see the big picture of what math is really about.
Tuesday, March 26, 2013
Friday, March 15, 2013
Making a difference
In the fall I had a student (Tony) who had serious issues with anger. He often butted heads with anyone in a position of authority, which lead to many suspensions, and even lead to expulsions from many other schools.
Tony responded well to me, because he could see that I cared for him and I wasn't just here to tell him what to do. In time, I saw him begin to change. First, he didn't explode as often. Then I saw him start to recognize when he was getting angry; when this happened, he would talk to me about needing to leave, so that he didn't blow up. In time he got better at first recognizing he was getting angry and then dealing with his anger.
After Tony had significantly improved, he transferred out to give another school a try. I didn't see anything of him for months; I just hoped that I had made an impact on his life in a way that would stay with him for the rest of his life.
I saw Tony again this morning, for the first time since he left. Tony was really excited to see me, and I could tell that he changed. He is holding a good job making decent money. I know that I made a difference in his life - which is the most satisfying feeling in the world.
Tony responded well to me, because he could see that I cared for him and I wasn't just here to tell him what to do. In time, I saw him begin to change. First, he didn't explode as often. Then I saw him start to recognize when he was getting angry; when this happened, he would talk to me about needing to leave, so that he didn't blow up. In time he got better at first recognizing he was getting angry and then dealing with his anger.
After Tony had significantly improved, he transferred out to give another school a try. I didn't see anything of him for months; I just hoped that I had made an impact on his life in a way that would stay with him for the rest of his life.
I saw Tony again this morning, for the first time since he left. Tony was really excited to see me, and I could tell that he changed. He is holding a good job making decent money. I know that I made a difference in his life - which is the most satisfying feeling in the world.
Thursday, March 14, 2013
How many above the mean?
Robert was challenged by a question regarding the following scenario: consider a skating rink with 200 skaters. How many are skating above the average speed? He first assumed that about half would be skating the average speed, and therefor a quarter (50) would be skating faster than the average speed.
I asked Robert to consider the batting average of a baseball player. If a player had an average of .285, how many games does he actually bat .285? Robert had to think this one through for a minute, but then suggested that the baseball player will not have any days of batting exactly .285; this is just the average of all his days.
After this, he had a better understanding of what the mean measures. He came to the conclusion that half of the skaters would be skating above the average speed (assuming an even distribution of speeds above and below the mean).
I asked Robert to consider the batting average of a baseball player. If a player had an average of .285, how many games does he actually bat .285? Robert had to think this one through for a minute, but then suggested that the baseball player will not have any days of batting exactly .285; this is just the average of all his days.
After this, he had a better understanding of what the mean measures. He came to the conclusion that half of the skaters would be skating above the average speed (assuming an even distribution of speeds above and below the mean).
Monday, March 11, 2013
"Irrational" Sequences
I had an idea for a visual metaphor for irrational numbers, that I would like to use to help students understand these mysterious numbers. First, I would ask students to imagine that there are an infinite number of videos on youtube (more on this later). Then I would ask student to pick out their favorite music video. I want them to call this video "1" and write "1" on a piece of paper.
Student will then click the 1st link on the sidebar. They will call this video "2", and write it on a piece of paper. They will then click the 1st link on the sidebar of this video. This third video will be "3". Students will continue this process for 25-50 videos.
When students are finished, we will discuss whether their video sequence fell into a looping pattern, or whether all the videos of their sequence were different. I would compare a video sequence that "loops" to a rational number - all rational numbers have a pattern of decimal digits that eventually repeats itself. I would then compare a sequence that never repeated itself to an irrational number, which never contains a repeating pattern of digits.
I would then tell students that it is impossible for a video sequence on youtube to be "irrational", because there is a finite number of videos on youtube, so eventually any sequence must fall into a loop.
What do you think?
Student will then click the 1st link on the sidebar. They will call this video "2", and write it on a piece of paper. They will then click the 1st link on the sidebar of this video. This third video will be "3". Students will continue this process for 25-50 videos.
When students are finished, we will discuss whether their video sequence fell into a looping pattern, or whether all the videos of their sequence were different. I would compare a video sequence that "loops" to a rational number - all rational numbers have a pattern of decimal digits that eventually repeats itself. I would then compare a sequence that never repeated itself to an irrational number, which never contains a repeating pattern of digits.
I would then tell students that it is impossible for a video sequence on youtube to be "irrational", because there is a finite number of videos on youtube, so eventually any sequence must fall into a loop.
What do you think?
Wednesday, March 6, 2013
Area in real life
I want to make geometry more interesting to my students by making it feel more "real". In an attempt to do so, I created the following exercise:
Part II: What is the area of the whole yard (the rectangle)?
Part III: What is the area of the lawn that needs to be reseeded? Show your work.
Lillian wanted to reseed her lawn after installing a new gazebo. The area that needs to be reseeded is represented by the shaded part of the diagram below. Help her find the area that she needs to reseed.
Part I: What is the area of the gazebo (the octagon)? Assume that the gazebo is the shape of a regular octagon. Remember, the apothem is the distance from the center to an edge. The distance of the gazebo from edge to edge is 4.8 meters.
In the future, I would like to develop this activity further by giving students a set yard size and allowing students to shop for their own hexagon or octagon-shaped gazebo to put in the yard (which will change the area of the lawn that needs to be reseeded). This will give them more freedom in the assignment and allow them to use their creativity.
Identifying the names of different sides of a right triangle
I have found that one reason my students have struggled with applying trigonometric ratios to mathematical scenarios is because they are unable to identify the differences between the "opposite" side, "adjacent" side, and the "hypotenuse". I decided to try to build up their vocabulary and boost their understanding of trigonometry at the same time.
First, I tackled the hypotenuse. My students were already familiar with the term, due to the Pythagorean Theorem. I reminded them that the hypotenuse is always the long side of a right triangle. I demonstrated that the long side is always the side opposite the right angle. Regardless of the way the triangle is "facing", the hypotenuse is always opposite the right angle.
I then told them that in trigonometry, they must think of themselves as looking at the triangle from the perspective of the angle they will be using in their trig function. I encouraged them to picture themselves standing at that angle and evaluating the triangle from there.
I asked my students what the word "adjacent" means. They seemed confused at first, so I asked them what it would mean if I said I was "adjacent" to my desk. One of my students suggested that I would be "next" to my desk. I encouraged this line of thought, and suggested that the "adjacent" side of the triangle is the side (other than the hypotenuse) that is touching the angle. The students were able to identify which side this is.
I then told them that the "opposite" side is the one that is opposite the angle - that is, that doesn't touch the angle at all. They were also able to identify this side.
After we discussed the names for the different sides of the triangles, it was much easier for the students to identify which trigonometric ratio they would use to find a missing side.
First, I tackled the hypotenuse. My students were already familiar with the term, due to the Pythagorean Theorem. I reminded them that the hypotenuse is always the long side of a right triangle. I demonstrated that the long side is always the side opposite the right angle. Regardless of the way the triangle is "facing", the hypotenuse is always opposite the right angle.
I then told them that in trigonometry, they must think of themselves as looking at the triangle from the perspective of the angle they will be using in their trig function. I encouraged them to picture themselves standing at that angle and evaluating the triangle from there.
I asked my students what the word "adjacent" means. They seemed confused at first, so I asked them what it would mean if I said I was "adjacent" to my desk. One of my students suggested that I would be "next" to my desk. I encouraged this line of thought, and suggested that the "adjacent" side of the triangle is the side (other than the hypotenuse) that is touching the angle. The students were able to identify which side this is.
I then told them that the "opposite" side is the one that is opposite the angle - that is, that doesn't touch the angle at all. They were also able to identify this side.
After we discussed the names for the different sides of the triangles, it was much easier for the students to identify which trigonometric ratio they would use to find a missing side.
Monday, March 4, 2013
Like a family
My favorite part of teaching in a small school is the close friendships that have developed between the staff. My fellow teachers and staff have become like sisters and brothers; we are a team working hard to help our students, and this effort has brought us together in a very special way. I know that they care about how I'm feeling and how my day has gone. I know that they are there for me when I'm stressed during a bad day and I need someone to vent to.
Last week we participated in a team-building exercise. We all wrote something we appreciated about each of our fellow staff. It warmed my heart to read the encouraging notes that they wrote for me, and to know how much they appreciate me.
Wherever life may take us, I know that this experience has built friendships that will always leave a mark. I am grateful for each person that I have had the opportunity to work beside.
Last week we participated in a team-building exercise. We all wrote something we appreciated about each of our fellow staff. It warmed my heart to read the encouraging notes that they wrote for me, and to know how much they appreciate me.
Wherever life may take us, I know that this experience has built friendships that will always leave a mark. I am grateful for each person that I have had the opportunity to work beside.
Infinite Series
Robert had trouble understanding how it is possible for an infinte series to converge. I asked him to consider an exponential expression with a base that is a fraction less than one. First I asked him to consider 1/2 squared. After he found 1/4, I asked him to consider whether this number is larger or smaller than 1/2. He wrestled with the concept for a moment, and then realized that 1/2 times 1/2 is the same as a half of 1/2, which is of course a smaller number.
I then asked him to consider 1/2 cubed. He was quickly able to understand that this number is smaller yet. I then asked him to consider 1/2 raised to the 1000th power. This number is very close to zero.
Robert was then able to understand that as x approaches infinity, (1/2)^x will approach 0. This enabled him to understand how an infinite series can converge.
I then asked him to consider 1/2 cubed. He was quickly able to understand that this number is smaller yet. I then asked him to consider 1/2 raised to the 1000th power. This number is very close to zero.
Robert was then able to understand that as x approaches infinity, (1/2)^x will approach 0. This enabled him to understand how an infinite series can converge.
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