Math Forum - Ask Dr. Math
However, there is a relation between sides of any triangle and the area and is known as Heron's formula: Area = Sqrt[s(s-a)(s-b)(s-c)] where s. We know that the area of our triangle is constant no matter which altitude we use to calculate. Wouldn't it be great if we could determine a relationship between the lengths Let's simplify this formula: where or half the perimeter of our triangle . . Because I did it already, so you need to trust me on this one), which poses a. A student, Stuart, related perimeter to pixels and the professor, Beth, moved back conjecture about the relationship between perimeter and circumference after . a new perspective, (c) trust the feedback they received, (d) feel accountable to . Specifically, Beth first led the students through finding a formula for the area of .
Sherin and Han recruited a group of four mathematics middle school teachers to participate in their study of video club. Teachers moved from talking about themselves or the other teachers in the videos to talking about the students in the videos.
Furthermore, the discussions of student thinking moved from simply restating student ideas to analyses of student thinking. For those reasons Sherin and Han chose to facilitate those kinds of conversations during video club meetings.
Participants in the study conducted by Seidel et al. Researchers wanted to find out the impact of teachers viewing videos of their own teaching or videos of other teachers teaching.
Triangle of Medians
However, Seidel et al. Another study with classroom teachers was realized by Tripp and Rich The teachers recorded classroom lessons for 2 months and reported that self-reflection with the use of video was more beneficial than when they were merely recalling what happened in the moment of teaching.
Three elementary preservice teachers participated in the Rosaen et al. They asked the preservice teachers to view two types of videos, of classroom teachers and of interviews with children.
In another study with preservice teacher participants, Star and Strickland used three video assessments. For the preassessment the preservice teachers watched a video in class and answered a questionnaire about the video. They then viewed a second video, this time outside of class, and wrote about it.
Finally, for the postassessment, the preservice teachers watched a third video in class and completed the same preassessment questionnaire. Findings suggested that preservice teachers did not begin methods coursework with proficient observation skills.
However, the course they took and the course assignments led to significant increases in their observations related to noticing aspects of the classroom environment, the mathematics of the lesson, and discourse between the teacher and students.
Researchers recommended that improving the skills of noticing should be an explicit aspect of initial coursework and assignments in tertiary mathematics methods classes. Using previous research findings, van Es et al. Their suggestions included the following: Again, this notion of teacher learning emerged as an important reason for the use of video.
Van Es et al. Beth was not consciously attempting either believing or doubting while the data were collected. However, we were attempting to answer the research question: How does a teacher professor play the believing game Elbow in a college mathematics class? The Course Setting and Context Beth was one of four mathematics educators in a mathematics department of a midsize metropolitan university in the midwestern United States, and she regularly taught the two courses observed for this study.
The three-credit required courses were designed specifically for students studying to become elementary teachers or middle grades mathematics teachers; about one fourth of them were preparing to become middle grades mathematics teachers. Typically, students in these courses were in their second or third year of college.Area and Perimeter of Triangles
The content of the first course included problem solving, number sense and numeration, number systems, number theory, and patterns and functions.
The content of the second course included problem solving, algebra, and geometry. They were held during consecutive 8-week sessions within a week semester.
How To Teach Perimeter And Area With Geoboards | Triumphant Learning
The overall goal of both courses was to introduce the students to mathematical concepts important to the understandings of elementary mathematics and to develop the art of problem solving. Students in both courses were expected to explain, justify, and apply mathematical concepts.
Class meetings occurred Mondays, Wednesdays, and Fridays, 1: During about one third of the time students worked in small groups, and the other two thirds of the time was devoted to lecture. Students used the following manipulatives throughout the semester: Data Collection and Analysis The phases of research for the original study were iterative and inductive as in Creswell, Shelly took field notes and videotaped 36 class sessions; she transcribed 12 class sessions verbatim in order to get a deeper understanding of the classroom conversations.
These 12 were chosen because as Shelly took field notes she thought Beth was, possibly, believing during these particular class sessions. As Shelly transcribed the video she used the field notes to help analyze the data, coding for a priori categories of believing or doubting. It was not the intent of this research to analyze the data for how many times the a priori categories occurred or for what percent of the time Beth seemed to believe or doubt.
As per our research question, we wanted to describe how Beth believed and to find episodes that showed the classroom interactions when she believed. Next, we identified more nuanced episodes when Beth seemed to be doubting what she perceived to be wrong answers and believing what she perceived to be wrong answers, and we came to consensus about the episodes that portrayed these categories. In order to increase complexity, we also considered context Flick, and used the transcripts of the interviews to help triangulate the data.
During the final interview we watched video when the segments of believing and doubting occurred to facilitate simulated recall.
It was during this process that we generated the categories of reserved believing and reserved doubting. Additionally, Shelly facilitated five audio-recorded conversations with Beth; four occurred throughout the semester while videotape data were collected, and one occurred a year later. Our goals with each conversation were twofold: During the final conversation we watched segments of video in which Beth seemed to either believe or doubt. The main work of transferability, however, is done by readers and consumers of research [emphasis added].
Their job is to evaluate the extent to which the findings apply to new situations. Furthermore, according to Connelly and Clandininteacher education is a process of learning to tell and retell educational stories.
Solve Right Triangle Given Perimeter and Area - Problem With Solution
Let the story begin. The Lesson The goal of this lesson was to use a regular polygon to derive the formula for the area of a circle. Specifically, Beth first led the students through finding a formula for the area of a regular hexagon. Her plan was then to use that formula, along with the idea that as n gets bigger the area of a regular n-gon approximates the area of a circle, in order to determine the formula for the area of a circle.
In order to meet this goal, Beth drew a regular hexagon with side length s and apothem r, with a triangle formed by the center of the hexagon and two adjacent vertices see Figure 1. Regular hexagon with apothem, r, and side length, s.
- How To Teach Perimeter And Area With Geoboards
- Heron's Formula
- Believing and Doubting a Student’s Intuitive Conjecture About Perimeter
Because there are six such congruent triangles within the hexagon, Beth talked about how to use the area of one of those triangles to find the area of the hexagon. Subsequently, along with the fact that as n gets bigger a regular n-gon approximates a circle, it is true that as n gets bigger the perimeter approximates the circumference of a circle.
At this point in a typical class, Beth would be ready for the punchline: Stuart had other ideas. The following conversation transpired S is Stuart and T is Beth: As you continue on the spectrum, you know, the circle becomes like a super high-resolution, super small pixel. So could you, like, essentially, like, you know, treat these polygons as …low resolution circles. And two, this relationship holds for circles, but what does that mean for a polygon? What does that mean… for a triangle?
Circumference, I guess you could say the measurement around. Okay, and just draw a line through it. That length is going to be the same throughout, because you may be losing ground on, on one. Class transcript from video, April 17, Stuart asked a higher order thinking question. Also, by using the analogy of pixels, Stuart gave her something else to think about: How are pixels related to the limit ideas present in this classroom discussion?
Beth ended the class meeting with a comment that implied Stuart was asking good questions and attempting to make mathematical connections. It ended there, however. There was no real follow up on his comments or his conjectures. Reflection with the Use of Video Beth did not think much more about this classroom episode until she and Shelly met to watch it and reflect on what had happened.
We were watching the video in order to identify episodes in the class where Beth was believing and doubting. That length is going to be the same throughout because you may be losing ground on, on one. What does he say? He says the center of the triangle. Well, there are lots of different centers of the triangle actually. So, what center are we talking about? Maybe we could get around that, but that could be…well what does center even mean?
It goes through a vertex and the center. We could call that the diameter. And then pi, because I think I heard myself say that this relationship holds for circles. So do I use pi? You know, so once I find that length….
Interview, April 4, This was a real learning moment for Beth. These same questions and the discussion that arose could have also furthered her own mathematical understanding.
Shelly noticed something about how this classroom scenario differed from other classroom scenarios in which Beth engaged in believing. During the interview in which she and Beth watched the classroom video with Stuart, Shelly noticed, Perhaps with the other segments we watched [where believing was observed], you followed up with a question instead of….
Does that make sense? In this particular scenario, Beth was either doubting or using reserved doubting. While it is regrettable that it was after the fact and the students could not be a part of the exploration, it still speaks to the power of video as a tool for noticing and reflecting. Watching the scenario play out again prompted us to engage in mathematical investigation.
The process that Shelly first used was to find the center of the equilateral triangle with sides of length 24 and a perimeter of Next, she drew line segments to simulate possible radii of the triangle. Using the Pythagorean Theorem to calculate the lengths of each of the 12 radii, she listed those lengths and then found the average length, which was about Finally, assuming the radii of this triangle to be Close, but would using a square see Figure 3 instead of a triangle give a more accurate approximation?
Using a slightly different process, Shelly first drew a square with sides of length 24 and a perimeter of Taking the average between the shortest radii from the center of the square and the longest radii from the center of the square again using the Pythagorean Theoremshe found the average radii to be approximately Using this radii and the formula for the circumference she found an approximate perimeter of the square to be about The difference between the approximate perimeter and the actual perimeter was about 5, which was a closer approximation.
Would the difference continue to decrease if she used a regular pentagon, a regular hexagon, etc.? At first, she used a similar approach as Shelly, but on Sketchpad. She wanted to consider lots of different radii of the triangle. Figure 4 shows her sketch. Using Sketchpad, Beth calculated the actual perimeter of this triangle to be She defined a radius as a segment connecting the circumcenter of the triangle E to a point F on a side of the triangle in this figure, side AC.
In an equilateral triangle the medians of the triangle are also the altitudes. Can we prove it? Thus, by the Side-Side-Side triangle congruence postulate. From the angle addition postulate, we know thatwhich in turn tells us that and are supplementary angles. If two angles are supplementary and congruent, then they must be right angles.
By similar argument we are able to show that each median is altitude of the triangle. We know that the area of our triangle is constant no matter which altitude we use to calculate. Calculating the area using each of the three altitudes we have: We know that all sides of an equilateral triangle are congruent and so we can call each side a.
Substituting into two of our three formulas we have: And then solving each for the median we generate the three following formulas: And now, it should be quite obvious that having generated the same formula for each median that the medians of an equilateral triangle are in fact congruent and therefore the triangle formed by these congruent medians is equilateral.
And so, we are now definitively convinced that the triangle formed by the medians of an equilateral triangle is itself equilateral.