This is a professional development blog primarily for teachers in Spring Branch ISD.
I think students will find the intro to trigonometry interesting because it does seem like magic that can be applied to so many different areas: space, land, math class, a playground...I also think that being able to use specific ratios that don't change even though the size of the triangle does just adds to more of the magic appeal.
I think the students will like the introduction to trigonometry because it builds on what they just learned. It starts out again with what sound very complicated and after the past chapter their minds will start to piece together what they know to figure if they can do more. It then reassures them that they do know Pythagorean Theorem and then jumps in to the hard part of knowing the side length. They comic people are saying what they might be thinking so they are reassured that they will be able to understand. Then they are shown the ratios. The student know that they can understand what is thought to be a difficult concept and are ready to put it in to use.
I think that gifted students would like the introduction to Trigonometry because Ed Zaccaro does such a good job of bringing a little historical background, the entymology of the word itself, and interesting facts about it. Not only this, but it also allows a mathematically gifted student to see and feel that trigonometry is something that they can do, and not need to wait till the end of high school to tackle. Finally, the little cartoon characters clarify information, and add to the interest of the students.
Response to NLopez:I agree with you about the "magical" feeling that Zaccaro imparts to students in this chapter. The idea that something is magical is very attractive to students, and adds to the interest. I have used this idea of magic when teaching some concepts for the first time, and when I change my tone of voice, and then used "magical words" as I explain, I notice that student interst rises, and students pay attention more closely. I also think that Harry Potter, and the references to magic, has made kids very interested in anything tied to magic.
Demonstrating the relationships between 'side lengths' and 'angles' in right triangles (tangent, p.170), (sine, p.177,178), and (cosine, p.183) facilitates and clarifies understanding for all ,including G.T., students.
In response to nlopez: I agree that the concepts presented in chapter 12 can seem like 'magic' because of the many applications that can be drawn from the information.
nlopez in response to MW on July 7th- yes, this chapter does build on what they have learned and I like your connection about the comic characters saying what the kids might be thinking in their heads, but that they might be afraid to actually voice.
nlopez in response to tatumt on July 10th- I agree that the pages you cited do clarify things, especially for those of us who have been out of highschool for so many years and don't have to teach that level of mathematics on a daily basis. Great refresher!!
I think that GT students would really enjoy learning about trigonometry due the way Zaccaro presents it. He started with sharing all the ways trig could be used, which caught my interest b/c I didn't know it was used in so many ways. But, he also gave history again on how it all began. Then he used prior knowledge of the Pythagorean Theorem. Cartoons also always help make things more enjoyable too. Great presentation to learn from.
In response to PKassir...I agree with you. GT students shouldn't have to wait until High School to learn math concepts like trig. This book allows them to start playing with these higher level concepts.
I think the students will enjoy the intro to trigonometry because it allows them to use their imagination of being on a cliff and relating it to a real world example they may be interested in finding the answer for. Also, the fact that the introduction includes some history on trigonometry the GT students will enjoy that connection.
In response to Pkassir on July 8th. I agree with you in the fact of letting elementary GT students work with trigonometry instead of waiting till high school to tackle. However, we need to make sure that the students are learning it the correct way with deep understanding and alignment. This book is a wonderful resource for teachers too!
GT students will like the way Zaccaro introduces trigonometry because he uses variety of non related examples where trigonometry can be used. He also clearly states the connection between Pythagorean theorem and trigonometry which makes it easier for students to understand.
In response to sluther on July 13th:I wholeheartedly agree that we need to ensure students are learning and understanding mathematics correctly so that they are ready for more advanced mathematics. You reminded me of a conversation a group of GT teachers had a few years back. Among the group was an experienced high school teacher that worked with GT students, and she said that most of her GT students were not ready for her classes because they had "been rushed through middle school in taking Algebra and Geometry, just to be ahead," and "they had not built a solid foundation in these courses," and therefore were having difficulty with Calculus and Trigonometry. She said very few students are truly mathematically gifted, and that we should not be in hurry to get GT students into the fast math track unless they are absolutely ready. She certainly gave the group much to think about.
I think gifted students would like how the author introduces Trigonometry because he compares it to magic which the children find very intriguing. The additional use of the cartoon characters would relly intrest almost any student.
In response to PKassir. I agree that the students will be interest by the connections to historical figures. It is also a great way to integrate science (scientists) with math.
In response to MW's statement "The comic people are saying what they might be thinking so they are reassured that they will be able to understand"--That is a wonderful observation-- The students would be maore at ease asking questions for deeper understanding since the door is already opened by the comics.
In response to NLopez on July 6th. I agree that introducing trigonometry as magic that can be applied in various problems is very appealing for GT students. Later in the chapter where sine, cosine and tangent for 1-90degrees are given, GT kids would easily memorize commonly used angles in triangles such 30,60 or 40,50 and apply the "magic" mentally.
I do think that gifted students would be intrigued by the examples used that would involvetrig to solve them. Also the cartoon charactersadd a lighthearted approach.
In response to ratliffb I think several strong pointswere made that the GT student would identify with as well.Especially all of the connections and background presentedwould add to this.
I think GT learners will enjoy the chapter on Trig because they'll be introduced to numerous ways to apply its content to the world, which may really get them thinking about neat things they never thought of, or from an entirely new perspective. It will be fun to hear the questions they come up with. I also think they'll enjoy this lesson because they have to combine new information to other information they've used, like a puzzle.
My GT students would like the way the author uses real world examples to explain trigonometry. He introduces the concept using interesting examples like finding the height of a building or the distance from the Earth to the moon. I like the way he describes trigonometry being like "magic instead of math".
I agree with ratliffb that trigonometry can be used in so many ways. I think the GT kids will appreciate the usefulness of trigonometry and all the different ways it can be applied.
I agree with PKassir on July 8 that introducing historical facts grabs the attention of the GT students. I also agree that students do not have to wait until high school to use trigonometry. My gifted students will be excited to try something so challenging!
In response to cynthiam on July 14th, I agree that lighthearted approach makes content much more likable for kids. My GT kids love it when I say what they did is geometry, algebra, trigonometry, or calculus. Some are intimidated by the name of the study , but lighthearted approach definitely helps.
I think GT students would enjoy the interesting examples of where trigonometry could be used. What I thought was even more interesting is that none of these problems would lead you to think of a triangle unless you had experience with trigonometry before. I wish that the author had gone back to these examples later in the section (as closure) to show pictures of how triangles and trigonometry could be used to solve these problems.
In response to MW:The cartoon characters are starting to grow on me the more I read the book. I think that the GT kids often have such a quirky sense of humor that enjoys puns and would enjoy these corny jokes more than the typical child.
Gifted students would like all of the background information the author includes about trigonometry. The rules are absolute and never change. Students can also relate to the real world examples that the author includes to make learning more meaningful. If higher level information is broken down for younger gifted students to relate to, they are more inclined to understand it at a deeper level.
This comment has been removed by the author.
I think the students would like the way Zaccaro introduces the Trig section because he compares math to magic and gives useful, conceptual examples on how to use Trigonometry to solve real life problems.
I really think that the kids will enjoy the way Zaccaro introduces this section because it builds on prior information learned, but is kindof mysterious in the fact that they will want to try to prove if it works or not. I believe it would encourage alot of discussion to share ideas on if they believe it to work or not.
In response to ndeans:I agree with you about GT students enjoying the way trigonometry is being presented in this book. Using it in the real world as magic instead of math.
I agree with ReneeR and PKassir in their responses that the students would enjoy the history behind it, and be curious to learn more about it on their own. The real life applications could be something they could discover more about and share with their classmates.
In response to PKassir’s July 14th response to SLuther, I agree that students are being rushed into Algebra I and Geometry in middle school and then struggle in high school. The high school teachers rarely feel that freshmen are ready for Algebra II. On occasion there is a truly gifted student that excels even as a freshman. When my G/T sister was a 7th grader, she and another G/T friend opted for the 2 year Algebra I class so they would have a good background in math going into high school. One of the math teachers absolutely ridiculed them for “taking the easy way out” and that their parents were making a huge mistake for letting them do that. Both G/T students went on to successful math careers in high school and both graduated from UT. The other student’s mother is a high level administrator here in SBISD- she knew exactly what she was doing encouraging her child to go deeper, not faster in math. I saw this issue a lot when I taught 1st grade. Sure, a 6 year old can memorize multiplication facts, but do they understand them?
I think GT students will like the way Zaccarro introduces trigonometry because of how concise and challenging is the subject with all of its 3 formulas including the Trigonometry table that work like magic in bringing the answer of different real life problems like finding the distance of a ship from shore, the height of buildings, etc. He doesn’t give a drill of the same type of simple exercises, but just a sweet doze of beginning exercises to understand the formula and then it jumps to the real life math problems.
In response to Sadlok on July 15, I agree with your response that “ I also think they'll enjoy this lesson because they have to combine new information to other information they've used, like a puzzle” allowing the student to experience “intellectual frustration”. The author makes the child remember the formula of the astronomy chapter about how fast the speed of light & sound travel to solve part of the problem and then using trigonometry like in the Einstein Level exercise 3 of page 187.
In response to sluther on July 13th, I agree about letting elementary GT students work with trigonometry instead of waiting till high school to tackle. Sometimes the GT student looses the thrill in math because they don’t get stimulate it beyond their capacity. They ask “’why should I learn this” who cares about a right or acute angle, I already know the geometry figures and I see them everywehere, so what? So by introducing a taste of trigonometry and liking it with 4th grade geometry and letting them see that it is more complex than what they think and how it could be use in the real world, it will light up a fire and nurture that mathematical gift they have.
As a student I never found trigonometry to be that interesting, unlike algebra and geometry. Perhaps it was because I had a teacher that didn't bring it to life the way it has been by Zaccaro. I think his initial explanation that trigonometry is like "magic" is an instant hook for any child learning from this material! We all like to think we are doing a little "magic", don't we? I also like the practical applications that Zaccaro explains can be solved using trigonometry. I think the student will see that trigonometry has relevance in this way. As before I continue to like the little cartoon characters' dialogue boxes because I can almost hear my kids asking questions and making comments such as these.
In response to both sluther and pkassir on 7/13 and 7/14, I agree with both of you regarding the pace with which GT students are learning math if they follow the GT math track starting in 6th grade. As a 5th grade teacher I am frequently asked by my GT parents whether their child should take GT Math or opt for Pre-AP instead. As pkassir said, there are very few instances of truly gifted math students that can process and move that quickly. Even if a student seems to have it at the time, I question how superficial an understanding they have and whether gaps will surface later as they take upper level math classes.