本節(jié)課選自《普通高中課程標準實驗教科書數(shù)學必修1本(A版)》第五章的5.5.1 兩角和與差的正弦、余弦和正切公式。本節(jié)的主要內(nèi)容是由兩角差的余弦公式的推導,運用誘導公式、同角三角函數(shù)的基本關系和代數(shù)變形,得到其它的和差角公式。讓學生感受數(shù)形結(jié)合及轉(zhuǎn)化的思想方法。發(fā)展學生數(shù)學直觀、數(shù)學抽象、邏輯推理、數(shù)學建模的核心素養(yǎng)。課程目標 學科素養(yǎng)1.了解兩角差的余弦公式的推導過程.2.掌握由兩角差的余弦公式推導出兩角和的余弦公式及兩角和與差的正弦、正切公式.3.熟悉兩角和與差的正弦、余弦、正切公式的靈活運用,了解公式的正用、逆用以及角的變換的常用方法.4.通過正切函數(shù)圖像與性質(zhì)的探究,培養(yǎng)學生數(shù)形結(jié)合和類比的思想方法。 a.數(shù)學抽象:公式的推導;b.邏輯推理:公式之間的聯(lián)系;c.數(shù)學運算:運用和差角角公式求值;d.直觀想象:兩角差的余弦公式的推導;e.數(shù)學建模:公式的靈活運用;
三個“二次”即一元二次函數(shù)、一元二次方程、一元二次不等式是高中數(shù)學的重要內(nèi)容,具有豐富的內(nèi)涵和密切的聯(lián)系,同時也是研究包含二次曲線在內(nèi)的許多內(nèi)容的工具 高考試題中近一半的試題與這三個“二次”問題有關 本節(jié)主要是幫助考生理解三者之間的區(qū)別及聯(lián)系,掌握函數(shù)、方程及不等式的思想和方法。課程目標1. 通過探索,使學生理解二次函數(shù)與一元二次方程,一元二次不等式之間的聯(lián)系。2. 使學生能夠運用二次函數(shù)及其圖像,性質(zhì)解決實際問題. 3. 滲透數(shù)形結(jié)合思想,進一步培養(yǎng)學生綜合解題能力。數(shù)學學科素養(yǎng)1.數(shù)學抽象:一元二次函數(shù)與一元二次方程,一元二次不等式之間的聯(lián)系;2.邏輯推理:一元二次不等式恒成立問題;3.數(shù)學運算:解一元二次不等式;4.數(shù)據(jù)分析:一元二次不等式解決實際問題;5.數(shù)學建模:運用數(shù)形結(jié)合的思想,逐步滲透一元二次函數(shù)與一元二次方程,一元二次不等式之間的聯(lián)系。
本節(jié)內(nèi)容是三角恒等變形的基礎,是正弦線、余弦線和誘導公式等知識的延伸,同時,它又是兩角和、差、倍、半角等公式的“源頭”。兩角和與差的正弦、余弦、正切是本章的重要內(nèi)容,對于三角變換、三角恒等式的證明和三角函數(shù)式的化簡、求值等三角問題的解決有著重要的支撐作用。 課程目標1、能夠推導出兩角和與差的正弦、余弦、正切公式并能應用; 2、掌握二倍角公式及變形公式,能靈活運用二倍角公式解決有關的化簡、求值、證明問題.數(shù)學學科素養(yǎng)1.數(shù)學抽象:兩角和與差的正弦、余弦和正切公式; 2.邏輯推理: 運用公式解決基本三角函數(shù)式的化簡、證明等問題;3.數(shù)學運算:運用公式解決基本三角函數(shù)式求值問題.4.數(shù)學建模:學生體會到一般與特殊,換元等數(shù)學思想在三角恒等變換中的作用。.
本節(jié)課選自《2019人教A版高中數(shù)學選擇性必修第一冊》第二章《直線和圓的方程》,本節(jié)課主要學習拋物線及其標準方程在經(jīng)歷了橢圓和雙曲線的學習后再學習拋物線,是在學生原有認知的基礎上從幾何與代數(shù)兩 個角度去認識拋物線.教材在拋物線的定義這個內(nèi)容的安排上是:先從直觀上認識拋物線,再從畫法中提煉出拋物線的幾何特征,由此抽象概括出拋物線的定義,最后是拋物線定義的簡單應用.這樣的安排不僅體現(xiàn)出《課程標準》中要求通過豐富的實例展開教學的理念,而且符合學生從具體到抽象的認知規(guī)律,有利于學生對概念的學習和理解.坐標法的教學貫穿了整個“圓錐曲線方程”一章,是學生應重點掌握的基本數(shù)學方法 運動變化和對立統(tǒng)一的思想觀點在這節(jié)知識中得到了突出體現(xiàn),我們必須充分利用好這部分教材進行教學
∵在△EFP中,|EF|=2c,EF上的高為點P的縱坐標,∴S△EFP=4/3c2=12,∴c=3,即P點坐標為(5,4).由兩點間的距離公式|PE|=√("(" 5+3")" ^2+4^2 )=4√5,|PF|=√("(" 5"-" 3")" ^2+4^2 )=2√5,∴a=√5.又b2=c2-a2=4,故所求雙曲線的方程為x^2/5-y^2/4=1.5.求適合下列條件的雙曲線的標準方程.(1)兩個焦點的坐標分別是(-5,0),(5,0),雙曲線上的點與兩焦點的距離之差的絕對值等于8;(2)以橢圓x^2/8+y^2/5=1長軸的端點為焦點,且經(jīng)過點(3,√10);(3)a=b,經(jīng)過點(3,-1).解:(1)由雙曲線的定義知,2a=8,所以a=4,又知焦點在x軸上,且c=5,所以b2=c2-a2=25-16=9,所以雙曲線的標準方程為x^2/16-y^2/9=1.(2)由題意得,雙曲線的焦點在x軸上,且c=2√2.設雙曲線的標準方程為x^2/a^2 -y^2/b^2 =1(a>0,b>0),則有a2+b2=c2=8,9/a^2 -10/b^2 =1,解得a2=3,b2=5.故所求雙曲線的標準方程為x^2/3-y^2/5=1.(3)當焦點在x軸上時,可設雙曲線方程為x2-y2=a2,將點(3,-1)代入,得32-(-1)2=a2,所以a2=b2=8.因此,所求的雙曲線的標準方程為x^2/8-y^2/8=1.當焦點在y軸上時,可設雙曲線方程為y2-x2=a2,將點(3,-1)代入,得(-1)2-32=a2,a2=-8,不可能,所以焦點不可能在y軸上.綜上,所求雙曲線的標準方程為x^2/8-y^2/8=1.
一、情境導學我國著名數(shù)學家吳文俊先生在《數(shù)學教育現(xiàn)代化問題》中指出:“數(shù)學研究數(shù)量關系與空間形式,簡單講就是形與數(shù),歐幾里得幾何體系的特點是排除了數(shù)量關系,對于研究空間形式,你要真正的‘騰飛’,不通過數(shù)量關系,我想不出有什么好的辦法…….”吳文俊先生明確地指出中學幾何的“騰飛”是“數(shù)量化”,也就是坐標系的引入,使得幾何問題“代數(shù)化”,為了使得空間幾何“代數(shù)化”,我們引入了坐標及其運算.二、探究新知一、空間直角坐標系與坐標表示1.空間直角坐標系在空間選定一點O和一個單位正交基底{i,j,k},以點O為原點,分別以i,j,k的方向為正方向、以它們的長為單位長度建立三條數(shù)軸:x軸、y軸、z軸,它們都叫做坐標軸.這時我們就建立了一個空間直角坐標系Oxyz,O叫做原點,i,j,k都叫做坐標向量,通過每兩個坐標軸的平面叫做坐標平面,分別稱為Oxy平面,Oyz平面,Ozx平面.
新知講授(一)——隨機試驗 我們把對隨機現(xiàn)象的實現(xiàn)和對它的觀察稱為隨機試驗,簡稱試驗,常用字母E表示。我們通常研究以下特點的隨機試驗:(1)試驗可以在相同條件下重復進行;(2)試驗的所有可能結(jié)果是明確可知的,并且不止一個;(3)每次試驗總是恰好出現(xiàn)這些可能結(jié)果中的一個,但事先不確定出現(xiàn)哪個結(jié)果。新知講授(二)——樣本空間思考一:體育彩票搖獎時,將10個質(zhì)地和大小完全相同、分別標號0,1,2,...,9的球放入搖獎器中,經(jīng)過充分攪拌后搖出一個球,觀察這個球的號碼。這個隨機試驗共有多少個可能結(jié)果?如何表示這些結(jié)果?根據(jù)球的號碼,共有10種可能結(jié)果。如果用m表示“搖出的球的號碼為m”這一結(jié)果,那么所有可能結(jié)果可用集合表示{0,1,2,3,4,5,6,7,8,9}.我們把隨機試驗E的每個可能的基本結(jié)果稱為樣本點,全體樣本點的集合稱為試驗E的樣本空間。
【設計意圖】以“新科技是一把雙刃劍”為主題開展討論活動。這個話題比較大,所以給了一定的限制,首先需要以智能家居為例來闡述科技發(fā)展對未來生活產(chǎn)生的影響,然后列舉其優(yōu)點和缺點,最后就以上現(xiàn)象發(fā)表自己的觀點。這個活動是對整個文本的創(chuàng)意性總結(jié)與意義的升華,留給學生一定的想象空間,學生借此機會可以運用文本內(nèi)容和語言表達自己對科技發(fā)展的看法和態(tài)度。Assignments:1. Finish Exercise 4 on Page 15 of the textbook.A smart home is one that integrated computers into the structure of the building itself. In this way, many of the things that we now do ourselves become automatic. For example, the smart home could control the air conditioning and lights so that you would no longer have to turn switches on and off, and so that your home would be more energy-efficient. The smart home could also monitor itself to make sure that everything is working as it should, and send you warnings if there is a problem. Such smart homes could even be programmed to detect your health problems, and then give you reasonable advice as to the food you should eat or if you should see a doctor. So, in a sense, smart homes will lead us to living smarter lives.2.Create a smart function for your home, and share your creation in the next period.
【設計意圖】 基于上述分析,學生總結(jié)新聞故事語篇類型的特征,即標題的省略性、導語的概括性、數(shù)據(jù)的支撐性和引語的重要性。在此過程中,學生對新聞故事語篇類型有了更深層次的了解,對于學生閱讀和寫作具有重要意義。Activity 7: Discussing to make a writing outline.本活動為實現(xiàn)課時目標3。1.Discuss and make an outline.Ask students to discuss in groups of four and make their own writing outline.To celebrate the 70th anniversary of the People’s Republic of China, our school newspaper starts a new column about “People of Achievement”. You plan to write a news story about one of the winners of the Medal of the Republic in 2019, including Yu Min, Shen Jilan, Sun Jiadong, Li Yannian, Zhang Fuqing, Yuan Longping, Huang Xuhua and Tu Youyou.【設計意圖】 教師創(chuàng)設情境,即為慶祝新中國成立70周年,校報新增有關卓有成就人物的欄目,你將從共和國勛章獲得者中,選擇一位寫一篇新聞故事,向該欄目投稿。學生以小組合作形式,展開討論,并寫出新聞故事框架,有助于學生課后收集人物信息,撰寫新聞故事。
2. Explore the significance of the establishment of Sarek National Park.Q1: Which event is the most important one in the park’s history?Ss: The establishment of Sarek national park in 1909 is the most important one.Q2: Is it worth making a place like Sarek a national park? Give your reasons.Ss: Yes. In this way, the place can be kept in its natural state and natural beauties and other rare and valuable resources can be preserved instead of being destroyed by endless exploitation driven by profits.Q3: How does the writer organize his introduction to the history of Sarek?Ss: The writer organizes his introduction in the sequence of time, using time indicators such as “used to”, “around 9,000 years ago”, “soon after”, “following the reindeer”, and “in 1909”.Q4: What is the feature of the language used to introduce the history of the park?Ss: The introduction to the park is to provide facts, using time indicators to organize the events. Sentences starts with “third person” and passive voice is used more often, feeling objective.【設計意圖】學生尋找和梳理公園歷史上的重要事件,體會人與自然的和諧關系,評鑒介紹性語言的特點。【核心素養(yǎng)提升點】發(fā)展自主提取、梳理文本信息能力,訓練邏輯思維和高階思維。Activity 3: Highlighting the secret of the text
本閱讀材料的話題是交際中的肢體語言,作者從三個方面講述了肢體語言的特征與作用,通過主題句和舉例闡述的方式讓讀者了解不同文化中肢體語言的相同或者不同的意義,并從更抽象、概括的維度深入認識肢體語言的特點,理解肢體語言的作用?;谥w語言的特點,作者提醒讀者在與人交流中,尤其是當文化背景有差異的時候,要使用得體的肢體語言,尊重、理解和包容不同的文化,進行有效、有素養(yǎng)的溝通。文本共由六個段落組成,篇章結(jié)構(gòu)為“總—分”。第一段用簡練的語言引出了話題,并且從我們自身表達的需要和了解他人感受兩個角度講述了肢體語言的作用。第二段闡述了肢體語言的第一個重要特點——肢體語言在不同的文化中有不同的內(nèi)涵——這也是文中寫作篇幅最大的一個要點,最為重要。通過講述肢體語言的這一特點,作者向讀者傳遞了要尊重不同的文化、要使用與所在文化相宜的肢體語言。
1. 完成文本主要信息填空(斜體字部分設空):As a witness to farmers’ poor harvests and even a severe shortage of food, Yuan Longping was determined to devote his life to tackling this crisis. After graduation, he realized that what farmers needed most was to boost yields in the fields. Yuan was convinced that the answer lay in the creation of hybrid rice, one characteristic of which is that it usually achieve higher output than conventional crops. However, it was no easy job. The first difficulty he needed to overcome was scientists’ general assumption that this could not be done. Through trial and error, Yuan managed to generate this incredible crop. It is estimated that about 60 percent of domestic rice consumption in China was comprised of crops generated from Yuan’s hybrid strains. His innovation has enabled Chinese farmers to considerably expand their output and helped feed the world. Unwilling to retire early to a life of leisure and unconcerned about celebrity or fortune, Yuan continues to turn one vision after another into realities.2. 模仿寫作:Do some research via the Internet and introduce another agricultural scientist, Chen Risheng(陳日勝), using the structure, expressions and writing techniques you have learnt from the text “A Pioneer for All People”.【設計意圖】任務1是對文本內(nèi)容和詞匯學習成效的檢測。任務2借助文本中學到的詞匯和寫作手法仿寫另一位科學家,提升學生遷移運用詞匯、文體結(jié)構(gòu)和寫作方法的能力。
本活動旨在落實課時教學目標2。 1.Think, discuss and share. Students form groups of 4, discuss about the given ending make comments. Q1: Do you like the ending? Q2: Was it a logical ending? Why so or why not? [設計意圖]通過引導學生思考、討論、評價,比較個人、同伴所預測的結(jié)局和聽力文本所給定的結(jié)局的異同點,深化對文本的認知,發(fā)展學生的評判性思維能力。 Activity 4: Exploring Asimov’s three laws of robotics and the purpose of the writing 本活動旨在落實課時教學目標3。 1. Get to know Isaac Asimov’s three laws of robotics. The teacher shares Isaac Asimov’s three laws of robotics. The three laws state that: ①A robot may not injure a human being or, through inaction, allow a human being to come to harm. ②A robot must obey any orders given to it by human beings, except where such orders would conflict with the First Law. ③A robot must protect its own existence as long as such protection does not conflict with the First or Second Law. Q: How does Tony’s story relate to the laws? 2. Figure out Isaac Asimov’s purpose of writing Satisfaction Guaranteed. The students express their opinions about the author’s writing purpose. Q: Why did Isaac Asimov write such a story? S: To explore the relationship between robots and humans. [設計意圖]通過了解艾薩克·阿西莫夫所制定的機器人三大定律,加深學生對文本的理解,深入探究文本的主題意義。推理作者的寫作目的,聯(lián)系生活實際,思考人類與機器人的關系。
[2] An important breakthrough…was… [3] Another innovation was … [4] the emphasis increasingly shifted from…to… [5] New ideas and values gradually replaced… [6] While his paintings still had religious themes… … T: All these expressions serve to show how Western painting has developed. Some of them share similar structure but with varied use of words, which makes the text vivid and more readable. 【設計意圖】主題類語言整理有助于學生類化語言應用,提高語言輸出的豐富性。處理完文本內(nèi)容信息后,進入語篇信息處理,進行主題相關的詞塊歸類。引導學生快速閱讀,尋找表達相同主題(發(fā)展或者藝術)的詞和短語,再根據(jù)詞性、用法和結(jié)構(gòu)進行歸類,儲備主題相關詞匯,豐富語言儲備,提升語言素養(yǎng)。 Assignment: Go online to gather more information about Chinese painting and write a short history of it. 【設計意圖】結(jié)合所學,遷移運用,根據(jù)實際語境,進行模仿性運用。在此過程中,學生嘗試借鑒已學的語言、內(nèi)容、語篇結(jié)構(gòu)和寫作手法來建構(gòu)新文本,實現(xiàn)語篇輸出,同時關注中西藝術文化的差別,加深對優(yōu)秀文化的認同,培養(yǎng)文化意識。
幫助學生通過討論與對主題的提升,反思自己的旅游方式,以便做出更優(yōu)化的安排,在今后的旅程中有更多的收益。 Activity 2: Further discussion of the 6 elements above and supplement of more background knowledge 本活動為實現(xiàn)課時教學目標2。 1.Target Q: The writer’s clear target of traveling, meeting the people and experiencing the culture, is closely related to his major in social studies. Then what is social studies? Social studies is a part of a school or college curriculum concerned with the study of social relationships and the functioning of society and usually made up of courses in history, government, economics, civics, sociology, geography, and anthropology. (Dictionary by Merriam-Webster) Reflection: When you go out to travel, what targets do you usually have in mind? 2.Research Q: Suppose you are traveling to Hangzhou during the school holidays, how will you do research on the city? (surf the internet, read books or travel brochures, consult friends, ...) What information will you be interested to know? (location, iconic sites, local cuisines, interesting customs, shopping malls ...) 3.Abandonment To make the most of time, we have to learn to abandon so that we can accomplish our plan. What will you be interested in doing if you go to Hangzhou if you have a week’s time? What if you only have 2 days? 4~5. Venturing & Experiencing Q: What did he venture to do during the trip? What new experiences did he have? In Sydney: attend his first open-air barbecue, enjoy many different but yummy meals In Catherine: observe the life and customs of the aborigines appreciate their music & try the musical instruments: the didgeridoo
Activity 4: Figuring out the structure and the writing purpose 本活動為實現(xiàn)課時目標2。 1.Read Paragraph 6 and think about its main idea and the writer’s writing purpose. Q1: If you were the author, how would you end your article? “For young people, there is plenty of time to change bad habits. However, there is no “magic pill” or delete button that will help you; you have to think about your bad habits and decide on some changes. You have the power to build a happy and healthy life full of good habits!” Q2: What is Paragraph 6 mainly about? (Possible answer: to appeal to young people including teenagers to change bad habits and live a happy and healthy life.) 2.Think about the writer’s writing purpose and share opinions. Q1: What is the writing purpose? Work in pairs and figure it out. (Possible answer: On the one hand, the passage is written to help teenagers change their bad habits and live a healthy lifestyle. On the other hand, it provides us with a scientific way to identify and analyse our problems objectively, thus strengthening our resolve to tackle the seemingly common yet tough problems in our lives.) 【設計意圖】 步驟1旨在預測和驗證文章最后一段主要內(nèi)容,梳理完整的語篇結(jié)構(gòu),步驟2旨在思考和討論作者的寫作目的。教師也可根據(jù)學生課堂反應情況融入對語篇人稱多次轉(zhuǎn)換的思考。
1、中國三大自然區(qū)的空間位置和基本特征。2、中國自然區(qū)域差異對人類活動的影響?!糁匾獔D釋圖1.1“三大自然區(qū)圖”三大自然區(qū)的界線(自然地理分界):西北干旱半干旱區(qū)與東部季風區(qū)之間大致以400mm等年降水量線為界,青藏高寒區(qū)與東部季風區(qū)約以3000米等高線為界,青藏高寒區(qū)與西北干旱半干旱區(qū)以昆侖山——阿爾金山——祁連山為界?!緦W習策略】1、讀圖分析:通過讀圖、分析、歸納的方法,識記三大自然區(qū)的空間位置、相互界線,理解各自然地理要素的特征和空間分布規(guī)律。2、綜合訓練:運用空白地圖,將地理事物落實在圖上,并進行比較分析、歸納整理,理解三大自然區(qū)的區(qū)域差異?!窘虒W內(nèi)容】一、三大自然區(qū)的劃分1.三大自然區(qū)的劃分依據(jù)(地貌、氣候的地域差異)
通過觀看課件引導學生對各種航天器的了 解,使學生直觀的了解宇宙環(huán)境的特點.體現(xiàn)教師為主導學生的主體的教學理念.培養(yǎng)學生歸納能力.通過分組討論澄清知識盲點―我國航天事業(yè)接近并超過發(fā)達國家.要求學生根據(jù)前后聯(lián)系自己得出結(jié)論.體現(xiàn)學生主體設疑,引進熱點主要是要學生明白:宇宙環(huán)境保護的重要性.例題練習2004年高考大綜合地36題展示習題學生做讓學生走近高考了解高考(三)課堂小結(jié)師生共同完成知識點總結(jié)、教師做思想方法的總結(jié)教師引導、學生說知識點使學生對知識有整體掌握.使學生能堅持用科學的觀點分析發(fā)現(xiàn)各種新問題.(四)布置作業(yè)用發(fā)展的觀點談談對中國加強航天事業(yè)發(fā)展的看法.鞏固所學知識:聯(lián)系實際為下節(jié)課學習做鋪墊.板書設計世界宇宙新探索中國航天事業(yè)的發(fā)展宇宙資源開發(fā)宇宙資源宇宙環(huán)境問題的產(chǎn)生宇宙環(huán)境保護
月球?qū)Φ厍虻囊饬x月球與地球形影相隨,關系密切。月球?qū)Φ厍虻挠绊?,主要有以下幾方面?.地球上夜晚的自然照明,主要靠月亮。2.遠在古代,人們就根據(jù)月相變化的周期,編訂歷法。我國農(nóng)歷月份的安排,是以月相變化 周期為標準的。農(nóng)歷規(guī)定朔為初一,這樣,望就在十五日或十六日。我國人民的傳統(tǒng)節(jié)日,像春節(jié)、中秋節(jié)就是按農(nóng)歷的月日計算安排的。3.由于地球表面各處所受月球和太陽引力的不同,地球上的水體產(chǎn)生了明顯的潮汐現(xiàn)象。4.月球是人類星際航行的第一站。已知月球表面有多種寶貴礦藏;重力較??;沒有大氣層,利用太陽能的條件較好。如何利用月球的特殊環(huán)境,開發(fā)月球的資源,以及如 何利用月球作“碼頭”,使人類更好地研究宇宙,這將是月地關系的新篇章。
5.對土星五號火箭和登月艙的質(zhì)疑現(xiàn)代航天飛機只能把20噸載荷送上低軌,而當 年的土星五號卻能輕而易舉地把100噸以上載荷送上地球軌道,將幾十噸物體推出地球重力圈,為什么后來卻棄而不用,據(jù)說連圖紙都沒有保存下來?6.溫度對攝影器材的影響月面白天可達到121°C,據(jù)圖片看,相機是露在宇航服外而沒有采用保溫措施的。膠卷在66°C就會受熱卷曲失效,怎么拍得了照片?這些人士認為,對以上這一切美國政府一直沒個 交代,而知情者由于擔心生活和安全受到影響,甚至可能直接遭到了脅迫,至今對此沉默不言。但相信不久的將來,誕生于美蘇太空競賽年代的“登月騙局”定會水落石出。反駁不過,也有許多人認為“阿波羅”登月計劃不可能造假:首先因為該計劃當時是在全球?qū)崨r轉(zhuǎn)播的,近億人親眼看到。另外,宇航員還從月球帶回了一些實物,如巖石。