問題導學類比橢圓幾何性質的研究,你認為應該研究雙曲線x^2/a^2 -y^2/b^2 =1 (a>0,b>0),的哪些幾何性質,如何研究這些性質1、范圍利用雙曲線的方程求出它的范圍,由方程x^2/a^2 -y^2/b^2 =1可得x^2/a^2 =1+y^2/b^2 ≥1 于是,雙曲線上點的坐標( x , y )都適合不等式,x^2/a^2 ≥1,y∈R所以x≥a 或x≤-a; y∈R2、對稱性 x^2/a^2 -y^2/b^2 =1 (a>0,b>0),關于x軸、y軸和原點都是對稱。x軸、y軸是雙曲線的對稱軸,原點是對稱中心,又叫做雙曲線的中心。3、頂點(1)雙曲線與對稱軸的交點,叫做雙曲線的頂點 .頂點是A_1 (-a,0)、A_2 (a,0),只有兩個。(2)如圖,線段A_1 A_2 叫做雙曲線的實軸,它的長為2a,a叫做實半軸長;線段B_1 B_2 叫做雙曲線的虛軸,它的長為2b,b叫做雙曲線的虛半軸長。(3)實軸與虛軸等長的雙曲線叫等軸雙曲線4、漸近線(1)雙曲線x^2/a^2 -y^2/b^2 =1 (a>0,b>0),的漸近線方程為:y=±b/a x(2)利用漸近線可以較準確的畫出雙曲線的草圖
二、直線與拋物線的位置關系設直線l:y=kx+m,拋物線:y2=2px(p>0),將直線方程與拋物線方程聯(lián)立整理成關于x的方程k2x2+2(km-p)x+m2=0.(1)若k≠0,當Δ>0時,直線與拋物線相交,有兩個交點;當Δ=0時,直線與拋物線相切,有一個切點;當Δ<0時,直線與拋物線相離,沒有公共點.(2)若k=0,直線與拋物線有一個交點,此時直線平行于拋物線的對稱軸或與對稱軸重合.因此直線與拋物線有一個公共點是直線與拋物線相切的必要不充分條件.二、典例解析例5.過拋物線焦點F的直線交拋物線于A、B兩點,通過點A和拋物線頂點的直線交拋物線的準線于點D,求證:直線DB平行于拋物線的對稱軸.【分析】設拋物線的標準方程為:y2=2px(p>0).設A(x1,y1),B(x2,y2).直線OA的方程為: = = ,可得yD= .設直線AB的方程為:my=x﹣ ,與拋物線的方程聯(lián)立化為y2﹣2pm﹣p2=0,
二、典例解析例4.如圖,雙曲線型冷卻塔的外形,是雙曲線的一部分,已知塔的總高度為137.5m,塔頂直徑為90m,塔的最小直徑(喉部直徑)為60m,喉部標高112.5m,試建立適當?shù)淖鴺讼担蟪龃穗p曲線的標準方程(精確到1m)解:設雙曲線的標準方程為 ,如圖所示:為喉部直徑,故 ,故雙曲線方程為 .而 的橫坐標為塔頂直徑的一半即 ,其縱坐標為塔的總高度與喉部標高的差即 ,故 ,故 ,所以 ,故雙曲線方程為 .例5.已知點 到定點 的距離和它到定直線l: 的距離的比是 ,則點 的軌跡方程為?解:設點 ,由題知, ,即 .整理得: .請你將例5與橢圓一節(jié)中的例6比較,你有什么發(fā)現(xiàn)?例6、 過雙曲線 的右焦點F2,傾斜角為30度的直線交雙曲線于A,B兩點,求|AB|.分析:求弦長問題有兩種方法:法一:如果交點坐標易求,可直接用兩點間距離公式代入求弦長;法二:但有時為了簡化計算,常設而不求,運用韋達定理來處理.解:由雙曲線的方程得,兩焦點分別為F1(-3,0),F2(3,0).因為直線AB的傾斜角是30°,且直線經過右焦點F2,所以,直線AB的方程為
二、探究新知一、點到直線的距離、兩條平行直線之間的距離1.點到直線的距離已知直線l的單位方向向量為μ,A是直線l上的定點,P是直線l外一點.設(AP) ?=a,則向量(AP) ?在直線l上的投影向量(AQ) ?=(a·μ)μ.點P到直線l的距離為PQ=√(a^2 "-(" a"·" μ")" ^2 ).2.兩條平行直線之間的距離求兩條平行直線l,m之間的距離,可在其中一條直線l上任取一點P,則兩條平行直線間的距離就等于點P到直線m的距離.點睛:點到直線的距離,即點到直線的垂線段的長度,由于直線與直線外一點確定一個平面,所以空間點到直線的距離問題可轉化為空間某一個平面內點到直線的距離問題.1.已知正方體ABCD-A1B1C1D1的棱長為2,E,F分別是C1C,D1A1的中點,則點A到直線EF的距離為 . 答案: √174/6解析:如圖,以點D為原點,DA,DC,DD1所在直線分別為x軸、y軸、z軸建立空間直角坐標系,則A(2,0,0),E(0,2,1),F(1,0,2),(EF) ?=(1,-2,1),
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)學的重要內容,具有豐富的內涵和密切的聯(lián)系,同時也是研究包含二次曲線在內的許多內容的工具 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,性質解決實際問題. 3. 滲透數(shù)形結合思想,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)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)形結合及轉化的思想方法。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)圖像與性質的探究,培養(yǎng)學生數(shù)形結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;
本節(jié)內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。.
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)這些可能結果中的一個,但事先不確定出現(xiàn)哪個結果。新知講授(二)——樣本空間思考一:體育彩票搖獎時,將10個質地和大小完全相同、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果?如何表示這些結果?根據(jù)球的號碼,共有10種可能結果。如果用m表示“搖出的球的號碼為m”這一結果,那么所有可能結果可用集合表示{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果稱為樣本點,全體樣本點的集合稱為試驗E的樣本空間。
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、文體結構和寫作方法的能力。
本單元閱讀文本介紹了幾種簡單的英語詩歌形式,旨在讓學生能理解和欣賞英語詩歌的美,包括形式美、內容美、意義美,培養(yǎng)審美情趣,并從作品的意義美中獲得積極的人生態(tài)度和價值觀念啟示。 該文本是一則典型的說明性文本,用簡單易懂的語言向英語詩歌的初學者們介紹了五種比較簡單的英語詩歌形式。文題“A Few Simple Forms of English Poems”是整個文本的核心話題。正文內容共七段,按“總—分—總”建構。第一段為總起段,概述了英語詩歌的創(chuàng)作原因以及主要特點,涉及了形式、內容、意義等方面賞析英語詩歌的幾個主要維度;第二段至第六段分別介紹了童謠、清單詩、五行詩、俳句和唐詩五種英語詩歌形式;第七段收尾,鼓勵讀者嘗試英語詩歌創(chuàng)作。 在進行文本教學設計時,要引導學生梳理關于與詩歌要素和詩歌賞析有關的話題類語言,既包括format、rhythm、subject、image等名詞,也包括have a strong rhythm、be made up of、contain、consist of、convey這樣的動詞和動詞短語。
本單元閱讀文本向中學生推薦職業(yè)能力測試(Career Aptitude Test),旨在建議學生利用職業(yè)傾向測試來發(fā)現(xiàn)自己更感興趣、更有潛力的學習或職業(yè)方向,并規(guī)劃自己的未來職業(yè)。 本文采用了建議性文本,全篇從職業(yè)生涯的重要意義講起,針對中學生對職業(yè)規(guī)劃比較迷茫的現(xiàn)狀,提出了職業(yè)傾向測試這一建議。全文共七段,其中第一段和第二段為第一部分,其余五段為第二部分。第一部分論述職業(yè)的重要性和職業(yè)生涯規(guī)劃的最佳時間是在校期間,第二部分提出解決職業(yè)選擇困惑可以通過完成職業(yè)能力測試,介紹了不同種類的職業(yè)傾向測試,結合圖表詳細說明其中一種操作步驟,并提醒職業(yè)建議也基于學歷和經驗,最后一段概括論述,建議學生通過職業(yè)能力測試這一有效的工具,找到自己真正熱愛的事物。 在文本教學設計時,要幫助學生梳理有關職業(yè)的話題語言。閱讀策略層面,指導學生通過學習圖表和圖形,在有限的空間內獲取廣泛信息,如閱讀圖表標題,圖表上的標簽,X軸Y軸上的數(shù)據(jù)所指。
(3)2004年6月末至7月初,廣東省出現(xiàn)罕見的大面積持續(xù)高溫炎熱天氣,全省有45個縣(市)的最高氣溫破歷史同期記錄。造成此次異常天氣的主要原因是()A.副熱帶高壓和熱帶氣旋外圍下沉氣流共同影響B(tài).大量使用汽車和制冷設備C.綠色植物呼吸作用釋放CO:D.全球溫室效應長期作用【解析】(1)根據(jù)“源于西太平洋洋面的臺風”判斷該臺風屬于北半球的氣旋型的天氣系統(tǒng),當臺風中心位于南海時,廣東沿海地區(qū)正好位于該氣旋的北部,故吹東北風。(2)夏秋季節(jié)的臺風主要影響我國的南部、東部沿海地區(qū),廣東沿海正好位于該地區(qū),受到的影響較大;廣東沿海是我國經濟非常發(fā)達的地區(qū),富含有N、P、K等營養(yǎng)元素的工廠生產、農業(yè)生產、生活廢水的排放導致海水中的藻類物質大量繁殖,水中的溶解氧減少,魚類因窒息而死亡的現(xiàn)象在該地區(qū)時常發(fā)生。(3)從題干中提供的信息分析可知,6月末至7月初為北半球的夏季,因是短期的高溫,與正常年份相比出現(xiàn)異常天氣是因副熱帶高壓和熱帶氣旋外圍下沉氣流共同影響。
(一)自然災害監(jiān)測系統(tǒng)1.概念:自然災害監(jiān)測系統(tǒng)是由國家、區(qū)域及地方等各級組織,通過不同平臺對自然災害進行監(jiān)測和分析的網絡系統(tǒng)。2.作用:災前預警、災中跟蹤、災后評估以及提出減災決策方案3.世界和我國災害監(jiān)測系統(tǒng)的發(fā)展情況①已經形成了遍布世界各地、相互交織的災害監(jiān)測和預警網絡。②我國已經運用現(xiàn)代科學技術建立起各種自然災害監(jiān)測系統(tǒng)(二)遙感技術在自然災害監(jiān)測中的作用1.遙感(RS)技術的特點:觀測范圍廣、信息獲取量大、獲取速度快、實時性好和動態(tài)性強等。從空間尺度看,遙感具有全球觀測能力,可從多波段、多時相和全天候角度獲得全球自然災害的觀測數(shù)據(jù);從時間尺度看,在遙感平臺上能夠對地球進行同步觀測,可獲得地球表層及其瞬間變化的災害信息。
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
【設計意圖】 基于上述分析,學生總結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。
【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.
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、同伴所預測的結局和聽力文本所給定的結局的異同點,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、用法和結構進行歸類,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、內容、語篇結構和寫作手法來建構新文本,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。
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容,梳理完整的語篇結構,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換的思考。
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