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小学数学人教版二年级上册《数学广角——搭配》说课稿

  • 人教版高中数学选修3离散型随机变量及其分布列(1)教学设计

    人教版高中数学选修3离散型随机变量及其分布列(1)教学设计

    4.写出下列随机变量可能取的值,并说明随机变量所取的值表示的随机试验的结果.(1)一个袋中装有8个红球,3个白球,从中任取5个球,其中所含白球的个数为X.(2)一个袋中有5个同样大小的黑球,编号为1,2,3,4,5,从中任取3个球,取出的球的最大号码记为X.(3). 在本例(1)条件下,规定取出一个红球赢2元,而每取出一个白球输1元,以ξ表示赢得的钱数,结果如何?[解] (1)X可取0,1,2,3.X=0表示取5个球全是红球;X=1表示取1个白球,4个红球;X=2表示取2个白球,3个红球;X=3表示取3个白球,2个红球.(2)X可取3,4,5.X=3表示取出的球编号为1,2,3;X=4表示取出的球编号为1,2,4;1,3,4或2,3,4.X=5表示取出的球编号为1,2,5;1,3,5;1,4,5;2,3,5;2,4,5或3,4,5.(3) ξ=10表示取5个球全是红球;ξ=7表示取1个白球,4个红球;ξ=4表示取2个白球,3个红球;ξ=1表示取3个白球,2个红球.

  • 人教版高中数学选修3离散型随机变量的方差教学设计

    人教版高中数学选修3离散型随机变量的方差教学设计

    3.下结论.依据均值和方差做出结论.跟踪训练2. A、B两个投资项目的利润率分别为随机变量X1和X2,根据市场分析, X1和X2的分布列分别为X1 2% 8% 12% X2 5% 10%P 0.2 0.5 0.3 P 0.8 0.2求:(1)在A、B两个项目上各投资100万元, Y1和Y2分别表示投资项目A和B所获得的利润,求方差D(Y1)和D(Y2);(2)根据得到的结论,对于投资者有什么建议? 解:(1)题目可知,投资项目A和B所获得的利润Y1和Y2的分布列为:Y1 2 8 12 Y2 5 10P 0.2 0.5 0.3 P 0.8 0.2所以 ;; 解:(2) 由(1)可知 ,说明投资A项目比投资B项目期望收益要高;同时 ,说明投资A项目比投资B项目的实际收益相对于期望收益的平均波动要更大.因此,对于追求稳定的投资者,投资B项目更合适;而对于更看重利润并且愿意为了高利润承担风险的投资者,投资A项目更合适.

  • 人教版高中数学选修3离散型随机变量的均值教学设计

    人教版高中数学选修3离散型随机变量的均值教学设计

    对于离散型随机变量,可以由它的概率分布列确定与该随机变量相关事件的概率。但在实际问题中,有时我们更感兴趣的是随机变量的某些数字特征。例如,要了解某班同学在一次数学测验中的总体水平,很重要的是看平均分;要了解某班同学数学成绩是否“两极分化”则需要考察这个班数学成绩的方差。我们还常常希望直接通过数字来反映随机变量的某个方面的特征,最常用的有期望与方差.二、 探究新知探究1.甲乙两名射箭运动员射中目标靶的环数的分布列如下表所示:如何比较他们射箭水平的高低呢?环数X 7 8 9 10甲射中的概率 0.1 0.2 0.3 0.4乙射中的概率 0.15 0.25 0.4 0.2类似两组数据的比较,首先比较击中的平均环数,如果平均环数相等,再看稳定性.假设甲射箭n次,射中7环、8环、9环和10环的频率分别为:甲n次射箭射中的平均环数当n足够大时,频率稳定于概率,所以x稳定于7×0.1+8×0.2+9×0.3+10×0.4=9.即甲射中平均环数的稳定值(理论平均值)为9,这个平均值的大小可以反映甲运动员的射箭水平.同理,乙射中环数的平均值为7×0.15+8×0.25+9×0.4+10×0.2=8.65.

  • 人教版高中数学选修3分类变量与列联表教学设计

    人教版高中数学选修3分类变量与列联表教学设计

    一、 问题导学前面两节所讨论的变量,如人的身高、树的胸径、树的高度、短跑100m世界纪录和创纪录的时间等,都是数值变量,数值变量的取值为实数.其大小和运算都有实际含义.在现实生活中,人们经常需要回答一定范围内的两种现象或性质之间是否存在关联性或相互影响的问题.例如,就读不同学校是否对学生的成绩有影响,不同班级学生用于体育锻炼的时间是否有差别,吸烟是否会增加患肺癌的风险,等等,本节将要学习的独立性检验方法为我们提供了解决这类问题的方案。在讨论上述问题时,为了表述方便,我们经常会使用一种特殊的随机变量,以区别不同的现象或性质,这类随机变量称为分类变量.分类变量的取值可以用实数表示,例如,学生所在的班级可以用1,2,3等表示,男性、女性可以用1,0表示,等等.在很多时候,这些数值只作为编号使用,并没有通常的大小和运算意义,本节我们主要讨论取值于{0,1}的分类变量的关联性问题.

  • 人教版高中数学选修3离散型随机变量及其分布列(2)教学设计

    人教版高中数学选修3离散型随机变量及其分布列(2)教学设计

    温故知新 1.离散型随机变量的定义可能取值为有限个或可以一一列举的随机变量,我们称为离散型随机变量.通常用大写英文字母表示随机变量,例如X,Y,Z;用小写英文字母表示随机变量的取值,例如x,y,z.随机变量的特点: 试验之前可以判断其可能出现的所有值,在试验之前不可能确定取何值;可以用数字表示2、随机变量的分类①离散型随机变量:X的取值可一、一列出;②连续型随机变量:X可以取某个区间内的一切值随机变量将随机事件的结果数量化.3、古典概型:①试验中所有可能出现的基本事件只有有限个;②每个基本事件出现的可能性相等。二、探究新知探究1.抛掷一枚骰子,所得的点数X有哪些值?取每个值的概率是多少? 因为X取值范围是{1,2,3,4,5,6}而且"P(X=m)"=1/6,m=1,2,3,4,5,6.因此X分布列如下表所示

  • 人教版高中数学选修3一元线性回归模型及其应用教学设计

    人教版高中数学选修3一元线性回归模型及其应用教学设计

    1.确定研究对象,明确哪个是解释变量,哪个是响应变量;2.由经验确定非线性经验回归方程的模型;3.通过变换,将非线性经验回归模型转化为线性经验回归模型;4.按照公式计算经验回归方程中的参数,得到经验回归方程;5.消去新元,得到非线性经验回归方程;6.得出结果后分析残差图是否有异常 .跟踪训练1.一只药用昆虫的产卵数y与一定范围内的温度x有关,现收集了6组观测数据列于表中: 经计算得: 线性回归残差的平方和: ∑_(i=1)^6?〖(y_i-(y_i ) ?)〗^2=236,64,e^8.0605≈3167.其中 分别为观测数据中的温度和产卵数,i=1,2,3,4,5,6.(1)若用线性回归模型拟合,求y关于x的回归方程 (精确到0.1);(2)若用非线性回归模型拟合,求得y关于x回归方程为 且相关指数R2=0.9522. ①试与(1)中的线性回归模型相比较,用R2说明哪种模型的拟合效果更好 ?②用拟合效果好的模型预测温度为35℃时该种药用昆虫的产卵数.(结果取整数).

  • 小水滴的诉说 说课稿

    小水滴的诉说 说课稿

    本节课我设计了四个教学环节 。 环节一:创设情境,激趣导入学生阅读教材第 34 页的绘本《爷爷家的洗脸水》,读完之后交 流感受,教师引导到水很珍贵的话题,由此导入新课,板书课题:小水滴的诉说。设计意图:激发学生的学习兴趣,并引出本节课要学习的内容, 为接下来的学习作好铺垫。环节二:自主探究,合作交流 这一环节我安排了三个活动。 活动一:我很宝贵首先,课件出示教材第 35页的小诗《小水滴的自我介绍》,并 播放画外音,学生说一说对水的认识。然后,学生阅读教材第 34 页和第 35 页的图文,结合课前查找收集到的有关水资源的资料,说说发现了什么?教师相机引导,板书:宝贵稀少。设计意图:了解水与人们生活的关系,感受水资源的珍贵。 活动二:我遭遇了不幸学生阅读教材第 36页的图文,找一找小水滴遭遇了哪些不幸。 小组内交流,生活中见到的小水滴遭遇的不幸,全班交流汇报,教 师相机引导,板书:珍惜。设计意图:发现生活中污染、浪费水资源的现象,明白可以靠 法律法规的作用,切实有效地保护水资源。

  • 《少年中国说(节选)》说课稿

    《少年中国说(节选)》说课稿

    说教学目标:? 1.认识6个生字,会写7个生字。? 2.正确、流利、有感情地朗读课文,能结合注释了解课文内容,背诵课文。? 3.反复诵读理解作者炽热的思想感情;体会句式整齐,气势磅礴的语言特点。三、说教学重难点:1.能结合注释了解课文内容,背诵课文。(重点)?2.正确理解作者炽烈的感情,领会文章表现手法的特点。(难点)四、说教学方法:? 根据设定的教学目标,这节课我采用的教学方法有:???1.朗读法? ??本课课文用语以四字韵文为主,读起来朗朗上口,让学生通过大量的朗读,感受作者语言上的特点,同时也加深对课文情感的了解。

  • 我的环保小搭档 说课稿

    我的环保小搭档 说课稿

    一、依标扣本,说教材本课是统编教材道德与法治二年级下册第三单元“绿色小卫士”的最后一课,通过“我的环保小搭档”主题活动,旨在引导学生了解环境保护的重要性,树立保护环境、低碳生活的意识,培养学生在生活中养成保护环境、低碳生活的习惯。本课主要有“环保有搭档”和“展示我的环保搭档”两个部分,计划分两个课时完成。本节课为第二课时,完成第二个部分“展示我的环保搭档”。二、以人为本,说学情二年级学生生活经验有限,知识面较窄,对环保的意识多停留在形式上、口号上,对绿色生活方式也了解不多。对于如何在生活践行环保并没有清晰的认识,甚至还有认识上的偏差。在本课教学中,根据二年级学生的心理和认知特点,结合学生的生活实际创设情景,用展示生活场景的方式帮助学生拓宽思维,丰富环保知识,在活动中促使其从身边的小事做起,养成良好的低碳生活方式。

  • 初二学生代表新学期第一次国旗下讲话

    初二学生代表新学期第一次国旗下讲话

    尊敬的老师们,亲爱的同学们,大家新年好!我是初二(9)班的xx,很荣幸在新的一学期里给大家做第一次国旗下演讲。春风吹来了温暖的气息,冬天也即将挥手而去,虽然空气中弥漫着些许的凉意,但是,同学们的心是火热的,我们正以更加蓬勃的姿态迎接新学期的到来。经过寒假的修整,新年的更迭,年岁的增长,我相信同学们的心智更加成熟,学识也在逐渐增长。同学们,我们一起憧憬更美好的明天,让我们再一次踏上征途!一寸光阴一寸金,寸金难买寸光阴!时间如小河淌水,不知不觉中,却流去了多年时光。珍惜时间,要记住,没有付出,便没有回报!在新学期中,让我们抛弃不思进取之心,在马年里,跨上属于自己的战马,我们一起向前,朝着属于我们自己的理想,策马奔腾,让一切的累赘化作路边的尘土。握紧手中的马鞭,我们一起绝尘而去!我们一起为自己的成功而努力,为安外的辉煌而拼搏,为祖国的昌盛而奋斗!

  • 春季期第二学期4月份国旗下讲话

    春季期第二学期4月份国旗下讲话

    主题 缅怀先烈,继承遗志尊敬的老师,亲爱的同学们:早上好!这个星期三就是清明节了,很多人都要回家祭祖扫墓了。“清明时节雨纷纷,路上行人欲断魂.”说的就是每当这个特殊的时候,人们都会倍加思念已经死去的亲人,心情难免伤感。今天在这庄严的升旗仪式上,我们凝视着烈士鲜血所染的国旗,不觉思绪又回到那艰苦峥嵘的岁月:旧中国饱受战争的创伤和帝国主义的欺凌,人民受尽磨难和屈辱。旧上海的公园门口曾立着一块醒目的牌子,上面写着“华人与狗不得入内。”外国轮船在黄浦江上肆意闯荡,撞翻中国民船,淹死中国人也不管。那时洋人在中国土地上为所欲为,国人已丧失了最起码的做人尊严。祖国危难之时,无数爱国志士挺身而出,为了祖国的独立统一和人民的解放,甘洒一腔热血。从不惜千金买宝刀,貂裘换酒也堪豪的女侠秋瑾,到一腔热血勤珍重,洒去犹能化碧涛的邹容;从江姐到刘胡兰;从黄继光到董存瑞……无数英烈用他们的生命,谱写了一曲曲壮烈的爱国乐章。难道他们不知道生命的可贵吗?不!他们非常珍惜自己的生命,但是他们深深懂得:在祖国危亡之时,作为一个中国人,保卫祖国,拯救祖国是自己的责任,为了祖国的解放,先烈们抛头颅洒热血,战火中出生入死,监牢里坚贞不屈,刑场上大义凛然。他们用生命、热血换来了华夏民族的新生。

  • 春季期第二学期6月份国旗下讲话

    春季期第二学期6月份国旗下讲话

    主题 孔子的“每事问”今天我国旗下讲话的题目是:孔子的“每事问”孔子名丘,是我国古代著名的思想家、政治家和教育家。孔子自幼勤奋好学,15岁时就确立了坚定的学习志向,开始走上了自学的道路。他不仅刻苦努力,认真读书,而且虚心好问,他的许多学问是他随时随地向别人请教得来的。他问过有名的学者,也问过普通的农夫;他问过白发苍苍的老人,也问过梳者小辫的孩童;他还愿意向不如自己的人请教,能够“不耻下问”。从这当中,他体会到一个道理:“三人行,必有我师焉”。意思说,几个人在一起,其中必定有值得我学习的人。有一次,孔子有机会进入鲁国的太庙。太庙是古代帝王祭祀祖先的地方,里面陈列着许多文物古器,还常举行祭祀活动,在这里,可以了解历史和有关的典章制度。孔子进太庙后,就下功夫认真地进行考察,对每一件不明白的事,都向别人请教。从庙里陈列的件件文物古器到举行仪式时伴奏的音乐,样样都要找人问个究竟。活动结束后,他还拉住别人的衣袖,继续问一些自己不明白的问题。他这样做,曾被人看不起。有人说:“谁说这个年轻人懂得礼呢?

  • 人教版高中语文必修3《马嵬(其二)》教案2篇

    人教版高中语文必修3《马嵬(其二)》教案2篇

    结合历史自古以来,江山美人历来都是引无数英雄豪杰竞折腰的,如果说英雄选择了美人却丢了江山,把所有罪名都归结于“女人是祸水”。但是,纣王无道,和有了妲己有必然的联系吗?有人说妲己坏透了,坏透了的妲己如果不是取得纣王的信任是坏不起来的,纣王听信了妲己的谗言,听与不听决定权在纣王,而不在妲己。强势永远在纣王一边。再来看看西施和杨贵妃:西施是作为越国贡献给吴国的供品来到吴王夫差的身边的,杨贵妃更是先是李隆基的儿媳妇被看中而得宠的。所以,西施和杨贵妃这两个可怜的女人根本没有自己的独立选择,没有独立的爱情,如果没有西施,就会有南施,或北施;如果没有杨贵妃,就会有李贵妃,王贵妃,总之什么施,什么妃是不能少的,因为那是吴王和李隆基的需要。

  • 新人教版高中英语必修2Unit 1 Cultural Heritage-Discovering Useful Structure教案二

    新人教版高中英语必修2Unit 1 Cultural Heritage-Discovering Useful Structure教案二

    This theme of the part is “ Describe people or things in greater detail”. Students have learned the grammar(restrictive relative clauses) in Book 1, and further review and consolidate its structure “prep+relative pronouns(which/whom)” and the relative adverbs(when, where and why), besides students should understand its form, meaning and functions. In this section, students should be able to express the grammar correctly in daily communication and in the writing. 1. Review the basic usages of relative pronouns and adverbs of attributive clauses . 2. Learn to use some special cases about restrictive relative clauses.3. Learn to write sentences with restrictive relative clauses flexibly according to the context.1. Review the basic usages of relative pronouns and adverbs of attributive clauses .2. Learn to use some special cases about restrictive relative clauses.3. Learn tow rite sentences with restrictive relative clauses flexibly according to the context.Step 1. Observe the following sentences, and mark the relative pronouns and the adverbs. 1. After listening to the scientists who had studied the problems, and citizens who lived near the dam, the government turned to the United Nations for help.2. Temples and other cultural sites were taken down piece by piece, and then moved and put back together again in a place where they were safe from the water.Step 2 PracticePlease complete these sentences with relative pronouns and relative adverbs and answer the following questions.Questions: 1. What is the head noun ?2. What relative words should be used ?3. What elements do they act in these sentences ?

  • 新人教版高中英语必修2Unit 2 Wildlife Protection-Reading for Writing教案二

    新人教版高中英语必修2Unit 2 Wildlife Protection-Reading for Writing教案二

    This lesson aims at making a poster about protecting wildlife after reading some posters. During reading students are guided to understand the content and try to summarize the posters with one sentence. Then students are guided to try to make a poster about protecting wildlife.1. Read the two posters and try to understand the summary sentences.2. Look at the two posters and try to understand what emotions they express.3. Try to summarize the features of posters4. Try to make a poster about wildlife.1. Look at the two posters and try to understand what emotions they express.2. Try to summarize the features of posters3. Try to make a poster about wildlife.Step 1 Lead inLook at the the posters on the textbook and ask:Which emotions do the posters communicate ?Step 2 Read the poster and answer the questions.1. What do you think of the animals in the poster on the left ?I think it is frightening and ugly.2. Why do we should protect the ugly animals ?All species--the good, the bad, and the ugly-- should be treated equally.The world needs all kinds--without variety, our planet cannot survive.3. Why are billions of trees being cut down every year ?To make paper for humans.4. What result will be lead to after the trees are cut down ?A lost of animal homes are being destroyed./The habitat of wildlife is being destroyed.Step 3 Find the feature of posters1. What does each poster use to stir up emotions ?On the left, it makes us a little frightened and it looks a little ugly, but it can activate our curiosity--What is it? And What is wrong with it?On the right, it makes us feel a little sad and want to protect them.

  • 新人教版高中英语必修2Unit 1 Cultural Heritage-Reading and Thinking教案二

    新人教版高中英语必修2Unit 1 Cultural Heritage-Reading and Thinking教案二

    1. This section focuses on "Understanding how a problem was solved”, which is aimed to guide students to analyze and discuss the challenges and problems faced by cultural heritage protection during the construction of Aswan Dam, as well as the solutions. On the basis of understanding, students should pay attention to the key role of international cooperation in solving problems, and attach importance to the balance and coordination between cultural heritage protection and social and economic development. Students are encouraged to face challenges actively, be good at cooperation, and make continuous efforts to find reasonable ways and means to solve problems.2. Enable students to understand the main information and text structure of the reading text;3. Motivate students to use the reading strategy "make a timeline" according to the appropriate text genre;4. Enable students to understand how a problem was solved;5. Enable students to understand the value of protecting cultural heritage by teamwork and global community;1. Guide students to pay attention to reading strategies, such as prediction, self-questioning and scanning.2. Help students sort out the topic language about protecting cultural relics and understand the narrative characteristics of "time-event" in illustrative style3. Lead students to understand the value of protecting cultural heritage by teamwork and global community;

  • 新人教版高中英语必修2Unit 2 Wildlife Protection-Discovering Useful Structure教案二

    新人教版高中英语必修2Unit 2 Wildlife Protection-Discovering Useful Structure教案二

    2.表示现阶段正在进行的被动动作(该动作在说话的瞬间未必正在进行)。Many interesting experiments are being carried out these days.(说话时,并不一定正在进行)3.表示一种经常性的被动行为,常和always,constantly 等表示频度的副词连用,这种用法常常带有赞扬或厌恶的感情色彩。He is always being praised by the leader.4.表示按计划或安排主语将要承受谓语动词所表示的动作(仅限于少数及物动词)。A party is being held tonight.Step 4 Special cases1.像take care of, look after, talk about, think of等动词与介词构成的短语用于现在进行时的被动语态时, 其中的介词不可省略。The ways to stop illegally hunting are being talked about. 2.可与部分情态动词连用,表示对正在发生的事情的推测。She may be being punished by her mother.3.有时可表示按计划或安排将要进行的一个被动动作。A celebration is being held this weekend for his success.4.某些表示“状态、心理活动、存在”等的动词,如have,want,need,love,一般不用现在进行时的被动语态,而常用一般现在时的被动语态。With the population increasing,more land is needed.5.“be+under/in+n.”可表示现在进行时的被动意义。My computer is under repair.=My computer is being repaired.

  • 新人教版高中英语必修2Unit 2 Wildlife Protection-Reading and Thinking教案二

    新人教版高中英语必修2Unit 2 Wildlife Protection-Reading and Thinking教案二

    The theme of this unit is human and nature, focusing on the theme of wildlife protection. Nature is a complex ecosystem, in which there are delicate balance between animals and plants. Because of the role of the food chain, the extinction of one species will produce influence, causing a series of chain reaction. Large scale extinction of species will have a serious and even irreversible impact on the ecosystem, resulting in immeasurable losses. Therefore, it is of great significance to protect wild species. To protect wild species is to protect human beings themselves. The motto of this unit is "when the buying stops, the killing can too,” which is a public service advertising slogan to protect wildlife. It tells people that every rhinoceros horn, every fur, every bowl of shark fin soup, every Ivory product, and every tiger bone product, etc. consumed by human beings, are innocent wild animals slaughtered behind them. The mission of wild aid is to ban illegal trade in endangered wildlife and mitigate climate change. It aims to educate the public to reduce the consumption demand for endangered wildlife products through public publicity and improve the awareness of environmental protection.1. Improve the awareness of wildlife protection by acquiring the knowledge of wildlife protection.2. Focus on environmental protection and protection of all lives.3. Analysis of the living environment of wild animals with appropriate thinking mode.4. Skillfully use the vocabulary and grammar knowledge of this unit to cultivate self-study ability according to the unit content5. Develop cooperative learning ability through discussion and other ways1. Enable the Ss to talk about the current situation of wild animals.2. Guide the Ss to summarize the main idea of each paragraph as well as the main idea of the text.

  • 新人教版高中英语必修2Unit 3 The Internet-Discovering Useful Structure教案二

    新人教版高中英语必修2Unit 3 The Internet-Discovering Useful Structure教案二

    This teaching period mainly deals with grammar “The Present Perfect Passive Voice.” To begin with, teachers should lead students to revise what they have learned about the Present Perfect Passive Voice. And then, teachers move on to stress more special cases concerning this grammar。This period carries considerable significance to the cultivation of students’ writing competence and lays a solid foundation for the basic appreciation of language beauty. The teacher is expected to enable students to master this period thoroughly and consolidate the knowledge by doing some exercises. 1. Guide students to review the basic usages of the Present Perfect Passive Voice2. Lead students to learn to use some special cases concerning the Present Perfect Passive Voice flexibly.2. Enable students to use the basic phrases structures flexibly.3. Strengthen students’ great interest in grammar learning.1. Help students to appreciate the function of the Present Perfect Passive Voice in a sentence2. Instruct students to write essays using the proper the Present Perfect Passive Voice.观察下列句子特点,总结共同点。1.(教材P28)Much has been written about the wonders of the World Wide Web.2.(教材P28)But the Internet has done much more for people than simply make life more convenient.3.(教材P28)Many people have been helped by the club.4.(教材P28)She no longer feels lonely, and her company has become quite successful.5.(教材P32)Today I thought I’d blog about a question that has been asked many times—how do you stay safe online and avoid bad experiences on the Internet?

  • 新人教版高中英语必修2Unit 3 The Internet-Listening &Speaking&Talking教案二

    新人教版高中英语必修2Unit 3 The Internet-Listening &Speaking&Talking教案二

    From the pictures in the text and the title--- choose the best app, we can know that this part is about how to save money by using apps.Step 2 While-listening1. Laura and Xiao Bo are talking about apps. Listen to their conversation and find out what apps they want.Xiao Bo is looking for a(n) exercise app to help him get in shape.Laura would like an app for getting rich and another that will make her grades better.2. Listen again. Are the sentences true T or false F?1). Both of Xiao Bo's apps keep track of the steps he takes._____2). Xiao Bo's second app can help him make a fitness plan._____3). Laura needs an app that will help her get discounts.______4). Laura needs an app that will add money to her bank account._______F T F T3. Listen once more and tick the sentence you hear. Underline the words used to express predictions, guesses, and beliefs.Predictions, Guesses, and Beliefs________It might help me walk more.________My guess is that it wouldn't work.________I imagine this app would help me get fit faster________I suppose that would be good.________I guess you could save a little with this app.________I suppose there would be some problems, too.________I believe this app could help me get thinner.

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