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Show that 2821 is a Carmichael number.

Discrete Mathematics and Its Applications | 7th Edition | ISBN: 9780073383095 | Authors: Kenneth Rosen ISBN: 9780073383095 37

Solution for problem 47E Chapter 4.4

Discrete Mathematics and Its Applications | 7th Edition

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Discrete Mathematics and Its Applications | 7th Edition | ISBN: 9780073383095 | Authors: Kenneth Rosen

Discrete Mathematics and Its Applications | 7th Edition

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Problem 47E

Show that 2821 is a Carmichael number.

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Total Physical Response (Chapter 8)  Total Physical Response (TPR) - is a teaching technique that enables students to acquire new English vocabulary by listening to and carrying out spoken commands. o In TPR activities, students are not required to speak.  The tutor models the commands and continually repeats and reviews them until the students can carry out the commands with no difficulty.  Students are more likely to be and feel successful when the tutor provides constant support and modeling and eliminates the pressure on students to speak the new words. o Although TPR can be used with students at all levels, it is most useful with beginning students who understand little or no English. o Since you will be modeling the action as you speak, it will be especially important to use gestures and facial expressions.  With beginning students, first teach commands that call for simple body movements and no props: o stand up o sit down o walk o And turn around. (See below.)  Starting with such basic commands gives students a welcome feeling of accomplishment and helps them become comfortable with TPR right away.  Students can go on to more advanced TPR activities in which they interact with props and people in the learning environment.  Examples of commands to use at this stage are: o touch the o point to o pick up o put down o And give me.  These are especially useful for teaching both the names of objects in pictures and of objects that are in the immediate environment, some of which may be out of reach.  You can also use TPR for the following purposes: o to review and reinforce vocabulary you have already taught using non-TPR methods :  as a “catch-up” at the beginning of a lesson for the benefit of students who have missed previous lessons in which new material was introduced. o To provide students with an enjoyable, relaxing break during a lesson.  What are the Basic Steps of TPR 1) Select the commands (and vocabulary) you are going to teach. 2) Before the teaching session, make a list of the complete commands in the order you plan to teach them. (The list will serve as a record of what you have taught, and will help you plan review activities for later lessons. Lists are also important when working with more advanced students who will be learning longer and more complex commands. The list will enable you to repeat your exact instructions in case a student asks you to do so.) 3) Gather any equipment, props, or pictures you will need to set the context or illustrate the meaning of the commands. 4) If you are working with a group, select two or three students for the demonstration. a) (Teaching more than one student at the same time takes the pressure off any individual student. The other students in the class will also be learning as they watch. When you finish the demonstration, you can invite other students to carry out the commands they saw you teach.) 5) Teach the commands. a. a. Model the action as you give the first command. a) As you do this, use gestures and facial expressions to help the students understand what you want them to do. b. Do the action with the students several times, and give the command each time you do the action. a) (If you are teaching the English words for objects, you will need to work with only one student at a time as the others watch—unless you have enough objects for more than one student.) c. Give the command without doing the action yourself. d. Model the action again as you say the command if the students have difficulty carrying out the command. a) Always be ready to help out if necessary. 1. Repeat steps a-d for each command you plan to teach. Before introducing each new command, review the commands you have already taught. Review them in the same order that you taught them. e. Finally, review all the commands in random order. 6) (Optional) for the advanced students:  Teach the students to read the commands they have just learned. Use the following steps: i. Write each command on a separate card. ii. Show and read the first command aloud as you model the action. iii. Show and read the same command aloud as you do the action with the students. iv. Show the card without reading or modeling it. Gesture for the students to carry out the action. (They should not read the card aloud.) v. Repeat these steps with each card. Before introducing each new card, review the written commands you have already taught. Review them in the same order that you taught them. vi. Finally, mix up the cards and review the commands in random order.  Teaching Action Words (Body Movements) With TPR:  For example: o Teach: Stand up. Sit down.  Stand up as you say, “Stand up.”  Gesture to the students to stand up.  Sit down as you say, “Sit down.”  Gesture to the students to sit down.  Stand up again as you say, “Stand up.”  Gesture to the students to stand up again.  Sit down again as you say, “Sit down.”  Gesture to the students to sit down again. o Look at each student in turn and then say, “Stand up.”  You should remain seated as you do this.  Use a gesture if necessary. (You are checking the students’ ability to respond to the command without your modeling it for them.)  If they still do not seem to understand, model the command again by standing up as you gesture to them to stand up. o Then teach: Walk. Stop. Turn around.  Begin walking forward as you say, “Walk.”  Gesture to the students to walk with you.  As you all walk forward, repeat the word “Walk.”  Stop walking suddenly and say, “Stop.”  Gesture to the students to stop.  Walk forward again as you say, “Walk.”(If necessary, gesture to the students to walk with you.)  After a few steps, stop walking again as you say, “Stop.” (If necessary, gesture to the students to stop.)  As you turn around to face the three chairs, say, “Turn around.”  Gesture to the students to turn around.  Say “Walk” as you gesture to the students to walk toward their chairs. Remain standing where you are as they walk forward. (If they do not seem to understand, model walking for them as you repeat the command Walk.)  After a few steps, say, “Stop.”  Walk forward to catch up with the students. Stand between them.  Say “Walk” without using any gesture. Remain standing where you are as the students again walk toward their chairs.  When the students reach their chairs, say, “Stop.”  Walk forward and stand between them, facing the chairs.  Say, “Turn around” as you turn around. (The students should turn around, too. Use gestures if needed.)  Say, “Sit down.” (Be ready to model this since the command sit down came in the early part of the instruction, and the students may not remember it very well.)  When the students sit down, make a point of clapping enthusiastically for them as you gesture to the rest of the students in the group to do the same. 7) You can introduce the vocabulary words slowly and quickly by adding them to the above commands (example: Walk slowly.) Then teach each new command according to the steps shown above. 8) If you are teaching a class or small group, do the above activity with two students. When you finish, you can invite other students to come to the front of the room and carry out the same commands.  Suggestions For Teaching TPR:  Go slowly. If you go too fast, students are likely to become confused and tense and make mistakes. They will learn best if they are relaxed and feel comfortable with the activity.  Do not try to teach too many commands at one time. Since you will also be reviewing previously taught vocabulary, introduce no more than four to six new commands. Call students by name as you give the commands. This helps students feel comfortable.  Tell students that it is ok to watch each other if they are unsure how to respond to a particular command. (There is no such thing as “cheating” in a TPR activity.)  Provide whatever support the students need in order to be successful. TPR activities should not be used to test students. They are supposed to ensure that students don’t fail. If students are not successful in carrying out a command, you have either gone too fast, included too much material, or asked them to do something you did not adequately teach and model.  You can also do a TPR lesson using the steps in a familiar task such as baking a cake or addressing an envelope.  Introducing Speaking to Students: o Although Total Physical Response is an excellent way to introduce new vocabulary with beginning level students, you will want them to start saying the words rather than just responding physically to commands.  There are several ways of doing this: A. Call on one student to give commands to another. After you have taught specific vocabulary using TPR, call on a student to give commands to another student. B. Have students give you commands. They will enjoy having the chance to tell you what to do. C. Throw a foam ball to students and encourage them to speak while catching and throwing. o Throw a soft foam ball to Student A, saying, “[Name], catch the ball.” When Student A catches the ball, say, “[Name], throw the ball back to me.” Use appropriate gestures to convey your meaning. o Throw the ball to Student B, saying, “[Name], catch the ball.” o Ask, “Who has the ball” Model the answer: “[Name] has the ball.” o Call on two or three students by name and gesture for them to repeat this response. (Model again if necessary.) o Tell Student B, “[Name], throw the ball to [name of Student C].” (If necessary, use gestures to convey your meaning. Point to Student B and make a throwing gesture in the direction of Student C.) o By making a beckoning gesture with your hand (and mouthing the statement if necessary), encourage Student B to say to Student C, “[Name], catch the ball.” o Call on other students and ask, “Who has the ball” Model the appropriate answer as needed. o Ask students to continue to throw the ball to each other while they say, “[Name], catch the ball.” Each time the ball is caught, you will ask two or three students, “Who has the ball” o Provide help as needed, modeling correct responses and encouraging the students to speak— not just throw and catch.  There are several unique and fun ways to apply the TPR method in the classroom, experimentation is the best way to find what is best for your ESL classroom.

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Chapter 4.4, Problem 47E is Solved
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Textbook: Discrete Mathematics and Its Applications
Edition: 7
Author: Kenneth Rosen
ISBN: 9780073383095

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