Enter the equations on the view screen calculator. Ask students to make a sketch of what they see from the projected image. This will produce a window that has equal spacing between tick marks on the x and y axes. Use the calculate function of the calculator to find the slope of each line and its y intercept. Challenge students to draw conclusions about the effect of m and b on the resulting graphs.

In Figure 3. Educational research supports the notion that students retain knowledge attained through experimentation and discovery. Consider an example at the high school level. More advanced students can be asked to make more sophisticated generalizations about the graph of the function and its relationship to the value of the discriminant. By completing the following table, students will be guided to discover the relationship between the number of roots and the value of the discriminant. When planning your lesson, give careful consideration to the role of the calculator.

Using it to replace repetitive and mundane tasks is appropriate and frees valuable time for discovery and inquiry-based learning. The graphing calculator is an extremely useful tool for exploring functions and their graphs. In studying quadratic functions, it is important to develop the equation of the axis of symmetry. This can be done by inspecting the graphs of a series of parabolas. You may want to pique student interest by telling them that there are little clues in the equations that provide a means to find the equation of the axis of symmetry and the coordinates of the vertex.

Thus, you might want to challenge the students to find the relationship between the coefficients of the quadratic equation and the equation of the axis of symmetry by presenting the graphs in Figure 3. At this point in the lesson, some students may wish to hypothesize the relationship between the coefficients of the quadratic and the equation of the axis of symmetry. To lead the discovery of the equation of the axis of symmetry from the given equation, the teacher must ask students to write the equations for the five axes of symmetry. Having students create a table of values is helpful as you ask them to discover relationships.

Technological advances enable students and classes to interact with each other from home. It is important to recognize that it is essential for you to employ technology as an enhancement, not a replacement for a solid mathematics lesson. With an LCD projector, the teacher can show animated images, movie clips, and computergenerated models to capture student interest.

The support material that accompanies many textbooks includes CDs and DVDs containing pictures and animations that can be shown to a class. These multimedia demonstrations can enliven your lessons. Are Scientific Calculators Obsolete? The scientific calculator is a valuable tool because of its simplicity and availability. Although students should possess strong mathematical skills, it might be wise to use scientific calculators for long and laborious calculations to free valuable class time for more meaningful activities.

Many states have assessments that require the use of calculators of one type or another. Examinations that include trigonometry require the use of a scientific or graphing calculator because trigonometric tables are no longer in vogue. Some veteran teachers continue to use traditional trigonometric charts along with scientific calculators so that students can better see the behavior of the trigonometric functions. It can be used in various of ways to deliver exciting presentations to students and facilitate learning through inquiry, discovery, and connections to the real world.

As software continues to develop, the technology choices available to educators will continue to increase dramatically. Dynamic software enables students to view mathematics in motion. Many different dynamic software packages are appropriate for mathematics, and each has features that can be used to enrich a lesson. Ask students if quadrilateral EFGH appears to have any special properties. Some students will suggest that EFGH is a parallelogram.

Use the slope tool of the software to test this hypothesis Figure 3. Dynamic software now becomes an invaluable tool. Dynamic software enables the class to view, in real time, the effect of moving one or more of the vertices. In this example, quadrilateral EFGH remains a parallelogram in every case.

This supports the conjecture that the line segments joining the consecutive midpoints of the sides of a quadrilateral form a parallelogram. The students can also experiment and then make conjectures regarding the circumstances for which this interior parallelogram will be a rectangle Figure 3. This is a great time to prove, using deductive geometry, that this conjecture is indeed a theorem. The proof is relatively short. Use auxiliary line segment AC and apply the triangle midsegment theorem that states that the line joining the midpoints of two sides of a triangle is parallel to the third side and half its length, so that we can establish that EF and HG are both parallel to and half the length of AC.

The teacher has the ability to project images that can be manipulated and examined to investigate and discover relationships. These can be elementary or complex relationships. Cabri Geometry is another dynamic software program that allows student investigation. Cabri Jr. Dynamic software can also be used effectively in a computer lab.

Guided activities allow students to discover certain properties or relationships. Students can be paired to follow the guided activity to discover relationships in algebra, geometry, trigonometry, precalculus, and calculus. Teachers now have the option of having students prove these conjectures.

Another effective use of the dynamic software is to have students use it at home with a guided activity sheet and share their discoveries with the class. Many companies provide student versions of their software for this purpose, particularly if your school has a site license for their product. Ask your supervisor to request student copies for their home computers for this purpose. In any of the three scenarios, dynamic software can enrich instruction and capture the interest of students.

Creating a Mathematics Library in the Classroom Most of this chapter has been devoted to technology that can be integrated into instruction in the mathematics classroom. Vignettes from the history of mathematics show mathematics in a context that many students find fascinating. For example, before presenting a proof of the Pythagorean theorem, you may wish to ask students to investigate some of the notable people who have devised proofs of this theorem. Garfield have fashioned proofs of the Pythagorean theorem. Using the history of mathematics to inspire students makes mathematics more meaningful to students and allows students to investigate and report back to the class.

The mathematics library is undergoing change, as technology becomes the rule rather than the exception. Print journals are being replaced by their digital electronic counterparts, as the Internet provides access to libraries across the street and around the world. Teachers should encourage students to view the collections in a print library as a tiny fraction of the resources available to them.

The Internet is now considered an integral part of any library and not a separate and distinct entity. One excellent resource can be found at www. By using Google or some other search engine, students should get instant responses to any issue they wish to pursue. It is important for students to select the appropriate keywords so that the document search can be as effective as possible. Allowing students the time and opportunity to browse leisurely through a book or journal with a wide variety of enrichment topics will undoubtedly pay huge dividends.

Teachers must give direction when students are given library time. Students should be given a library lesson to make their time most productive. Many students enjoy the challenge of comprehending and mastering some topics slightly beyond the scope of the syllabus and the average ability level of the class. A classroom library can begin with just a few titles and a few magazines. These monthly journals have many interesting explorations for teachers and students. The Canadian Mathematical Society also publishes monthly journals that contain articles about mathematics and math education for middle school and high school teachers.

The magazine Chance, published by Springer-Verlag, contains stimulating articles that are based on statistics. These real-life studies provide outstanding motivation for mathematics lessons. Scientific American is another magazine that would be appropriate for your math library. By subscribing to these magazines and periodicals, your library will continually grow; in a few years, you will have a substantial collection. Finding appropriate books for your library is not a difficult task. There are many books on recreational mathematics that are suitable see References and Resources.

You want to expose students to a variety of topics at this time, and they can then use the Internet to delve into these topics in more depth. Because the mathematics standards require students to express themselves mathematically, the math library presents a wonderful opportunity for students at all ability levels to investigate and write about mathematics. Teachers are advised to reserve some time so that students can share their findings with each other. Most curricula and syllabi are too restrictive to allow for student research outside the scope of the syllabus.

In addition, new teachers have to concern themselves with following the prescribed syllabus or risk falling behind. A mathematical library in each classroom gives students the opportunity to investigate many interesting topics in mathematics. Assigning projects for extra credit on the history of mathematics similarly promotes the goal of students learning to appreciate mathematics as the foundation of science and discovery.

Using your classroom as a mathematics laboratory encourages students to think beyond the textbook and make connections between the mathematics taught in the classroom and the world around them. There is an extensive list of books at the end of this book. It is an important component of your teaching in a middle or high school because often the textbook essentially serves as the curriculum guide for the course you are teaching.

Many teachers tend to rely on the textbook to determine what to teach, how to teach it, and the best order in which to teach the various topics. How can you make the best use of this valuable aid? Your textbook is a guide. Think of it as a blueprint or a road map that can help you decide what to teach and provide at least one way to teach it. In fact, we recommend that you acquire several textbooks from various publishers and compare the different approaches to developing the same topics. Some of these approaches may be new to you and should enhance your understanding of the topics.

Keep in mind that the traditional textbook is designed for the mainstream teacher, one who believes that following the textbook and its approaches is the best way to succeed. It is difficult to put a discussion in a textbook. This would make the teaching too rigid and would not compensate for individual teaching styles.

A good textbook must allow you to exercise freedom to exhibit professional judgment in presenting lessons in a fashion consistent with your teaching style. You can read across the Scope and Sequence to see what your students should know when they reach your class and what they will have learned after they leave you. This helps put the topics you will teach into their proper perspective. Everything you teach is merely a link in a long chain of mathematical topics.

You should be aware of what came before and what comes afterward. In addition, the Scope and Sequence permits you to see how closely your textbook actually follows the curriculum from which you will be teaching. The Becoming a Better Test Taker section provides support for student assessment. A quick glance at this section provides an overview of what your lesson might look like.

Introduction The introductory section may provide a motivation for the lesson. Mike is the captain of the Storms. Should he call heads or tails? It advises you to accept all answers that are supported by a reason and to discuss them with the class. The textbook for such an arrangement might provide students with directions for performing an experiment in coin tossing. One student will actually toss the coin, one will announce heads or tails, one will record the data in a table, and the fourth will report the results to the class.

Begin with the original problem, discuss whether it matters if Mike calls heads or tails, and why. They should carry out the experiment and decide which number occurs most often and then substitute their decision. Here is the problem suggested: A commuter looks at the tickets in his tunnel-toll ticket booklet.

He sees that Number 11 is the top ticket, and number 20 is the last ticket. How many tickets does he have left? This practice may not always be a page of practice problems similar to those done in class. For example, having taught a lesson on exponents, the following question is suggested: Are the values of 34 and 43 the same? Why or why not? Notice that this problem provides the student with practice in using exponents, but at the same time puts the practice into a situation that is not mere drill.

Becoming a Better Test Taker This section delivers exactly what its name promises. In this day of frequent testing and assessment see Chapter 9 , it is more important than ever that your students improve their test-taking abilities. Students should be provided with an excellent example of a response to the problem as a model, a suggestion for thinking it through, and some basic tips for avoiding common errors. In addition, the students should be given practice in writing out their thought processes as they solve problems.

Some text series now even have a section for the student to read about how to avoid careless mistakes when taking a test. This is a fine motivating device. Is the writing based on the latest research? The mathematics classroom is changing rapidly; it is important to keep up with what we are finding out about how children learn mathematics. For example, does the daily lesson include something for everyone? Is there material designed for slower learners? For rapid learners who finish early?

For gifted and talented learners? This will help you when looking for test and review questions for your students. These questions might also be useful later in the school year for reviewing something taught earlier. It has limitations and may not always offer the best way to teach a topic. As a result, consider the following ideas.

Most lessons in a textbook encompass two pages. The first page usually introduces the topic and explains the algorithm or skill. The textbook is merely a guide, and you are the best judge of how much time a particular class requires to properly learn the material. The textbook can provide an excellent source of problems for review or homework assignment.

Select those activities, problems, and exercises that best suit your goals, both instructional and content. You can easily skip problems and even pages that you think are not appropriate for your students. Remember that clever students are wise to the teacher who, sometimes out of a lack of caring, assigns merely the odd- or even numbered exercises for homework. This gives the students the impression that little time and effort was put into examining the problems and selecting appropriate ones for the homework.

So, why should they spend a lot of time doing it? Take the time to carefully read each exercise, work several of them out, and select those that provide direct practice in the various skills you would like your students to have learned in that lesson. Try to eliminate exercises with difficult computation, thus focusing on the real intent of the practice on the topic taught.

You will want the students to spiral back and review topics periodically. To do this, you should save some problems from each topic to assign throughout the year. The textbook should be used for more than a set of problems to be assigned as homework. It is hoped the material will be written at a reading level that is appropriate to the students who will use the book. They need to understand that the best way to solidify their learning is to review it at their own pace, which means reading the expository material in the textbook at home and in a quiet setting. Besides their class notes which sometimes are not the easiest to follow , the textbook is an excellent resource for student test preparation.

They can reread the sections on which they will be tested and try some of the exercise examples which you might assign as a practice test. As mentioned, each textbook often has a different approach to teaching a particular topic. Thus, if you examine how something is taught in more than one textbook, then you might find a more interesting way to teach the topic to your classes. The first thing you should do is see which textbooks are already available in your school.

Often, different teachers use different texts. Ask your supervisor. Another approach you might consider is attending meetings of the local, state, or national professional organizations held in your local area. These meetings are usually accompanied by exhibits from many various publishers. In most cases, publishers are happy to do so.

Believe it or not, this is an inexpensive way for publishers to advertise their books. They believe that the more copies that get into the hands of teachers, the more sales they will be able to realize.

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You might even find some gems in a used book store. Old books sometimes provide interesting alternatives and background information that you can use to enhance your lesson. Remember, the Internet offers an almost boundless way to search for common books as well as out-of-print ones. Selecting a New Text You may be asked to serve on a committee to select a textbook to use in your school. In some schools, a panel of teachers and the supervisor select the texts from an approved list of books. In others, you may find that there are virtually no restrictions on which book to select.

In either case, however, there are some key points to consider. First, do the topics in the textbook under consideration match the topics in your curriculum guide? In such a case, it is advisable not to consider that book further, even if it appears to be attractive. Second, read through and check the mathematical accuracy presented. Are there any conceptual errors? Are the definitions and algorithms expressed in good, clear mathematical language and easily understood? For the longest time, most textbooks included only skill-building topics; this is no longer true. Today, most textbooks include problem solving and reasoning as topics to be taught, and these topics appear as lessons throughout the text.

If the textbook shows only separate lessons for problem solving and reasoning, however, a red flag should go up.

They not only should be taught as separate lessons, but also should be appropriately integrated throughout the textbook. Fourth, is the book at a reading level appropriate for your students? The text must be written at the grade level for which it was intended. Fifth, is the textbook based on up-to-date research on how children learn mathematics? Or will you have to redo each lesson to make certain your students can understand the topic? Sixth, what are the teacher support materials that come with the textbook? Current research supports the notion that students learn better with a hands-on approach, even in middle school and in high school.

Do the materials come with the text series? Or are they an additional expense? Does the publisher offer free workshops for the teachers on some of the ways to make best use of their text? Check on this; it can be an important part of your decision making. Finally, what does the book cost? This may affect your decision. Be careful about the copyright date. When a book is more than two years beyond the copyright date, it is generally already three years old because publishers usually issue a book with a copyright date of the following year.

As a book approaches the fifth year, the publisher will already be preparing a new edition, making the current one obsolete and ordering replacement copies difficult and sometimes impossible. If you are asked to serve on a textbook selection committee, then examine all the textbooks carefully. In some cases, these may only add to the cost without adding to usability. It has been said that too much glitz on a page can distract from the purpose of the book: presenting mathematics in a pedagogically sound manner.

Examining the professional practice of effective teachers reveals two commonalities: great organizational skills and the thoughtful design of the lesson plan. A lesson plan serves at least two main functions. The obvious purpose is to provide the teacher with a guide or notes for conducting a lesson. The less obvious but no less important purpose is to give the teacher an opportunity to mentally rehearse the lesson while writing the plan. Just as an actor would hardly ascend a stage without rehearsal, so, too, should a teacher not teach a lesson without a rehearsal: the writing of the lesson plan.

Lesson plans must be designed to reflect continuity of purpose from one day to the next. In general, the teacher should begin planning a lesson by identifying the student learning outcomes: exactly what the student is expected to learn, and what the teacher will use as evidence of a successful learning outcome. What Are the Components of a Lesson Plan? There are nine major components that might be included in the design of an effective lesson plan: 1. Prior skills inventory preassessment. The aim of the lesson, or its purpose. A start-up activity. A motivational activity.

The body of the lesson discovery, developmental, application of new concepts, pivotal questions, etc. The planning of differentiated instructional paths for the gifted, average, and weaker students. The generalizations and conclusions to be modified, if necessary, based on the progress of the lesson. The homework assignment. If time permits. These basic elements are designed to be both universal and flexible and lend themselves to a variety of instructional approaches.

Prior Skills Inventory Most lessons have a set of prerequisite skills that will be used in the lesson. For instance, students must know how to find the least-common multiple of two numbers before learning how to add fractions with unlike denominators. Relevant skills obtained from previous years should be reviewed so that your lesson proceeds smoothly. Otherwise, you may be forced to break the continuity of the lesson to provide review for only a few students.

Effective teachers anticipate potential student weaknesses and address them well in advance of the lesson. Of course, this clearly implies that you should begin a rough plan a few weeks ahead of time. You can refine your lesson a few days before presenting it to ensure that it suits the actual class needs and abilities. The Aim of the Lesson The aim of the lesson should be clear to you and to all members of the class. It can either be written on the board before students enter the room or be elicited from members of the class during the development of the lesson.

In either case, it clarifies for the students exactly what is to be learned in class. In some districts, the aim will be requested to be phrased in the form of a question. This is a style issue. The Start-up Activity Most mathematics lessons begin with a start-up exercise. The purpose of the lesson-starting activity is to engage students in something that will contribute to the lesson. This activity can include review of past work and previously learned concepts, foreshadow topics to be learned, or motivate students in some novel way. Anticipating student skill deficiencies and providing appropriate support and review in the start-up exercises can be part of the design of an effective lesson.

What should you do while the class works on the motivational start-up exercise? Here are several options: 1. Circulate among the students, offering some assistance when necessary. Check student homework at their seats. This is merely a cursory spot-check and should not be considered as anything comprehensive. The Motivational Activity The motivational activity addresses the question, Why do we have to learn this? One motivational technique is to highlight the utility of learning a topic. Students are surprised to learn that mathematics has been important from ancient civilizations through modern times.

See Chapter 7 for more ideas on starting a lesson. After they struggle with it for a while, tell them that the puzzle will be very easy to solve with the mathematical techniques they will learn that day. The Body of the Lesson The development of the lesson. Now that you have students motivated to learn, you have to lead the class through a series of discoveries and the development of the lesson. You now must create a series of activities that will artfully lead them to your intended goal.

This is sometimes referred to as guided discovery. Students must be active participants in the lesson and provide explanations, conjectures, and thoughtful questions. The development and design of a good lesson include anticipating opportunities to embrace students in the instructional process. Sending students to the board while other students work at their seats promotes a lively classroom where a weaker student can seek help from the teacher, work with their ordered pair,1 or sneak a peek at the board to overcome a small hurdle in negotiating the task at hand.

The foregoing can easily be adapted to a workshop-type class organization model. Application s of newly acquired concepts. Allow time in each lesson for students to apply what they have just learned. These application problems exercises present an opportunity to solidify the concepts, methods, and rationale developed in the body of the lesson and to extend the concepts through a series of increasingly complex problems. This is the time for the teacher to present differentiated instruction to advanced students and to provide support for weaker students.

It is also an ideal time to have students exhibit their work on the chalkboard, overhead projector, or projection device and to provide supporting explanations. Planning pivotal questions. Students are likely to remain actively engaged if challenged with questions that effectively lead them through the discovery and development of the lesson. Effective questioning clarifies and validates learning for the student, and it informs the teacher of student progress.

Pivotal questions should be written out beforehand and be a part of every well-designed lesson plan. The structure of the question is important, and most methods books deal with this in great detail see Posamentier et al. Planning Paths of Differentiated Instruction Because many mathematics classes have students with varying skill levels, design lessons to engage learners of all abilities. This means you must come to class prepared to present advanced students with some challenging problems and somewhat weaker students with a series of simpler exercises that gradually lead to the point at which the average student is functioning.

Weaving these plans into the instructional design is tricky but can be done by finding a time when most of the class is working on a similar parallel problem of an easier nature. Having students place solutions to the problems on the board for a side-by-side comparison shows the similarity in their solution, although the second problem requires more algebraic symbolic manipulation and factoring skills than the first problem. Without drawing attention to the differences, this differentiated instructional model supports the goals of the lesson in a seamless fashion but allows students to work at their appropriate levels of sophistication.

For homework, problems at all levels can be assigned because students can refer to their notes for clarification see Chapter 2 for levels of student abilities. Generalizations and Conclusions Every lesson should end with a summary and conclusion to clarify the concepts and skills presented. Either summarize the lesson or ask student volunteers to do it.

Encourage students to communicate about mathematics. When students articulate what they understand, either verbally or in written form, the main points of the lesson are reinforced. Listening to student summaries serves as an informal assessment of both student comprehension and the effectiveness of your instruction. If Time Permits: Some Ideas Because teaching and learning are individual aspects, it is important to recognize that, for a variety of reasons, some lessons may finish early. You may find that a lesson has gone more smoothly than the same lesson did in previous classes.

These do not have to relate to the theme of the lesson and thus can be saved for an appropriate time. These problems typically reward students who think creatively and fashion elegant solutions. Explain how you arrived at your answer. Planning Group Work in the Mathematics Classroom Much research has been devoted to the efficacy of using group work in the mathematics classroom.

In addition to the ordered-pair construct, there are many other ways to design effective lessons that employ group work. Students should know exactly what is expected of them during group work. Basic principles that make group work successful are as follows: 1. Groups should consist of three or four students. Students should have a clear understanding of how to form groups. Students are to complete the entire assignment, in the same order, at the same time. How can students help one another, if they are not working on the same problem? Each student in the group must assume all the roles in the group, including that of the facilitator, the recorder, the contributor, and the presenter.

Thus, everyone should be writing, calculating, and contributing. Anyone from the group may be called on to present their findings to the class. Students must understand that they are responsible for each other. All members of the group should benefit from the learning activity.

There can be no leisure learners in group work. This grade will be assigned to all of the members of the group. Teacher-Centered Versus Student-Centered Instructional Models As you plan your lesson, decide the extent to which the lesson will be student centered. Teacher-dominated lessons sometimes referred to as chalk and talk are usually not too effective because they do not adequately engage students. However, students should not be expected to learn everything without your structure and guidance. An effective lesson includes opportunities for student discovery and for teacher-guided learning.

Some topics lend themselves to one instructional model over another; consider this as you design your lessons. We discuss the developmental and workshop models. The Developmental Lesson Model The developmental lesson model relies primarily on the teacher as the main source of information, if not by lecturing, then as the guide who elicits the information from students. Until recently, this was the predominant lesson format, and it still remains popular, particularly with veteran teachers.

The teacher typically controls the flow of the lesson and the pace of student learning. When done properly, this format can be effective and engaging for students. Students feel comfortable when they are led, step by step, through a difficult concept. Some teachers misinterpret their role in the presentation of the developmental lesson and simply lecture the students, providing them with little or no opportunity to discover or make meaningful contributions to the lesson.

The Workshop Model The student-centered workshop model, also known as discovery learning, has had great success in science and humanity classes. It can also be adapted for use in a math class with thoughtful planning. An explicit statement of what is going to be taught and modeling it. Active engagement that allows students to try it out during the lesson. Provision of a link to independent or group work. As the students work independently or in groups, the teacher circulates among them, providing guidance and encouragement. The teacher then restates these results to reinforce the aim of the lesson.

Obviously, if you plan to use the workshop model, you have to plan accordingly. Teachers can distribute circular objects to group members and ask them to measure and record the circumference and diameter of each, using string. Using a calculator, each group computes the ratio of the circumference to the diameter for each object. Encourage groups to identify the patterns they notice and use them to predict the circumference of the sun given its diameter , miles.

Students get great satisfaction from discovering mathematical patterns. On the other hand, this model has several pitfalls. Planning a lesson using the workshop model often requires a significant amount of time. As is the case with most group work, you may find it difficult to assess individual members of a group.

For example, developing a formula for the sum of an arithmetic series or finding a relationship between the number of sides of a polygon and the sum of its interior angles are examples of lessons that can be effective in a workshop model class. A middle school lesson on order of operations would be difficult to present using the workshop model and is more suitable as a developmental lesson. A high school lesson on the derivation of the quadratic formula would probably be most effective if presented using the developmental lesson format. The lesson aim is to solve systems of equations by eliminating a variable.

Start-up activities. The sum of two numbers is 10, and their difference is 4. Find the two numbers. Although the first start-up problem reinforces previously learned skills, the second exercise is a more obvious look at a system of equations. Many students will use trial and error and arrive at the correct answer, 7 and 3. The teacher will now present the class with a problem that cannot easily be solved by trial and error and will work with the students to develop a universal method that can solve all systems of linear equations. The expert teacher will solicit ideas from the students and direct them to look at an easier problem of the same type as the third start-up exercise.

The two major concepts that should be introduced are writing verbal expressions algebraically and combining two linear equations to form a third one. They are an algebraic representation of the second start-up question. When you combined these equations, how did the answer differ from the two sets of equations above? The y term drops out when we add the two equations. Now that we have solved for x, how can we solve for y?

Now that you have solved for x and y, how can you check your answers? You must substitute your answers into both equations. However, to check you must substitute your answers into both equations. Students will work either individually or in small groups to solve for x and y and check: 1. The sum of two numbers is 15, and their difference is 5. Answer: 10 and 5 Comment: During this activity, circulate among the groups of students and send students to the board to share their solutions with the class. After having students explain their solutions to these problems, ask students to summarize the lesson in their math journals or notebooks.

Figure 5. The aim of the lesson is the introduction to solving systems of linear equations.

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Start-up activity. Students are seated at desks arranged in groups of three or four. Submissions must be handed in by Friday, October 20, in class for extra credit. Homework assignment, developmental model. Activity 1. Each group is asked to investigate and answer the following: 1. Find five pairs of numbers with a sum of Are these the only five pairs? Find five pairs of numbers with a difference of 8. Find one pair of numbers with a sum of 20 and a difference of 8. Is this the only pair? The teacher should circulate among the groups and identify those groups that have made significant progress.

Students may arrive at their answers in various ways, including making charts, guessing and checking, or taking an algebraic approach. The teacher should give opportunities for all of the different approaches to be shared. After this sharing, the teacher should guide the class to use variables to express the relationship of the pair of numbers with a sum of 20 in Activity 1. Ask students to do the following: 1. Write an algebraic expression to represent the pair of numbers whose sum is Write an algebraic expression to represent the pair of numbers whose difference is 8.

The teacher takes a few minutes to make certain that the class understands that, in this instance, when these two algebraic expressions are combined in a certain fashion, one variable will drop out; they are then left with a simple equation in one variable to solve. The class then returns to their small groups to begin Activity 2. Activity 2. Find 10 pairs of numbers with a sum of Find 10 pairs of numbers with a difference of Find 1 pair of numbers with a sum of 50 and a difference of Express the relationship between the numbers in Questions 1 and 2 algebraically.

Use the algebraic method from Activity 1 to find the pair of numbers with a sum of 50 and a difference of Use an algebraic method to solve the following: The sum of two numbers is , and their difference is Answer: and 41 Comment: Notice that the scope of this lesson is much smaller than that of the developmental lesson.

However, in this lesson, students are challenged to make conjectures and discover relationships for these and similar problems. Share time. Each group will share the method they used to solve the problems in Activities 1 and 2. Write a paragraph about how to use an algebraic method to solve Question 4 in Activity 2. Solve using an algebraic method: 1. Find two numbers with a sum of 83 and a difference of Find two numbers with a sum of 2, and a difference of 1, Find two numbers with a sum of 7, and a difference of 2, Activity 3 if time permits.

Make up three problems similar to those in Activities 1 and 2. Use the algebraic method developed to find the numbers. Exchange your three problems with another group and compare your answers. Designing the Homework Assignment Designing a series of meaningful homework assignments requires thoughtful planning and outstanding organizational skills. Homework assignments should be designed to practice, reinforce, and extend the concepts presented in class, as well as review or revisit previously learned material.

Homework provides the teacher valuable information about student progress and ideas for future lessons. Homework must be assigned thoughtfully and be limited to enable completion in a reasonable amount of time. The homework assignment is your communication to the student about what you view as important, what you value conceptually, and which specific tasks you expect students to be able to master.

Although the majority of your homework problems should be within the reach of your students, you may include some problems that present an extra challenge. If you choose to assign problems that you know will be a bit difficult for many students, then you may wish to inform them of the difficulty ahead of time. This is not to excuse them from attempting the problem but rather to communicate the difficulty level so that they do not become frustrated if they fail to complete it.

Consider this homework assignment and determine if it supports your educational objectives: Homework: Pages —, do odd-numbered exercises 1— This homework assignment sends a negative message.

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An item that foreshadows what is to come in future lessons also is missing. Teachers may wish to design spiraled homework assignments. As the name indicates, these assignments spiral back to previously learned materials. With spiraled assignments, students can spend time reinforcing the current material while taking a small amount of time every night refreshing their memories of prior lessons. You can choose some or all of these features as you design your homework assignments to reflect your teaching style and tastes.

Submissions must be handed in by Monday, December 19, in class for extra credit. Homework assignment reflecting best practices. The elements of the homework assignment that make it effective are as follows: 1. The assignment is clearly numbered, with a date, a title, and a due date. It contains a specific reading assignment. Specific examples from the textbook are highlighted for the students to review. The homework assignment spirals. Notice that the class is reminded of an examination in 1 week, which begins the reviewing process. Notice how problems are grouped and labeled with a capital letter, A through G, to the right of each grouping.

This eliminates much of the commotion at the beginning of the lesson as students have clearly defined roles and responsibilities with respect to the homework. An added advantage of this method is that the homework is likely to be placed on the board correctly because students are assigned the problems in advance. In addition, students can easily refer to a specific problem because it is placed in a designated area. Homework can also be prepared on transparencies for display on an overhead projector or on a laptop computer. Of course, as new technologies evolve, there will be more options available to students and teachers.

The upcoming test is announced more than one week ahead of time. The material that the students are responsible for is clearly defined by both the textbook sections and the homework assignments. After-school tutoring schedules are included in the homework as well. More importantly, it initiates an online relationship with a respected mathematics organization that can serve as an outstanding resource in the future. Select several problems from the textbook that are aligned with this lesson.

Often, a problem that appears elementary has hidden complexities that make it unsuitable for homework. In such instances, you may wish to include these problems in future spiraled assignments because students will have had more time to master the concepts. Routines for Checking Homework Teachers must develop routines for checking homework. Students should understand that homework is an essential component of learning, and that you value their time and efforts in completing the assignment. Students prepare homework with increasing care when they receive positive feedback for their efforts.

There are many ways to check homework; it is important to adopt one or more of these techniques to validate student achievement.

## Exemplary Practices for Secondary Math Teachers

The model that is most efficient for checking homework is to circulate among the students during the start-up exercises and spot-check certain problems in the assignment. The third clearly identifies those students who require remediation or those who did not seriously attempt to do the homework. Remind students that because the homework assignment contains a significant amount of review, they are always expected to be successful with a portion of the assignment. A student who continually comes to class unprepared may be experiencing problems that require a referral to a guidance counselor.

You should ask students who are unprepared to see you after class for a private chat about the importance of doing homework and to ascertain the reason for not having done the homework. Another model that you can use is to collect the homework and spot-check it for accuracy. Some teachers find it appropriate to randomly select a row of students and ask them to hand in their homework for inspection. Of course, you can mix and match these methods and collect homework on some days but check homework at their seats on others.

Every now and then, you can collect a homework assignment and grade it as a quiz. Regardless of which method you choose, you must make students realize that homework is important, and it must be done regularly to be successful at learning mathematics. You do not have to ask each student to explain the problem that he placed on the board.

## Further Education in the Balkan Countries

However, students should have the opportunity to check their solutions against those placed on the board. It is a good practice to come to class with a notebook that contains the solutions to the homework problems, which makes it easy for you to compare the solutions on the board with your solutions. Most important, when you actually do the homework assignment, you can choose an appropriate subset for explanation in class to ensure that students see the solution to a variety of problems without redundancy.

Planning for Students Who Are Absent It is highly recommended that each student have the phone numbers and e-mail addresses of three other students in the class. You may wish to put aside a few minutes of class time during the first week of school for students to choose their partners. Encourage students to call their absentee partners to keep them abreast of class activities.

Should an absent student find the homework assignment daunting, they can always do the part that is a review of previously learned material. When students return to class after an absence, you may want to lend them your homework solution book. Many teachers use technology to communicate with their students. Homework assignments can be posted on your Web site, and your school may have educational chat rooms for student communication. As new technologies evolve, there will be more opportunities for members of a class to communicate with the teacher and each other.

It is advisable to be extremely organized when you plan your board work or the use of a technological alternative. You should know exactly where the aim, the start-up, and homework will be written. This is typically done at the beginning of the period and can be placed on a single panel of the board.

Some teachers, in an effort to save time, have these items prepared on an overhead projector transparency. Although it may save time at the beginning of the lesson, this practice has many drawbacks, the most serious of which is that it may not remain projected. Thus, students arriving late to class cannot record the motivational start-up exercise or copy the homework assignment. One effective technique is to have the aim, the start-up activity, and the homework prepared in advance on a small portable whiteboard.

This is particularly valuable when teachers are not teaching the same class back to back or are scurrying from one room to another for instruction. The time saved at the beginning of the period is significant and allows teachers to attend to other classroom routines, such as greeting students, taking attendance, checking homework, and providing support to students as they work on the start-up exercise.

Dealing with the Alternative Solutions Be ready to deal with unexpected strokes of genius or creative solutions. Although some of these may be invalid, they often provide valuable insights into how students think. You need to judge the validity of the statement and its value to the lesson. Do not feel pressured to draw an immediate conclusion.

Incorporating Technology into Your Lessons Teachers are encouraged to integrate technology into the mathematics classroom. To be successful, you must carefully plan the use of technology with your other classroom activities. The NCTM also provides a variety of lesson plans that are technology based. Let us consider presenting a lesson on the properties of the medians of a triangle.

In each scenario, an exciting lesson can be presented. Scenario 1: Low Tech Compasses, Straightedge, Cardboard, Scissors Students should either be familiar with the geometric construction to determine the perpendicular bisector of a given line segment or be taught it at the beginning of this lesson. Give students a piece of cardboard 8. Lead students through the construction of the perpendicular bisector of each of the sides of the triangle to find each midpoint. Students should draw the three medians to the sides of the triangle. At this point, students should make some observations about the medians of the triangle.

For each of the three medians, students should measure the distance from the vertex to the centroid and the distance from the centroid to the midpoint. Ask students to make conjectures about these lengths. They will be surprised to find that, for each of the three medians, these distances maintain a ratio. Instruct students to use their scissors to cut out their triangle from the rectangular cardboard.

Ask them to experiment and attempt to balance the triangle on the tip of a pen or pencil. Students will discover that the centroid is the balance point or center of gravity of the triangle. Using a dynamic geometry software program, perform the same constructions as in Scenario 1 Figure 5. Students will see a series of diagrams that will allow them to discover the concurrency of the medians.

The dynamic software will also reveal the ratio of the segments of the medians. As the software computes the lengths of the segments, students should be asked to make conjectures and draw conclusions. While there is certainly a value for teachers to create their own innovative lessons and units, the results of the multitude of Teacher Enhancement and Local Systemic Change projects supported by the NSF in the last two decades suggest that the use of exemplary comprehensive mathematics curricula is critical to the success of systemic reform.

Teachers also need to learn about high quality software and other technological tools if they are to implement mathematical learning experiences consistent with the most recent calls for reform. Teachers need to become proficient users of these technologies and to learn to consider how using these tools could affect not only their teaching practices but also their instructional goals. Understanding equity issues and their implications for the classroom. Because the new instructional goals and teaching practices articulated in the NCTM Standards are meant to recognize and respond to student diversity, researchers and policy makers are confident they will help bridge the achievement gap.

Our vignette is evidence of how mathematical tasks can be designed to provide access to students with diverse learning styles, strengths and background experiences. Multiple forms of assessment, as exemplified in our vignette by the combination of a group performance assessment and more traditional paper-and-pencil tests, may also help students with different strengths and learning styles to show more easily what they know.

However, taking on new instructional goals and teaching practices will not be enough for teachers to fully address equity issues in school mathematics. Each teacher must first gain a good understanding of the many issues related to equity and diversity and their implications for mathematics instruction Darling-Hammond, Teachers must also become aware of their own biases and privileges and learn how these may affect their relationship with students who are different with respect to race, class, gender, primary language, sexual orientation, etc. Weissglass, Teachers must also believe that all students can learn mathematics when they are provided with ample opportunities, conditions conducive to learning and high teacher expectations.