After completing this chapter, you should

**Solving Equations **

- be able to identify various types of equations
- understand the meaning of solutions and equivalent equations
- be able to solve equations of the form (x+a=b) and (x−a=b).
- be familiar with and able to solve linear equation

**Solving Equations of the Form (ax=b) and (dfrac{x}{a}=b)**

- understand the equality property of addition and multiplication
- be able to solve equations of the form (ax=b) and (dfrac{x}{a}=b)

**Further Techniques in Equation Solving**

- be comfortable with combining techniques in equation solving
- be able to recognize identities and contradictions

**Applications I - Translating from Verbal to Mathematical Expressions**

- be able to translate from verbal to mathematical expressions

**Applications II - Solving Problems**

- be able to solve various applied problems

**Linear Inequalities in One Variable**

- understand the meaning of inequalities
- be able to recognize linear inequalities
- know, and be able to work with, the algebra of linear inequalities and with compound inequalities

**Linear Inequalities in Two Variables**

- be able to identify the solution of a linear equation in two variables
- know that solutions to linear equations in two variables can be written as ordered pairs

## Tough Math Riddle: Solve Equations 0 0 0=6, 1 1 1=6

Your objective is to insert mathematical operations between the numbers on the left hand side in such a way that it equals to the number on the right hand side.

In the space between the numbers you can use any kind of functions like addition subtraction division etc.

But you cannot bring up new numbers

Let me solve one for you as an example

So I add addition sign to make the equation correct as follows

Similarly solve the remaining equations.

So were you able to solve the riddle? Leave your answers in the comment section below.

You can check if your answer is correct by clicking on show answer below. If you get the right answer, please do share the riddle with your friends and family on WhatsApp, Facebook and other social networking sites.

To get the answers you can solve them as following

The Only way to make a zero 1 without adding any other number is to use factorial because 0! = 1

## Key stage 1 - years 1 and 2

The principal focus of mathematics teaching in key stage 1 is to ensure that pupils develop confidence and mental fluency with whole numbers, counting and place value. This should involve working with numerals, words and the 4 operations, including with practical resources [for example, concrete objects and measuring tools].

At this stage, pupils should develop their ability to recognise, describe, draw, compare and sort different shapes and use the related vocabulary. Teaching should also involve using a range of measures to describe and compare different quantities such as length, mass, capacity/volume, time and money.

By the end of year 2, pupils should know the number bonds to 20 and be precise in using and understanding place value. An emphasis on practice at this early stage will aid fluency.

Pupils should read and spell mathematical vocabulary, at a level consistent with their increasing word reading and spelling knowledge at key stage 1.

## 5.1: Objectives - Mathematics

1 iv) The sum of the series given by is equal to

v) Geometrical meaning of scalar triple product of three vectors is the

a) Volume of parallelepiped formed by as adjacent sides

5. Find a unit vector perpendicular to the plane containing points P(1, -1, 0), Q(2, 1, -1) and R(-1, 1, 2).

The given points P(1, -1, 0), Q(2, 1, -1) and R(-1, 1, 2) lies in the plane PQR.

Accordingly, the vectors the plane PQR .

Finally, the required unit vector will be

Finally, the desired unit vector is

1 v) If the order of matrix A is m x n and order of matrix B is n x m, then the order of the matrix AB is,

a) n x n b) m x n c) n x m d) m x m

vi) If a, b, c is in H.P then, what is the value of b?

6. In how many ways can be letter of words “Sunday” be arranged? How many of these arrangement begin with S? How many begin with S and don’t end with y?

The word SUNDAY be arranged in 6!=6*5*4*3*2=720 ways

When a word begins with S. Its position is fixed, i.e. the first position. Now rest 5 letters are to be arranged in 5 places. So,

No. of arrangement begin with S = 5! = 120

The are 5! ways of ordering the word SUNDAY if the first letter is restricted to S, because only the 5 remaining ones are allowed to change.

There are 4! ways of ordering the word SUNDAY if the first letter is restricted to S "and" the last letter is restricted to Y

Therefore, there are 5!-4! ways that the word SUNDAY can be arranged such that the first letter is S and the last letter "is not" Y

5!-4! = 120-24 =96

1 vi) Length of the Latus rectum of the parabola 2y2 - 9x = 0 is,

vii) Which of the following is the rank of the Matrix ?

7. If then show that x 2 + y 2 = 1.

1 vii) Which of the following statement is not true?

a) is a vector b) is a vector

c) is a vector c) is a vector

viii) In how many ways 6 persons can seat in a round table?

1 viii) How many triangles can be formed by joining six non-collinear points.

ix) Let , and a map defined by T(x)=A(x) then what is the image of under T?

1 ix) Real part of the complex number is,

x) If then, this is the equation of..

a) Parabola b) Hyperbola c) Ellipse d) Circle

8. a) Define conic section. Find the coordinates of vertices, eccentricity and foci of the ellipse

9x 2 + 4y 2 - 18x - 16y - 11 = 0. (**1+5)**

A conic section (or simply conic) is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic sections are the hyperbola, the parabola, and the ellipse.

9x 2 + 4y 2 - 18x - 16y - 11 = 0

Converting to standard form of ellipse:

** Coordinate of Vertices:**

x) If S_{1} is the sum of first n natural numbers and S_{2} is the sum of cubes of first n numbers then,

b) If T(x_{1}, x_{2}) = (x_{1} + x_{2}, x_{2}, x_{1})defined by be the linear transformation, then find matrix associated with linear map T. **(4)**

T(e _{1} ) = T(1, 0) = (1+0, 0, 1) = (1, 0, 1)

T(e _{2} ) = T(0, 1) = (0+1, 1, 0) = (1, 1, 0)

9. a) Define irrational number. Prove that √2 is an irrational number. (**1+4)**

An **irrational number** is a number that cannot be expressed as a fraction for any integers and .

## Oklahoma Priority Academic Student Skills

Please be aware that all below information may not reflect current standards and should be used only as secondary reference.

### Pre-Kindergarten and Kindergarten

### Priority Academic Student Skills

### Codified Rules of thePriority Academic Student SkillsClick here to visit the Rules Pages

Pre-Kindergarten* (Adopted July 24, 2003, Revised Spring 2011)

*Includes Language Arts, Mathematics, Health, Safety and Physical Development, Science, Social and Personal Skills, and Social Studies.

**Includes Language Arts, Mathematics, Motor Skills and Lifetime Activity Development, Science, Social and Personal Skills, Social Studies, and the Arts.

### Content Area Priority Academic Student Skills

### Codified Rules of thePriority Academic Student SkillsClick here to visit the Rules Pages

### Integrated Curriculum by Subject

### Priority Academic Student Skills

### Codified Rules of thePriority Academic Student SkillsClick here to visit the Rules Pages

OAC 210:15-3-147—210:15-3-152

OAC 210:15-3-153—210-15-3-162

### Integrated Curriculum by Grade

### Priority Academic Student Skills

### Codified Rules of thePriority Academic Student SkillsClick here to visit the Rules Pages

Pre-Kindergarten (Adopted July 24, 2003, Science Revised Spring 2011)

Kindergarten (Revisions: Math Summer 2009 Language Arts March 2010 Science Spring 2011)

Grade 1 (Revisions: Math Summer 2009 Language Arts March 2010 Science Spring 201

OAC 210:15-3-12

OAC 210:15-3-40.2

OAC 210:15-3-41

OAC 210:15-3-71

OAC 210:15-3-91

OAC 210:15-3-115

OAC 210:15-3-134

Grade 2 (Revisions: Math Summer 2009 Language Arts March 2010 Science Spring 2011)

OAC 210:15-3-13

OAC 210:15-3-40.2

OAC 210:15-3-42

OAC 210:15-3-72

OAC 210:15-3-92

OAC 210:15-3-116

OAC 210:15-3-134

OAC 210:15-3-14

OAC 210:15-3-40.2

OAC 210:15-3-43

OAC 210:15-3-73

OAC 210:15-3-93

OAC 210:15-3-117

OAC 210:15-3-134

Grade 4 (Revisions: Math Summer 2009 Language Arts March 2010 Science Spring 2011)

OAC 210:15-3-15

OAC 210:15-3-40.2

OAC 210:15-3-44

OAC 210:15-3-74

OAC 210:15-3-94

OAC 210:15-3-118

OAC 210:15-3-135

Grade 5 (Revisions: Math Summer 2009 Language Arts March 2010 Science Spring 2011)

OAC 210:15-3-16

OAC 210:15-3-40.2

OAC 210:15-3-45

OAC 210:15-3-75

OAC 210:15-3-95

OAC 210:15-3-119

OAC 210:15-3-135

Grade 6 (Revisions: Math Summer 2009 Language Arts March 2010 Science Spring 2011)

OAC 210:15-3-17

OAC 210:15-3-46.1

OAC 210:15-3-47

OAC 210:15-3-76

OAC 210:15-3-96

OAC 210:15-3-120

OAC 210:15-3-135

Grade 7 (Revisions: Math Summer 2009 Language Arts March 2010 Science Spring 2011)

OAC 210:15-3-18

OAC 210:15-3-46.1

OAC 210:15-3-48

OAC 210:15-3-77

OAC 210:15-3-97

OAC 210:15-3-121

OAC 210:15-3-135

Grade 8 (Revisions: Math Summer 2009 Language Arts March 2010 Science Spring 2011)

OAC 210:15-3-19

OAC 210:15-3-46.1

OAC 210:15-3-49

OAC 210:15-3-78

OAC 210:15-3-98

OAC 210:15-3-122

OAC 210:15-3-135

## Technology in Mathematics Education: Preparing Teachers for the Future

The preparation of preservice teachers to use technology is one of the most critical issues facing teacher education programs. In response to the growing need for technological literacy, the University of Northern Colorado created a second methods course, Tools and Technology of Secondary Mathematics. The goals of the course include (a) providing students with the opportunity to learn specific technological resources in mathematical contexts, (b) focusing student attention on how and when to use technology appropriately in mathematics classrooms, and (c) giving opportunities for students to apply their knowledge of technology and its uses in the teaching and learning of mathematics. Three example activities are presented to illustrate these instructional goals of the course.

The preparation of tomorrow’s teachers to use technology is one of the most important issues facing today’s teacher education programs (Kaput, 1992 Waits & Demana, 2000). Appropriate and integrated use of technology impacts every aspect of mathematics education: what mathematics is taught, how mathematics is taught and learned, and how mathematics is assessed (National Council of Teachers of Mathematics [NCTM], 2000). Changes in the mathematics curriculum, including the use of technology, have been advocated for several years. The Mathematical Sciences Education Board (MSEB) and the National Research Council maintain that “the changes in mathematics brought about by computers and calculators are so profound as to require readjustment in the balance and approach to virtually every topic in school mathematics” (MSEB, 1990, p. 2). Future mathematics teachers need to be well versed in the issues and applications of technology.

Technology is a prominent feature of many mathematics classrooms. According to the National Center for Education Statistics (NCES, 1999), the percentage of public high school classrooms having access to the Internet jumped from 49% in 1994 to 94% in 1998. However, the use of computers for instructional purposes still lags behind the integration of technology in the corporate world and is not used as frequently or effectively as is needed. One way to close the gap and bring mathematics education into the 21st century is by preparing preservice teachers to utilize instructional tools such as graphing calculators and computers for their future practice.

In the past at our campus, technology issues and “training” in mathematics education were addressed within the confines of a regular three-semester-hour mathematics methods course, taught by a professor of mathematics education within the College of Arts and Sciences. With increasing demands placed on the teacher preparation program by state legislation, which has become common throughout all of education, the amount of content in the methods course was becoming overwhelming. As a result, little time was available to address the issue of the technology required for effective mathematics instruction.

Even before the additional state requirements, relatively little time was spent providing preservice teachers with hands-on experience using graphics calculators and mathematics software. Secondary mathematics majors occasionally used a computer algebra system (CAS) for different projects within their calculus courses, as well as spreadsheets and software applications in their statistics course. Additionally, most teacher candidates have had experience using graphics calculators at different points within various mathematics courses. However, little time was spent preparing preservice mathematics teachers to use technology in their future classrooms. Our program has required all secondary education majors to take two one-credit general education technology courses that address spreadsheet, word processing, and Web-page development, but none of these college technology experiences provided them with content specific or classroom specific experiences they will need as future mathematics teachers.

Our response to the growing need for technological literacy was to create a second methods course entitled, Tools and Technology of Secondary Mathematics. This course supplements the content and methods of our existing methods course, but focuses on the utilization of technology in secondary mathematics classrooms. In keeping with the philosophy of our Secondary Professional Teacher Education Program, the course has three broad aims. First, teacher candidates receive hands-on training in using software tools, graphing calculators, and the Internet for mathematics instruction focused at the secondary school level. Second, they learn how and when to use appropriate technology to enhance their mathematics instruction of topics that are taught at the middle and high school grades. Third, they develop and teach lessons to their peers with equipment available to a typical public school mathematics classroom, using the technology learned in this course.

One purpose of the technology methods course is to provide the opportunity for preservice teachers to use specific technological resources in mathematical contexts. That is, teacher candidates are presented with a task involving some mathematical problem or situation and are required to learn to use and apply an appropriate piece of technology in completing the task. For example, one activity used in the methods course is found on the NCTM (2004) *Illuminations* Web site (available at http://illuminations.nctm.org/lessonplans/9-12/webster/index.html). The activity, titled “The Devil and Daniel Webster” and adapted from Burke, Erickson, Lott, and Obert (2001), has teacher candidates explore recursive functions using technology. The undergraduates are presented a scenario in which each person earns an initial salary of $1,000 on the first day, but pays a commission of $100 at the end of the day. On subsequent days, both amount earned and commission are doubled. Preservice teachers complete a chart using either handheld or computer technology to determine if it is profitable to work for one month under these conditions. Additional questions require the undergraduates to graph the data from the chart. In this way, teacher candidates not only learn to use the kinds of technological tools that are available for use in instruction, but also learn them in the context of examining mathematics, which helps increase their content knowledge.

In addition to learning to use the technology, pedagogical issues associated with the instructional tools are emphasized. Specifically, the course focuses attention on how and when to use technology appropriately in mathematics classrooms. Misuses of technology are discussed and discouraged, such as using calculators as a way to avoid learning multiplication skills and using computers to practice procedural drills rather than to address conceptual understanding. Rather, preservice teachers discuss the uses and benefits of commercial software and handheld devices to explore different content topics that have become possible with technology and consider pedagogical issues. Some time is also spent previewing national curriculum projects that have a high involvement with technology (e.g., Key Curriculum Press, 2002). As a result, preservice teachers address and discuss issues of teaching prior to their clinical experience, which helps these students focus attention on these matters when participating in their practicum.

Teacher candidates in the technology methods course apply their knowledge of technology and its uses in the teaching and learning of mathematics. These future mathematics teachers create several lesson plans using technology as an instructional tool. Lesson plans center around concepts and skills found in pre-algebra, algebra, geometry, precalculus and calculus that are enhanced using technology. Once a topic is selected for the lesson plan, preservice teachers determine an appropriate piece of technology that facilitates instruction. They develop and write instructional lessons using graphing calculators, an interactive mathematics computer environment, an interactive geometry application, computer spreadsheets, and the Internet. However, based on a selection of specific mathematics topics, each teacher candidate creates lessons using additional forms of technology examined in the course, including dynamic statistical software and a CAS. As a result, each teacher candidate has a unique experience of using technology to enhance mathematics instruction at the secondary school level.

Depending on time constraints, preservice teachers teach at least one of their lessons with their peers as students. Our course ensures that these future mathematics teachers are able to write and deliver lesson plans that incorporate appropriate technology for mathematics courses at the level for which they are seeking licensure.

It is important that teachers are able to develop well-conceived lesson plans that are structured and detailed, focusing on specific mathematics topics and using multiple representations, such as the examples in the appendices. Open-ended exploration and inquiry-driven mathematics lessons using such software as interactive, dynamic geometry or algebra software are also developed after the teacher candidates are able to develop a detailed lesson that explores the topic with some depth. For students to experience a mathematics topic in depth, specific “guided” discovery lesson planning is required. Part of the objective is to counter a pervasive disposition of the mathematics curriculum in this country as being a mile wide and an inch deep.

Since the creation of the technology methods course, we believe that our program adequately addresses the needs of many preservice teachers to be competent at integrating these instructional tools for teaching and learning mathematics. The growth of future teachers’ ability to use technology appropriately in the mathematics classroom during the course becomes evident in observations. The following illustrations provide detailed descriptions of the process in which preservice teachers engage as they learn, analyze, and apply a particular piece of technology in the course.

Interactive Computer Environment

One important feature of the course is to introduce future teachers to the world of possibilities open to instruction when computers are used effectively. The vast majority of our preservice teachers have had some experience using computers within and outside their high school mathematics courses, but few have had the opportunity to learn mathematics in an interactive computer environment. Providing this experience for our teacher candidates has created a template on which they can draw as future teachers.

**Figure 1.** *Changing the value of v0 in the function v(t) is apparent in the graphs and tables.* (Click anywhere on figure to view the enlarged image.)

For one activity, the preservice teachers use an interactive mathematics computer environment as an electronic textbook. Embedded in the text is the derivation, using calculus, of the velocity of an object under the influence of earth’s gravity as a function of time (i.e., *v(t)* = *gt* + *v0*). Through this interactive environment preservice teachers manipulate parameters and see, in real time, the effects of those changes on the graphs and data tables of the function. For example, after explaining that the value of the gravitational constant, *g*, is 9.8 meters per second per second, teacher candidates integrate the gravitational constant with respect to time, *t*, to obtain the velocity function: *v(t)* = –*gt* + v0. This function illustrates the physics principle that the velocity of an object is the integral of its acceleration. In Figure 1, the result of changing the initial velocity from 49 meters per second to 4.9 meters per second is apparent by the graphs and tables. After completing this assignment, preservice teachers learn how to create an activity using the interactive computer environment.

The potential of such an instructional tool is readily apparent to teacher candidates. Instead of using a static textbook in which authors determine examples and illustrations, using an interactive computer environment in instruction allows the preservice teachers to choose their own examples and participate in dynamic illustrations. Additionally, the undergraduates can type and check spelling, as in any common word processor, respond to problems and questions embedded in the computer application, and print copies for classroom use or assessment purposes by the teacher.

Teacher candidates then develop lessons or activities using this technology that are appropriate for their future middle school or high school students. One possible activity applies the knowledge gained in the initial experience with the interactive computer environment. Appendix A contains an example of one such activity used in our program as a guide for preservice teacher generated work that uses the height of an object acted upon only by the force of gravity as an application of quadratic equations. The scenario involves the launch of a model rocket into the air and requires high school students to model the height of the object as a function of time in tabular and graphical form. Such an activity demonstrates the multiple uses of important components of the interactive computer environment within an appropriate context of secondary mathematics.

Interactive Geometry Application

One way to introduce teacher candidates to a particular piece of technology is through classroom-ready, published materials. This is particularly useful when the software is well established and used regularly in classrooms, because teachers could adopt the activity for future classroom use. In one case, we use Bennett (2002) to introduce undergraduates to interactive geometry software on the computer. For example, the following problem could be posed at the beginning of a class session: How could you determine the height of a tree without measuring it directly? At the time they take the technology methods course, teacher candidates typically have an extensive cache of techniques to solve such a problem from prior geometry and trigonometry courses. Bennett (2002) utilizes interactive geometry software to find such indirect measures using lengths that are easy to measure and proportions in similar triangles. Specifically, the worksheet directs the learner to create line segments to represent the tree’s height and the learner’s height in the application then learners construct parallel lines to simulate the rays of the sun. Finding the tree’s height is a matter of calculating the unknown length (tree’s height) in the proportion of ratios of object height to shadow length. Although preservice teachers often know this technique, constructing the solution in the interactive environment helps clarify concepts and procedures learned in prior courses.

After becoming familiar with the software from the activity, discussions take place on the appropriate uses of the technology. In the case of the interactive geometry software, teacher candidates should recognize several potential uses of the software in a high school geometry course. For example, appropriate use of the software can reinforce properties of similar triangles in students’ minds. The preservice teachers should also recognize that the interactive component of the software allows their students to see that corresponding angle measures remain equal and that corresponding ratios of sides remain equal during actions that change the dimensions of the similar triangles. Preservice teachers reflect on the ability of the software to have students discover these properties, rather than simply telling their students, thus creating a more student-centered classroom environment. These future teachers should also recognize the need to transfer the knowledge gained from the interactive domain to problem situations away from the technology, which leads to discussions of how this might be accomplished.

As a culminating experience with the technology, preservice teachers create lessons using the software that are applicable to a secondary mathematics course. Often, ideas for these activities are generated by recognizing alternative solution methods for problems already considered. After exploring the interactive geometry software while solving the tree problem, teacher candidates are encouraged to develop alternative solution methods for solving indirect heights. Appendix B presents a follow-up activity for finding indirectly unknown heights of objects. The problem involves finding the height of a flagpole when a mirror is placed on the ground between an observer and the flagpole. The activity leads learners to find an indirect height using similar triangles formed by the reflection in the mirror because the angle of incidence equals the angle of reflection for light. Additionally, the solution plan requiring learners to reflect a ray across a line demonstrates the principles involved, as well as a more sophisticated feature of the interactive geometry environment.

One of the easiest technologies for preservice teachers to learn, and yet one of the most adaptable for classroom instruction, is graphing calculator technology. Still, too few secondary school mathematics teachers are comfortable using graphing calculators or know how to use them effectively for classroom instruction. A primary goal of the technology methods course is to provide instruction and experience with handheld technology. Utilizing graphing calculators in a statistical application is one way to meet this goal.

Recording, graphing, and analyzing data are important skills in mathematics, as well as in everyday life. The notion that data exist everywhere in the world is important for students to realize. Additionally, the ability to organize data provides a person with quick numerical and visual representations of the data and the power to predict, to within a predetermined degree of accuracy, future related events based on the data. An introductory lesson for managing data using handheld technology is to enter and graph party affiliations of the presidents of the United States. Two common representations of the data are bar graphs and circle graphs (see Figure 2).

**Figure 2**. Circle graph and bar graph of presidential party affiliations in TRACE mode.

One of the issues that should be raised by preservice teachers involves the best visual representation of the political parties of the presidents. They should discuss the advantages and disadvantages of their bar graphs and circle graphs, as well as other common graphical representations. Although the graphs can be obtained from computer spreadsheet technology, students must recognize the importance of being familiar with handheld technology as well. We want our teacher candidates to be capable and experienced with various technological tools so that they are comfortable using the technology available to them in the schools in which they will be teaching.

One required activity of the course is to develop a problem involving the collection, graphing, and analysis of data for middle school or high school mathematics students to complete. Appendix C contains an authors’ example of one such activity used in the technology methods course but applicable for a high school class. This activity uses presidential data, similar to the introductory activity, but involves the ages of the presidents at the time of inauguration. The activity extends the relatively simple task of representing data using handheld technology and includes more statistically rigorous analysis of the presidential ages. The activity highlights the mathematical power available to most students to make sense of the world around them using statistical analysis.

Teachers will use technology appropriately and effectively in their mathematics classrooms if they are familiar and comfortable with the technology and, especially, if they have had successful experiences with the technology in an instructional environment. Additionally, teachers who are able to use today’s technology in the classroom will be prepared to learn and utilize tomorrow’s technology. This core course for the secondary teacher education program provides that experience. After this course, the teacher candidates integrate the technology in their field experiences conducted in one of the university’s partner schools. In one instance, preservice teachers use technology during their first clinical teaching experience. At another time, during their semester-long student-teaching experience, host teachers and university faculty members evaluate student teachers on their ability to integrate technology in the classroom. Upon graduation, these future teachers should not only be knowledgeable as to which mathematics concepts are best learned through technology, but also will have had many successful experiences in developing and carrying out lesson plans that involve a variety of different technologies.

Since the creation of our technology-based methods course, its need is apparent. Although technology in typical secondary schools is sparse, several of our partnership schools are dedicated to utilizing technology in mathematics education. From interactive chalkboards to data-sharing hubs for handheld devices, our preservice teachers are beginning to experience these instructional tools during their field experiences. Consequently, we think it is important to prepare them for these eventualities. Our preservice teachers’ experience with technology in our program makes them attractive to secondary school selection committees.

The quality of our preservice teachers since our program emphasized technology in the mathematics classroom is apparent. As university supervisors, we often hear from the host teachers that our graduates are highly knowledgeable in dealing with technological instructional tools. Many host teachers admit to learning valuable teaching strategies using technology from individuals in our program. Although most of our preservice teachers receive favorable technology evaluations, we think we can do better. Our preservice teachers continue to think pedagogically in ways that they were taught rather than to think of the potential learning gains using technology. This course does lay the foundation for these teachers as they become more comfortable with their teaching practices and different ways to educate their students.

Today’s middle school and high school students were born into a world with technology. Using technology during mathematics instruction is natural for them, and to exclude these devices is to separate their classroom experiences from their life experiences. One objective in preparing teachers for the future is to ensure that their classrooms will include the technology that will be commonplace for a future generation of mathematics learners, thus ensuring that the mathematicians, mathematics educators, and citizens of tomorrow experience harmony between their world of mathematics and the world in which they live.

Bennett, D. (2002). *Exploring geometry with Geometer’s Sketchpad.* Emeryville, CA: Key Curriculum Press.

Burke, M, Erickson, D., Lott, J. W., & Obert, M. (2001). *Navigating through algebra in grades 9 – 12.* Reston, VA: National Council of Teachers of Mathematics.

Kaput, J. J. (1992). Technology and mathematics education. In D. A. Grouws (Ed.), *Handbook of research on mathematics teaching and learning*, (pp. 515–556). New York: MacMillan Publishing Company.

Key Curriculum Press. (2002). *IMP sample activities*. Retrieved November 15, 2004, from http://www.mathimp.org/curriculum/samples.html

Mathematical Sciences Education Board. (1990). *Reshaping school mathematics: A philosophy and framework for curriculum.* Washington, DC: National Academy Press.

National Center for Education Statistics. (1999). *Digest of education statistics 1998.* Washington, DC: U.S. Department of Education.

National Council of Teachers of Mathematics. (2000). *Principles and standards for school mathematics*. Reston, VA: Author.

National Council of Teachers of Mathematics. (2004). *Illuminations*. Retrieved November 8, 2005, from http://illuminations.nctm.org/

Waits, B. K., & Demana F. (2000). Calculators in mathematics teaching and learning: Past, present, and future. In M. J. Burke & F. R. Curcio (Eds.), *Learning mathematics for a new century* (pp. 51–66). Reston, VA: National Council of Teachers of Mathematics.

Robert Powers

University of Northern Colorado

[email protected]

William Blubaugh

University of Northern Colorado

[email protected]

**Figure 1**. Changing the value of *v0* in the function *v*(*t*) is apparent in the graphs and tables.

Students can download NCERT Maths Class 10 Chapter 5 Exercise 5.1 PDF from Vedantu for free. The PDF contains solutions to the sums given in the exercise with clearly defined steps for the understanding of students. The familiarity with these concepts can be achieved through practice. Class 10 Chapter 5 Exercise 5.1 introduces the concepts of Arithmetic progression. It is one of the foundational concepts in mathematics. With a firm grasp of the topic and an in-depth understanding of the concepts governing progressions, students can score well in their upcoming exams and also in their higher studies. Our curated solution for CBSE NCERT books for Class 10 Maths has a specific focus on exam preparation. With these Solutions of 10th class maths , students can acquire in-depth knowledge of all the chapters. Candidates can download NCERT solution PDF from Vedantu and continue with their exam preparation.

Download class 10 maths ch 5 ex 5.1 pdf here and begin your preparations. These notes are provided free of cost with the objective of imparting knowledge to everyone. Students can also download Class 10 Science Solutions for free from Vedantu.

## Meaningful Connections: Objectives and Standards

As a new teacher, you are probably being asked how your learning objectives are linked to standards. You might even be asked to display your objectives and/or standards for each lesson. On top of taking attendance, learning student names, classroom management . . . are you wondering how you will accomplish that? Don't despair, this is not as daunting as it seems!

### Why Do We Link Objectives to Standards?

Hopefully, you are using the standards as a foundation for what you teach so that your students are learning the material they should be learning that's the science of teaching. Then you take the standards and create objectives for your students that's the art of teaching. You think about the question: "What do I want students to learn, and how will they demonstrate that learning?" Look at the example below where we have taken the standard for "solving problems" and made it creative by having students "create a blueprint." That's how we make a meaningful connection between the standards and objectives. That's how we link the science of teaching to the art of teaching. We have also included writing, which is a focus of Common Core Standards. Yet it is not just writing an explanation it is a persuasive essay.

#### Example

**Standard:** Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. (This is a Common Core mathematics standard for seventh grade.)

**Objective:** Students will compute lengths and areas of a classroom to create a blueprint of the classroom indicating the scale used. When finished, students will write a "sales pitch" to a person explaining why their blueprint is accurate and should be purchased.

Within the objective, we have included the "what" and the "how." This will keep us on task in the classroom and will tell the students what the task is. When we post this objective for the students, we are letting them know the task at hand and that it is important enough to post. We have also included multiple levels of Bloom's Taxonomy, which is important to ensure that our students are critical thinkers.

### Creating Objectives

So here is the challenge. Take the standards below and create objectives for your classroom. Choose a grade level, or several grade levels. The standards are listed by grade levels and are taken directly from the Common Core Standards.

**Kindergarten:** Correctly name shapes regardless of their orientations or overall size.

**Grade 1:** Partition circles and rectangles into two and four equal shares, describe the shares using the words *halves, fourths* and *quarters,* and use the phrases *half of, fourth of* and *quarter of*. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

**Grade 2:** Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons and cubes.

**Grade 3:** Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. *For example, partition a shape into four parts with equal area, and describe the area of each part as 1/4 of the area of the shape.*

**Grade 4:** Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.

**Grade 5:** Classify two-dimensional figures in a hierarchy based on properties.

**Grade 6:** Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes apply these techniques in the context of solving real-world and mathematical problems.

**Grade 7:** Know the formulas for the area and circumference of a circle, and use them to solve problems give an informal derivation of the relationship between the circumference and area of a circle.

**Grade 8:** Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

Once you've met this challenge, post your objectives in the comments section below, and let's help each other take the science of teaching and connect it to the art of teaching.

## Maths Targets

Here you can find the target sheets for Maths. The targets:

- link to the end of year and key stage expectations set out in the 2014 National Curriculum
- include additional targets that have been developed by staff at Parkfield to ensure high expectations
- are used by teachers regularly to assess what a pupil can do and identify what a child needs to work on to improve
- provide the basis for teacher assessment each term
- are used when a child has completed an independent piece of writing
- are used to demonstrate the progress a pupil makes across the year.

**What are the target sheets for?**

The reading, writing and maths target sheets include the age related objectives that each child is expected to meet by the end of the academic year. Teachers use these target sheets when assessing to regularly record what a pupil can do and identify what a child needs to work on to improve.

We're incredibly proud of our targets because they've been personalised by our staff to ensure that we have high expectations at the school. Our targets have been requested and used by schools up and down the country because of their robustness and ease of use.

**Why do you send out the targets?**

Many schools don't share what's expected by the end of the year. However, we believe that showing a whole year's worth of objectives helps parents understand the expectations in that year group. We send the targets out at the end of Autumn and Spring in Years 1-6 so you can see the progress already made and what they need to work on next.

**Why don't you send out a level?**

Levels are not helpful. Although they may help you compare against others or give a snapshot measurement they don't tell you what your child can and can't do. This system helps identify the actual learning objective they need to learn in order to make progress.

**What do the 'ticks' mean in Years 1-6?**

One tick on a target means that they've shown some understanding but they may not understand fully or be able to complete independently. Two ticks means that the target has been met. Three ticks means that they have a greater understanding of that target and can confidently use it in different contexts. For example, in maths, they will be able to solve problems and reason confidently on a target which has three ticks.

**Why aren't many of the targets ticked?**

In Autumn and Spring it is highly unlikely that all targets have been ticked. This is because many of the targets haven't been taught and we don't assess until after teaching a particular topic. Teachers focus on different objectives across each term so that by the end of the year all objectives have been covered in depth. For example, some of the maths curriculum won't be covered until the summer term.

**What is the highlighting for?**

The highlighting helps teachers identify the progress made by your child each term and ensure that they're on track.

**My child isn't making the same progress as others in their class, why?**

Every child makes progress at a different rate and has different starting points so please don't compare. We regularly track and monitor rates of progress so that we can intervene when necessary and ensure that each child achieves their very best.

**When do you report how well they are doing?**

We will only report on this in the summer term because during Autumn and Spring the vast majority of children will still be working towards the expected level. In their end of year report each child will be assessed as either 'Working towards', 'Working at' or 'Working at a greater depth/above' the age related expectations in reading, writing and maths.

**How many targets does my child need to achieve to be at the expected level?**

When the vast majority of the targets on a sheet are ticked twice the class teacher will assess your child as at the expected level. If a high number of targets are ticked three times, they maybe assessed by the teacher to be working at a greater depth.

**How do I help my child?**

Focus your support by helping with the targets that have only one tick or no ticks at all. When writing the sheets, we've tried to make the targets child and parent friendly. However, if you're unsure of what a target means please contact the class teacher and they'll be happy to assist.

## Intermediate Algebra

Math 0110 is a preparatory course for college algebra that carries no credit towards any baccalaureate degree. However, the grade received in Math 0110 does count towards a student’s overall GPA. The course covers operations with real numbers, graphs of functions, domain and range of functions, linear equations and inequalities, quadratic equations operations with polynomials, rational expressions, exponents and radicals equations of lines. Emphasis is also put on problem-solving.

**Textbook and Course Materials:**

- MML AUTO ACCESS -
**Required** - INTERMEDIATE ALGEBRA WORKBOOK – Custom Edition -
**Required**– A manual containing an outline of class notes. - TEXTBOOK –
**Recommended**- Intermediate Algebra by Martin-Gay, 7th edition.

**Sections Covered**

**Section 1.2 Algebraic expressions and Sets of Numbers**

- Identify and evaluate algebraic expressions.
- Identify natural numbers, whole numbers, integers, and rational and irrational numbers.
- Find the absolute value of a number.
- Find the opposite of a number.
- Write phrases as algebraic expressions.

**Section 1.3 Operations on Real Numbers and Order of Operations**

- Add and subtract real numbers.
- Multiply and divide real numbers.
- Evaluate expressions containing exponents.
- Find roots of numbers.
- Use the order of operations.
- Evaluate algebraic expressions.

**Section 1.4 Properties of Real Numbers and Algebraic Expressions**

- Use operation and order symbols to write mathematical sentences.
- Identify identity numbers and inverses.
- Identify and use the commutative, associative, and distributive properties.
- Write algebraic expressions.
- Simplify algebraic expressions.

**Section 2.1 Linear Equations in One Variable**

- Solve linear equations using properties of equality.
- Solve linear equations that can be simplifies by combining like terms.
- Solve linear equations containing fractions or decimals.
- Recognize identities and equations with no solutions.

**Section 2.2 An Introduction to Problem Solving**

- Write algebraic expressions that can be simplified.
- Apply the steps for problems solving.

**Section 2.3 Formulas and Problem Solving**

**Section 2.4 Linear Inequalities and Problem Solving**

- Use interval notation.
- Solve linear inequalities using the addition property of inequality.
- Solve linear inequalities using the multiplication and the addition properties of inequality.
- Solve problems that can be modeled by linear inequalities.

**Section 2.5 Compound Inequalities**

- Find the intersections of two sets.
- Solve compound inequalities containing
**and**. - Find the union of two sets.
- Solve compound inequalities containing
**or**.

**Section 2.6 Compound Inequalities**

**Section 2.7 Absolute Value Inequalities**

- Solve absolute value inequalities of the form |x|<a.
- Solve absolute value inequalities of the form |x|>a.

**Section 3.1 Graphing Equations**

- Plot ordered pairs.
- Determine whether an ordered pair of numbers is a solution of an equation in two variables.
- Graph linear equations.
- Graph nonlinear equations.

**Section 3.2 Introduction to Functions**

- Define relation, domain, and range.
- Identify functions.
- Use the vertical line test for functions.
- Use function notation.

**Section 3.3 Graphing Linear Functions**

- Graph linear functions.
- Graph linear functions by using intercepts.
- Graph vertical and horizontal lines.

**Section 3.4 The Slope of a Line**

- Find the slope of a line given two points on the line.
- Find the slope of a line given the equation of the line.
- Interpret the slope-intercept form in an application.
- Find the slopes of horizontal and vertical lines.
- Compare the slopes of parallel and perpendicular lines.

**Section 3.5 Equations of Lines**

- Graph a line using its slope and intercept.
- Use the slope-intercept form to write the equation of a line.
- Use the point-slope form to write the equations of a line.
- Write equations of vertical and horizontal lines.
- Find equations of parallel and perpendicular lines.

**Section 4.1 Solving Systems of Linear Equations in Two Variables**

- Determine whether an ordered pair is a solution of a system of two linear equations.
- Solve a system by graphing.
- Solve a system by substitution.
- Solve a system by elimination.

**Section 4.3 Systems of Linear Equations and Problem Solving**

- Solve problems that can be modeled by a system of two linear equations.
- Solve problems with cost and revenue functions.
- Solve problems that can be modeled by a system of three linear equations.

**Section 5.1 Exponents**

- Use the product rule for exponents.
- Evaluate expressions raised to the 0 power.
- Use the quotient rule for exponents.
- Evaluate expressions raised to the negative nth power.
- Convert between scientific notation and standard notation.

**Section 5.2 More work with exponents**

- Use the power rules for exponents.
- Use exponent rules and definitions to simplify exponential expressions.
- Compute using scientific notation.

**Section 5.3 Polynomials and Polynomial Functions**

- Identify term, constant, polynomial, monomial, binomial, trinomial, and the degree of a term and of a polynomial.
- Define polynomial functions.
- Review combining like terms.
- Add polynomials.
- Subtract polynomials.
- Recognize the graph of a polynomial function from the degree of the polynomial.

**Section 5.4 Multiplying Polynomials**

- Multiply two polynomials.
- Multiply binomials.
- Square binomials.
- Multiply the sum and difference of two terms.
- Multiply three or more polynomials.
- Evaluate polynomial functions.

**Section 5.5 The Greatest Common Factoring and Factoring by Grouping**

- Identify the GCF.
- Factor out the GCF of a polynomial’s terms.
- Factor polynomials by grouping.

**Section 5.6 Factoring Trinomials**

- Factor trinomials of the form .
- Factor trinomials of the form .
- Factor by substitution.

S**ection 5.7 Factoring by Special Products**

- Factor a perfect square trinomial.
- Factor the difference of two squares.
- Factor the sum or difference of two cubes.

**Section 5.8 Solving Equations by Factoring**

- Solve polynomial equations by factoring.
- Solve problems that can be modeled by polynomial equations.
- Find the x-intercept of a polynomial function.

**Section 6.1 Rational Functions and Multiplying and Dividing Rational Expressions**

- Find the domain of a rational expression.
- Simplify rational expressions.
- Multiply rational expressions.
- Divide rational expressions.
- Use rational functions in applications.

**Section 6.2 Adding and subtracting Rational Expressions**

- Add or subtract rational expressions with a common denominator.
- Identify the least common denominator (LCD) of two or more rational expressions.
- Add or subtract rational expressions with unlike denominators.

**Section 6.3 Simplifying Complex Fractions**

- Simplify complex fractions by simplifying the numerator and denominator and then dividing.
- Simplify complex fractions by multiplying by a common denominator.
- Simplify expressions with negative exponents.

**Section 6.5 Solving Equations containing Rational Expressions**