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Thank you for taking the time to download our Middle School Math Digital Kit. Enclosed in this kit are the following materials:

-‐ Motivating Middle School Students: The Critical Part of Lesson Planning in Mathematics

-‐ Graphing Calculator Lessons for Students • Getting Started • Working with Fractions, Mixed Numbers, and Decimals • Evaluating Algebraic Expressions and Absolute Value • Powers and Exponents

We hope you save time with these Middle School Mathematic resources! -‐-‐Sadlier

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Motivating Middle School Students: The Critical Part

of Lesson Planning in Mathematics

by

Alfred S. Posamentier, Ph.DDean, School of Education

The City College of The City University of New York

nspiring students to learn is the

cornerstone of successful teaching.

A teacher’s skill in engaging a class in

its opening moments can set the tone

for an entire lesson and contribute

to its success. Regardless of the

approach—whole class, small groups, or

individuals—a key planning objective

is to determine ways to draw students

in at the outset. Teachers should strive

to develop motivating activities that

will not only introduce the lesson, but

also hold students’ attention throughout

the class period.

Positive learning environments make

the best use of students’ attitudes,

abilities, and experiences. Teachers can

create a successful learning environment

by crafting activities that appeal to

students motivated in one of two ways:

extrinsically or intrinsically.

Extrinsic motivation stimulates action

in pursuit of tangible rewards or set

goals. Sometimes extrinsic methods

of motivation may work well. These

methods include: grades, charts

with personal goals, competition,

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existing skills and knowledge, resulting in

a feeling of competence (Wolters, 2004).

Often, a student’s interest is stimulated

when his or her curiosity is piqued.

Teachers can whet students’ curiosity by

bringing, for example, an unusual item to

class—a large ball to demonstrate a geometric

principle or to explore Earth’s properties.

A teacher might also stir curiosity by

demonstrating a mathematical trick with

an unusual result that prompts students to

wonder why and how this trick works. Try

this “trick”: invite students to pick a number

(x), then multiply it by 2 [2x]. Then have

them add 9 to that [2x + 9], add in the

original number [3x + 9], and divide by

3 [x + 3]. To this students will add 4 [x +

7], and as a final step, subtract the original

number (7). No matter what original number

is chosen, the final result will always be 7.

contextualizing tasks that relate to students’

experiences, economic rewards for good

performance, peer acceptance of good

performance, avoidance of “punishment” by

performing well, and praise for good work

(Guild and Garger, 1998). Extrinsic methods

are effective for students in varying forms:

they often demonstrate extrinsic goals in

their desire to understand a topic or concept

(task-related), to outperform others (ego-

related), or to impress others (social-related).1

Intrinsic motivation—learning for its own

sake—results from the internal drives already

present in learners, such as the following:

l curiosityl the learner’s need to understand his or

her immediate environmentl a need to acquire a more complete

understanding of a topic or subjectl a need to improve one’s positionl a need to be entertained

This last drive may be affected by a teacher’s

classroom behavior, the content of material,

or the style in which it is presented.

Sources of MotivationAll learners, whether extrinsically or

intrinsically motivated, possess basic needs

and desires. Among them are:

Innate Curiosity It is a natural human trait to seek out

challenges that can be conquered by using

1 The socially-related goal can apply to both extrinsic

and intrinsic motivation.

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This drive to know more seems to go hand

in hand with achievement. According to

Gottfried, “Academic intrinsic motivation

was found to be significantly and

positively correlated with children’s school

achievement and perceptions of academic

competence and negatively correlated with

academic anxiety” (Gottfried, 1985).

Helping students achieve intrinsic

motivation contributes to the positive

learning environment. Here, teachers can

introduce the lesson with an example that

will challenge these students. In addition,

the lesson presentation can be followed

with an enriched problem to further allow

students to develop their competencies.

Need for Acceptance Students’ social needs, particularly by

middle school, influence their relationship

with the teacher, as well as with peers,

thereby affecting how students learn. For

instance, students tend to seek the approval

of teachers, and tend to view higher grades

as marks of approval, and conversely, lower

grades as disapproval (Fehr, 2007).

Students also develop a need to perform well in

front of their peers (Wolters, 2004). A teacher’s

Invite students to try several numbers to see

that this “trick” does, in fact, work every

time. Algebra will then reveal the “trick.”

A Choice to Learn The desire to act on something as a result of

one’s own volition is often a motivating factor

in the general learning process. Students will

be more motivated if they can determine for

themselves what is to be learned, rather than

learning merely to satisfy someone else or to

attain an extrinsic reward (Reeve, 2006).

There are motivational activities to support

autonomy and encourage students to want

to learn (Reeve, 2006). To do this, a teacher

can provide a problem that students may

not know how to solve, such as factoring

a binomial or a trinomial. Students can be

reminded that every math skill they develop

is based on knowledge and strategies they

already know. In this case, the teacher can

suggest that they factor two- and three-digit

sums and differences to find the strategies

they already know. Then students can be

encouraged to use these known strategies to

factor the binomial or trinomial.

Desire for Challenge Some students are more eager to do a

challenging problem than a routine one.

It is not uncommon to see this type of student

begin homework assignments with the most

challenging problem. If a test has an “extra

credit” item, these students might tackle it

before looking at the remainder of the test,

even if the time spent on the item prevents

them from completing the required portion.

SOURCES OF MOTIVATION FOR A LESSON INTRODUCTION

l Innate Curiosityl A Choice to Learnl Desire for Challengel Need for Acceptance

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awareness of these developing attitudes is

integral in planning effective motivation and

positive interactions with students.

Techniques for Engaging StudentsActive engagement helps students to make

knowledge their own and enables them

to think about new situations. There are

many techniques the teacher can use in the

mathematics classroom to engage students,

including:

Find Patterns Patterns are inherent in mathematics; they

are fundamental to algebra. Selected properly

and presented enthusiastically, patterns

can be an effective device for generating a

concept and stimulating students’ interest.

For example, consider a lesson on multiplying

two negative numbers. This concept can be

introduced by presenting a pattern similar to

the one below by having students predict the

next three numbers—in this case, 5, 10, and

15. Students can be asked to specify a general

rule about this pattern and then complete a

table to demonstrate the rule.

Factor 1 X Factor 2 = Product-5 X 3 = -15

-5 X 2 = -10

-5 X 1 = -5

-5 X 0 = 0

-5 X -1 = ?

-5 X -2 = ?

-5 X -3 = ?

Teachers can use the table as a basis for a

discussion on the pattern in the “Product”

column, engaging them as active learners.

The teacher can guide students’ thinking

through modeling or questioning to help

them understand the underlying rule and

justify it through common usage: “two

negatives imply a positive.”

For example, “I will not withdraw (two

negatives) money from the bank.” To further

engage students, the pattern can be extended—

in this example, to –15, –10, –5, 0. Students

then work to predict the next three numbers.

A similar table can lead students to generalize

that the product of two negative numbers is a

positive number. They may also discover that

their responses are to be addressed as the focus

of the next lesson, thus increasing anticipation

and interest. In this activity, the chart might

appeal to field-dependent learners, who, states

Whitefield (1985), “love to graph, map,

illustrate, draw, role-play, create charts,

invent games, make things, etc.”

Patterns can also bring students closer to having

a clearer understanding of negative exponents.

Presented with a pattern such as 81, 27, 9, 3,

students can be asked to predict what comes

next. Students can also be asked to consider the

first four powers of 3 in reverse order (from 34 to

31) and be guided to see the pattern as dividing

by 3 each time to get the next number:

34 = 81

33 = 27

32 = 9

31 = 3

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Extending this pattern of subtracting 1 from

the exponent and dividing the value by 3,

the sequence continues:

30 = 1

3–1 = 1—3

3–2 = 1—9

3–3 = 1—27

Some students may quickly see that they can

continue the pattern by subtracting 1 from

the exponent and dividing the preceding value

by 3. This example can be a good lead-in to a

deeper discussion of negative exponents.

The above examples show how patterns can

serve as motivators if selected strategically

and presented appropriately.2 In addition,

this type of problem solving might appeal

to students who need the support of specific

teacher directions, concrete solutions, and clear

instructions on what they are expected to learn.

2 A word of caution: there are patterns that appear to lead in one way but do not necessarily follow that anticipated direction. Teachers must be careful to select those that will not lead the class into an ambiguous situation. One example is the sequence: 1, 2, 4, 8, 16, …, which can follow at least two perfectly correct mathematical patterns: 1, 2, 4, 8, 16, 32, 64, 128, … or 1, 2, 4, 8, 16, 31, 57, 99, …. To find out more about this kind of problem see: 101+ Great Ideas for Introducing Key Concepts in Mathematics by Alfred S. Posamentier and Herbert A. Hauptman (Thousand Oaks, CA: Corwin Press, 2006). In addition, the National Mathematics Advisory Panel (2008) found that patterns are not emphasized in high-achieving countries. Because the prominence given to patterns in PreK-8 in this country is not supported by comparative analyses of curricula or mathematical considerations (Wu, 2007), the Panel strongly recommended that “algebra” problems involving patterns should be greatly reduced in state and NAEP assessments, textbooks, and curriculum expectations (p.59).

Present a Challenge A challenge can be rewarding, particularly

for those students who see a challenge as

a way to become a better thinker, a better

problem solver. For students for whom

challenges might invoke anxiety, offering

a guided challenge can instill a sense of

support and confidence.

For example, a teacher might say, “I’ll show

you that 2 = 1,” and write the following proof.

1. Let a=b2. Multiply both sides by a.

a2=ab3. Subtract b2 from both sides.

a2–b2=ab–b2

4. Factor. (a+b)(a–b)=b(a–b)

5. Divide both sides by (a–b). (a+b)=b

6. Since a=b, then

2b=b7. Divide both sides by b.

2=1

The students can then be challenged to find

the error (Posamentier, Letourneau, Quinn,

2009). If they cannot see it, point out in

step 5 that the original equation shows that

a and b are equal terms. In step 5, then,

(a – b) = 0. Division by zero is undefined,

or simply not permitted! At this point, the

rest of the proof falls apart.

Two important goals can be accomplished

using the above example: (1) it can provide

a powerful illustration of the role of

definitions, or rules, in mathematics and

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(2) introduces the concept that division by

zero cannot be permitted because doing so

leads to contradictions such as 2 = 1. Such

a challenge might inspire students to think

critically, challenge their assumptions, and

engage with mathematics in new ways.

Challenge Student’s Thinking Another form of motivation can be described

as enticing students with “amazing and

unexpected” aspects of mathematics (Sobel,

Maletsky, 1988). This may include, as a lead-

in, showing the class some counterintuitive

mathematical results.

For example, when introducing concepts in

probability, a teacher can show the totally

unexpected results of the birthday problem. To

do this, a teacher would have the class announce

their birth dates aloud one at a time and ask the

remaining students in the class to raise their

hand if they hear their own birthday mentioned.

Students will observe that in a room

of 30 people, the probability of two of

them sharing the same birthday is about

70%—far greater than anyone would

intuitively expect it to be. To further add

to the “amazing and unexpected” aspect, a

teacher could tell students that among the

first 34 American presidents, two had the

same birthday of November 2—the 11th

president, James K. Polk, and Warren G.

Harding, the 29th president (Posamentier,

Letourneau, Quinn, 2009).

Engaging students in discussions about

mathematics can “promote their active

sense-making” (Jansen, 2006). It can also

foster a sense of community, contributing

not only to students’ motivation, but

also to their success (Lewis, Schaps, and

Watson, 1996).

Connect to the Real World Pointing out a topic’s usefulness can also

be motivating (Boyer, 2002). For example,

students might be asked how they would

measure the height of the Empire State

Building. A discussion of applying

trigonometry, rather than dropping a tape

measure down from the top of the structure

(though possible, highly impractical),

would enhance students’ appreciation of

trigonometry’s real-world applications

(Posamentier, Letourneau, Quinn,

2009). It might help students to see this

problem exemplified in words, numbers,

and pictures on the board, overhead, or

interactive whiteboard.

Tell a Story A well-told story can be motivating, can

reduce math anxiety by activating the

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imagination, and can provide a relatable

connection from the story’s context to the

topic (Schiro, 2004). Storytelling also appeals

to the nature of field-dependent students,

as it puts mathematics in a framework that

is realistic or that relates to students’ lives

(Whitefield, 1995).

For example, there is a well-known story

about one of the greatest mathematicians,

Carl Friedrich Gauss (1777–1855). When

he was in elementary school, Gauss found

the sum of the first 100 natural numbers

much faster than his teacher anticipated he

would. Rather than add the numbers 1 + 2

+ 3 + … + 98 + 99 + 100 in the order in

which they are written, he decided to add

them in pairs:

1 + 100 = 1012 + 99 = 1013 + 98 = 1014 + 97 = 101...50 + 51 = 101

Then he multiplied 101 by 50 (the number

of pairs) and found the product 5,050.

After providing this relevant anecdote to

the class, the teacher can use the procedure

to help the students generate the formula

for the sum of an arithmetic sequence. The

style in which this story is presented is

key—not as a rush to get to the derivation

of the formula, but as story with interesting

embellishments.

Make Math Fun Often, a recreational feature of mathematics

that is related to a topic can be motivating.

For example, a teacher might inspire an

algebraic discussion of number properties by

asking each student to select a three-digit

number in which the hundreds digit and

the ones digit are not the same (for example,

847). The students then write their selected

number in reverse order (748) and subtract

the lesser number from the greater number

(847 – 748 = 99). Having them reverse the

digits in the result (099 to 990) and add

these two numbers (099 + 990) yields a

result they could share with the class.

Those who did not make an arithmetic

error should all have arrived at the same

answer: 1,089—no matter the number

with which they started. Students may

be amazed at this unusual number

characteristic, and motivated to determine

why it works; at the same time, the

teacher has produced a thought-provoking

introduction to an algebraic investigation.

TECHNIQUES FOR ENGAGING STUDENTSl Find Patternsl Present a Challengel ChallengeStudents’Thinkingl Connect to the Real Worldl TellaStoryl MakeMathFunl Discuss Surprising Relationshipsl PresenttheUnknownl IntegrateTechnology

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Discuss Surprising Relationships The teacher can further cultivate students’

interest by discussing the many surprising

relationships in mathematics. These

curiosities may pique the interest of students

and motivate them to determine why they

are true. Here, a teacher can have students

draw any quadrilateral: the purpose to show

the same result among varied shapes. The

class can be instructed to join the midpoints

of the sides and to observe each other’s

drawings. In every case, they will get a

parallelogram. This can lead to a discussion

of the properties of parallelograms.

Present the Unknown A motivational technique for more advanced

learners can be to make them aware of a

lack or void in their knowledge of a subject.

Students should be given an opportunity to

discover this lack on their own.

An activity for presenting the unknown can

be found in trigonometry. Students begin

their study of trigonometry with the various

trigonometric ratios (sine, cosine, tangent)

on the right triangle. Eventually they will be

led to consider angles of measure greater than

90°. To spark interest in this topic, a teacher

might have students find values such as:

sin 30°, cos 60°, and cos 120°.

Students familiar with the 30-60-90

triangle will likely be able to figure out the

first two values fairly easily. The third value

may confuse some students, since they are

unfamiliar with trigonometric functions

of angles that measure more than 90°.

Students will realize that they can find the

trigonometric functions of acute angles, but

not of obtuse angles. In this sense, students

might draw on their curiosity and be

motivated to extend their knowledge to find

the answer.

The technique of presenting the unknown

can also be employed in geometry when

introducing the measures of angles having

vertices outside a given circle. To illustrate

this, suppose students have learned the

relationships between the measures of arcs of

a circle and the measure of an angle (whose

rays subtend these arcs) with its vertex in or

on the circle, but not outside the circle.

The teacher might present the class with the

examples like the following, asking for the

value of x:

1.

x

x°

x°

15 92

30

25

36°

2. 3.

Because students are familiar with how

to find the value for x in the first two

diagrams, their confidence may grow. Then

the knowledge may set in that they cannot

find the measure of the angle formed by two

secants intersecting outside the circle. If

asked (appropriately) what they would like to

learn during the ensuing lesson, they would

likely ask to learn how to find the measure

of an angle formed outside the circle. This

shows they have been motivated.

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Integrate Technology Many teachers have begun employing

presentation stations, interactive whiteboards,

and other technologies to present lesson

introductions and engaging activities like

those above. These can provide or activate

important background knowledge, as well

as stimulate student interest. Such tools can

make technology-based teaching resources

such as virtual manipulatives, videos,

animations, and software tools available (with

oversight) to an entire class.

Studies report that students’ attitudes toward

learning improve when technology is used

in instruction (Silvin-Kachala 1998, Kulik

1994). The illustration mentioned earlier,

where the midpoints of a quadrilateral are

joined to form a parallelogram, can be very

dramatically demonstrated with geometric

software such as Geometer’s Sketchpad™.

Selecting a Motivational ActivityCareful selection of a motivational beginning

is the most creative, if not perhaps also the

most difficult aspect of planning a lesson. Some

helpful guidelines for selecting and presenting

a motivational activity include the following:

l Brevity

So as to allow time for teaching the

content of the lesson, teachers should

keep the length of the motivational

activity to a minimum.l Focus

The motivational activity should not

become the lesson. It should be a

means to an end, not an end in itself.

l Appropriateness

The motivational activity should

match the students’ level of ability

and interest. l Resourcefulness


draw on interest already present in

the learner. l Transparency


clearly connect to the content of the

lesson, as well as reveal the lesson’s

goal. Success here will determine how

effective the motivational activity was.

ConclusionA teacher can use a host of motivational

activities that have the ability to invigorate

the first few moments of a lesson—that

critical time when students’ attention

and interest might be won or lost. Using

activities like those discussed here, a teacher

can capture his or her students not only in

those first moments, but also throughout the

lesson, making learning a joy, not a chore.

The rewards can be boundless.

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About the AuthorAlfred S. Posamentier, Ph.D., Dean of the School of Education

and Professor of Mathematics Education at The City College of

the City University of New York, is the author of more than 40

mathematics books for teachers, secondary and elementary students,

and general readership. He is also a frequent commentator in

newspapers on topics relating to education. Dr. Posamentier

has been a frequent speaker at National Council of Teachers of

Mathematics (NCTM) conventions and has been a long-term

reviewer of new publications for The Mathematics Teacher

journal. Dr. Posamentier is also co-author of Sadlier-Oxford’s

Progress in Mathematics program for Grades K–8.

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Boyer, K.R. (2002). “Using Active Learning Strategies to Motivate Students.” Mathematics Teaching in the Middle School, 8 (1), 48–51.

Fehr, H.F. (2007). “Psychology of Learning in the Junior High School.” Mathematics Teaching in the Middle School, 12 (5), 283–287.

Friedel, J.M., Cortina, K.S., Turner, J.C., & Midgley, C. (2007). “Achievement Goals, Efficacy Beliefs and Coping Strategies in Mathematics: The Roles of Perceived Parent and Teacher Goal Emphases.” Contemporary Educational Psychology, 32, 434–458.

Gottfried, A.E., (1985). “Academic Intrinsic Motivation in Elementary and Junior High Students.” Journal of Educational Psychology, 77 (6), 631–645.

Guild, P.B. & Garger, S. (1998). Marching to Different Drummers, Second Edition. Alexandria, VA: Association for Supervision & Curriculum Development.

Harmin, M. (1998). Strategies to Inspire Active Learning: Complete Handbook. White Plains, NY: Inspiring Strategy Institute.

Jansen, A. (2006). “Seventh Graders’ Motivations for Participating in Two Discussion-Oriented Mathematics Classrooms.” The Elementary School Journal, 106 (5), 409–428.

Kulik, J. (1994). Meta-analytic studies of findings on computer-based instruction. In Baker, E. L. and O’Neil, H. F. Jr. (Eds.), Technology assessment in education and training. (pp. 9-33) Hillsdale, NJ: Lawrence Erlbaum.

Lewis, C.C., Schaps, E., & Watson, M. (1996). “The Caring Classroom’s Academic Edge.” Educational Leadership, 54, 16-21.

ReferencesMiddleton, J.A. (1999). “Curricular

Influences on the Motivational Beliefs and Practice of Two Middle School Mathematics Teachers: A Follow-up Study.” Journal for Research in Mathematics Education, 30 (3), 349–358.

National Mathematics Advisory Panel Final Report. Foundations for Success. 2008. U.S. Department of Education.

Posamentier, A.S. (2003). Math Wonders to Inspire Teachers and Students. Alexandria, VA: Association for Supervision and Curriculum Development.

Posamentier, A.S., Jaye, D., & Krulik, S. (2007). Exemplary Practices for Secondary Math Teachers. Alexandria, VA: Association for Supervision and Curriculum Development.

Posamentier, A.S., & Hauptman, H.A. (2006). 101+ Great Ideas for Introducing Key Concepts in Mathematics. Thousand Oaks, CA: Corwin Press.

Posamentier, A.S., Smith, B.S., & Stepelman, J. (2009). Teaching Secondary School Mathematics: Techniques and Enrichment Units, Eighth Edition. Upper Saddle River, NJ: Pearson/Merrill/Prentice-Hall.

Posamentier, A.S., Letourneau, C.D., & Quinn, E.W. (2009). Foundations of Algebra. New York, NY: Sadlier-Oxford.

Preckel, F., Goetz, T., Pekrun, R., Klein, M. (2008). “Gender Differences in Gifted and Average-Ability Students: Comparing Girls’ and Boys’ Achievement, Self-Concept, Interest, and Motivation in Mathematics.” Gifted Child Quarterly, 52 (2), 146–159.

Reeve, J. (2006). “Teachers as Facilitators: What Autonomy-Supportive Teachers Do and Why Their Students Benefit.” The Elementary School Journal, 106 (3), 225–236.

Schiro, M.S. (2004). Oral Storytelling and Teaching Mathematics: Pedagogical and Multicultural Perspectives. Thousand Oaks, CA: Sage Publications, Inc.

Silvin-Kachala, J. (1998). Report on the effectiveness of technology in schools, 1990-1997. Software Publishers Association.

Sobel, M.A., & Maletsky, E.M. (1988). Teaching Mathematics: A Sourcebook of Aids, Activities, and Strategies, Second Edition. Edgewood Cliffs, NJ: Prentice Hall.

Threadgill, J.A. (1979). “The Relationship of Field-Independent/Dependent Cognitive Style and Two Methods of Instruction in Mathematics Learning.” Journal for Research In Mathematics Education, 10 (3), 219–222.

Vaidya, S., & Chansky, N. (1980). “Cognitive Development and Cognitive Style as Factors in Mathematics Achievement.” Journal of Educational Psychology, 72 (3), 326–330.

Witkin, H.A., Moore, C.A., Goodenough, D.R., & Cox, P.W. (1977). “Field Dependent and Independent Cognitive Styles and Their Educational Implications.” Review of Educational Research, 47, 1–64.

Wolters, C.A. (2004). “Advancing Achievement Goal Theory: Using Goal Structures and Goal Orientations to Predict Students’ Motivation, Cognition, and Achievement.” Journal of Educational Psychology, 96 (2), 236–250.

Wu, H.H. (2007). Fractions, decimals, and rational numbers. University of California, Department of Mathematicc. Retrieved on February 1, 2008 from http://math.berkeley.edu/-wu/.

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• Checkthe menu to be sure all items on the left are highlighted. If necessary, use the arrow key to highlight your selection, then press .

• TogetbacktotheHOMEscreen: Press , since QUIT appears in blue above the key.

• ChecktheFORMAT menu.Press . All items on the left should be highlighted. Use the arrow key to highlight your selection, then press .

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• TurnallSTATPLOTSoff.

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Select 4:PlotsOFF. Press . You will return to the HOME screen.

• Tomakethescreendarker, press .

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GettingStarted

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Working with Fractions

To use the graphing calculator with fractions, remember the fraction bar means “division”.

Working on the HOME screen,

enter 3 __ 4 as .

Press . The calculator displays .75.

Working with Mixed Numbers

To use the graphing calculator with mixed numbers, remember to use a plus (1) sign.

Enter 5 3 __ 4 as .

Press . The calculator displays 5.75.

Objective To add and subtract fractions and mixed numbers • To change non-repeating and repeating decimals to fractions

Working with Fractions, Mixed Numbers, and Decimals

HintTo clear screen display, press .

Adding Fractions

Add: 7 __ 8 1 2 __ 16

The solution is 1.

Subtracting Mixed Numbers

Subtract: 5 3 __ 4 2 4 1 __ 4

Be sure to use parentheses. The parentheses prevent the calculator from misinterpreting the expression.

The solution is 1.5.

Changing Nonrepeating Decimals to Fractions

Write the decimal 0.875 as a fraction.

Step 1 Enter .

Step 2 Step 3 Step 4 Step 5

Press the Select 1. Frac. The word Frac appears Press . key. on the HOME screen.

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Changing Repeating Decimals to Fractions

Write the repeating decimal 0. _ 3 as a fraction.

Step 1 Enter 0.3333333333333….. (type across the screen until the line is full)

Step 2 Step 3 Step 4 Step 5

Press the Select 1. Frac. The word Frac appears Press . key. on the HOME screen.

Not all decimals can be expressed as fractions. Decimals that do not repeat and do not terminate are called irrational numbers. These cannot be expressed as fractions.

Subtract: 4 1 __ 2 23 1 __ 3 . Express your answer as a fraction.

Step 1 Enter the number. If the second set of parentheses was not used, only the 3 would have been subtracted (by the order of operations).

Step 2 Press . The decimal solution is 1.1 _ 6 .

Step 3 Press . Select 1. Frac.

Step 4 Press . Solution: 7 __ 6

Add or subtract. Express your answer as a fraction.

1. 5 __ 4 1 2 __ 5 2. 1 __

5 2 18 __ 20 3. 7 __ 2 1 10 __ 3

4. 8 4 __ 7 1 6 2 __ 9 5. 22 27 5 __ 6 6. 21 11 __

12 2(3 3 __ 4 )

Write each decimal as a fraction.

7. 0.12 8. 0. _ 4 9. 4.

_ 1 10. 0.

___ 235

11. Examine your answer for exercise 10. Is there a pattern? Use your calculator to test your conclusion.


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Objective To evaluate algebraic expressions and absolute value signs

Evaluating Algebraic Expressions and Absolute Value

Method 1:

You can substitute directly on the HOME screen. Enter the expression replacing the variable with the given numerical value. The absolute value sign (1:abs) is located under the MATH key’s NUM menu.

Method 2:

You can store the value of 5 as the variable x using the STO key. To access any letter variable written in green above the key, first press then the key. For “x” you can

press or then , since “x” appears in green above the STO key.

Evaluate: 2 __ 3 (x 2 2)2 1 2x, when x 5 5

Solution: 11

Evaluate: 2x 1 4.1 1 5x, when x 5 2.7

Store 2.7 as the variable x.

Press .

Enter the expression using x on the HOME screen.

Solution: 6.7

Let x 5 22 and y 5 3. Use the STO key to store each variable’s value.

Notice since “y” is written in green above the key, press then .

and

Evaluate each expression.

1. xy2 2. 3(xy)2 3. 6x 1 y

4. 2x22y 5. 2(5x 1 3) 23y 6. 20.31x 1 2.5y


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Objective To evaluate expressions with exponents

Powers and Exponents

Raising to a Power

To evaluate a number in the form baseexponent, use the key in row 4, column 5. Notice that it is placed in the same column as the four basic operation keys.

Simplify: 67

Press .

Solution: 279,936

Squaring a Number

You can use to square a number. (The exponent is 2.) You can also use the “square” key .

Simplify: 92

Press .

Solution: 81

Cubing a Number

You can use to cube a number. (The exponent is 3.) You can also use the “cube” function.

Simplify: 73

Press .

Press to return to the HOME screen.

Press again for the solution: 343.

Simplify.

1. 246 2. (24)6 3. 252 4. 62 5. 23 6. 33(322)

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