Middle School Cover - William H. Sadlier · by crafting activities that appeal to ... To this...
Transcript of Middle School Cover - William H. Sadlier · by crafting activities that appeal to ... To this...
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,
I
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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
The motivational activity should
draw on interest already present in
the learner. l Transparency
The motivational activity should
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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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)