Download Rational Numbers

Rational Numbers
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Fractions
Decimals
Percents
It is important for students to know how
these 3 concepts relate to each other
and how they can be interchanged.
Fraction Vocabulary
•  It is important that vocabulary terms are
taught to students.
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Fraction - part of a whole - all parts are equal.
Numerator - top number (number of parts considered).
Denominator - bottom number (number of equal parts).
Greatest Common Factor (GCF) - largest common
factor for both the numerator and denominator that is
larger than 1.
–  Least Common Multiple (LCM) - the smallest common
multiple of two or more denominators.
–  Improper Fraction - fraction that is greater than 1.
–  Mixed Number - whole number and a fraction.
Fraction Models
•  Region or Area
–  Surface is divided into equal parts
–  Rectangular or circular
•  Set
–  Whole is a set of objects and subsets of
the whole make up the fractional parts.
–  Example: You have 12 pieces of candy.
You give your friend 5 pieces. They have
5/12 of the candy.
Fraction Models
•  Length or Measurement
–  Length is compared instead of area.
–  Number lines
–  Line segments
Reading and Writing Fractions
•  Students must know the following terms to be able to
use fractions both orally and in written form.
–  Halves, thirds, fourths, fifths, sixths, sevenths, etc.
•  Students will need to read and write fractions with
guidance from teachers and quick reinforcement if
they are reading and/or writing fractions incorrectly.
•  Writing fractions
–  Show students fractions with either concrete objects or
pictures. Instruct them that the denominator is the
number of parts and the numerator is the number of
shaded parts.
Beginning Fraction Concepts
•  When students are beginning to learn about
fractions they should be working with
concrete materials.
–  Fraction tiles
–  Fraction circles
•  Fractions should also be related to real life
experiences. Example: Give half of your
candy to your brother.
Beginning Fraction Concepts
•  Identifying/Counting Fractional Parts
–  Identify the number of equal parts.
–  Count the number of the parts being
named.
–  This will assist students in understanding
that 1/8 is smaller than 1/4, which is a
concept that is difficult for students to
understand at first.
Beginning Fraction Concepts
•  Relationship to 1
–  Students must quickly identify if a fraction is less
than 1 (1/2), equal to 1 (5/5), or greater than 1 (7/5).
–  Best demonstrated by using concrete fraction pieces
that can be manipulated.
–  Denominator larger than the numerator - less than 1
(proper fraction).
–  Numerator and denominator are the same number equal to 1.
–  Numerator larger than denominator - greater than 1
(improper fraction).
Beginning Fraction Concepts
•  Mixed Numbers - 7 1/8
–  Students must understand that it is the
same as 7 + 1/8.
–  Reading mixed numbers - 7 and 1/8.
–  Writing mixed numbers - 7 1/8.
Beginning Fraction Skills
•  Comparing Fractions
–  Students must understand that the more parts in a
whole, the smaller the fraction.
–  Compare using <,>,= signs.
•  1/2 > 1/4
4/8 < 2/5 4/8 = 1/2
–  Students must also realize that 7/8 is closer to 1
than 3/4.
–  Concrete fraction pieces will assist students, as
they can put pieces on top of each other and
compare the differences.
Error Analysis of Beginning
Fraction Concepts
•  Students should complete pages 137-150 of
the Error Analysis.
–  Identify the error
–  Complete two problems following the student s
method
–  Provide 2 strategies for teaching the student.
–  Check your answers compared to the author s
Adding & Subtracting
Fractions
•  Like Denominators
–  This should be fairly simple if students
have a good understanding of the
beginning fraction concepts.
–  3/4 - 1/4 = 2/4
7 4/9 + 1 1/9 = 8 5/9
–  If students are having difficulty, a review of
beginning concepts is necessary.
Adding & Subtracting
Fractions
•  Unlike Denominators
–  Students should first explore with fraction pieces.
(See handouts for examples)
–  Students need to be taught the concept of
common multiples. Students must know their
multiplication facts to be able to do this accurately
and quickly.
•  For students who have difficulty, fraction bars may be of
assistance. (See handouts for example).
–  Once students determine the least common
denominator they write equivalent fractions and
solve the problem.
Multiplying Fractions
•  Begin with concrete objects. (see handouts for
examples).
•  It is important to emphasize that the denominators
do not have to be the same in multiplication
problems.
•  Algorithm - should only be taught after students
have add ample opportunities to practice
conceptually.
–  Multiply the numerators - write the answer.
–  Multiply the denominators - write the answer.
–  Read the problem.
Dividing Fractions
•  Dividing fractions is the inverse of multiplying
fractions. Most are taught to invert the divisor
and multiply. While students can do this with
little difficulty, most do not understand what
they are doing or why they are doing it.
•  Providing students with story problems may
assist them in understanding. Example: Maria
has 1 1/2 hours to complete her test. If the test
has 6 questions, how much time can Maria
spend on each question?
Writing Fractions in Simplest
Terms
•  Teach the concept of finding the greatest
common factor (GCF).
–  Determine factors of the numerator.
–  Determine factors of the denominator.
–  Determine the greatest common factor.
•  4 = 1,2,4
12 = 1,2,3,4,6,12
GCF = 4
4 x? = 4
4 x1= 4
4 x ? = 12
4 x 3 = 12
4 = 1
12 3
Website
•  Students should explore the following
website that shows how fractions can
be taught.
•  http://nlvm.usu.edu/en/nav/
category_g_2_t_1.html
•  There are several fraction activities to
explore.
Error Analysis
•  Students should complete pages 151-166
AND 172-180 of the Error Analysis.
–  Identify the error
–  Complete two problems following the student s
method
–  Provide 2 strategies for teaching the student.
–  Check your answers compared to the author s
Decimals
•  Equal parts of a whole.
•  The use of Base 10 Blocks will assist
students in conceptually understanding
decimals.
–  Flat = Whole numbers
–  Rod = Tenths
–  Cube = Hundredths
Reading and Writing Decimals
•  As with fractions, it is important that students
are able to understand decimals both in
written form and verbally.
–  Tenths, hundredths, thousandths, etc.
–  .7 = seven tenths
–  .77 = seventy-seven hundredths
–  .501 = Five-hundred one thousandths
–  4.53 = Four and fifty-three hundredths
Comparing Decimal Values
•  Students often think the more numbers
to the right of the decimal, the larger the
number.
•  Using manipulatives to show decimals
will assist students with this (see
handouts for examples).
•  .4 > .06
.754 < .75
Adding & Subtracting
Decimals
•  Students should have mastered addition and
subtraction up to 4-digit numbers (with and
without regrouping) prior to working with
decimals.
•  When beginning the problems should have
the same place value.
•  When adding and subtracting students should
be taught to first bring down the decimal point
and then compute the problem.
Multiplying Decimals
•  Proficiency in multiplying whole multi-digit numbers is
essential before beginning to work with decimals.
•  The use of Base 10 Blocks will assist students in
understanding.
•  Algorithm
–  Multiply problem.
–  Starting from the right, count the number of decimal points
and place the decimal point in the problem.
–  Important to provide examples with zero in both the factors
and the product.
Dividing Decimals
•  Proficiency in dividing whole numbers is
necessary prior to working with
decimals.
•  Algorithm
–  Bring the decimal point directly up.
–  Solve as a regular division problem.
Percents
•  Can be introduced once students have a
strong sense of fractions and decimals.
•  What part of 100% belong to a certain group.
(see handouts for examples)
•  Common uses of percents:
–  Sales tax of 6.25%
–  Worth 25% of your grade
–  Put 50% of your money in a savings account
–  We have raised 84% of our goal.
Relating Fractions, Decimals,
Percents
•  It is important to be able to switch
between fractions, decimals, and
percents and know which one is easiest
to use in different situations.
Relating Fractions, Decimals,
Percents
•  A shirt cost $35.00. The sign says it is
33% off. Will you multiply 35 x. 33 or 35
x 1/3?
•  If you have 60 baseball cards and you
have to share half with your brother, will
you multiply 60 x .50 or 60 x 1/2?
Common Fraction, Decimal,
Percent Equivalents
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3/4
1/2
1/3
1/4
1/5
1/10
.75
.50
.33
.25
.20
.10
75%
50%
33%
25%
20%
10%
Error Analysis in Decimals
•  Students should complete pages 167-171
AND 181-184 of the Error Analysis.
–  Identify the error
–  Complete two problems following the student s
method
–  Provide 2 strategies for teaching the student.
–  Check your answers compared to the author s
References
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Ashlock, R.B. (2005). Error patterns in computation. Upper Saddle, NJ:
Prentice Hall.
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Hudson, P., & Miller, S.P. (2006). Designing and implementing
mathematics instruction for students with diverse learning needs.
Boston: Allyn & Bacon.
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Kennedy, L.M., Tipps, S., & Johnson, A. (2004). Guiding children s
learning of mathematics (10th ed.). Belmont, CA: ThomsonWadsworth
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Sovchik, R.J. (1996). Teaching mathematics to children. New York: Longman.
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Van De Walle, J.A. (2004). Elementary and middle school
mathematics: Teaching developmentally (5th ed.). Boston:
Allyn & Bacon.