Format
of Guidelines GRADE 5
STRAND
|
SEQUENCE
|
SKILL
|
OBJECTIVE |
VOCABULARY
|
EXAMPLE |
PROBLEM SOLVING
EXAMPLE |
CONNECTIONS,
INTEGRATION |
RESOURCES |
|
|
TOPIC |
Description of the extent to which a skill is developed,
or of what behaviors the students will exhibit in completing the activities
used to develop the skill. |
Vocabulary with which the students should be able to
communicate (speak, write) mathematically. |
Computational or conceptual representation of skill
indicating the extent to which the skill is to be presented at this level. |
Application of skill to problem-solving/real-life
situation. |
Inter-relatedness of skill to real-life or applications
of skill to other areas of the curriculum or to other skills in mathematics. |
This space is intended to be utilized by the teacher to
list the references and materials appropriate to the development of the
skill: e.g. teacher guide, technology, supplementary materials,
manipulatives, etc. |
STRAND
refers to the general area of mathematics into which each topic is
categorized. Strands include READINESS
SKILLS, NUMBER CONCEPTS, OPERATIONS, PROBLEM SOLVING, MENTAL MATHEMATICS,
ESTIMATION, GEOMETRY, MEASUREMENT, REASONING SKILLS, DATA , TECHNOLOGY,
RATIO-PROPORTION-PERCENT, AND PRE-ALGEBRA SKILLS.
A
section of these guidelines is also designated as ENRICHMENT. This is intended for the horizontal
expansion of topics rather than the vertical progression of skills.
TOPIC –
specific area within each strand. (For
example, USING A PROBLEM-SOLVING MODEL is a topic in the strand of PROBLEM
SOLVING).
READINESS CONCEPTS, NUMBER CONCEPTS –
GRADE 5
|
SEQUENCE
|
SKILL |
OBJECTIVE |
VOCABULARY |
EXAMPLE |
PROBLEM
SOLVING EXAMPLE |
CONNECTIONS,
INTEGRATION |
RESOURCES
|
|
M - D
|
Number
Sense
Place
Value
|
M-5.1: The students will
maintain understanding of place value of whole numbers through billions and
decimals through hundredths. M-5.2: The students will
determine the effect that changing a digit will have on the value of the
number. |
units tens hundreds thousands ten-thousands hundred-thousands millions ten-millions hundred-millions billions ten-billions hundred-billions numeral digit |
7,824,315 reads seven million, eight
hundred twenty four thousand, three hundred, fifteen. Caution: Do not allow the students to read the word “and” when reading
whole numbers. “And” indicates the
placement of a decimal point. (e.g.
8.4 is “eight and four tenths.”) |
Draw a place value chart for forty nine
billion Write a ten-digit number with a six in the
millions place and a four in the hundred-thousands place. |
Study Skills –
-use of reference materials. Language Arts –
Writing large numbers within a paragraph
sentence Social Studies, Science – Larger and smaller numbers used in data. Physical
Education – number
used to record speed and distances Technology –
limitations
of calculator display |
NCTM
ADDENDA SERIES – Number
sense |
|
M
|
READING AND WRITING NUMBERS (NUMERALS)
|
M.5.3
The
students will continue to develop skills in reading and writing numbers.
|
number
numeral |
Reading: Stress use of commas as an aid in reading whole numbers and the use of the decimal point in separating whole numbers and decimals. Writing: Stress correct formation
of numerals and |
Write in words: 4,003,016.024 (four million, three thousand, sixteen and twenty-four thousandths) Write the standard
numeral for:) |
Language Arts: Handwriting Reading- vocabulary enrichment (number-related words). Spelling-correct use the –th at the end of words indicating decimals: tenths, hundredths, etc. Correct writing of numbers in sentences outlines, tables, etc. |
|
|
READINESS CONCEPTS, NUMBER CONCEPTS –
GRADE 5
SEQUENCE
|
SKILL
|
OBJECTIVE
|
VOCABULARY
|
EXAMPLE
|
PROBLEM SOLVING
EXAMPLES
|
CONNECTIONS
INTEGRATION
|
RESOURCES
|
|
|
|
|
|
mathematical symbols;
correct size of fractions and exponents. |
Sixteen billion, twenty-two thousand, one
hundred fifty-three and twenty-one hundredths. (16,000,022,153.21 |
Social Studies, Science: reading and writing numerical data. Technology: interpreting |
|
|
D
|
NUMBER
SENSE
Fractions
|
M-5.4: The students will define a
fraction as 1)
equal parts of a whole; or 2)
equal parts of a set M-5.5: The students will relate
fractions to a point on a number line |
fraction fraction
bar numerator denominator |
˜ ˜ ˜ 1 ˜ ˜ ˜ 2 |
Mrs.
Ricardo cut the pizza into eight pieces and gave Ricky two pieces. Draw a picture to show this. What fraction does Ricky’s share
represent? |
Real Life – give the students practice
dividing items into fractional parts. |
|
SEQUENCE
|
SKILL
|
OBJECTIVE
|
VOCABULARY
|
EXAMPLE
|
PROBLEM SOLVING EXAMPLES
|
CONNECTIONS
INTEGRATION
|
RESOURCES
|
D
|
NUMBER SENSE Decimals |
M.5.6 The
students will understand the concept of decimals. M.5.7The students will be able to change decimals to fractions and percents. |
decimal decimal
point decimal
fraction decimal
square percent |
Use
graph paper/decimal squares to illustrate concept of a decimal as part of the
whole represented in units divisible by ten-tenths, hundredths, thousandths,
etc. The
decimal square has 100 parts. Twenty
two of these are covered. This can be
represented by the decimal numeral 0.22. |
Real Life: Money-a penny is one one-hundredth of a
dollar; a dime is one-tenth of a dollar. Time—a
year is one-one-hundredth of a century; a year is one-tenth of a decade. Language Arts: Vocabulary-note
words with dec-as a root- December, decade, decagon, etc. |
|
|
|
D |
NUMBER
PATTERNS
|
M-5.8: The students will discover
and complete number patterns. |
multiples skip
counting patterns number
sequence ordinal
numbers cardinal
numbers |
2,
5, 8, 11, 14 (
+ 3) 1,
5, 7, 11, 13,
17 (
+ 4, then + 2) |
Create
a pattern that adds 1, then 2, then 3, then 4, and so on to get to succeeding
numbers in the pattern. |
Science – patterns in nature Social Studies –
Election
years Real Life – patterns in architecture,
industry, sports Math – Geometry- Using pattern
block or cubes, have the students discover patterns with rectangular
arrangement. |
|
READINESS CONCEPTS, NUMBER CONCEPTS –
GRADE 5
SEQUENCE
|
SKILL
|
OBJECTIVE
|
VOCABULARY
|
EXAMPLE
|
PROBLEM SOLVING
EXAMPLES
|
CONNECTIONS
INTEGRATION
|
RESOURCES
|
||
|
D
|
READING AND WRITING MATHEMATICAL SYMBOLS
|
M-5.9: The students will recognize, read, and write
mathematical symbols. M-5.10 The students will
recognize the fraction bar as a symbol of division. |
symbol +, -, x, ¸, =, <,
> fraction
bar |
Provide
practice using symbols in context. |
Write
an equation for the following: Sister
Jean had four and one-half pretzel left over. She gave two and one-half to her helpers. How many did she have left? |
Real Life: reading signs Language Arts: using abbreviations |
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M M D
I
|
COMPARING
AND ORDERING NUMBERS Whole Numbers Fractions Decimals
through Hundredths
|
M-5.11: Given real-life
situations, the students will compare numbers. M.5.12 Use a number line as an
aid in ordering whole numbers, decimals and fractions. |
greater than less
than equal equivalent |
Compare
on number line: number to the right is always the greater number. Use
place-value chart to compare whole numbers. Using
graph paper or fraction bars, create models of fractions to compare. |
Items
to compare: -
sports statistics - gas mileage - nutritional value of food - interest rates |
Social Studies: distances on maps Study Skills: reading charts, tables,
etc. Math: problem solving,
measurement |
Newspapers
and magazines provide rich sources of material for number comparisons. |
||
SEQUENCE
|
SKILL
|
OBJECTIVE
|
VOCABULARY
|
EXAMPLE
|
PROBLEM SOLVING
EXAMPLES
|
CONNECTIONS
INTEGRATION
|
RESOURCES
|
||
|
M |
ROUNDING
WHOLE NUMBERS AND DECIMALS |
M.5.13 The students will round
whole numbers and decimals to any place-value position. |
rounding |
Round 53,769.217
to the nearest hundredth. (The
digit one is in hundredths place. The
number immediately to the right is 7 which is greater than 5, so 53,769.217
rounded to the nearest hundredth is 53,769.22 |
Locate
sizes of countries in an atlas or almanac. Round
these numbers to the nearest thousand. Order
these from greatest to least. |
Mathematics: measurement,
estimating |
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|
D |
EXPANDED
NOTATION
|
M-5.14 The students will express
numbers thorough billions using expanded notation. |
standard
form exponential
notation |
805,602,000,000 800,000,000,000 + 5,000,000,000 + 600,000,000 + 2,000,000 |
Write
the numeral 745,274,103,204
using expanded notation. |
Study Skills: outlining skills; finding
the smaller components of the whole. |
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|
|
POWERS
AND EXPONENTS
|
M-5.15: The students will
understand that when a number is multiplied by itself it can be expressed as
a base with an exponent. |
factors
exponent base powers |
exponent 100,000
= 105 base Write
23 as a product. (23
= 2 · 2 · 2) 3 factors Write
5 · 5· 5 · 5 using
exponents (54) |
Which
of the following statements is correct: A.
32 = 23 B.
42 = 24 B
is correct. A.
9 ¹ 8 B.
16 = 16 Express
a billion as a power of ten. (109) |
Science, Social Studies: Use of larger numbers |
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ESTIMATING |
Estimation
skills are developed in various places throughout these guidelines. |
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READINESS CONCEPTS, NUMBER
CONCEPTS GRADE - 5
SEQUENCE
|
SKILL
|
OBJECTIVE
|
VOCABULARY
|
EXAMPLE
|
PROBLEM SOLVING
EXAMPLES
|
CONNECTIONS,
INTEGRATION
|
RESOURCES
|
D
|
FACTORS Common
Factors
|
M-5.16: The students will identify
common factors of a set of numbers. M-5.17: The students will
determine the greatest common factor of two or more numbers. |
factors common
factors greatest
common factor |
Find
the common factors of 12 and 16. Factors
of 12 – 1,
2, 3, 4, 6, 12 Factors
of 16 – 1,
2, 4, 8, 16 1,
2, and 4 are common factors. 4
is the greatest common factor. |
One
group of chairs has 16. Another group
has 12. The chairs are arranged in
rows of the same number. What is the
greatest amount that could be in each row? |
Mathematics: patterns, problem solving Real Life: careers in art and design |
|
|
D
|
MULTIPLES Common
Multiples Least
Common Multiple
|
M-5.18: The students will identify
common multiples of a set of numbers; they will determine the least common
multiple of two or more numbers. |
multiple common
multiple least
common multiple |
Multiples
of 8 – 0,8,
16, 24, 32, 40, 48, …… Multiples
of 12 – 0,12,
24, 36, 48, …. Common
multiples of 8 and 12 are 0,
24, and 48. The least common multiple
is 24. |
What
is the smallest number of chairs that would fit into rows of 12 or rows of 8
without having any left over? |
Mathematics: least common denominator of a fraction; problem
solving – make a model, draw a picture strategies |
|
|
M |
ROUNDING
WHOLE NUMBERS AND DECIMALS |
M.5.19 The students will round whole
numbers and decimals to any place-value position. |
rounding |
Round 53,769,217 |
|
|
|
READINESS CONCEPTS, NUMBER CONCEPTS
GRADE - 5
SEQUENCE
|
SKILL
|
OBJECTIVE
|
VOCABULARY
|
EXAMPLE
|
PROBLEM SOLVING
EXAMPLES
|
CONNECTIONS,
INTEGRATION
|
RESOURCES
|
|
I |
PRIMES
AND COMPOSITES
|
M-5.20: The students will become
familiar with the concepts of prime and composite numbers. M-5.21: The students will
explore divisibility and develop
rules for divisibility by 2, 3, 5, & 10. |
prime composite divisibility |
A
prime number is a number that has 2 factors – itself and one. 2,
3, 5, 7, & 11 are prime numbers. Composite
numbers have more than two factors. 4,
6, 8, 9, & 10 are composite numbers. Note: 0 and 1 are neither prime
nor composite. |
Find
the prime numbers that fall between 15 and 30. Explain
why each of the following is either prime or composite. 32,
17, 26, 41 |
Mathematics: patterns, problem solving |
|
|
I |
RATIOS
|
M-5.22: The students will express
a fraction as a ratio of two numbers. |
ratio |
3/5
can be expressed as “3 out of 5”. |
|
Language Art: writing ads: 4 out of
every 6 people choose Spiffy Peanut Butter. Mathematics: graphs |
|
READINESS CONCEPTS, NUMBER CONCEPTS
GRADE - 5
SEQUENCE
|
SKILL
|
OBJECTIVE
|
VOCABULARY
|
EXAMPLE
|
PROBLEM SOLVING
EXAMPLES
|
CONNECTIONS,
INTEGRATION
|
RESOURCES
|
|
I
|
PROPORTION
|
M-5.23. The students will express
proportions in terms of equivalent fractions |
proportion equivalent
fractions |
2/3
= 6/9 |
A pie was cut into six equal pieces. Mike got two pieces. If the same pie were cut into 12 equal pieces and Mike got the same amount of pie, how many pieces did he get? |
Mathematics: problem solving – act it
out, make a model, draw a picture strategies. |
|
|
I
|
RECIPROCALS
|
M-5.24: The students will identify
the reciprocal or mutiplicative inverse of fractions. |
|
The
reciprocal of 3/4 is 4/3 because 3/4 x 4/3 = 1. |
|
Mathematics: inverse operations Language Arts: antonyms |
|
|
I |
DECIMAL
FRACTIONS
|
M-5.25: The students will
understand that a fraction with a denominator, which is a multiple of ten,
can be written as a decimal. M-5.26: The students will express
fractions as decimals and percents. |
decimal fraction decimal
point percent |
To
write 7/10 as a decimal: keep the numerator and write it proceeded by a
decimal point. 0.7 To
write 3/4 as a fraction, recognize that a fraction bar is a division sign: 3
|
Joe
answered 9/10 of the questions on his science quiz correctly. Write this as a decimal. |
Real Life: careers which use decimals |
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READINESS CONCEPTS, NUMBER CONCEPTS
GRADE - 5
SEQUENCE
|
SKILL
|
OBJECTIVE
|
VOCABULARY
|
EXAMPLE
|
PROBLEM SOLVING
EXAMPLES
|
CONNECTIONS,
INTEGRATION
|
RESOURCES
|
|
D |
ROMAN
NUMERALS
|
M-5.27: The students will
understand the structure of Roman numerals and will apply their use in other
areas. |
I
= 1 V
= 5 X
= 10 L
= 50 C
= 100 C
= 500 M
= 1000 Lines
connecting the Roman numerals indicate multiplication by 1000. Students should be instructed that they
should not draw lines connecting the letters at top and bottom. |
Explain:
A Roman numeral of lesser value preceding one of greater value indicates
subtraction. e.g.
XL means 50
– 10 = 40. A
Roman numeral of lesser value following one of greater value indicates
addition. CIII
= 103. |
Write
the following equations using Roman Numerals: 35
· 27 = 243
+ 169 + 146 = 843
- 116 = |
Real Life: clocks, outlines,
copyright dates, dates in film credits, building cornerstones |
|
|
I |
INTEGERS
|
M-5.28: The students will become
familiar with the concept of positive and negative numbers. |
integer positive negative |
Use
in real life context: degrees below zero, feet below sea level, yards lost on
a football play, etc. Represent
on a number line. |
Write
a number to show: eighteen
degrees below zero; a
loss of 20 points in a card game. |
Real Life: point out the use of
positive and negative numbers on the stock market page of the newspaper. |
|
READINESS CONCEPTS, NUMBER CONCEPTS
GRADE 5
SEQUENCE
|
SKILL
|
OBJECTIVE
|
VOCABULARY
|
EXAMPLE
|
PROBLEM SOLVING
EXAMPLES
|
CONNECTIONS,
INTEGRATION
|
RESOURCES
|
|
D - M |
USE
CONCRETE OBJECTS TO MODEL OPERATIONS
|
M-5.29: Manipulative material =
connecting cubes, blocks, centimeter cubes, beans, etc. should be used
frequently when modeling concepts. Understanding
is built when skills are presented following the concept continuum. |
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|
M
|
OPERATIONS
INTRODUCED AND DEVELOPED IN PROBLEM SOLVING CONTEXT
|
M-5.30: When operations are
presented, it is important to give purpose to the learning by presenting the
material in a problem solving/real life context. Equations should never be presented isolation until mastery has
occurred. |
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|
M |
USE
SYMBOLS FOR OPERATIONS
|
M-5.31: Refer to “symbols” in
Readiness and Number Concepts section of these guidelines. |
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|
M |
ADD,
SUBTRACT, MULTIPLY, AND DIVIDE SINGLE-DIGIT AND MULTI-DIGIT WHOLE NUMBERS
|
M-5.32: Review and maintain
material presented at earlier levels.
Extend through millions.
Students should make frequent use of the calculator to check material
that they have already mastered. |
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READINESS CONCEPTS, NUMBER
CONCEPTS GRADE - 5
SEQUENCE
|
SKILL
|
OBJECTIVE
|
VOCABULARY
|
EXAMPLE
|
PROBLEM SOLVING
EXAMPLES
|
CONNECTIONS,
INTEGRATION
|
RESOURCES
|
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|
M M M I |
PROPERTIES
OF OPERATIONS |
M-5.33: Review the following
properties of operations: |
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Properties + x |
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Commutative |
a + b = b + a |
ab = ba |
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Associative |
(a + b) + c = a + (b + c) |
(a x b) c = a (b x c) |
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Identity |
a + 0 = a |
1 x a =
a 0 x a =
0 |
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Distributive |
a (b + c) = a x b + a x c |
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Chart used here is for
teacher reference only at this level. Replace variables with numbers in
presenting to students. |
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M M D D
|
ESTIMATION
SUMS DIFFERENCES PRODUCTS QUOTIENTS |
M-5.34: Estimation in all operations and with all sets of
numbers should be reviewed and maintained.
Estimation is a key skill and should be interwoven into every lesson
involving operations. |
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OPERATIONS – GRADE 5
|
SEQUENCE
|
SKILL
|
OPERATIONS |
VOCABULARY |
EXAMPLE |
PROBLEM SOLVING EXAMPLES |
CONNECTIONS, INTEGRATION |
RESOURCES |
|
D |
FRACTIONS
|
M-5.35: The students will have a definite procedure for writing equivalent
fractions. |
equivalent fractions |
To change halves to
eighths: 1 =
n . 2 8 1.
Decide which fraction value for one to use by dividing the original
denominator into the new denominator. 8 ÷ 2 = 4 2. Multiply original
numerator by the result. n x 4 = 4 . 2 4 8 3. Write the product over the new
denominator. |
Six out of every nine
fifth graders chose pizza for their special lunch. If twenty-seven students were served, how many chose pizza? |
Real Life: Equal parts of the whole. |
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OPERATIONS – GRADE 5
|
SEQUENCE
|
SKILL
|
OBJECTIVES
|
VOCABULARY |
EXAMPLES
|
PROBLEM SOLVING EXAMPLE |
CONNECTIONS, INTEGRATION |
RESOURCES |
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|
D
|
FRACTIONS
Comparing
fractions
|
M-5.36: The students will examine various methods for comparing fractions. (Emphasize: common
denominator method is best.) |
regions common denominator number line lowest common denominator |
Method 1: Use regions. 1 1 1 1 1 1 1 1 1 1 10 10
10 10 10
10 10 10
10 10 1 1 1 1 1 5 5 5 5 5 It is easy to see
inequalities using this method. Method 2: Use a number line. · · · · · · · · · · 0 1 2 3 4 5 6 7 8 9 10 10 10
10 10 10
10 10 10
10 0 1 2 3 4 . 5 5 5 5 5 Three fifths is greater
than four-tenths since three-fifths equals six-tenths. Six-tenths is greater that four-tenths. Emphasize: Position to the right on
the number line indicates the greater value. Method 3: Common Denominator Find equivalent fractions. |
Real Life: Show a candy bar or a cake divided into equal parts. Divide some pieces into smaller
sections. In recipes, show how three
fourths of a cup is more than a half cup.
Compare amounts used in recipes. Folded paper: Fold paper
in half. Label each half. Fold in half again. Label.
Fold again and label. Show
equivalent fractions. |
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OPERATIONS – GRADE 5
|
SEQUENCE
|
SKILL
|
OBJECTIVE |
VOCABULARY |
EXAMPLE
|
PROBLEM
SOLVING EXAMPLE |
CONNECTIONS, INTEGRATION |
RESOURCES |
|
D
I
I
|
FRACTIONS
Addition and
Subtraction
--like
denominators
--unlike denominators
--with whole numbers |
M-5.37: The students will add and subtract fractions with like denominators. M-5.38: The students will add and subtract fractions with unlike
denominators. M-5.39: The students will add and subtract a fraction and a whole number. |
|
3
. 8 - 1 . 8 . 2 = 1 . 8 4 3 = 3 . 8
8 4 8
. 5 . 8 7 = 6 3 . 3 - 2 = 2 . 3 3 . 6 1 . 3 |
Two-thirds of Harry’s
birthday cake was left. One-third was
eaten for a snack. How much is left
now? Three-fourths of the
students at St. Mary’s stay for lunch.
Three-eighths of these buy a hot lunch. What part of the student body brings their own lunch? Mrs. Green bought four
cantaloupes. One-half of a cantaloupe
was eaten for breakfast. How much was
left? |
Real Life: consumer topics. Mathematics: problem solving operations on whole numbers, measurement, geometry. |
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OPERATIONS – GRADE 5
|
|
SKILL |
OBJECTIVE |
VOCABULARY |
EXAMPLE
|
PROBLEM
SOLVING EXAMPLE |
CONNECTIONS, INTEGRATION |
RESOURCES |
|
I
I
I |
MIXED NUMBERS Addition
and Subtraction
|
M-5.40: The students will add and subtract like mixed numbers. M-5.41: The students will add and subtract unlike mixed numbers. M-5.42: The students will subtract a mixed number from a whole number or a
mixed number by renaming the whole number and subtracting. |
mixed number whole number renaming |
9 3 8 + 1 1 8 10 4 = 10 1 8
2 4 3 = 4 12 5 10 - 2 1 = 2 5 . 4 10 = 2 7 . 10 8 = 7 3 3 - 6 2 = 6 2 3 3 = 1 1 3 2 1 = 2 2 = 1 6 2 4 4 - 3 =
3 = 3 4 4 4 = 1 3 4 |
Mrs. McDonald told her
students to read 5 ½ pages in their Science book. Kelly read 1 ½ pages. How
many more pages will she have to read? 3 3/8 pizzas were left
after the fifth grade party. In order
to share with the sixth grade, Miss Lynne would need 5 and one-half
pizzas. How much more would she need? If Miss Lynne purchased
three more pizzas, how much would she have left over? |
Mathematics: problem solving, operations Real Life: Provide ample practice in subtracting fractional parts of real life
objects – cake, oranges, sets of cubes, etc. |
|