Trenton Public Schools 1
Grade 5 Mathematics
2008-2009
Standard 4.1: NUMBER AND NUMERICAL OPERATIONS.
All students will develop number sense and will perform standard numerical operations and estimations on all types of numbers in a variety of ways.
Goals
By the end of Grade 5, students will:
demonstrate how different types of numbers such as fractions and decimals are related to each other and how each can be used to describe a particular situation.
demonstrate mastery of place value, number base, magnitude, approximations and estimation.
select and apply various computational methods including mental math, paper-and-pencil technique and the use of calculators.
demonstrate how to add, subtract, multiply an divide whole numbers, fractions, decimals and other kinds of numbers.
demonstrate estimation skills and mental math skills.
demonstrate understanding of when a computational answer is reasonable.
demonstrate an understanding of when an approximated answer is as good as or better than an exact one.
Strand
A. Number Sense
Number sense is an intuitive feel for numbers and a common sense approach to using them. It is a comfort with what numbers represent that comes from investigating their
characteristics and using them in diverse situations. It involves an understanding of how different types of numbers, such as fractions and decimals, are related to each other, and
how each can best be used to describe a particular situation. It subsumes the more traditional category of school mathematics curriculum called numeration and thus includes the
important concepts of place value, number base, magnitude, and approximation and estimation.
Enduring Understandings
One representation may sometimes be more helpful than another; and, used together, multiple representations give a fuller understanding of a problem.
A quantity can be represented numerically in various ways. Problem solving depends upon choosing wise ways.
Numeric fluency includes both the understanding of and the ability to appropriately use numbers.
Essential Questions
How do mathematical ideas interconnect and build on one another to produce a coherent whole? (4.5C1; 4.5C6)
How can we compare and contrast numbers? (4.5A4)
How can counting, measuring, or labeling help to make sense of the world around us?
Focal points at this grade level are bolded.
Process Standard 4.5 is imbedded in Enduring Understandings and Essential Questions.
11/30/2011 1:16:23 PM
Trenton Public Schools 2
Grade 5 Mathematics
2008-2009
Refer to Section 9 for
Refer to Section 5 for Refer to Section 7 for Refer to Section 8 for Instructional Resources and
Assessment Options Differentiated Strategies Technology Support Cross-Curricular Integration
Cumulative Progress Indicators (CPI) Objectives: Students will be able to… Content
4.1.5.A.1 Use real-life experiences, physical use a variety of number types in different mathematical situations. Integers
materials, and technology to construct meanings for identify fractional parts of regions and sets and locations on a number Fractions
numbers (unless otherwise noted, all indicators for line. Decimals
grade 5 pertain to these sets of numbers as well): show fractional parts of regions and sets and locations on a number
All integers; line.
All fractions as part of a whole, as subset of a set, use division to divide objects into equal parts where the parts are
as a location on a number line, and as divisions of fractions of a whole.
whole numbers; express fractions greater than 1as mixed numbers or improper
All decimals. fractions.
identify fractions and mixed numbers on a number line.
locate fractions and mixed numbers on a number line.
4.1.5.A.2 Recognize the decimal nature of United perform the four basic math operations involving money. Money
States currency and compute with money. represent money as a decimal to the hundredths place value.
find quotients of money amounts divided by one-digit divisors.
use the standard algorithm to find quotients of money amounts divided
by 2-digit divisors.
4.1.5.A.3 Demonstrate a sense of the relative develop an understanding of very small numbers. Whole numbers
magnitudes of numbers. develop an understanding of very large numbers. Decimals
Fractions
Focal points at this grade level are bolded.
Process Standard 4.5 is imbedded in Enduring Understandings and Essential Questions.
11/30/2011 1:16:23 PM
Trenton Public Schools 3
Grade 5 Mathematics
2008-2009
Cumulative Progress Indicators (CPI) Objectives: Students will be able to… Content
4.1.5.A.4 Use whole numbers, fractions and work flexibly with fractions to solve problems. Fractions
decimals to represent equivalent forms of the same work flexibly with decimals to solve problems. Whole numbers
number. express fractions greater than 1as mixed numbers or improper Decimals
fractions. Equivalencies
identify equivalent fractions.
write equivalent fractions.
find fractions equivalent to a given fraction using models and/or a
computational procedures.
identify fraction that are in simplest form.
find the simplest form of a fraction.
represent decimals(tenths and hundredths) as fractions and simple
fractions as decimals.
4.1.5.A.5 Develop and apply number theory identify numbers as prime or composite. Prime numbers
concepts in problem solving situations. find the multiples for any given number. Factors
Primes, factors, multiples determine common factors and the greatest common factor of numbers. Multiples
determine if numbers are divisible by 2,3,4,5,6,9, and 10.
identify fraction that are in simplest form and find the simplest form of
a fraction.
find a common denominator for two fractions.
4.1.5.A.6 Compare and order numbers. compare numbers of all types; fractions and mixed numbers, percents, Rational numbers
whole numbers through millions, decimals through thousandths. Percents
order numbers from least to greatest and vice versa for numbers of all Whole numbers (millions)
types; rational, percents and whole numbers through millions. Decimals (thousandths)
identify and locate fractions and mixed numbers on a number line.
determine which of two fractions is greater or less and write a
comparison.
label a point on a number line using a fraction and a decimal.
write a fraction and decimal for a point on a number line.
Focal points at this grade level are bolded.
Process Standard 4.5 is imbedded in Enduring Understandings and Essential Questions.
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Trenton Public Schools 4
Grade 5 Mathematics
2008-2009
Strand
B. Numerical Operations
Numerical Operations are an essential part of the mathematics curriculum, especially in the elementary grades. Students must be able to select and apply various computational
methods, including mental math, pencil-and-paper techniques, and the use of calculators. Students must understand how to add, subtract, multiply, and divide whole numbers,
fractions, decimals, and other kinds of numbers. With the availability of calculators that perform these operations quickly and accurately, the instructional emphasis now is on
understanding the meanings and uses of these operations, and on estimation and mental skills, rather than solely on the development of paper-and-pencil proficiency.
Enduring Understandings
Computational fluency includes understanding the meaning and the appropriate use of numerical operations.
The magnitude of numbers affects the outcome of operations on them.
In many cases, there are multiple algorithms for finding a mathematical solution, and those algorithms are frequently associated with different cultures.
Essential Questions
What makes a computational strategy both effective and efficient? (4.5D1)
How do operations affect numbers?
How do mathematical representations reflect the needs of society across cultures? (An essential question with broad applicability across multiple standards) (4.5C5)
Refer to Section 9 for
Refer to Section 5 for Refer to Section 7 for Refer to Section 8 for Instructional Resources and
Assessment Options Differentiated Strategies Technology Support Cross-Curricular Integration
Cumulative Progress Indicators (CPI) Objectives: Students will be able to… Content
4.1.5.B.1 Recognize the appropriate use of each choose the correct math operation needed to solve a problem. Addition
arithmetic operation in problem situations. Subtraction
Multiplication
Division
Problem solving
Focal points at this grade level are bolded.
Process Standard 4.5 is imbedded in Enduring Understandings and Essential Questions.
11/30/2011 1:16:23 PM
Trenton Public Schools 5
Grade 5 Mathematics
2008-2009
Cumulative Progress Indicators (CPI) Objectives: Students will be able to… Content
4.1.5.B.2 Construct, use, and explain procedures for compute sums of decimals involving tenths, hundredths, and Fractions
performing calculations with fractions and decimals thousandths using paper and pencil. Decimals
with: compute sums of decimals involving tenths, hundredths, and
Pencil-and-paper thousandths using mental math.
Mental math compute sums of decimals involving tenths, hundredths, and
Calculator thousandths using a calculator.
compute differences of decimals involving tenths, hundredths, and
thousandths using paper and pencil.
compute differences of decimals involving tenths, hundredths, and
thousandths using mental math.
compute differences of decimals involving tenths, hundredths, and
thousandths using a calculator.
explain strategy and processes for solving problems involving fractions
and decimals using paper/pencil, mental math and a calculator.
add fractions with like denominators.
add fractions with unlike denominators.
add mixed numbers with and without renaming.
subtract fractions with like denominators.
subtract fractions with unlike denominators.
subtract mixed numbers with and without renaming.
find a common denominator for two fractions using fraction strips.
4.1.5.B.3 Use an efficient and accurate pencil-and- use the standard algorithm to divide 3-digit whole numbers by 2-digit Division
paper procedure for division of a 3-digit number by divisors.
a 2-digit number.
4.1.5.B.4 Select pencil-and-paper, mental math, or a choose the appropriate tool to use (paper/pencil, mental math, or Problem solving
calculator as the appropriate computational method in a calculator) when problem solving, depending on the context and
given situation depending on the context and numbers. numbers.(adding, subtracting, multiplying ,and dividing).
Focal points at this grade level are bolded.
Process Standard 4.5 is imbedded in Enduring Understandings and Essential Questions.
11/30/2011 1:16:23 PM
Trenton Public Schools 6
Grade 5 Mathematics
2008-2009
Cumulative Progress Indicators (CPI) Objectives: Students will be able to… Content
4.1.5.B.5 Check the reasonableness of results of determine whether my answer is reasonable or not. Reasonableness
computations.
4.1.5.B.6 Understand and use the various mentally compute products of whole numbers using patterns and Math operations
relationships among operations and properties of multiplication properties. Math properties
operations. mentally multiply numbers of up to three places by numbers of up to
two places.
solve problems involving the commutative property.
(Properties of operations-commutative, associative, solve problems involving the associative property.
distributive, identity elements, reciprocals, solve problems involving the identity property.
multiplication or division by zero.) solve problems involving the distributive property.
use reciprocals when solving problems.
multiply by zero.
divide by zero.
draw a picture or use objects to show a division situation and find
quotients.
Focal points at this grade level are bolded.
Process Standard 4.5 is imbedded in Enduring Understandings and Essential Questions.
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Trenton Public Schools 7
Grade 5 Mathematics
2008-2009
Strand
C. Estimation
Estimation is a process that is used constantly by mathematically capable adults, and one that can be easily mastered by children. It involves an educated guess about a quantity
or an intelligent prediction of the outcome of a computation. The growing use of calculators makes it more important than ever that students know when a computed answer is
reasonable; the best way to make that determination is through the use of strong estimation skills. Equally important is an awareness of the many situations in which an
approximate answer is as good as, or even preferable to, an exact one. Students can learn to make these judgments and use mathematics more powerfully as a result.
Enduring Understandings
Context is critical when using estimation.
Essential Questions
How can we decide when to use an exact answer and when to use an estimate?
Refer to Section 9 for
Refer to Section 5 for Refer to Section 7 for Refer to Section 8 for Instructional Resources and
Assessment Options Differentiated Strategies Technology Support Cross-Curricular Integration
Cumulative Progress Indicators (CPI) Objectives: Students will be able to… Content
4.1.5.C.1 Use a variety of estimation strategies for both use rounding, frond-end estimation, and front-end estimation with Estimating
number (quantities) and computation. adjusting to estimate sums and differences of whole numbers and Problem solving
decimals.
use rounding and compatible numbers to estimate products of whole
numbers and decimal numbers, and identify estimates as overestimates
or underestimates.
use rounding, compatible numbers and multiplication to estimate
quotients of whole numbers and money.
estimate quotients of whole numbers, decimals, and money divided by
2-digit whole numbers.
estimate fractional parts of a region.
estimate sums and differences of mixed numbers.
4.1.5.C.2 Recognize when an estimate is appropriate, recognize when an estimate is appropriate. Estimating
and understand the usefulness of an estimate as distinct explain when an approximation is more/less useful than an exact Problem solving
from an exact answer. answer.
Focal points at this grade level are bolded.
Process Standard 4.5 is imbedded in Enduring Understandings and Essential Questions.
11/30/2011 1:16:23 PM
Trenton Public Schools 8
Grade 5 Mathematics
2008-2009
Cumulative Progress Indicators (CPI) Objectives: Students will be able to… Content
4.1.5.C.3 Determine the reasonableness of an determine if my answer is reasonable by comparing it to an estimated Estimating
answer by estimating the result of operations. answer. Problem solving
4.1.5.C.4 Determine whether a given estimate is an determine whether an estimate is over or under the exact answer by Estimating
overestimate or an underestimate. how I estimated. Problem solving
use rounding to estimate products of whole numbers and decimal
numbers.
use compatible numbers to estimate products of whole numbers and
decimal numbers.
use rounding to estimate quotients of whole numbers and money.
use compatible numbers to estimate quotients of whole numbers and
money.
use multiplication to estimate quotients of whole numbers and money.
Focal points at this grade level are bolded.
Process Standard 4.5 is imbedded in Enduring Understandings and Essential Questions.
11/30/2011 1:16:23 PM
Trenton Public Schools 9
Grade 5 Mathematics
2008-2009
Standard 4.2: GEOMETRY AND MEASUREMENT.
All students will develop spatial sense and the ability to use geometric properties, relationships, and measurement to model, describe, and analyze phenomena.
Goals
By the end of Grade 5, students will:
identify, describe and classify standard geometric objects.
describe and compare properties of geometric objects.
demonstrate an understanding of symmetry, congruence and similarity.
locate and name points on a coordinate grid.
demonstrate an understanding of how we attach numbers to real-world phenomena.
demonstrate familiarity with common measurement units.
demonstrate practical knowledge of measurement tools and techniques.
apply knowledge and understanding of units of measurement in order to actually perform measurement.
develop and apply strategies that derive from their understanding of the attributes of formulas.
apply formulas.
demonstrate an ability to measure objects directly and indirectly (similar triangles).
Strand
A. Geometric Properties.
This includes identifying, describing and classifying standard geometric objects, describing and comparing properties of geometric objects, making conjectures concerning them,
and using reasoning and proof to verify or refute conjectures and theorems. Also included here are such concepts as symmetry, congruence, and similarity.
Enduring Understandings
Geometric properties can be used to construct geometric figures. (4.5D1; 4.5D2; 4.5E3)**
Geometric relationships provide a means to make sense of a variety of phenomena.
Essential Understandings
How can spatial relationships be described by careful use of geometric language?
How do geometric relationships help to solve problems and/or make sense of phenomena?
Focal points at this grade level are bolded.
Process Standard 4.5 is imbedded in Enduring Understandings and Essential Questions.
11/30/2011 1:16:23 PM
Trenton Public Schools 10
Grade 5 Mathematics
2008-2009
Refer to Section 9 for
Refer to Section 5 for Refer to Section 7 for Refer to Section 8 for Instructional Resources and
Assessment Options Differentiated Strategies Technology Support Cross-Curricular Integration
Cumulative Progress Indicators (CPI) Objectives: Students will be able to… Content
4.2.5.A.1 Understand and apply concepts involving define different angles-acute, right, obtuse, straight Lines
lines and angles. recognize different angles-acute, right, obtuse, straight. Angles
Notation for line, ray, angle, line segment use proper notation for line.
Properties of parallel, perpendicular, and use proper notation for ray.
intersecting lines use proper notation for angle.
Sum of the measures of the interior angles of a use proper notation for line segment
triangle is 180° define parallel lines.
define perpendicular lines.
define intersecting lines.
recognize parallel lines.
recognize perpendicular lines.
recognize intersecting lines.
find the value of a missing angle measure in a triangle.
identify important geometric terms relating to lines, parts of a line,
angles, and planes.
measure angles.
draw angles.
classify angles according to their measures.
4.2.5.A.2 Identify, describe, compare, and classify identify polygons based on their properties Polygons
polygons: describe polygons based on their properties Circles
Triangles by angles and sides classify polygons based on their properties
Quadrilaterals, including squares, rectangles, compare polygons.
parallelograms, trapezoids, rhombi compare circles.
Polygons by number of sides. prove that all points are equidistant from any given point form a
Equilateral, equiangular, regular circle.
All points equidistant from a given point form a
circle.
Focal points at this grade level are bolded.
Process Standard 4.5 is imbedded in Enduring Understandings and Essential Questions.
11/30/2011 1:16:23 PM
Trenton Public Schools 11
Grade 5 Mathematics
2008-2009
Cumulative Progress Indicators (CPI) Objectives: Students will be able to… Content
4.2.5.A.3 Identify similar figures. determine whether figures are similar to each other or not. Similarity
4.2.5.A.4 Understand and apply the concepts of identify congruent figures. Congruence
congruence and symmetry (line and rotational). determine how many lines of symmetry a shape has. Symmetry
determine the angle of symmetry (rotational) a shape has.
identify symmetrical figures.
make symmetrical figures.
draw a line or lines of symmetry.
Focal points at this grade level are bolded.
Process Standard 4.5 is imbedded in Enduring Understandings and Essential Questions.
11/30/2011 1:16:23 PM
Trenton Public Schools 12
Grade 5 Mathematics
2008-2009
Strand
B. Transforming Shapes. Analyzing how various transformations affect geometric objects allows students to enhance their spatial sense. This includes combining shapes to
form new ones and decomposing complex shapes into simpler ones. It includes the standard geometric transformations of translation (slide), reflection (flip), rotation (turn), and
dilation (scaling). It also includes using tessellations and fractals to create geometric patterns.
Enduring Understandings
Shape and area can be conserved during mathematical transformations.
Essential Questions
What situations can be analyzed using transformations and symmetries? (4.5E1; 4.5E2; 4.5E3)**
Refer to Section 9 for
Refer to Section 5 for Refer to Section 7 for Refer to Section 8 for Instructional Resources and
Assessment Options Differentiated Strategies Technology Support Cross-Curricular Integration
Cumulative Progress Indicators (CPI) Objectives: Students will be able to… Content
4.2.5.B.1 Use a translation, a reflection, or a rotation map one figure onto another congruent figure by translating, Transformations
to map one figure onto another congruent figure. map one figure onto another congruent figure by reflecting
map one figure onto another congruent figure by rotating.
determine whether a pair of congruent figures are related by a slide
(translation), flip (reflection), or turn (rotation).
4.2.5.B.2 Recognize, identify, and describe geometric recognize geometric properties that can be found in real-world Geometric properties
relationships and properties as they exist in nature, art, settings.
and other real-world settings. identify geometric properties that can be found in real-world
settings.
describe geometric properties that can be found in real-world
settings.
identify important geometric terms relating to lines, parts of a line,
angles, and planes.
Focal points at this grade level are bolded.
Process Standard 4.5 is imbedded in Enduring Understandings and Essential Questions.
11/30/2011 1:16:23 PM
Trenton Public Schools 13
Grade 5 Mathematics
2008-2009
Strand
B. Coordinate Geometry.
Coordinate geometry provides an important connection between geometry and algebra. It facilitates the visualization of algebraic relationships, as well as an analytical
understanding of geometry.
Enduring Understandings
Reasoning and/or proof can be used to verify or refute conjectures or theorems in geometry (4.5D1; 4.5D3; 4.5D4; 4.5D5; 4.5F5)**
Coordinate geometry can be used to represent and verify geometric/algebraic relationships.
Essential Understandings
How can geometric/algebraic relationships best be represented and verified? (4.5C2; 4.5D2; 4.5E1; 4.5E2; 4.5F5)**
Refer to Section 9 for
Refer to Section 5 for Refer to Section 7 for Refer to Section 8 for Instructional Resources and
Assessment Options Differentiated Strategies Technology Support Cross-Curricular Integration
Cumulative Progress Indicators (CPI) Objectives: Students will be able to… Content
4.2.5.C.1 Create geometric shapes with specified create a geometric shape in quadrant I by following specified Coordinate geometry
properties in the first quadrant on a coordinate grid. coordinates.
Focal points at this grade level are bolded.
Process Standard 4.5 is imbedded in Enduring Understandings and Essential Questions.
11/30/2011 1:16:23 PM
Trenton Public Schools 14
Grade 5 Mathematics
2008-2009
Strand
D. Units of Measurement.
Measurement helps describe our world using numbers. An understanding of how we attach numbers to real-world phenomena, familiarity with common measurement units (e.g.,
inches, liters, and miles per hour), and a practical knowledge of measurement tools and techniques are critical for students' understanding of the world around them.
Enduring Understandings
Everyday objects have a variety of attributes, each of which can be measured in many ways.
What we measure affects how we measure it. (4.5A4; 4.5A6)**
Measurements can be used to describe, compare, and make sense of phenomena.
Essential Questions
How can measurements be used to solve problems? (4.5A6)**
Refer to Section 9 for
Refer to Section 5 for Refer to Section 7 for Refer to Section 8 for Instructional Resources and
Assessment Options Differentiated Strategies Technology Support Cross-Curricular Integration
Cumulative Progress Indicators (CPI) Objectives: Students will be able to… Content
4.2.5.D.1 Select and use appropriate units to determine the appropriate units of measure when measuring angles and Measurement
measure angles and area. area.
measure angles.
draw angles.
classify angles according to their measures.
Focal points at this grade level are bolded.
Process Standard 4.5 is imbedded in Enduring Understandings and Essential Questions.
11/30/2011 1:16:23 PM
Trenton Public Schools 15
Grade 5 Mathematics
2008-2009
Cumulative Progress Indicators (CPI) Objectives: Students will be able to… Content
4.2.5.D.2 Convert measurement units within a convert a measurement into a smaller or larger unit within a system. Measurement- Standard and Metric
system (e.g., 3 feet = ___ inches). change between one customary unit of length and another.
add customary units of length.
subtract customary units of length.
change from one unit of time to another.
change among the customary units of capacity.
add customary units of capacity.
subtract customary units of capacity.
estimate capacity using metric measures.
measure capacity using metric measures
change millimeters to liters and vice versa.
change between customary units of weight
add customary units of weight.
subtract customary units of weight.
estimate mass using metric measures.
measure mass using metric measures.
4.2.5.D.3 Know approximate equivalents between develop approximate measurement equivalencies between standard Measurement- Standard and Metric
the standard and metric systems (e.g., one kilometer and metric systems.
is approximately 6/10 of a mile).
4.2.5.D.4 Use measurements and estimates to use measurements and estimates to describe facts. Measurement
describe and compare phenomena. use measurements and estimates to compare facts.
write to explain how a broken ruler can be used to measure.
Focal points at this grade level are bolded.
Process Standard 4.5 is imbedded in Enduring Understandings and Essential Questions.
11/30/2011 1:16:23 PM
Trenton Public Schools 16
Grade 5 Mathematics
2008-2009
Strand
E. Measuring Geometric Objects.
This area focuses on applying the knowledge and understandings of units of measurement in order to actually perform measurement. While students will eventually apply
formulas, it is important that they develop and apply strategies that derive from their understanding of the attributes. In addition to measuring objects directly, students apply
indirect measurement skills, using, for example, similar triangles and trigonometry.
Enduring Understandings
Everyday objects have a variety of attributes, each of which can be measured in many ways.
What we measure affects how we measure it. (4.5A4; 4.5A6)**
Measurements can be used to describe, compare, and make sense of phenomena.
Essential Questions
How can measurements be used to solve problems? (4.5A6)**
Refer to Section 9 for
Refer to Section 5 for Refer to Section 7 for Refer to Section 8 for Instructional Resources and
Assessment Options Differentiated Strategies Technology Support Cross-Curricular Integration
Cumulative Progress Indicators (CPI) Objectives: Students will be able to… Content
4.2.5.E.1 Use a protractor to measure angles. measure angles. Measurement-Angles
draw angles.
classify angles according to their measures.
4.2.5.E.2 Develop and apply strategies and develop strategies for finding area of various shapes. Measurement-Perimeter,
formulas for finding perimeter and area. develop strategies for finding perimeter of various shapes. Circumference, and Area
Square apply strategies for finding area of various shapes.
Rectangle apply strategies for finding perimeter of various shapes.
find the area of irregular shapes by counting square units.
find the area of a rectangle or square by using a formula.
find the area of a parallelogram by using a formula
find the length when the area and other side length are known.
4.2.5.E.3 Recognize that rectangles with the same prove that shapes with the same perimeter do not necessarily have the Measurement-Area and Perimeter
perimeter do not necessarily have the same area and same area.
vice versa. prove that shapes with the same area do not necessarily have the same
perimeter.
Focal points at this grade level are bolded.
Process Standard 4.5 is imbedded in Enduring Understandings and Essential Questions.
11/30/2011 1:16:23 PM
Trenton Public Schools 17
Grade 5 Mathematics
2008-2009
4.2.5.E.4 Develop informal ways of approximating use informal ways to measure familiar objects. Measurement-Approximations
the measures of familiar objects (e.g., use a grid to
approximate the area of the bottom of one’s foot).
Focal points at this grade level are bolded.
Process Standard 4.5 is imbedded in Enduring Understandings and Essential Questions.
11/30/2011 1:16:23 PM
Trenton Public Schools 18
Grade 5 Mathematics
2008-2009
Standard 4.3: PATTERNS AND ALGEBRA
All students will represent and analyze relationships among variable quantities and solve problems involving patterns, functions, and algebraic concepts and processes.
Goals
By the end of Grade 5, student will:
investigate patterns in numbers, shapes and expressions to make math discoveries.
analyze, extend and create variety of patterns.
use pattern-based thinking to understand math and other real-world phenomena.
form rules, display information in a table or chart, and write equations which express the relationships they have observed in patterns.
represent data in tables, pictures, graphs, equations or inequalities, and rules.
model situations by writing simple number sentences.
use functions to describe the relationship between two variables.
understand and apply properties of numbers and operations.
use symbols and variables appropriately.
work with and solve expressions and equations.
Strand
A. Patterns.
Algebra provides the language through which we communicate the patterns in mathematics. From the earliest age, students should be encouraged to investigate the patterns that
they find in numbers, shapes, and expressions, and, by doing so, to make mathematical discoveries. They should have opportunities to analyze, extend, and create a variety of
patterns and to use pattern-based thinking to understand and represent mathematical and other real-world phenomena.
Enduring Understandings
The symbolic language of algebra is used to communicate and generalize the patterns in mathematics.
Algebraic representation can be used to generalize patterns and relationships.
Essential Questions
How can change be best represented mathematically? (4.5C1; 4.5F1; 4.5F2; 4.5F3; 4.5F4)**
How can patterns, relations, and functions be used as tools to best describe and help explain real-life situations? (4.5C1)**
Focal points at this grade level are bolded.
Process Standard 4.5 is imbedded in Enduring Understandings and Essential Questions.
11/30/2011 1:16:23 PM
Trenton Public Schools 19
Grade 5 Mathematics
2008-2009
Refer to Section 9 for
Refer to Section 5 for Refer to Section 7 for Refer to Section 8 for Instructional Resources and
Assessment Options Differentiated Strategies Technology Support Cross-Curricular Integration
Cumulative Progress Indicators (CPI) Objectives: Students will be able to… Content
4.3.5.A.1 Recognize, describe, extend, and create mentally compute products of whole numbers using patterns and Patterns
patterns involving whole numbers. multiplication properties.
Descriptions using tables, verbal rules, graphs, use place value ideas to write multiples of 100, 1000, and 10,000 in
simple equations different ways.
identify patterns and write a rule for the pattern.
find the quotient of a division problem whose dividend is a multiple of
10, where the division involves a basic fact.
find the quotient of a division problem whose dividend and divisors are
multiples of 10, where the division involves a basic fact.
give missing numbers or figures in a pattern.
create a table of values for a rule and a graph based on the table, and
use the table or graph to give the output for an input.
Focal points at this grade level are bolded.
Process Standard 4.5 is imbedded in Enduring Understandings and Essential Questions.
11/30/2011 1:16:23 PM
Trenton Public Schools 20
Grade 5 Mathematics
2008-2009
Strand
B. Functions and Relationships.
The function concept is one of the most fundamental unifying ideas of modern mathematics. Students begin their study of functions in the primary grades, as they observe and
study patterns. As students grow and their ability to abstract matures, students form rules, display information in a table or chart, and write equations which express the
relationships they have observed. In high school, they use the more formal language of algebra to describe these relationships.
Enduring Understandings
Patterns and relationships can be represented graphically, numerically, symbolically, or verbally. (4.5E1)**
Essential Questions
How are patterns of change related to the behavior of functions? (4.5F1; 4.5F2; 4.5F3; 4.5F4)**
Refer to Section 9 for
Refer to Section 5 for Refer to Section 7 for Refer to Section 8 for Instructional Resources and
Assessment Options Differentiated Strategies Technology Support Cross-Curricular Integration
Cumulative Progress Indicators (CPI) Objectives: Students will be able to… Content
4.3.5.B.1 Describe arithmetic operations as determine the input value given the output value of a function machine. Functions
functions, including combining operations and determine the output value given the input value of a function machine.
reversing them.
4.3.5.B.2 Graph points satisfying a function from create a table of values for a rule and a graph based on the table, and Functions
T-charts, from verbal rules, and from simple use the table or graph to give the output for an input.
equations.
Focal points at this grade level are bolded.
Process Standard 4.5 is imbedded in Enduring Understandings and Essential Questions.
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C. Modeling.
Algebra is used to model real situations and answer questions about them. This use of algebra requires the ability to represent data in tables, pictures, graphs, equations or
inequalities, and rules. Modeling ranges from writing simple number sentences to help solve story problems in the primary grades to using functions to describe the relationship
between two variables, such as the height of a pitched ball over time. Modeling also includes some of the conceptual building blocks of calculus, such as how quantities change
over time and what happens in the long run (limits).
Enduring Understandings
Mathematical models can be used to describe and quantify physical relationships. (4.5E2)**
Physical models can be used to clarify mathematical relationships. (4.5E3)**
Essential Questions
How can mathematical models be used to describe physical relationships? (4.5E2)**
How physical models be used to clarify mathematical relationships? (4.5E3)**
Refer to Section 9 for
Refer to Section 5 for Refer to Section 7 for Refer to Section 8 for Instructional Resources and
Assessment Options Differentiated Strategies Technology Support Cross-Curricular Integration
Cumulative Progress Indicators (CPI) Objectives: Students will be able to… Content
4.3.5.C.1 Use number sentences to model situations: use patterns, relations, algebra and functions to model situations. Linear functions
Using variables to represent unknown quantities use manipulatives, tables, graphs, rules, equations to represent Patterns
Using concrete materials, tables, graphs, verbal situations. Symbolic algebra
rules, algebraic expressions/equations use variables to write algebraic expressions. Relations
write number expressions for phrases.
solve equations using mental math and by guessing and testing values
for the variable.
4.3.5.C.2 Draw freehand sketches of graphs that draw a graph that represents changes over time. Graphing
model real phenomena and use such graphs to draw a graph that shows the relationship between two quantities.
predict and interpret events. make line graphs to represent data.
Changes over time make stem-and-leaf plots.
Rates of change (e.g., when is plant growing
slowly/rapidly, when is temperature dropping
most rapidly/slowly)
Focal points at this grade level are bolded.
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D. Procedures.
Techniques for manipulating algebraic expressions – procedures – remain important, especially for students who may continue their study of mathematics in a calculus program.
Utilization of algebraic procedures includes understanding and applying properties of numbers and operations, using symbols and variables appropriately, working with
expressions, equations, and inequalities, and solving equations and inequalities.
Enduring Understandings
Algebraic and numeric procedures are interconnected and build on one another to produce a coherent whole.
Reasoning and/or proof can be used to verify or refute conjectures or theorems in algebra. (4.5D1; 4.5D3; 4.5D4; 4.5D5)**
Essential Questions
What makes an algebraic algorithm both effective and efficient? (4.5D1)**
Refer to Section 9 for
Refer to Section 5 for Refer to Section 7 for Refer to Section 8 for Instructional Resources and
Assessment Options Differentiated Strategies Technology Support Cross-Curricular Integration
Cumulative Progress Indicators (CPI) Objectives: Students will be able to… Content
4.3.5.D.1 Solve simple linear equations with solve simple equations using manipulatives. Equations
manipulatives and informally. solve simple equations with a variable on both sides of the equal sign.
Whole-number coefficients only, answers also solve equations using mental math and by guessing and testing values
whole numbers for the variable.
Variables on one or both sides of equation
Focal points at this grade level are bolded.
Process Standard 4.5 is imbedded in Enduring Understandings and Essential Questions.
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Standard 4.4: DATA ANALYSIS, PROBABILITY,
AND DISCRETE MATHEMATICS
All students will develop an understanding of the concepts and techniques of data analysis, probability, and discrete mathematics, and will use them to model situations, solve
problems, and analyze and draw appropriate inferences from data.
Goals
By the end of Grade 5, students will:
read, understand, and interpret data in order to make informed decisions.
collect, organize and display data using tables, charts, and graphs.
gather data using sampling.
make inferences from data and analyze data and inferences made by others.
predict and determine probabilities in a variety of situations (intuitive, experimental and theoretical).
develop strategies for listing and counting to represent all possibilities in a given situation.
follow and devise lists of instructions (algorithms) and use them to find the best solution to a problem.
Strand
A. Data Analysis
In today's information-based world, students need to be able to read, understand, and interpret data in order to make informed decisions. In the early grades, students should be
involved in collecting and organizing data, and in presenting it using tables, charts, and graphs. As they progress, they should gather data using sampling, and should increasingly
be expected to analyze and make inferences from data, as well as to analyze data and inferences made by others.
Enduring Understandings
The message conveyed by the data depends on how the data is collected, represented, and summarized. (4.5A6; 4.5D6; 4.5E1; 4.5E2; 4.5E3)**
The results of a statistical investigation can be used to support or refute an argument. (4.5D1; 4.5D3; 4.5D5; 4.5E2; 4.5E3; 4.5F6)**
Essential Questions
How can the collection, organization, interpretation, and display of data be used to answer questions? (4.5A4; 4.5A6; 4.5E1; 4.5E2; 4.5F1; 4.5F6)**
Focal points at this grade level are bolded.
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Refer to Section 9 for
Refer to Section 5 for Refer to Section 7 for Refer to Section 8 for Instructional Resources and
Assessment Options Differentiated Strategies Technology Support Cross-Curricular Integration
Cumulative Progress Indicators (CPI) Objectives: Students will be able to… Content
4.4.5.A.1 Collect, generate, organize, and display data. collect and generate data form surveys. Data collecting and displaying
Data generated from surveys organize and display data from surveys.
identify a statement as fact or opinion.
interpret a line plot and a frequency table.
write a survey question.
4.4.5.A.2 Read, interpret, select, construct, analyze, read and interpret information from line graphs. Data
generate questions about, and draw inferences from make a double bar graph to represent data.
displays of data. make a line graph to represent data.
Bar graph, line graph, circle graph, table, complete stem-and-leaf plots.
Range, median, and mean make stem-and-leaf plots.
interpret stem-and-leaf plots.
make inferences from displays of data.
understand and use a variety of graphs.
find the range for data sets.
find the median for data sets.
find the mean for data sets
choose the measure(mean, median or range) that best represents a
given set of data.
complete circle graphs based on data given.
interpret given circle graphs.
choose the most appropriate type of graph to represent a given set
of data.
4.4.5.A.3 Respond to questions about data, generate answer questions about data. Data
their own questions and hypotheses. create their own questions and hypotheses about data.
Focal points at this grade level are bolded.
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B. Probability
Students need to understand the fundamental concepts of probability so that they can interpret weather forecasts, avoid unfair games of chance, and make informed decisions
about medical treatments whose success rate is provided in terms of percentages. They should regularly be engaged in predicting and determining probabilities, often based on
experiments (like flipping a coin 100 times), but eventually based on theoretical discussions of probability that make use of systematic counting strategies. High school students
should use probability models and solve problems involving compound events and sampling.
Enduring Understandings
Experimental results tend to approach theoretical probabilities after a large number of trials.
Essential Questions
How can experimental and theoretical probabilities be used to make predictions or draw conclusions? (4.5D5; 4.5D6)**
Refer to Section 9 for
Refer to Section 5 for Refer to Section 7 for Refer to Section 8 for Instructional Resources and
Assessment Options Differentiated Strategies Technology Support Cross-Curricular Integration
Cumulative Progress Indicators (CPI) Objectives: Students will be able to… Content
4.4.5.B.1 Determine probabilities of events. determine the probability a certain event will happen or not happen. Probability
Event, probability of an event predict the chance an event will happen on a number line 0-1.(0=will
Probability of certain event is 1 and of impossible not happen, 1=will definitely happen)
event is 0 use fraction to represent the possibilities of events.
use probability to decide if a game is fair or unfair.
4.4.5.B.2 Determine probability using intuitive, use fraction to represent the possibilities of events. Probability
experimental, and theoretical methods (e.g., using use probability to decide if a game is fair or unfair.
model of picking items of different colors from a determine how many of each type in a bag, given the number pulled
bag). from an experiment.
Given numbers of various types of items in a bag, identify events and favorable outcomes, and determine if an outcome
what is the probability that an item of one type is equally likely, impossible, less likely, more likely, or certain.
will be picked
Given data obtained experimentally, what is the
likely distribution of items in the bag.
Focal points at this grade level are bolded.
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Cumulative Progress Indicators (CPI) Objectives: Students will be able to… Content
4.4.5.B.3 Model situations involving probability using conduct probability experiments with spinners and dice. Probability
simulations (with spinners, dice) and theoretical models. predict probability of an event based on theory.
use fraction to represent the possibilities of events.
use probability to decide if a game is fair or unfair.
Focal points at this grade level are bolded.
Process Standard 4.5 is imbedded in Enduring Understandings and Essential Questions.
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C. Discrete Mathematics—Systematic Listing and Counting
Development of strategies for listing and counting can progress through all grade levels, with middle and high school students using the strategies to solve problems in
probability. Primary students, for example, might find all outfits that can be worn using two coats and three hats; middle school students might systematically list and count the
number of routes from one site on a map to another; and high school students might determine the number of three-person delegations that can be selected from their class to visit
the mayor.
Enduring Understandings
Grouping by attributes (classification) can be used to answer mathematical questions. (4.5E1; 4.5E3)**
Algorithms can effectively and efficiently be used to quantify and interpret discrete information.
Essential Questions
How can attributes be used to classify data/objects?
What is the best way to solve this? What counting strategy works best here?
Refer to Section 9 for
Refer to Section 5 for Refer to Section 7 for Refer to Section 8 for Instructional Resources and
Assessment Options Differentiated Strategies Technology Support Cross-Curricular Integration
Cumulative Progress Indicators (CPI) Objectives: Students will be able to… Content
4.4.5.C.1 Solve counting problems and justify that all solve counting problems using organized lists. Systematic listing
possibilities have been enumerated without solve counting problems using charts.
duplication. solve counting problems using tree diagrams.
Organized lists, charts, tree diagrams, tables solve counting problems using tables.
find all possible outcomes of an event by making a tree diagram or
by multiplying.
4.4.5.C.2 Explore the multiplication principle of find all possible outcomes of an event by making a tree diagram or Systematic counting
counting in simple situations by representing all by multiplying.
possibilities in an organized way (e.g., you can make
3 x 4 = 12 outfits using 3 shirts and 4 skirts).
Focal points at this grade level are bolded.
Process Standard 4.5 is imbedded in Enduring Understandings and Essential Questions.
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D. Discrete Mathematics—Vertex-Edge Graphs and Algorithms
Vertex-edge graphs, consisting of dots (vertices) and lines joining them (edges), can be used to represent and solve problems based on real-world situations. Students should learn
to follow and devise lists of instructions, called "algorithms," and use algorithmic thinking to find the best solution to problems like those involving vertex-edge graphs, but also
to solve other problems.
Enduring Understandings
Optimization is finding the best solution within given constraints.
Algorithms can effectively and efficiently be used to quantify and interpret discrete information.
Essential Questions
How can visual tools such as networks (vertex-edge graphs) be used to answer questions? (4.5E1; 4.5E3)**
How can algorithmic thinking be used to solve problems?
Refer to Section 9 for
Refer to Section 5 for Refer to Section 7 for Refer to Section 8 for Instructional Resources and
Assessment Options Differentiated Strategies Technology Support Cross-Curricular Integration
Cumulative Progress Indicators (CPI) Objectives: Students will be able to… Content
4.4.5.D.1 Devise strategies for winning simple games develop strategies for winning simple games. Algorithms
(e.g., start with two piles of objects, each of two players use fraction to represent the possibilities of events.
in turn removes any number of objects from a single use probability to decide if a game is fair or unfair.
pile, and the person to take the last group of objects
wins) and express those strategies as sets of directions.
Focal points at this grade level are bolded.
Process Standard 4.5 is imbedded in Enduring Understandings and Essential Questions.
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Standard 4.5: MATHEMATICAL PROCESSES
All students will use mathematical processes of problem solving, communication, connections, reasoning, representations, and technology to solve problems and
communicate mathematical ideas.
The mathematical processes described here highlight ways of acquiring and using the content knowledge and skills delineated in the first four mathematics standards.
Goals
By the end of Grade 5, students will:
solve problems in math and other contexts using a variety of strategies.
share the mathematical understandings in oral and written form with their classmates, teachers and parents.
make connections between different topics and draw on those relationships in the future in daily life.
recognize that math makes sense and can be understood through reasoning.
evaluate situations, select problem-solving strategies, draw logical conclusions, develop and describe solutions. and recognize how those solutions can be applied.
use physical objects, drawings, charts, graphs, and symbols to represent mathematical concepts and problem situations.
explore new and familiar concepts with calculators and computers to enhance their understanding of math and the power to use math.
Enduring Understandings
All previously listed enduring understandings reference the process standard that is embedded in its content.
Essential Questions
All previously listed enduring understandings reference the process standard that is embedded in its content.
Focal points at this grade level are bolded.
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A. Problem Solving
Problem posing and problem solving involve examining situations that arise in mathematics and other disciplines and in common experiences, describing these situations
mathematically, formulating appropriate mathematical questions, and using a variety of strategies to find solutions. Through problem solving, students experience the power
and usefulness of mathematics. Problem solving is interwoven throughout the grades to provide a context for learning and applying mathematical ideas.
CPI and Descriptor Objectives: Students will be able to…
4.5.5.A.1 Learn mathematics through problem solving, inquiry, and discovery. build new mathematical knowledge through problem solving.
solve problems that arise in mathematics and in other contexts.
apply and adapt a variety of appropriate strategies to solve problems.
4.5.5.A.2 Solve problems that arise in mathematics and in other contexts (cf. monitor and reflect on the process of mathematical problem solving.
workplace readiness standard 8.3).
Open-ended problems
Non-routine problems
Problems with multiple solutions
Problems that can be solved in several ways.
4.5.5.A.3 Select and apply a variety of appropriate problem-solving strategies
(e.g., “try a simpler problem” or “make a diagram”) to solve problems.
4.5.5.A.4 Pose problems of various types and levels of difficulty.
4.5.5.A.5 Monitor their progress and reflect on the process of their problem
solving activity.
4.5.5.A.6 Distinguish relevant from irrelevant information, and identify
missing information.
Focal points at this grade level are bolded.
Process Standard 4.5 is imbedded in Enduring Understandings and Essential Questions.
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B. Communication
Communication of mathematical ideas involves students’ sharing their mathematical understandings in oral and written form with their classmates, teachers, and parents.
Such communication helps students clarify and solidify their understanding of mathematics and develop confidence in themselves as mathematics learners. It also enables
teachers to better monitor student progress.
CPI and Descriptor Objectives: Students will be able to…
4.5.5.B.1 Use communication to organize and clarify their mathematical organize and consolidate my mathematical thinking through communication.
thinking. communicate my mathematical thinking coherently and clearly to peers,
Reading and writing teachers, and others.
Discussion, listening, and questioning analyze and evaluate the mathematical thinking and strategies of others.
use the language of mathematics to express mathematical ideas precisely.
4.5.5.B.2 Communicate their mathematical thinking coherently and clearly to
peers, teachers, and others, both orally and in writing.
4.5.5.B.3 Analyze and evaluate the mathematical thinking and strategies of others.
4.5.5.B.4 Use the language of mathematics to express mathematical ideas precisely.
Focal points at this grade level are bolded.
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C. Connections
Making connections involves seeing relationships between different topics, and drawing on those relationships in future study. This applies within mathematics, so that
students can translate readily between fractions and decimals, or between algebra and geometry; to other content areas, so that students understand how mathematics is used
in the sciences, the social sciences, and the arts; and to the everyday world, so that students can connect school mathematics to daily life.
CPI and Descriptor Objectives: Students will be able to…
4.5.5.C.1 Recognize recurring themes across mathematical domains (e.g., patterns recognize and use connections among mathematical ideas.
in number, algebra, and geometry). understand how mathematical ideas interconnect and build on one another to
produce a coherent whole.
recognize and apply mathematics in contexts outside of mathematics.
4.5.5.C.2 Use connections among mathematical ideas to explain concepts (e.g., two
linear equations have a unique solution because the lines they represent intersect at
a single point).
4.5.5.C.3 Recognize that mathematics is used in a variety of contexts outside of
mathematics.
4.5.5.C.4 Apply mathematics in practical situations and in other disciplines.
4.5.5.C.5 Trace the development of mathematical concepts over time and across
cultures (cf. world languages and social studies standards).
4.5.5.C.6 Understand how mathematical ideas interconnect and build on one
another to produce a coherent whole.
Focal points at this grade level are bolded.
Process Standard 4.5 is imbedded in Enduring Understandings and Essential Questions.
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D. Reasoning
Mathematical reasoning is the critical skill that enables a student to make use of all other mathematical skills. With the development of mathematical reasoning, students
recognize that mathematics makes sense and can be understood. They learn how to evaluate situations, select problem-solving strategies, draw logical conclusions, develop
and describe solutions, and recognize how those solutions can be applied.
CPI and Descriptor Objectives: Students will be able to…
4.5.5.D.1 Recognize that mathematical facts, procedures, and claims must be recognize reasoning and proof as
justified. fundamental aspects of mathematics.
make and investigate mathematical
conjectures.
4.5.5.D.2 Use reasoning to support their mathematical conclusions and develop and evaluate mathematical
problem solutions. arguments and proofs.
select and use various types of reasoning and
methods of proof.
4.5.5.D.3 Select and use various types of reasoning and methods of proof. This may be included in classroom
enrichment activities at this grade
level, but is more of a focus at
secondary grade levels.
4.5.5.D.4 Rely on reasoning, rather than answer keys, teachers, or peers, to
check the correctness of their problem solutions.
4.5.5.D.5 Make and investigate mathematical conjectures.
Counterexamples as a means of disproving conjectures.
Verifying conjectures using informal reasoning or proofs.
4.5.5.D.6 Evaluate examples of mathematical reasoning and determine whether This may be included in classroom
they are valid. enrichment activities at this grade
level, but is more of a focus at
secondary grade levels.
Focal points at this grade level are bolded.
Process Standard 4.5 is imbedded in Enduring Understandings and Essential Questions.
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E. Representations
Representations refer to the use of physical objects, drawings, charts, graphs, and symbols to represent mathematical concepts and problem situations. By using various
representations, students will be better able to communicate their thinking and solve problems. Using multiple representations will enrich the problem solver with alternative
perspectives on the problem. Historically, people have developed and successfully used manipulatives (concrete representations such as fingers, base ten blocks, geoboards,
and algebra tiles) and other representations (such as coordinate systems) to help them understand and develop mathematics.
CPI and Descriptor Objectives: Students will be able to…
4.5.5.E.1 Create and use representations to organize, record, and communicate create and use representations to organize, record, and
mathematical ideas. communicate mathematical ideas.
Concrete representations (e.g., base-ten blocks or algebra tiles) select, apply, and translate among mathematical representations to
Pictorial representations (e.g., diagrams, charts, or tables) solve problems.
Symbolic representations (e.g., a formula) use representations to model and interpret physical, social, and
Graphical representations (e.g., a line graph) mathematical phenomena.
4.5.5.E.2 Select, apply, and translate among mathematical representations to solve problems.
4.5.5.E.3 Use representations to model and interpret physical, social, and mathematical
phenomena.
Focal points at this grade level are bolded.
Process Standard 4.5 is imbedded in Enduring Understandings and Essential Questions.
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F. Technology
Calculators and computers need to be used along with other mathematical tools by students in both instructional and assessment activities. These tools should be used, not to
replace mental math and paper-and-pencil computational skills, but to enhance understanding of mathematics and the power to use mathematics. Students should explore
both new and familiar concepts with calculators and computers and should also become proficient in using technology as it is used by adults (e.g., for assistance in solving
real-world problems).
CPI and Descriptor Objectives: Students will be able to…
4.5.5.F.1 Use technology to gather, analyze, and communicate mathematical information. use technology to gather, analyze and communicate math
information.
use spreadsheets, software and graphing calculators and programs to
4.5.5.F.2 Use computer spreadsheets, software, and graphing utilities to organize and display display data.
quantitative information. use graphing calculators and computer software to work with
functions and their graphs.
use calculators to solve problems.
4.5.5.F.3 Use graphing calculators and computer software to investigate properties of use computer software to make and verify assumptions about
functions and their graphs. geometric objects.
use technology to apply math in science.
4.5.5.F.4 Use calculators as problem-solving tools (e.g., to explore patterns, to validate
solutions).
4.5.5.F.5 Use computer software to make and verify conjectures about geometric objects.
4.5.5.F.6 Use computer-based laboratory technology for mathematical applications in the
sciences.
Focal points at this grade level are bolded.
Process Standard 4.5 is imbedded in Enduring Understandings and Essential Questions.
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