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Proprietary Math, Grade 5
Scope and Sequence
COURSE OVERVIEW
This course investigates whole numbers through practical situations in rounding,
exponents and powers, and elementary number theory. Much of the course is devoted to
all four operations with fractions, decimals, and percents and equivalencies among the
three representations. Students will also begin addition and subtraction of integers and
apply all of their work with rational numbers to problem-solving experiences. The study
of algebra includes work with variables, solving equations and inequalities, using
formulas within geometry and measurement, and work within the coordinate system.
Students will also analyze and display data with a variety of additional graphic
representations. The study of geometry encompasses properties of lines, angles, twoand three-dimensional figures, and formal constructions and transformations.
Note: This course meets many national and state standards. California state standards
are shown here as a reference.
COURSE OUTLINE
SEMESTER 1
Unit 1: Whole Numbers and Powers (10 lessons/13 instructional days)
Big Ideas:
 Estimation is a useful tool in problem solving.
High-Priority Master Objectives:
 n/a
Master Objectives:
 Round whole numbers in a story problem.
 Estimate or calculate a sum or a difference in a whole-number problem.
 Estimate or calculate a sum or a difference in a whole-number story problem.
 Estimate or calculate a product or a quotient in a whole-number problem.
 Estimate or calculate a product or quotient in a whole-number story problem.
 Represent and compute a power by using repeated multiplication.
 Solve a problem that involves powers.
 Determine the prime factorization of a composite number.
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Standards:
This unit meets California Content Standards:
 Gr. 5: Number Sense
o 1.1: Estimate, round, and manipulate very large (e.g., millions) and very small
(e.g., thousandths) numbers; 1.3: Understand and compute positive integer
powers of nonnegative integers; compute examples as repeated
multiplication; 1.4: Determine the prime factors of all numbers through 50 and
write the numbers as the product of their prime factors by using exponents to
show multiples of a factor (e.g., 24 = 2  2  2  3 = 23  3); and 2.2:
Demonstrate proficiency with division, including division with positive decimals
and long division with multidigit divisors.
Unit 2: Geometry (9 lessons/13 instructional days)
Big Ideas:
 A right angle forms a square corner that measures 90 degrees; an acute angle is
less than a right angle and an obtuse angle is greater than a right angle.
High-Priority Master Objectives:
 n/a
Master Objectives:
 Identify, measure, and draw angles with appropriate math tools.
 Identify and draw perpendicular or parallel lines with appropriate math tools.
 Construct rectangles or triangles with appropriate math tools.
 Identify that the sum of the interior angles of any triangle is 180º and solve
related problems.
 Identify that the sum of the interior angles of any quadrilateral is 360º and solve
related problems.
 Predict, describe, and perform transformations on two-dimensional shapes.
 Identify or draw a two-dimensional view of a three-dimensional object.
Standards:
This unit meets California Content Standards:
 Gr. 5: Measurement and Geometry
o 2.1: Measure, identify, and draw angles, perpendicular and parallel lines,
rectangles, and triangles by using appropriate tools (e.g., straightedge, ruler,
compass, protractor, drawing software); 2.2: Know that the sum of the angles
of any triangle is 180° and the sum of the angles of any quadrilateral is 360°
and use this information to solve problems; 2.3: Visualize and draw twodimensional views of three-dimensional objects made from rectangular solids.
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Unit 3: Fractions: Multiplication and Division (6 lessons/10 instructional days)
Big Ideas:
 Fractions can be added, subtracted, multiplied, and divided.
High-Priority Master Objectives:
 Multiply fractions and explain a step-by-step approach.
 Divide fractions and explain a step-by-step approach.
Master Objectives:
 see High-Priority Master Objectives
Standards:
This unit meets California Content Standards:
 Gr. 5: Number Sense
o 2.4: Understand the concept of multiplication and division of fractions.
Unit 4: Problems Involving Fractions (4 lessons/9 instructional days)
Big Ideas:
 Fractions can be added, subtracted, multiplied, and divided.
High-Priority Master Objectives:
 Solve a story problem involving multiplication or division of fractions.
 Solve a simple problem involving addition or subtraction of fractions.
Master Objectives:
 see High-Priority Master Objectives
Standards:
This unit meets California Content Standards:
 Gr. 5: Number Sense
o 2.3: Solve simple problems, including ones arising in concrete situations,
involving the addition and subtraction of fractions and mixed numbers (like
and unlike denominators of 20 or less), and express answers in the simplest
form; and 2.5: Compute and perform simple multiplication and division of
fractions and apply these procedures to solving problems.
Unit 5: Decimals: Addition and Subtraction (6 lessons/8 instructional days)
Big Ideas:
 Estimation is a useful tool in problem solving.
 Addition, subtraction, multiplication, and division can be represented by models
and by using math symbols.
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High-Priority Master Objectives:
 Estimate the sum or difference in a problem involving decimal numbers.
 Solve a story problem involving addition or subtraction of decimal numbers.
Master Objectives:
 Round a decimal number to any place through hundredths.
Solve an addition or subtraction problem involving decimal numbers.
 Verify that the calculated result of a problem involving addition or subtraction of
decimal numbers is reasonable.
Standards:
This unit meets California Content Standards:
 Gr. 5: Number Sense
o 1.1: Estimate, round, and manipulate very large (e.g., millions) and very small
(e.g., thousandths) numbers; 2.1: Add, subtract, multiply, and divide with
decimals; add with negative integers; subtract positive integers from negative
integers; and verify the reasonableness of the results.
Unit 6: Decimals: Multiplication and Division (5 lessons/11 instructional days)
Big Ideas:
 Estimation is a useful tool in problem solving.
 Multiplication and division can be represented by models and by using math
symbols.
High-Priority Master Objectives:
 Estimate the product or quotient of a computation problem involving decimal
numbers.
 Solve a story problem involving multiplication or division of decimal numbers.
Master Objectives:


Solve a computation problem involving multiplication or division of decimal
numbers.
Verify that the calculated result of a problem involving multiplication or division of
decimal numbers is reasonable.
Standards:
This unit meets California Content Standards
 Gr. 5: Number Sense
o 1.1; 2.1: Add, subtract, multiply, and divide with decimals; add with negative
integers; subtract positive integers from negative integers; and verify the
reasonableness of the results; and 2.2: Demonstrate proficiency with division,
including division with positive decimals and long division with multidigit
divisors.
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Unit 7: Integers (5 lessons/9 instructional days)
Big Ideas:
 Any integer or rational number can be plotted on a number line.
High-Priority Master Objectives:
 Identify and represent decimal numbers, fractions, mixed numbers, and positive
and negative integers on a number line.
 Solve a problem involving addition or subtraction of integers.
Master Objectives:
 Verify that the calculated result of a problem involving addition or subtraction of
integers is reasonable.
Standards:
This unit meets California Content Standards:
 Gr. 5: Number Sense
o 1.5: Identify and represent on a number line decimals, fractions, mixed
numbers, and positive and negative integers; 2.1: Identify and represent on a
number line decimals, fractions, mixed numbers, and positive and negative
integers.
Unit 8: Semester Review and Checkpoint (2 lessons/2 instructional days)
SEMESTER 2
Unit 9: Percents and Probability (8 lessons/13 instructional days)
Big Ideas:
 Ratios, fractions, percents, and decimals can be used to compare one value to
another, or through models, to compare properties of two things or situations.
 Equivalence is a fundamental property of rational numbers; equivalent fractions,
percents, and decimals all name the same relationship between two values.
 Independent events are events in which the occurrence of one event does not
affect the occurrence of the other.
 Knowing the probability of an event can help you predict the likelihood that it will
occur again.
High-Priority Master Objectives:
 Determine the decimal and percent equivalents for a common fraction and
explain why they represent the same value.
Master Objectives:
 Interpret a percent as a part of a hundred.
 Compute a given percent of a whole number.
 Represent probabilities as fractions, decimals, and percents.
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

Identify events that are dependent or independent.
Use probability to predict future events.
Standards:
This unit meets California Content Standards:
 Gr. 5: Number Sense
o 1.2: Interpret percents as a part of a hundred; find decimal and percent
equivalents for common fractions and explain why they represent the same
value; compute a given percent of a whole number.
Unit 10: Algebra (6 lessons/10 instructional days)
Big Ideas:
 A variable is a symbol, usually a letter, that is used to stand for a number or a set
of numbers.
 The distributive property illustrates how to multiply a specific multiplier by a series
of numbers being added or subtracted.
 An expression represents a value that can be a number, a variable, or a group of
numbers, variables, and operation symbols. Some examples of expressions are
10 – 4 + 1, 3 + x, 5y + 2, b, and 5.
High-Priority Master Objectives:
 Identify and apply the distributive property in an equation or an expression with
variables.
 Identify or use an expression or an equation to answer questions about a
problem.
Master Objectives:
 Use a letter to represent an unknown value in an expression or an equation.
 Evaluate a simple algebraic expression in one variable by using substitution.
Standards:
This unit meets California Content Standards:
 Gr. 5: Algebra and Functions
o 1.1: Use information taken from a graph or equation to answer questions
about a problem situation; 1.2: Use a letter to represent an unknown number;
write and evaluate simple algebraic expressions in one variable by
substitution; and 1.3: Know and use the distributive property in equations and
expressions with variables.
Unit 11: Coordinate Graphs (5 lessons/8 instructional days)
Big Ideas:
 Any point in a coordinate plane can be described by an ordered pair of
coordinates.
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High-Priority Master Objectives:
 Graph or write an equation to solve a problem that involves a linear function.
Master Objectives:
 Identify and graph ordered pairs in all quadrants of a coordinate plane.
 Use the situation presented in a problem to describe the meaning of each
coordinate of an ordered pair displayed on a graph.
Standards:
This unit meets California Content Standards:
 Gr. 5: Statistics, Data Analysis, and Probability
o 1.5: Know how to write ordered pairs correctly; for example, (x, y).
 Gr. 5: Algebra and Functions
o 1.4: Identify and graph ordered pairs in the four quadrants of the coordinate
plane.
Unit 12: Perimeter, Area, and Volume (9 lessons/12 instructional days)
Big Ideas:
 The perimeter of any polygon is the sum of the lengths of its sides.
 Area is a measure of how much material is needed to cover a plane figure.
 Volume is a measure of the amount of space a solid figure occupies.
 Measurement is the process of repeatedly using a unit over a quantity to
determine how much you have.
High-Priority Master Objectives:
 n/a
Master Objectives:
 Determine the perimeter of a plane figure and use appropriate units.
 Derive and use the formula for the area of a parallelogram and use appropriate
units.
 Derive and use the formula for the area of a triangle and use appropriate units.
 Construct a cube or a rectangular box from a two-dimensional pattern and
determine the surface area.
 Use squares to approximate the area of an irregular shape.
 Explain and determine the volume of a solid figure and use appropriate units.
 Differentiate among appropriate units to measure perimeter, area, and volume.
Standards:
This unit meets California Content Standards:
 Gr. 5: Measurement and Geometry
o 1.1: Derive and use the formula for the area of a triangle and of a
parallelogram by comparing it with the formula for the area of a rectangle (i.e.,
two of the same triangles make a parallelogram with twice the area; a
parallelogram is compared with a rectangle of the same area by cutting and
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pasting a right triangle on the parallelogram); 1.2: Construct a cube and
rectangular box from two-dimensional patterns and use these patterns to
compute the surface area for these objects; 1.3: Understand the concept of
volume and use the appropriate units in common measuring systems (i.e.,
cubic centimeter [cm3], cubic meter [m3], cubic inch [in3], cubic yard [yd3]) to
compute the volume of rectangular solids; and 1.4: Differentiate between, and
use appropriate units of measures for, two-and three-dimensional objects (i.e.,
find the perimeter, area, volume).
Unit 13: Math Reasoning: Methods and Strategies (7 lessons/13 instructional days)
Big Ideas:
 n/a
High-Priority Master Objectives:
 Use a variety of methods, such as words, numbers, symbols, charts, graphs,
tables, diagrams, and models, to explain mathematical reasoning in non-routine
or complex problems.
 Identify and generalize methods for solving problems that are similar to each
other.
Master Objectives:
 Prioritize and sequence the information in a story problem that involves
multiplication or division of decimal numbers.
 Determine when and how to break a multistep whole-number story problem or
money problem into simpler parts.
 Apply strategies and results from simple story problems involving fractions to
more complex problems.
Standards:
This unit meets California Content Standards:
 Gr. 5: Mathematical Reasoning
o 1.1: Analyze problems by identifying relationships, distinguishing relevant from
irrelevant information, sequencing and prioritizing information, and observing
patterns; 1.2: Determine when and how to break a problem into simpler parts;
2.2: Apply strategies and results from simpler problems to more complex
problems; 2.3: Use a variety of methods, such as words, numbers, symbols,
charts, graphs, tables, diagrams, and models, to explain mathematical
reasoning; 3.2: Note the method of deriving the solution and demonstrate a
conceptual understanding of the derivation by solving similar problems; and
3.3: Develop generalizations of the results obtained and apply them in other
circumstances.
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Unit 14: Math Reasoning: Solutions (6 lessons/7 instructional days)
Big Ideas:
 Estimation is a useful tool in problem solving.
High-Priority Master Objectives:
 n/a
Master Objectives:
 Express clear and logical solutions to equal measures problems and rate
problems.
 Use estimation in addition or subtraction of fractions to verify whether calculated
results are reasonable.
 Make precise calculations and use the situation presented in a problem involving
decimal-number operations to check the validity of the result.
 Evaluate whether a solution for a problem is reasonable.
Standards:
This unit meets California Content Standards:
 Gr. 5: Mathematical Reasoning
o 2.1: Use estimation to verify the reasonableness of calculated results; 2.4:
Express the solution clearly and logically by using the appropriate
mathematical notation and terms and clear language; support solutions with
evidence in both verbal and symbolic work; 2.6: Make precise calculations and
check the validity of the results from the context of the problem; 3.1: Evaluate
the reasonableness of the solution in the context of the original situation.
Unit 15: Data Analysis and Representation (9 lessons/10 instructional days)
Big Ideas:
 Mean, median, and mode are all measures of where the center of a data set lies.
 Graphs and charts are useful ways to represent and compare numerical data.
High-Priority Master Objectives:
 n/a
Master Objectives:
 Explain and compute the mean, median, or mode for a set of data.
 Compare the mean, median, and mode of a data set and explain the differences
and appropriate uses.
 Organize and display single-variable data in a histogram.
 Organize and display single-variable data in a circle graph.
 Interpret information displayed in a graph or table.
 Use fractions and percents to compare different data sets.
 Explain which types of graphs are appropriate for various data sets.
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Standards:
This unit meets California Content Standards:
 Gr. 5: Statistics, Data Analysis, and Probability
o 1.1: Know the concepts of mean, median, and mode; compute and compare
simple examples to show that they may differ; 1.2: Organize and display
single-variable data in appropriate graphs and representations (e.g.,
histogram, circle graphs) and explain which types of graphs are appropriate
for various data sets; and 1.3: Use fractions and percentages to compare data
sets of different sizes.
 Gr. 5: Algebra and Functions
o 1.1: Use information taken from a graph or equation to answer questions
about a problem situation.
Unit 16: Semester Review and Checkpoint (2 lessons/2 instructional days)
LESSON TIME AND SCHEDULING
Total lessons: 150 instructional days and 30 “Your Choice” days (99 lessons; lessons
may span several days)
Lesson time: 60 minutes
Standard curriculum items:
 textbook
 lesson guide book
 protractor
 common items that may be found in a typical home
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