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Proprietary Math, Grade 4
Scope and Sequence
COURSE OVERVIEW
This course moves quickly through whole number operations with greater numbers into
applications and properties of operations. Students will begin to work with simple fraction
and decimal operations, which are applied in the study of measurement, probability, and
data, and mathematical reasoning techniques. Rational numbers are extended as
students begin the study of equivalencies between fractions and decimals on the
number line, as well as early work with integers. Algebraic thinking is developed as
students begin to work with variables and coordinate graphing and also begin to use
formulas in problems involving perimeter, area, and rate. Geometry is extended into
greater classification of shapes and work with lines, angles and rotations.
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 Number Sense (7 lessons/8 instructional days)
Big Ideas:
 Place-value notation makes it easier to write and operate on large numbers.
High-Priority Master Objectives:
 n/a
Master Objectives:
 Identify the place value for each digit in whole numbers through 100,000,000.
 Read numerals and number words through 100,000,000.
 Write numerals through 100,000,000.
 Use expanded form to represent numbers through 100,000,000.
 Compare and order numbers through 100,000,000.
 Round a whole number.
Standards:
This unit meets California Content Standards:
 Gr. 4: Number Sense
o 1.1: Read and write whole numbers in the millions; and 1.3: Round whole
numbers through the millions to the nearest ten, hundred, thousand, ten
thousand, or hundred thousand.
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Unit 2: Whole Number Operations (8 lessons/11 instructional days)
Big Ideas:
 Any integer or rational number can be plotted on a number line.
 Inverses undo each other. Addition and subtraction are inverse operations, and
multiplication and division are inverse operations.
High-Priority Master Objectives:
 n/a
Master Objectives:
 Estimate sums and differences on a number line.
 Explain and apply standard step-by-step approaches for addition.
 Use an inverse relationship to simplify a computation or check a result.
 Explain and apply standard step-by-step approaches for multiplication.
 Explain and apply standard step-by-step approaches for division.
 Define and identify a prime number.
Standards:
This unit meets California Content Standards:
 Gr. 4: Number Sense
o 3.1: Demonstrate an understanding of, and the ability to use, standard
algorithms for the addition and subtraction of multidigit numbers; 3.2:
Demonstrate an understanding of, and the ability to use, standard algorithms
for multiplying a multidigit number by a two-digit number and for dividing a
multidigit number by a one-digit number; use relationships between them to
simplify computations and to check results; and 4.2: Know that numbers such
as 2, 3, 5, 7, and 11 do not have any factors except 1 and themselves and
that such numbers are called prime numbers.
Unit 3: Applications of Operations (6 lessons/10 instructional days)
Big Ideas:
 The order of operations dictates the order in which operations are to be
performed. The order of operations ensures that any numerical expression has
exactly one correct value.
 The distributive property illustrates how to multiply a specific multiplier by a series
of numbers that are being added or subtracted.
 Inverses undo each other. Addition and subtraction are inverse operations, and
multiplication and division are inverse operations.
 Rates of growth and decrease, comparisons of size, and rules for financial
transactions can all be explained through use of fractions, percents, and
decimals.
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High-Priority Master Objectives:
 Demonstrate how and when to use the distributive property.
 Solve a story problem involving rate.
Master Objectives:
 Use parentheses and the order of operations to write or evaluate an expression.
 Solve a story problem involving whole numbers.
 Use an inverse relationship to simplify a computation or check a result.
 Check the computation of a solution to a story problem.
Standards:
This unit meets California Content Standards:
 Gr. 4: Algebra and Functions
o 1.3: Use parentheses to indicate which operation to perform first when writing
expressions containing more than two terms and different operations; and 3.2:
Demonstrate an understanding of, and the ability to use, standard algorithms
for multiplying a multidigit number by a two-digit number and for dividing a
multidigit number by a one-digit number; use relationships between them to
simplify computations and to check results.
 Gr. 4: Mathematical Reasoning
o 2.6: Make precise calculations and check the validity of the results from the
context of the problem.
Unit 4: Lines, Angles, and Rotations (5 lessons/5 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 lines that are parallel or intersecting.
 Identify lines that are perpendicular.
 State and recognize the definitions of a right angle, an acute angle, an obtuse
angle, and a straight angle.
 Demonstrate understanding of relative angle measures.
 Recognize that 90°, 180°, 270°, and 360° are associated respectively with a ¼,
½, ¾, and full turn.
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Standards:
This unit meets California Content Standards:
 Gr. 4: Measurement and Geometry
o 3.1: Identify lines that are parallel and perpendicular; and 3.5: Know the
definitions of a right angle, an acute angle, and an obtuse angle. Understand
that 90°, 180°, 270°, and 360° are associated, respectively with ¼, ½, ¾, and
full turns.
Unit 5: Fraction Sense (8 lessons/14 instructional days)
Big Ideas:
 Fractions represent the ratio of a part to a whole, including a part of a set to the
whole set.
 a/a = 1 when a is not equal to 0.
 Equivalence is a fundamental property of rational numbers; equivalent fractions,
percents, and decimals all name the same relationship between two values.
Equivalent fractions are created by multiplying or dividing by a/a.
High-Priority Master Objectives:
 Explain and give examples of different interpretations of fractions.
 Explain why two given fractions are equivalent.
 Recognize and determine equivalent fractions.
 Find a fraction between two numbers.
Master Objectives:
 Identify the fraction represented by a part of a whole figure.
 Explain why a/a = 1.
 Represent a fraction with a sketch.
Standards:
This unit meets California Content Standards:
 Gr. 4: Number Sense
o 1.5: Explain different interpretations of fractions, for example, parts of a whole,
parts of a set, and division of whole numbers by whole numbers; explain
equivalents of fractions; and 1.7: Write the fraction represented by a drawing
of parts of a figure; represent a given fraction by using drawings; and relate a
fraction to a simple decimal on a number line.
Unit 6: Measurement (7 lessons/8 instructional days)
Big Ideas:
 Estimation is a useful tool in problem solving.
 Measurement is a process that assigns a number to the magnitude of a quantity.
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High-Priority Master Objectives:
 Solve a story problem involving equal measures.
Master Objectives:
 Estimate the length of a line segment to the nearest inch or centimeter.
 Solve a measurement-conversion problem by using multiplication or division.
 Identify the appropriate metric and English units and tools to measure
temperature.
 Read a thermometer that measures temperature in Fahrenheit degrees.
 Read a thermometer that measures temperature in Celsius degrees.
 Identify the appropriate Fahrenheit or Celsius temperature for a given practical
setting.
Standards:
This unit meets California Content Standards:
 Gr. 4: Number Sense
o 3.3: Solve problems involving multiplication of multidigit numbers by two-digit
numbers.
Unit 7: Fraction Operations (8 lessons/13 instructional days)
Big Ideas:
 Rational numbers and math operations can describe many physical events and
relationships in our world.
 Rational numbers and arithmetic operations can describe many physical events
and relationships in our world.
 Fractions can be added, subtracted, multiplied, and divided.
High-Priority Master Objectives:
 Simplify factors in fraction multiplication problems in which numerators and
denominators have common factors.
 Multiply a fraction by a whole number to solve a story problem.
 Divide a whole number by a fraction to solve a story problem.
Master Objectives:
 Use objects or sketches to solve a story problem that involves addition or
subtraction of fractions.
 Solve and simplify a problem that involves addition or subtraction of fractions with
unlike denominators.
 Write equations to demonstrate that whole numbers can be factored in multiple
ways.
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Standards:
This unit meets California Content Standards:
 Gr. 4: Number Sense
o 4.1: Understand that many whole numbers break down in different ways (e.g.,
12 = 4 × 3 = 2 × 6 = 2 × 2 × 3).
Unit 8: Decimals and Equality with Fractions (9 lessons/10 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.
 Any integer or rational number can be plotted on a number line.
High-Priority Master Objectives:
 n/a
Master Objectives:
 Compare decimal numbers.
 Order three or more decimal numbers.
 Round a decimal number.
 Judge the accuracy of a rounded decimal number.
 Identify and explain when rounding is useful.
 Estimate the sum or difference of positive decimal numbers.
 Compute the sum or difference of positive decimal numbers.
 Write tenths and hundredths in decimal and fraction notation and show that the
representations are equivalent.
 Identify fraction and decimal-number equivalents for halves and fourths.
 Relate a decimal number to a fraction on a number line.
Standards:
This unit meets California Content Standards:
 Gr. 4: Number Sense
o 1.2: Order and compare whole numbers and decimals to two decimal places;
1.4: Decide when a rounded solution is called for and explain why such a
solution may be appropriate; 1.6: Write tenths and hundredths in decimal and
fraction notations, and know the fraction and decimal equivalents for halves
and fourths (e.g., ½ = 0.5 or .50; 7/4 = 1 ¾ = 1.75); 1.7: Write the fraction
represented by a drawing of parts of a figure; represent a given fraction by
using drawings; and relate a fraction to a simple decimal on a number line;
2.1: Estimate and compute the sum or difference of whole numbers and
positive decimals to two places; and 2.2: Round two-place decimals to one
decimal or the nearest whole number and judge the reasonableness of the
rounded answer.
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Unit 9: Semester Review and Checkpoint (2 lessons/2 instructional days)
SEMESTER 2
Unit 10: Probability and Data (9 lessons/9 instructional days)
Big Ideas:
 Probability is a measure of how likely it is that some event will occur.
 Graphs and charts are useful ways to represent and compare numerical data.
 Mean, median, and mode are all measures of where the center of a data set lies.
High-Priority Master Objectives:
 n/a
Master Objectives:
 Represent a probability as a fraction.
 Organize all possible outcomes for a simple probability situation.
 Determine the number of possible combinations of objects from three sets.
 Design survey questions and systematically collect and represent the data.
 Answer questions about one- and two-variable data graphs.
 Recognize appropriate representations of survey data.
 Identify the mode or modes for a set of numerical data or a set of categorical
data.
 Identify the median and outliers for a numerical data set.
Standards:
This unit meets California Content Standards:
 Gr. 4: Statistics, Data Analysis, and Probability
o 1.1: Formulate survey questions; systematically collect and represent data on
a number line and coordinate graphs, tables, and charts; 1.2: Identify the
mode(s) for sets of categorical data and the mode(s), median, and any
apparent outliers for numerical data sets; 1.3: Interpret one- and two-variable
data graphs to answer questions about a situation; 2.1: Represent all possible
outcomes for a simple probability situation in an organized way (e.g., tables,
grids, tree diagrams); and 2.2: Express outcomes of experimental probability
situations verbally and numerically (e.g., 3 out of 4; ¾).
Unit 11: Mathematical Reasoning (10 lessons/14 instructional days)
Big Ideas:
 Estimation is a useful tool in problem solving.
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High-Priority Master Objectives:
 Analyze a story problem by identifying the question, recognizing relevant
information, sequencing and prioritizing information, and developing a solution
strategy.
 Use estimation to predict a solution to a story problem and to verify the
reasonableness of the calculated result.
 Express the solution to a story problem clearly and logically.
Master Objectives:
 Determine when and how to break a multistep story problem into simpler
problems.
 Identify different story problems that can be solved by using the same
procedures.
 Apply strategies or results from a simpler problem to a similar or more complex
problem.
 Explain mathematical reasoning in a story problem by using multiple
representations.
 Evaluate a strategy or strategies used in a story problem.
 Explain the advantages and disadvantages of exact and approximate solutions to
story problems.
 Answer a story problem to a specified degree of accuracy, such as hundredths.
Standards:
This unit meets California Content Standards:
 Gr. 4: 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;
2.1: Use estimation to verify the reasonableness of calculated results; 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; 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; and 2.5: Indicate the
relative advantages of exact and approximate solutions to problems and give
answers to a specified degree of accuracy.
Unit 12: Geometry (9 lessons/10 instructional days)
Big Ideas:
 Two figures are defined to be congruent if and only if one figure can be moved
using rigid motions (reflection, translation, rotation) so that one is exactly on top
of the other, matching every point with another point.
 Geometric figures can be described and classified by the shapes of their faces
and by how many faces, sides, edges, or vertices they have.
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High-Priority Master Objectives:
 n/a
Master Objectives:
 Define and sketch different types of triangles and identify their attributes.
 Know how to define and sketch different quadrilaterals.
 Identify the diameter and radius of a circle.
 Identify and explain why given figures are congruent.
 Identify figures that have bilateral symmetry and draw the line or lines of
symmetry.
 Identify figures that have rotational symmetry.
 Describe a geometric solid in terms of the shapes of its faces and the number of
faces, edges, and vertices it has.
 Recognize and sketch a two-dimensional representation of a three-dimensional
object.
Standards:
This unit meets California Content Standards:
 Gr. 4: Measurement and Geometry
o 3.2: Identify the radius and diameter of a circle; 3.3: Identify congruent figures;
3.4: Identify figures that have bilateral and rotational symmetry; and 3.6:
Visualize, describe, and make models of geometric solids (e.g., prisms,
pyramids) in terms of the number and shape of faces, edges, and vertices;
interpret two-dimensional representations of three-dimensional objects; and
draw patterns (of faces) for a solid that, when cut and folded, will make a
model of the solid; 3.7: Know the definitions of different triangles (e.g.,
equilateral, isosceles, scalene) and identify their attributes; 3.8: Know the
definition of different quadrilaterals (e.g., rhombus, square, rectangle,
parallelogram, trapezoid).
Unit 13: Rational Numbers (6 lessons/9 instructional days)
Big Ideas:
 Any integer or rational number can be plotted on a number line.
High-Priority Master Objectives:
 Identify relative positions of rational numbers on a number line.
Master Objectives:
 Identify and place negative numbers on a number line.
 Determine missing negative numbers in counting sequences.
 Use a negative number to represent a temperature.
 Use negative numbers in story problems that involve owing money.
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Standards:
This unit meets California Content Standards:
 Gr. 4: Number Sense
o 1.8: Use concepts of negative numbers (e.g., on a number line, in counting, in
temperature, in “owing”); and 1.9: Identify on a number line the relative
position of positive fractions, positive mixed numbers, and positive decimals to
two decimal places.
Unit 14: Algebra Thinking (9 lessons/13 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.
 Solving an equation means finding all the possible values of certain symbols, the
variables, within the allowed domain that make the equation true.
 Any point in a coordinate plane can be described by an ordered pair of
coordinates.
High-Priority Master Objectives:
 Demonstrate that when equal quantities are added to equal quantities the
resulting quantities are equal.
 Demonstrate that when equal quantities are multiplied by equal quantities the
resulting quantities are equal.
Master Objectives:
 Use symbols to stand for variables in simple expressions or equations.
 Solve for one variable in a two-variable equation when the value of the other
variable is given.
 Locate and plot points on a coordinate plane.
 Find the length of a horizontal line segment by finding the difference of the xcoordinates.
 Find the length of a vertical line segment by finding the difference of the ycoordinates.
 Plot a linear relationship in the first quadrant of a coordinate plane.
Standards:
This unit meets California Content Standards:
 Gr. 4: Algebra and Functions
o 1.1: Use letters, boxes, or other symbols to stand for any number in simple
expressions or equations (e.g., demonstrate an understanding and the use of
the concept of a variable); 1.5: Understand that an equation such as y = 3x +
5 is a prescription for determining a second number when a first number is
given; 2.1: Know and understand that equals added to equals are equal.
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
Gr. 4: Measurement and Geometry
o 2.1: Draw the points corresponding to linear relationships on graph paper
(e.g., draw 10 points on the graph of the equation y = 3x and connect them by
using a straight line); 2.2: Understand that the length of a horizontal line
segment equals the difference of the x-coordinates; and 2.3: Understand that
the length of a vertical line segment equals the difference of the y-coordinates.
Unit 15: Perimeter and Area Formulas (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.
High-Priority Master Objectives:
 n/a
Master Objectives:
 Define and demonstrate understanding of the perimeter of any polygon.
 Use a formula to find the perimeter of a rectangle or a square.
 Interpret and use formulas to answer questions about quantities and their
relationships.
 Use a formula to find the perimeter of a rectangle or a square.
 Define and demonstrate understanding of the area of any plane figure.
Find the area of a rectangular shape and use the appropriate unit.
 Use a formula to find the area of a rectangle, a square, or a figure that can be
divided into rectangles or squares.
 Interpret and use formulas to answer questions about quantities and their
relationships.
 Solve a story problem that requires finding rectangular area.
 Demonstrate understanding that rectangles that have the same area can have
different perimeters.
 Demonstrate understanding that rectangles that have the same perimeter can
have different areas.
Standards:
This unit meets California Content Standards:
 Gr. 4: Measurement and Geometry
o 1.1: Measure the area of rectangular shapes by using appropriate units such
as square centimeter (cm2), square meter (m2), square kilometer (km2),
square inch (in. 2), square yard (yd2), or square mile (mi2); 1.2: Recognize that
rectangles that have the same area can have different perimeters; 1.3:
Understand that rectangles that have the same perimeter can have different
areas; and 1.4: Understand and use formulas to solve problems involving
perimeters and areas of rectangles and squares. Use those formulas to find
the areas of more complex figures by dividing the figures into basic shapes.
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
Gr. 4: Algebra and Functions
o 1.4: Use and interpret formulas (e.g., area = length × width or A = lw) to
answer questions about quantities and their relationships.
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 (115 lessons; lessons
may span several days)
Lesson time: 60 minutes
Standard curriculum items:
 textbook
 color tiles set
 lesson guide book
 protractor
 common items that may be found in a typical home
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