Proprietary Math, Grade 3

Proprietary Math, Grade 3

Scope and Sequence

Course Overview

This course provides a quick overview of whole number addition and subtraction, but has a much greater focus on whole number multiplication and division, encompassing early algebraic thinking. The concepts of both decimals and fractions are introduced in greater detail than in earlier courses. Decimals are primarily studied in relationship to place value and money, whereas fractions are addressed through multiple representations and probability. All work with whole numbers, decimals, and fractions is used in the study of mathematical reasoning where students are introduced to specific methods and strategies to help them become more effective problem solvers. Geometry and measurement are addressed through the study of two- and three-dimensional shapes, early work with perimeter, area, and volume, and applying measuring techniques to time, length, capacity, and weight.

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(10 lessons/11 instructional days)

Big Ideas:

Place-value notation makes it easier to write and operate on large numbers.

High-Priority Master Objectives:

Master Objectives:

Identify the place value for each digit in whole numbers through 10,000.
Count aloud whole numbers through 10,000.
Identify odd and even numbers and describe their characteristics.
Read whole numbers through 10,000.
Write numerals through 10,000.
Write number words through 10,000.
Use expanded form to represent numbers through 10,000.
Demonstrate understanding of place values as cumulative multiples of 10.
Compare whole numbers through 10,000.
Order three or more whole numbers through 10,000.
Round numbers through 10,000.

Standards:

This unit meets California Content Standards:

Gr. 3: Number Sense

1.1 Count, read, and write whole numbers to 10,000; 1.2: Compare and order whole numbers to 10,000; 1.3: Identify the place value for each digit in numbers to 10,000; 1.4: Round off numbers to 10,000 to the nearest ten, hundred, and thousand; and 1.5: Use expanded notation to represent numbers (e.g., 3,206 = 3,000 + 200 + 6).

Unit 2: Whole Number Addition and Subtraction(6 lessons/6 instructional days)

Big Ideas:

Inverses undo each other. Addition and subtraction are inverse operations, and multiplication and division are inverse operations.
The use of letters, numbers, and mathematical symbols makes possible the translation of complex situations or long word statements into concise mathematical sentences or expressions.

High-Priority Master Objectives:

Master Objectives:

Demonstrate an understanding of how addition and subtraction affect whole numbers.
Determine the sum or difference of two whole numbers.
Recognize and solve a story problem in which two quantities are combined.
Recognize and solve a story problem in which a quantity changes by addition or subtraction.
Recognize and solve a story problem in which one quantity must be changed to equal another quantity.

Standards:

This unit meets California Content Standards:

Gr. 3: Number Sense

2.1: Find the sum or difference of two whole numbers between 0 and 10,000.

Unit 3: Algebra Thinking(9 lessons/14 instructional days)

Big Ideas:

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.
An equation is a statement that two expressions are equal.
Addition, subtraction, multiplication, and division can be represented by models and by using math symbols.
The equals sign denotes an equivalent relationship.
Rules can be used to generate number patterns.

High-Priority Master Objectives:

Determine a missing number in an equation or an inequality.

Master Objectives:

Use a mathematical expression to represent a relationship between quantities.
Use an equation to represent a relationship between quantities.
Use an inequality to represent a relationship between quantities.
Select the appropriate symbol to show an operation or a relationship that makes a number sentence true.
Recognize and describe a linear pattern, such as counting by 5s or multiplying 5 times a number to reach 100, by its rule.
Extend a linear pattern, such as stating what number comes next in a series.
Solve a simple story problem that involves a function.

Standards:

This unit meets California Content Standards:

Gr. 3: Algebra and Functions

1.1: Represent relationships of quantities in the form ofmathematical expressions, equations, or inequalities; 1.2: Solve problems involving numeric equations or inequalities; 1.3: Select appropriate operational and relational symbols to make an expression true (e.g., if 4 ___ 3 = 12, what operational symbol goes in the blank?); 2.1: Solve simple problems involving a functional relationship between two quantities (e.g., find the total cost of multiple items given the cost per unit); and 2.2: Extend and recognize a linear pattern by its rules (e.g.,the number of legs on a given number of horses may be calculated by counting by 4s or by multiplying the number of horses by 4).

Unit 4: Whole Number Multiplication Sense(9 lessons/13 instructional days)

Big Ideas:

Multiplication and division can be represented by models and by using math symbols.
Multiplication can be understood as repeated addition or can be represented through area models.
Inverses undo each other. Addition and subtraction are inverses, and multiplication and division are inverses.
The commutative and associative properties can be used to simplify expressions.
Multiplying any number by 1, the multiplicative identity, results in a product that is the given number.
The commutative and associative properties can be used to simplify expressions.

High-Priority Master Objectives:

Use an area model to explain multiplication.

Master Objectives:

Use objects or sketches to solve a multiplication problem.
Use a model to explain multiplication as repeated addition of the same quantity.
Demonstrate an understanding of how multiplication affects whole numbers.
Explain and apply the commutative property of multiplication.
Explain and apply the zero property of multiplication.
Explain and apply the multiplication property of 1.
Demonstrate automatic recall of multiplication facts.
Explain and apply the associative property of multiplication.

Standards:

This unit meets California Content Standards:

Gr. 3: Algebra and Functions

1.5: Recognize and use the commutative and associative properties of multiplication (e.g., if 5 × 7 = 35, then what is 7 × 5? and if 5 × 7 × 3 = 105, then what is 7 × 3 × 5?).

Gr. 3: Number Sense

2.2: Memorize to automaticity the multiplication table for numbers between 1 and 10; and 2.6: Understand the special properties of 0 and 1 in multiplication and division.

Unit 5: Whole Number Multiplication (7 lessons/9 instructional days)

Big Ideas:

Multiplication and division can be represented by models and by using math symbols.
The use of letters, numbers, and mathematical symbols makes possible the translation of complex situations or long word statements into concise mathematical sentences or expressions.

High-Priority Master Objectives:

Create a story problem that can be represented by a multiplication number sentence.

Master Objectives:

Use objects or sketches to solve a multiplication story problem.
Solve a multiplication problem involving a multidigit factor and a one-digit factor.
Use multiplication to solve a story problem that involves equal groups.
Use multiplication to solve a story problem that involves equal measures.

Standards:

This unit meets California Content Standards:

Gr. 3: Number Sense

2.4: Solve simple problems involving multiplication of multidigit numbers by one-digit numbers (3,671 × 3 = ___).

Unit 6: Whole Number Division Sense(8 lessons/8 instructional days)

Big Ideas:

Multiplication and division can be represented by models and by using math symbols.
Division can be understood as repeated subtraction or as division of a quantity into equal groups.
Addition, subtraction, multiplication and division can be represented by models and by using math symbols.
Inverses undo each other. Addition and subtraction are inverse operations, and multiplication and division are inverse operations.

High-Priority Master Objectives:

Demonstrate an understanding of the inverse relationship between multiplication and division.
Use the inverse relationship of multiplication and division to compute and check results.

Master Objectives:

Use objects or sketches to solve a division problem.
Explain division as repeated subtraction.
Explain the meaning of the  symbol.
Explain and apply the division property of 1.
Demonstrate understanding that division by zero is undefined.
Recognize the meaning of the three symbols for division.
Explain division as the sharing of a quantity into equal groups.
Demonstrate an understanding of the effects of division on whole numbers.

Standards:

This unit meets California Content Standards:

Gr. 3: Number Sense

2.3: Use the inverse relationship of multiplication and division to compute and check results; and 2.6: Understand the special properties of 0 and 1 in multiplication and division.

Unit 7: Whole Number Division (8 lessons/10 instructional days)

Big Ideas:

The use of letters, numbers, and mathematical symbols makes possible the translation of complex situations or long word statements into concise mathematical sentences or expressions.

High-Priority Master Objectives:

Create a story problem that can be represented by a division number sentence.

Master Objectives:

Use objects or sketches to solve a division story problem.
Solve a division problem that has a multidigit dividend, a one-digit divisor, and no remainder.
Use division to solve a story problem that involves equal groups.
Use division to solve a story problem that involves equal measures.
Determine unit cost.

Standards:

This unit meets California Content Standards

Gr. 3: Number Sense

2.5: Solve division problems in which a multidigit number is evenly divided by a one-digit number (135  5 = ___); and 2.7: Determine the unit cost when given the total cost and number of units.

Unit 8: Semester Review and Checkpoint(2 lessons/2 instructional days)

SEMESTER 2

Unit 9: Whole Numbers and Multiple Operations(5 lessons/7 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 use of letters, numbers, and mathematical symbols makes possible the translation of complex situations or long word statements into concise mathematical sentences or expressions.

High-Priority Master Objectives:

Determine whether addition, subtraction, multiplication, or division is the appropriate operation to use to solve a story problem and solve the problem.
Solve a story problem involving two or more operations.

Master Objectives:

Use the order of operations to evaluate an expression.

Standards:

This unit meets California Content Standards:

Gr. 3: Number Sense

2.8: Solve problems that require two or more of the skills mentioned above.

Unit 10: Geometry(8 lessons/8 instructional days)

Big Ideas:

Geometric figures can be described and classified by the shapes of their faces and by how many faces, sides, edges, or vertices they have.
Shapes can be constructed from other shapes.

High-Priority Master Objectives:

Master Objectives:

Identify right angles in geometric figures or everyday objects.
Identify the measure of an angle in a geometric figure or an everyday object as greater than or less than a right angle.
Identify, describe, and classify a polygon according to the number of its sides.
Identify attributes of isosceles, equilateral, and right triangles.
Identify attributes of parallelograms, rectangles, and squares.
Identify and describe common solid geometric figures.
Classify common solid geometric figures.
Determine solid objects that could be combined to create a given solid object.

Standards:

This unit meets California Content Standards:

Gr. 3: Measurement and Geometry

2.1: Identify, describe, and classify polygons (includingpentagons, hexagons, and octagons); 2.2: Identify attributes of triangles (e.g., two equal sides for the isosceles triangle, three equal sides for the equilateral triangle, right angle for the right triangle); 2.3: Identify attributes of quadrilaterals (e.g., parallel sides for the parallelogram, right angles for the rectangle, equal sides and right angles for the square); 2.4: Identify right angles in geometric figures or in appropriate objects and determine whether other angles are greater or less than a right angle; 2.5: Identify, describe, and classify common three-dimensional geometric objects (e.g., cube, rectangularsolid, sphere, prism, pyramid, cone, cylinder); and 2.6: Identify common solid objects that are the components needed to make a more complex solid object.

Unit 11: Decimals and Money(5 lessons/7 instructional days)

Big Ideas:

The use of letters, numbers, and mathematical symbols makes possible the translation of complex situations or long word statements into concise mathematical sentences or expressions.

High-Priority Master Objectives:

Master Objectives:

Identify decimal place values through thousandths.
Solve a story problem involving addition or subtraction of money amounts in decimal notation.

Standards:

This unit meets California Content Standards:

Gr. 3: Number Sense

3.3: Solve problems involving addition, subtraction, multiplication, and division of money amounts in decimal notation and multiply and divide money amounts in decimal notation by using whole-number multipliers and divisors.

Unit 12: Fractions and Probability(10 lessons/13 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 rational number (a fraction) is any number that can be written as a ratio of one integer to another integer.
The use of letters, numbers, and mathematical symbols makes possible the translation of complex situations or long word statements into concise mathematical sentences or expressions.
Fractions can be added, subtracted, multiplied, and divided.
Equivalence is a fundamental property of rational numbers; equivalent fractions, percents, and decimals all name the same relationship between two values.
Probability is a measure of how likely it is that some event will occur.
Knowing the probability of an event can help you predict the likelihood that it will occur again.

High-Priority Master Objectives:

Compare and order unit fractions, such as 1/4, and fractions with like denominators, such as 2/5 and 4/5, by using objects or sketches.

Master Objectives:

Explain that a fraction can be used to represent part of a set, the relationship of a part to a whole, and a rational number on the number line.
Write the fraction represented by a drawing that shows parts of a set or parts of a whole.
Use a sketch to represent a fraction.
Use objects or sketches to solve a simple story problem involving addition or subtraction of fractions.
Solve and simplify an addition or subtraction problem involving fractions with like denominators.
Explain that a simple fraction and a decimal amount can represent the same quantity.
Identify whether specific events are certain, likely, unlikely, or impossible.
Identify and systematically record the possible outcomes for a simple event.
Summarize and display the results of a probability experiment in a clear and organized way.
Use the results of a probability experiment to predict future events.

Standards:

This unit meets California Content Standards:

Gr. 3: Number Sense

3.1: Compare fractions represented by drawings or concrete materials to show equivalency and to add and subtract simple fractions in context (e.g., 1/4 of a pizza is the same amount as 2/4 of another pizza that is the same size; show that 3/8 is larger than 1/4); 3.2: Add and subtract simple fractions (e.g., determine that 1/8 + 3/8 is the same as 1/2); and 3.4: Know and understand that fractions and decimals are two different representations of the same concept (e.g., 50 cents is 1/2 of a dollar, 75 cents is 3/4 of a dollar).

Gr. 3: Statistics, Data Analysis, and Probability

1.1: Identify whether common events are certain, likely, unlikely, or improbable; 1.2: Record the possible outcomes for a simple event (e.g., tossing a coin) and systematically keep track of the outcomes when the event is repeated many times; 1.3: Summarize and display the results of probability experiments in a clear and organized way (e.g., use a bargraph or a line plot); and 1.4: Use the results of probability experiments to predict future events (e.g., use a line plot to predict the temperatureforecast for the next day).

Unit 13: Measurement: Length and Time(8 lessons/9 instructional days)

Big Ideas:

Measurement is the process of repeatedly using a unit over a quantity to determine how much you have.

High-Priority Master Objectives:

Estimate and measure the length of an object to the nearest centimeter.
Estimate the length of an object to the nearest 1/2 inch and measure the length to the nearest 1/4 inch.

Master Objectives:

Identify the appropriate tools for measuring the length of an object.
Identify the appropriate metric or English units for measuring the length of an object.
Tell time to the nearest minute.
Determine elapsed time to the nearest minute.
Use a calendar to determine elapsed time.

Standards:

This unit meets California Content Standards:

Gr. 3: Measurement and Geometry

1.1: Choose the appropriate tools and units (metric and U.S.) and estimate and measure the length, liquid volume, and weight/mass of given objects.

Unit 14: Measurement: Capacity and Weight(10 lessons/11 instructional days)

Big Ideas:

Measurement is the process of repeatedly using a unit over a quantity to determine how much you have.

High-Priority Master Objectives:

Master Objectives:

Identify the appropriate tools for measuring liquid volume.
Identify the appropriate metric and English units for measuring liquid volume.
Estimate and measure liquid volume to the nearest liter.
Estimate and measure liquid volume to the nearest cup.
Identify the appropriate tools for measuring the weight of an object.
Identify the appropriate metric and English units for measuring the weight of an object.
Estimate and measure the weight of an object to the nearest gram.
Estimate and measure the weight of an object to the nearest ounce.
Write a simple unit conversion, such as inches to feet, as an expression or an equation.
Use a simple unit conversion, such as centimeters to meters, to solve a problem.
Solve a unit-conversion story problem by using multiplication or division.

Standards:

This unit meets California Content Standards:

Gr. 3: Measurement and Geometry

1.1: Choose the appropriate tools and units (metric and U.S.) and estimate and measure the length, liquid volume, and weight/mass of given objects; and 1.4: Carry out simple unit conversions within a system of measurement (e.g., centimeters and meters, hours and minutes).

Gr. 3: Algebra and Functions

1.4: Express simple unit conversions in symbolic form (e.g., inches = _____feet × 12).

Unit 15: Mathematical Reasoning(10 lessons/14 instructional days)