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# Math

K-8

Course Outline

Kindergarten:

Course Overview

This research-based course focuses on computational fluency, conceptual understanding, and problem-solving. The engaging course features new graphics, learning tools, and games; adaptive activities that help struggling students master concepts and skills before moving on; and more support for Learning Coaches to guide their students to success. The course introduces Kindergarten students to numbers through 30. Students learn through reading, writing, counting, comparing, ordering, adding, and subtracting. They experience problem solving and encounter early concepts in place value, time, length, weight, and capacity. They learn to gather and display simple data. Students also study two- and three-dimensional figures—they identify, sort, study patterns, and relate mathematical figures to objects within their environment.

Course Outline

SEMESTER 1

Unit 1: Shapes and Sorting

This unit focuses on describing, sorting, and classifying objects according to attributes or features. Students investigate the attributes of geometric shapes, such as circles, triangles, squares, and rectangles. They also use everyday objects, such as beads, stuffed animals, or types of fruit to improve their skills.

• Identify Plane Figures

• Circle, Triangle, Square, Rectangle

• Compare Shapes

• Compare Colors

• Compare Sizes

• Sizes, Shapes, and Colors

• Sort by Color

• Sort by Shape

• Sort by Size

• Sorting Different Ways

Unit 2: Shapes and Patterns

Students identify which object from a group does not belong according to the color, shape, or size of the objects. Students also learn what a pattern and a pattern core are and view different pattern types. Students begin using the letters A, B, and C to describe pattern rules AB, ABB, AAAB, AAB, ABB, ABC, and ABCC. Students identify and extend these patterns with attribute blocks and other objects.

• Which Object Is Different?

• AB and ABB Patterns

• AAAB and AAB Patterns

• ABCC and ABC Patterns

Unit 3: Numbers Through 5 and Plane Figures

Students begin by counting and grouping objects into groups of up to five, and then learn to write the numerals 1 through 5. They review triangles, squares, rectangles, and circles; learn to identify and count sides and corners—and find out that circles do not have sides or corners.

• Count Through 5

• Count and Show 0 Through 5

• Write Numerals Through 5

• Sides of a Shape

• Corners of a Shape

• Sides and Corners of Shapes

Unit 4: Numbers Through 10

Students begin by hearing the counting sequence of numbers from 1 through 10, and then they count aloud on their own. They represent quantities through 10 by using objects and drawings. Students then move to counting sets of 10 or fewer objects, learning that they can count the objects in any order if they count each item exactly one time. Students read the numbers 1 through 10 to prepare them for writing the numbers. By watching a virtual pencil, and practicing with pencil and paper, they learn to form the numbers themselves. Students learn to compare and order groups of objects, learning that numbers with greater values describe groups with more objects than numbers with lesser values. They move to comparing and ordering numbers that describe groups of objects.

• Count Through 10

• Show an Amount Through 10

• Represent Amounts

• Count Aloud Through 10

• Show Amounts in Different Ways

• Write Numerals 1 Through 10

• More, Fewer, and Equal

• Compare and Order Groups

• Describe and Order Groups by Number

• Write Numbers to Describe Groups

Unit 5: Calendar and Time

This unit is a review of calendars and time lessons taught throughout the first quarter. In the online Calendar/Time activities, students have learned about calendar concepts such as the names and numbers of days in a week; the notion of yesterday, today, and tomorrow; how to find a specific date on a calendar; and how many months are in a year. Students have also learned about concepts of time by studying the hours and minutes on a clock; parts of the day such as morning, afternoon, and evening; and the typical times in a day when certain activities occur.

• Understanding Concepts of Time

Unit 6: Data and Graphs

Students learn how to compare groups of objects to determine which group has more or fewer objects, and that greater numbers are used to describe groups with more objects. Students learn to collect data and represent that data with objects, pictures, and picture graphs. They pose questions, collect data, record the results, compare, and answer questions. Students also learn to compare and answer questions about data in graphs that they have not prepared themselves.

• Collect Data and Pose Questions

• Ways to Show Data

• Compare Data in a Picture Graph

• Interpret Picture Graphs

• Analyze Data in Picture Graphs

Unit 7: Numbers Through 20

Students begin by counting groups of up to 20 objects, learning that they can count in any order if each item is counted once. They use models, drawings, and finally numerals to represent groups of up to 20 objects. Students then compare groups having 20 or fewer objects to determine which has more, fewer, the most, or the fewest objects. They compare numbers from 1 through 20 to determine which is more or greater, and which is lesser or fewer. They learn to write the numerals from 1 through 20.

• Count Aloud Through 20

• Represent an Amount Through 20

• Count Through 20

• Show Amounts Through 20

• Compare Sets Through 20

• Write Numerals Through 20

• Compare Numbers and Sets Through 20

• Write Numerals from 1 Through 20

Students learn the meaning of addition by combining two groups of objects to find the total. By experimenting with groups of objects, they learn that changing the order in which the numbers are added does not change the sum. Students then learn to add with sums through 20 by using number lines, models, and sketches. They also learn to add by counting on by 1 and by 2 from a number.

• Count On

Unit 9: Problem Solving With Addition

Students use concrete objects to solve addition story problems that involve combining groups and explain how they are solving them. They move on to problems in which one of the groups is unknown (missing addend problems), and to checking answers to addition word problems. Students also learn how to estimate to find a sum.

• Combine to Find Totals

• Recognize Combine Problems

• Missing Parts Problems

• Estimate Sums Through 20

• Check the Accuracy of Calculations

SEMESTER 2

Unit 10: Introduction to Subtraction

In this unit, students are introduced to subtraction as taking away objects from a group of objects. They learn to take away objects and to tell how many are left, and then use sketches and countable objects to model subtraction problems and story problems. Students learn that through subtraction, they can also find out the amount of a mystery addend in an addition problem. Students learn to apply their knowledge by using benchmarks of 5, 10, 15, and 20 to make reasonable estimates for solutions to subtraction problems. They learn that counting principles and numbers can be used to solve addition and subtraction problems and use models or sketches to check the accuracy of their solutions to subtraction word problems.

• Take Away to Subtract

• Subtraction as Taking Away

• Subtract with Objects

• Model Subtraction

• Subtract with Pictures

• Estimate and Check Differences

Unit 11: Problem Solving with Subtraction

Students learn to recognize and solve subtraction story problems by using concrete objects and sketches. They create their own story problems, and then solve those problems using models or sketches. They apply that knowledge to make reasonable estimates for solutions and check the accuracy of subtraction calculations.

• Model Subtraction Stories

• Sketch Subtraction Stories

• Take-Away Stories

• Compare Take-Away and Combine

• Recognize and Solve Problems

• Make Estimates and Check Answers

Unit 12: Subtraction as Comparison

Students use models and sketches to solve comparison subtraction problems using one-to-one-correspondence. They use pairs of numbers to create addition and subtraction problems, exploring the differences between comparing, combining, and take-away problems.

• Compare and Subtract

• Sketch Subtraction Problems

• Take Away, Combine, and Compare

• Compare to Subtract

• Subtraction as Comparing

• Comparison Subtraction

Unit 13: Comparison Subtraction: Story Problems

In this unit, students learn to use concrete objects to explain how to solve addition and subtraction problems involving numbers up to 10. Students also learn to recognize and solve word problems involving numbers up to 10 in which two quantities are compared using addition or subtraction. Students then make estimates for solutions to subtraction problems and check the accuracy of those solutions.

• What’s the Difference?

• Add and Subtract Story Problems

• Compare Quantities to 10

• Compare: More or Fewer?

• Compare in Everyday Situations

• Estimate and Check Subtraction

Unit 14: Add or Subtract: Problem Solving

Students learn to recognize and solve a variety of story problems in which two quantities are combined, two quantities are compared, or one quantity changes through addition or subtraction.

• Different Types of Problems

• Combine and Change Problems

• Compare and Combine Problems

• Change and Compare Problems

• Add or Subtract: More Exploration

Unit 15: Measurement

Students begin by finding lengths of objects using various nonstandard units, such as paperclips or beads. Then they explore measuring and comparing length, weight, and capacity.

• Measure Objects Introduction

• Compare Length Introduction

• Compare Weight Introduction

• Compare Capacity Introduction

Unit 16: Numbers Through 30

Students extend their counting ability to be able to count through 30 objects in a group. They learn to represent amounts through 30 with objects and with sketches, and to compare groups of objects. Students determine which of two groups has more or fewer objects. They then can tell which of three groups has the fewest or the most objects. Finally, students learn to write numbers from 1 through 30.

• Count and Show Numbers Through 30

• Count Objects Through 30

• Represent Amounts Through 30

• Compare Groups Through 30

• Groups in a Picture Graph

• Write Numerals Through 30

• Compare Groups and Numbers

• Write Numerals From 1 Through 30

Unit 17: Solid Figures

Students begin by reviewing plane figures and then are introduced to some solid figures: cubes, cones, and spheres. They learn to identify these figures and to recognize them in everyday objects such as boxes and marbles. They compare the attributes of the various solids, such as the number of corners, the roundness, the color, or the size. Students identify which solid in a set of solids does not belong according to color, size, or shape.

• Identify Solid Figures

• More Exploration with Identifying Solid Figure

• Compare Solid Figures by Shape or Size

• More Exploration with Attributes of Solid Figures

• Sort Solid Figures

• Put Together Shapes

• Combine Shapes: More Exploration

• Take Apart Shapes

• More Exploration with Taking Apart Plane Figures

Course Overview

This research-based course focuses on computational fluency, conceptual understanding, and problem-solving. The engaging course features new graphics, learning tools, and games; adaptive activities that help struggling students master concepts and skills before moving on; and more support for Learning Coaches to guide their students to success. This course for students in Grade 1 extends their work with place value to numbers through 100, emphasizing fluency of addition and subtraction facts, and focusing on number sentences and problem solving with addition and subtraction. Students begin work with money, telling time, ordering events, and measuring length, weight, and capacity with non-standard units. Students identify attributes of geometric figures and extend their work with patterns and data, including representing and comparing data.

Course Outline

SEMESTER 1

Unit 1: Read, Write, Count, and Compare Numbers

This unit focuses on counting, comparing, and ordering numbers. Students explore reading and writing whole numbers, which prepares them to later add and subtract numbers. The skill of skip counting by 2s, 5s, and 10s assist students in comparing and ordering numbers. Skip counting also serves as an important introduction to number patterns as a first step to learning algebraic concepts.

• Numbers Through 50

• Write Numerals Through 50

• Count by 10s and 5s Through 50

• Count by 2s Through 50

• Numbers Through 100

• Write Numerals Through 100

• Count by 10s and 5s Through 100

• Count by 2s Through 100

• Compare Numbers Through 100

• Order Numbers Through 100

Unit 2: Time and Position

Students learn about the hour hand and how to tell time to the nearest hour. Then they learn about the minute hand and how to tell time to the nearest half hour. They learn to identify when it is exactly, a little before, or a little after the hour and the half hour, and to draw these scenarios on a clock. Then students relate time to events and compare events by how long they take to complete and by the order in which they occur. Students learn to compare, arrange, and describe the position of objects using words such as up, down, behind, in front of, next to, to the left of and to the right of.

• Time to the Nearest Hour

• Time to the Nearest Half Hour

• Arrange and Describe Position

• Use Direction Words

The concept of part-part-whole is foundational to many topics in math from addition and subtraction to measurement and geometry. Students learn that the meaning of addition is putting together groups of objects, and that the order in which groups are added does not affect the sum. They use sketches, counting chips, and snap blocks to model addition problems, learning that addition is combining, or putting together, groups of objects. They are introduced to the plus sign (+) and the equals sign (=) and learn how to write number sentences using numbers, the plus sign, and the equals sign.

• The Plus Symbol

• The Equals Symbol

• Number Sentences: The Equals Symbol

Unit 4: Addition Facts for Sums Through 12

Students begin by learning different ways to add numbers to make 8, and what happens when adding 0. Students then learn the addition facts for sums through 8, and then sums through 12. They use online and offline flash cards and other activities to help them develop automatic recall, and they complete a chart to document their progress.

• Facts Through 8

• Sums Through 8

• Facts Through 12

• Sums Through 12

Unit 5: Addition Facts for Sums Through 20

Students review addition facts with sums through 12 and learn the remaining facts through 16. Then they continue to learn addition facts through 20. Students use online and offline flash cards and other materials and tools to help them develop automatic recall. They complete a chart to document their progress.

• Facts Through 16

• Sums Through 16

• Facts Through 20

• Sums Through 20

Students learn to find one more than and ten more than another number. They practice finding one more and ten more using hundred charts and number lines. This leads to learning how to add two numbers by counting on. Students use counting chips and number lines to assist with counting on to add. They learn that counting on from the greater number is easier than counting on from the lesser number. Students are introduced to the associative property, learning that they can group three numbers in different ways to make it easier to find their sum. Finally, students use the various addition strategies they have learned—counting on, using doubles, using memorized facts and similar facts, and grouping to solve addition problems with sums through 30.

• One More, 10 More

Students identify and practice showing numbers in various ways—with models, sketches, and with addition expressions. They then represent equivalent forms of the same number in multiple ways. Students use a balance to help identify equivalent forms of a number, including two addition expressions that are equal. Then they find a missing number in an addition sentence using the balance. They identify missing sums and missing addends, using the balance and snap cubes as tools. They learn that you can add two numbers in any order and the sum will not change. They explore this property with balance snap cubes, then use this knowledge to find missing addends in number sentences with addition expressions on each side.

• Different Forms of Numbers

• Ways to Show Numbers

• Missing Numbers in Addition Sentences

Unit 8: Introduction to Subtraction

Students learn that subtraction means to take away, and they demonstrate the meaning of subtraction by taking away objects. They learn that when they subtract, the number that is left is the difference. They learn the meaning of the minus symbol (-) and review the meaning of the equals symbol (=) so they can read and write subtraction sentences. Students explore the relationship between addition and subtraction, learning that they are opposite operations. They demonstrate the opposite operations with objects and drawings. They learn that subtraction is not commutative, and that subtracting zero from a given number results in a difference of the original number. Students then learn about using subtraction in comparing numbers, which allows them to find how much greater or lesser a number is than another number. They use pairing, modeling, and drawings to compare numbers.

• Understand Subtraction

• The Minus Symbol

• Equal Expressions

• More Equal Expressions

• Put Together, Take Away

• Order and Zero in Subtraction

• Subtract to Compare

• Use Pairs to Subtract

Unit 9: Subtraction Facts Through 20

Students explore different strategies to solve subtraction problems with minuends through 20, including using models, counting back, using patterns, and using addition facts and fact families. They practice the subtraction facts, working toward automatic recall: first through 12, then through 16, and finally through 20.

• Subtraction Facts Through 8

• Subtraction Facts Through 12

• Count Back Subtraction Facts

• Subtraction Facts Through 16

• Facts Using Subtraction

• Subtraction through 20

• All the Subtraction Facts

Unit 10: Subtraction Strategies

Students learn various strategies to use in solving subtraction problems. They start by learning how to find one less than and 10 less than a given number. Then students use hundred charts and number lines to count back, use counting chips to model subtraction, and use facts that they know to help them find differences.

• One Less, Ten Less

• Counting Back and Other Strategies

• Use Strategies to Subtract

Unit 11: Semester Review and Checkpoint

SEMESTER 2

Unit 12: Subtraction Number Sentences

Students model and draw the same number in different ways. They also write different expressions for the same number. They use various strategies to find the missing number in a subtraction number sentence. The missing number may be a difference or a subtrahend.

• Same Number Different Ways

• Represent Numbers Different Ways

• Missing Parts in Subtraction Sentences

• Subtract with Missing Numbers

Unit 13: Money and Measurement

Students recognize, identify by name, and learn the value of pennies, nickel, dimes, and quarters. They identify how many of a named coin are in a group and learn to find the value of a group of one type of coin. They identify groups of coins that have the same value. Although the coin names are shown on the screen, students are required to read the coin names. Students learn to measure with nonstandard units and to compare length, weight, capacity, and volume.

• Coins

• Identify Coins

• Equal Money Amounts

• Measure and Compare Length

• Weight

• Capacity and Volume

Unit 14: Place Value, Addition, and Subtraction

In this unit, students learn about place value through 100. They learn how to count and group objects in tens and ones, how to estimate quantities and number of objects, and how to use base-10 blocks to model and write a two-digit number as tens and ones. Students model two-digit numbers different ways as an introduction to regrouping tens as ones and ones as tens. Students apply regrouping to add and subtract with sums and minuends through 100. Students learn several addition and subtraction strategies to help them find sums and differences of two-digit numbers.

• Tens, Ones, and Estimation

• Place Value

• Represent Numbers

• Place Value for Numbers

• Model Numbers Different Ways

• Addition with Sums through 100

• Use Objects to Subtract

• Use Sketches to Subtract

• Subtraction with Regrouping

• More Subtraction with Regrouping

• Different Ways to Subtract

Unit 15: Add or Subtract: Problem Solving

Students solve story problems about the number of stars on the US flag. They learn that as we added states to our country, we added stars to the flag. They use base-10 blocks, models, or sketches to solve addition and subtraction story problems. Students work with the concept of parts and total, how to recognize a problem in which amounts are combined, and that they can use subtraction to solve a problem in which one part and the total are given and the other part is missing. Students solve combine problems in which the total is missing as well as problems in which a part is missing.

• Adding Stars to the Flag

• Compare and Change Stories

• Story Problems to 100

• Part, Part, Total Problems

• Problems with Parts and Total

• More Exploration with Combine Story Problems

• Change Problems

• Missing Numbers in Story Problems

• More Exploration with Solving Change Word Problems

• More Exploration with Solving Change Word Problems

• Comparison Story Problems

• Story Problems that Compare

• More Exploration with Compare Story Problems

Unit 16: Add or Subtract: More Problem Solving

Students check the accuracy of an answer to a word problem, look at two-word problems to see how they are similar, and use a problem and its solution to solve a similar problem. They learn about creating story problems that represent number sentences.

• Equalize Story Problems

• Make Them Equal

• More Story Problems

• Explore Number Sentences

• Number Sentences

• Write and Solve Number Sentences

• Explain Solution Strategies

• Justify Selected Procedures

• Justify Different Solutions

• Story Problems That Are Alike

• Write Story Problems

Unit 17: Geometric Figures, Data, and Attributes

This unit focuses on shape, color, size, and number patterns. Students identify and describe plane shapes and compare them to the faces of solid figures. They demonstrate taking apart and putting together shapes to make other shapes. They learn how to recognize and describe the pattern core and then extend the pattern. Students identify what the shapes in a given group have in common, sort objects by common attributes, and represent and compare data in a tally chart.

• Plane Figures

• Put Together and Take Apart Shapes

• Group Shapes Different Ways

• Classify Objects and Data

• Patterns

• Tally Charts and Bar Graphs

• Data in Pictures and Graphs

Unit 18: Semester Review and Checkpoint

Course Overview

This research-based course focuses on computational fluency, conceptual understanding, and problem-solving. The engaging course features new graphics, learning tools, and games; adaptive activities that help struggling students master concepts and skills before moving on; and more support for Learning Coaches to guide their students to success. This course for students in Grade 2 focuses primarily on number concepts, place value, and addition and subtraction of numbers through 1,000. Special emphasis is given to problem solving, inverse operations, properties of operations, decomposition of numbers, and mental math. Students study money, time, and measurement; geometric figures; analyzing and displaying data with new representations; and determining the range and mode of data. Early concepts about multiplication, division, and fractions are introduced.

Course Outline

SEMESTER 1

Unit 1: Numbers Through 500

In this unit, students investigate three different ways to represent numbers: concrete models, numerals, and number words. Students use models to build numbers through 500 while focusing on counting, reading, and writing numbers. For example, they use ones cubes, tens rods, and hundreds flats to model numbers and demonstrate their understanding of place value. From this concrete foundation, students move more easily into the abstract representations of numerals and number words.

• Count Aloud Through 500

• Read Whole Numbers Through 500

• Write Numerals Through 500

• Identify Place Value

• Use Expanded Form: Numbers Through 500

• Place Value and Regrouping

• Compare Numbers Through 500

• Comparing and Ordering

• Order Whole Numbers Through 500

• Read Number Words Through 500

Unit 2: Time and Money

Students learn how to tell when the time is exactly or about a quarter past, half past, or quarter ‘til the hour. They learn about relationships between units of time and how to compare them using an equivalency chart. They learn the following units of time relationships: 60 seconds = 1 minute, 60 minutes = 1 hour, 24 hours = 1 day, 7 days = 1 week, 52 weeks = 1 year, and 12 months = 1 year. They then learn about a.m. and p.m. and how to find elapsed time in hours. Students learn to find the value of groups of coins by counting on from the coin with the greatest value to the coin with the least value. They use the same technique for counting bills, up to and including the twenty-dollar bill. They are introduced to the cent sign, the dollar sign, and using a decimal in writing money amounts. They practice counting groups of coins and bills and writing the amount using the correct notation. Students practice trading coins or bills of lesser value for coins or bills of greater value to use the fewest number of coins or bills.

• Time to the Nearest Quarter Hour

• Time Relationships

• Elapsed Time

• Find the Value of Coins or Bills

• Dollar and Cent Symbols for Money

• Decimal Notation for Money

• Fewest Bills and Coins

• How Much Money?

Unit 3: Addition, Subtraction, and Number Composition

Students already know how to add and subtract numbers through 100. In this unit, they learn how to add and subtract with numbers through 500, strategies for addition and subtraction, and how to identify and correct errors in addition and subtraction. They begin by using base-10 blocks, drawings, and place-value charts to solve addition and subtraction problems with sums or minuends up through 500 with and without regrouping. They learn the meaning of the equals sign, and write various equivalent expressions, including exploring fact families. They learn how numbers are composed of other numbers, and how to decompose numbers in various ways. They use this knowledge to solve addition and subtraction problems mentally.

• Finding the Difference

• Subtraction and the Equals Symbol

• Decompose Numbers

• Make and Break Numbers

• Break Up Numbers

• Breaking Numbers to Subtract

• Decompose to Subtract

• Choose Friendly Numbers

Unit 4: Inverse Operations: Add and Subtract

Students observe and use models to explore how addition and subtraction are related. They use fact triangles to show the inverse relationship between addition and subtraction. Later, they will use that knowledge to solve missing addend or missing subtrahend problems. Students learn strategies for using mental math to calculate sums and differences of two-digit numbers and explain which strategies they used. They explore strategies for computing sums and differences of numbers through 500. They explain which strategies they used in their computations.

• Opposite Operations: + and -

• Mental Math: Addition and Subtraction

• Strategies to Add and Subtract Through 500

• Subtraction Strategies Up Through 500

• Addition and Subtraction Are Related

Unit 5: Measurement

After measuring length with nonstandard units, students are introduced to a ruler as a tool for measuring length with inches and centimeters. They learn to use the ruler and other objects (1-inch tiles and centimeter cubes) to measure the length of objects. Then they learn to estimate measurements and to recognize when a measurement estimate is reasonable. They use different measurement units (nonstandard and standard) to compare the length of objects, finding that measurements should be in the same unit for easy comparing. Students also learn to add and subtract measurements of the same unit. They learn about capacity and how to use a standard measuring cup to measure and compare volumes of objects.

• Inches

• Centimeters

• Estimate Length

• Compare Measurements

• Capacity

Unit 6: Add or Subtract: Problem Solving

Students use models and sketches to solve situations that involve addition. Using models to help represent regrouping, they solve problems in which they are combining groups, including some with missing addends. They move on to solve subtraction problems using models and sketches, and then learn to write number sentences. The problems include combining, comparing, take away, and change problems.

• Subtraction Problem Solving

• Modeling Story Problems

• Problem Solving

• Problem Solving with Combining

• Problem Solving with Change

• Solve Change Story Problems

• Compare to Solve Story Problems

• Compare Amounts to Solve Problems

• Make Equal Amounts to Solve Problems

• Equalize Story Problems

Unit 7: Problem Solving: Reason and Connect

Students learn about addition and subtraction story problems. They analyze a problem to check for errors and determine if the answer is correct. They explain and justify solutions and learn that there can be more than one way to find the answer. They compare story problems and learn to recognize story problems that are solved the same way. They also write and solve their own story problems.

• Story Problems

• More Story Problems

• Explain Problem Solutions

• Justify Procedures Selected

• Justify Solutions

• Create Story Problems

• Make Your Own Story Problems

• Similar Story Problems

• Classify Story Problems

• Different Kinds of Problems

Unit 8: Semester Review and Checkpoint

SEMESTER 2

Unit 9: Numbers Through 1,000

This unit focuses on counting, representing, comparing, and ordering numbers from 500 through 1,000. Although students can count aloud and write numbers, they now extend their understanding by modeling greater numbers with base-10 blocks and learning to read and write number words, using their number and symbol card deck. Students learn to see the connections between the number 325, the number words three hundred twenty-five, and the representation of the number in expanded form (325 = 3 hundreds + 2 tens + 5 ones or 325 = 300 + 20 + 5). As students work with these representations, they develop a deeper understanding of how our base-10 number system works. Fully understanding the number system builds confidence and skills that will help them solve problems involving greater numbers. The understanding of multiple representations of numbers and place value also leads students to create strategies for comparing numbers through 1,000 and for properly using the greater-than (>), less-than (<), and equal-to (=) symbols.

• Count Aloud Through 1,000

• Write Number Words Through 1,000

• Represent Numbers Through 1,000

• Work With Numbers Through 1,000

• Model Numbers Through 1,000

• Place Value Through 1,000

• Standard to Expanded Form

• Expanded to Standard Form

• Compare and Order Numbers

Unit 10: Plane and Solid Figures

First, students learn to identify and describe plane figures by the number of sides and vertices the figures have, and to describe solid figures by the number and shape of faces. Students then learn that shapes can be put together and taken apart to form other geometric shapes.

• Plane Figures

• Solid Figures

• Build and Take Apart Shapes

Unit 11: Add or Subtract Numbers Through 1,000

Students extend their knowledge of adding and subtracting with sums and minuends up through 1,000. They use addition and subtraction in working with problems that involve combining two groups, a change in a quantity, comparing two groups, and problems in which two groups must be equalized. They learn to solve the problems, write number sentences for the problems, find similarities between problems, and check answers to word problems.

• Sums and Differences

• Story Problems Through 1,000

• Compare and Equalize Story Problems

• Write Sentences for Story Problems

• Identify Similarities and Differences

• Check Story Problem Solutions

• Explain Operations to Solve Problems

• Which is the Addition/Subtraction Problem?

Unit 12: Multiplication and Number Patterns

Students learn about multiplication. They explore arrays as a way to model multiplication. They relate multiplication to repeated addition and equal groups and use these to solve multiplication computations. They learn about number patterns, pattern rules involving multiplication and addition, and applying the rule to extend patterns. They use drawings, models, and symbols to represent multiplication. Lastly, they explore the 2s, 5s, and 10s facts and work on automatic recall of these basic multiplication facts.

• Model Multiplication with Arrays

• Different Types of Problems

• Linear Patterns

• Number Patterns

• Represent Multiplication

• Multiply By 2

• Multiplication: 2s Facts

• Multiply By 10

• Multiplication: 10s Facts

• Multiply By 5

• Multiplication: 5s Facts

Unit 13: Multiplication and Addition Properties

Students learn the Commutative and Associative Properties of addition and multiplication. They also learn how to multiply with 0 and 1. Students use properties to make problems easier to solve and learn how to use the properties to check calculations.

• Multiplication Order and Rules

• The Commutative Property

• The Associative Property

• Use Properties

Unit 14: Introduction to Division

Students explore division. They solve division problems using repeated subtraction and equal sharing. Students first model repeated subtraction with counting chips and number lines and learn how to record repeated subtraction. Then they model equal sharing and solve division word problems. Lastly, they learn about the division sign, using symbols to record division, and division with remainders.

• Division as Repeated Subtraction

• Division with Repeated Subtraction

• Division with Equal Sharing

• Equal Share Division

• Represent Division

• Remainders in Division

Unit 15: Data Representations and Analysis

Students explore different ways to represent data and analyze data. They make horizontal and vertical bar graphs and learn how to read them. Students show the same set of data multiple ways, in charts, tables, and graphs. They ask and answer questions and solve addition and subtraction problems using data from tally charts, picture graphs, and bar graphs. Lastly, students find the range and mode of data sets.

• Represent Data

• Data Questions

• Use Data to Solve Problems

• Range and Mode of Data Sets

Unit 16: Introduction to Fractions

Students learn that they can write fractions to describe parts of a whole and parts of a set and how to create models that represent fractions, including unit fractions and fractions equal to 1. Students also learn how to compare unit fractions and to identify a few simple equivalent fractions. They learn that fractions are numbers that can be plotted on the number line.

• Fractional Parts of a Whole

• Fractional Parts of a Group

• Fractional Relationships

• Fractional Parts and One Whole

• Fractions and Whole Numbers

• Fractions and Mixed Numbers

• Fractions

• Equivalent Fractions

Unit 17: Semester Review and Checkpoint

Course Overview

This research-based course focuses on computational fluency, conceptual understanding, and problem solving. This engaging course features new graphics, learning tools, and games; adaptive activities that help struggling students master concepts and skills before moving on; and more support for Learning Coaches to guide their students to success. This course emphasizes conceptual understanding of the mathematical operations: addition, subtraction, multiplication, and division. Students make connections between the operations, as well as practice through problem solving, to achieve fluency. The use of problem solving and representing problem situations with equations, which include symbols for unknown values, introduces algebraic thinking. The course addresses fractions through multiple representations, as well as solving real-world problems, which gives students the ability to connect the use of fractions with problem situations in a way that makes sense and creates deeper understanding. The courses address geometry and measurement through introductory work on perimeter, area, and attributes of two-dimensional geometric figures, and applying measuring techniques to solving problems involving time, length, capacity, and mass. Throughout the course, problem solving connects individual mathematical skills and concepts in a useful and in-depth way. This course includes standards-based tasks, digital literacy skills, and assessment questions.

Course Outline

SEMESTER 1

Unit 1: Whole Number Sense

Students build and use place value understanding to identify whole numbers, order whole numbers by using the symbols <, =, >, compare whole numbers, and round numbers.

• Numbers Through 1,000

• Compare and Order Numbers Through 1,000

• Round Numbers Through 1,000

• Core Focus

Unit 2: Whole Number Addition and Subtraction

Students learn how addition and subtraction affect whole numbers and how to determine the sum or difference of two whole numbers. They practice solving both one- and two-step story problems in which two quantities are combined, quantities change by addition or subtraction, two quantities are compared by the use of addition or subtraction, and one quantity must be changed to equal another quantity.

• Effects of Addition and Subtraction

• Combine and Change Problems

• Compare and Equalize Story Problems

• Core Focus

Unit 3: Whole Number Multiplication Sense

Students use objects or sketches to solve multiplication problems. They use models to explain multiplication as repeated addition of the same quantity. They learn how to explain and apply the commutative, associative, and zero properties of multiplication. They demonstrate automatic recall of multiplication facts and an understanding of how multiplication affects whole numbers. They learn how to apply the multiplication property of 1.

• Model and Explain Multiplication

• Area Models for Multiplication (parts A, B)

• Understand Multiplication

• Commutative Property of Multiplication

• Multiplication Facts

• Multiplication Facts (parts A–D)

• Associative Property

• Core Focus

Unit 4: Whole Number Multiplication

Students use objects or sketches to solve multiplication story problems. They solve multiplication problems involving a multidigit factor and a one-digit factor. They use multiplication to solve story problems that involve equal groups and equal measures and learn how to create story problems that can be represented by a multiplication number sentence.

• Multiplication Story Problems

• Multiply Multidigit by 1-Digit Numbers

• Multiply Equal Groups (parts A, B)

• Multiplication with Equal Measures

• Write Multiplication Stories (parts A, B)

• Core Focus

Unit 5: Whole Number Division Sense

Students use objects or sketches to solve division problems. They learn that division is repeated subtraction and the sharing of a quantity into equal groups. They learn the meaning of the ÷ symbol, the division property of 1, division by zero is undefined, and the inverse relationship between multiplication and division. Students use objects or sketches to solve division story problems. They learn to solve division problems with a multidigit dividend, a one-digit divisor, and no remainder. They practice solving story problems that involve equal groups and equal measures.

• Model and Explain Division

• Applying Division Symbols and Rules

• Division as Sharing

• Relating Multiplication and Division

• Use Inverse Relationships

• Effects of Division

• Division Story Problems with Equal Groups and Equal Measures (parts A, B)

• Core Focus

Unit 6: Algebra Thinking

Students learn to use mathematical expressions, equations, and inequalities to represent relationships between quantities. They learn to select the appropriate symbol to show an operation or a relationship that makes a number sentence true, to determine a missing number in an equation or an inequality, and to recognize and describe a linear pattern, such as counting by 5s or multiplying 5 times a number to reach 100, by its rule. They extend linear patterns and solve simple story problems that involve functions.

• Mathematical Expressions

• Expressions and Number Sentences (parts A, B)

• Expression Comparison (parts A, B)

• Missing Symbols

• Missing Values (parts A–C)

• Number Patterns

• Story Problems and Patterns (parts A, B)

• Core Focus

Unit 7: Geometry

Students learn how to identify right angles and the measure of angles greater than or less than a right angle. They learn to classify polygons according to the number of sides; the attributes of isosceles, equilateral, and right triangles; and the attributes of parallelograms, rectangles, and squares.

• Right Angles and Other Angles

• Identify and Classify Polygons

• Triangles

• Parallelograms

• Core Focus

Unit 8: Semester Review and Checkpoint

SEMESTER 2

Unit 9: Whole Numbers and Multiple Operations

Students learn how to determine whether addition, subtraction, multiplication, or division is the appropriate operation to use to solve a story problem. They practice solving story problems involving two or more operations and using the order of operations to evaluate an expression.

• The Order of Operations

• Choose the Correct Operation (parts A, B)

• Use More Than One Operation (parts A, B)

• Core Focus

Unit 10: Fractions and Probability

Students learn about fractions as the relationship of a part to a whole, and as a rational number on the number line. They learn to write the fraction represented by a drawing that shows parts of a whole. They compare and order unit fractions and use objects or sketches to solve simple story problem involving addition or subtraction of fractions. They solve and simplify addition and subtraction problems involving fractions with like denominators and learn that multiple simple fractions can represent the same quantity. They learn to identify whether specific events are certain, likely, unlikely, or impossible; record the possible outcomes for a simple event; summarize and display the results of a probability experiment; represent data on scaled graphs; and use the results of a probability experiment to predict future events.

• Represent and Name Fractions (parts A, B)

• Equivalent Fractions

• Compare and Order Fractions (parts A, B)

• Probability

• Identify, Record, and Display Outcomes

• Drawing Scaled Graphs (parts A, B)

• Interpreting Scaled Graphs

• Use Data to Make Predictions

• Core Focus

Unit 11: Measurement: Length and Time

Students learn the appropriate tools and metric and English units for measuring the length of objects. They practice estimating and measuring the length of an object to the nearest centimeter, 1/2 inch, and 1/4 inch. They learn to tell time to the nearest minute, determine elapsed time to the nearest minute, and solve word problems involving time intervals.

• Tools and Units for Measuring Length

• Estimate and Measure Centimeters

• Estimate and Measure Inches (parts A, B)

• Display Measurement Data in Line Plots

• Tell Time in Minutes

• Determine Elapsed Time in Minutes

• Measuring and Displaying Time Intervals

• Core Focus

Unit 12: Measurement: Capacity and Weight

Students learn the appropriate tools and metric and English units for measuring liquid volume and weight. They practice estimating and measuring liquid volume to the nearest liter and cup, and weight of an object to the nearest gram and ounce. They write simple unit conversions as expressions and equations and use simple unit conversions, such as centimeters to meters, to solve problems.

• Capacity

• Measure to the Nearest Liter

• English Units of Capacity

• Measure in English and Metric Units

• Measure in Grams

• Measure Weight in Ounces and Pounds

• Unit Conversions

• Measurement Conversions (parts A, B)

• Core Focus

Unit 13: Mathematical Reasoning

Students practice analyzing story problems by identifying the question, recognizing relevant information, and developing a solution strategy. They learn how to break a multistep story problem into simpler steps, predict solutions to story problems, and apply strategies and results from simpler problems to similar or more complex problems. They practice mathematical reasoning in story problems by using words, numbers, symbols, charts, graphs, tables, diagrams, and models; learn how to express solutions with appropriate mathematical notation, terms, and accurate language; and check the accuracy of a solution to a story problem.

• Analyze Story Problems (parts A–C)

• Understand Multistep Problems

• Estimate to Predict Solutions

• Strategies to Solve Complex Problems

• Story Problem Reasoning (parts A, B)

• Exact and Approximate Solutions

• Core Focus

Unit 14: Perimeter and Area

Students learn to determine the perimeter of a polygon with whole-number side lengths. They practice using multiplication and division to solve story problems involving rectangular area. They learn to estimate or determine the number of squares or cubes required to cover the area of a solid figure.

• Find the Perimeter of Objects

• Finding the Missing Side Length

• Practical Perimeter Problems with Missing Length

• Rectangular Area (parts A, B)

• Core Focus

Unit 15: Semester Review and Checkpoint

Course Overview

This research-based course focuses on computational fluency, conceptual understanding, and problem solving. This engaging course features new graphics, learning tools, and games; adaptive activities that help struggling students master concepts and skills before moving on; and more support for Learning Coaches to guide their students to success. This course continues to emphasize the understanding of numbers and operations. There is a focus on computational fluency in addition, subtraction, multiplication, and division of whole numbers. The course enhances fluency of operations through application in the solving of measurement, geometry, and data analysis problems using mathematical problem-solving techniques. Students make connections between fraction and decimal representation of numbers. Students study equivalences and relationships between fractions and decimals on the number line and with other models. Students develop algebraic thinking as they work with variables and formulas to solve multistep word problems and as they study patterns and rules. They extend their knowledge of geometry through more in-depth classification of shapes and work with lines, angles, and rotations and the connection of geometric concepts to measurement and problem solving. This course includes standards-based tasks, digital literacy skills, and assessment questions.

Course Outline

SEMESTER 1

Unit 1: Whole Number Sense

Students learn to read and write numerals and number words and identify the place value in whole numbers through 1,000,000. They learn to compare, order, and round numbers.

• Place Value Through 1,000,000

• Numbers Through 1,000,000

• Expanded Form Through 1,000,000

• Compare and Order Greater Numbers (parts A, B)

• Use Boundary Numbers for Rounding

• Core Focus

Unit 2: Whole Number Operations

Students learn to estimate sums and differences on a number line and apply standard step-by-step approaches for addition, subtraction, multiplication, and division. They use inverse relationships to simplify computations and check their results. They learn to identify prime numbers and practice plotting whole numbers on a number line. They use models to explain multiplication as repeated addition of the same quantity and division as repeated subtraction. They use concrete objects or sketches of arrays to model multiplication problems. They learn to find all factor pairs of a whole number.

• Estimate Sums and Differences (parts A, B)

• Subtract Whole Numbers

• Multiply by 2-Digit Numbers (parts A, B)

• Model and Explain Multiplication

• Area Models for Multiplication (parts A, B)

• Multiply Multidigit by 1-Digit Numbers

• Model and Explain Division

• Division as Sharing

• Different Ways to Divide (parts A, B)

• Dividing with Remainders

• Divide Greater Numbers

• Prime Numbers Less Than 100

• Prime Factorization

• Core Focus

Unit 3: Applications of Operations

Students use parentheses and the order of operations to write and evaluate expressions. They learn about the distributive property and solve story problems involving whole numbers. They apply standard step-by-step approaches for multiplication and addition; use the order of operations to evaluate expressions; and determine whether addition, subtraction, multiplication, or division is the appropriate operation to use to solve a story problem.

• Order of Operations (parts A, B)

• The Distributive Property (parts A, B)

• Story Problems: Solve and Check (parts A, B)

• Core Focus

Unit 4: Lines, Angles, and Rotations

Students identify lines that are parallel, intersecting, or perpendicular. They learn about right, acute, obtuse, and straight angles and relative angle measures. They learn to identify right angles in geometric figures or everyday objects. They learn the attributes of parallelograms, rectangles, and squares. They measure angles to a whole number of degrees and sketch angles.

• Line Pairs

• Types of Angles

• Angles and Rotation

• Angles (parts A, B)

• Core Focus

Unit 5: Fraction Sense

Students learn to represent fractions with a sketch, explain why two given fractions are equivalent, and recognize and determine equivalent fractions. They practice finding fractions between two numbers, writing fractions represented by drawings that show parts of a set or parts of a whole, and identifying a few simple equivalent fractions, such as 1/2 = 2/4. They learn that fractions 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. They learn how fractions and whole numbers can be plotted on a number line.

• Fractions

• Sketch Fractions

• Different Meanings of Fractions (parts A–D)

• Explain Equivalent Fractions (parts A, B)

• Determine Equivalent Fractions (parts A, B)

• Find a Fraction (parts A, B)

• Core Focus

Unit 6: Measurement

Students learn to estimate the length of a line segment to the nearest inch or centimeter and solve measurement-conversion problems using multiplication and division. They solve story problems involving measurement.

• Estimating Lengths

• Change Measurement

• Measurements in Stories (parts A, B)

• Core Focus

Unit 7: Semester Review and Checkpoint

SEMESTER 2

Unit 8: Fraction Operations

Students use objects and sketches to solve story problems that involve addition and subtraction of fractions. They practice writing equations, simplify factors in fraction multiplication problems, and multiply and divide with fractions. They demonstrate automatic recall of multiplication facts, represent fractions with sketches, explain why two given fractions are equivalent, and find a fraction between two numbers.

• Add and Subtract Fractions (parts A, B)

• Unlike Denominators (parts A, B)

• Different Ways to Write Products

• Fraction Factors (parts A, B)

• Fraction and Whole Number Products (parts A, B)

• Fraction Division (parts A, B)

Unit 9: Decimals and Equality with Fractions

Students compare and order decimal numbers. They estimate and 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, and identify fraction and decimal-number equivalents for halves and fourths. They relate decimal numbers to fractions on a number line.

• Compare Decimals

• Add and Subtract Decimal Numbers

• Equivalent Decimals and Fractions

• Halves and Fourths

• Fractions and Decimals (parts A, B)

• Relate Decimal Numbers to Fractions (parts A, B)

• Core Focus

Unit 10: Mathematical Reasoning

Students practice analyzing story problems by identifying the question, recognizing relevant information, and developing a solution strategy. They use estimation to predict a solution to a story problem and verify the reasonableness of the calculated result. They learn to express solutions clearly and logically, answer to a specified degree of accuracy, and identify different story problems that can be solved by using the same procedures.

• Analyze Story Problems (parts A, B)

• Multistep Problems

• Estimate to Predict and Verify (parts A, B)

• Represent and Explain Story Problems

• State Solutions Clearly (parts A, B)

• Problem-Solving Strategies

• Core Focus

Unit 10: Mathematical Reasoning

Students practice analyzing story problems by identifying the question, recognizing relevant information, and developing a solution strategy. They use estimation to predict a solution to a story problem and verify the reasonableness of the calculated result. They learn to express solutions clearly and logically, answer to a specified degree of accuracy, and identify different story problems that can be solved by using the same procedures.

• Analyze Story Problems (parts A, B)

• Multistep Problems

• Estimate to Predict and Verify (parts A, B)

• Represent and Explain Story Problems

• State Solutions Clearly (parts A, B)

• Problem-Solving Strategies

• Core Focus

Unit 12: Algebra Thinking

Students learn to use symbols to stand for variables in simple expressions or equations. They learn that when equal quantities are added to or multiplied by equal quantities, the resulting quantities are equal. They practice solving for one variable in a two-variable equation when the value of the other variable is given. They learn to locate and plot points on a coordinate plane, find the length of horizontal and vertical line segments, and plot linear relationships in the first quadrant of a coordinate plane.

• Expressions and Equations

• Addition Property of Equality (parts A, B)

• Multiply by Equal Quantities (parts A, B)

• Two-Variable Equations (parts A, B)

• The Coordinate Plane

• Line Segments in the Coordinate Plane

• Linear Relationships (parts A, B)

• Core Focus

Unit 13: Perimeter and Area Formulas

Students learn how to find the perimeter of rectangles and squares. They learn to interpret and use formulas to answer questions about quantities and their relationships. They learn how to find the area of rectangles, squares, or figures that can be divided into rectangles or squares. They practice solving story problems that require finding rectangular area.

• Perimeters of Polygons

• Formulas for Perimeters (parts A, B)

• Understand Area

• Areas of Rectangular Shapes

• Formulas for Area (parts A, B)

• Area and Perimeter Story Problems (parts A, B)

• Compare Area and Perimeter

• How Many Squares Does It Take?

• Core Focus

Unit 14: Semester Review and Checkpoint

Course Overview

This research-based course focuses on computational fluency, conceptual understanding, and problem solving. This engaging course features new graphics, learning tools, and games; adaptive activities that help struggling students master concepts and skills before moving on; and more support for Learning Coaches to guide their students to success. This course builds on student understanding of numbers and operations by making connections between place value, decimals, and fractions; introducing multiplication and division of decimal numbers; and extending understanding of fraction operations. The course focuses on computational fluency in multiplication and division of whole numbers using standard algorithms. The course enhances fluency of operations with whole numbers, fractions, and decimals through application in the solving of measurement, geometry, and data-analysis problems using mathematical problem-solving techniques. Students continue to develop algebraic thinking as they work with variables and formulas to solve multistep word problems; they further study patterns and rules; and they are introduced to representing problems graphically using the coordinate plane. They extend their knowledge of geometry using the classification of shapes into hierarchies based on their attributes, the introduction of three-dimensional figures and volume, and the connection of geometric concepts to measurement and problem solving. This course includes standards-based tasks, digital literacy skills, and assessment questions.

Course Outline

SEMESTER 1

Unit 1: Whole Numbers and Powers

Students learn to estimate or calculate sums, differences, products, and quotients in whole-number problems. They apply standard step-by-step approaches for addition, subtraction, multiplication, and division; use estimation to predict solutions to story problems; learn patterns of place values; and are introduced to bases and powers.

• Round Whole Numbers in Story Problems

• Estimate and Find Sums and Differences

• Estimate Sums and Differences (parts A, B)

• Multiply Multidigit Whole Numbers

• Divide Multidigit Whole Numbers

• Multiply and Divide Whole Numbers

• Place-Value Patterns

• Bases and Exponents (parts A, B)

• Core Focus

Unit 2: Geometry

Students learn to identify, measure, and draw angles, perpendicular and parallel lines, rectangles, and triangles with appropriate math tools. They predict, describe, and perform transformations on two-dimensional shapes. They learn about right, acute, obtuse, and straight angles; lines that are parallel, intersecting, and perpendicular; and different types of triangles and quadrilaterals. They learn the attributes of isosceles, equilateral, and right triangles, parallelograms, rectangles, and squares.

• Angles (parts A, B)

• Perpendicular and Parallel Lines

• Define and Sketch Triangles

• Define and Sketch Quadrilaterals (parts A, B)

• Angles and Triangles (parts A, B)

• Angles in a Quadrilateral (parts A, B)

• Core Focus

Unit 3: Fractions: Multiplication & Division

Students learn to multiply and divide fractions and explain a step-by-step approach. They simplify factors in fraction multiplication problems in which numerators and denominators have common factors. They multiply and divide fractions by whole numbers to solve story problems.

• Use Models to Multiply Fractions

• Multiply Fractions (parts A–C)

• Multiplication as Scaling

• Different Meanings of Fractions

• Understand Division of Fractions

• Fraction Division (parts A, B)

• Core Focus

Unit 4: Problems Involving Fractions

Students learn to solve story problems involving addition, subtraction, multiplication, and division of fractions. They use objects or sketches to solve story problems that involve addition or subtraction of fractions. They solve and simplify problems that involve addition or subtraction of fractions with unlike denominators.

• Solve Fraction Story Problems (parts A, B)

• Add and Subtract Fractions (parts A–D)

• Core Focus

Unit 5: Decimals: Addition and Subtraction

Students learn to compare, order, and expand decimals. They learn to round decimal numbers to any place through hundredths, estimate the sum or difference in problems involving decimal numbers, and solve addition or subtraction problems involving decimal numbers. They learn how to verify that the calculated result of a problem involving addition or subtraction of decimal numbers is reasonable. They solve story problems involving addition or subtraction of decimal numbers.

• Compare Decimals

• Compare and Expand Decimals

• Order Three Decimal Numbers

• Round Decimals Through Hundredths

• Estimate Decimal Sums/Differences (parts A, B)

• Reasonable Answers and Decimal Problems

• Solve Story Problems with Decimals (parts A, B)

• Core Focus

Unit 6: Decimals: Multiplication and Division

Students practice solving multiplication and division problems that involve decimal numbers and verify that the calculated results are reasonable.

• Estimate Decimal Products, Quotients (parts A–C)

• Multiply and Divide Decimals (parts A–C)

• Compute Decimal Story Problems (parts A–C)

• Core Focus

Unit 7: Semester Review and Checkpoint

SEMESTER 2

Unit 8: Algebra

Students learn to use letters to represent unknown values in expressions and equations. They learn to apply the distributive property in equations or expressions with variables. They evaluate simple algebraic expressions and use expressions or equations to answer questions about a problem.

• Understand Variables in Algebra (parts A, B)

• Use the Distributive Property (parts A, B)

• One Variable in Algebraic Expressions

• Expression and Equation Problems (parts A–C)

• Core Focus

Unit 9: Coordinate Graphs

Students learn to identify, and graph ordered pairs in all quadrants of a coordinate plane. They learn to use the situation presented in a problem to describe the meaning of each coordinate of an ordered pair displayed on a graph. They practice graphing and writing equations to solve problems that involve a linear function.

• Quadrants in the Coordinate Plane

• Ordered Pairs

• Graph or Write an Equation (parts A–D)

• Core Focus

Unit 10: Perimeter, Area, and Volume

Students learn to find the perimeter of plane figures. They connect area to surface area using nets. They learn to use squares to approximate the area of an irregular shape. They learn to determine the volume of a solid figure. They practice constructing cubes and rectangular boxes from two-dimensional patterns and determining the surface area. They learn to differentiate among appropriate units to measure perimeter, area, and volume.

• Perimeter of a Plane Figure

• Nets, Solids, and Surface Area

• Area of Irregular Shapes

• How Many Cubes Does It Take?

• Volume of Solid Figures (parts A, B)

• Units of Perimeter, Area, and Volume

• Core Focus

Unit 11: Math Reasoning: Methods and Strategies

Students learn to prioritize and sequence the information in a story problem that involves multiplication or division of decimal numbers. They use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning in nonroutine or complex problems. Students learn to apply strategies and results from simple story problems involving fractions to more complex problems and how to break a multistep whole-number story problem or money problem into simpler parts. They learn how to identify and represent decimal numbers, fractions, mixed numbers, and positive and negative integers on a number line.

• Steps to Solve Story Problems (parts A, B)

• Break Down Multistep Problems

• Mathematical Reasoning Methods (parts A, B)

• Choose and Use Strategies (parts A–C)

• Solve Simple to Complex Problems (parts A, B)

• Core Focus

Unit 12: Math Reasoning: Solutions

Students learn to express clear and logical solutions to equal-measures problems and rate problems. They learn to use estimation in addition and subtraction of fractions to verify whether calculated results are reasonable. They learn the advantages of exact solutions and approximate solutions to problems involving addition or subtraction of decimal numbers, and give answers to a specified degree of accuracy, such as hundredths. They convert among units within a given measurement system. They learn to make precise calculations and use the situation presented in a problem involving decimal-number operations to check the validity of the result.

• Solve Problems Logically

• Change Measurement

• Measurements in Story Problems

• Decimal Solutions

• Reasonable Solutions

• Core Focus

Unit 13: Data Analysis and Representation

Students practice organizing and displaying single-variable data in histograms, line plots, and circle graphs and learn how to interpret information displayed in a graph or table. They learn how to use fractions to compare different data sets. They learn which types of graphs are appropriate for various data sets.

• Organize Data to Draw Histograms (parts A, B)

• Create Circle Graphs

• Line Plots (parts A, B)

• Interpret Graphs and Tables

• Fractions, and Graphs

• Choose an Appropriate Graph

• Core Focus

Unit 14: Semester Review and Checkpoint

Intermediate Mathematics A:

Course Overview

Intermediate Mathematics A is the first of a three-year middle school math sequence. This research-based course focuses on computational fluency, conceptual understanding, and problem solving and expands more deeply into concepts of geometry, algebra, and statistics. The course also features new graphics and learning tools. Students solve expressions and equations in the context of perimeter, area, and volume problems while further developing computational skills with fractions and decimals. Also, in the context of problem solving, students add, subtract, multiply, and divide positive and negative numbers and work with problems addressing net gains and losses. Students solve problems involving ratios, proportions, and percents with an emphasis on both unit rates and constant rates, as well as problems involving direct variation. They learn multiple representations for communicating information, such as graphs on the coordinate plane, measures of center with statistical data, and a variety of data displays. This course also includes standards-based tasks, digital literacy skills, and multiple question types for assessments.

Course Outline

Unit 1: Problem Solving

• Semester 1 Introduction

• Foundations for Unit 1

• On the Number Line

• Order of Operations

• Number Properties

• Core Focus: Distributive Property Factoring

• Translation Between Words and Math

• Translating Mixed Operations

• Problem-Solving Strategies

• Core Focus: Problem Solving

• Foundations for Unit 2

• Units of Distance

• Polygons and Perimeter

• Applications of Addition and Subtraction Equations

• Core Focus: Addition and Subtraction

• Negative Numbers

• Absolute Value and Distance

• Addition and Subtraction with Negative Numbers

• Core Focus: Negative Numbers

• Solving Addition Equations with Negative Numbers

Unit 3: Area: Multiplication Equations

• Foundations for Unit 3

• Areas of Rectangles

• Core Focus: Similar Parallelograms

• Areas of Triangles

• Figures Made up of Triangles and Parallelograms

• Unknown Side Lengths: Division

• Core Focus: Modeling by Restructuring

Unit 4: Working with Rational Numbers

• Foundations for Unit 4

• Primes and Composites

• Using Prime Factorization

• Equivalent Fractions

• Representing Rational Numbers

• Comparing Rational Numbers

• Perimeters with Fractions

• Areas with Fractions

• Core Focus: Factoring Fractions

• Dividing Fractions

• Solving Problems with Fraction Division

• Core Focus: Fraction Division

Unit 5: Solids

• Foundations for Unit 5

• Cubes and Cube Roots

• Volumes of Prisms

• Nets of Solids

• Core Focus: Measuring Volume

• Surface Area: Prisms and Pyramids

• Properties of Volume and Surface Area

• Core Focus: Volumes and Surface Areas

Unit 6: Comparisons: Ratios

• Foundations for Unit 6

• Ratios as Comparisons

• Percent

• Finding Percents of Numbers

• Core Focus: Understanding Ratio and Percent

Unit 7: Semester 1 Review and Checkpoint

• Semester Review 1

• Semester Review 2

• Semester Review 3

• Semester Checkpoint 1

• Semester Checkpoint 2

Unit 8: The Second Dimension

• Semester 2 Introduction

• Foundations for Unit 8

• Points on a Coordinate Plane

• Using Points to Solve Problems

• Equations with Two Variable

• Core Focus: Reflection in the Coordinate Plane

• Core Focus: Coordinate Plane Applications

• Figures on a Coordinate Plane

• Core Focus: Polygons in the Coordinate Plane

Unit 9: Statistical Displays

• Foundations for Unit 9

• More Statistical Graphs

• Histograms

• Scatter Plots

• Interpreting Scatter Plots

• Core Focus: Understanding Data Display

Unit 10: Statistical Measures

• Foundations for Unit 10

• Measures of Center

• Box-and-Whisker Plots

• Core Focus: Distribution of Data

• Measures of Variation

• Core Focus: Interpreting Data Sets

Unit 11: Rates

• Foundations for Unit 11

• Rates as Comparisons

• Unit Rates

• Solving Unit Rate Problems

• Core Focus: Unit Rates

• Average-Speed Problems

Unit 12: Rates and Direct Variation

• Foundations for Unit 12

• Constant-Rate Problems

• Core Focus: Constant Rates

• Direct Variation

• Interpreting Direct Variation

• Core Focus: Direct Variation Applications

Unit 13: Working with Positives and Negatives

• Foundations for Unit 13

• Adding and Subtracting Signed Numbers

• Net Gains and Losses

• Core Focus: Add and Subtract Signed Numbers

• Multiplying Signed Numbers

• Dividing Signed Numbers

• Core Focus: Multiply and Divide Signed Numbers

Unit 14: Signed Numbers: Properties and Inequalities

• Foundations for Unit 14

• Properties of Signed Numbers

• Core Focus: Simplifying with Properties

• Core Focus: Factoring with Signed Numbers

• Inequalities

• Core Focus: Applications of Inequalities

Unit 15: Semester 2 Review and Checkpoint

• Semester Review 1

• Semester Review 2

• Semester Review 3

• Semester Checkpoint 1

• Semester Checkpoint 2

Math 6: Fundamentals of Geometry and Algebra:

Course Overview

Students enhance computational and problem-solving skills while learning topics in algebra, geometry, probability, and statistics. They solve expressions and equations in the context of perimeter, area, and volume problems while further developing computational skills with fractions and decimals. The study of plane and solid figures includes construction and transformations of figures. Also, in the context of problem solving, students add, subtract, multiply, and divide positive and negative integers and solve problems involving ratios, proportions, and percents, including simple and compound interest, rates, discount, tax, and tip problems. They learn multiple representations for communicating information, such as graphs on the coordinate plane, statistical data and displays, as well as the results of probability and sampling experiments. They investigate patterns involving addition, multiplication, and exponents, and apply number theory and computation to mathematical puzzles.

Course Outline

SEMESTER 1

Unit 1: Problem Solving

Mountain climbing involves solving different kinds of problems. Just like solving math problems, climbing requires tools and a solid strategy. In this unit, you will learn about number lines, the order of operations, and problem solving. To solve problems, you will learn how to translate between words and math symbols, and you will use strategies such as drawing figures, estimating, and breaking a problem down into smaller parts. You will also learn how to handle precision and reasonableness.

• Semester 1 Introduction

• Foundations

• On the Number Line

• Order of Operations

• Number Properties

• Translating Between Words and Math

• Translating Mixed Operations

• Problem-Solving Strategies

• Getting to the Core: Problem Solving

• Identifying Information in Word Problems

• Precision

• Unit Review 1

• Unit Review 2

• Unit Checkpoint 1

• Unit Checkpoint 2

Unit 2: Distance: Addition and Equations

If a farmer has painted part of a fence, how much more does she need to paint? Addition equations can help the farmer solve a problem like that one. In this unit, you will learn how to use units to measure distance and perimeter. You will also solve addition and subtraction equations and discover how those equations can give rise to the idea of negative numbers. Finally, you will use absolute value and operations with positive and negative numbers to solve problems.

• Foundations

• Units of Distance

• Polygons and Perimeter

• Applications of Addition and Subtraction Equations

• Getting to the Core: Addition and Subtraction

• Negative Numbers

• Absolute Value and Distance

• Addition and Subtraction with Negative Numbers

• Getting to the Core: Negative Numbers

• Unit Review 1

• Unit Review 2

• Unit Checkpoint 1

Unit 3: Area: Multiplication Equations

A general contractor needs to calculate area to determine the amount of wood for a floor. In this unit, you will learn how to compute the areas of squares, triangles, rectangles, and other polygons. You will also learn how to divide to find an unknown side length and how a square root relates a side length to the area of a square.

• Units of Area

• Areas of Rectangles

• Getting to the Core: Similar Parallelograms

• Areas of Triangles

• Figures Made Up of Triangles and Parallelograms

• Unknown Side Lengths: Division

• Getting to the Core: Modeling by Restructuring

• Unit Review 1

• Unit Review 2

• Unit Checkpoint 1

• Unit Checkpoint 2

Unit 4: Working with Rational Numbers

Most two-by-fours are actually about 1-1/2 inches by 3-1/2 inches. Any carpenter working with lumber is also working with rational numbers. In this unit, you will learn how to change between various representations of rational numbers including equivalent fractions and decimals. You will also add, subtract, multiply, and divide rational numbers and use these skills to solve practical problems.

• Foundations

• Primes and Composites

• Using Prime Factorization

• Equivalent Fractions

• Representing Rational Numbers

• Comparing Rational Numbers

• Perimeters with Fractions

• Areas with Fractions

• Dividing Fractions

• Solving Problems with Fraction Division

• Getting to the Core: Fractions

• Unit Review 1

• Unit Review 2

• Unit Checkpoint 1

• Unit Checkpoint 2

Unit 5: Solids

When shipping merchandise, you need to know the volume of the container to determine how much it will hold. In this unit, you will learn how to find the volume and surface area of shapes such as prisms and pyramids. You will also find out how a cube root connects the volume of a cube to its side length.

• Foundations

• Cubes and Cube Roots

• Volumes of Prisms

• Nets of Solids

• Getting to the Core: Measuring Volume

• Surface Area: Prisms and Pyramids

• Properties of Volume and Surface Areas

• Getting to the Core: Volumes and Surface Areas

• Unit Review 1

• Unit Review 2

• Unit Checkpoint 1

• Unit Checkpoint 2

Unit 6: Comparisons: Ratios

In southern Asia and South America, some mosquitoes carry a disease called malaria. How can you compare how efforts to fight the disease are progressing in various countries? Scientists and doctors use ratios to understand many problems. In this unit, you will use ratios and proportions to solve many different problems. For instance, you will compute interest on loans, as well as calculate taxes, tips, and discounts.

• Foundations

• Ratios as Comparisons

• Percent

• Finding Percents of Numbers

• Getting to the Core: Understanding Ratio and Percent

• Unit Review 1

• Unit Review 2

• Unit Checkpoint 1

• Unit Checkpoint 2

Unit 7: Semester Review and Test

• Semester Review

• Semester Test

SEMESTER 2

Unit 8: Statistics

Every jellybean can be described. Each one has color, flavor, mass, and number of calories. The language and tools of statistics help to describe buckets full of data. In this unit, you will learn how to create and interpret statistical graphs including circle graphs, bar graphs, line plots, line graphs, box-and-whisker plots, and histograms. You will also learn how to calculate and interpret measures of center and variation. Finally, you will learn how sampling can help you make decisions about a population.

• Semester 2 Introduction

• Foundations

• More Statistical Graphs

• Histograms

• Getting to the Core: Understanding Data Displays

• Measures of Center

• Box-and-Whisker Plots

• Getting to the Core: Distribution of Data

• Measures of Variation

• Statistical Claims

• Getting to the Core: Interpreting Data Sets

• Unit Review 1

• Unit Review 2

• Unit Checkpoint 1

• Unit Checkpoint 2

Unit 9: The Second Dimension

Scientists can use data to figure out how tall someone was from a single bone. When you have two variables, such as femur length and overall height, a two-dimensional plot can help you see patterns and make predictions. In this unit, you will learn how to identify and plot points on a coordinate plane. You will then identify points that are solutions to equations with two variables and create and interpret scatter plots.

• Foundations

• Points on a Coordinate Plane

• Using Points to Solve Problems

• Equations with Two Variables

• Getting to the Core: Reflecting Points on a Coordinate Plane

• Getting to the Core: Coordinate Plane

• Scatter Plots

• Interpreting Scatter Plots

• Figures on a Coordinate Plane

• Getting to the Core: Polygons on the Coordinate Plane

• Unit Review 1

• Unit Review 2

• Unit Checkpoint 1

• Unit Checkpoint 2

Unit 10: Rates

On average, about 1088 cubic meters of water flow over southern Africa’s Victoria Falls every second. That is more than 1,000,000 liters, or enough to fill 26 Olympic-sized swimming pools every minute! In this unit, you will calculate and use rates to solve many types of problems including pricing, speed, and work problems. You will also use direct variation and see how rates affect graphs of relationships.

• Foundations

• Rates as Comparisons

• Unit Rates

• Solving Unit-Rate Problems

• Getting to the Core: Another Look at Unit Rates

• Average-Speed Problems

• Constant-Rate Problems

• Getting to the Core: Another Look at Constant Rates

• Direct Variation

• Interpreting Direct Variation

• Getting to the Core: Another Look at Direct Variation

• Unit Review 1

• Unit Review 2

• Unit Checkpoint 1

• Unit Checkpoint 2

Unit 11: Working with Positives and Negatives

In the stock market positive and negative numbers are key to understanding how companies’ stocks are valued. In this unit, you will learn how to add, subtract, multiply, and divide positive and negative numbers including decimals. You will also work with inequalities.

• Foundations

• Adding and Subtracting Signed Numbers

• Net Gains and Losses

• Getting to the Core: Addition/Subtraction of Signed Numbers

• Multiplying Signed Numbers

• Dividing Signed Numbers

• Exponents and Patterns

• Getting to the Core: Multiplication/Division of Signed Numbers

• Properties of Signed Numbers

• Inequalities

• Getting to the Core: Number Properties and Inequalities

• Unit Review 1

• Unit Review 2

• Unit Checkpoint 1

• Unit Checkpoint 2

Unit 12: Probability

People who play on and coach sports teams, like baseball, as well as those who follow the teams, deal with uncertainty all the time. Probability provides the tools to understand and communicate this uncertainty. In this unit, you will learn how to use Venn and tree diagrams to count the number of ways a trial can be conducted. You can use a diagram to calculate a theoretical probability. You will also learn how to use experimental probability and the law of large numbers. Finally, you will learn about independent, dependent, and complementary events.

• Foundations

• Counting

• Probability and Experiments

• Experimental Probability

• Theoretical Probability

• The Law of Large Numbers

• Independent and Dependent Events

• Complementary Events

• Unit Review

• Unit Checkpoint

Unit 13: Making and Moving Figures

Two men from Southampton, England, say that they used only planks, rope, hats, and wire to make the first crop circles in the 1970s. Crop circle designs range from the simple to the complex, but anyone who makes crop circles needs to know about circles and transformations. In this unit, you will construct and transform figures. For constructions, you will use paper folding as well as a compass and a straightedge. For transformations, you will use coordinates and other methods.

• Foundations

• Folded-Paper Construction

• Compass and Straightedge Construction

• Translation

• Reflection

• Rotation

• Translating with Coordinates

• Reflecting with Coordinates

• Unit Review

• Unit Checkpoint

Unit 14: Semester Review and Test

• Semester Review

• Semester Test

Intermediate Mathematics B:

## Course Overview

Intermediate Mathematics B is the second of a three-year middle school math sequence that prepares students for success in high school algebra. The course begins by developing an understanding of operations with rational numbers, which is applied to working with algebraic expressions and linear equations. This course also helps students develop understanding of proportional relationships and the use of these relationships to solve problems. Geometry topics focus on constructions of two-dimensional figures; properties of circles; scale factors; and problems involving area, surface area, and volume. Finally, students use the tools of probability and statistics to solve basic probability problems and to make inferences based on population samples. This course aligns to national standards and is designed to focus on critical skills and knowledge needed for success in further mathematical studies, including high school algebra.

## Course Outline

#### SEMESTER 1

##### Unit 1: The Basics

In this unit, students focus on the building blocks of basic algebra. Building on their understanding of numbers and operations, students use the order of operations to evaluate numerical expressions. Students also interpret, write, and evaluate expressions to solve real-world problems.

• Semester 1 Introduction

• Order of Operations

• Variable Expressions

• Writing Expressions for Word Phrases

• Related Equations

• Solving Problems

• Core Focus: Word Problems

• Core Focus: Interpreting Expressions

• Unit Review

• Unit Test

##### Unit 2: Addition and Subtraction on a Number Line

Number lines are a great way to show how to add and subtract numbers. In this unit, students look at how addition and subtraction can also be done with negative numbers, using the number line. Problem solving with integers is also emphasized.

• Integers on a Number Line, Part 1

• Integers on a Number Line, Part 2

• Subtracting Integers

• Core Focus: Distance

• Decimals on a Number Line

• Core Focus: Opposites in the Real World

• Unit Review

• Unit Test

##### Unit 3: Addition and Subtraction Properties

This unit focusses on facts, or properties, that are true when it comes to adding integers. For example, the commutative property states that the order in which numbers are added does not change the sum. Students explore this and other properties that will help them simplify expressions and solve equations.

• Subtracting Decimals, Part 1

• Subtracting Decimals, Part 2

• Core Focus: Absolute Value and Distance

• Equations Involving Addition and Subtraction

• Core Focus: Distances Between Rationals

• Unit Review

• Unit Test

##### Unit 4: Multiplication and Division

In this unit, students extend their previous understanding of multiplication and division to signed (positive and negative) numbers as well as other rational numbers. This unit also covers rounding and estimation as well as using equations to solve problems.

• Multiplying Integers and Decimals, Part 1

• Multiplying Integers and Decimals, Part 2

• Dividing Integers and Decimals

• Multiplication and Division Properties

• Core Focus: Closure

• Rounding and Estimation

• Equations Involving Multiplication and Division

• Multiplication and Division Applications, Part 1

• Multiplication and Division Applications, Part 2

• Core Focus: Modeling with Multiplication and Division

• Core Focus: Decimal Forms of Rational Numbers

• Unit Review

• Unit Test

##### Unit 5: Fractions

This unit covers addition, subtraction, multiplication, and division with fractions. This work is also extended to mixed numbers and other forms of rational numbers. These skills are then applied to solving equations and word problems involving rational numbers.

• Equivalent Fractions

• Multiplying Fractions

• Dividing Fractions

• Core Focus: Rational Numbers

• Common Denominators

• Working with Mixed Numbers

• Multiplying and Dividing with Mixed Numbers

• Equations with Fractions

• Core Focus: Fractions and Mixed Numbers

• Core Focus: Applications with Rational Numbers

• Unit Review

• Unit Test

##### Unit 6: Combined Operations

The distributive property provides a powerful tool for working with expressions and equations that have both multiplication and addition. In this unit, the distributive property is used to work with numerical expressions as well as variable expressions and equations. Also, inequalities are used to solve problems.

• The Distributive Property

• Like Terms

• Core Focus: Variable Expressions

• Expressions with Mixed Operations

• Core Focus: Algebraic Expressions

• Equations with Mixed Operations

• Core Focus: Multistep Equations

• Inequalities

• Core Focus: Applications of Inequalities

• Unit Review

• Unit Test

##### Unit 7: Semester Review and Test
• Semester Review

• Semester Test

#### SEMESTER 2

##### Unit 8: Ratio, Proportion, and Percent

In this unit, students work with ratios and proportions. After work calculating and converting ratios, students also solve proportions, and use ratios, proportions, and percents to solve real-world problems.

• Semester 2 Introduction

• Ratios, Part 1

• Ratios, Part 2

• Word Problems with Ratios

• Core Focus: Unit Rates

• Proportion, Part 1

• Proportion, Part 2

• Percents, Fractions, and Decimals

• Working with Percent

• Core Focus: Identifying Proportions

• Unit Review

• Unit Test

##### Unit 9: Proportion Applications

Proportional thinking is important in many real-world applications. In this unit, students focus on many applications of and strategies for working with proportions. Applications include markup and discount, percent problems, simple interest, and problems from science.

• Similarity and Scale

• Proportion Problems

• Direct Linear Variation

• Core Focus: Graphing Proportions

• Percent Problems

• Percent of Increase and Decrease

• Core Focus: Percent Error

• Simple Interest

• Core Focus: Multistep Ratio and Percent Problems

• Core Focus: Constant of Proportionality

• Unit Review

• Unit Test

##### Unit 10: Plane Figures

In this unit, students focus on geometry. Angles, triangles, and quadrilaterals are the focus Special attention is given to constructing triangles and to calculating areas of triangles, quadrilaterals, and other polygons.

• Parallel Lines and Transversals

• Triangles

• Constructing Triangles

• Areas of Rectangles and Triangles

• Areas of Regular Polygons

• Core Focus: How Many Triangles?

• Unit Review

• Unit Test

##### Unit 11: Circles

Circles are some of the most useful geometric shapes. In this chapter, students learn how to compute the circumference and area of a circle and use circles to solve real-world problems.

• Circles

• Circumference

• Areas of Circles

• Core Focus: Circumference and Area

• Unit Review

• Unit Test

##### Unit 12: Solid Figures

This unit starts with finding volumes of prisms and then looking at cross-sections of solid figures. Next, students learn about surface area and then use properties of volume and surface area to solve problems.

• Volume and Capacity

• Volumes of Prisms

• Slicing Solids

• Surface Area

• Surface Areas of Prisms

• Properties of Volume and Surface Area

• Core Focus: Applications of Volume and Surface Area

• Unit Review

• Unit Test

##### Unit 13: Probability and Statistics

In this unit, students learn about simple probability and then probability of compound events. Next, the tools of statistics allow students to use samples to make predictions. Then, students use statistical graphs and measures of center and spread to compare populations.

• Probability

• Combined Probability

• Mutually Exclusive Events

• Core Focus: Compound Events

• Samples and Prediction

• Measures of Center

• Frequency Tables and Histograms

• Measures of Variability

• Comparing Populations

• Core Focus: Samples and Simulations

• Unit Review

• Unit Test

##### Unit 14: End-of-Year Project
• Project Day 1

• Project Day 2

• Project Day 3

• Project Day 4

• Project Day 5

##### Unit 15: Semester Review and Test
• Semester Review

• Semester Test

Math 7: Pre-Algebra:

Course Overview

Students take a broader look at computational and problem-solving skills while learning the language of algebra. Students translate word phrases and sentences into mathematical expressions; analyze geometric figures; solve problems involving percentages, ratios, and proportions; graph different kinds of equations and inequalities; calculate statistical measures and probabilities; apply the Pythagorean theorem; and explain strategies for solving real-world problems. Online lessons provide demonstrations of key concepts, as well as interactive problems with contextual feedback. A textbook supplements the online material. Students who take Pre-Algebra are expected to have mastered the skills and concepts presented in the K12 Fundamentals of Geometry and Algebra course (or equivalent).

Course Outline

SEMESTER ONE

Unit 1: The Basics

Let us start at the very beginning; it is an incredibly good place to start. Just as you need to know basic grammar and vocabulary as you begin to learn any language, you need to know some basic building blocks as you begin to learn algebra.

• Order of Operations

• Variable Expressions

• Writing Expressions for Word Phrases

• Comparing Expressions

• Replacement Sets

• Related Equations

• Solving Problems

If you have two oranges and a friend gives you three oranges, how many do you have? If you then give four oranges to your friend, how many are you left with? This sort of addition and subtraction problem with passing fruit back and forth is the type of simple math you have done since you were young. When you expand your addition and subtraction skills to negative numbers and decimals, you can solve many more complicated problems.

• Integers on a Number Line

• Subtracting Integers

• Decimals on a Number Line

• Subtracting Decimals

• Equations Involving Addition and Subtraction

Unit 3: Multiplication and Division

Isaac Newton's third law of motion is often paraphrased as “for every action, there is an equal and opposite reaction.” Just as forces come in pairs, so can mathematical operations. Multiplication and division are inverse operations. They undo each other and can both be used to solve many types of problems.

• Multiplying Integers and Decimals

• Dividing Integers and Decimals

• Multiplication and Division Properties

• Rounding and Estimation

• Equations Involving Multiplication and Division

• Multiplication and Division Applications

Unit 4: Fractions

Every fraction can be written as a decimal and every decimal can be written as a fraction. As a result, you could do just about all math with only fractions or only decimals, but decimals are used for certain applications just as fractions are used for others. For example, carpenters use fractions and mixed numbers quite a bit; anybody building a house or a deck deals with lots of fractions.

• Equivalent Fractions

• Multiplying Fractions

• Dividing Fractions

• Common Denominators

• Working with Improper Fractions and Mixed Numbers

• Multiplying and Dividing Mixed Numbers

• Equations with Fractions and Mixed Numbers

Unit 5: Combined Operations

Many yachts can be powered by the wind, by a gas engine, or both. A hybrid automobile can run on gasoline or electric power. These combinations are powerful. Combining addition or subtraction with multiplication or division is powerful as well. You can use equations and expressions with mixed operations to solve many complex problems.

• The Distributive Property

• Like Terms

• Expressions with Mixed Operations

• Equations with Mixed Operations

• Error Analysis

• Inequalities

Unit 6: Number Properties

Astronomers study things that are extremely far away. For example, the Horsehead Nebula is about 14,000 trillion kilometers away. On the other extreme, molecular geneticists study things that are ridiculously small. A double helix of DNA has a diameter of about one nanometer (a billionth of a meter.) With exponents, you can describe very great or exceedingly small distances.

• Positive Exponents

• Factors and Primes

• GCF and Relative Primes

• Negative Exponents

• Powers of Ten

• Scientific Notation

Unit 7: Geometry Basics

Shapes such as polygons and circles provide us with shelter, art, and transportation. Some artists use geometric shapes in their art, but most painters and photographers use rectangular frames to surround their art. Look at any art museum, and you will see triangles, rectangles, and other polygons in the structure of the building and in the artwork inside.

• Points, Lines and Planes

• Rays and Angles

• Parallel Lines and Transversals

• Triangles

• Polygons

• Circles

• Transformations

• Congruence

Unit 8: Semester Review and Test

• Semester Review

• Semester Test

SEMESTER TWO

Unit 1: Ratio, Proportion and Percent

Model builders use ratios and percents to describe how their models compare to real objects. They can use proportions to figure out the length of every item in the model.

• Ratio

• Proportion

• Percents, Fractions and Decimals

• Similarity and Scale

• Working with Percent

• Percent of Increase or Decrease

• Simple Interest

Unit 2: Analytic Geometry

A pilot uses numbers to locate the airport she is flying to. An air traffic controller uses numbers on a radar screen to locate each airplane approaching the airport. Without a system of locating points, airplanes would have a hard time getting anywhere safely.

• Points on the Plane

• Two-Variable Equations

• Linear Equations and Intercepts

• Slope

• Problem Solving

• Relations and Functions

• Systems of Linear Equations

Unit 3: Perimeter and Area

You can find geometric shapes in art. Whether determining the amount of leading or the amount of glass for a piece of stained-glass art, stained-glass artists need to understand perimeter and area to solve many practical problems.

• Types of Polygons

• Perimeter

• Areas of Rectangles and Triangles

• Circumference

• Areas of Circles

Unit 4: Square Roots and Right Triangles

Since ancient times, people have used right triangles to survey land and build structures. Even before Pythagoras was born, the relationship between the side lengths of a right triangle has been essential to anyone building just about any structure, including pyramids, houses, skyscrapers, and bridges.

• Rational Square Roots

• Irrational Square Roots

• The Pythagorean Theorem

• The Distance Formula

• Special Types of Triangles

• Trigonometric Ratios

Unit 5: Solid Figures

Gas-powered engines are driven by little explosions that move pistons up and down in cylinders. When you add up the volume of all the cylinders, you get the displacement of the engine. For instance, each cylinder in a four-cylinder, 1000 cc engine has a volume of 250 cubic centimeters. Engineers and mechanics must accurately compute volume when they build or maintain engines.

• Volume and Capacity

• Volumes of Prisms and Cylinders

• Volumes of Pyramids and Cones

• Surface Area

• Surface Areas of Prisms and Cylinders

Unit 6: Counting and Probability

How many apples have mass between 100 and 200 grams? How many are bruised? How many are not yet ripe? Checking every single apple would probably be impractical, but if you understand probability and sampling, you could make a good estimate.

• Counting Principles

• Permutations

• Combinations

• Probability

• Mutually Exclusive Events

• Samples and Prediction

Unit 7: Statistics

Data are everywhere. When you look at a group of people, you could use many numbers to describe them. How tall are they? How long is their hair? How old are they? What is their gender? What color are their eyes? Statistics helps you make sense of data.

• Graphs

• Measures of Center

• Stem-and-Leaf Plots

• Box-and-Whisker Plots

• Frequency Tables and Histograms

Unit 8: Semester Review and Test

• Semester Review

• Semester Test

Intermediate Mathematics C:

Course Overview

Intermediate Mathematics C is the third of a three-year middle school math sequence that prepares students for success in high school algebra. The course begins with properties of numbers, including exponents, as well as measurement and precision with scientific notation. After using transformations to solve linear equations with one variable, the course presents linear equations and systems with two variables. The course emphasizes modeling with linear relationships, including the use of linear functions to model relationships between bivariate statistical data. Geometry topics include distances, angles, similarity, and congruence with two-dimensional figures and volumes of three-dimensional figures. Finally, students use irrational numbers and the Pythagorean theorem to solve mathematical and real-world problems. This course aligns to national standards and is designed to focus on critical skills and knowledge needed for success in further mathematical studies, including high school algebra. After completing this course, students will be ready to take Algebra I in high school.

Course Outline

SEMESTER 1

Unit 1: Number Properties

In this unit, students learn about number properties including the order of operations and properties of positive and negative exponents. Then, students use scientific notation and orders of magnitude to make sense of exceptionally large or small numbers.

• Semester 1 Introduction

• Expressions

• Distributive Property

• Positive Exponents

• Negative Exponents

• Core Focus: Working with Exponents

• Scientific Notation

• Orders of Magnitude

• Core Focus: Precision

• Core Focus: Comparing Big and Small Numbers

• Unit Review

• Unit Test

Unit 2: Equations

In this unit, students learn how to solve equations with one variable. Students start with equations with only addition or subtraction and then only multiplication or division before moving to more complicated equations that have more than one operation.

• Core Focus: Addition and Subtraction Equations

• Multiplication and Division Equations, Part 1

• Multiplication and Division Equations, Part 2

• Multiple Transformations

• Variables on Both Sides of an Equation

• Strange Solutions

• Core Focus: Transforming Equations

• Core Focus: Tougher Equations

• Unit Review

• Unit Test

Unit 3: Slope and Proportional Thinking

In this unit, students start with the basics of graphs of equations in two variables. Then, students focus on using proportional thinking as a way to understand slope of a line. By the end of the unit, students see how to connect slope to rates, similar triangles, and proportional relationships.

• Equations in Two Variables

• Graphs

• Lines and Intercepts

• Slope

• Simple Linear Graphs

• Using Slope as a Rate

• Slope and Similar Triangles

• Comparing Proportional Relationships

• Core Focus: Graphs of Proportional Relationships

• Unit Review

• Unit Test

Unit 4: Lines

In this unit, students learn how to use several methods to graph linear equations in two variables. Students also learn how to figure out an equation of a line when given its graph. Finally, they see how to write an equation that models a problem situation, and then use that equation to solve the problem.

• Slope-Intercept Form

• Point-Slope Form

• Equations from Graphs

• Core Focus: Sketching Lines

• Applications: Linear Models

• Core Focus: Linear Models

• Core Focus: Interpreting Linear Models

• Unit Review

• Unit Test

Unit 5: Systems of Equations

In this unit, students learn how to solve systems of linear equations using graphs and substitution and by inspection. They also use systems of linear equations to solve several types of real-world problems.

• Systems of Equations

• Using Graphs to Solve Systems

• Solving Systems Using Inspection

• Substitution Method

• Core Focus: Methods of Solving Systems

• Applications: Systems of Linear Equations

• Core Focus: Applications of Linear Systems

• Core Focus: Mixture Problems

• Unit Review

• Unit Test

Unit 6: Function Basics

In this unit, students learn about mathematical relations and functions. Then, students see how to work with functions in graphs, equations, and other forms.

• Relations

• Functions, Part 1

• Functions, Part 2

• Function Equations, Part 1

• Function Equations, Part 2

• Interpreting Function Graphs

• Linear Function Models

• Function Representations

• Core Focus: Functions

• Core Focus: Sketching Function Graphs

• Unit Review

• Unit Test

Unit 7: Semester Review and Test

• Semester Review

• Semester Test

SEMESTER 2

Unit 8: Linear Models

In this unit, students use lines as simple models of real-world phenomena. Students also use lines to understand plots of two-dimensional statistical data.

• Semester 2 Introduction

• Direct Linear Variation 1

• Direct Linear Variation 2

• Core Focus: Interpreting Slope and Intercepts

• Patterns in Two-Way Tables

• Scatter Plots

• Clustering and Outliers

• Associations in Scatter Plots

• Lines of Best Fit

• Core Focus: Model Fit

• Unit Review

• Unit Test

Unit 9: Basic Geometric Shapes

In this unit, students learn how to define and use geometric figures made up of points, lines, and angles. Students also see the relationships between angles formed when lines intersect. Finally, they learn about triangles and other polygons.

• Points, Lines, and Angles

• Parallel Lines and Transversals

• Pairs of Angles

• Triangles, Part 1

• Triangles, Part 2

• Core Focus: Angles in a Triangle

• Polygons

• Core Focus: Exterior Angles

• Unit Review

• Unit Test

Unit 10: Volume

In this unit, students learn formulas for the volumes of cylinders, cones, and spheres. They also apply these formulas to real-world problems.

• Volumes of Cylinders

• Applications of Cylinders

• Volume of Cones

• Applications of Cones

• Volume of Spheres

• Applications of Spheres

• Core Focus: Comparing Volumes

• Unit Review

• Unit Test

Unit 11: Congruence and Similarity

In this unit, students learn the difference between congruence and similarity, and then explore each idea on its own. First, they see the relationship between similarity and scale. Then, students look at congruence in terms of transformations.

• Congruence and Similarity

• Similarity and Scale

• Core Focus: Similarity

• Transformations

• Verifying Properties of Transformations

• Transformations and Congruence

• Transformations and Similarity

• Transformations in the Coordinate Plane

• Core Focus: Preserving Geometric Relationships

• Unit Review

• Unit Test

Unit 12: Irrational Numbers

In this unit, students work with irrational numbers. First, they figure out what makes a number irrational, and then they learn to simplify and approximate irrational numbers that arise from square roots. Finally, students find irrational solutions to equations.

• Rational Numbers

• Terminating and Repeating Numbers

• Irrational Numbers

• Rational Square Roots

• Irrational Square Roots 1

• Irrational Square Roots 2

• Core Focus: Approximations of Irrationals

• Higher Roots

• Using Square Roots to Solve Equations

• Core Focus: Irrational Solutions

• Unit Review

• Unit Test

Unit 13: The Pythagorean Theorem

In this unit, students learn about one of the most famous and useful mathematical theorems. They learn how to use the theorem to solve problems, learn proofs of the theorem, and even use the theorem to solve problems in three dimensions.

• Pythagorean Theorem

• Proofs of the Pythagorean Theorem

• Applications of the Pythagorean Theorem

• Distances with the Pythagorean Theorem

• Core Focus: Pythagorean Theorem in 3D

• Core Focus: More Pythagorean Applications

• Unit Review

• Unit Test

Unit 14: End-of-Year Project

• Project Day 1

• Project Day 2

• Project Day 3

• Project Day 4

• Project Day 5

Unit 15: Semester Review and Test

• Semester Review

• Semester Test

Math 8: Algebra:

Course Overview

Students develop algebraic fluency by learning the skills needed to solve equations and perform manipulations with numbers, variables, equations, and inequalities. They also learn concepts central to the abstraction and generalization that algebra makes possible. Students learn to use number properties to simplify expressions or justify statements; describe sets with set notation and find the union and intersection of sets; simplify and evaluate expressions involving variables, fractions, exponents, and radicals; work with integers, rational numbers, and irrational numbers; and graph and solve equations, inequalities, and systems of equations. They learn to determine whether a relation is a function and how to describe its domain and range; use factoring, formulas, and other techniques to solve quadratic and other polynomial equations; formulate and evaluate valid mathematical arguments using various types of reasoning; and translate word problems into mathematical equations and then use the equations to solve the original problems. Students who take Algebra are expected to have mastered the skills and concepts presented in the K12 Pre-Algebra course (or equivalent).

Course Outline

SEMESTER ONE

Unit 1: Algebra Basics

The English word algebra and the Spanish word algebrista both come from the Arabic word al-jabr, which means "restoration". A barber in medieval times often called himself an algebrista. The algebrista also was a bonesetter who restored or fixed bones. Mathematicians today use algebra to solve problems. Algebra can find solutions and "fix" certain problems that you encounter.

• Semester Introduction

• Expressions

• Variables

• Translating Words into Variable Expressions

• Equations

• Translating Words into Equations

• Replacement Sets

• Problem Solving

Unit 2: Properties of Real Numbers

There are many kinds of numbers. Negative numbers, positive numbers, integers, fractions, and decimals are just a few of the many groups of numbers. What do these varieties of numbers have in common? They all obey the rules of arithmetic. They can be added, subtracted, multiplied, and divided.

• Number Lines

• Sets

• Comparing Expressions

• Number Properties

• Measurement, Precision, and Estimation

• Distributive Property

• Algebraic Proof

• Opposites and Absolute Value

Unit 3: Operations with Real Numbers

There are many kinds of numbers. Negative numbers, positive numbers, integers, fractions, and decimals are just a few of the many groups of numbers. What do these varieties of numbers have in common? They all obey the rules of arithmetic. They can be added, subtracted, multiplied, and divided.

• Subtraction

• Multiplication

• Reciprocals and Division

• Applications: Number Problems

Unit 4: Solving Equations

The Greek mathematician Diophantus is often called "the father of algebra." His book Arithmetica described the solutions to 130 problems. He did not discover all of these solutions himself, but he did collect many solutions that had been found by Greeks, Egyptians, and Babylonians before him. Some people of long ago obviously enjoyed doing algebra. It also helped them—and can help you—solve many real-world problems.

• Multiplication and Division Equations

• Patterns

• Multiple Transformations

• Variables on Both Sides of an Equation

• Transforming Formulas

• Estimating Solutions

• Cost Problems

Unit 5: Solving Inequalities

Every mathematician knows that 5 is less than 7, but when is y < x? An inequality symbol can be used to describe how one number compares to another. It can also indicate a relationship between values.

• Inequalities

• Solving Inequalities

• Combined Inequalities

• Absolute Value Equations and Inequalities

• Applications: Inequalities

Unit 6: Applying Fractions

What do a scale drawing, a bicycle's gears, and a sale at the local store all have in common? They all present problems that can be solved using equations with fractions.

• Ratios

• Proportions

• Unit Conversions

• Percents

• Applications: Percents

• Applications: Mixture Problems

Unit 7: Linear Equations and Inequalities

You have probably heard the phrase, "That's where I draw the line!" In algebra, you can take this expression literally. Linear functions and their graphs play an important role in the never-ending quest to model the real world.

• Equations in Two Variables

• Graphs

• Lines and Intercepts

• Slope

• Using Slope as a Rate

• Slope-Intercept Form

• Point-Slope Form

• Parallel and Perpendicular Lines

• Equations from Graphs

• Applications: Linear Models

• Graphing Linear Inequalities

• Inequalities from Graphs

Unit 8: Systems of Equations

When two people meet, they often shake hands or say "hello" to each other. Once they start talking to each other, they can find out what they have in common. What happens when two lines meet? Do they say anything? Probably not, but whenever two lines meet, you know they have at least one point in common. Finding the point at which they meet can help you solve problems in the real world.

• Systems of Equations

• Substitution Method

• Linear Combination

• Linear Combination with Multiplication

• Applications: Systems of Linear Equations

• Systems of Linear Inequalities

Unit 9: Semester Review and Test

• Semester Review

• Semester Test

SEMESTER TWO

Unit 1: Relations and Functions

A solar cell is a little machine that takes in solar energy and puts out electricity. A mathematical function is a machine that takes in a number as an input and produces another number as an output. There are many kinds of functions. Some have graphs that look like lines, while others have graphs that curve like a parabola. Functions can take other forms as well. Not every function has a graph that looks like a line or a parabola. Not every function has an equation. The important thing to remember is that if you put any valid input into a function, you will get a single result out of it.

• Semester Introduction

• Relations

• Functions

• Function Equations

• Order of Operations

• Absolute Value Functions

• Direct Linear Variation

• Inverse Variation

• Translating Functions

Unit 2: Rationals, Irrationals, and Radicals

Are rational numbers very levelheaded? Are irrational numbers hard to reason with? Not really, but rational and irrational numbers have things in common and things that make them different.

• Rational Numbers

• Terminating and Repeating Numbers

• Square Roots

• Dimensional Analysis

• Irrational Numbers

• Evaluating and Estimating Square Roots

• Using Square Roots to Solve Equations

• The Pythagorean Theorem

• Higher Roots

Unit 3: Working with Polynomials

Just as a train is built from linking railcars together, a polynomial is built by bringing terms together and linking them with plus or minus signs. You can perform basic operations on polynomials in the same way that you add, subtract, multiply, and divide numbers.

• Overview of Polynomials

• Multiplying Monomials

• Multiplying Polynomials by Monomials

• Multiplying Polynomials

• FOIL

Unit 4: Factoring Polynomials

A polynomial is an expression that has variables that represent numbers. A number can be factored, so you should be able to factor a polynomial, right? Sometimes you can and sometimes you cannot. Finding ways to write a polynomial as a product of factors can be quite useful.

• Factoring Integers

• Dividing Monomials

• Common Factors of Polynomials

• Dividing Polynomials by Monomials

• Factoring Perfect Squares

• Factoring Differences of Squares

• Factoring Completely

• Finding Roots of a Polynomial

Solving equations can help you find answers to many kinds of problems in your daily life. Linear equations usually have one solution, but what about quadratic equations? How can you solve them and what do the solutions look like?

• Solving Perfect Square Equations

• Completing the Square

• Scientific Notation

• Equations and Graphs: Roots and Intercepts

• Applications: Area Problems

• Applications: Projectile Motion

Unit 6: Rational Expressions

A fraction always has a number in the numerator and in the denominator. However, those numbers can actually be expressions that represent numbers, which means you can do all sorts of interesting things with fractions. Fractions with variable expressions in the numerator and denominator can help you solve many kinds of problems.

• Simplifying Rational Expressions

• Multiplying Rational Expressions

• Dividing Rational Expressions

• Like Denominators

• Adding and Subtracting Rational Expressions

Unit 7: Logic and Reasoning

Professionals use logical reasoning in a variety of ways. Just as lawyers use logical reasoning to formulate convincing arguments, mathematicians use logical reasoning to formulate and prove theorems. Once you have mastered the uses of inductive and deductive reasoning, you will be able to make and understand arguments in many areas.

• Reasoning and Arguments

• Hypothesis and Conclusion

• Forms of Conditional Statements

• Using Data to Make Arguments

• Inductive and Deductive Reasoning

• Algebraic Proof

• Counter Example

Unit 8: Semester Review and Test

• Semester Review

• Semester Test

SUPPLEMENTAL UNITS

Two supplemental units provide additional coursework. Measurement and Geometry provides some of the essentials for beginning geometry students and Counting, Probability, and Statistics provides a solid foundation for further studies in statistics and probability.

A–1: Measurement and Geometry

A tessellation is a way of repeating a shape over and over again to cover a plane surface. The artist Maurits Cornelis (M.C.) Escher was fascinated with tessellations. He used tessellations and geometric ideas such as points, segments, angles, and congruence to make lots of beautiful, interesting art.

• Points, Lines, and Angles

• Pairs of Angles

• Triangles

• Polygons

• Congruence and Similarity

• Area

• Volume

• Scale

A–2: Counting, Probability, and Statistics

How much corn can a farmer get from an acre of land? Which countries export the most corn? How has the price of corn changed over time and how will it change moving forward? Data are all around us. With a good understanding of probability and statistics, people can make better decisions.

• Counting

• Permutations and Combinations

• Probability

• Combined Probability

• Graphs

• Summary Statistics

• Frequency Distributions

• Samples and Prediction

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