Foundations 5 Curriculum Guide

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1 1. Review: Natural Numbers Reading and Writing Natural Numbers Lines, Rays, and Line Segments Comparing Natural Numbers Rounding Numbers Adding Natural Numbers Properties of Addition, Convenient Calculation and Polygons Column Addition Subtracting Natural Numbers Column Subtraction Numerical Expressions with Parentheses Multiplying Natural Numbers Properties of Multiplication Column Multiplication by a One-Digit Number Column Multiplication by a Two-Digit Number Formula for the Area of Squares and Rectangles Dividing Natural Numbers Dividing Round Numbers and Finding How Many Times More/Less Division with a Remainder Long Division Order of Operations Fractions Fractions and Division, Improper Fractions, and Comparing Fractions : Mixed Numbers and Improper Fractions Trapezoids and Parallelograms Comparing Fractions Adding and Subtracting Fractions with Like Denominators Adding Mixed Numbers with Like Denominators Subtracting Mixed Numbers with Like Denominators Review Factors, Prime and Composite Numbers, GCFs Review: Multiples and LCMs The Equivalency Property of a Common Fraction, Reducing Fractions Bringing Fractions to a Common Denominator Comparing Fractions with Different Denominators Adding and Subtracting Fractions with Unlike Denominators Adding and Subtracting Mixes Numbers Multiplying Common Fractions Reasoning Mind 1 Curriculum Guide

2 38. Dividing Fractions and Mixed Numbers Review Angles Types of Triangles Review: Decimals Review: Comparing Decimals Review: Rounding Decimals Review: Adding Decimals Review: Subtracting Decimals Multiplying a Decimal by a Whole Number Multiplying a Decimal by a Decimal Dividing a Decimal by a Whole Number Dividing a Decimal by a Decimal Reasoning Mind 2 Curriculum Guide

3 1. Review: Natural Numbers Approximate Length: 1.8 student study hours Overview Prior to this Objective, your students will work the Smarter Solving Lesson 1. This objective gives a review of basic facts concerning Natural Numbers, or Whole Numbers, including writing numbers using the decimal system. In Chapter 1 the natural numbers are defined and how natural numbers are formed is discussed. Chapter 2 continues with how 1-, 2-, and 3-digit natural numbers are written and understanding the sum of place values. The special case of having a zero as one of the digits in a 2-, and 3-digit number is shown in Chapter 3. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lesson 2. Related Objectives Objective 2. Writing Numbers and Counting Within 1,000 (Foundations 4 Curriculum) Objective 1. Review: Natural Numbers Objective 2. Reading and Writing Natural Numbers Objective 6. Adding Natural Numbers Objective 8. Column Addition Objective 9. Subtracting Natural Numbers Vocabulary New Terms: Glossary Terms: natural numbers, whole numbers, one-digit number, two-digit number, three-digit number natural number, one s place, places, sum, summand, ten, thousand, thousand s place, whole number Lesson Notes Main Idea #1 - Students must understand the unit place values in order to decompose a number into sums of the values in each place, as well as to add and subtract with carrying and borrowing. Write This Down Screen (Theory 8) Notes Test Item 1 Reasoning Mind 3 Curriculum Guide

4 Main Idea #2 - Students need to learn the names of the places in a three digit number in order to understand how the decimal system works and the rules that will be developed in future theory. Write This Down Screen (Theory 11) Notes Test Item 2 Main Idea #3 - It is important to understand that a zero in one of the places means there are zero unit place values in that number.. Write This Down Screen (Theory 15) Notes Test Item 3 Main Idea #4- The ability to decompose a number into the sum of its place values will be important when learning mental math techniques. Write This Down Screen (Theory 17) Notes Test Item 4 Theory Block Chapter 1 Natural Numbers (Theory 2) Gives the definition of Natural Numbers, or Whole Numbers. (Theory 3) Explains the ordering of Natural Numbers, that they come one after another. We get the next natural number by adding 1. We get the previous Natural Number by subtracting 1. Chapter 2 The Decimal System: Places and Three-Digit Numbers (Theory 5) We can write any natural number with just 10 digits using what is called the decimal system. (Theory 6, 7) Discusses further defining natural numbers by the how many digits are used. (Theory 8) WTD See Lesson Notes, Main Idea 1. (Theory 10) Gives the names of the places for a three digit number. (Theory 11) WTD See Lesson Notes, Main Idea 2. Explains how to compose a number as the sum of place values. (Theory 15) WTD See Lesson Notes, Main Idea 3. Reasoning Mind 4 Curriculum Guide

5 Chapter 3 Showing a Number as a Sum of Place Values (Theory 17) WTD See Lesson Notes, Main Idea 4. (Theory 19) In 206, the 0 is in the ten s place. Therefore, we add no tens. A-Level Problems 1 Name the first natural number. Related to Theory 6 2 Name the natural number right after a given Natural number. Theory Problem 1 3 Name the natural number right before a given Natural number. Theory Problem 2 SG 1 Speed Game: Compose a given natural number as another natural number and an unknown natural number. 4 Write a three-digit natural number with a zero in a place value as the sum of place values. B-Level Problems 1 Find two-digit number from a known place value relationship to the unknown place value. 2 Find how many two-digit natural numbers have a certain digit in specified place. Review from Foundations grades 2-4. Theory Problem 14 Similar, but more complicated than A-Level Problem 4 Combination of Theory 3, Theory 7, and Theory 10 3 Drag the numbers to give the largest sum of 2 two-digit numbers. Review of ordering of natural numbers using place value, Theory 10 and Theory 13. C-Level Problems 1 List the three-digit numbers with a given place value and a relationship between the remaining two digits. 2 Find how many 1-, 2-, and 3-digit numbers can be made using only 2 given digits. 3 Find how many 3-digit numbers have a given relationship between two of the digits. Combination of B-Level Problems 1 and 2 B-Level Problem 2 Combination of B-Level Problems 1 and 2 Diagnostic Block 1 Write given number as sum of hundreds, tens and ones. Theory 13, Theory Problem 10 2 Determine which, in a list of numbers, have a given digit in a specified place. (1 hint) 3 Choose equality which shows a given number as the sum of its place values. Similar to Theory Problem 8 Theory Problem 12 Reasoning Mind 5 Curriculum Guide

6 2. Reading and Writing Natural Numbers Approximate Length: 1.4 student study hours Overview The objective covers how to handle natural numbers with more than four digits. Student s learn about periods and how to name numbers based on the period and place value in that period. Related Objectives Objective 1. Review: Natural Numbers Objective 2. Reading and Writing Natural Numbers Objective 6. Adding Natural Numbers Objective 8. Column Addition Objective 9. Subtracting Natural Numbers Vocabulary New Terms: Glossary Terms: period hundred s place, period, ten s place, tenth s place Lesson Notes Main Idea #1 - Students should be able to take a natural number with any value of digits and structure the number into periods.. Write This Down Screen (Theory 4) Notes Test Item 1 Theory Block Chapter 1 Periods (Theory 1) The thousand s place is defined. (Theory 3) Since a natural number can have any amount of digits, the concept of periods is defined to make reading and writing natural numbers easier. (Theory 4) WTD See Lesson Notes, Main Idea 1. (Theory 5) The names of the first 4 periods are given. Students learn that in each period, there is a place name and how to read and write numbers using these definitions. (Theory 7 and 8) Problems from Objective 1 are expended to numbers with 4 and 5 digits. Reasoning Mind 6 Curriculum Guide

7 A-Level Problems 1 Name the place containing a specified digit. (1 hint) See Theory 5 2 Choose numbers from a list which have a certain digit in a specific place. (1 hint) See Theory 5 3 Name the digit in a certain place and period. See Theory 5 4 Write a four-digit number as the sum of place values. Theory Problem 3 B-Level Problems 1 Drag numbers to get the smallest sum from two three-digit numbers. Find the sum. (1 hint) Review of ordering of natural numbers using place value, and Theory 5. 2 Find the period with the smallest sum of place values. (1 hint) Review of ordering sums and Theory 5 C-Level Problems 1 Find how much a three-digit number is increased by writing a specified two-digit number in from of it. (1 hint) 2 Find how many four-digit numbers can be written with the digits in the one s period specified. Theory 5 and comparing numbers from previous grades. Builds on Theory Problem 4, where one digit is not specified. Diagnostic Block 1 Determine how many periods in a five-digit number Related to Theory 3 2 Determine what digit is in the thousand s place. Related to Theory 5 3 Select the correct way to show a five-digit number as the sum of its place values. Theory Problem 4 Reasoning Mind 7 Curriculum Guide

8 3. Lines, Rays, and Line Segments Approximate Length: 2 student study hours Overview This objective defines the line, ray and line segment culminating with the development of the number ray. In Chapter 1 lines are defined, named and characteristics investigated. Chapter 2 builds the understanding of a ray and line segment as part of a line, with a only the line segment being finite in length. Chapter 3 refines the concept of the length of a line segments by determining the length using a line segment of one unit in length. The number ray with coordinate point locations is built in Chapter 4 by defining its major features and then locating points with specific coordinates. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lesson 3. Related Objectives Objective 6. Segment Chains (Foundations 2 Curriculum) Objective 3. Lines, Rays and Line Segments Objective 4. Comparing Natural Numbers Objective 5. Rounding Natural Numbers Objective 7. Properties of Addition, Convenient Calculations, and Polygons Objective 25. Fractions Objective 26. Fractions and Division, Improper Fractions, and Comparing Fractions Objective 27. Comparing Fractions Objective 28. Adding and Subtracting Fractions with Like Denominators Objective 34. Comparing Fractions with Different Denominators Objective 35. Adding and Subtracting Fractions with Unlike Denominators Vocabulary New Terms: Glossary Terms: line, Intersecting lines, point of intersection, ray, number ray adjacent segments, coordinates, curve, endpoints, number ray, ray, ruler, segment, supplementary rays, unit Lesson Notes Main Idea #1 - It is important for students to understand the difference between lines, rays and line segments. The ray is important for introducing the number ray. Write This Down Screen (Theory 8) Notes Test Item 1 Reasoning Mind 8 Curriculum Guide

9 Main Idea #2 - This is important because defining the segment this way leads to thinking in terms of lengths of line segments. This will be used in understanding the number ray and defining fractions. Write This Down Screen (Theory 14) Notes Test Item 2 Main Idea #3 - The concept, that line segments endpoints can be given in any order, reinforces the understanding that line segments have finite length. Relates to Theory 16 Notes Test Item 3 Main Idea #4 - It is important for students to understand the relationship between distance of a point from the Origin, or point with coordinate 0, and number associated with that point. This will be used to define fractions, which are represented by points between points with whole number coordinates Write This Down Screen (Theory 24) Notes Test Item 4 Reasoning Mind 9 Curriculum Guide

10 Theory Block Chapter 1 Lines and Intersections (Theory 2) Lines, the most important type of a curve, are defined. (Theory 3) The ways to name a line discussed. Chapter 2 Rays (Theory 4) A ray, and its endpoint are defined. (Theory 5) The relationship for a line, ray and line segment, the numbers of endpoints, and the directions in which each is infinite is discussed. (Theory 6) Definition screen: A ray is a straight curve that has one endpoint and extends infinitely in one direction. (Theory 7) When we name a ray we say the ray s endpoint first. (Theory 8) WTD See Lesson Notes, Main Idea 1. (Theory 9) Rays can have different names depending on which point, other than the endpoint, is used. Chapter 3 Line Segments and the Length of a Segment (Theory 11) Defines the part of a line between two points as a line segment, and shows how to name the line segment. (Theory 13) The two endpoints of a segment determine the segment. If you draw two points and connect them, you have a line segment. (Theory 14) WTD See Lesson Notes, Main Idea 2. (Theory 15, 16) Describe ways to name a line segment. (Theory 17) The concept of points on or off a line segment is introduced. (Theory 18) Introduces the concept of measurement using a units of length. (Theory 19) Talks about units of measurement for length. Chapter 4 The Number Ray (Theory 21) Shows how to draw a number ray with the important components such as; the endpoint named O and labeled with a zero the unit segment labeled with a 1, and finding successive unit segments labeled with successive natural numbers. (Theory 22) Definition: A ray with its endpoint labeled zero and call the origin, and with a given unit segment is called a number ray. (Theory 24) WTD See Lesson Notes, Main Idea 4. A-Level Problems 1 Find the length of a line segment in centimeters using a ruler. (1 hint) 2 Find the names of two points on a number ray with given coordinates Similar to Theory Problem 12 Similar Theory Problem 18 3 Drag the name to the point with the given coordinate. (1 hint) See Theory Problem 15 B-Level Problems 1 Give the name and coordinate of a point whose coordinate is an amount more than the coordinate of a specified point s coordinate. 2 Give the name and coordinate of a point whose coordinate is an amount less than the coordinate of a specified point s coordinate. C-Level Problems 1 Give name and coordinate of a point whose coordinate is equal to the difference between two given numbers. 2 Give name and coordinate of a point whose coordinate is equal to the sum of two given numbers. Similar to A-Level problem 4, but with added difficulty Similar to A-Level problem 4, but with added difficulty Builds on B-Level problem 2 Builds on B-Level problem 1 Reasoning Mind 10 Curriculum Guide

11 Diagnostic Block 1 Determine how many periods in a five-digit number Related to Theory 3 2 Determine what digit is in the thousand s place. Related to Theory 5 Reasoning Mind 11 Curriculum Guide

12 4. Comparing Natural Numbers Approximate Length: 1.8 student study hours Overview This objective discusses what an inequality is, and gives rules for comparing numbers with and without using the number ray. In Chapter 1, an inequality is reviewed with the notation. Chapter 2 shows rules for comparing natural numbers. Finally, Chapter 3, numbers are compared by position on the number ray with the double inequality defined. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lesson 4. Related Objectives Objective 3. Lines, Rays and Line segments Objective 4. Comparing Natural Numbers Objective 4. Comparing Natural Numbers Objective 5. Rounding Natural Numbers Objective 7. Properties of Addition, Convenient Calculations, and Polygons Vocabulary New Terms: Glossary Terms: inequality, double inequality inequality, number ray Lesson Notes Main Idea #1 - This is important because it shows the inequality symbols and the symmetry of the inequality. Write This Down Screen (Theory 2) Notes Test Item 1 Theory Block Chapter 1 Review: Inequality (Theory 1) An inequality is a statement that one quantity is greater than or less than another. (Theory 2) WTD See Lesson Notes, Main Idea 1. Chapter 2 Rules for Comparing Natural Numbers (Theory 4,) Gives the rule to compare two natural numbers with a different number of digits. (Theory 7) Gives the rule to compare two numbers that have the same number of digits. (Theory 10) Listing numbers from least to greatest means start with the smallest number, then list next smallest and so on. When listing numbers from greatest to least means start with the largest number, then next greatest and so on. Reasoning Mind 12 Curriculum Guide

13 Chapter 3 Comparing Natural Numbers Using the Number Ray (Theory 11) Points on the number line are arranged in increasing order. The point with the smaller coordinate is to the left of the point with the greater coordinate. (Theory 13) An inequality comparing three different numbers, such as 2 < 5 < 7, is called a double inequality. Numbers in a double inequality can be place in increasing order: 2 < 5 < 7 or decreasing order: 7 > 5 > 2. Both signs in each double inequality are the same. A-Level Problems 1 Of two named points with coordinates, determine which is to the right on the number ray. 2 Mark a point on the number ray whose coordinate is between to given numbers. Similar to Theory Problem 12 Similar to Theory 14 3 Compare two numbers with the same number of digits. Similar to Theory Problem 2 4 Compare two numbers with a different number of digits. (1 hint) Similar to Theory Problem 1 5 Compare two distances to determine which is shorter. Similar to Problem 4, with review that shorter means less. 6 See the hidden puzzle pieces by solving 4 inequalities. Similar to Problems 3 and 4 with larger numbers 7 Find the smallest 4-digit number and greatest 2-digit number from a list. (1 hint) Uses Theory 7 Rule 8 Put four numbers in order from least to greatest. Similar to Theory Problem 3 B-Level Problems 1 Compare two numbers with different number of digits when one number is unknown. 2 Find the smallest seven digit number that has all different digits. (1 hint) 3 Write all the natural numbers greater than a given natural number and less than another natural number. 4 Find the 4-digit number you can add 7 to, to get the smallest 5-digit number. (1 hint) 5 Find the greatest and smallest 5-digit number with a given number in specified place. C-Level Problems 1 Put numbers in order from greatest to least when some of the digits are unknown. (1 hint) 2 Compare two numbers with the same number of digits when some of the digits are unknown. (1 hint) 3 Give all numbers between two numbers with one place value digit given. A-Level Problem 4 A-Level Problem 7 A-Level Problem 2 A-Level Problem 7 and A-Level Problem 7 Rules in Theory 4 and 7 Rule in Theory 4 Rule in Theory 7, B-Level Problem 3 Reasoning Mind 13 Curriculum Guide

14 4 Write in order from greatest to least all two-digit numbers with a given relationship between the two digits. 5 Choose the number that can be substituted as given place value to make an inequality true. Rule in Theory 4, Theory Problem 4 Rule from Theory 7 Diagnostic Block 1 Compare two natural numbers with different number of digits. (1 hint) Theory Problem 1 2 Compare two natural numbers with the same number of digits Theory Problem 2 3 Determine which of two named points with coordinates is to the right on the number ray. 4 Determine which of two numbers with the same number of digits is greater. Similar to Theory 12 Speed Game 1 Reasoning Mind 14 Curriculum Guide

15 5. Rounding Numbers Approximate Length: 2.3 student study hours Overview The objective covers rounding natural numbers to an approximate round number for a given place value. Chapter 1 begins by defining round numbers and approximate values, or estimates. In Chapter 2, the number ray is used to begin the concept of rounding by finding two round numbers closest to a given natural number on the number ray. Chapter 3 continues with finding round numbers closest to a given value according to place value of the zero farthest to the left. Chapter 4 culminates in the Rounding Rule to round any natural number to any associated place value. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lesson 5. Related Objectives Objective 3. Lines, Rays, and Line Segments Objective 4. Comparing Natural Numbers Objective 5. Rounding Numbers Objective 7. Properties of Addition, Convenient Calculation and Polygons Vocabulary New Terms: Glossary Terms: round number, rounding a number, estimate, approximate value, approximately equal approximate value, approximately equal, estimate, hundred, hundred s place, round numbers, rounding, ten, ten s place, thousand, thousand s place Theory Block Chapter 1 Round Numbers (Theory 1) Any number ending with one or more zeros is called a round number. (Theory 2) Discusses using a round number as an estimate, or an approximate value when the exact value cannot be known. (Theory 3) Gives an example of when a round number is also an exact number. Chapter 2 Rounding Numbers to an Approximate Value using the Number Ray (Theory 4) To round a number means to replace the number with a round number close to it. (Theory 5) Shows how to use the number ray to round a number. (Theory 7) We use the approximately equal sign,, and write: 13 10, and Chapter 3 Rounding Numbers to a Place Using the number Ray (Theory 10) Numbers can be rounded to different places. An example of rounding 1,274 to the hundred s place is shown. Chapter 4 The Whole Number Rounding (Theory 13) Rule screen for the Rounding Rule. A-Level Problems 1 Round a 4-digit number to the underlined place. (1 hint) Theory 14 and Theory Problem 7 Reasoning Mind 15 Curriculum Guide

16 2 Round a 3-digit number down to the ten s place. (1 hint) Theory Problem 8 3 Round a 3-digit number up to the ten s place. (1 hint) Similar to Theory Problem 6 4 Round a 4-digit number up to the hundred s place. (1 hint) Theory Problem 9 5 Round a 4-digit number down to the hundred s place. (1 hint) Theory Problem 7 B-Level Problems 1 Word problem given an approximate value that has been rounded to the ten s place. Choose the possible original number. (1 hint) 2 Round a 3-digit number up and 4-digit number down to the ten s place. (1 hint) 3 Round a 6-digit number down and a 6-digit number up to the thousand s place. (1 hint) 4 Determine which place value a 4-digit number was rounded to. (1 hint) 5 Complete a table rounding a 4-digit number to different places. (1 hint) C-Level Problems 1 Fill in the greatest number that could have been rounded to each round number. (1 hint) 2 For a given 3-digit number, determine place value rounded to for an approximate value. (1 hint) 3 Word problem to find the least number that when rounded, first to the ten s place, then to the hundred s place would result in a given 5-digit round number. (1 hint) 4 Complete a table of two values rounded to different place values. (1 hint) 5 Word problem to find the greatest and smallest original value of a number that was rounded to its highest value, then 4 times the number was rounded to the ten s place. (2 hints) Rounding Rule in Theory 13, similar to Theory 15 A-Level Problem 3 and similar to Theory 15 Rounding Rule in Theory 13 Rounding Rule in Theory 13 Rounding Rule in Theory 13, A- Level Problems 1, 4, and 5 Builds on B-Level Problem 1 Builds on B-Level Problem 4 Builds on B-Level Problem 1 Builds on B-Level Problem 5 Builds on B-Level Problem 1 Diagnostic Block 1 Round 3-digit number up to the nearest hundred. Theory Problem 6 2 Choose the approximate number after rounding a 4-digit number down to the hundred s place. Theory Problem 7 3 Round a 3-digit number down to the tens place. Theory Problem 8 4 Round a 4-digit number up to the hundreds place. Theory Problem 9 Reasoning Mind 16 Curriculum Guide

17 6. Adding Natural Numbers Approximate Length: 1.9 student study hours Overview This objective starts with situations, in the form of word problems, that require addition to solve the problem, and then defines the names of the terms of an addition equality. In Chapter 1 two types of word problems are discussed: finding the total of two values, and finding a number as a certain amount more than a known value. Chapter 2 looks at an equality with addition as the operation and defines the parts of an addition equality. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lessons 6-7. Related Objectives Objective 1. Review: Natural Numbers Objective 2. Reading and Writing Natural Numbers Objective 6. Adding Natural Numbers Objective 8. Column Addition Objective 9. Subtracting Natural Numbers Objective 10. Column Subtraction Vocabulary New Terms: Glossary Terms: shorthand, summands, sum addition, equality, summand, sum Lesson Notes Main Idea #1 - Learning the names of the parts of an addition equality will be needed to understand the rules for solving certain types of equations. Write This Down Screen (Theory 6) Notes Test Item 1 Theory Block Chapter 1 Solving Word Problems with Addition (Theory 1) Shows an example of a word problem finding how many altogether with writing the shorthand and finding the answer. (Theory 2) Shows a similar type word problem where the student fills in blanks to get the shorthand. (Theory 3) Looks at using addition to find a number that is a given amount more than a known value. (Theory 4) Shows a word problem finding a number that is a given amount more than a known value. Chapter 2 The Operation of Addition (Theory 5) Gives the names of the parts of an addition equality. (Theory 6) WTD See Lesson Notes, Main Idea 1. A-Level Problems 1 Word problem to find the total of two quantities. (1 hint) Theory Problem 3 Reasoning Mind 17 Curriculum Guide

18 2 Find the number a given amount more than a known value. Theory Problem 1 3 Find the sum of two numbers. Theory Problem 2 4 Word problem to find a number given amount more than a known number and. Theory Problem 4 5 Word Problem to find the total of two quantities. Theory Problem 3 6 Word Problem to find the total of two quantities. Theory Problem 3 7 Find next number in a sequence when each number is a certain amount more than the previous number. 8 Find two natural numbers that have a sum equal to a given number. Theory Problem 5 B-Level Problems 1 Fill in values as the sum of three numbers. (1 hint) Builds on A-Level Problems 3 and 2 2 Choose the smallest sum from a list of summands. (1 hint) Builds on A-Level Problem 3 3 Word problem to find the total from a known value and a quantity a given amount more than the known value. (1 hint) Combines A-Level Problems 4 and 5 4 Give three summands that equal a given sum. Builds on A-Level Problem 8 5 Find the number in a sequence that is between two specified values. (1 hint) Builds on A-Level Problem 7 C-Level Problems 1 Find the sum of the greatest number with specified number of digits and smallest number of another specified number of digits. 2 Find how much the sum changes if each summand is increased by a certain number. (1 hint) 3 Sort 6 numbers into two groups so the sum of the numbers in each group is equal. (1 hint) 4 Find the sum of numbers in a sequence that are between to given numbers. (1 hint) 5 Find the next three numbers in a sequence where each number is the sum of the two previous numbers. Builds on A-Level Problem 8 Builds on B-Level Problem 1 Builds on B-Level Problem 4 Builds on B-Level Problem 5 Builds on A-Level Problem 7 Diagnostic Block 1 Find the number great than a given number by a certain amount. Theory Problem 1 2 Word problem to find the total of two quantities. (1 hint) A-Level Problem 1, Theory Problem 3 Reasoning Mind 18 Curriculum Guide

19 7. Properties of Addition, Convenient Calculation and Polygons Approximate Length: 2.4 student study hours Overview The first exposure to the properties of addition is used for convenient calculations. This will allow students to more deeply understand the concepts when experienced in algebra. Chapter 1 reviews the Commutative and Associative Properties of Addition. In Chapter 2 the properties are used to add three numbers in a convenient way to allow the calculations to be done mentally. Chapter 3 defines the perimeter of a polygon and how the perimeter is calculated. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lessons 8. Related Objectives Objective 3. Lines, Rays, and Line Segments Objective 4. Comparing Natural Numbers Objective 5. Rounding Natural Numbers Objective 5. Rounding Natural Numbers Objective 7. Properties of Addition, convenient Calculations and Polygons Vocabulary New Terms: Glossary Terms: side of a polygon, vertices of a polygon, hexagon, perimeter algebraic expressions, Associative Property of Addition, Commutative Property of Addition, hexagon, pentagon, perimeter, polygon, quadrilateral, side of a polygon, segment, vertices of a polygon Lesson Notes Main Idea #1.. This important property of addition will be used for convenient calculations and prepare students for algebra. Write This Down Screen (Theory 2) Notes Test Item 1 Main Idea #2 - This important property of addition will be used for convenient calculations and to prepare students for algebra. Write This Down Screen (Theory 6) Notes Test Item 2 Reasoning Mind 19 Curriculum Guide

20 Main Idea #3 - This important property of addition will be used with the other properties of addition for convenient calculations and to prepare students for algebra. Write This Down Screen (Theory 8) Notes Test Item 3 Main Idea #4 - The perimeter is an important concept to practice the use of addition in real world applications. Write This Down Screen (Theory 17) Notes Test Item 4 Main Idea #5 - The perimeter of objects with a greater number of sides will let students expand the idea. Write This Down Screen (Theory 18) Notes Test Item 5 Reasoning Mind 20 Curriculum Guide

21 Theory Block Chapter 1 The Properties of Addition (Theory 2) WTD See Lesson Notes, Main Idea 1. (Theory 6) WTD See Lesson Notes, Main Idea 2. (Theory 8) WTD See Lesson Notes, Main Idea 3. Chapter 2 Using the Properties of Addition to Simplify Calculations (Theory 10) Shows finding the value of 7 + (13 + 8) by first finding using the Associative Property of Addition. (Theory 11) Shows finding the value of by first finding using the Commutative Property of Addition. Chapter 3 Polygons and Perimeter (Theory 13) Shows shapes that are polygons. (Theory 14) Defines the sides and the vertices of a polygon. Shows several polygons named by the numbers of sides. (Theory 16) Discussed the perimeter of a polygon. (Theory 17) WTD See Lesson Notes, Main Idea 4. (Theory 18) WTD See Lesson Notes, Main Idea 5. A-Level Problems 1 Calculate the sum of three numbers using the associative property of addition to make the calculation easier. (1 hint) Theory Problem 2 2 Calculate the sum of three numbers in a convenient way. (1 hint) Theory Problem 3 3 Calculate the sum of four numbers, where one of the numbers is 0, in a convenient way. (1 hint) 4 Calculate the perimeter of a triangle given the lengths of each side. (1 hint) 5 Calculate the perimeter of a polygon given the lengths of each side. (1 hint) B-Level Problems 1 Calculate the perimeter of a pentagon, where the lengths of four sides are given, and a relationship of the fifth side to one of the other sides. (1 hint) Theory Problem 4 Similar to Theory Problem 6 Theory Problem 7 Builds on A-Level Problem 5 2 Calculate the sum of four numbers in a convenient way. (1 hint) Builds on A-Level Problems 1 and 2 3 Calculate in a convenient way a sum of 9 numbers with consecutive digits. (1 hint) 4 Find the length of a segment chain where all but one side is known, and there is a relation between the length of the unknown side and a known side. (1 hint) C-Level Problems 1 Find the perimeter of a pentagon if three sides are known and a relationship between the unknown sides and a known side. (1 hint) Builds on A-Level Problems 1 and 2 Builds on A-Level Problem 5 Builds on B-Level Problem 1 2 Calculate the sum of 5 numbers in a convenient way where the Builds on B-Level Problem 2 Reasoning Mind 21 Curriculum Guide

22 numbers are 2-, 3- and 4-digit numbers.(1 hint) 3 Arrange 1-digit numbers around the side of a triangle so each side sums to the same number.(1 hint) 4 Find the length of a line segment, divided into three parts, given the length of one part and a relationship to the lengths of the unknown parts with the known part. Builds on B-Level Problem 3 Builds on B-Level Problem 4 Diagnostic Block 1 Connect the card with the equal expression. Theory Problem 1 2 Use the associative property of addition to add three numbers in a convenient way. (1 hint) Theory Problem 2 3 Calculate the sum three numbers in a convenient way. (1 hint) Theory Problem 3 4 Calculate the sum of four numbers, where one of the numbers is 0, in a convenient way. (1 hint) Theory Problem 4 Reasoning Mind 22 Curriculum Guide

23 8. Column Addition Approximate Length: 1.7 student study hours Overview This objective looks at column addition which is necessary when addition cannot be done mentally. Chapter 1 looks at rules for forming column addition and rules for carrying. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lesson 9. Related Objectives Objective 1. Review: Natural Numbers Objective 2. Reading and Writing Natural Numbers Objective 7. Adding Natural numbers Objective 8. Column Addition Vocabulary New Terms: Glossary Terms: bar graph, carry over, column addition Lesson Notes Main Idea #1 - Column addition is used for calculations that cannot be done mentally. Students need to understand how to correctly line up the columns and to carry when the result of a column calculation has two digits. Write This Down Screen (Theory 3) Notes Test Item 1 Theory Block Chapter 1 Column Addition (Theory 2) Shows the rule for carrying during column addition. (Theory 3) WTD See Lesson Notes, Main Idea 1. (Theory 5) Steps for column addition are given. A-Level Problems 1 Find the sum of two numbers using column addition with carrying. (1 hint) Theory Problem 1 Reasoning Mind 23 Curriculum Guide

24 2 Add two 4-digit numbers. (1 hint) Similar to Theory 4 and Theory Problem 2, adding two amounts with same number of digits 3 Add a 2-digit number to a 4-digit number. (1 hint) Builds on Theory Problem 1 and 3 4 Word problem using addition to find a number a certain amount more than a given number.(1 hint) Builds on Theory Problem 3 5 Fill in table to find totals of sales. (1 hint) Similar to Theory Problem 5 B-Level Problems 1 Reveal picture by adding numbers using column addition. Builds on A-Level Problems 2 and 3 2 Word problem to find total of three numbers and round to a specified nearest place value. (1 hint) Builds on A-Level Problem 4 3 Use a bar graph to find the sum of two 3-digit numbers. (1 hint) Builds on A-Level Problem 5 C-Level Problems 1 Fill in the blanks for column addition of two 5-digit numbers with carrying. (1 hint) 2 Word problem to find the sum of three 5-digit numbers and round the sum to a specified place.(1 hint) 3 Find and add values from a graph to find a total of three 5-digit numbers. Builds on Theory Problem 1, B- Level Problem 1 Builds on B-Level Problem 2 Similar to B-Level Problem 3 Diagnostic Block 1 Add a 3-digit number to a 4-digit number using column addition. (1 hint) 2 Word problem using addition to find a number a certain amount more than a given number.(1 hint) Similar to Theory Problem 1 and A-Level Problem 1 A-Level Problem 4 3 Fill in totals in a table. (1 hint) Theory Problem 5 Reasoning Mind 24 Curriculum Guide

25 9. Subtracting Natural Numbers Approximate Length: 1.2 student study hours Overview This objective starts with examples of situations where subtraction is used to find the answer, then looks at the names of the parts of a subtraction equality and properties of subtraction. In Chapter 1 two situations are shown; finding an unknown summand, and finding a number from a known quantity and a certain amount less. Then, Chapter 2 covers the names of a subtraction equality and the Zero Property of Subtraction and subtracting a number from itself. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lessons Related Objectives Objective 1. Review: Natural Numbers Objective 2. Reading and Writing Natural Numbers Objective 6. Adding Natural Numbers Objective 9. Subtracting Natural Numbers Objective 10. Column Subtraction Vocabulary New Terms: Glossary Terms: subtraction, minuend difference, minuend, subtraction, subtrahend, sum, summand Lesson Notes Main Idea #1 - Using words, like decreasing, to indicate subtraction will help students understand word problems where subtraction is needed. Write This Down Screen (Theory 5) Notes Test Item 1 Main Idea #2 - Knowing the names of the parts of a subtraction equality will help students understand rules for solving equations that involve subtraction Write This Down Screen (Theory 8) Notes Test Item 2 Reasoning Mind 25 Curriculum Guide

26 Main Idea #3 - This important property of subtraction will be used for convenient calculations and to prepare students for algebra. Write This Down Screen (Theory 10) Notes Test Item 3 Main Idea #4 - This important property of subtraction will be used for convenient calculations and to prepare students for algebra. Write This Down Screen (Theory 12) Notes Test Item 4 Theory Block Chapter 1 Solving Word Problems Using Subtraction (Theory 1) Shows solving a word problem by finding the missing summand. (Theory 2) Gives the definition of subtraction with the minus sign, and what we say and write when we subtract one number from another. (Theory 5) WTD See Lesson Notes, Main Idea 1. (Theory 6) Shows a word problem to find a number a certain amount less than a known quantity. Chapter 2 The Operation of Subtraction (Theory 7) Defines the parts of a subtraction equality. (Theory 8) WTD See Lesson Notes, Main Idea 2. (Theory 9) Discusses the zero property of subtraction. (Theory 10) WTD See Lesson Notes, Main Idea 3. (Theory 12) WTD See Lesson Notes, Main Idea 4. A-Level Problems 1 Find the difference of a 1-digit number and a 2-digit numbers. Similar to operations shown in Theory 7, Speed Game 1 2 Find the difference of two 2-digit numbers Similar to operations shown in Theory 7 3 Find the difference when the minuend and subtrahend are given. (1 hint) 4 Find a number given a certain amount less than a known value. (1 hint) 5 Solve a word problem to find an amount that is a certain quantity less than a known amount. (1 hint) Theory 7 Theory 4 Theory Problem 3 Reasoning Mind 26 Curriculum Guide

27 6 Solve a word problem to find an amount that is a certain quantity less than a known amount. (1 hint) B-Level Problems 1 Find the total, given one part and a relationship of the second part to the first part. (1 hint) 2 Find the next three numbers in a sequence when each number is a certain amount less than the previous number. (1 hint) C-Level Problems 1 Determine the effect on the difference if the subtrahend, in a given subtraction problem, is decreased by an amount. Write the resulting expression and calculate its value. (1 hint) 2 Word problem to find the total length of a rope and the length of two parts given the length of one part, and the relationship to other parts. (1 hint) Theory Problem 3 Builds on A-Level Problems 5 and 6 Builds on A-Level Problem 1 Builds on B-Level Problem 1 Builds on B-Level Problem 1 Diagnostic Block 1 Find the difference by subtracting a 1-digit number from a 2-digit number. Speed Game 1 2 Find a number that is a given amount less than a known number. Theory 4 3 Word Problem to find a quantity that is a certain amount less than a known quantity. (1 hint) Theory Problem 3 Reasoning Mind 27 Curriculum Guide

28 10. Column Subtraction Approximate Length: 2.1 student study hours Overview This objective looks at column subtraction which is necessary when subtraction cannot be done mentally. Chapter 1 looks at how to use column subtraction including with borrowing and trading. Then in Chapter 2, the idea of using subtraction to compare numbers is discussed. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lessons Related Objectives Objective 1. Review: Natural Numbers Objective 2. Reading and Writing Natural Numbers Objective 6. Adding Natural Numbers Objective 9. Subtracting Natural Numbers Objective 10. Column Subtraction Vocabulary New Terms: Glossary Terms: borrowing, column subtraction, price Lesson Notes Main Idea #1 - This is important because it shows how to compare numbers using subtraction and the symmetry of the comparison. Write This Down Screen (Theory 10) Notes Test Item 1 Theory Block Chapter 1 Column subtraction (Theory 1) Shows an example of subtracting a 3-digit number from a 4-digit number. (Theory 2) Shows an example of column subtraction with borrowing and trading. (Theory 3, 4) Students can set up a subtraction problem by lining up the minuend and subtrahend and borrowing if necessary. (Theory 5) Shows the steps for Column Subtraction. (Theory 6, 7) Shows column subtraction with borrowing from two places to the left. Chapter 2 How Many More, How Many Fewer (Theory 8) Subtracting two numbers tells us how much more the larger number is, and how much less the smaller number is. (Theory 10) WTD See Lesson Notes, Main Idea 1. Reasoning Mind 28 Curriculum Guide

29 A-Level Problems 1 Subtract using column subtraction borrowing from the ten s place for a problem with the minuend and subtrahend already lined up. 2 Subtract using column subtraction to find the difference of 3- and 4-digit numbers with borrowing from the thousand s place. (1 hint) Similar to Theory Problem 3 Similar to Theory Problem 3 3 Find the difference of two 4-digit numbers. (1 hint) Similar to Theory Problem 3 4 Word problem to find how much less the smaller number is than the bigger number. 5 Word problem to find how much less a smaller number is than a bigger numbers by reading values from a picture. Theory Problem 8 Similar to Theory Problem 8 B-Level Problems 1 Evaluate expressions with addition and subtraction to determine which are equal in value. (1 hint) 2 Word problem to find the missing summand, then find how much more the greater summand is than the lesser summand. (1 hint) 3 Find the difference between a number and the sum of other numbers by reading the values from a pie chart. (1 hint) Builds on A-Level Problem 2 and Objective 8 A-Level Problem 1 Objective 9 Theory 1 and builds on A-Level Problem 5 from this objective. Builds on Objective 8 C-Level Problem 3 and A-Level Problem 5 from this objective C-Level Problems 1 Fill in the blanks to make correct column subtraction. (1 hint) Builds on Theory Problem 2 2 Word problem to find a missing third summand, by first finding the sum, then finding another summand based on the relationship to a known summand.(1 hint) 3 Read sales from a graph, then compare total sales for three products to find the worst total sales over a specified time period. Then, compare the product with lowest sales to company goal for sales. (1 hint) Diagnostic Block 1 Fill in blanks to subtract a 2-digit number from a 3-digit number without borrowing. 2 Subtract two 3-digit numbers with borrowing from the ten s place, then from the hundred s place. (1 hint) Builds on B-Level Problem 3 Builds on B-Level Problem 3 Theory Problem 1 Theory Problem 2 3 Subtract two 3-digit numbers with borrowing from the ten s place. (1 hint) Theory Problem 3 4 Subtract two 3-digit numbers with borrowing from the ten s place. (1 hint) Theory Problem 4 Reasoning Mind 29 Curriculum Guide

30 11. Numerical Expressions with Parentheses Approximate Length: 1.9 student study hours Overview This objective covers numerical expressions containing one or two sets of parentheses and with only addition and subtraction as the operations. Chapter 1 covers expressions and evaluating expression with the rule for doing parentheses first. In Chapter 2 word problem solutions which involve using parentheses in forming an expression are discussed. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lessons Related Objectives Objective 10. Column Subtraction within 1,000 Borrowing Once (Fourth Grade) Objective 15. Expressions for Solving Word Problems Part 2(Fourth Grade) Objective 21. Order of Operations in Expressions with Parentheses (Fourth Grade) Objective 11. Numerical Expressions with Parentheses Vocabulary New Terms: Glossary Terms: value, evaluate, numerical expressions evaluate, expression, numerical expression, parentheses, value of an expression Lesson Notes Main Idea #1 - This is important to remember in order of operations to get the correct answer. Write This Down Screen (Theory 4) Notes Test Item 1 Theory Block Chapter 1 Evaluating Numerical Expressions (Theory 1) A numerical expressions tells us what operations to perform and in what order. (Theory 2) Looks at order of operations, and defines finding the value of a numerical expression. (Theory 3) A remember screen combining ideas from the previous two Theories. (Theory 4) WTD See Lesson Notes, Main Idea 1. (Theory 7) If there are two sets of parentheses, we have to find the values within each set of parentheses, from left to right. Chapter 2 Solving Problems Using Numerical Expressions (Theory 9, 10) Shows a word problem that has a sequence of two different actions. A-Level Problems 1 Evaluate an expression with one set of parentheses. Theory Problem 1 2 Evaluate an expression with one set of parentheses. Theory Problem 2, 3 3 Evaluate an expression with one set of parentheses. Similar to Theory Problem 1 4 Evaluate an expression with two sets of parentheses. Similar to Theory Problems 4,5 Reasoning Mind 30 Curriculum Guide

31 5 Evaluate an expression with two sets of parentheses. Similar to Theory Problems 4,5 6 Find the expression to solve a word problem with two actions in sequence, then evaluate the expression. Similar to Theory Problem 7 B-Level Problems 1 Evaluate an expression with one set of parentheses. Builds on A-Level Problem 2 2 Evaluate an expression with two sets of parentheses. Builds on A-Level Problems 4, 5 3 Word problem to find how much is left after buying two items. Theory 10 C-Level Problems 1 Give an expression that is the difference between a given expression and a number. 2 Fill in the blank to make an equality with one set of parentheses true. 3 Choose the expression to find the answer to a word problem to find the sum of two expressions, each showing the relationship of two numbers to a given number. 4 Determine the total length of a line segment given the length of one part and the relationship of the lengths of the two other parts to the known length. Builds on B-Level Problem 2 Builds on B-Level Problem 1 Builds on Theory Problem 6 Builds on C-Level Problem 3 Diagnostic Block 1 Evaluate an expression with one set of parentheses. Theory Problem 1 2 Evaluate an expression with one set of parentheses. Theory Problem 2 3 Evaluate an expression with two sets of parentheses. Theory Problem 4 4 Find the expression to show the result of two actions in sequence. Theory Problem 6 Reasoning Mind 31 Curriculum Guide

32 12. Multiplying Natural Numbers Approximate Length: 1.2 student study hours Overview In this objective the topic of multiplication as a short form of addition is introduced. Word problem situations are shown that use multiplication to find the answer. Chapters 1 and 2 give the definition of multiplication and the names of the parts of a multiplication equality. Then, Chapter 3 shows two different word problem situations that use multiplication; finding the total of the same number of objects in a certain number of containers, and finding a value that is a certain multiple of a given number. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lesson 19. Related Objectives Objective 12. Multiplying Natural Numbers Objective 13. Properties of Multiplication Objective 16. Dividing Natural Numbers Objective 17. Dividing Round Numbers and Finding How Many Times More/Less Vocabulary New Terms: Glossary Terms: multiplication, factor, product, times more factor, multiplication, product, summand, times more Lesson Notes Main Idea #1 - The definition of multiplication is important to understand the multiplication tables and to prepare students for algebra. Write This Down Screen (Theory 4) Notes Test Item 1 Main Idea #2 - Knowing the names of the parts of a multiplication equality will be needed to understand the properties of multiplication and to solve equations with multiplication. Write This Down Screen (Theory 6) Notes Test Item 2 Reasoning Mind 32 Curriculum Guide

33 Theory Block Chapter 1 The Definition of Multiplication (Theory 1) Gives the definition of multiplication with examples. (Theory 4) WTD See Lesson Notes, Main Idea 1. Chapter 2 Factors and Products (Theory 5) Gives the names of the parts of a multiplication equality. (Theory 6) WTD See Lesson Notes, Main Idea 2. Chapter 3 Solving Word Problems Using Multiplication (Theory 7) Shows a word problem, with the same amount of items in a certain number of containers in the problem story, then includes the shorthand. (Theory 8) Shows a word problem, with the situation that there are a certain number times more of one quantity than another quantity. A-Level Problems 1 Write a sum of equal summands as multiplication. (1 hint) Theory Problem 1 2 Find the value of an expression with multiplication. Review from Multiplication Basics SG 1 Speed Game to find the products Builds on A-Level Problem 1 3 Find the number a given amount more than a given number. (1 hint) 4 Find the number a given amount more than a given number. (1 hint) 5 Word problem to find total in certain number of groups, given the amount in one group. 6 Write expression to find answer to word problem to find an amount a certain times more than a given amount. (1 hint) Theory Problem 5 Theory Problem 5 Theory Problem 2 Theory Problem 6 B-Level Problems 1 Find how many total objects are in three boxes given the number of objects in a one box and a relationship to the number in the first box for the other two boxes. C-Level Problems 1 Find how many total people are in four groups given the number of people in a one group and a relationship to the number in the other groups. (1 hint) Builds on A-Level Problem 6 Builds on B-Level Problem 1 Diagnostic Block 1 Write the sum as multiplication. (1 hint) Theory Problem 1 2 Find a number a certain amount times more than a given number. Theory Problem 5 3 Word problem given how many are in one container, find the total in a certain number of containers with same number of objects. (1 Theory Problem 3 Reasoning Mind 33 Curriculum Guide

34 hint) 4 Word problem to find how many are a certain number times more than a given amount. (1 hint) Theory Problem 4 Reasoning Mind 34 Curriculum Guide

35 13. Properties of Multiplication Approximate Length: 2.3 student study hours Overview This objective investigates several important properties of multiplication. In Chapter 1, the Commutative Property is discussed. This is followed in Chapter 2 by multiplication by 10, 100 and 1,000. Chapter 3 covers the Associative Property of Multiplication. Next, in Chapter 4, the properties of multiplying by 0 and 1 are shown. Finally, Chapter 5 ends by showing multiplying by any round number. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lesson 20 Related Objectives Objective 12. Multiplying Natural Numbers Objective 13. Properties of Multiplication Objective 18. Dividing round Numbers and Finding How Many Times More/Less Vocabulary New Terms: Glossary Terms: Commutative Property of Multiplication, Associative Property of Multiplication, Zero Property of Multiplication, One Property of Multiplication Associative Property of Multiplication, Commutative Property of Multiplication, One Property of Multiplication, Zero Property of Multiplication Lesson Notes Main Idea #1 - This is an important property for convenience calculation and to prepare students for algebra. Write This Down Screen (Theory 2) Notes Test Item 1 Main Idea #2 - Multiplying by 10, 100 and 1,000 is important in mental math and later in multiplying decimals by these numbers. Write This Down Screen (Theory 6) Notes Test Item 2 Reasoning Mind 35 Curriculum Guide

36 Main Idea #3 - Understanding the associative property of multiplication will help with mental math, solving equations and prepare students for algebra. Write This Down Screen (Theory 10) Notes Test Item 3 Main Idea #4 - The zero property of multiplication will be helpful with mental math, solving equations and preparing students for algebra. Write This Down Screen (Theory 14) Notes Test Item 4 Main Idea #5 - The one property of multiplication will be helpful with mental math, solving equations and preparing students for algebra. Write This Down Screen (Theory 18) Notes Test Item 5 Main Idea #6 - Multiplying round numbers can be confusing in how to handle the zeros at the end of each number. This rule allows students to do it mentally and with fewer mistakes. Write This Down Screen (Theory 22) Notes Test Item 6 Reasoning Mind 36 Curriculum Guide

37 Theory Block Chapter 1 The Commutative Property of Multiplication Theory 1 Shows that changing the order of factors does not change the product. (Theory 2) WTD See Lesson Notes, Main Idea 1. Chapter 2 Multiplying by 10; 100; 1,000; and so on (Theory 4) Shows multiplying by 10. (Theory 5) Shows a remember screen for multiplying by 10, 100, and 1,000. (Theory 6) WTD See Lesson Notes, Main Idea 2. Chapter 3 The Associative Property of Multiplication (Theory 10) WTD See Lesson Notes, Main Idea 3. Chapter 4 The Zero Property of Multiplication and the One Property of Multiplication (Theory 14) WTD See Lesson Notes, Main Idea 4. (Theory 18) WTD See Lesson Notes, Main Idea 5. Chapter 5 Multiplying Round Numbers (Theory 22) WTD See Lesson Notes, Main Idea 6. (Theory 24, 25) Uses the commutative and associative properties of multiplication for convenient calculations. A-Level Problems 1 Multiply a 2-digit number by ten. See Theory 7 2 Multiply a 3-digit number by 100. See Theory 8 3 Multiply a round numbers 3-digit number by ten. Similar in calculation to Theory Problem 2, 3 4 Multiply a 2-digit number by 100. Similar in calculation to Theory 7 5 Multiply a 3-digit number by 10, then by 100. (1 hint) Theory Problem 3 SG 1 Speed Game to multiply with 1 as a factor. Theory Speed Game 2 6 Multiply a number by a round number. Theory Problem 5 7 Multiply two round numbers. Similar in calculation to Theory Problem 6 8 Multiply three numbers in a convenient way. Theory Problem 9 9 Word problem to find product of two number, one of which is a round number. (1 hint) B-Level Problems 1 Two step word problem to find the sum of two products, or a product by first finding a sum for the total in one group, then how many in a certain number of groups. 2 Multiple step word problem where one step includes multiplying two round numbers. Objective 13, A-Level Problem 5 Lead to seeing different ways to make calculations Builds on A-Level Problem 9 3 Two step word problem to find the difference of two products. Builds on A-Level Problem 9 Reasoning Mind 37 Curriculum Guide

38 C-Level Problems 1 Word problem to find different ways to put a certain number of items into equal groups. 2 Solve a two-step word problem involving adding, subtracting and multiplying round numbers. Builds on Theory 1-3 Builds on B-Level 3 Diagnostic Block 1 Multiply a 2-digit number by 10. Theory 7 2 Multiply two 3-digit round numbers. Theory Problem 8 Reasoning Mind 38 Curriculum Guide

39 14. Column Multiplication by a One-Digit Number Approximate Length: 1.8 student study hours Overview This objective reviews column multiplication of 2-, 3- and 4-digit numbers by a 1-digit number. In Chapter 1, students see how to set up column multiplication and carry over when necessary. Evaluating Equations with letters is reviewed that involve column multiplication by a 1-digit number. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lessons Related Objectives Objective 27. Column Multiplication of a Three-Digit Number: Part 2 (Grade 4) Objective 14. Column Multiplication by a One-Digit Number Objective 15. Column Multiplication by a Two-Digit Number Vocabulary New Terms: Glossary Terms: carry over, column multiplication, substitution Lesson Notes Main Idea #1 - Column multiplication is important when the answer cannot be done mentally. Carrying correctly is necessary to get the right answer. Write This Down Screen (Theory 3) Notes Test Item 1 Theory Block Chapter 1 Column Multiplication by a One-Digit Number (Theory 1) Review a problem without carrying. (Theory 2) Introduces carrying. (Theory 3) WTD See Lesson Notes, Main Idea 1. (Theory 8) Discusses multiplying a round number by a one-digit number. (Theory 8, 9) Shows evaluating an expression for a certain value of the letter(s). A-Level Problems 1 Find the product by filling in the blanks. Theory Problem 1 2 Find the product of a 3-digit number and a 1-digit number by setting up the column multiplication. Similar to Theory Problem 1 Reasoning Mind 39 Curriculum Guide

40 3 Find the product of a 3-digit number and a 1-digit number by setting up the column multiplication. 4 Find the product of a 3-digit number and a 1-digit number by setting up the column multiplication.(1 hint) 5 Find the product of a 3-digit number and a 1-digit number by setting up the column multiplication. 6 Find the product of a 4-digit number and a 1-digit number by setting up the column multiplication. 7 Find the value of an expression with a letter for a given value of the letter. (1 hint) 8 Find the value of an expression with two letters for a given value of each letter. Similar to Theory Problems 1, 2 Similar to Theory Problems 1, 2 Similar to Theory Problems 1, 2 Similar to Theory Problems 1, 2 Theory Problem 5 Theory Problem 6 B-Level Problems 1 Word problem to find amount in a certain times more than another number. (1 hint) 2 Choose the expression, that includes multiplication of a 3-digit number by a 1-digit, that has the greater value. 3 Find the value of an expression with a letter that includes multiplication of a 3-digit number by a 1-digit number. Similar to Theory Problem 4 Builds on A-Level Problems 1-4, review for Order of Operations Builds on A-Level Problem 7, reviews for Order of Operations 4 Drag numbers to make the column multiplication correct. Builds on A-Level Problem 1 C-Level Problems 1 Determine which expressions have an equal value. Builds on B-Level Problem 2 2 Fill in the blanks so the column multiplication is correct. Builds on B-Level Problem 4 Diagnostic Block 1 Find the product. Theory 1, 2 and A-Level Problem 3 2 Find the product. Theory 5 3 Find the value of an expression with one letter, given a value of the letter. (1 hint) 4 Word problem to find cost of a certain number of items given the cost of 1 item. Theory 6, Theory Problem 5, A- Level Problem 7 Theory Problem 3 Reasoning Mind 40 Curriculum Guide

41 15. Column Multiplication by a Two-Digit Number Approximate Length: 3.1 student study hours Overview This objective is a continuation of the previous objective and extends column multiplication to situations where both factors have more than one digit. Chapter 1 shows how to set up column multiplication by a two-digit number, multiplying round numbers and evaluating expressions with letters. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lessons and Smarter Solving Offline Lessons 1-2. Related Objectives Objective 27. Column Multiplication of a Three-Digit Number: Part 2 (Grade 4) Objective 14. Column Multiplication by a One-Digit Number Objective 15. Column Multiplication by a Two-Digit Number Objective 46. Multiplying a Decimal by a Whole Number Objective 47. Multiplying a Decimal by a Decimal Vocabulary New Terms: Glossary Terms: carry over, column multiplication, substitution Lesson Notes Main Idea #1 - Column multiplication becomes more complicated when the number of digits in the second factor increases. Students are expected to be able to build on this example to include multiplying by numbers with any number of digits, including with carrying over. Write This Down Screen (Theory 2) Notes Test Item 1 Theory Block Chapter 1 Column Multiplication by a Two-Digit Number (Theory 1) Reviews how to set up and calculate a column multiplication calculation where both factors have more than one digit. (Theory 2) WTD See Lesson Notes, Main Idea 1. (Theory 8) Shows applying column multiplication to round numbers. (Theory 11) Find the value of an expression with a letter when the calculation involves multiplication by a two-digit number. Reasoning Mind 41 Curriculum Guide

42 A-Level Problems 1 Find the product by a 1-digit number. (1 hint) Review of Objective 14 2 Complete the calculation of multiplication by a 2-digit number by filling in the blanks. 3 Find the product by multiplying a 3-digit number by a 2-digit number. (1 hint) 4 Find the product by multiplying a 3-digit number by a 2-digit number, when the 3-digit number has zero tens. (1 hint) 5 Find the product by multiplying a 4-digit number by a 2-digit number, when the 4-digit number has zero tens. (1 hint) 6 Multiply two round numbers where one is a 3-digit number. (1 hint) 7 Multiply two round numbers where one is a 4-digit number. (1 hint) 8 Find the value of an expression with one letter and the calculation requires multiplying by a 2-digit number. (1 hint) 9 Find the value of an expression with one letter and the calculation requires multiplying by a 2-digit number. 10 Find the value of an expression with two letters and the calculation requires multiplying by a 2-digit number. 11 Find the value of an expression with two letters and the calculation requires multiplying by a 2-digit number. 12 Word problem to find total in a given number of groups from number in one group. Similar to Theory Problem 1 Theory 3, Theory Problem 3 Theory 6 Similar to Theory 6 Theory Problem 4 Theory Problem 5 Theory 11 Theory 11 Builds on Theory 11 Builds on Theory 11 Theory Problem 2 B-Level Problems 1 Find the value of an expression with two letters where one operation includes multiplying by a 2-digit number. 2 Evaluate expressions that include a step with multiplication by a 2- digit number. List the results in decreasing order. (1 hint) 3 Word problem to find total in a given number of groups from number in one group, but with extra unnecessary information. (1 hint) A-Level Problems 8-12 Builds on A-Level Problem 2 and review of order of operations Builds on A-Level Problem 12 C-Level Problems 1 Find the value of an expression with 2 letters. Builds on B-Level Problem 2 2 Fill in blanks to make a correct multiplication problem. (1 hint) Builds on A-Level Problem 2 Reasoning Mind 42 Curriculum Guide

43 3 Word problem to find total in of two different groups from number in one of each group. (1 hint) Builds on B-Level Problem 3 Diagnostic Block 1 Complete the calculation by filling in the blanks. Theory 3, 4 and A-Level Problem 2 2 Multiply two round numbers. (1 hint) Similar to Theory 10 3 Find the value of an expression with two letters and the calculation requires multiplying by a 2-digit number. 4 Word problem to find total in certain number of groups from number in one group. (1 hint) A-Level Problem 10 Theory Problem 2 Reasoning Mind 43 Curriculum Guide

44 16. Formula for the Area of Squares and Rectangles Approximate Length: 1.5 student study hours Overview This objective looks at finding measurements for area as well as developing formulas for the area of a square and a rectangle. In Chapter 1, measuring area based a unit area square is discussed. Chapter 2 discusses different units for measuring area. Finally, in Chapter 3, formulas for the areas of a square and rectangle are given. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lessons Related Objectives Objective 33. Area and Perimeter of a Rectangle (Grade 3) Objective 16. Formula for the Area of Squares and Rectangles Vocabulary New Terms: Glossary Terms: square centimeter, square meter area, formula, rectangle, square, unit Lesson Notes Main Idea #1 - Converting from a larger unit measurement to a smaller unit of measurement is an important concept. In particular, for area the numbers can get very large. Write This Down Screen (Theory 8) Notes Test Item 1 Main Idea #2 - The area of a rectangle is an important formula. If we think of length as representing the number of unit square in a row, then width tells us how many rows. This is an application of multiplication as finding the sum with like summands. Write This Down Screen (Theory 11) Notes Test Item 2 Reasoning Mind 44 Curriculum Guide

45 Main Idea #3 - The area of a square is an important concrete way to understand multiplying a number by itself. Write This Down Screen (Theory 15) Notes Test Item 3 Theory Block Chapter 1 Area Basics (Theory 3) Looks at measuring area using a unit area measurement. Chapter 2 Units of Area (Theory 4) Defines a square centimeter. (Theory 5) Shows how to use the square centimeter to measure a surface. (Theory 7) Defines a square meter. (Theory 8) WTD See Lesson Notes, Main Idea 1. (Theory 22 Chapter 3 Formulas for the Area of a Rectangle and a Square (Theory 11) WTD See Lesson Notes, Main Idea 2. (Theory 15) WTD See Lesson Notes, Main Idea 3. A-Level Problems 1 Give the formula for the area of a square and then calculate the area of a given square. 2 Give the formula for the area of a rectangle and then calculate the area of a given rectangle. B-Level Problems 1 Find the area of a rectangle with known length and relationship between length and width. C-Level Problems 1 Word problem to find the perimeter of a square with side lengths the same as the calculated length of a rectangle. Theory Problem 5 and Theory 15 Theory Problem 3 and Theory 11 Builds on A-Level Problem 2 Builds on B-Level Problem 1 Diagnostic Block 1 Find the area of a rectangle with given length and width. Theory Problem 3 2 Find the area of a square with given side length. Theory Problem 5 Reasoning Mind 45 Curriculum Guide

46 17. Dividing Natural Numbers Approximate Length: 1.7 student study hours Overview This objective reviews dividing natural numbers and important properties and applications. In Chapter 1, the operation of division is reviewed as well as the names of the parts of a division equality. Chapter 2 continues with important properties and facts about special cases of division equalities. Finally, Chapter 3 looks at the situation of finding a number a certain amount times less than a given number. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lesson 29. Related Objectives Objective 12. Multiplying Natural Numbers Objective 17. Dividing Natural Numbers Vocabulary New Terms: Glossary Terms: dividend, divisor, quotient, Division Property of One, times less dividend, division, Division Property of One, divisor, factor, quotient Lesson Notes Main Idea #1 - Understanding the connection between multiplication and division allows students to use the multiplication table to solve division problems and understand why we cannot divide by zero. Later, in working with equations, it will be important to recognize division as the inverse operation of multiplication. Write This Down Screen (Theory 2) Notes Test Item 1 Main Idea #2 - It is important to know the names of the parts of a division equality to understand the rules for solving equations with division as the operation. Write This Down Screen (Theory 5) Notes Test Item 2 Reasoning Mind 46 Curriculum Guide

47 Main Idea #3 - This very important concept in math will be used in algebra as a cautionary check when finding solutions for equations and evaluating rational expressions. Write This Down Screen (Theory 7) Notes Test Item 3 Main Idea #4 -The One Property of Division will be extremely useful in solving and simplifying equations with multiplication and division as the operation. Write This Down Screen (Theory 9) Notes Test Item 4 Main Idea #5 - This concept will be extremely useful in solving equations with multiplication and division as the operation, as well as finding equivalent fractions. This explains why multiplying the numerator and denominator by the same number does not change the value of the fraction. Write This Down Screen (Theory 11) Notes Test Item 5 Main Idea #6 - This concept will be extremely useful in solving equations with multiplication and division as the operation. Write This Down Screen (Theory 13) Notes Test Item 6 Reasoning Mind 47 Curriculum Guide

48 Theory Block Chapter 1 The Operation of Division (Theory 2) WTD See Lesson Notes, Main Idea 1. (Theory 3) Reviews the shorthand for finding a missing factor. (Theory 4) Defines the names of a division equality. (Theory 5) WTD See Lesson Notes, Main Idea 2. Chapter 2 Properties of Division (Theory 6) Explains why we cannot divide by zero. (Theory 7) WTD See Lesson Notes, Main Idea 3. (Theory 9) WTD See Lesson Notes, Main Idea 4. (Theory 11) WTD See Lesson Notes, Main Idea 5. (Theory 13) WTD See Lesson Notes, Main Idea 6. Chapter 3 Solving Word Problems Using Division (Theory 15) Defines times less. (Theory 16) Reviews shorthand for situation of finding a value that is a certain amount times less than is a given number. (Theory 17) Shows a word problem where we need to find a value a certain amount less than a given number, and a certain amount times less than the same number. A-Level Problems SG1 Review of division facts. Theory Speed Game 1 1 Find the quotient when the divisor is 1. Theory 8 2 Find the quotient when dividing a number by itself. Theory 10 3 Find the quotient when the dividend is zero and the divisor is a non-zero number. Theory 12 4 Find the quotient given the divisor and the dividend. (1 hint) Theory 4 5 Word problem to find how many in one container given the total in a certain number of containers. (1 hint) Theory Problem 1 6 Find a number a certain times less than a given number. Theory Problems 2, 3 7 Word problem with situation of finding a number a certain times less than a given number. Theory 16 B-Level Problems 1 Find the quotients to reveal a picture. A-Level Speed Game 1 2 Use multiplication to check the answer to a division problem. (1 hint) 3 Choose the relationship when the dividend or divisor is made a certain amount smaller with correct change to quotient. 4 Find the total of a value divided into two parts from knowing that the unknown part is a certain amount times less than the known part. Builds on Theory 1 Builds on Theory 15 Builds on A-Level Problem 7 C-Level Problems 1 Choose the correct relationship after changing the dividend and divisor with correct change to quotient. (1 hint) Builds on B-Level Problem 3 Reasoning Mind 48 Curriculum Guide

49 2 Find the perimeter of a triangle with one known side length, one length is a certain amount times shorter than the known length and the third side is a certain amount shorter than the known length. (1 hint) Builds on Theory 17 Diagnostic Block 1 Find the quotient given the divisor and the dividend. (1 hint) A-Level Problem 4 and Theory 4 2 Find the number a certain amount times less than a given number. Theory Problem 2 3 Word problem to find how many in one container given the total in a certain number of containers. (1 hint) 4 Word problem with situation of finding a number a certain times less than a given number. Theory Problem 1 Theory 16 Reasoning Mind 49 Curriculum Guide

50 19. Dividing Round Numbers and Finding How Many Times More/Less Approximate Length: 2.1 student study hours Overview This objective looks at division with round numbers with an emphasis on how to do it mentally. Chapter 1 looks at dividing by 10, 100, and 1,000, then by general 2-,3- and 4-digit round numbers. In Chapter 2, the concepts of times more and times less for comparing numbers using division is reviewed. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lessons and Smarter Solving Offline Lesson 3. Related Objectives Objective 12. Multiplying Natural Numbers Objective 13. Properties of Multiplication Objective 18. Dividing Round Numbers and Finding How Many Times More/Less Vocabulary New Terms: Glossary Terms: average, dividend, division, divisor, mean, round number, quotient Lesson Notes Main Idea #1 - Understanding dividing a round number by a round number is important as a quick way to mentally find quotients in special situations. Write This Down Screen (Theory 3) Notes Test Item 1 Main Idea #2 - This screen is important because it gives students an easy way to approach dividing a round number by a round number when the number of zeros on the right side is not the same with each round number. Write This Down Screen (Theory 9) Notes Test Item 2 Reasoning Mind 50 Curriculum Guide

51 Main Idea #3 - It is important to recognize the symmetry of comparing two numbers with division. We find how many times more the larger number is than the smaller number, and how many times less the smaller number is than the larger number. Write This Down Screen (Theory 15) Notes Test Item 3 Theory Block Chapter 1 Dividing Round Numbers by Round Numbers (Theory 1) Reviews how to divide a round number by ten. (Theory 3) WTD See Lesson Notes, Main Idea 1. (Theory 9) WTD See Lesson Notes, Main Idea 2. Chapter 2 How Many Times More, How Many Times Less (Theory 14) Develops the understanding of times more and times less as expressing a comparison. (Theory 15) WTD See Lesson Notes, Main Idea 3. A-Level Problems 1 Find the quotient of a round number divided by 10. Theory Problem 2 2 Find the quotient of a round number divided by 100. Theory Problem 4 3 Find the quotient of a round number divided by 1,000. Theory Problem 5 4 Find the quotient of a round number divided by 100. Theory Problem 3 5 Find how many times more a larger round number is than smaller round number. 6 Find how many times less a smaller round number is than a larger one. 7 Word problem to determine how many times smaller one round number is than another.(1 hint) Theory Problem 10 Theory Problem 11 Similar to Theory Problem 12 B-Level Problems 1 Uncover a picture by dividing a round number by another round number in three situations. (1 hint) 2 Word problem to find how many times heavier one thing is than another. (1 hint) 3 Word problem to find, after rounding numbers, how many times bigger the sum of two quantities is than the third quantity. Builds on A-Level Problems 1-4 Builds on A-Level Problem 7 Builds on A-Level Problem 7 Reasoning Mind 51 Curriculum Guide

52 C-Level Problems 1 Uncover picture by finding the values of expressions given by descriptions of the equations. 2 Solve a word problem in which the needed operations include dividing a round number by two different round numbers. (1 hint) 3 Determine the relationship between a change in the dividend and divisor and a change in the quotient. Builds on B-Level Problem 1 Builds on B-Level Problem 2 Builds on C-Level Problem 1 in Objective 18 Diagnostic Block 1 Divide a round number by ten. Theory 4, 5 2 Divide a round number by 100. Theory 4 3 Divide a round number by a round number. (1 hint) Theory Problem 8 4 Find how many times less a smaller round number is than a larger one. Theory Problem 11 Reasoning Mind 52 Curriculum Guide

53 19. Division with a Remainder Approximate Length: 2.5 student study hours Overview This objective looks at division with a remainder which allows for division when the divisor is not a factor of the dividend. Chapter 1 Defines division with a remainder, how to write a the equality, names for parts of a division with a remainder equality, and what it means for a number to be divisible by another number. In Chapter 2, the student is shown, using a multiplication table, how to find the quotient when there will be a remainder. Finally, in Chapter 3, the student sees how to guess the quotient of two 2-digit numbers and check the answer. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lessons Related Objectives Objective 18. Division with a Remainder (Grade 4) Objective 20. Division with a Remainder Vocabulary New Terms: Glossary Terms: divisible, remainder dividend, divisible, division, divisor, quotient, remainder Lesson Notes Main Idea #1 - Dividing with a remainder extends the concept of dividing whole numbers to cases where the divisor is not a factor of the dividend. Write This Down Screen (Theory 3) Notes Test Item 1 Main Idea #2 - It is important for students to understand the remainder is less than the divisor, otherwise the quotient should be bigger. Write This Down Screen (Theory 8) Notes Test Item 2 Reasoning Mind 53 Curriculum Guide

54 Main Idea #3 - Students can begin to understand two different mathematical phrases can mean the same thing. This allows the students to equate concepts in a similar way to equating expressions and numbers. Write This Down Screen (Theory 10) Notes Test Item 3 Theory Block Chapter 1 What is Division with a Remainder? (Theory 1) Looks at an example of when a number is divisible by another number and when it is not. (Theory 2) Gives an example of division with a remainder, and shows the symbols to show it. (Theory 3) WTD See Lesson Notes, Main Idea 1. (Theory 4) The names of the parts of a division equality are the same, but now we call what is left over a remainder. (Theory 7) Looks at the comparison between the divisor and the remainder in examples of division with a remainder. (Theory 8) WTD See Lesson Notes, Main Idea 2. (Theory 9) Looks again at the idea of divisible introduced in Theory 1. (Theory 10) WTD See Lesson Notes, Main Idea 3. (Theory 12) Explores dividing a smaller number by a larger number using the idea of division with a remainder. Chapter 2 Division with the Multiplication Table (Theory 13) Gives an example of using the multiplication table to divide with a remainder. Chapter 3 Division with Guess and Check (Theory 16) Gives an example of guessing the quotient of two 2-digit number and checking to see if the remainder is less than the divisor. A-Level Problems 1 Fill in the blanks to give the quotient and remainder. Theory Problem 1 2 Fill in the blanks to give the quotient with the remainder given. (1 hint) Theory Problem 1 SG1 Speed Game to fill in the blanks to give the quotient and remainder. Theory Speed Game 1 B-Level Problems 1 Fill in the blank with the correct dividend given the divisor, quotient and remainder. (1 hint) 2 Reveal the picture by finding a sum, or difference, then taking that number and dividing with a remainder. Builds on A-Level Problem 1 Builds on A-Level Problem 1 3 Divide a 3-digit number by a 2-digit number. Theory 17 4 Word problem using division with a remainder to find. (1 hint) Builds on A-Level Problems 1, 2 Reasoning Mind 54 Curriculum Guide

55 C-Level Problems 1 Fill in a table for values of x and y given the relationship between x and y, then find x y by dividing with a remainder. Builds on B-Level Problem 2 2 Calculate the dividend, then divide with a remainder. Builds on B-Level Problem 1 3 Word problem using division to find the missing number in each group given the total and number of groups. Then answer questions about the groups. Builds on Theory 1 and 2 Diagnostic Block 1 Fill in the blanks to give the quotient and remainder. Theory Problem 3 2 Given the divisor, quotient and remainder, find the dividend. (1 hint) Builds on Theory Problem 1 3 Fill in the blanks to give the quotient and remainder. (1 hint) Theory Problem 4 Reasoning Mind 55 Curriculum Guide

56 20. Long Division Approximate Length: 3.5 student study hours Overview This objective reviews the steps in various long division problem situations. Chapter 1 deals with long division by a 1-digit number including situations with the dividend evenly dividing the first digit of the dividend, and a dividend with a zero place value. In Chapter 2, long division with by a 2-digit number is shown. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lessons Related Objectives Objective 27. Column Multiplication of a Three-Digit Number: Part 2 (Grade 4) Objective 20. Long Division Objective49. Dividing a Decimal by a Whole Number Objective50. Dividing a Decimal by a Decimal Vocabulary New Terms: Glossary Terms: dividend, divisor, long division, remainder, quotient Lesson Notes Main Idea #1 - This example is a reference for student to understand and work a long division problem. Write This Down Screen (Theory 3) Notes Test Item 1 Theory Block Chapter 1 Long Division: Dividing by One-Digit Numbers (Theory 1) Describes how to write a long division problem. (Theory 3) WTD See Lesson Notes, Main Idea 1. (Theory 7) Shows a dividend with 4-digits. (Theory 8) Shows long division in which there is no remainder when dividing the hundreds place. (Theory 11) Shows dividing a 3-digit dividend with a zero in the tens place. Chapter 2 Long Division: Dividing by Two-Digit Numbers (Theory 12) Shows the steps for dividing a 3-digit number by a 2-digit number. (Theory 13) Shows a 4-digit dividend. Reasoning Mind 56 Curriculum Guide

57 A-Level Problems 1 Divide a 2-digit number by a 1-digit number using long division. Theory 1 2 Divide a 3-digit number by a 1-digit number using long division. (1 hint) 3 Divide a 3-digit number by a 2-digit number using long division. (1 hint) 4 Evaluate an expression with two letters by substituting in given values. (1 hint) 5 Word problem which requires dividing a 3-digit number by a 1-digit number in order to find a number a certain amount times less than another value. 6 Find the length of a part of a whole divided into a certain number of pieces. (1 hint) Theory 2, Theory Problem 1 Theory Problem 6 Builds on Theory 1 Theory Problem 3 Builds on Theory Problem 4 B-Level Problems 1 Uncover a picture by solving a variety of long division problems. Builds on A-Level Problems Solve a two-part problem where each part involves long division and other operations. 3 Find the missing part of a division problem, then match the value with a letter with same given value to spell a word. (1 hint) 4 Word problem to find a value a certain amount times less than a given value using long division with a 2-digit number. (1 hint) Builds on A-Level problems 1-4. Builds on A-Level Problems 1-4 Builds on A-Level Problem 5 C-Level Problems 1 Uncover a picture by working two-step problems with each step using long division. 2 Three-part problem where each step involves using long division and other operations. 3 Solve a series of problems to find the missing part of a division equality, then match a letter with a given value to spell a word. (1 hint) 4 Word problem that uses long division with a 2-digit number to find how a whole is divided into parts. (1 hint) 5 Solve a word problem in which long division is used as well as other operations to get the answer.(1 hint) Diagnostic Block 1 Divide a 3-digit number by a 1-digit number using long division. (1 hint) Builds on B-Level Problem 1 Builds on B-Level Problem 2 Builds on B-Level Problem 3 Builds on A-Level Problem 6 and Objective 20, C-Level Problem 3 Builds on B-Level Problem 2 A-Level Problem 1 and similar to Theory Problem 1 Reasoning Mind 57 Curriculum Guide

58 2 Find the length of a part of a whole divided into a certain number of pieces. (1 hint) 3 Word problem to find a number a certain amount times less than another given number. 4 Divide a 3-digit number by a 2-digit number using long division.(1 hint) A-Level Problem 6 and builds on Theory problem 4 Theory Problem 3 Theory Problem 6 Reasoning Mind 58 Curriculum Guide

59 21. Order of Operations Approximate Length: 1.8 student study hours Overview This Objective reviews the order of operations for increasingly complicated expressions and introduces the square and the cube of a number. In Chapter 1, The Order of Operations Rule is developed with expressions of increasing difficulty. Chapter 2 Introduces the concepts of the square and cube of a number with notation and sample calculations. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lessons Related Objectives Objective 23. Order of Operations in Expressions with Parentheses (Grade 4) Objective 21. Order of Operations Vocabulary New Terms: Glossary Terms: square, cube (of a number) addition, cube, division, evaluating a numerical expression, expressions, multiplication, numerical expressions, square (of a number), subtraction Theory Block Chapter 1 Review: Order of Operations (Theory 1) Reviews the order of operations in a numerical expression with only addition and subtraction. (Theory 2) Looks at how parentheses should be handled in a numerical expression with just addition and subtraction. (Theory 3, 4) Reviews the order of operations in a numerical expression with parentheses, but only multiplication and division as the operations. (Theory 5) Looks at the Order of Operations Rule with all operations, but no parentheses. (Theory 7) Looks at the Order of Operations Rule with all operations, and with parentheses. Chapter 2 The Square of a Number and the Cube of a Number (Theory 11) Defines a square and shows notation to express the square of a number. (Theory 14) Defines a cube and shows notation to express the cube of a number. A-Level Problems 1 Defines a square and shows how to express the square of a number. (1 hint) 2 Evaluate an expression with only multiplication and division. (1 hint) Theory Problem 6 Theory Problem 1 3 Evaluate an expression with parentheses and division. (1 hint) Theory Problem 3 4 Drag the cards to show the order of operations in a problem with three operations and parentheses. (1 hint) Theory 7 5 Calculate the cube of a number. Theory 16 6 Calculate the square of a number. Theory Problems 9, 10 Reasoning Mind 59 Curriculum Guide

60 B-Level Problems 1 Evaluate an expression with three operations and parentheses. Builds on A-Level Problem 4 2 Choose the operation done last in an expression with parentheses and several operations. Builds on A-Level Problem 4 C-Level Problems 1 Drag the cards to give the order of operations for an expression with several sets of parentheses and operations. 2 Find the original expression given the calculations and order of calculations to evaluate it. 3 Drag the correct operations for an expression to make the equality true. Diagnostic Block 1 Evaluate an expression with three operations and parentheses. (1 hint) 2 Evaluate an expression with three operations and parentheses. (1 hint) 3 Evaluate an expression with two different expressions and parentheses. Builds on B-Level Problem 2 Builds from Theory 8 Builds on B-Level Problem 1 Theory 2 Theory Problem 4 Theory Problem 6 Reasoning Mind 60 Curriculum Guide

61 22. Fractions Approximate Length: 2.4 student study hours Overview This objective introduces the idea of parts of a whole. In Chapter 1 fractions as representing part of a whole are shown with the names of the parts of a fraction. Chapter 2 looks at how fractions are read and introduces the fractions a half and a quarter. Finally, in Chapter 3, showing fractions on the number ray is discussed. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lessons Related Objectives Objective 22. Fractions Objective 23. Fractions and Division, Improper Fractions, and Comparing Fractions Objective 26. Comparing Fractions Objective 27. Adding and Subtracting Fractions with Like Denominators Objective 34. Comparing Fractions with Different Denominators Objective 35. Adding and Subtracting Fractions with Unlike Denominators Objective 40. Review: Decimals and Percentages Vocabulary New Terms: Glossary Terms: numerator, denominator, half, quarter denominator, even number, half, number ray, numerator, odd number, quarter, unit segment Lesson Notes Main Idea #1 - Students need to know the names and what they mean for all mathematical objects. Knowing the parts of a fraction will help students understand rules for working with fractions. Write This Down Screen (Theory 5) Notes Test Item 1 Main Idea #2 - These fractions are important because they represent frequently used fraction names that do not fit the usual rule for naming fractions. Also, they begin the connection of fractions with division. Write This Down Screen (Theory 18) Notes Test Item 2 Reasoning Mind 61 Curriculum Guide

62 Theory Block Chapter 1 Fraction Parts (Theory 1, 2) Reviews fractions as representing a part of a whole in different situations. (Theory 4) Defines the numerator and denominator. (Theory 5) WTD See Lesson Notes, Main Idea 1. (Theory 6-9) Shows how to represent part of the whole as a fraction. (Theory 10, 11) Relates the numerator and denominator of a fraction to the location of the number in the fraction. Chapter 2 Reading Fractions (Theory 12, 13) Shows how to read a fraction. (Theory 17) Introduces important fractions of a half and a quarter. (Theory 18) WTD See Lesson Notes, Main Idea 2. (Theory 19) Shows a word problem using a half to describe the situation. (Theory 20) Shows a word problem using a quarter to describe the situation. Chapter 3 Fractions on the Number Ray (Theory 21, 22) Shows when parts of a line segment can, or cannot be shown as a fraction. (Theory 23) Shows the idea of equal parts for rectangles and circles. (Theory 25) Shows dividing the unit segment on the number ray into six parts and labeling each division as a fraction. A-Level Problems 1 Express the shaded part of a figure as a fraction. (1 hint) Theory Problem 6 2 Express the shaded part of a figure as a fraction. (2 hint) Theory Problem 6 3 Color in the part of a circle to represent a certain fraction. (2 hints) Theory Problem 15, 17 4 Given the value of the parts of the fraction, write the fraction. (1 hint) 5 Given the value of the parts of the fraction, write the fraction. (1 hint) 6 Word problem using a quarter to understand the situation. (1 hint) Theory Problem 11, 12, 13 Theory Problem 11, 12, 13 Theory 20, Theory Problem 14 B-Level Problems 1 Mark a fraction on the unit segment of the number ray. (1 hint) Theory Problem 19 2 Give a fraction that has certain properties for the numerator and denominator. (1 hint) Builds on A-Level Problems 4, 5 C-Level Problems 1 Give a fraction that has certain properties for the numerator and denominator. (2 hints) 2 Find the next number in a sequence of fractions, then find that fraction on the unit segment of a number ray. (1 hint) Builds on B-Level Problem 2 Builds on B-Level Problem 1 Diagnostic Block 1 Fill in the blanks to show the shaded portion of a figure. (2 hints) A-Level Problem 2 2 Color in the parts to represent the given fraction. (2 hints) A-Level Problem 3 Reasoning Mind 62 Curriculum Guide

63 3 Given the value of the parts of the fraction, write the fraction. (1 hint) 4 Given the value of the parts of the fraction, write the fraction. (1 hint) A-Level Problem 4 A-Level Problem 5 Reasoning Mind 63 Curriculum Guide

64 23. Fractions and Division, Improper Fractions, and Comparing Fractions Approximate Length: 3 student study hours Overview This objective looks at the relationship between fractions and division and applications in extending the understanding of fractions. Chapter 1 begins with showing how situations of splitting a certain amount into equal shares can be thought of in terms of division and in terms of writing fractions, then shows how to write a quotient as a fraction. In Chapter 2 the concept of proper and improper fractions is introduced. The process of comparing fractions is started in Chapter 3 with writing fractions on the number ray to show proper fractions are always less than improper fractions. Chapter 4 concludes this objective with looking at how to write natural numbers as fractions. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lessons Related Objectives Objective 3. Lines, Rays and Line Segments Objective 22. Fractions Objective 23. Fractions and Division, Improper Fractions, and Comparing Fractions Vocabulary New Terms: Glossary Terms: improper fraction, proper fraction common fraction, denominator, dividend, divisor, improper fraction, natural number, number ray, numerator, proper fraction, quotient, ratio, whole number Lesson Notes Main Idea #1 - This relationship is very helpful in expressing a quotient in such a way that it can be found in various ways. Write This Down Screen (Theory 4) Notes Test Item 1 Main Idea #2 - It is important to have this explanation of how to rewrite a quotient as a fraction. Write This Down Screen (Theory 9) Notes Test Item 2 Reasoning Mind 64 Curriculum Guide

65 Main Idea #3 - It is important to be able to write a fraction as a division problem in order to express it as a decimal and change improper fractions to mixed numbers. Write This Down Screen (Theory 12) Notes Test Item 3 Main Idea #4 - Recognizing proper and improper fractions is necessary for performing operations on fractions as well comparing fractions. Write This Down Screen (Theory 17) Notes Test Item 4 Main Idea #5 - This screen helps in understanding the concept of less than one as a point on the number ray. Students will know that these numbers are coordinates for points to the left of 1 on the unit segment. Write This Down Screen (Theory 22) Notes Test Item 5 Main Idea #6 - This screen helps in understanding the concept of greater than or equal to one on the number ray. Students will know that these numbers are coordinates for points to the right of the unit segment. Write This Down Screen (Theory 27) Notes Test Item 6 Reasoning Mind 65 Curriculum Guide

66 Main Idea #7 - This is an important simple test to compare any improper fraction to any proper fraction Write This Down Screen (Theory 32) Notes Test Item 7 Main Idea #8 - Writing whole numbers as a fraction with a given denominator will be needed in applications like subtracting fractions from whole numbers. Write This Down Screen (Theory 38) Notes Test Item 8 Main Idea #9 - These examples demonstrate the simplicity of the rule for writing the number 1 as an improper fraction. Write This Down Screen (Theory 40) Notes Test Item 9 Theory Block Chapter 1 Fractions and Division (Theory 3) Discusses how a fraction bar is another way of writing the division sign. (Theory 4) WTD See Lesson Notes, Main Idea 1. (Theory 7) Begins connecting the writing of a quotient as a fraction to the case when there is or is not a remainder. (Theory 9) WTD See Lesson Notes, Main Idea 2. (Theory 12) WTD See Lesson Notes, Main Idea 3. (Theory 13-15) Shows writing a fraction as a quotient and uses the relationship to show fractions that are equal to whole numbers. Reasoning Mind 66 Curriculum Guide

67 Chapter 2 Proper and Improper Fractions (Theory 16) Defines an improper fraction. (Theory 17) WTD See Lesson Notes, Main Idea 4. Chapter 3 Comparing Common Fractions (Theory 21) Shows marking a proper fraction on the number ray and comparing its position to where the number 1 is marked. (Theory 22) WTD See Lesson Notes, Main Idea 5. (Theory 24) Shows marking a fraction in which the numerator equals the denominator on the number ray. (Theory 26) Shows marking a fraction in which the numerator is greater than the denominator on the number ray. (Theory 27) WTD See Lesson Notes, Main Idea 6. (Theory 31) Shows marking two fractions with the same denominator, one proper and one improper, on the number ray. The conclusion is reached that all improper fractions are greater than all proper fractions. (Theory 32) WTD See Lesson Notes, Main Idea 7. Chapter 4 Writing a natural number as a fraction with a given denominator (Theory 35) Shows a natural number as the quotient in a division equality, then the quotient equal to an improper fraction. (Theory 37) Shows using a rule to find a fraction with a given denominator equal to a given whole number. (Theory 38) WTD See Lesson Notes, Main Idea 8. (Theory 40) WTD See Lesson Notes, Main Idea 9. A-Level Problems 1 Write a quotient as a fraction. (1 hint) Theory Problems Find the value of an improper fraction as a whole number. (1 hint) Theory Problems 6, 8 3 Write a quotient as a fraction. (1 hint) Theory Problems Find the value of an improper fraction as a whole number. (1 hint) Theory Problems 6, 8 5 Find the value of an improper fraction as a whole number. (1 hint) Theory Problems 6, 8 6 Determine if the given fractions are proper or improper. (1 hint) Theory Problems 9-11 SG1 Speed Game to compare proper and improper fractions to 1. Theory Speed Game 3 7 Write a whole number as a fraction with a given denominator. (1 hint) Theory Problem 17, 18 8 Write the number 1 as a fraction with a given denominator. (1 hint) Speed Game 5 9 Write all the proper fractions with a given denominator. (1 hint) Theory 21, Theory 10 Write all the proper fractions with a given denominator. (1 hint) Theory 21, Theory B-Level Problems 1 Solve a word problem in which the answer is expressed as a fraction. (2 hints) 2 Determine the portions of the value of one expression to another. (1 hint) Builds on A-Level Problems 1, 3 Builds on A-Level Problems 1, 3 C-Level Problems 1 Solve a word problem in which the answer is a fraction. (2 hints) Builds on B-Level Problem 1,2 2 Find the fraction made from two numbers with the sum as the numerator and the product as the denominator. Builds on B-Level Problem 2 Reasoning Mind 67 Curriculum Guide

68 Diagnostic Block 1 Determine if the given fractions are proper or improper. (1 hint) A-Level Problem 6 2 Write a whole number as a fraction with a given denominator. (1 hint) A-Level Problem 7 3 Compare a proper and improper fraction. (2 hints) Theory Problems 15, 16 4 Compare a proper fraction to 1. (2 hints) Theory Problem 12 Reasoning Mind 68 Curriculum Guide

69 24: Mixed Numbers and Improper Fractions Approximate Length: 2.7 student study hours Overview This objective defines mixed numbers and the whole part and fractional part of a mixed number. Then looks at working with mixed numbers and the equivalent improper fractions. Chapter 1, defines and shows models of mixed numbers. Chapter 2 looks at converting an improper fraction to a mixed number using a model to understand the relationship including division with a remainder. Then, in Chapter 3, students see how to convert a mixed number into an equivalent improper fraction. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lessons Related Objectives Objective 22 Fractions Objective 24. Mixed Numbers and Improper Fractions Objective 27. Adding and Subtracting Mixed Numbers Vocabulary New Terms: Glossary Terms: mixed number, whole part, fractional part circle, denominator, division, equivalent, fraction, fractional part, improper fraction, mixed number, numerator, proper fraction, remainder, whole part Lesson Notes Main Idea #1 - It is important to understand the parts of a mixed number in order to know how to work with them. Write This Down Screen (Theory 5) Notes Test Item 1 Main Idea #2 - It will be necessary to add and subtract mixed numbers to be able to write a mixed number as a sum of its whole and fractional parts. Write This Down Screen (Theory 14) Notes Test Item 2 Reasoning Mind 69 Curriculum Guide

70 Main Idea #3 - This is an important example since many operations on mixed numbers require this skill. Write This Down Screen (Theory 31) Notes Test Item 3 Theory Block Chapter 1 Definition of a Mixed Number (Theory 1-3) Shows two ways of dividing an amount between certain number of people which will not go evenly, then shows the equality between the two expressions. (Theory 4) Shows how to write mixed numbers and defines mixed number, fractional part, and whole part. (Theory 5) WTD See Lesson Notes, Main Idea 1. (Theory 6) Notices the fractional part of a mixed number is a proper fraction. (Theory 7) Shows an example of measuring a length that is a mixed number. (Theory 8) Models a mixed number with shaded circles. (Theory 11, 12) Shows an example of choosing the mixed number modeled by the shaded area. (Theory 13) Shows representing a mixed number as the sum of its whole and fractional parts. (Theory 14) WTD See Lesson Notes, Main Idea 2. (Theory 16) Shows writing the sum of a whole number and a decimal as a mixed number. Chapter 2 Converting Improper Fractions to Mixed Numbers (Theory 18) Reviews the definition of an improper fraction. (Theory 19) Represents an improper fraction as parts of a circle, then suggests a method to know how many wholes are represented by an improper fraction. (Theory 20) Looks at dividing the numerator by the denominator as division with a remainder. (Theory 23) Gives a rule to convert an improper fraction to a mixed number. (Theory 24) Gives practice using the rule. (Theory 25-27) Gives students practice. Chapter 3 Converting Mixed Numbers to Improper Fractions (Theory 28) Shows a model of a mixed number and how to show the number of the same size pieces. Then shows the mixed number as the sum of an improper fraction, representing the whole number with the same denominator, and a proper fraction to find the correct improper fraction equal to the mixed number. (Theory 29) Gives a rule to convert a mixed number to an improper fraction. (Theory 30) Shows an example. (Theory 31) WTD See Lesson Notes, Main Idea 3. A-Level Problems 1 Express a mixed number as the sum of its whole and fractional parts. (1 hint) Theory Problems 1, 2 2 Express a sum as a mixed number. (1 hint) Theory Problems 3, 4 3 Express an improper fraction as a mixed number. (1 hint) Theory Problems Express an improper fraction as a mixed number. (1 hint) Theory Problems Express an improper fraction as a mixed number. (1 hint) Theory Problems Express a quotient as a mixed number. (2 hints) Theory Problems Express a quotient as a mixed number. (2 hints) Theory Problems 8-11 Reasoning Mind 70 Curriculum Guide

71 8 Write the mixed number as an improper fraction. (1 hint) Theory Problems Write the mixed number as an improper fraction. (1 hint) Theory Problems Write the mixed number as an improper fraction. (1 hint) Theory Problems B-Level Problems 1 Write the mixed number as an improper fraction. (1 hint) Builds on A-Level Problems Express an improper fraction as a mixed number. (1 hint) Builds on A-Level Problems Solve a word problem by dividing with a remainder and expression the quotient as a mixed number. Builds on A-Level problems 6,7 C-Level Problems 1 Write the mixed number as an improper fraction. (1 hint) Builds on B-Level Problem 1 2 Express an improper fraction as a mixed number. (1 hint) Builds on B-Level Problem 2 3 Fill in the blanks to give the whole numbers an improper fraction lies between (1 hint) Builds on A-Level Problem 1 Diagnostic Block 1 Express the mixed number as an improper fraction. (1 hint) Theory Problem 15 2 Express an improper fraction as a mixed number. (1 hint) A-Level Problem 5 3 Express a quotient as a mixed number. (2 hints) A-Level Problem 6 4 Express a sum as a mixed number. (1 hint) A-Level Problem 2 Reasoning Mind 71 Curriculum Guide

72 25. Trapezoids and Parallelograms Approximate Length: 3.3 student study hours Overview This objective covers the definition and properties of parallelograms and several special parallelograms. Chapter 1 starts with a definition of trapezoid as a special quadrilateral with only one pair of parallel sides. Chapter 2 contrasts the trapezoid with a parallelogram that has two parallel sides and gives the special properties of parallelogram. The process of adding properties to define new special parallelograms is continued in Chapter 3 with the rectangle, then in Chapter 4 with the square. Finally, in Chapter 5, the rhombus is defined with it s special properties. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lesson Vocabulary New Terms: Glossary Terms: trapezoid, bases of a trapezoid, legs of a trapezoid, isosceles trapezoid acute angle, angle, base of a trapezoid, degree, isosceles trapezoid, obtuse angle, parallel lines, proportion, protractor, rhombus, right angle, trapezoid Theory Block Chapter 1 The Trapezoid (Theory 1-3) Looks at quadrilaterals with one or two pairs of parallel sides. (Theory 4) Definition of a trapezoid. (Theory 5) Definition of the bases and legs of a trapezoid. (Theory 7) Gives examples that are and are not trapezoids. (Theory 8) Shows naming a trapezoid, the bases, and the legs. (Theory 9) Defines an isosceles trapezoid. Chapter 2 The Parallelogram (Theory 9) Gives the definition of a parallelogram. (Theory 10) Shows naming the two pairs of parallel sides. (Theory 11) Shows examples that are and are not parallelograms. (Theory 12) The opposite parallel sides of a parallelograms are of equal length. (Theory 14) Opposite angles in a parallelogram are equal. (Theory 15) Reviews that the opposite sides and angles of a parallelograms are equal. (Theory 17) Gives an example of finding the perimeter of a parallelogram given the length of two adjacent sides. Chapter 3 The Rectangle: a Special Parallelogram (Theory 18) Shows a rectangle as a special parallelogram. (Theory 19) Shows the properties of a parallelogram, such as opposite sides and angles being equal, apply to a rectangle. (Theory 21) Reviews the special properties of a rectangle. Chapter 4 The Square: a special Parallelogram (Theory 22) Shows a square as a special rectangle. (Theory 24) Notes that all squares are also parallelograms, so the properties of parallelograms will apply to squares. Chapter 5 The Rhombus: A Special Parallelogram (Theory 25) Shows a rhombus as a special quadrilateral with all sides equal. (Theory 26) Shows that a rhombus is also a parallelogram. (Theory 27) Shows finding the perimeter of a rhombus given the length of one side. (Theory 28) Summarizes the shapes discussed in this objective, with added definitions. An isosceles trapezoid is defined. Reasoning Mind 72 Curriculum Guide

73 A-Level Problems 1 Given the perimeter of a rhombus, find the length of one side. (2 hints) 2 Compare the given perimeter of a trapezoid with the perimeter of a parallelogram with adjacent sides shown. (2 hints) Builds on Theory Problem 25 Similar to Theory Problem Select the untrue statement about the properties of certain shapes. From Theory Select the true statement about a rectangle. (1 hint) Theory Find the shape that could not have parallel sides. Theory 18, 22, Find the correct statement about a trapezoid. (1 hint) Theory 4, Find the correct statement about a trapezoid.(2 hints) Theory Find the correct statement about a trapezoid.(2 hints) Theory 2, Find the correct statement about a trapezoid.(2 hints) Theory Find the correct statement about a trapezoid.(2 hints) Theory 2, 28 B-Level Problems 1 Find the length of one side of a parallelogram given the perimeter and length of the other side. (1 hint) Builds on Theory Problem 6 C-Level Problems 1 Write the mixed number as an improper fraction. (1 hint) Builds on B-Level Problem 1 Diagnostic Block 1 Find the length of the side of a rhombus given the perimeter of the rhombus, 2 Find the length of a parallelogram given the length of two adjacent sides. 3 Find the length of the side of a square given the perimeter of the square. 4 Reveal the picture by finding the perimeters of several rhombuses given the length of one of the sides. (2 hints) A-Level Problem 1 Theory Problem 6 Theory Problem 21 Theory Problem 25 Reasoning Mind 73 Curriculum Guide

74 26. Comparing Fractions Approximate Length: 1.8 student study hours Overview This objectives looks at comparing fractions with like denominators and fractions with a one in each numerators and different denominators. Chapter 1 begins the process with looking at fractions with like denominators using shapes as models. Chapter 2 continues the theory from Chapter 1 using the number ray as a model and the rule for comparing numbers on the number ray. Chapter 3 looks at dividing an object into different numbers of equal pieces and what that means in comparing fractions with each having a one in the numerator. Finally, Chapter 4 begins looking at simple cases of fractions with different numerators and denominators being equal. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lessons Related Objectives Objective 22. Fractions Objective 24. Mixed Numbers Objective 26. Comparing Fractions Objective 32. Comparing Fractions with Different Denominators Vocabulary New Terms: Glossary Terms: like denominators denominator, like denominators, number ray, numerator Lesson Notes Main Idea #1 - This rule begins the process of comparing fractions. It is easy to see that if the pieces are the same size, the fraction which represents more pieces is the greater fraction. Write This Down Screen (Theory 9) Notes Test Item 1 Main Idea #2 - This is another way to see that the greater of two fractions with like denominators will have the greater numerator. This is because we count more segments to the right. Write This Down Screen (Theory 15) Notes Test Item 2 Reasoning Mind 74 Curriculum Guide

75 Main Idea #3 - This rule is important in understanding that, since the denominator represents the total number of pieces, the larger denominator means the piece size is smaller. The one in the numerator means both fractions represent the same number of pieces. Write This Down Screen (Theory 22) Notes Test Item 3 Theory Block Chapter 1 Comparing Fractions with Like Denominators (Theory 1, 2) Review of what the denominator and numerator represent. (Theory 5) Defines and gives examples of like denominators. (Theory 8) Gives a rule for comparing fractions with like denominators. (Theory 9) WTD See Lesson Notes, Main Idea 1. Chapter 2 Using the Number Ray to Compare Fractions with Like Denominators (Theory 12-14) Models using the number ray to compare two proper fractions with like denominators. (Theory 15) WTD See Lesson Notes, Main Idea 2. Chapter 3 Comparing fractions with 1 in the Numerator (Theory 21, 22) Gives an example of comparing one whole cut into more and fewer pieces to see which is larger. (Theory 22) WTD See Lesson Notes, Main Idea 3. Chapter 4 Equal Fractions (Theory 24, 25) Uses a model of shaded parts to show two fractions with different numerators and denominators can be equal. (Theory 26) Two fractions are equal if they represent the same part of the whole. (Theory 27) Shows how to see that two fractions are equal on the number ray. (Theory 28) Two fractions are equal if they are located on the same spot on the number ray. A-Level Problems 1 Compare two fractions with like denominators, with one of the fractions having 1 in the numerator. (1 hint) Theory Problems 1 2 Compare two fractions with like denominators.(1 hint) Theory Problem 2 3 Compare two fractions with like denominators.(1 hint) Theory Problem 2 4 Compare fractions with like denominators by locations on the number ray. (1 hint) 5 Compare fractions with like denominators by locations on the number ray. (1 hint) Theory Problem 3-5 Theory Problem 3-5 Reasoning Mind 75 Curriculum Guide

76 6 Place the fractions in the correct location on the number ray using the rule to compare fractions with like denominators. (2 hints 7 Place the fractions in the correct location on the number ray using the rule to compare fractions with like denominators. (2 hints B-Level Problems 1 Put three fractions with like denominators in descending order. (2 hints) 2 Put three fractions in descending order with two having like denominators and the third fraction equal to 1. (1 hint) 3 Mark four fractions on the number ray and determine which is greatest and which is smallest. (1 hint) C-Level Problems 1 Compare two fractions with the same denominator by first evaluating the numerical expression in each numerator. 2 Compare two fractions by first evaluating the numerical expression in the numerator and denominator of one of the fractions. 3 Choose the correct comparison of fractions in which some of the digits in the numerators are missing. Theory Problem 7 Theory Problem 7 A-Level Problems 2, 3 Builds on A-Level Problems 2, 3 Builds on A-Level Problem 7 Builds on A-Level Problems 1-3 Builds on A-Level Problems 1-3 Builds on A-Level Problems 1-3 Diagnostic Block 1 Compare two fractions with like denominators.(1 hint) A-Level Problem 2 2 Compare fractions with like denominators by locations on the number ray. (1 hint) A-Level Problem 4 3 Compare two fractions with like denominators.(1 hint) A-Level Problem 3 4 Compare two fractions with 1 in each numerator and different denominators. Theory Problem 8 Reasoning Mind 76 Curriculum Guide

77 27. Adding and Subtracting Fractions with Like Denominators Approximate Length: 1.8 student study hours Overview This objective looks at the rules and applications for adding and subtracting fractions with like denominators. In Chapter 1, the addition of fractions with like denominators is investigated first using models of shapes cut into equal size pieces. Chapter 2 continues with subtracting fractions with like denominators in the same way. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lessons Related Objectives Objective 22. Fractions Objective 27. Adding and Subtracting Fractions with Like Denominators Objective 35. Adding and subtracting Fractions with Unlike Denominators Vocabulary New Terms: Glossary Terms: addition, cross terms, denominators, difference, fractional expression, numerators, subtraction, sum Lesson Notes Main Idea #1 - The rule for adding fractions with like denominators will be helpful in solving more complicated equations and word problems. Write This Down Screen (Theory 6) Notes Test Item 1 Main Idea #2 - The rule for subtracting fractions with like denominators will be helpful in solving more complicated equations and word problems Write This Down Screen (Theory 14) Notes Test Item 2 Reasoning Mind 77 Curriculum Guide

78 Theory Block Chapter 1 Adding Fractions with like Denominators (Theory 2-4) Looks at modeling adding fractions with a shape cut into equal pieces. (Theory 5) Develops a rule to add fractions with like denominators. (Theory 6) WTD See Lesson Notes, Main Idea 1. (Theory 7) Represents the rule for adding fractions with like denominators with letters. Chapter 2 Subtracting Fractions with Like Denominators (Theory 10-12) Models subtracting fractions with like denominators using shapes cut into equal pieces. (Theory 14) WTD See Lesson Notes, Main Idea 2. (Theory 15) Represents the rule for subtracting fractions with like denominators with letters. A-Level Problems 1 Subtract two fractions with like denominators.(1 hint) Theory Problems Add two fractions with like denominators. (1 hint) Theory Problems Add two fractions with like denominators. (1 hint) Theory Problems Subtract two fractions with like denominators.(1 hint) Theory Problems Add two fractions with like denominators. (1 hint) Theory Problems Subtract two fractions with like denominators.(1 hint) Theory Problems Solve a word problem which requires adding and subtracting fractions with like denominators.(2 hints) Theory Problem 11 B-Level Problems 1 Word problem to compare two fractions with like denominators using subtraction. (1 hint) 2 Evaluate an expression with adding and subtracting fractions with like denominators. (2 hints) 3 Evaluate an expression with adding and subtracting fractions with like denominators. (2 hints) C-Level Problems 1 Evaluate an expression with adding and subtracting fractions with like denominators and parentheses. (3 hints) 2 Compare two improper fractions by expressing each as a mixed number. (2 hints) 3 Solve a word problem using adding and subtracting fractions with like denominators. (1 hint) A-Level Problem 7 Builds on A-Level problems 1-6 Builds on A-Level problems 1-6 Builds on B-Level Problems 2, 3 Builds on Theory Problems Builds on B-Level problem 1 Diagnostic Block 1 Subtract two fractions with like denominators.(1 hint) A-Level Problem 1 2 Add two fractions with like denominators. (1 hint) A-Level Problem 2 Reasoning Mind 78 Curriculum Guide

79 3 Add two fractions with like denominators. (1 hint) A-Level Problem 3 4 Subtract two fractions with like denominators.(1 hint) A-Level Problem 4 Reasoning Mind 79 Curriculum Guide

80 28. Adding Mixed Numbers with Like Denominators Approximate Length: 1.95 student study hours Overview This objective gives the rules for adding mixed numbers with like denominators, as well as adding mixed numbers and whole numbers and mixed numbers and fractions. Related Objectives Objective 24. Mixed Numbers Objective 26. Comparing Fractions with Like Denominators Objective 28. Adding Mixed Numbers with Like Denominators Objective 29. Subtracting Mixes Numbers with Like Denominators Vocabulary New Terms: Glossary Terms: equation, fraction, fractional part, mixed number, whole part Lesson Notes Main Idea #1 - The rule for adding mixed numbers is important because it will be similar to the rule for subtracting mixed numbers. It will also lead to understanding for adding decimals by lining up the decimal point. Write This Down Screen (Theory 4) Notes Test Item 1 Theory Block Chapter 1 Adding Mixed Numbers with Like Denominators (Theory 1) Shows a diagram to see how to add mixed numbers. (Theory 2) Gives the first part of the rule to add mixed numbers. (Theory 4) WTD See Lesson Notes Main Idea 1. (Theory 5-7) Examples of adding mixed numbers. (Theory 8) Shows adding a mixed number and a whole number. (Theory 10) Shows adding a fraction to a mixed number. Reasoning Mind 80 Curriculum Guide

81 A-Level Problems 1 Add two mixed numbers. (1 hint) Theory Problems 1, 2 2 Add a whole number and a mixed number. (2 hints) Theory Problems 4, 5 3 Add two mixed numbers. (1 hint) Theory Problems 1, 2 4 Add a fraction and a mixed number. (2 hints) Theory Problems 6, 7 5 Add two mixed numbers. (1 hint) Theory Problems 1, 2 6 Solve a word problem that requires adding a mixed number and a whole number. (2 hints) 7 Solve a word problem that requires adding a mixed number and a fraction. (2 hints) Theory Problems 3, 4, 5 Theory Problems 3, 6, 7 B-Level Problems 1 Evaluate expression with three summands. (1 hint) Builds on A-Level Problems 1, 2, 3 2 Evaluate a word problem which involves adding three summand. (1 hint) Builds on A-Level Problems 1, 3, 4, 7 3 Find the perimeter of a triangle. (2 hints) Builds on A-Level Problems 1, 3, 4, 7 C-Level Problems 1 Evaluate expression with three summands. (1 hint) Builds on B-Level Problem 1 2 Solve word problem by adding two mixed numbers. (1 hint) Builds on Theory Problem 3 3 Find the perimeter of a triangle. (2 hints) Builds on B-Level Problem 3 Diagnostic Block 1 Add two mixed numbers. (1 hint) Theory Problem 1 2 Add two mixed numbers. (1 hint) A-Level Problem 1 3 Add a whole number and a mixed number. (2 hints) A-Level Problem 2 4 Add a fraction and a mixed number. (2 hints) Theory Problem 6 Reasoning Mind 81 Curriculum Guide

82 29. Subtracting Mixed Numbers with Like Denominators Approximate Length: 2.4 student study hours Overview This objective shows various ways in which mixed numbers are subtracted from other numbers, as well as different types of numbers are subtracted from mixed number. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lessons Related Objectives Objective 24. Mixed Numbers Objective 26. Comparing Fractions with Like Denominators Objective 28. Adding Mixed Numbers with Like Denominators Objective 29. Subtracting Mixes Numbers with Like Denominators Vocabulary New Terms: Glossary Terms: difference, equation, fraction, fractional part, mixed number, subtrahend, whole part Lesson Notes Main Idea #1 - This rule is important to understand how mixed numbers are subtracted Write This Down Screen (Theory 4) Notes Test Item 1 Theory Block Chapter 1 Subtracting Mixed Numbers with Like Denominators (Theory 1-3) Shows two ways to represent subtracting two mixed numbers. (Theory 4) WTD See Lesson Notes, Main Idea 1. (Theory 5-7) Shows examples of subtracting mixed numbers. (Theory 6) Shows subtracting a whole number from a mixed number. (Theory 10) Shows subtracting a whole number from a mixed number where the whole number is equal to the whole part of the mixed number. (Theory 12) Shows subtracting a fraction from a mixed number. (Theory 14) Shows subtracting a fraction from a mixed number where the fraction is equal to the fractional part of the mixed number. Reasoning Mind 82 Curriculum Guide

83 A-Level Problems 1 Subtract the two mixed numbers. (1 hint) Theory Problems Subtract a fraction from a mixed number. (1 hint) Theory Problems 8, 9 3 Subtract a whole number, equal to the whole part of the of the mixed number, from the mixed number. (1 hint) Theory Problem 7 4 Subtract the two mixed numbers. (1 hint) Theory Problems Subtract a fraction, equal to the fractional part of the of the mixed number, from the mixed number. (1 hint) 6 Word problem with subtract a whole number, equal to the whole part of the of the mixed number, from the mixed number. (2 hints) Theory Problems 11, 12 Theory Problem 8 7 Word problem with subtracting two mixed numbers. (2 hints) Theory Problem 4 B-Level Problems 1 Word problem involved adding two mixed numbers, then subtracting two mixed numbers. (2 hints) C-Level Problems 1 Evaluate two expressions with parentheses and mixed operations. (1 hint) 2 Find the perimeter of a triangle given that one side is a certain amount shorter than another side. (2 hints) Builds on A-Level Problem 7 Builds on A-Level Problems 1, 2 Builds on B-Level Problem 1 Diagnostic Block 1 Subtract two mixed numbers. (1 hint) A-Level Problem 1 2 Subtract a fraction from a mixed number. (1 hint) A-Level Problem 2 3 Subtract a whole number, equal to the whole part of the of the mixed number, from the mixed number. (1 hint) 4 Subtract a fraction, equal to the fractional part of the of the mixed number, from the mixed number. (1 hint) A-Level Problem 3 Theory Problem 11 Reasoning Mind 83 Curriculum Guide

84 30. Review Factors, Prime and Composite Numbers, GCFs Approximate Length: 2.5 student study hours Overview This objective looks at finding the factors of a natural number, as well as defining prime and composite numbers culminating in determining the GCF and defining relatively prime numbers. In Chapter 1 the concept of divisibility is investigated with the connection between divisors and factors of a number. Chapter 2 continues with a method for finding all the factors of a given number. Then, Chapter 3 gives the definition of prime and composite numbers. In Chapter 4, the Greatest Common Factor (GCF) is defined with a method for finding the GCF is shown. Finally, in Chapter 5, the definition of relatively prime numbers is given with a method to check that two numbers are relatively prime. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lessons Vocabulary New Terms: Glossary Terms: composite number, prime number, Greatest Common Factor (GCF), Greatest Common Divisor (GCD), relatively prime numbers composite number, divides, divisible, divisor, factor, Greatest Common Divisor (GCD), Greatest Common Factor (GCF), natural number, prime number, relatively prime numbers, remainder Theory Block Chapter 1 Divisibility (Theory 1) Reviews what it means for one number to be divisible by another. (Theory 2) Defines a divisor and factor of a natural number. (Theory 4, 5, 6) Give some properties of divisors and factors. Chapter 2 Finding all the Factors of a Number (Theory 7, 8, 9) Gives an example of finding the factors of a natural number by trying different natural numbers less than the number. (Theory 10) Shows the notation for the factors of a natural numbers as the set of all factors. Chapter 3 Prime and Composite Numbers (Theory 12) Defines a prime and composite number. (Theory 13) An animation of examples of numbers that are prime and composite, then explaining why the number 1 is neither prime nor composite. (14) A screen stating that 1 is not prime or composite, but all other natural numbers are. Chapter 4 The Greatest Common Factor (Theory 15) Shows how to find the greatest common factor of two numbers. (Theory 16) Defines the Greatest Common Factor (GCF) and Greatest Common Divisor (GCD). (Theory 17) Shows the notation for GCF of two numbers. (Theory 18) Gives a rule for finding the GCF of two numbers. Chapter 5 Relatively Prime Numbers (Theory 20) Gives the definition of relatively prime numbers. (Theory 21) Shows checking that two numbers are relatively prime. Reasoning Mind 84 Curriculum Guide

85 A-Level Problems 1 Find the GCF of two numbers. (1 hint) Theory Problems Choose the pair of relatively prime numbers.(2 hints) Theory 21, Theory Problem 9 3 Find the factors of a given composite number.(1 hint) Theory Problems 1, 2 4 Give the greatest factor of a natural number.(1 hint) Theory 6 5 Give the smallest factor of a natural number.(1 hint) Theory 6 6 Find all the factors of a composite number.(1 hint) Theory Problems 1, 2 7 Find all the common factors of three numbers.(2 hints) Theory 15 B-Level Problems 1 Find the GCF of three numbers. (1 hint) Builds on A-Level Problem 1 2 Solve a word problem to find the GCF of two numbers. (2 hints) Builds on A-Level Problem 1 C-Level Problems 1 Given one number as product of a list of factors, and a second number, find the GCF of the two numbers.(1 hint) 2 Express a given composite number as the product of two relatively prime numbers factors with the difference between the factors as small as possible. (1 hint) 3 Defines the term twin primes and asks the student to list the first 5 pairs of twin primes. (1 hint) Builds on A-Level Problems 1 and 6 Builds on A-Level Problem 6 Builds on Theory 12 Diagnostic Block 1 Choose the correct statements. Theory Problem 3 2 Give all the factors of a number and determine if it is prime or composite. Theory Problem 4 3 Find the GCF of two numbers. Theory Problem 5 4 Choose the true statement. Theory problem 10 Reasoning Mind 85 Curriculum Guide

86 31. Review: Multiples and LCMs Approximate Length: 4.4 student study hours Overview This objective reviews the concepts of multiple of a natural number and Least Common Multiple between several natural numbers. In Chapter 1, the definition of a multiple and connection to divisibility is made along with methods for finding multiples of a natural number. Chapter 2 defines the Least Common Multiple (LCM) and how to find the LCM for two natural numbers. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lesson 74 Vocabulary New Terms: Glossary Terms: multiple, Least Common Multiple (LCM) divides, Least Common Multiple (LCM), multiple, natural number Theory Block Chapter 1 Multiples of a Number (Theory 1) Introduces the idea of one number being a multiple of another number if the first number divides the second number. (Theory 2) Gives definition of a multiple and shows the smallest multiple of any natural number. (Theory 5) Gives a method to find multiples of a natural number. (Theory 6) Gives examples of finding multiples and expands ways to find multiples using the fact that multiplication is repeated addition. Chapter 2 The Least Common Multiple (Theory 7) Gives the definition of the Least Common Multiple of several natural numbers. (Theory 8) Shows an example of finding the LCM of two numbers. (Theory 10) Gives a rule to quickly find the LCM of two numbers. (Theory 11) Shows an example of finding the LCM of two numbers. A-Level Problems 1 Select all of the numbers listed that are multiples of a given number. (2 hints) Theory 2 2 Solve a word problem to find the LCM of two numbers. (2 hints) Theory Problem 9 3 Find the LCM of two numbers. (1 hint) Theory Problem 5 4 Find the LCM of two numbers. (1 hint) Theory Problem 5 5 Find the LCM of two consecutive number. (1 hint) Theory Problem 5 6 Find the first three multiples of a given number.(1 hint) Theory Problem 2 B-Level Problems 1 Find the relatively prime numbers from a list and then find their least common multiple. (2 hints) 2 Word problem to find the least common multiple of two numbers. (2 hints) Builds A-Level Problem 5 Builds on A-Level Problem 2 Reasoning Mind 86 Curriculum Guide

87 C-Level Problems 1 Choose the two relatively prime numbers from a list and find their LCM. (2 hints) 2 Word problem to find the least common multiple of two numbers. (1 hints) Builds on B-Level Problem 1 Builds on B-Level Problem 2 Diagnostic Block 1 Give the first three multiples of a number. Theory Problem 4 2 Find the LCM of two numbers. Theory Problem 6 Reasoning Mind 87 Curriculum Guide

88 32. The Equivalency Property of a Common Fraction, Reducing Fractions Approximate Length: 4.1 student study hours Overview This objective introduces the Equivalency Property of Fractions and uses it to simplify fractions by dividing the numerator and denominator by the GCF and by reducing the fraction in steps using any common factors. Chapter 1 starts with motivating the Equivalency Property of Fractions by showing how two fractions with different numerators and denominators can represent the same value. Then in Chapter 2, this property is used to understand simplifying a fraction into lowest terms. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lessons Related Objectives Objective 32. The Equivalency Property of a Common Fraction, Reducing Fractions Objective 33. Bringing a Fraction to a Common Denominator Vocabulary New Terms: Glossary Terms: equivalent fractions, reducing a fraction, irreducible fraction, lowest terms, simplified denominator, equivalent fractions, irreducible fraction, numerator Theory Block Chapter 1 The Equivalency Property of a Common Fraction (Theory 1, 2) Shows that two fractions are equivalent by showing they can represent the same shaded part of a figure and represent the same point on the number ray. (Theory 4) Gives a formula for the idea that if the numerator and denominator are multiplied by the same non-zero number, the resulting fraction is equal to the original fraction. (Theory 5, 6) Look at getting an equivalent fraction by dividing the numerator and denominator by the same non-zero number. (Theory 7) Gives the Equivalency Property of Common Fractions. Chapter 2 Reducing Fractions (Theory 10) Defines reducing a fraction. (Theory 11) Uses the terminology of reducing a fraction by cancelling out a common factor. (Theory 13) Gives a definition of an irreducible fraction and a fraction in lowest terms. (Theory 14) Gives alternate terminology to say a fraction is irreducible. (Theory 16) Gives the rule for reducing fractions to lowest terms using the GCF. (Theory 17, 18) Shows an alternative way by reducing a fraction in steps and gives a rule for this process. A-Level Problems 1 Reduce a fraction, with the numerator and denominator expressed as products, to its lowest terms. (2 hints) 2 Determine if an equality is true using the Equivalency Property of Fractions. (1 hint) 3 Determine if an equality is true using the Equivalency Property of Fractions. (1 hint) Theory Problem 9 Theory 4 Theory 4 Reasoning Mind 88 Curriculum Guide

89 4 Determine if an equality is true using the Equivalency Property of Fractions. (1 hint) 5 Determine if an equality is true using the Equivalency Property of Fractions. (1 hint) Theory 4 Theory 4 6 Reduce a fraction to simplest terms. Theory Problem 10 7 Reduce a fraction to simplest terms. Theory Problem 11 8 Reduce a fraction to simplest terms. Theory Problem 10 B-Level Problems 1 Reduce a fraction, with the numerator and denominator expressed as products, to its lowest terms. (1 hint) Theory Problem 9 2 Select the two fractions that are equal. (1 hint) Builds on A-Level Problems 2-5 C-Level Problems 1 Write reducible and irreducible fractions based on requirements in the form of a word problem.(2 hints) B-Level Problem 2 2 Reduce a list of problems to lowest terms. (1 hint) Theory Problems 9-13 Diagnostic Block 1 Get an equivalent fractions by dividing the numerator and denominator by the same given number. (2 hints) 2 Fill in the blank to make the equality true for two fractions with different denominators. Theory Problem 3 Theory Problem 5 3 Reduce a fraction to lowest terms. Theory Problem 11 4 Reduce a fraction to lowest terms. Theory Problem 12 Reasoning Mind 89 Curriculum Guide

90 33. Bringing Fractions to a Common Denominator Approximate Length: 5.2 student study hours Overview This objective looks at bringing fractions to a common denominator and shows how to do this by finding the least common multiple of the denominators. Chapter 1 shows how to bring a fraction to a new denominator using the Equivalency Property of Fractions. Then, in Chapter 2, finding a common denominator for two fractions is shown with a rule to find the least common denominator. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lessons 77, 78. Related Objectives Objective 33. The Equivalency Property of a Common Fraction, Reducing Fractions Objective 34. Bringing Fractions to a Common Denominator Vocabulary New Terms: Glossary Terms: complementary factor, least common denominator complementary factor, least common denominator Theory Block Chapter 1 Converting a Fraction to a Fraction with a New Denominator (Theory 1) Shows using the equivalency property of fractions to get an equal fraction with a new denominator. (Theory 4) Gives a reminder what denominators can be used for an equivalent fraction. (Theory 6) Gives a rule for converting a fraction. (Theory 8) Looks at the case in which a fraction is irreducible. (Theory 9) Looks at a short way to convert a fraction using a complementary factor. Chapter 2 Bringing Fractions to a Common Denominator (Theory 11) Shows bringing two fractions to a common denominator by using the product of the denominators as the common denominator. (Theory 13) Gives a remember screen on what numbers can be used as the common denominator. (Theory 15) Defines the least common denominator. (Theory 16-21) Shows bringing two fractions to the least common denominator using the complementary factors. (Theory 22) Gives a rule for bringing fractions to their least common denominator. (Theory 23, 24) Shows bringing two fractions with prime denominators to their least common denominator. A-Level Problems 1 Convert a fraction to a fraction with a given denominator. (1 hint) Theory Problems Convert a fraction to a fraction with a given denominator. (1 hint) Theory Problems Convert a fraction to a fraction with a given denominator. (1 hint) Theory Problems From a list, select the numbers that could be common denominators for two given fractions. (1 hint) Theory Problem 5 5 Bring two fractions to their least common denominator. (2 hints) Theory Problems Bring two fractions to their least common denominator. (2 hints) Theory Problems 8-11 Reasoning Mind 90 Curriculum Guide

91 B-Level Problems 1 Choose the fractions that can be brought to a given denominator and convert those fractions to the new denominator. (1 hint) 2 Reduce a fraction to lowest terms, then bring it to a new denominator. 3 Reduce two fractions to lowest terms, then bring them to their least common denominator. (2 hints) Builds on A-Level Problem 4 Builds on A-Level Problems 1-3 Builds on A-Level Problems 1-3 and 5, 6 C-Level Problems 1 Bring three fractions to their least common denominator. (2 hints) Builds on B-Level Problem 2. Diagnostic Block 1 Convert a fraction to a fraction with a given denominator. (2 hints) Theory Problem 1 2 Find the complementary factor to convert a fraction to a given denominator. (2 hints) Theory Problem 3 3 Bring two fractions to their least common denominator. (2 hints) Theory Problem 11 Reasoning Mind 91 Curriculum Guide

92 34. Comparing Fractions with Different Denominators Approximate Length: 4.1 student study hours Overview This objective looks at several cases of comparing fractions and mixed numbers with unlike denominators. In Chapter 1, the case of comparing proper fractions is discussed. Chapter 2 continues with comparing mixed numbers with unlike whole parts. Chapter 3 builds on comparing mixed numbers by looking at mixed numbers with equal whole parts. Finally, Chapter 4 talks about comparing two improper fractions and comparing a proper fraction with an improper fraction. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lessons Related Objectives Objective 26. Comparing Fractions Objective 34. Comparing Fractions with Different Denominators Vocabulary New Terms: Glossary Terms: Common fraction, denominators, Equivalency Property of Common Fractions, fractional part, like denominators, mixed numbers, numerators, whole part Theory Block Chapter 1 Comparing Proper Fractions with different Denominators (Theory 1) Reviews the rule for comparing fractions with like denominators. (Theory 3) To compare fractions with different denominators, we have to bring the fractions to a common denominator. (Theory 4-7) Gives an example of comparing two fractions with unlike denominators. (Theory 8) Gives a screen with the steps to compare two fractions with unlike denominators. Chapter 2 Comparing Mixed Numbers with Different Whole Parts (Theory 14) Gives a rule to compare mixed numbers with different whole parts. Chapter 3 Comparing Mixed Numbers with Equal Whole Parts (Theory 16) Gives a rule for comparing two whole numbers with equal whole parts. (Theory 18) Gives a rule for comparing mixed numbers with equal and unequal whole parts with example when the whole parts are unequal. Chapter 4 Comparing Improper Fractions with Different Denominators (Theory 19-27) Give examples of comparing two improper fractions. (Theory 28) Notes that any improper fraction is greater than any proper fraction. A-Level Problems 1 Select all the correct inequalities of proper fractions. (3 hints) Similar to Theory Problems 1, 2 2 Word problem involving the comparison of two proper fractions. (2 hints) 3 Compare two mixed numbers with different whole parts. (1 hint) Theory 18 Builds on Theory Problems 1, 2 Reasoning Mind 92 Curriculum Guide

93 4 Compare two mixed numbers with equal whole parts. (1 hint) Theory Problems 6, 7, 9 B-Level Problems 1 Check whether a list of inequalities is done correctly. (3 hints) Builds on A-Level Problem 1 2 Word problem to compare three fractions to see which is greater. (2 hints) Builds on A-Level Problem 2 3 Reduce to lowest terms, then compare two proper fractions. Builds on Theory Problems 1,2 4 Find all natural number solutions that make an equality true. Builds on Theory Problems 1,2 C-Level Problems 1 Reduce two fractions and compare them.(2 hints) Builds on B-Level Problem 3 2 Solve word problem to compare three fractions.(3 hints) Builds on B-Level Problem 2 3 Choose the number ray with the points correctly marked. (2 hints) Builds on B-Level Problem 2 Diagnostic Block 1 Compare the fractions by bringing them to a common denominator. Theory Problem 2 2 Compare two mixed numbers with unequal whole parts. (1 hint) Theory Problem 5 3 Compare two mixed numbers with equal whole parts. (2 hints) Theory Problem 9 4 Compare two improper fractions. (2 hints) Theory Problem 10 Reasoning Mind 93 Curriculum Guide

94 35. Adding and Subtracting Fractions with Unlike Denominators Approximate Length: 1 student study hours Overview This objective looks at adding, subtracting and comparing fractions with unlike denominators. In Chapter 1, each operation is modelled starting with bringing the fractions to a common denominator. The answer should always be in lowest terms. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lessons Related Objectives Objective 27. Adding and Subtracting Fractions with Like Denominators Objective 35. Adding and Subtracting Fractions with Unlike Denominators Vocabulary New Terms: Glossary Terms: denominator, fraction, like denominator, improper fraction, irreducible fraction, mixed number Theory Block Chapter 1 Adding and Subtracting Fractions with Unlike Denominators (Theory 1) Reviews the rules for adding and subtracting fractions with like denominators. (Theory 2-5) Gives an example of adding two fractions with unlike denominators by bringing both fractions to the least common denominator. (Theory 6-10) Gives an example of subtracting two fractions with unlike denominators by bringing both fractions to the least common denominator. (Theory 11-15) Gives an example of comparing two fractions with unlike denominators by bringing both fractions to the least common denominator. (Theory 16) Reviews the operations on fractions with unlike denominators and concludes each time the fractions must be brought to a common denominator. (Theory 17) Notes that if the answer is a reducible fraction, the answer should be simplified. (Theory 19-21) Shows a word problem in which three fractions need to be added, then compared to a whole number. (Theory 22) Notes that if an answer is an improper fraction, the answer should be simplified to a mixed number. A-Level Problems 1 Find the sum of two fractions with unlike denominators. (2 hints) Theory Problems Find the difference of two fractions with unlike denominators. (2 hints) 3 Find the difference of two fractions with unlike denominators. (2 hints) 4 Find the difference of two fractions with unlike denominators and put the answer in lowest terms. (2 hints) 5 Perform the operations and give the answer in lowest terms. (2 hints) Theory Problem 5-7 Theory Problem 5-7 Theory Problem 11, 12 Theory Problem 11, 12 Reasoning Mind 94 Curriculum Guide

95 6 Perform the operations and give the answer in lowest terms. (2 hints) Theory Problem 11, 12 7 Solve a word problem to find the sum of three fractions. (3 hints) Theory Problem 10 B-Level Problems 1 Find the perimeter of a rectangle with one side given and a relationship to the other side. (3 hints) Builds on A-Level Problem 7 2 Calculate in a convenient way. (3 hints) Builds on A-Level Problem 1 3 Evaluate an expression with one letter for a given value. (2 hints) Builds on A-Level Problem 5, 6 C-Level Problems 1 Find the perimeter of a rectangle with one side given and a relationship to the other side. (3 hints) Builds on A-Level Problem 7 2 Calculate in a convenient way. (3 hints) Builds on A-Level Problem 1 Diagnostic Block 1 Find the sum of two fractions with unlike denominators. (1 hint) Theory Problem 2 2 Find the difference of two fractions with unlike denominators. (1 hint) 3 Find the difference of two fractions with unlike denominators. (1 hint) 4 Find the difference of two fractions with unlike denominators and give the answer in lowest terms. (2 hint) Theory Problem 6 Theory Problem 7 Theory Problem 11 Reasoning Mind 95 Curriculum Guide

96 36. Adding and Subtracting Mixes Numbers Approximate Length: 1.6 student study hours Overview This objective covers adding and subtracting mixed numbers with unlike denominators by building on adding and subtracting mixed numbers with like denominators, including reducing to lowest terms and borrowing from the whole part. In Chapter 1, adding mixed numbers is addressed beginning with finding the LCD. Chapter 2 looks at subtraction with borrowing from the whole part. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lesson Related Objectives Objective 22. Fractions Objective 26. Comparing Fractions Objective 29. Adding and Subtracting Mixed Numbers with Like Denominators Objective 36. Adding and Subtracting Mixed Numbers Vocabulary New Terms: Glossary Terms: difference, fractional part, minuend, mixed numbers, subtrahend, whole part, whole number Theory Block Chapter 1 Adding Mixed Numbers with Different Denominators (Theory 1, 2) Reviews adding mixed numbers with like denominators. (Theory 3) Addresses unlike denominators. (Theory 7) Reminds students to put the answer in lowest terms. (Theory 6) Gives the rule for adding mixed numbers with different denominators and a reminder to give the answer in lowest terms. Chapter 2 Subtracting Mixed Numbers with Different Denominators (Theory 9-11) Gives a word problem that requires subtracting two mixed numbers with different denominators. (Theory 12) Discusses borrowing from the whole part by showing a problem of subtracting a proper fraction from a whole number. (Theory 13) Shows subtracting two mixed numbers with different denominators which will require borrowing. (Theory 17) Gives the rule for subtracting mixed numbers A-Level Problems 1 Add two mixed numbers where one denominator is a multiple of the other denominator. (2 hints) Theory Problem 2 2 Subtract two mixed numbers with different denominators. (2 hints) Theory Problem 11 3 Add two mixed numbers where one denominator is a multiple of the other denominator. (2 hints) 4 Add two mixed numbers and put the answer with a proper fractional part in lowest terms. (2 hints) 5 Subtract mixed numbers with borrowing from the whole part of the minuend. (3 hints) Theory Problem 2 Theory Problems 8, 9 Theory Problem 15 Reasoning Mind 96 Curriculum Guide

97 B-Level Problems 1 Add three mixed numbers with parentheses. (3 hints) Builds on A-Level Problems 1, 3 2 Solve word problem that requires subtracting and adding mixed numbers. (2 hints) 3 Calculate adding three mixed numbers in a convenient way. (3 hints) Builds on Theory Problems 3, 16 Builds on A-Level Problems 1, 3 C-Level Problems 1 Word problem that requires adding and subtracting mixed numbers. (2 hints) 2 Calculate the coordinates of points representing mixed numbers from given relationships to known coordinates. (1 hint) 3 Evaluate an expression with the value of the letter given as a mixed number. (2 hints) Builds on B-Level Problem 2 Builds on Theory Problems 10, 13 Builds on B-Level Problem 2, 3 Diagnostic Block 1 Add two mixed numbers. (2 hints) Theory Problem 1 2 Find the difference between a whole number and a proper fraction. (2 hints) Theory Problem 10 3 Subtract a mixed number from a whole number. (2 hints) Theory Problem 14 4 Subtract a proper fraction from a mixed number with borrowing from the whole part. (3 hints) Similar to Theory Problem 15 Reasoning Mind 97 Curriculum Guide

98 37. Multiplying Common Fractions Approximate Length: 2.3 student study hours Overview This objective develops the rule for multiplying common fractions including reducing the answer before completing the multiplication. Chapter 1 starts with reviewing multiplying a fraction by a natural number. In Chapter 2, the concept of multiplying fractions is shown by a model, then the rule for multiplying common fractions is developed. Chapter 3 finishes developing the rule by showing how to simply the product by canceling out common factors in the numerator and denominator before multiplying. In Chapter 4, multiplying mixed numbers is addressed. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lessons Vocabulary New Terms: Glossary Terms: area, denominator, factor, fractions, lowest terms, numerator, product, square Theory Block Chapter 1 Multiplying a Common Fraction by a Natural Number (Theory 2) Gives the definition of multiplying a whole number n by a common fraction f. (Theory 4, 5) Give a rule for multiplying a whole number by a common fraction with the rule expressed with letters. Chapter 2 Multiplying a Common Fraction by a Common Fraction (Theory 6) Introduces the idea of multiplying two common fractions using the area of a rectangle. (Theory 7) Gives the rule for multiplying a fraction by a fraction with the rule expressed with letters. (Theory 9) Show finding the area of a square with side length equal to a fraction. Chapter 3 Multiplying Common Fractions Rule (Theory 10) Always reduce the product to lowest terms after multiplying two fractions. Shows how to do this before multiplying the numerators and denominators. (Theory 11) Shows the rule for multiplying fractions with cancelling out common factors in the products in the numerator and denominator. Chapter 4 Multiplying Mixed Numbers (Theory 12) Gives the rule for multiplying mixed numbers. (Theory 13) Shows an example. (Theory 14) Reminds students to simplify the answer to a mixed number. A-Level Problems 1 Multiply a fraction by a whole number. (1 hint) Theory Problems 2, 3 2 Multiply a fraction by a whole number. (1 hint) Theory Problems 2, 3 3 Multiply a mixed number by a fraction. (1 hint) Theory Problem 12 4 Multiply two mixed numbers (2 hints) Theory Problem 13 5 Multiply two mixed numbers (2 hints) Theory Problem 14 6 Find the area of a square with a side length equal to a fraction. (2 hints) Theory 9 Reasoning Mind 98 Curriculum Guide

99 B-Level Problems 1 Calculate the product of three fractions and given the answer in lowest terms. ( hint) 2 Evaluate an expression with two letters and the values of the letters represented by fractions. (3 hints) 3 Find the area of a rectangle based on the relationship of the length and width to another triangle. (2 hints) Builds on Theory Problems 8-11 Builds on Theory Problems 8-11 Builds on A-Level Problem 6 C-Level Problems 1 Find the value of a product of two numerical expressions with addition and multiplication of fractions. (2 hints) Builds on B-Level Problem 2 2 Find the volume of a rectangular prism. (3 hints) Builds on B-Level Problem 3 3 Solve a word problem that requires multiplying mixed numbers with a whole number. A-Level Problem 2 and Theory Problem 17 Diagnostic Block 1 Multiply a fraction by a whole number. Theory Problem 2 2 Multiply a fraction by a fraction number. Theory Problem 4 3 Multiply a fraction by a whole number and put the answer into lowest terms. Theory Problem 8 4 Multiply a fraction by a mixed number. (2 hints) Theory Problem 15 Reasoning Mind 99 Curriculum Guide

100 38. Dividing Fractions and Mixed Numbers Approximate Length: 2 student study hours Overview This objective defines the reciprocal of a common fraction and uses the reciprocal to give a rule for dividing by a fraction. In Chapter 1, the definition of a reciprocal is developed. Chapter 2 uses the reciprocal of a fraction to give a rule for dividing by a fraction. Chapter 3 extends these ideas to mixed numbers by changing the mixed numbers into improper fractions. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lessons Related Objectives Objective 37. Multiplying Common Fractions Objective 38. Dividing Fractions Vocabulary New Terms: Glossary Terms: reciprocal denominator, dividend, division, divisor, fraction, improper fraction, mixed number, numerator, reciprocal Theory Block Chapter 1 Reciprocals (Theory 1) Develops the idea of two fractions that are reciprocals. (Theory 2) Defines reciprocals. (Theory 3) Gives examples of reciprocals. Chapter 2 Dividing Common Fractions (Theory 5) Gives a rule to divide by a common fractions. Chapter 3 Dividing Mixed Numbers (Theory 7) Starts the discussion with changing the mixed numbers into improper fractions and using the rule for dividing fractions. (Theory 8) Gives the rule for dividing mixed numbers. (Theory 10) Shows a word problem where a mixed number is divided by a whole number. A-Level Problems 1 Use rule for dividing by a fraction to divide a fraction by a whole number. (3 hints) 2 Use rule for dividing by a fraction to divide a fraction by a whole number. (3 hints) 3 Find the length of a rectangle given the area and width are given as common fractions. (3 hints) 4 Use rule for dividing by a mixed number to divide a mixed number by a whole number. (3 hints) 5 Find the length of a rectangle given the area and width are given as common fractions. (3 hints) Builds on Theory Problems 4-7 Builds on Theory Problems 4-7 Builds on Theory Problems 4-7 Builds on Theory problems 16, 17 Builds on Theory Problems 4-7 Reasoning Mind 100 Curriculum Guide

101 B-Level Problems 1 Solve problems with multiplying and dividing fractions. (1 hint) Builds on Theory Problems 4-7 C-Level Problems 1 Evaluate and expression with two letters whose values are given as common fractions. (2 hints) 2 Evaluate and expression with a letter whose value is given as a mixed number. (2 hints) Builds on B-Level Problem 1 Builds on B-Level Problem 1 Diagnostic Block 1 Find the reciprocals of the given numbers. (1 hint) Theory Problem 2 2 Find the quotient of two fractions. (2 hints) Theory Problem 5 3 Find the number that is the quotient of a fraction divided by a whole number. (3 hints) 4 Find the number that is the quotient of a fraction divided by a whole number. (3 hints) A-Level Problem 1 A-Level Problem 1 Reasoning Mind 101 Curriculum Guide

102 39. Review Angles Approximate Length: 2.6 student study hours Overview Angles are defined and the properties of special angles are shown. In Chapter 1, angles are defined with notation for an angle shown. Chapter 2 shows how to compare angles by lining the vertex and one side of one angle over the other. Finally, Chapter 3 discusses special types of angles with their properties. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lesson 103. Vocabulary New Terms: Glossary Terms: angle, vertex of an angle, side of an angle, straight angle, supplementary rays acute angle, angle, angle bisector, drafting triangle, false, obtuse angle, right angle, side of an angle, straight angle, supplementary rays, true, vertex of an angle Theory Block Chapter 1 Angles (Theory 2) Talks about how angles are formed and names the parts of an angle. (Theory 3) Covers how angles are named and notation for an angle. Chapter 2 Comparing Angles (Theory 7) Shows comparing angles by matching up the vertices and one side, then gives notation and a standard for saying one angle is bigger than another angle. (Theory 8) Shows two angles that are the equal. Chapter 3 Types of Angles (Theory 10) Defines straight angle and supplementary rays. (Theory 12, 13) Define a right angle. (Theory 14) Uses the hands on a clock to find right angles. (Theory 15) Introduces the drafting triangle and it s uses. (Theory 17) Discusses acute and obtuse angles. (Theory 18) Shows a comparison of angle types discussed in the objective. (Theory 19) Gives the definitions of acute and obtuse angles. A-Level Problems 1 Select the figure that has four obtuse angles.(3 hints) Theory Problems 4-6 B-Level Problems 1 Count how many of each different type of angle are shown in the given figure. (2 hints) Builds on A-Level Problem 1 C-Level Problems 1.1 Count the number of acute angles in the picture.(2 hints) Builds on B-Level Problem Count the number of obtuse angles in the picture.(2 hints) Builds on B-Level Problem Count the number of right angles in the picture.(1 hint) Builds on B-Level Problem 1 Reasoning Mind 102 Curriculum Guide

103 Diagnostic Block 1 Write the name of the given angles in different ways. Theory Problem 1 2 Select the straight angle. Theory Problem 2 Reasoning Mind 103 Curriculum Guide

104 40. Types of Triangles Approximate Length: 2.2 student study hours Overview This objective looks at different types of triangles the special characteristics of each. In Chapter 1, the three basic types of triangles are defined by angles. Chapter 2 defines the three basic triangles according to the relationship of the sides. Chapter 3 continues by looking at the isosceles triangle. In Chapter 4, the base and vertex angles are defined and notes that the two base angles are equal. Finally in Chapter 5, the equilateral triangle in investigated. Vocabulary New Terms: Glossary Terms: right triangle, obtuse triangle, acute triangle, scalene triangle, isosceles triangle, equilateral triangle, base of an isosceles triangle, base angles of an isosceles triangle. Vertex angle of an isosceles triangle acute angle, acute triangle, equilateral triangle, isosceles triangle, leg of a right triangle, obtuse angle, obtuse triangle, perpendicular, regular polygon, regular triangle, right angle, right triangle, scalene triangle, triangle Theory Block Chapter 1 Triangles by Angles (Theory 1) Defines the right triangle, obtuse triangle, and acute triangle. Chapter 2 Triangles by Sides (Theory 3) Defines the scalene triangle. (Theory 4) Defines an isosceles triangle. (Theory 5, 6) Discusses an isosceles right triangle. (Theory 9) Defines an equilateral triangle. (Theory 10) Word problem to find the type of triangle formed from three designated points. Chapter 3 The sides of an Isosceles Triangle (Theory 12) Defines the base of an isosceles triangle. (Theory 13) Shows isosceles triangles with different views based on the location of the base. (Theory 14-18) Shows solving problems with isosceles triangles. Chapter 4 Angles of an Isosceles Triangle (Theory 17) Defines the base and vertex angles of an isosceles triangle. (Theory 19) Shows that the base angles of an isosceles triangle are equal. Chapter 5 Sides and Angles of an Equilateral Triangle (Theory 18) Shows that an equilateral triangle, unlike an isosceles triangle, looks the same no matter how it is turned. It also has three equal angles. This means an equilateral triangle is also a regular triangle. A-Level Problems 1 Find the length of a side of an isosceles triangle given the perimeter and the length of the base.(3 hints) 2 Find the length of each side of an equilateral triangle given its perimeter. (2 hints) Theory Problem 5 Theory problem 8 Reasoning Mind 104 Curriculum Guide

105 B-Level Problems 1 Find the length of each side of an equilateral triangle given it s perimeter. (2 hints) C-Level Problems 1 Problem to compare the sides of an isosceles and equilateral triangle given they have equal perimeter and the base of the isosceles triangle. (2 hints) A-Level Problem 2 Builds on B-Level Problem 1 Diagnostic Block 1 Paint the isosceles triangle. (1 hint) Theory Problem 4 2 Find the length of each side of an isosceles triangle given the perimeter and the base. (1 hint) 3 Find the length of each side of an equilateral triangle given the perimeter. (2 hints) Theory Problem 5 Theory Problem 8 Reasoning Mind 105 Curriculum Guide

106 41. Review: Decimals Approximate Length: 2.5 student study hours Overview This objective introduces decimals as having a whole and a fractional part, how to write decimals and how to read decimals. Chapter 1 the understanding of mixed numbers with the fractional part being a multiple of 10 is shown as understanding what a decimal represents. Then, Chapter 2 continues with a discussion of the three decimal places to the right of the decimal point. Chapter 3 shows how a decimal is read. Finally, in Chapter 4, rules are given for writing a decimal from its name, with and without a zero in any decimal place. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lessons Related Objectives Objective 22. Fractions Objective 41. Review: Decimals Objective 42. Review: Comparing Decimals Objective 43. Review: Rounding Decimals Objective 44. Review: Adding Decimals Objective 45. Review Subtracting Decimals Vocabulary New Terms: Glossary Terms: decimal, decimal point, decimal system, denominator, fractional part, hundred, hundred s place, hundredths place, mixed numbers, numerator, one s place, ten, ten s place, tenths place, thousand, thousand s place, thousandths place, whole part Lesson Notes Main Idea #1 -These are important examples to develop the understanding of how to write the fractional part of the mixed number as places to the right of the decimal point. It will also help develop the understanding of decimal places. Write This Down Screen (Theory 6) Notes Test Item 1 Reasoning Mind 106 Curriculum Guide

107 Main Idea #2 - These examples show putting zeros in the decimal that are not written as part of the mixed number or proper fraction. Write This Down Screen (Theory 14) Notes Test Item 2 Main Idea #3 - It is important to remember, when reading decimals, to take care to use the last place for the fractional part reading, especially when there are zeros in the fractional part. Write This Down Screen (Theory 32) Notes Test Item 3 Main Idea #4 - Students need to be able to write a decimal from its name. It is important to remember to put a zero to the left of the decimal point when the whole part is zero when working with decimals. Write This Down Screen (Theory 38) Notes Test Item 4 Theory Block Chapter 1 Writing Fractions as Decimals (Theory 1, 2) Shows writing a mixed number, with the fractional part having a denominator of ten and one hundred, as a decimal with steps to write a mixed number as a decimal. (Theory 6) WTD See Lesson Notes, Main Idea 1. (Theory 7, 8) Look at a fractions in which the decimal equivalent will require at least one zero after the decimal point. (Theory 10) Shows the steps to write a mixed number with denominator 10 or 100 or 1,000 (and so on) as a decimal. (Theory 12) Shows writing a proper fraction as a decimal. (Theory 13) Rules for writing any proper fraction or mixed number with a 10 or 100 or 1,000 (and so on) as a decimal. (Theory 14) WTD See Lesson Notes, Main Idea 2. Chapter 2 Decimal Places (Theory 17) Shows place values for whole numbers. (Theory 20) Gives the name of the three places to the right of the decimal point. (Theory 23) Shows writing a decimal as a sum of place values. Chapter 3 Reading Decimals (Theory 24, 25) Covers the case when the whole part is zero. (Theory 27) Covers reading decimals when the whole part is not zero. (Theory 31) Gives the steps for reading a decimal. (Theory 32) WTD See Lesson Notes, Main Idea 3. Reasoning Mind 107 Curriculum Guide

108 Chapter 4 Writing Decimals (Theory 33) Shows how to write a decimal from its name when there are no zeros to the right of the decimal point. (Theory 35) Shows how to write a decimal from its name when there are zeros to the right of the decimal point. (Theory 36) Shows how to write a decimal when the whole part is zero. (Theory 38) WTD See Lesson Notes, Main Idea 4. (Theory 39) Shows writing a decimal from the name with a zero whole part and a zero to the right of the decimal point. A-Level Problems 1 Write a proper fraction as a decimal. (1 hint) Theory Problem 17 2 Write a mixed number as a decimal with no zero to the right of the decimal point. (1 hint) Theory Problem 7, 11 3 Write a proper fraction as a decimal. (1 hint) Theory Problem 18 4 Choose the correct way to write a mixed number as a decimal with a zero to the right of the decimal point. (1 hint) Theory Problem 8 5 Choose the correct place value name. (1 hint) Theory Problems 21, 22 6 Choose the number with the given digit in a certain place. Builds on Theory Problems 21, 22 7 Choose the number with the given digit in a certain place. Builds on Theory Problems 21, 22 8 Name the place in each number the digit 8 represents. Builds on Theory Problem 23 9 Select the correct way to read a decimal with a zero whole part. (2 hints) Theory Select the correct way to read a decimal. (2 hints) Theory Problem Select the correct way to read a decimal with a zero whole part. (2 hints) Theory Problem Select the correct way to read a decimal. (2 hints) Theory Problem Select the correct way to read a decimal with a zero whole part. (2 hints) Theory Problem Write a decimal with a zero whole part in digits. (1 hint) Theory Problem Write a decimal in digits. (1 hint) Theory Problem Write a decimal with a zero whole part in digits. (1 hint) Theory Problem Write a decimal in digits. (1 hint) Theory Problem Write the given numbers as decimals. Builds on Theory Problems B-Level Problems 1 Reveal the picture by writing proper fractions as decimals. (1 hint) Builds on A-Level Problems 1, 3 2 Choose all the equivalent ways to write a decimal. (1 hint) Builds on A-Level Problem 4 and Theory 30 Reasoning Mind 108 Curriculum Guide

109 C-Level Problems 1 Drag the cards to match the name with the decimal. (1 hint) Builds on A-Level Problems Fill in the table to show the decimal, the place occupied by a given digit, and the mixed number. (2 hints) Builds on A- Level problems 2, 4 and 6-8. Diagnostic Block 1 Write a proper fraction as a decimal. (1 hint) A-Level Problem 1 2 Write a mixed number as a decimal with no zero to the right of the decimal point. (1 hint) A-Level Problem 2 3 Choose the correct place value name. (1 hint) A-Level Problem 5 4 Choose the number with the given digit in a certain place. A-Level Problem 6 5 Select the correct way to read a decimal. (2 hints) A-Level Problem 10 6 Write a decimal with a zero whole part in digits.(1 hint) A-Level Problem 16 Reasoning Mind 109 Curriculum Guide

110 42. Review: Comparing Decimals Approximate Length: 2 student study hours Overview This objective looks at comparing decimals by developing steps to expand the steps for comparing whole numbers. In Chapter 1, understanding changing the appearance of a decimal by adding zeros, or removing zeros at the end of the decimal is developed. Chapter 2 begins the process by comparing decimals by their whole parts. Finally, in Chapter 3, general rules for comparing decimals by expanding the rules for comparing whole numbers is given. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lessons Related Objectives Objective 22. Fractions Objective 41. Review: Decimals Objective 42. Review: Comparing Decimals Objective 43. Review: Rounding Decimals Objective 44. Review: Adding Decimals Objective 45. Review: Subtracting decimals Vocabulary New Terms: Glossary Terms: decimal, decimal place, place value, whole number Lesson Notes Main Idea #1 -Being able to recognize equivalent decimals will be helpful with comparing decimals as well as adding and subtracting decimals. Two decimals are only equivalent if they are different in the number of zeros at the end of the decimal. This screen shows the symmetry of adding, or taking away zeros at the end of a decimal. Write This Down Screen (Theory 6) Notes Test Item 1 Reasoning Mind 110 Curriculum Guide

111 Main Idea #2 - Comparing decimals by their whole parts starts the process of understanding the importance of place value of the digits in the decimal. Write This Down Screen (Theory 17) Notes Test Item 2 Main Idea #3 -This screen shows the steps for comparing decimals using the rules for comparing whole numbers with the same number of digits. This helps the students recognize the importance of place values in comparing numbers and gives a simple method for comparing all numbers on the number ray. Write This Down Screen (Theory 31) Notes Test Item 3 Theory Block Chapter 1 Equivalent Decimals (Theory 1) Looks at writing a whole number as a decimal. (Theory 2) Points out that decimals that look different can mean the same. (Theory 5) Shows how to mark decimal places on the unit segment on a number ray. The observation is made that placing a zero at the end of a decimal does not change the value of the number. Decimals that look different, but mean the same are called equivalent. (Theory 6) WTD See Lesson Notes, Main Idea 1. (Theory 7) Students practice finding an equivalent decimal by removing zeros in the decimal places. (Theory 8) Shows finding an equivalent decimal by adding zeros to the end of the decimal to give a decimal with a specified number of decimal places. Chapter 2 Comparing Decimals by their Whole Parts (Theory 16) Shows comparing two decimals by finding the equivalent fraction, then comparing the fractions. (Theory 17) WTD See Lesson Notes, Main Idea 2. Chapter 3 General Rules for Comparing Decimals (Theory 23) Gives an example of comparing decimals with equal whole parts. (Theory 24) Gives the steps for comparing decimals. (Theory 30) Gives steps that relate comparing decimals to comparing whole numbers. (Theory 31) WTD See Lesson Notes, Main Idea 3. A-Level Problems 1 Determine if an equality of two decimals is true. (2 hints) Theory 11, 2 Determine if an equality of two decimals is true. (2 hints) Theory 11, 3 Determine if an equality of two decimals is true. (2 hints) Theory 11, Reasoning Mind 111 Curriculum Guide

112 4 Determine if an equality of two decimals is true. (2 hints) Theory 11, 5 Compare two decimals. (1 hint) Theory 33 6 Compare two decimals. (1 hint) Theory 32 7 Compare two decimals. (1 hint) Theory Problems 25, 26 8 Compare two decimals. (1 hint) Theory Problems 25, 26 9 Compare two decimals. (2 hints) Theory 23, Theory Problems 25, 26 B-Level Problems 1 Arrange decimals from largest to smallest. (2 hints) Builds on A-Level problems Fill in the blanks to make the inequalities true. (1 hint) Builds on A-Level problems Reveal the picture by filling in a digit that will make the inequalities true. (1 hint) Builds on A-Level problems 5-9 C-Level Problems 1 Take values from a table and arrange in increasing order. (2 hints) Builds on B-Level Problem 1 2 Choose all the digits that would make an inequality true. Builds on B-Level Problems 2, 3 3 Reveal a picture by choosing all the digits that would make each inequality true. Builds on B-Level Problems 2, 3 Diagnostic Block 1 Determine if an equality of two decimals is true. (2 hints) A-Level Problem 1 2 Compare two decimals. (1 hint) A-Level Problem 5 3 Compare two decimals. (1 hint) A-Level Problem 7 4 Compare two decimals. (2 hint) A-Level Problem 9 Reasoning Mind 112 Curriculum Guide

113 43. Review: Rounding Decimals Approximate Length: 1.7 student study hours Overview This objective takes the rules for rounding whole numbers and expands it to the rules for rounding decimals. In Chapter 1, rounding whole numbers is reviewed. Chapter 2 shows rounding a decimal to a given decimal place by using the number ray. Then, in Chapter 3, a rule for rounding decimals is given. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lessons Related Objectives Objective 22. Fractions Objective 41. Review : Decimals Objective 24. Review: Comparing Decimals Objective 43. Review: Rounding Decimals Objective 44. Review: Adding Decimals Objective 45. Review: Subtracting decimals Vocabulary New Terms: Glossary Terms: approximate value, ones place, rounding, tenths place Theory Block Chapter 1 Review: Rounding Whole Numbers (Theory 1) Reviews rounding whole numbers to a certain place. Chapter 2 Rounding Decimals to a Place (Theory 3) Shows rounding a decimal to a given decimal place by finding the decimal on the number ray. Chapter 3 Decimal Rounding Rule (Theory 5) Gives the decimal rounding rule. (Theory 6) Gives a reminder that removing zeros at the end of the decimal does not change the number. (Theory 8) Shows an example of using the decimal rounding rule. A-Level Problems 1 Round the decimal to the nearest tenth. (1 hint) Theory Problem 2 2 Round the decimal to the nearest one. (1 hint) Theory 9 3 Round the decimal to the nearest hundredth. (1 hint) Theory Problem 5 4 Round the decimal to the nearest tenth. (1 hint) Theory Problem 6 5 Round the decimal to the nearest thousandth. (1 hint) Builds on Theory Problem 5 6 Round the decimal to the nearest tenth. (1 hint) Theory Problem 10 7 Round decimals in the table to the nearest whole number. (1 hint) Builds on Theory Problem 8 Reasoning Mind 113 Curriculum Guide

114 B-Level Problems 1 Reveal the picture by rounding each decimal to the specified place. (1 hint) 2.1 Word problem to round two decimals and then find the different of the approximate values. (2 hints) 2.2 Word problem to round three decimals, sum the two smaller values and then find the different of the sum and greatest value. (3 hints) C-Level Problems 1 Reveal the picture by rounding each decimal to the specified place. (1 hint) Builds on A-Level Problems 1-7 Builds on A-Level Problem 2 Builds on A-Level Problem 2 Similar to B-Level Problem 1 Diagnostic Block 1 Round the decimal to the nearest tenth. (1 hint) A-Level Problem 1 2 Round the decimal to the nearest hundredth. (1 hint) A-Level Problem 3 3 Round the decimal to the nearest thousandth. (1 hint) A-Level Problem 5 4 Round the decimal to the nearest tenth. (1 hint) A-Level Problem 6 Reasoning Mind 114 Curriculum Guide

115 44. Review: Adding Decimals Approximate Length: 2.7 student study hours Overview This objective looks at adding decimals using column addition by lining up the decimal point to line up the place values, and adding zeros to the end of decimal if necessary to completely line up the numbers. Chapter 1 starts with adding two decimals using the mixed number equivalents, then gives rules for adding decimals. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lessons and Smarter Solving Offline Lesson 4. Related Objectives Objective 22. Fractions Objective 41. Review : Decimals Objective 42. Review: Comparing Decimals Objective 42. Review: Rounding Decimals Objective 44. Review: Adding Decimals Objective 45. Review: Subtracting decimals Vocabulary New Terms: Glossary Terms: column addition, decimal, decimal place, decimal point, decimal place, evaluate, expression, mixed number, sum, summand, whole number Lesson Notes Main Idea #1 - This example is important because it shows lining up the decimal points and adding zeros to the end of one summand to have the same number of digits after the decimal place in both decimals. Write This Down Screen (Theory 6) Notes Test Item 1 Theory Block Chapter 1 Adding Decimals (Theory 2) Shows adding two decimals by adding the equivalent mixed numbers. (Theory 3) Shows how to add the decimals using column addition. (Theory 4) Gives the steps to add two decimals. (Theory 6) WTD See Lesson Notes, Main Idea 1. (Theory 6) Gives practice lining up decimals for column addition. (Theory 9) Shows evaluating an expression with a letter for a decimal value for the letter. (Theory 10) Shows solving a wording problem by adding two decimals. (Theory 11) Shows adding a decimal to a whole number. (Theory 13, 14) Shows evaluating expressions by adding a decimal to a whole number. (Theory 15) Shows adding two decimals in which no zeros need to be added to the end of either number, but the answer has zeros that can be removed. Reasoning Mind 115 Curriculum Guide

116 A-Level Problems 1 Find the sum of two decimals. (1 hint) Theory Problem 12 2 Find the sum of two decimals. (1 hint) Theory Problem 3 3 Find the sum of two decimals. (1 hint) Theory Problem 3 4 Find the sum of two decimals. (1 hint) Theory Problem 4 5 Add a decimal to a whole number. (2 hints) Theory Problems 9, 10 6 Word problem that requires adding two decimals. (2 hints) Theory Problem 7 7 Find the value of an expression with one letter with a given value that is a decimal. (3 hints) 8 Evaluate a numerical expression with two operations, the second of which is adding a whole number to a decimal. (2 hints) Theory problem 6 Builds on Theory Problems 9, 10 9 Word problem to add a whole number to a decimal. (3 hints) Theory Problem 11 B-Level Problems 1 Review a picture by adding the decimals. (1 hint) Builds on A-Level Problems Find the sum of three decimals. (1 hint) Builds on A-Level Problems Word problem which requires adding a decimal to a whole number. Builds on A-Level Problem 9 C-Level Problems 1 Check the addition equalities with decimal summands and give the correct answer to any that are incorrect. (2 hints) Builds on B-Level Problem 1 2 Word problem which requires adding three decimals. (3 hints) Builds on B-Level Problem 2 3 Word problem to add three decimals and compare the sum to another decimal. (1 hint) Builds on B-Level Problem 2 Diagnostic Block 1 Find the sum of two decimals. (1 hint) A-Level Problem 1 2 Find the sum of two decimals. (1 hint) A-Level Problem 2 3 Find the sum of two decimals. (1 hint) A-Level Problem 3 4 Add a decimal to a whole number. (2 hints) A-Level Problem 5 Reasoning Mind 116 Curriculum Guide

117 45. Review: Subtracting Decimals Approximate Length: 3 student study hours Overview This objective looks at subtracting decimals using column subtraction with lining up the place values using the decimal point. Chapter 1 starts with modeling subtracting decimals by first looking at subtracting the equivalent mixed numbers, then giving the rule for column subtraction of two decimals. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lessons Related Objectives Objective 22. Fractions Objective 41. Review : Decimals Objective 42. Review: Comparing Decimals Objective 43. Rounding Decimals Objective 44. Adding Decimals Objective 45. Subtracting decimals Vocabulary Glossary Terms: column subtraction, decimal, decimal place, decimal point, difference, equality, evaluate, expression, minuend, mixed number, order of operations, subtrahend, whole number Lesson Notes Main Idea #1 - This example is important because it shows adding zeros to the subtrahend to get the same number of zeros as in the minuend, as well as lining up the places by lining up the decimal point. Write This Down Screen (Theory 8) Notes Test Item 1 Theory Block Chapter 1 Subtracting Decimals (Theory 2) Shows subtracting a decimal from a decimal by changing them to the equivalent mixed numbers. (Theory 3) Shows subtracting the same decimals using column subtraction. (Theory 5) Gives practice in subtracting one decimal from another. (Theory 6) Gives the rule for subtracting one decimal from another decimal. (Theory 8) WTD See Lesson Notes, Main Idea 1. (Theory 9) Gives practice in setting up column subtraction including adding zeros if needed. (Theory 11) Shows solving a word problem by subtraction one decimal from another. (Theory 12) Shows an expression with two subtraction operations. (Theory 13) Shows subtracting a decimal from a whole number. (Theory 16) Gives an example of a problem where no zeros need to be added to either the minuend or subtrahend and zeros need to be removed from the answer. (Theory 17) Shows evaluating an expression with subtraction in parentheses. Reasoning Mind 117 Curriculum Guide

118 A-Level Problems 1 Find the difference of two decimals. (1 hint) Theory Problem 1 2 Find the difference of two decimals. (1 hint) Theory Problem 3 3 Find the difference of two decimals. (1 hint) Theory Problem 3 4 Find the difference of two decimals. (1 hint) Theory Problem 5 5 Find the difference of a whole number and decimal. (2 hints) Theory Problem 11 6 Find the difference of a whole number and decimal. (2 hints) Theory problem 10 7 Word problem which requires finding the difference between two decimals. (2 hints) Theory Problem 7 8 Word problem to compare two decimals using subtraction. (2 hints) Theory Problem 12 9 Evaluate expression with letter for given value. (2 hints) Theory Evaluate the expression with parentheses. (3 hints) Similar to Theory Problem 8 B-Level Problems 1 Reveal the picture by finding the differences. (2 hints) Builds on A-Level Problems Word problem with three steps to compare two decimals. (2 hints) Builds on A-Level Problem 8 3 Solve a word problem by subtracting two decimals. (2 hints) Builds on A-Level problem 7 C-Level Problems 1 Find the equalities that are correct. Find the correct answer to the ones that are incorrect. (2 hints) 2 Word problem which requires solving an expression with two operations. (3 hints) 3 Solve multiple step word problem with subtracting decimals. (2 hints) Builds on B-Level Problem 1 Builds on B-Level Problem 3 Builds on B-Level Problem 2 Diagnostic Block 1 Find the difference of two decimals. (1 hint) Theory Problem 1 2 Find the difference of two decimals. (1 hint) A-Level Problem 1 3 Find the difference of two decimals. (1 hint) Theory Problem 6 4 Subtract a decimal from a whole number. (2 hints) A- Level Problem 5 Reasoning Mind 118 Curriculum Guide

119 46. Multiplying a Decimal by a Whole Number Approximate Length: 2.7 student study hours Overview This objective looks at multiplying a decimal by a whole number by modeling the product as repeated addition, then looking at the special case of multiplying by multiples of 10. In Chapter 1, multiplying a decimal by a whole number is defined as repeated addition of the decimal, with a rule given in Chapter 2. Then in Chapter 3, the special case of multiplying by 10, 100, etc. is discussed. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lesson 117 Related Objectives Objective 27. Column Multiplication of a Three-Digit Number: Part 2 (Grade 4 Objective 15. Column Multiplication by a Two-Digit Number Objective 46. Multiplying a Decimal by a Whole Number Objective 47. Multiplying a Decimal by a Decimal Vocabulary New Terms: Glossary Terms: decimal, decimal point, factor, hundred, product, natural number, ten, thousand, whole number Lesson Notes Main Idea #1 - Multiplication of a decimal by a whole number begins the process of multiplying by a decimal. It is easily modeled expanding the definition of multiplication of whole numbers. Write This Down Screen (Theory 4) Notes Test Item 1 Main Idea #2 - Multiplying a whole number by a decimal does not have the same definition as multiplying a whole number by a decimal. This example shows the rule follows the same pattern Write This Down Screen (Theory 8) Notes Test Item 2 Reasoning Mind 119 Curriculum Guide

120 Theory Block Chapter 1 Multiplying Decimals by Natural Numbers (Theory 1, 2) Shows finding a perimeter of a square with a decimal side length as an example of multiplying a decimal by a natural number. (Theory 3) Gives a definition of multiplying a decimal by a whole number as repeated addition of the decimal. (Theory 4) WTD See Lesson Notes, Main Idea 1. Chapter 2 Rule for Multiplying a Decimal by a Natural Number. (Theory 5, 6) Looks at adding decimals using column addition and builds an understanding for similarities that will be used for column multiplication. (Theory 7) Gives the rule for multiplying a decimal by a natural number. (Theory 8) WTD See Lesson Notes, Main Idea 2. (Theory 9-11) Works examples using the rule. Chapter 3 Multiplying a Decimal by 10, 100, 1,000, etc. (Theory 12, 13) Looks at an example of multiplying a decimal by 10, then 100. (Theory 14) Gives a Rule for multiplying a decimal by 10, 100, 1,100,. (Theory 16) Shows examples of multiplying decimals by 100 and 1,100. A-Level Problems 1 Find the product of a decimal and a whole number. (1 hint) Theory Find the product of a decimal and a whole number. (1 hint) Theory Find the product of a decimal and a whole number. (1 hint) Theory Find the product of a decimal and a whole number. (1 hint) Theory Find the product in a expression by substituting 10, 100, and 1,000 for the letter. (2 hints) Theory Word problem to find the product of a decimal and a natural number. (2 hints) Theory Problems Word problem to find the product of a decimal and a natural number. (2 hints) Theory Problems Word problem to find the product of a decimal and a natural number. (2 hints) Theory Problems Word problem to find the product of a decimal and a natural number. (2 hints) Theory Problems Word problem to find the product of a decimal and a natural number. (2 hints) Theory Problems 2-4 B-Level Problems 1 Reveal the picture by evaluating expressions which include finding the products from multiplying a decimal by a whole number. 2 Evaluate the expression for given values which includes multiplying a decimal by a whole number. C-Level Problems 1 Evaluate the expressions and choose the two whose values add up to a certain amount. (1 hint) 2 Compare the evaluations of an expression for two sets of values for the letters, and choose which one is the smaller value. (2 hints) Builds on A-Level Problems 1-5 Builds on A-Level Problems 1-5 Builds on B-Level Problem 2 Builds on B-Level Problem 2 Reasoning Mind 120 Curriculum Guide

121 Diagnostic Block 1 Word problem to find the product of a decimal and a natural number. (2 hints) Theory Problem 3 2 Find the product of a decimal and a whole number. (1 hint) Theory Problem 7 3 Find the product of a decimal and 100. Theory Problem 9 Reasoning Mind 121 Curriculum Guide

122 47. Multiplying a Decimal by a Decimal Approximate Length: 2.5 student study hours Overview This objective looks at multiplying a decimal by a decimal by developing rules for the simple case where on decimal is 0.1, 0.01, 0.001, etc, then a rule for all decimals. Chapter 1 begins with a rule for multiplying a decimal by 0.1, 0.01, etc., then, gives a rule for multiplying all decimals. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lessons Related Objectives Objective 27. Column Multiplication of a Three-Digit Number: Part 2 (Grade 4) Objective 15. Column Multiplication by a Two-Digit Number Objective 46. Multiplying a Decimal by a Whole Number Objective 47. Multiplying a Decimal by a Decimal Vocabulary New Terms: Glossary Terms: decimal, decimal point, power Lesson Notes Main Idea #1 - This rule starts the process by looking at multiplication by unit decimal place values as a similar rule for multiplying by multiples of 10. Write This Down Screen (Theory 2) Notes Test Item 1 Main Idea #2 This example shows using column multiplication and determining the decimal point in the product by summing the number of decimal points in each factor. Write This Down Screen (Theory 6) Notes Test Item 2 Reasoning Mind 122 Curriculum Guide

123 Theory Block Chapter 1 Multiplying a Decimal by a Decimal (Theory 2) Gives a Rule for multiplying a decimal by 0.1, 0.01, 0.001, etc. (Theory 2) WTD See Lesson Notes, Main Idea 1. (Theory 3) Gives examples of multiplying a decimal by 0.1, And (Theory 5) Gives a rule for multiplying a decimal by a decimal. (Theory 6) WTD See Lesson Notes, Main Idea 2. (Theory 7) Shows examples of multiplying a decimal by a decimal with a zero whole part. A-Level Problems 1 Find the products of a decimal and 0.1 and (1 hint) Theory 3 2 Evaluate an expression for a given value of the letter resulting in multiplying a decimal by 0.1 or (2 hints) 3 Solve a word problem with the multiplication of two decimals. (2 hints) Theory Problems 2, 3 Builds on Theory Problems 6-11 B-Level Problems 1 Reveal the picture by evaluating expressions with several steps that include multiplying a decimal by 0.1 and (1 hint) 2 Fill in the table of areas by multiplying a decimal by 0.1, 0.01, and (2 hints) 3 Evaluate an expression for several given values of the letter resulting in multiplying a decimal by a decimal. (2 hints) C-Level Problems 1 Evaluate an expression for several given values of the letter resulting in multiplying a decimal by a decimal. (2 hints) 2 Solve a word problem using values from a graph which results in multiplying a decimal by 0.1. Builds on A-Level Problem 1 Builds on A-Level Problem 2 Builds on A-Level Problem 3 Builds on B-Level Problem 3 Builds on B-Level Problem 2 Diagnostic Block 1 Find the area of a rectangle which involves multiplying a decimal by 0.1 or (2 hints) Theory Problem 4 2 Find the product of two decimals. Theory Problem 7 Reasoning Mind 123 Curriculum Guide

124 48. Dividing a Decimal by a Whole Number Approximate Length: 2.8 student study hours Overview In this objective, dividing a decimal by a whole number is developed by first dividing a decimal by a multiple of ten, then applying this to dividing a decimal by a whole number. This leads to finding the decimal equivalent of a proper fraction. In Chapters 1 and 2, multiplying by a multiple of ten means moving the decimal point to the right. Chapter 3 develops the rule for dividing a decimal by a whole number. Then, in Chapter 4, the understanding is expanded to finding the decimal equivalent of some common proper fractions. Upon completion of this objective, the following Smarter Solving content is automatically released to students: Smarter Solving Lessons Related Objectives Objective 30. Long Division of 3-Digit Numbers (Grade 4) Objective 19. Long Division Objective 48. Dividing a Decimal by a Whole Number Objective 49. Dividing a Decimal by a Decimal Vocabulary New Terms: Glossary Terms: decimal, dividend, divisor, natural number, quotient, whole number Lesson Notes Main Idea #1 - Understanding dividing by 10, 100, 1,000 and so on is the first step toward dividing decimals by whole numbers since these represent the unit place values. Write This Down Screen (Theory 7) Notes Test Item 1 Reasoning Mind 124 Curriculum Guide

125 Main Idea #2 - This is an important example to show the place values in the quotient when dividing a decimal by a whole number. Write This Down Screen (Theory 11) Notes Test Item 2 Main Idea #3 - This is an example showing a quotient with a zero whole part. This will be used to find the decimal equivalent for a common fraction. Write This Down Screen (Theory 15) Notes Test Item 3 Theory Block Chapter 1 Dividing Decimals by Natural Numbers (Theory 1) Reminder we know how to divide a whole number by a whole number. Reasoning Mind 125 Curriculum Guide

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