Rectangles and Factors


 Cecil Tyler
 1 years ago
 Views:
Transcription
1 Rectangles and Factors Background information: Unlike addition, in multiplication, each factor has a different purpose; one identifies the number in each group and the other, the number of groups. Therefore we don t just want to introduce multiplication as repeated addition but also explore an array or rectangle. Properties of arithmetic provide the conceptual underpinnings for computational strategies and foundation for algebraic thinking. It is important to introduce these properties at the basic fact level so that students can apply them easily as multiplication examples become more complex. The array or area model of multiplication is particularly powerful for modeling the commutative and distributive properties of multiplication. By turning the rectangle or looking at it sideways, students can see that the total stays the same, that a x b = b x a. The distributive property of multiplication over addition states that a(b + c) = ab + ac. It is this property that allows us to find 9 x 6 or 9 x (3 + 3) by realizing that this is the same as (9 x 3) = (9 x 3) or 2(9 x 3). So if students know the product when multiplying by three, they can double it to multiply by six. Though not helpful to learning of basic facts, multiplication is also associative, that is (a x b) x c = a x(b x c). This property is useful when multiplying three numbers, for example, 7 x 2 x 5. We can choose to start with 2 x 5, rather than 7 x 2. Square numbers, like doubles in addition, should be emphasized. Once a student knows a fact such as 7 x 7 = 49, he or she can use this fact to find 8 x 7, by adding one more 7 to 49 to get 56. Having such anchors is essential, as unlike addition and subtraction facts, most basic multiplication facts cannot be determined quickly by counting on or back. Think about divisibility rules! A prime number is a positive integer that has two and only two unique positive factors. The two most pervasive misconceptions that students demonstrate when discussing prime and composite numbers are that one is a prime number that all prime numbers are odd. It is important to be very explicit that one is a special number, as it has only one unique factor, and that two is the only even prime number. DPI 35 Mathematics Fall REAS, 2011 Page 1
2 Materials needed: one inch color tiles cm grid paper Task: (this task could be 13 days) Make as many different rectangles as you can using 12 squareinch color tiles. Complete a table like the one below to record your rectangles. Rectangle Possible with 12 Tiles Number of Rows Number in each Row Multiplication Sentence x 6 = 12 What did you notice? Were the rectangles the same? Were the rectangles different? How would you describe your rectangle? Does that description fit someone else's rectangle? Possible Classroom Discussion: Does the orientation of the rectangles matter? Is a 2 x 6 rectangle the same as a 6 x 2 rectangle? The rectangles are different in that they show two distinct arrays, but they cover the same area: a 2 x 6 rectangle can be rotated to represent a 6 x 2 rectangle. Third grade (If fourth grade) For this activity we are interested in the dimensions of the rectangles (or the factors of the numbers), not in their vertical or horizontal orientation. What is important to note is that 2 and 6 are both factors of 12. Now let s find all the rectangles you can make with 18 tiles. Record your rectangles on grid paper. DPI 35 Mathematics Fall REAS, 2011 Page 2
3 Now.. Let s work together to make a class table from 125 Which numbers have rectangles with 2 rows? List them from smallest to largest. Which numbers have rectangles with 3 rows? List them from smallest to largest. Which numbers on the chart are multiples of 4? (have rectangles with 4 rows) Which numbers on the chart are multiples of 5? How many different rectangles can you make with 5 tiles How many with 7 tiles? List the prime numbers between 1 and 25 Are all odd numbers prime? Explain. Look at the number nine, what do you notice? What other numbers have rectangles that are squares? What is the next largest square after 25? Things to think about: A number is a multiple of 2 if it equals 2 times another whole number. If you can make a rectangle with 2 rows for a number then it is a multiple of 2. Numbers that are multiples of two (2, 4, 6, 8, etc ) are called even numbers. Numbers that are not multiplies of 2 (1, 3, 5, 7 etc ) are called odd numbers. When you skip counts you say the multiples of a number. For example, skip counting by 3 gives the multiples of 3. The multiples of 3 are 3, 6, 9, 12 and so on. They are all the numbers that have rectangles with 3 rows. Numbers that are larger than one and have only one rectangle have a special name. They are called prime numbers. For example, 5, 7, are prime numbers. Prime numbers are defined as numbers that are larger than one and have only on rectangle. One is Not prime, since a prime number is conventionally taken to be number with exactly two factors (itself and one), one is not considered to be prime. Note that two is prime since its only factors are itself and one. Two is the only even prime number. During the discussion of prime numbers, the question sometimes arises of what to call numbers that are not prime numbers, the ones that have more than one rectangular array. Tell students that these are called composite numbers. The definition of a composite number is one DPI 35 Mathematics Fall REAS, 2011 Page 3
4 that has more than two distinct factors. All numbers that are not prime and are not one are composite numbers. Possible Classroom Discussion What did you find out about the number of factors? The number 1 has one factor (itself) and forms one rectangle (a 1by1 square); it is classified by mathematicians as a special number and is neither prime nor composite. Many numbers have only two factors and make just one rectangle: 2, 3, 5, 7, 1, 13, 17, 19, 23. These numbers are the prime numbers. Prime numbers are composite numbers and have more than two factors. All the other numbers are composite numbers and have more than two factors; composite numbers can be represented by at least two unique rectangular arrays. Examine the rectangles representing the numbers 1, 4, 9, 16, and 25. Did you notice that in each case one of the rectangles that can be found is also a square? Ancient Greek mathematicians thought of number relationships in geometric terms and called numbers like this square numbers, because of the rectangular arrays they can be represented by a square. The square numbers have an odd number of factors, whereas the other numbers examined have an even number of factors. Numbers that are not square always have factor pairs. For example 12, the factor pairs are 1 and 12, 2, and 6, and 3 and 4. But square numbers always have one factor that has no partner other than itself. For example 9, 1 and 9 are factor pair, but 3 is its own partner because 3 x 3 = 9. The factor of a square number that has no partner 3 for the square number 9 isn t listed twice. Therefore, the factors of 9 are 1, 3, and 9, and the factors of 16 are 1, 2, 4, 8, and 16 an odd number of factors. DPI 35 Mathematics Fall REAS, 2011 Page 4
5 Number Factors Number of Rectangles Prime or Composite Neither 2 1, 2 2 Prime 3 1, 3 2 Prime 4 1, 2, 4 3 Composite 5 1, 5 2 Prime 6 1, 2, 3, 6 4 Composite 7 1, 7 2 Prime 8 1, 2, 4, 8 4 Composite 9 1, 3, 9 3 Composite 10 1, 2, 5, 10 4 Composite 11 1, 11 2 Prime 12 1, 2, 3, 4, 6, 12 6 Composite 13 1, 13 2 Prime 14 1, 2, 7, 14 4 Composite 15 1, 3, 5, 15 4 Composite 16 1, 2, 4, 8, 16 5 Composite 17 1, 17 2 Prime 18 1, 2, 3, 6, 9, 18 6 Composite 19 1, 19 2 Prime 20 1, 2, 4, 5, 10, 20 6 Composite 21 1, 3, 7, 21 4 Composite 22 1, 2, 11, 22 4 Composite 23 1, 23 2 Prime 24 1, 2, 3, 4, 6, 8, 12, 24 8 Composite 25 1, 5, 25 3 Composite DPI 35 Mathematics Fall REAS, 2011 Page 5
6 Possible Extension: If my rectangle has a total of 18 square tiles and 3 rows of tiles How many tiles are in each row? Write a number sentence for this rectangle. Is 34 a multiple of 2? Explain why or why not. Is 3 a factor of 35? Explain why or why not. Name ten numbers that have 3 as a factor. Common Core State Standards Third Grade: Represent and solve problems involving multiplication and division. 3.OA.1 Interpret products of whole numbers, e.g., interpret 5 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as OA.2 Interpret wholenumber quotients of whole numbers, e.g., interpret 56 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as OA.3 Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.1 3.OA.4 Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8? = 48, 5 = 3, 6 6 =?. Understand properties of multiplication and the relationship between multiplication and division. 3.OA.5. Apply properties of operations as strategies to multiply and divide.2 Examples: If 6 4 = 24 is known, then 4 6 = 24 is also known. (Commutative property of multiplication.) can be found by 3 5 = 15, then 15 2 = 30, or by 5 2 = 10, then 3 10 = 30. (Associative property of multiplication.) Knowing that 8 5 = 40 and 8 2 = 16, one can find 8 7 as 8 (5 + 2) = (8 5) + (8 2) = = 56. (Distributive property.) 3.OA.6. Understand division as an unknownfactor problem. For example, find 32 8 by finding the number that makes 32 when multiplied by 8. Fourth Grade: Gain familiarity with factors and multiples. 4.OA.4 Find all factor pairs for a whole number in the range Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range is a multiple of a given onedigit number. Determine whether a given whole number in the range is prime or composite. DPI 35 Mathematics Fall REAS, 2011 Page 6
7 This task came from these resources: 1. Math Matters Grade K8 Understanding the Math You Teach by Suzanne Chapin and Art Johnson 2. A collection of Math Lessons from Grades 3 through 6 by Marilyn Burns 3. Trailblazer 4 th Grade, Unit 4: Product and Factors A TIMS Curriculum from University at Chicago 4. Grades 34 Zeroing in on Number and Operations Key Ideas and Common Misconceptions by Linda Dacey and Anne Collins Possible starting task: Many people have a number that they think is interesting. Choose a whole number between 1 and 100 that you think is special. Record your number. Explain why you chose that number. List three or four mathematical facts about your number. List three or four connections you can make between your number and your world. Revisit the task above after the Rectangle and Factor lesson, what mathematical facts about the selected number do students think about now! DPI 35 Mathematics Fall REAS, 2011 Page 7
8 Rectangles and Factors DPI 35 Mathematics Fall REAS, 2011 Page 8
9 cm grid paper DPI 35 Mathematics Fall REAS, 2011 Page 9
10 DPI 35 Mathematics Fall REAS, 2011 Page 10
11 Factor Game DPI 35 Mathematics Fall REAS, 2011 Page 11
12 Table 2 Common multiplication and division situations. 7 7 The first examples in each cell are examples of discrete things. These are easier for students and should be given before the measurement examples. DPI 35 Mathematics Fall REAS, 2011 Page 12
13 NCTM Illuminations The Factor Game Factor Game: Factor Game Player A chooses a number on the game board by clicking on it. The square will be colored blue, as shown for 12. Player A receives 12 points for this choice. Player B then clicks on all the proper factors of Player A s number. The proper factors of a number are all the factors of that number, except the number itself. For example, the proper factors of 12 are 1, 2, 3, 4, and 6. Although 12 is also a factor of 12, it is not DPI 35 Mathematics Fall REAS, 2011 Page 13
14 considered a proper factor. All of the proper factors that Player B selects will be colored green, as shown to the left. Player B will receive = 16 points for selecting all of the proper factors. Players reverse roles. On the next turn, Player B colors a new number and gets that many points, and Player A colors all the factors of the number that are not already colored and receives the sum of those numbers in points. The players take turns choosing numbers and coloring factors. If a player chooses a number with no uncolored factors remaining, that player loses a turn and does not get the points for the number selected. The game ends when there are no numbers remaining with uncolored factors. The player with the greater total when the game ends is the winner. The Factor Game applet was adapted with permission and guidance from "Prime Time: Factors and Multiples," Connected Mathematics Project, G. Lappan, J. Fey, W. Fitzgerald, S. Friel and E. Phillips, Dale Seymour Publications, (1996), pp DPI 35 Mathematics Fall REAS, 2011 Page 14
15 Product Game Product Game: Object of the Game: To get four squares in a row vertically, horizontally, or diagonally. 1. To begin the game, Player 1 moves a marker to a number in the factor list of numbers 19 along the bottom of the game screen. 2. Player 2 then moves the other marker to any number in the factor list (including the number marked by Player 1). The product of the two marked numbers is determined, and that product is colored red for Player Player 1 moves either marker to another number, and the new product is colored blue for Player Players take turns moving a marker, and each product is marked red or blue, depending on which player made the product. However, if a product is already colored, the player does not get a square for that turn. 5. Play continues until one player wins, or until all squares have been colored. This Product Game Investigation was adapted with permission and guidance from Prime Time: Factors and Multiples, Connected Mathematics Project, G. Lappan, J. Fey, W Fitzgerald, S. Friel and E. Phillips, Dale Seymour Publications (1996), pp DPI 35 Mathematics Fall REAS, 2011 Page 15
16 What does Computational Fluency mean? Developing fluency requires a balance and connection between conceptual understanding and computational proficiency. On the one hand, computational methods that are over practiced without understanding are often forgotten or remembered incorrectly On the other hand; understanding without fluency can inhibit the problemsolving process. (PSSM, Page 35 How do students demonstrate Computational Fluency? Students exhibit computational fluency when they demonstrate flexibility in the computational methods they choose, understand and can explain these methods, and produce accurate answers efficiently. The computational methods that a student uses should be based on mathematical ideas that the student understands well, including the structure of the baseten number system properties of multiplication and division (and of course, of the other operations, as well), and the number relationships. (PSSM, page 152) Is there such a thing as effective drill? There is little doubt that strategy development and general number sense (number relationship and operation meanings) are the best contributors to fact mastery. Drill in the absence of these factors has repeatedly been demonstrated as ineffective. However, the positive value of drill should not be completely ignored. Drill of nearly any mental activity strengths memory and retrieval capabilities. (Van de Walle) What about Timed Tests? Teachers who use timed test believe that the test help children learn basic facts. This makes no instructional sense. Children who perform well under time pressure display their skills. Children who have difficulty with skills, or who work more slowly, run the risk of reinforcing wrong learning under pressure. In addition, children can become fearful and negative toward their math learning. (Burns 2000, p.157) DPI 35 Mathematics Fall REAS, 2011 Page 16
Math Journal HMH Mega Math. itools Number
Lesson 1.1 Algebra Number Patterns CC.3.OA.9 Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. Identify and
More informationCommon Core State Standards for Math Grades K  7 2012
correlated to the Grades K  7 The Common Core State Standards recommend more focused and coherent content that will provide the time for students to discuss, reason with, reflect upon, and practice more
More informationTYPES OF NUMBERS. Example 2. Example 1. Problems. Answers
TYPES OF NUMBERS When two or more integers are multiplied together, each number is a factor of the product. Nonnegative integers that have exactly two factors, namely, one and itself, are called prime
More informationSolve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.
Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. Solve word problems that call for addition of three whole numbers
More informationGrade 2. M4: and M:5 Addition and Subtraction of Numbers to 1000. M3: Place Value, Counting, and Comparison of Numbers to 1000
Grade 2 Key Areas of Focus for Grades K2: Addition and subtractionconcepts, skills and problem solving Expected Fluency: Add and Subtract within 20 Add and Subtract within 100 Module M1: Mastery of Sums
More informationPlanning Guide. Grade 6 Factors and Multiples. Number Specific Outcome 3
Mathematics Planning Guide Grade 6 Factors and Multiples Number Specific Outcome 3 This Planning Guide can be accessed online at: http://www.learnalberta.ca/content/mepg6/html/pg6_factorsmultiples/index.html
More informationDescribing and Solving for Area and Perimeter
Grade 3 Mathematics, Quarter 2,Unit 2.2 Describing and Solving for Area and Perimeter Overview Number of instruction days: 810 (1 day = 90 minutes) Content to Be Learned Distinguish between linear and
More informationNC DEPARTMENT OF PUBLIC INSTRUCTION
Rectangle Riddles Common Core Standard: Work with equal groups of objects to gain foundations for multiplication. 2.G.2 Partition a rectangle into rows and columns of samesize squares and count to find
More informationFactor Game. Overview of Lesson. Duration. Materials. Vocabulary. Math Solutions Lesson from the Classroom. A Lesson for Grade 5
Math Solutions Lesson from the Classroom Factor Game A Lesson for Grade 5 Featured in Math Solutions Online Newsletter, Issue 36 The lesson was adapted from the Math Solutions publications Enriching Your
More informationSwavesey Primary School Calculation Policy. Addition and Subtraction
Addition and Subtraction Key Objectives KS1 Foundation Stage Say and use number names in order in familiar contexts Know that a number identifies how many objects in a set Count reliably up to 10 everyday
More information1) Make Sense and Persevere in Solving Problems.
Standards for Mathematical Practice in Second Grade The Common Core State Standards for Mathematical Practice are practices expected to be integrated into every mathematics lesson for all students Grades
More informationBPS Math Year at a Glance (Adapted from A Story Of Units Curriculum Maps in Mathematics K5) 1
Grade 4 Key Areas of Focus for Grades 35: Multiplication and division of whole numbers and fractionsconcepts, skills and problem solving Expected Fluency: Add and subtract within 1,000,000 Module M1:
More informationBasic Facts INTENSIVE INTERVENTION. National Center on. at American Institutes for Research
National Center on INTENSIVE INTERVENTION at American Institutes for Research Basic Facts 1000 Thomas Jefferson Street, NW Washington, DC 20007 Email: NCII@air.org While permission to reprint this publication
More informationComposing and Decomposing Whole Numbers
Grade 2 Mathematics, Quarter 1, Unit 1.1 Composing and Decomposing Whole Numbers Overview Number of instructional days: 10 (1 day = 45 60 minutes) Content to be learned Demonstrate understanding of mathematical
More informationMinnesota Academic Standards
A Correlation of to the Minnesota Academic Standards Grades K6 G/M204 Introduction This document demonstrates the high degree of success students will achieve when using Scott Foresman Addison Wesley
More informationMath Content by Strand 1
Math Content by Strand 1 Number and Operations: Whole Numbers Addition and Subtraction and the Number System Kindergarten Young students develop their understanding of the operations of addition and subtraction
More informationMathUSee Correlation with the Common Core State Standards for Mathematical Content for Fourth Grade
MathUSee Correlation with the Common Core State Standards for Mathematical Content for Fourth Grade The fourthgrade standards highlight all four operations, explore fractions in greater detail, and
More informationOverview. Essential Questions. Precalculus, Quarter 3, Unit 3.4 Arithmetic Operations With Matrices
Arithmetic Operations With Matrices Overview Number of instruction days: 6 8 (1 day = 53 minutes) Content to Be Learned Use matrices to represent and manipulate data. Perform arithmetic operations with
More informationThe Crescent Primary School Calculation Policy
The Crescent Primary School Calculation Policy Examples of calculation methods for each year group and the progression between each method. January 2015 Our Calculation Policy This calculation policy has
More informationMultiplication and Division Fact Families
Multiplication and Division Fact Families Objectives To review fact families and the Multiplication/Division Facts Table; and to guide children as they practice multiplication and division facts. www.everydaymathonline.com
More informationA Concrete Introduction. to the Abstract Concepts. of Integers and Algebra using Algebra Tiles
A Concrete Introduction to the Abstract Concepts of Integers and Algebra using Algebra Tiles Table of Contents Introduction... 1 page Integers 1: Introduction to Integers... 3 2: Working with Algebra Tiles...
More informationReady, Set, Go! Math Games for Serious Minds
Math Games with Cards and Dice presented at NAGC November, 2013 Ready, Set, Go! Math Games for Serious Minds Rande McCreight Lincoln Public Schools Lincoln, Nebraska Math Games with Cards Close to 20 
More informationCurrent California Math Standards Balanced Equations
Balanced Equations Current California Math Standards Balanced Equations Grade Three Number Sense 1.0 Students understand the place value of whole numbers: 1.1 Count, read, and write whole numbers to 10,000.
More informationTopic: 1  Understanding Addition and Subtraction
8 days / September Topic: 1  Understanding Addition and Subtraction Represent and solve problems involving addition and subtraction. 2.OA.1. Use addition and subtraction within 100 to solve one and twostep
More informationOperations and Algebraic Thinking Represent and solve problems involving multiplication and division.
Performance Assessment Task The Answer is 36 Grade 3 The task challenges a student to use knowledge of operations and their inverses to complete number sentences that equal a given quantity. A student
More informationMath Content by Strand 1
Math Content by Strand 1 Number and Operations with Whole Numbers Multiplication and Division Grade 3 In Grade 3, students investigate the properties of multiplication and division, including the inverse
More informationGrade 5 Math Content 1
Grade 5 Math Content 1 Number and Operations: Whole Numbers Multiplication and Division In Grade 5, students consolidate their understanding of the computational strategies they use for multiplication.
More informationGrade 4 MultiDigit Multiplication and Division with One Divisor Unit of Instruction
Grade 4 MultiDigit Multiplication and Division with One Divisor Unit of Instruction This is a progressive unit of instruction beginning with students investigating the concrete area model of fraction
More informationPUBLIC SCHOOLS OF EDISON TOWNSHIP DIVISION OF CURRICULUM AND INSTRUCTION ELEMENTARY MATH GRADE 2 MATH IN FOCUS
PUBLIC SCHOOLS OF EDISON TOWNSHIP DIVISION OF CURRICULUM AND INSTRUCTION ELEMENTARY MATH GRADE 2 MATH IN FOCUS Length of Course: Term Elective / Required: Required Schools: Elementary Student Eligibility:
More informationCCSS Mathematics Implementation Guide Grade 5 2012 2013. First Nine Weeks
First Nine Weeks s The value of a digit is based on its place value. What changes the value of a digit? 5.NBT.1 RECOGNIZE that in a multidigit number, a digit in one place represents 10 times as much
More informationYearataGlance: 5 th Grade
September: UNIT 1 Place Value Whole Numbers Fluently multiply multidigit numbers using the standard algorithm Model long division up to 2 digit divisors Solve real world word problems involving measurement
More informationMultiplication Fact Power and Shortcuts
Multiplication Fact Power and Shortcuts Objectives To discuss multiplication facts and the importance of fact power; and to review fact shortcuts. www.everydaymathonline.com epresentations etoolkit Algorithms
More informationA Correlation of. to the. Common Core State Standards for Mathematics Grade 4
A Correlation of to the Introduction envisionmath2.0 is a comprehensive K6 mathematics curriculum that provides the focus, coherence, and rigor required by the CCSSM. envisionmath2.0 offers a balanced
More informationChapter 11 Number Theory
Chapter 11 Number Theory Number theory is one of the oldest branches of mathematics. For many years people who studied number theory delighted in its pure nature because there were few practical applications
More informationNumbers and Operations in Base 10 and Numbers and Operations Fractions
Numbers and Operations in Base 10 and Numbers As the chart below shows, the Numbers & Operations in Base 10 (NBT) domain of the Common Core State Standards for Mathematics (CCSSM) appears in every grade
More informationCommutative Property Grade One
Ohio Standards Connection Patterns, Functions and Algebra Benchmark E Solve open sentences and explain strategies. Indicator 4 Solve open sentences by representing an expression in more than one way using
More informationTeaching Basic Facts: Considerations for Instruction
Teaching Basic Facts: Considerations for Instruction Purpose and Overview of Guide The purpose of this guide is to provide strategies and materials for developing and implementing lessons for students
More informationPlanning For Success Mathematics: Numeration Inquiry Investigations. Operations: Multiplication and Division. Number Sense and Numeration
Planning For Success Mathematics: Numeration Inquiry Investigations Operations: Multiplication and Division Number Sense and Numeration OVERALL EXPECTATIONS By the end of Grade 4, students will: solve
More informationYear 1. Use numbered number lines to add, by counting on in ones. Encourage children to start with the larger number and count on.
Year 1 Add with numbers up to 20 Use numbered number lines to add, by counting on in ones. Encourage children to start with the larger number and count on. +1 +1 +1 Children should: Have access to a wide
More information7 th Grade Integer Arithmetic 7Day Unit Plan by Brian M. Fischer Lackawanna Middle/High School
7 th Grade Integer Arithmetic 7Day Unit Plan by Brian M. Fischer Lackawanna Middle/High School Page 1 of 20 Table of Contents Unit Objectives........ 3 NCTM Standards.... 3 NYS Standards....3 Resources
More informationLesson/Unit Plan Name: Multiplication Facts Made Easy
Grade Level/Course: Grades 36 Lesson/Unit Plan Name: Multiplication Facts Made Easy Rationale/Lesson Abstract: Teachers can eliminate the stress associated with developing fluency with multiplication
More informationPrime and Composite Numbers
Prime and Composite Numbers Student Probe Is 27 a prime number or a composite number? Is 17 a prime number or a composite number? Answer: 27 is a composite number, because it has factors other than 1 and
More informationMath Common Core Standards Fourth Grade
Operations and Algebraic Thinking (OA) Use the four operations with whole numbers to solve problems. OA.4.1 Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 7 as a statement
More informationINTRODUCTION CONTENTS
INTRODUCTION Algebra for All and No Child Left Behind are phrases in the education community that suggest, and in many cases require, action. They give impetus for mathematics teachers at all levels to
More informationCommon Core Standards for Mathematics Grade 4 Operations & Algebraic Thinking Date Taught
Operations & Algebraic Thinking Use the four operations with whole numbers to solve problems. 4.OA.1. Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 7 as a statement that 35
More informationSecond Grade Math Standards and I Can Statements
Second Grade Math Standards and I Can Statements Standard CC.2.OA.1 Use addition and subtraction within 100 to solve one and twostep word problems involving situations of adding to, taking from, putting
More informationFIRST GRADE MATH Summer 2011
Standards Summer 2011 1 OA.1 Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in
More information4 th Grade Mathematics Unpacked Content
4 th Grade Mathematics Unpacked Content This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these
More informationProgressing toward the standard
Report Card Language: add, subtract, multiply, and/or divide to solve multistep word problems. CCSS: 4.OA.3 Solve multistep work problems posed with whole numbers and having wholenumber answers using
More informationWhat Is Singapore Math?
What Is Singapore Math? You may be wondering what Singapore Math is all about, and with good reason. This is a totally new kind of math for you and your child. What you may not know is that Singapore has
More information1 BPS Math Year at a Glance (Adapted from A Story of Units Curriculum Maps in Mathematics P5)
Grade 5 Key Areas of Focus for Grades 35: Multiplication and division of whole numbers and fractionsconcepts, skills and problem solving Expected Fluency: Multidigit multiplication Module M1: Whole
More informationELEMENTARY MATH GRADE 1
PUBLIC SCHOOLS OF EDISON TOWNSHIP DIVISION OF CURRICULUM AND INSTRUCTION ELEMENTARY MATH GRADE 1 Length of Course: Term Elective / Required: Required Schools: Elementary Student Eligibility: Grade 1 Credit
More informationPrime Factorization 0.1. Overcoming Math Anxiety
0.1 Prime Factorization 0.1 OBJECTIVES 1. Find the factors of a natural number 2. Determine whether a number is prime, composite, or neither 3. Find the prime factorization for a number 4. Find the GCF
More informationDecimals and Percentages
Decimals and Percentages Specimen Worksheets for Selected Aspects Paul Harling b recognise the number relationship between coordinates in the first quadrant of related points Key Stage 2 (AT2) on a line
More informationSue. Rubric for the Critical Area of Mathematics Grades K  2
Rubric for the Critical Area of Mathematics Grades K  2 The intent of this document is to provide support as teachers transition to Common Core s. It draws attention to the most critical skills for their
More informationPythagorean Theorem Differentiated Instruction for Use in an Inclusion Classroom
Pythagorean Theorem Differentiated Instruction for Use in an Inclusion Classroom Grade Level: Seven Time Span: Four Days Tools: Calculators, The Proofs of Pythagoras, GSP, Internet Colleen Parker Objectives
More informationVocabulary, Signs, & Symbols product dividend divisor quotient fact family inverse. Assessment. Envision Math Topic 1
1st 9 Weeks Pacing Guide Fourth Grade Math Common Core State Standards Objective/Skill (DOK) I Can Statements (Knowledge & Skills) Curriculum Materials & Resources/Comments 4.OA.1 4.1i Interpret a multiplication
More informationComputation Strategies for Basic Number Facts +, , x,
Computation Strategies for Basic Number Facts +, , x, Addition Subtraction Multiplication Division Proficiency with basic facts aids estimation and computation of multidigit numbers. The enclosed strategies
More informationALGEBRA. sequence, term, nth term, consecutive, rule, relationship, generate, predict, continue increase, decrease finite, infinite
ALGEBRA Pupils should be taught to: Generate and describe sequences As outcomes, Year 7 pupils should, for example: Use, read and write, spelling correctly: sequence, term, nth term, consecutive, rule,
More informationAdding and Subtracting Integers Unit. Grade 7 Math. 5 Days. Tools: Algebra Tiles. FourPan Algebra Balance. Playing Cards
Adding and Subtracting Integers Unit Grade 7 Math 5 Days Tools: Algebra Tiles FourPan Algebra Balance Playing Cards By Dawn Meginley 1 Objectives and Standards Objectives: Students will be able to add
More informationCOMMON CORE STATE STANDARDS FOR MATHEMATICS 35 DOMAIN PROGRESSIONS
COMMON CORE STATE STANDARDS FOR MATHEMATICS 35 DOMAIN PROGRESSIONS Compiled by Dewey Gottlieb, Hawaii Department of Education June 2010 Operations and Algebraic Thinking Represent and solve problems involving
More informationUnit 3 Multiplication and Division
WEEK DATES TEK DESCRIPTION 9 Oct 1923 3.4F, 3.4E, 3.4D Unit 3 Multiplication and Division Unit 3 Multiplication and Division 4 district biweekly scanned tests 10 Oct 2630 3.4G 2x1 digit multiplication
More information1 Mathematics Curriculum
New York State Common Core 1 Mathematics Curriculum G R A D E GRADE 1 MODULE 1 Topic J Development of Subtraction Fluency Within 10 1.OA.6 Focus Standard: 1.OA.6 Add and subtract within 20, demonstrating
More informationNCTM Curriculum Focal Points for Grade 5. Everyday Mathematics, Grade 5
NCTM Curriculum Focal Points and, Grade 5 NCTM Curriculum Focal Points for Grade 5 Number and Operations and Algebra: Developing an understanding of and fluency with division of whole numbers Students
More information1 Mathematics Curriculum
New York State Common Core 1 Mathematics Curriculum G R A D E GRADE 1 MODULE 1 Topic F Development of Addition Fluency Within 10 1.OA.3, 1.OA.6 Focus Standard: 1.OA.3 Apply properties of operations as
More informationGrade 1. M3: Ordering and Expressing Length Measurements as Numbers
Grade 1 Key Areas of Focus for Grades K2: Addition and subtractionconcepts, skills and problem solving Expected Fluency: Add and Subtract within 10 Module M1: Addition and Subtraction of Numbers to 10
More informationEXTENSION: Students may draw two times before they make a rectangle. Are the two # s they combined odd or even? Is their sum odd or even?
TITLE: ODD or EVEN SOURCE: NCTM Addenda Series/Grades K6 OBJECIVE; Students will be able to demonstrate an understanding of the relationships between odds and evens. Appropriate for grades 12, extensions
More informationFactor Game (Number Theory)
The Middle School Math Project Factor Game (Number Theory) Objective Students will use a game setting to identify the properties of prime, composite, abundant, deficient and perfect numbers. Overview of
More informationMath Games For Skills and Concepts
Math Games p.1 Math Games For Skills and Concepts Original material 20012006, John Golden, GVSU permission granted for educational use Other material copyright: Investigations in Number, Data and Space,
More informationRectangles with the Same Numerical Area and Perimeter
About Illustrations: Illustrations of the Standards for Mathematical Practice (SMP) consist of several pieces, including a mathematics task, student dialogue, mathematical overview, teacher reflection
More informationMultiplication Arrays
Multiplication Arrays Objective To provide opportunities to use arrays, multiplication/ division diagrams, and number models to represent multiplication number stories. www.everydaymathonline.com epresentations
More informationIndicator 2: Use a variety of algebraic concepts and methods to solve equations and inequalities.
3 rd Grade Math Learning Targets Algebra: Indicator 1: Use procedures to transform algebraic expressions. 3.A.1.1. Students are able to explain the relationship between repeated addition and multiplication.
More informationAn Introduction to Number Theory Prime Numbers and Their Applications.
East Tennessee State University Digital Commons @ East Tennessee State University Electronic Theses and Dissertations 82006 An Introduction to Number Theory Prime Numbers and Their Applications. Crystal
More informationFourth Grade Math Standards and "I Can Statements"
Fourth Grade Math Standards and "I Can Statements" Standard  CC.4.OA.1 Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 x 7 as a statement that 35 is 5 times as many as 7 and
More informationGrade 4 Level. Math Common Core Sampler Test
Grade 4 Level Math Common Core Sampler Test This test sample is made to be used give students and teachers a basic overview of key Grade 4 Common Core grade level work. All questions are aligned to the
More informationVerbal Phrases to Algebraic Expressions
Student Name: Date: Contact Person Name: Phone Number: Lesson 13 Verbal Phrases to s Objectives Translate verbal phrases into algebraic expressions Solve word problems by translating sentences into equations
More information1 ST GRADE COMMON CORE STANDARDS FOR SAXON MATH
1 ST GRADE COMMON CORE STANDARDS FOR SAXON MATH Calendar The following tables show the CCSS focus of The Meeting activities, which appear at the beginning of each numbered lesson and are taught daily,
More information4TH GRADE FIRST QUARTER MATHEMATICS STANDARDS. Vocabulary. answers using mental computation and estimation strategies including rounding.
4TH GRADE FIRST QUARTER MATHEMATICS STANDARDS Critical Area: Developing understanding and fluency with multidigit multiplication, and developing understanding of dividing to find quotients involving multidigit
More informationPROGRESSION THROUGH CALCULATIONS FOR MULTIPLICATION
PROGRESSION THROUGH CALCULATIONS FOR MULTIPLICATION THE FOLLOWING ARE STANDARDS THAT WE EXPECT THE MAJORITY OF CHILDREN TO ACHIEVE BY THE END OF THE YEAR. YR Related objectives: Count repeated groups of
More informationEXPONENTS. To the applicant: KEY WORDS AND CONVERTING WORDS TO EQUATIONS
To the applicant: The following information will help you review math that is included in the Paraprofessional written examination for the Conejo Valley Unified School District. The Education Code requires
More informationMultiplication Facts Practice
Multiplication Facts Practice Objectives To introduce the 5facts test; and to provide practice with multiplication facts. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM Facts
More informationMD526 Stacking Blocks Pages 115 116
MD526 Stacking Blocks Pages 115 116 STANDARDS 5.MD.C.4 Goals Students will find the number of cubes in a rectangular stack and develop the formula length width height for the number of cubes in a stack.
More informationA Story of Units: A Curriculum Overview for Grades P5
New York State Common Core P5 Mathematics Curriculum GRADE A Story of Units: A Curriculum Overview for Grades P5 Table of Contents: Introduction... 2 Curriculum Map... 3 PreKindergarten... 4 Kindergarten...
More informationJust want the standards alone? You can find the standards alone at http://corestandards.org/thestandards
4 th Grade Mathematics Unpacked Content For the new Common Core State Standards that will be effective in all North Carolina schools in the 201213 school year. This document is designed to help North
More informationMultiplication Facts Survey
Multiplication Facts Survey Objective To guide children as they determine which multiplication facts they still need to learn. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM
More informationAssessment Management
Area Objectives To provide experiences with the concept of area, distinguishing between area and perimeter, and finding areas of rectangular figures by partitioning and counting squares. www.everydaymathonline.com
More informationVISUAL ALGEBRA FOR COLLEGE STUDENTS. Laurie J. Burton Western Oregon University
VISUAL ALGEBRA FOR COLLEGE STUDENTS Laurie J. Burton Western Oregon University VISUAL ALGEBRA FOR COLLEGE STUDENTS TABLE OF CONTENTS Welcome and Introduction 1 Chapter 1: INTEGERS AND INTEGER OPERATIONS
More informationMaths Targets Year 1 Addition and Subtraction Measures. N / A in year 1.
Number and place value Maths Targets Year 1 Addition and Subtraction Count to and across 100, forwards and backwards beginning with 0 or 1 or from any given number. Count, read and write numbers to 100
More informationMathematics. Mathematical Practices
Mathematical Practices 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with
More informationFirst Grade Exploring TwoDigit Numbers
First Grade Exploring TwoDigit Numbers http://focusonmath.files.wordpress.com/2011/02/screenshot20110217at31019pm.png North Carolina Department of Public Instruction www.ncdpi.wikispaces.net
More informationMEMORY WORK  MATH FACTS 1
MEMORY WORK  MATH FACTS ADDITION BOARD (aka Strip Board) Addition with Golden Bead materials Addition with Colored Beads To memorize Addition Tables Linear structure of addition Addition Board MATERIALS:
More informationPool Border Task. The Mathematics of the Task
Pool Border Task How many unit square tiles does it take to build a border around a square pool? The length of the sides of the pool is a natural number of units (1, 2, 3,...) and the border is just one
More informationGrade 1 and 2: Basic Addition and Subtraction Facts Series 2: First and Second Grade
Grade 1 and 2: Basic Addition and Subtraction Facts Series 2: First and Second Grade Texas Essential Knowledge and Skills (TEKS): (1.3) Number, operation and quantitative reasoning. The student recognizes
More informationMental Math Addition and Subtraction
Mental Math Addition and Subtraction If any of your students don t know their addition and subtraction facts, teach them to add and subtract using their fingers by the methods taught below. You should
More informationCCSSM Critical Areas: Kindergarten
CCSSM Critical Areas: Kindergarten Critical Area 1: Represent and compare whole numbers Students use numbers, including written numerals, to represent quantities and to solve quantitative problems, such
More informationSue Fine Linn Maskell
FUN + GAMES = MATHS Sue Fine Linn Maskell Teachers are often concerned that there isn t enough time to play games in maths classes. But actually there is time to play games and we need to make sure that
More informationCharlesworth School Year Group Maths Targets
Charlesworth School Year Group Maths Targets Year One Maths Target Sheet Key Statement KS1 Maths Targets (Expected) These skills must be secure to move beyond expected. I can compare, describe and solve
More informationAlgebra Tiles Activity 1: Adding Integers
Algebra Tiles Activity 1: Adding Integers NY Standards: 7/8.PS.6,7; 7/8.CN.1; 7/8.R.1; 7.N.13 We are going to use positive (yellow) and negative (red) tiles to discover the rules for adding and subtracting
More informationHooray for the Hundreds Chart!!
Hooray for the Hundreds Chart!! The hundreds chart consists of a grid of numbers from 1 to 100, with each row containing a group of 10 numbers. As a result, children using this chart can count across rows
More informationExploring Mathematics Through ProblemSolving and Student Voice
Exploring Mathematics Through ProblemSolving and Student Voice Created By: Lisa Bolduc, Aileen BurkeTsakmakas, Shelby Monaco, Antonietta Scalzo Contributions By: Churchill Public School Professional
More information