Top Resource Types
 Multiplication and Division Worksheets: This site offers printable worksheets and interactive practice for basic multiplication and division facts.
 All About Multiplication: Exploring Equal Sets: Students explore the set model for multiplication. They find products using equal sets and present their results in a table. Next students apply their knowledge about multiplication in the creation of pictographs (from NCTM's Illuminations website).
 All About Multiplication: Bibliography: List of five children's books with a multiplication focus (found on NCTM Illuminations site under "All About Multiplication").
 Number Line Bars: A versatile tool that can be used to illustrate the operations of addition, subtraction, multiplication, and division.
 The Product Game: Playing the Product Game: Students learn to play the Product Game to better understand factors and products, and the relationships between them.
 Factor Trail Game: Students determine factors of designated numbers and play a board game for practice. Students use a game board with numbered squares and write down what the factors of a given number are. If they are correct, the sum of the factors is how many points they get. If they are incorrect, their opponent gets 10 points for each mistake they find. The score sheet itself is an assessment tool that the teacher can use to evaluate the students' understanding of factors.

Factor Game:
In this lesson students will start off in pairs, playing a factorfinding game for two players (it also has a player vs. computer mode). Students must find winning strategies, including finding numbers with the fewest factors and knowing which factors were already discounted from the previous rounds. Afterwards, the students will analyze game strategies, and be introduced to prime and composite numbers.
(from NCTM Illuminations)  Area Model of Multiplication Using Base 10 Manipulatives (Double digit multiplication): "Students will use base 10 manipulatives to build a rectangular array to represent double digit multiplication. Students will make the connection between the standard algorithm and the rectangular array" from Alabama Learning Exchange).
 Arithmetic: The students will be given mutiplication and division problems which they must answer. They also have the option of being given a number then stating the factors of how that number was attained using either multiplication or division.
 Who Wants to be a Mathionaire?: This online game replicates the popular "Who Wants to be a Millionaire?" format, using multiplication facts. It's a fun way for kids to practice multiplication!
 Multiply and Conquer: In this lesson, students will segue from decomposing twodigit by onedigit into twodigit by twodigit problems. First, the problem 14 x 6 is written out using an array model (14 boxes by 6 boxes). Next, the 14 box side is decomposed into a 10x6 section and a 4x6 section, and then multiplication is carried out using the distributive property. After several of these types of problems, a twodigit by twodigit array is modeled for students. This time, both sides are decomposed into four grids. This lesson will provide conceptual knowledge for what happens in a 2 twodigit multiplication problem.
 Multiplication  It's in the Cards: Looking for Patterns: This lesson has students using a 099 grid paper to skip count in multiples of two, three, five and ten. Then, students are given 3 crayons to color in the twos in red, the threes in yellow, and fives in blue. This will result in multiples of sixes being orange, tens being purple, and fifteens being green.

All About Multiplication:
This fourlesson unit from NCTM's Illuminations has students explore several meaning and representations of multiplication (number line, equal sets, arrays, and balanced equations). Other multiplication topics covered include: the commutative (order) property, the results of multiplying by 1 and 0, and the inverse property. Students will write and solve multiplication story problems, and convert word problems into equations. A bibliography of children's literature with a focus on multiplication is provided, which can be integrated so that students can connect through literature.
Individual Lessons
 Lesson 1: Hopping on the Number Line
 Students are shown how to generate products on a number line, highlighting the measurement aspect of addition. After some practice, they are asked to predict and then verify sums, and are also introduced to the commutative property.
 Lesson 2: Exploring Equal Sets
 Students explore the set model for multiplication. They find products using equal sets and present their results in a table. Next students apply their knowledge about multiplication in the creation of pictographs.
 Lesson 3: Modeling Multiplication With Streets and Avenues
 The array model of multiplication lays the foundation for the later study of area, and in this lesson, students will create arrays of the factors of 4 and 6 using toothpicks as representations of streets and avenues. The teacher will ask how many stoplights would be needed to fill each intersection. Zero and one will also be used as factors. Finally they will write their results as multiplication problems.
 Lesson 4: Balance Beam Discoveries
 The balance beam model of multiplication is examined in this lesson, along with another important relationship, the inverse of multiplication. This will lead naturally into the representation of multiplication facts as equations. Students will also practice multiplication facts by playing the Product Game.
 Jeopardy Labs: This is a math review using interactive jeopardy games.
 Math Vocabulary Bingo: This lesson activity is an opportunity to assess and reinforce students' understanding of mathematical vocabulary. Students will play Bingo, except instead of calling numbers out, the teacher will call out clues that utilize math vocabulary to indicate numbers, such as: "the number of inches in one yard." This will make students listen for details, perform calculations mentally, associate math vocabulary to specific examples, and to communicate their mathematical thinking. Bingo cards will need to be revised to be aligned with the grade level's Standards.
 Estimator Quiz: In this activity, students are quizzed on their ability to estimate sums, products, and percentages. The student can adjust the difficulty of the problems and how close they have to be to the actual answer. This activity allows students to practice estimating addition, multiplication, and percentages of large numbers. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.
 Estimator Four: In this activity, students play a game of connect four, but to place a piece on the board they have to correctly estimate an addition, multiplication, or percentage problem. Students can adjust the difficulty of the problems as well as how close the estimate has to be to the actual result. This activity allows students to practice estimating addition, multiplication, and percentages of large numbers (100s). This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.
 Estimate: This interactive applet helps develop number sense. The user mentally estimates a number that is represented by an arrow on a number line and then checks the estimate by clicking to have the exact number revealed. Users can choose a number range for whole numbers (between 0 and 10, 100, 1000 or 10,000) or decimals (tenths or hundredths between 0 and 1). An optional scale of tick marks provides guidance. The activity does not provide a scoring component.

Decimal Place Value:
Students learn about decimal place value and the relationship between tenths, hundredths, and thousandths.
Students will explore decimal place value, read and write decimals using tenths, hundredths, and thousandths, and compare decimals using greaterthan and lessthan notation.
 Fraction Models: An interactive tool to represent a fraction circle, rectangle, or set model with numerators and denominators ranging from 1 to 100. The decimal and percent equivalents of the created fraction are also displayed.
 Fraction Strips in Black and White: This worksheet models a whole as well as fractions with denominators 2 through 12. Teachers could use these, as appropriate to grade level's benchmark, to have students color various fractions, model equivalent fractions, or begin the instruction on operations with fractions.
 Clipart ETC Fractions: Illustrations that can be used for teaching and demonstrating fractions. Fractional representations are modeled in wedges of circles ("pieces of pie") and parts of polygons. There are also clipart images of numerical fractions, both proper and improper, from halves to twelfths. Fraction charts and fraction strips found in this collection can be used as manipulatives and are ready to print for classroom use.
 Fraction/Decimal War: This lesson plan introduces a game that will reinforce fractions and decimals.
 Shopping Mall Math: Parking at the Mall (Area): Students estimate and determine the area for parking spaces at a mall. They are also introduced to determining percents.

Ordering numeric expressions :
Ordering decimals, fractions and percents.
 Awesome Area: "Students will explore the concept of area in a realistic application. They will develop their understanding of area through constructing, drawing, and labeling spaces." (from National Security Agency Central Security Service)

Sidewalk Capers:
"Students will explore spatial relationships by determining the area of various shapes and creating tessellation patterns. They use a variety of manipulatives in this lesson to explore spatial relationships of squares and their transformations.
Students create a square with 36 square tiles and determine that by sliding several of the pieces both vertically and horizontally, they form an irregular shape that can be tessellated. Students are given a variety of curved shapes and encouraged to use tape and scissors to reconfigure their shape back into a square. Given a variety of measurement tools, students are asked to estimate the number of squares tiles (24" x 24") needed to cover an assigned section of the school's sidewalk. Finally, they create a pattern from the square and tessellate it on the sidewalk using chalk." from PBS Mathline. 
Junior Architects: Finding Perimeter and Area:
In this worksheet, students are directed to find the perimeter and area for a clubhouse in the form of rectangles, composite figures, and other polygons. The second sheet urged them to make their own designs for a clubhouse and find the perimeter and area. This resource is recommended as an introduction or review of perimeter and area.
(Found under "Finding Perimeter and Area" on NCTM's Illuminations)  The Area of Tangram Pieces: The student uses tangrams to compute the area of polygons without formulas.
 Junior Architects: Finding Perimeter and Area: This lesson's focus is to have students consider perimeter and area while designing a clubhouse. The teacher will mark out a 10x10 "room" with masking tape to help students consider the actual size of such a small room. Students will use a handout and geoboards to find area and perimeter and problemsolve their designs.
 Makeshift Measurements: Students will be presented with an array of objects that all come in standard sizes, such as envelopes, dollar bills, credit cards, etc. They will be instructed to bring in other standardsized objects in order to measure them and compare with other students' objects. They will then use these common objects as references to determine the size of other objects.
 Perfect Patterns: This unit begins with nonnumeric patterns. The students use linkage cubes, pattern blocks and hundred square paper on day 1 to create visual (nonnumeric) patterns. Days 2 and 3 of the unit include increasing and decreasing numeric skip patterns. Students begin by identifying the pattern, extending it, then creating their own patterns in multiple ways using the same rule.
 Counting Embedded Figures: Students will focus on counting shape patterns within a given shape, and then will be engaged to form generalizations about the problem, which will sharpen their algebraic skills.
 Fractal Tool: Students investigate shapes that grow and change using an iterative process. Fractals are characterized by selfsimilarity, smaller sections that resemble the larger figure. From NCTM's Illuminations.
 Guess My Rule: Function Machine card game is used to reinforce student understanding of functions using an in/out table.
 Chips ahoy!: This lesson will help children recognize, continue, and create number patterns, as well as find the rules for the patterns. The activities progress from concrete to semiconcrete to abstract.
 True, False, and Open Sentences: "Students first explore arithmetic sentences to decide whether they are true or false. The lesson then introduces students to sentences that are neither true nor false but are algebraic equations, also called open sentences, such as x + 3 = 7 or 2 x = 12." from Math Solutions.
 Chairs Around the Table: This lesson allows exploration for students to discover effects of changes in seating when tables are moved together. Since pushing tables together makes the sides of the tables inaccessible, the teacher can use this to get students to come up with rules for linear patterns. These rules can be written on the board using variables to lay the groundwork for understanding the use of variables.
 Clipart: Geometric Shapes: In this lesson, you will find clip art and various illustrations of polygons, circles, ellipses, star polygons, and inscribed shapes.
 Paper Quilts: Exploring Flips and Slides: This lesson focuses on the geometric transformations of reflection and translation, or informally, flips and slides respectively. Students make a design, and then use flips and slides to make a "miniquilt" of that design.
 Paper Quilts: Analyzing Designs : The geometric transformations of rotation, reflection, and translation are explored in this lesson. Students create a design and copy it three times, then place the squares into the quadrants of their "miniquilt" and use turns, flips and slides to create a design that can be analyzed in terms of symmetry.

Transformations  Translation:
The user may drag or rotate a shape created with pattern block pieces, using or not using a coordinate axes and lattice points background.
(source: NLVM grade 68 "Transformations  Translation")  Using Cubes and Isometric Drawings: Building Using the FrontRightTop View: Students explore drawing the frontrighttop view when given a threedimensional figure built from cubes, and explore building a threedimensional figure when given the frontrighttop view.
 Count on Math: Making Your First Million: This lesson explores the concept of very large numbers by having students contextualize the magnitude of 1,000,000. For example, students are asked questions such as: "Do you think you've been alive for a million days, hours, minutes, or seconds?" and "How long would it take a person to count to a million?" Students will be also be asked to consider getting to a million by repeated addition and multiplication.
 Place Value Made Simple: Students will read, write, and represent whole numbers from 0  1,000,000 using symbols, models, expanded form, and words. Base ten blocks, place value pocket charts, folded paper strips for expanded form accordions, and checks are used.
 The MegaPenny Project: "Visualizing huge numbers can be very difficult. People regularly talk about millions of miles, billions of bytes, or trillions of dollars, yet it's still hard to grasp just how much a "billion" really is. The MegaPenny Project aims to help by taking one small everyday item, the U.S. penny, and building on that to answer the question: "What would a million, billion (or a trillion) pennies look like?" " from The Mega Penny Project.

Football Finances:
This activity has students analyze a photo of a football stadium in order to make estimates such as:
 how many people will attend the Super Bowl
 how many hot dogs will be eaten
 how many gallons of soft drinks will be drunk
 Estimation: The students will be presented with two shapes and must estimate how many times the smaller will fit in the larger. They will be surprised at some of the results but will quickly learn and make adjustments.
 Expanded Notation: With this stepbystep, interactive video from StudyJams students can earn more about expanded notation. Expanded notation, also called expanded form, is a handy way to write numbers that shows the place value of each digit. It looks like an addition problem and will help you break numbers down and understand their place value. A short, selfchecking quiz is also included with this link.
 Hand Spans, A Lesson for Third and Fourth Graders (Division and Measurement): A measurement activity to give students experience with the grouping model of division and practice with rulers and tape measures.
 Fun with Fractions: Making and Investigating Fraction Strips: This lesson has students discover relationships between different fractions. From equally cut pieces of paper (a whole), fraction strips are made into halves, thirds, fourths, sixths, and eighths. Students then compare them to one another, discovering their relationships, including equivalent fractions.
 Fun with Fractions: Investigating Equivalent Fractions with Relationship Rods: This lesson is to follow Fun with Fractions  Investigating Fraction Relationships with Relationship Rods (see Related Resources.) Students will be asked to make the various differentcolored rods the "whole" in relation to the other rods, and then give the relative fraction name for the other nine rods, based on how many of each color it would take to "fill in" the whole. Students will develop problemsolving skills and reasoning as they explain the equivalencies of length.
 Equivalent Fractions: "This lesson helps students discover how to obtain equal fractions by using both fraction strips and playing a fraction matching game. Students will learn that to obtain equal fractions they may multiply the numerator and denominator by the same number." (from ALEX  Alabama Learning Exchange)
 Learning and Teaching Ratio and Proportion: Research Implications: This is a chapter describing the theory and lesson planning for teaching ratio and rate problems. The authors describe how students use reasoning about multiplication and division to solve ratio and rate problems.

Factor Trees :
This virtual manipulative allows students to find the prime factorization of numbers by completing factor trees. Then they may use a venn diagram to sort the prime factors to find the greatest common factor and the least common multiple.
 Factortris (FactorTris): This game is an adaptation of Tetris, in which the student is tasked with creating an arrays product and using the arrays to play Tetris. The game increases in difficulty as the student progresses, but unfortunately, the teacher can not adjust the difficulty of the game at the onset.
 Factor Tree: This virtual manipulative will help the students in exploring the prime factorization of numbers and see how to use the factorization of a pair of numbers to find the greatest common factor (GCF) and the least common factor (LCM). In the manipulative the number pairs are presented randomly, so that a student returning to the factor tree will most likely begin with a pair of numbers not seen before.
 Fun with Estimation: This activity is a timed game that allows teams of students to practice their estimation skills while shopping for groceries (empty containers acquired for classroom use).

Fractional Clothesline:
This lesson has the teacher stretch a lengthy string across the room, and then ask 5 students to equally distribute cards with the numbers 0, 1, 2, 3, and 4 written on them. Once this number line is done, students will use it to place cards with mixed numbers, proper fractions and improper fractions, which will help them realize the difference between these kinds of rational numbers. They will also discuss their strategies for placing the cards. Finally, students will play an estimation game practicing combining numbers from two cards. Exposure to looking at fractions in different ways will help their developing conceptual fraction sense.
 Reasonable Estimates: Students are taught to estimate, to identify appropriate situations using estimation, to calculate estimated sums and differences, and to evaluate answers for reasonableness.

"Bar Model Math"  "Twice" as Nice:
In this lesson students will solve real world problems that have multiplicative comparisons in them. They will use the strategy of bar models to solve the problems.

Are You Ready for a Hurricane?:
This activity allows students to determine the types of items that should be in a hurricane survival kit, use a budget and calculations to determine the items to include in the kit and gain an understanding of hurricanes and the need to prepare for them.
 More Bang for your Buck!: In this openended MEA, students will work in teams to determine a procedure for ranking recycling companies. Students will need to calculate their return on recycling, make decisions based on a table of data, and write a letter to the client providing evidence for their decisions.
 Secret NumberPlace Value Game: Students will be able to use decimals through the thousandths place to name numbers between whole numbers through the use of number generators and critically thinking about where to place numbers on a place value chart.
 Cookies and Treats: Fourth graders will help Cookies and Treats find ecofriendly packaging for its cookies. Students will work with decimals and data in order to develop a procedure for ranking and choosing packaging for cookies.
 Dynamic Decimals, Fractions and Money!: In this lesson, students will realize the connection between fractions, decimals and money through the use of a 100 grid.
 Happy Healthy School Lunch: In this MEA students are asked by the school cafeteria manager to assist her in creating healthier school lunch menus. The students need to keep in mind both nutritional and cost guidelines. Students will develop a procedure to select school lunches.
 Area Isn't Just for Squares: This lesson helps students make the connections between area and multiplication using square tiles.
 Celebrity Floor Plan Frenzy: Students will help an architect find the area of each room in a celebrity home and then determine the best location to build the home based on qualitative data about the locations.

Banana County Public SchoolPainters MEA:
This Model Eliciting Activity (MEA) is written at a 4th grade level.
This activity allows students to think critically using information provided. Students will write a procedure on how they determined which painting company would be suitable for the client.  Triangles on a Lattice: In this activity, students will use a 3x3 square lattice to study transformations of triangles whose vertices are part of the lattice. The tasks include determining whether two triangles are congruent, which transformations connect two congruent triangles, and the number of noncongruent triangles (with vertices on the lattice) that are possible.
 Runway Rotations: Students will use small paper airplanes to model rotations required to turn onto a runway. Students will rotate planes 45, 90, 180, 270, and 360 degrees. Students will identify and describe the results of rotations using benchmark angles.
 All Things Being Equal: In this game, learners strategize to win the most cards by building number equations. Learners practice operations (addition, subtraction, multiplication, and division) to construct their equations. This activity guide contains sample questions to ask, literary connections, extensions, and alignment to local and national standards.
Related Resources

Are You Ready for a Hurricane?:
This activity allows students to determine the types of items that should be in a hurricane survival kit, use a budget and calculations to determine the items to include in the kit and gain an understanding of hurricanes and the need to prepare for them.
 More Bang for your Buck!: In this openended MEA, students will work in teams to determine a procedure for ranking recycling companies. Students will need to calculate their return on recycling, make decisions based on a table of data, and write a letter to the client providing evidence for their decisions.
 Cookies and Treats: Fourth graders will help Cookies and Treats find ecofriendly packaging for its cookies. Students will work with decimals and data in order to develop a procedure for ranking and choosing packaging for cookies.
 Happy Healthy School Lunch: In this MEA students are asked by the school cafeteria manager to assist her in creating healthier school lunch menus. The students need to keep in mind both nutritional and cost guidelines. Students will develop a procedure to select school lunches.
 Celebrity Floor Plan Frenzy: Students will help an architect find the area of each room in a celebrity home and then determine the best location to build the home based on qualitative data about the locations.

Banana County Public SchoolPainters MEA:
This Model Eliciting Activity (MEA) is written at a 4th grade level.
This activity allows students to think critically using information provided. Students will write a procedure on how they determined which painting company would be suitable for the client.  All About Multiplication: Exploring Equal Sets: Students explore the set model for multiplication. They find products using equal sets and present their results in a table. Next students apply their knowledge about multiplication in the creation of pictographs (from NCTM's Illuminations website).
 Factor Trail Game: Students determine factors of designated numbers and play a board game for practice. Students use a game board with numbered squares and write down what the factors of a given number are. If they are correct, the sum of the factors is how many points they get. If they are incorrect, their opponent gets 10 points for each mistake they find. The score sheet itself is an assessment tool that the teacher can use to evaluate the students' understanding of factors.

Factor Game:
In this lesson students will start off in pairs, playing a factorfinding game for two players (it also has a player vs. computer mode). Students must find winning strategies, including finding numbers with the fewest factors and knowing which factors were already discounted from the previous rounds. Afterwards, the students will analyze game strategies, and be introduced to prime and composite numbers.
(from NCTM Illuminations)  Area Model of Multiplication Using Base 10 Manipulatives (Double digit multiplication): "Students will use base 10 manipulatives to build a rectangular array to represent double digit multiplication. Students will make the connection between the standard algorithm and the rectangular array" from Alabama Learning Exchange).
 Multiply and Conquer: In this lesson, students will segue from decomposing twodigit by onedigit into twodigit by twodigit problems. First, the problem 14 x 6 is written out using an array model (14 boxes by 6 boxes). Next, the 14 box side is decomposed into a 10x6 section and a 4x6 section, and then multiplication is carried out using the distributive property. After several of these types of problems, a twodigit by twodigit array is modeled for students. This time, both sides are decomposed into four grids. This lesson will provide conceptual knowledge for what happens in a 2 twodigit multiplication problem.
 Multiplication  It's in the Cards: Looking for Patterns: This lesson has students using a 099 grid paper to skip count in multiples of two, three, five and ten. Then, students are given 3 crayons to color in the twos in red, the threes in yellow, and fives in blue. This will result in multiples of sixes being orange, tens being purple, and fifteens being green.

"Bar Model Math"  "Twice" as Nice:
In this lesson students will solve real world problems that have multiplicative comparisons in them. They will use the strategy of bar models to solve the problems.

Are You Ready for a Hurricane?:
This activity allows students to determine the types of items that should be in a hurricane survival kit, use a budget and calculations to determine the items to include in the kit and gain an understanding of hurricanes and the need to prepare for them.
 More Bang for your Buck!: In this openended MEA, students will work in teams to determine a procedure for ranking recycling companies. Students will need to calculate their return on recycling, make decisions based on a table of data, and write a letter to the client providing evidence for their decisions.
 Secret NumberPlace Value Game: Students will be able to use decimals through the thousandths place to name numbers between whole numbers through the use of number generators and critically thinking about where to place numbers on a place value chart.

Decimal Place Value:
Students learn about decimal place value and the relationship between tenths, hundredths, and thousandths.
Students will explore decimal place value, read and write decimals using tenths, hundredths, and thousandths, and compare decimals using greaterthan and lessthan notation.
 Cookies and Treats: Fourth graders will help Cookies and Treats find ecofriendly packaging for its cookies. Students will work with decimals and data in order to develop a procedure for ranking and choosing packaging for cookies.
 Dynamic Decimals, Fractions and Money!: In this lesson, students will realize the connection between fractions, decimals and money through the use of a 100 grid.
 Shopping Mall Math: Parking at the Mall (Area): Students estimate and determine the area for parking spaces at a mall. They are also introduced to determining percents.
 Happy Healthy School Lunch: In this MEA students are asked by the school cafeteria manager to assist her in creating healthier school lunch menus. The students need to keep in mind both nutritional and cost guidelines. Students will develop a procedure to select school lunches.

Sidewalk Capers:
"Students will explore spatial relationships by determining the area of various shapes and creating tessellation patterns. They use a variety of manipulatives in this lesson to explore spatial relationships of squares and their transformations.
Students create a square with 36 square tiles and determine that by sliding several of the pieces both vertically and horizontally, they form an irregular shape that can be tessellated. Students are given a variety of curved shapes and encouraged to use tape and scissors to reconfigure their shape back into a square. Given a variety of measurement tools, students are asked to estimate the number of squares tiles (24" x 24") needed to cover an assigned section of the school's sidewalk. Finally, they create a pattern from the square and tessellate it on the sidewalk using chalk." from PBS Mathline.  Junior Architects: Finding Perimeter and Area: This lesson's focus is to have students consider perimeter and area while designing a clubhouse. The teacher will mark out a 10x10 "room" with masking tape to help students consider the actual size of such a small room. Students will use a handout and geoboards to find area and perimeter and problemsolve their designs.
 Area Isn't Just for Squares: This lesson helps students make the connections between area and multiplication using square tiles.
 Celebrity Floor Plan Frenzy: Students will help an architect find the area of each room in a celebrity home and then determine the best location to build the home based on qualitative data about the locations.
 Makeshift Measurements: Students will be presented with an array of objects that all come in standard sizes, such as envelopes, dollar bills, credit cards, etc. They will be instructed to bring in other standardsized objects in order to measure them and compare with other students' objects. They will then use these common objects as references to determine the size of other objects.

Banana County Public SchoolPainters MEA:
This Model Eliciting Activity (MEA) is written at a 4th grade level.
This activity allows students to think critically using information provided. Students will write a procedure on how they determined which painting company would be suitable for the client.  Counting Embedded Figures: Students will focus on counting shape patterns within a given shape, and then will be engaged to form generalizations about the problem, which will sharpen their algebraic skills.
 Chips ahoy!: This lesson will help children recognize, continue, and create number patterns, as well as find the rules for the patterns. The activities progress from concrete to semiconcrete to abstract.
 Chairs Around the Table: This lesson allows exploration for students to discover effects of changes in seating when tables are moved together. Since pushing tables together makes the sides of the tables inaccessible, the teacher can use this to get students to come up with rules for linear patterns. These rules can be written on the board using variables to lay the groundwork for understanding the use of variables.
 Paper Quilts: Exploring Flips and Slides: This lesson focuses on the geometric transformations of reflection and translation, or informally, flips and slides respectively. Students make a design, and then use flips and slides to make a "miniquilt" of that design.
 Paper Quilts: Analyzing Designs : The geometric transformations of rotation, reflection, and translation are explored in this lesson. Students create a design and copy it three times, then place the squares into the quadrants of their "miniquilt" and use turns, flips and slides to create a design that can be analyzed in terms of symmetry.
 Triangles on a Lattice: In this activity, students will use a 3x3 square lattice to study transformations of triangles whose vertices are part of the lattice. The tasks include determining whether two triangles are congruent, which transformations connect two congruent triangles, and the number of noncongruent triangles (with vertices on the lattice) that are possible.
 Runway Rotations: Students will use small paper airplanes to model rotations required to turn onto a runway. Students will rotate planes 45, 90, 180, 270, and 360 degrees. Students will identify and describe the results of rotations using benchmark angles.
 Using Cubes and Isometric Drawings: Building Using the FrontRightTop View: Students explore drawing the frontrighttop view when given a threedimensional figure built from cubes, and explore building a threedimensional figure when given the frontrighttop view.
 Count on Math: Making Your First Million: This lesson explores the concept of very large numbers by having students contextualize the magnitude of 1,000,000. For example, students are asked questions such as: "Do you think you've been alive for a million days, hours, minutes, or seconds?" and "How long would it take a person to count to a million?" Students will be also be asked to consider getting to a million by repeated addition and multiplication.

Football Finances:
This activity has students analyze a photo of a football stadium in order to make estimates such as:
 how many people will attend the Super Bowl
 how many hot dogs will be eaten
 how many gallons of soft drinks will be drunk
 Fun with Fractions: Making and Investigating Fraction Strips: This lesson has students discover relationships between different fractions. From equally cut pieces of paper (a whole), fraction strips are made into halves, thirds, fourths, sixths, and eighths. Students then compare them to one another, discovering their relationships, including equivalent fractions.
 Fun with Fractions: Investigating Equivalent Fractions with Relationship Rods: This lesson is to follow Fun with Fractions  Investigating Fraction Relationships with Relationship Rods (see Related Resources.) Students will be asked to make the various differentcolored rods the "whole" in relation to the other rods, and then give the relative fraction name for the other nine rods, based on how many of each color it would take to "fill in" the whole. Students will develop problemsolving skills and reasoning as they explain the equivalencies of length.
 Equivalent Fractions: "This lesson helps students discover how to obtain equal fractions by using both fraction strips and playing a fraction matching game. Students will learn that to obtain equal fractions they may multiply the numerator and denominator by the same number." (from ALEX  Alabama Learning Exchange)
 Learning and Teaching Ratio and Proportion: Research Implications: This is a chapter describing the theory and lesson planning for teaching ratio and rate problems. The authors describe how students use reasoning about multiplication and division to solve ratio and rate problems.

Fractional Clothesline:
This lesson has the teacher stretch a lengthy string across the room, and then ask 5 students to equally distribute cards with the numbers 0, 1, 2, 3, and 4 written on them. Once this number line is done, students will use it to place cards with mixed numbers, proper fractions and improper fractions, which will help them realize the difference between these kinds of rational numbers. They will also discuss their strategies for placing the cards. Finally, students will play an estimation game practicing combining numbers from two cards. Exposure to looking at fractions in different ways will help their developing conceptual fraction sense.

All About Multiplication:
This fourlesson unit from NCTM's Illuminations has students explore several meaning and representations of multiplication (number line, equal sets, arrays, and balanced equations). Other multiplication topics covered include: the commutative (order) property, the results of multiplying by 1 and 0, and the inverse property. Students will write and solve multiplication story problems, and convert word problems into equations. A bibliography of children's literature with a focus on multiplication is provided, which can be integrated so that students can connect through literature.
Individual Lessons
 Lesson 1: Hopping on the Number Line
 Students are shown how to generate products on a number line, highlighting the measurement aspect of addition. After some practice, they are asked to predict and then verify sums, and are also introduced to the commutative property.
 Lesson 2: Exploring Equal Sets
 Students explore the set model for multiplication. They find products using equal sets and present their results in a table. Next students apply their knowledge about multiplication in the creation of pictographs.
 Lesson 3: Modeling Multiplication With Streets and Avenues
 The array model of multiplication lays the foundation for the later study of area, and in this lesson, students will create arrays of the factors of 4 and 6 using toothpicks as representations of streets and avenues. The teacher will ask how many stoplights would be needed to fill each intersection. Zero and one will also be used as factors. Finally they will write their results as multiplication problems.
 Lesson 4: Balance Beam Discoveries
 The balance beam model of multiplication is examined in this lesson, along with another important relationship, the inverse of multiplication. This will lead naturally into the representation of multiplication facts as equations. Students will also practice multiplication facts by playing the Product Game.
 Awesome Area: "Students will explore the concept of area in a realistic application. They will develop their understanding of area through constructing, drawing, and labeling spaces." (from National Security Agency Central Security Service)
 Perfect Patterns: This unit begins with nonnumeric patterns. The students use linkage cubes, pattern blocks and hundred square paper on day 1 to create visual (nonnumeric) patterns. Days 2 and 3 of the unit include increasing and decreasing numeric skip patterns. Students begin by identifying the pattern, extending it, then creating their own patterns in multiple ways using the same rule.
 Place Value Made Simple: Students will read, write, and represent whole numbers from 0  1,000,000 using symbols, models, expanded form, and words. Base ten blocks, place value pocket charts, folded paper strips for expanded form accordions, and checks are used.
 Learning and Teaching Ratio and Proportion: Research Implications: This is a chapter describing the theory and lesson planning for teaching ratio and rate problems. The authors describe how students use reasoning about multiplication and division to solve ratio and rate problems.
 Reasonable Estimates: Students are taught to estimate, to identify appropriate situations using estimation, to calculate estimated sums and differences, and to evaluate answers for reasonableness.
 Multiplication and Division Worksheets: This site offers printable worksheets and interactive practice for basic multiplication and division facts.
 All About Multiplication: Exploring Equal Sets: Students explore the set model for multiplication. They find products using equal sets and present their results in a table. Next students apply their knowledge about multiplication in the creation of pictographs (from NCTM's Illuminations website).
 The Product Game: Playing the Product Game: Students learn to play the Product Game to better understand factors and products, and the relationships between them.
 Factor Trail Game: Students determine factors of designated numbers and play a board game for practice. Students use a game board with numbered squares and write down what the factors of a given number are. If they are correct, the sum of the factors is how many points they get. If they are incorrect, their opponent gets 10 points for each mistake they find. The score sheet itself is an assessment tool that the teacher can use to evaluate the students' understanding of factors.
 Multiply and Conquer: In this lesson, students will segue from decomposing twodigit by onedigit into twodigit by twodigit problems. First, the problem 14 x 6 is written out using an array model (14 boxes by 6 boxes). Next, the 14 box side is decomposed into a 10x6 section and a 4x6 section, and then multiplication is carried out using the distributive property. After several of these types of problems, a twodigit by twodigit array is modeled for students. This time, both sides are decomposed into four grids. This lesson will provide conceptual knowledge for what happens in a 2 twodigit multiplication problem.
 Multiplication  It's in the Cards: Looking for Patterns: This lesson has students using a 099 grid paper to skip count in multiples of two, three, five and ten. Then, students are given 3 crayons to color in the twos in red, the threes in yellow, and fives in blue. This will result in multiples of sixes being orange, tens being purple, and fifteens being green.
 Math Vocabulary Bingo: This lesson activity is an opportunity to assess and reinforce students' understanding of mathematical vocabulary. Students will play Bingo, except instead of calling numbers out, the teacher will call out clues that utilize math vocabulary to indicate numbers, such as: "the number of inches in one yard." This will make students listen for details, perform calculations mentally, associate math vocabulary to specific examples, and to communicate their mathematical thinking. Bingo cards will need to be revised to be aligned with the grade level's Standards.
 Secret NumberPlace Value Game: Students will be able to use decimals through the thousandths place to name numbers between whole numbers through the use of number generators and critically thinking about where to place numbers on a place value chart.
 Fraction Strips in Black and White: This worksheet models a whole as well as fractions with denominators 2 through 12. Teachers could use these, as appropriate to grade level's benchmark, to have students color various fractions, model equivalent fractions, or begin the instruction on operations with fractions.
 Dynamic Decimals, Fractions and Money!: In this lesson, students will realize the connection between fractions, decimals and money through the use of a 100 grid.
 Shopping Mall Math: Parking at the Mall (Area): Students estimate and determine the area for parking spaces at a mall. They are also introduced to determining percents.
 Awesome Area: "Students will explore the concept of area in a realistic application. They will develop their understanding of area through constructing, drawing, and labeling spaces." (from National Security Agency Central Security Service)

Sidewalk Capers:
"Students will explore spatial relationships by determining the area of various shapes and creating tessellation patterns. They use a variety of manipulatives in this lesson to explore spatial relationships of squares and their transformations.
Students create a square with 36 square tiles and determine that by sliding several of the pieces both vertically and horizontally, they form an irregular shape that can be tessellated. Students are given a variety of curved shapes and encouraged to use tape and scissors to reconfigure their shape back into a square. Given a variety of measurement tools, students are asked to estimate the number of squares tiles (24" x 24") needed to cover an assigned section of the school's sidewalk. Finally, they create a pattern from the square and tessellate it on the sidewalk using chalk." from PBS Mathline. 
Junior Architects: Finding Perimeter and Area:
In this worksheet, students are directed to find the perimeter and area for a clubhouse in the form of rectangles, composite figures, and other polygons. The second sheet urged them to make their own designs for a clubhouse and find the perimeter and area. This resource is recommended as an introduction or review of perimeter and area.
(Found under "Finding Perimeter and Area" on NCTM's Illuminations)  Junior Architects: Finding Perimeter and Area: This lesson's focus is to have students consider perimeter and area while designing a clubhouse. The teacher will mark out a 10x10 "room" with masking tape to help students consider the actual size of such a small room. Students will use a handout and geoboards to find area and perimeter and problemsolve their designs.
 Area Isn't Just for Squares: This lesson helps students make the connections between area and multiplication using square tiles.
 Makeshift Measurements: Students will be presented with an array of objects that all come in standard sizes, such as envelopes, dollar bills, credit cards, etc. They will be instructed to bring in other standardsized objects in order to measure them and compare with other students' objects. They will then use these common objects as references to determine the size of other objects.
 Perfect Patterns: This unit begins with nonnumeric patterns. The students use linkage cubes, pattern blocks and hundred square paper on day 1 to create visual (nonnumeric) patterns. Days 2 and 3 of the unit include increasing and decreasing numeric skip patterns. Students begin by identifying the pattern, extending it, then creating their own patterns in multiple ways using the same rule.
 Counting Embedded Figures: Students will focus on counting shape patterns within a given shape, and then will be engaged to form generalizations about the problem, which will sharpen their algebraic skills.
 Paper Quilts: Analyzing Designs : The geometric transformations of rotation, reflection, and translation are explored in this lesson. Students create a design and copy it three times, then place the squares into the quadrants of their "miniquilt" and use turns, flips and slides to create a design that can be analyzed in terms of symmetry.
 Triangles on a Lattice: In this activity, students will use a 3x3 square lattice to study transformations of triangles whose vertices are part of the lattice. The tasks include determining whether two triangles are congruent, which transformations connect two congruent triangles, and the number of noncongruent triangles (with vertices on the lattice) that are possible.
 Runway Rotations: Students will use small paper airplanes to model rotations required to turn onto a runway. Students will rotate planes 45, 90, 180, 270, and 360 degrees. Students will identify and describe the results of rotations using benchmark angles.
 Count on Math: Making Your First Million: This lesson explores the concept of very large numbers by having students contextualize the magnitude of 1,000,000. For example, students are asked questions such as: "Do you think you've been alive for a million days, hours, minutes, or seconds?" and "How long would it take a person to count to a million?" Students will be also be asked to consider getting to a million by repeated addition and multiplication.
 Place Value Made Simple: Students will read, write, and represent whole numbers from 0  1,000,000 using symbols, models, expanded form, and words. Base ten blocks, place value pocket charts, folded paper strips for expanded form accordions, and checks are used.

Football Finances:
This activity has students analyze a photo of a football stadium in order to make estimates such as:
 how many people will attend the Super Bowl
 how many hot dogs will be eaten
 how many gallons of soft drinks will be drunk
 Fun with Fractions: Investigating Equivalent Fractions with Relationship Rods: This lesson is to follow Fun with Fractions  Investigating Fraction Relationships with Relationship Rods (see Related Resources.) Students will be asked to make the various differentcolored rods the "whole" in relation to the other rods, and then give the relative fraction name for the other nine rods, based on how many of each color it would take to "fill in" the whole. Students will develop problemsolving skills and reasoning as they explain the equivalencies of length.

Fractional Clothesline:
This lesson has the teacher stretch a lengthy string across the room, and then ask 5 students to equally distribute cards with the numbers 0, 1, 2, 3, and 4 written on them. Once this number line is done, students will use it to place cards with mixed numbers, proper fractions and improper fractions, which will help them realize the difference between these kinds of rational numbers. They will also discuss their strategies for placing the cards. Finally, students will play an estimation game practicing combining numbers from two cards. Exposure to looking at fractions in different ways will help their developing conceptual fraction sense.
 Reasonable Estimates: Students are taught to estimate, to identify appropriate situations using estimation, to calculate estimated sums and differences, and to evaluate answers for reasonableness.
 Multiplication and Division Worksheets: This site offers printable worksheets and interactive practice for basic multiplication and division facts.
 Number Line Bars: A versatile tool that can be used to illustrate the operations of addition, subtraction, multiplication, and division.
 The Product Game: Playing the Product Game: Students learn to play the Product Game to better understand factors and products, and the relationships between them.

Factor Game:
In this lesson students will start off in pairs, playing a factorfinding game for two players (it also has a player vs. computer mode). Students must find winning strategies, including finding numbers with the fewest factors and knowing which factors were already discounted from the previous rounds. Afterwards, the students will analyze game strategies, and be introduced to prime and composite numbers.
(from NCTM Illuminations)  Area Model of Multiplication Using Base 10 Manipulatives (Double digit multiplication): "Students will use base 10 manipulatives to build a rectangular array to represent double digit multiplication. Students will make the connection between the standard algorithm and the rectangular array" from Alabama Learning Exchange).
 Arithmetic: The students will be given mutiplication and division problems which they must answer. They also have the option of being given a number then stating the factors of how that number was attained using either multiplication or division.
 Multiply and Conquer: In this lesson, students will segue from decomposing twodigit by onedigit into twodigit by twodigit problems. First, the problem 14 x 6 is written out using an array model (14 boxes by 6 boxes). Next, the 14 box side is decomposed into a 10x6 section and a 4x6 section, and then multiplication is carried out using the distributive property. After several of these types of problems, a twodigit by twodigit array is modeled for students. This time, both sides are decomposed into four grids. This lesson will provide conceptual knowledge for what happens in a 2 twodigit multiplication problem.

All About Multiplication:
This fourlesson unit from NCTM's Illuminations has students explore several meaning and representations of multiplication (number line, equal sets, arrays, and balanced equations). Other multiplication topics covered include: the commutative (order) property, the results of multiplying by 1 and 0, and the inverse property. Students will write and solve multiplication story problems, and convert word problems into equations. A bibliography of children's literature with a focus on multiplication is provided, which can be integrated so that students can connect through literature.
Individual Lessons
 Lesson 1: Hopping on the Number Line
 Students are shown how to generate products on a number line, highlighting the measurement aspect of addition. After some practice, they are asked to predict and then verify sums, and are also introduced to the commutative property.
 Lesson 2: Exploring Equal Sets
 Students explore the set model for multiplication. They find products using equal sets and present their results in a table. Next students apply their knowledge about multiplication in the creation of pictographs.
 Lesson 3: Modeling Multiplication With Streets and Avenues
 The array model of multiplication lays the foundation for the later study of area, and in this lesson, students will create arrays of the factors of 4 and 6 using toothpicks as representations of streets and avenues. The teacher will ask how many stoplights would be needed to fill each intersection. Zero and one will also be used as factors. Finally they will write their results as multiplication problems.
 Lesson 4: Balance Beam Discoveries
 The balance beam model of multiplication is examined in this lesson, along with another important relationship, the inverse of multiplication. This will lead naturally into the representation of multiplication facts as equations. Students will also practice multiplication facts by playing the Product Game.
 Estimator Four: In this activity, students play a game of connect four, but to place a piece on the board they have to correctly estimate an addition, multiplication, or percentage problem. Students can adjust the difficulty of the problems as well as how close the estimate has to be to the actual result. This activity allows students to practice estimating addition, multiplication, and percentages of large numbers (100s). This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.
 Estimate: This interactive applet helps develop number sense. The user mentally estimates a number that is represented by an arrow on a number line and then checks the estimate by clicking to have the exact number revealed. Users can choose a number range for whole numbers (between 0 and 10, 100, 1000 or 10,000) or decimals (tenths or hundredths between 0 and 1). An optional scale of tick marks provides guidance. The activity does not provide a scoring component.
 Fraction Models: An interactive tool to represent a fraction circle, rectangle, or set model with numerators and denominators ranging from 1 to 100. The decimal and percent equivalents of the created fraction are also displayed.
 Junior Architects: Finding Perimeter and Area: This lesson's focus is to have students consider perimeter and area while designing a clubhouse. The teacher will mark out a 10x10 "room" with masking tape to help students consider the actual size of such a small room. Students will use a handout and geoboards to find area and perimeter and problemsolve their designs.
 Fractal Tool: Students investigate shapes that grow and change using an iterative process. Fractals are characterized by selfsimilarity, smaller sections that resemble the larger figure. From NCTM's Illuminations.
 Chairs Around the Table: This lesson allows exploration for students to discover effects of changes in seating when tables are moved together. Since pushing tables together makes the sides of the tables inaccessible, the teacher can use this to get students to come up with rules for linear patterns. These rules can be written on the board using variables to lay the groundwork for understanding the use of variables.

Transformations  Translation:
The user may drag or rotate a shape created with pattern block pieces, using or not using a coordinate axes and lattice points background.
(source: NLVM grade 68 "Transformations  Translation")  Using Cubes and Isometric Drawings: Building Using the FrontRightTop View: Students explore drawing the frontrighttop view when given a threedimensional figure built from cubes, and explore building a threedimensional figure when given the frontrighttop view.
 Estimation: The students will be presented with two shapes and must estimate how many times the smaller will fit in the larger. They will be surprised at some of the results but will quickly learn and make adjustments.
 Expanded Notation: With this stepbystep, interactive video from StudyJams students can earn more about expanded notation. Expanded notation, also called expanded form, is a handy way to write numbers that shows the place value of each digit. It looks like an addition problem and will help you break numbers down and understand their place value. A short, selfchecking quiz is also included with this link.

Factor Trees :
This virtual manipulative allows students to find the prime factorization of numbers by completing factor trees. Then they may use a venn diagram to sort the prime factors to find the greatest common factor and the least common multiple.
 Factor Tree: This virtual manipulative will help the students in exploring the prime factorization of numbers and see how to use the factorization of a pair of numbers to find the greatest common factor (GCF) and the least common factor (LCM). In the manipulative the number pairs are presented randomly, so that a student returning to the factor tree will most likely begin with a pair of numbers not seen before.
 Multiply and Conquer: In this lesson, students will segue from decomposing twodigit by onedigit into twodigit by twodigit problems. First, the problem 14 x 6 is written out using an array model (14 boxes by 6 boxes). Next, the 14 box side is decomposed into a 10x6 section and a 4x6 section, and then multiplication is carried out using the distributive property. After several of these types of problems, a twodigit by twodigit array is modeled for students. This time, both sides are decomposed into four grids. This lesson will provide conceptual knowledge for what happens in a 2 twodigit multiplication problem.

"Bar Model Math"  "Twice" as Nice:
In this lesson students will solve real world problems that have multiplicative comparisons in them. They will use the strategy of bar models to solve the problems.

Are You Ready for a Hurricane?:
This activity allows students to determine the types of items that should be in a hurricane survival kit, use a budget and calculations to determine the items to include in the kit and gain an understanding of hurricanes and the need to prepare for them.
 More Bang for your Buck!: In this openended MEA, students will work in teams to determine a procedure for ranking recycling companies. Students will need to calculate their return on recycling, make decisions based on a table of data, and write a letter to the client providing evidence for their decisions.
 Cookies and Treats: Fourth graders will help Cookies and Treats find ecofriendly packaging for its cookies. Students will work with decimals and data in order to develop a procedure for ranking and choosing packaging for cookies.

Sidewalk Capers:
"Students will explore spatial relationships by determining the area of various shapes and creating tessellation patterns. They use a variety of manipulatives in this lesson to explore spatial relationships of squares and their transformations.
Students create a square with 36 square tiles and determine that by sliding several of the pieces both vertically and horizontally, they form an irregular shape that can be tessellated. Students are given a variety of curved shapes and encouraged to use tape and scissors to reconfigure their shape back into a square. Given a variety of measurement tools, students are asked to estimate the number of squares tiles (24" x 24") needed to cover an assigned section of the school's sidewalk. Finally, they create a pattern from the square and tessellate it on the sidewalk using chalk." from PBS Mathline. 
Banana County Public SchoolPainters MEA:
This Model Eliciting Activity (MEA) is written at a 4th grade level.
This activity allows students to think critically using information provided. Students will write a procedure on how they determined which painting company would be suitable for the client.  Counting Embedded Figures: Students will focus on counting shape patterns within a given shape, and then will be engaged to form generalizations about the problem, which will sharpen their algebraic skills.
 Triangles on a Lattice: In this activity, students will use a 3x3 square lattice to study transformations of triangles whose vertices are part of the lattice. The tasks include determining whether two triangles are congruent, which transformations connect two congruent triangles, and the number of noncongruent triangles (with vertices on the lattice) that are possible.
 Hand Spans, A Lesson for Third and Fourth Graders (Division and Measurement): A measurement activity to give students experience with the grouping model of division and practice with rulers and tape measures.
 Learning and Teaching Ratio and Proportion: Research Implications: This is a chapter describing the theory and lesson planning for teaching ratio and rate problems. The authors describe how students use reasoning about multiplication and division to solve ratio and rate problems.
 Fun with Estimation: This activity is a timed game that allows teams of students to practice their estimation skills while shopping for groceries (empty containers acquired for classroom use).
 Math Vocabulary Bingo: This lesson activity is an opportunity to assess and reinforce students' understanding of mathematical vocabulary. Students will play Bingo, except instead of calling numbers out, the teacher will call out clues that utilize math vocabulary to indicate numbers, such as: "the number of inches in one yard." This will make students listen for details, perform calculations mentally, associate math vocabulary to specific examples, and to communicate their mathematical thinking. Bingo cards will need to be revised to be aligned with the grade level's Standards.
 Estimator Quiz: In this activity, students are quizzed on their ability to estimate sums, products, and percentages. The student can adjust the difficulty of the problems and how close they have to be to the actual answer. This activity allows students to practice estimating addition, multiplication, and percentages of large numbers. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.
 Dynamic Decimals, Fractions and Money!: In this lesson, students will realize the connection between fractions, decimals and money through the use of a 100 grid.
 Awesome Area: "Students will explore the concept of area in a realistic application. They will develop their understanding of area through constructing, drawing, and labeling spaces." (from National Security Agency Central Security Service)
 Perfect Patterns: This unit begins with nonnumeric patterns. The students use linkage cubes, pattern blocks and hundred square paper on day 1 to create visual (nonnumeric) patterns. Days 2 and 3 of the unit include increasing and decreasing numeric skip patterns. Students begin by identifying the pattern, extending it, then creating their own patterns in multiple ways using the same rule.
 Place Value Made Simple: Students will read, write, and represent whole numbers from 0  1,000,000 using symbols, models, expanded form, and words. Base ten blocks, place value pocket charts, folded paper strips for expanded form accordions, and checks are used.
 Reasonable Estimates: Students are taught to estimate, to identify appropriate situations using estimation, to calculate estimated sums and differences, and to evaluate answers for reasonableness.
 The Product Game: Playing the Product Game: Students learn to play the Product Game to better understand factors and products, and the relationships between them.
 Factor Trail Game: Students determine factors of designated numbers and play a board game for practice. Students use a game board with numbered squares and write down what the factors of a given number are. If they are correct, the sum of the factors is how many points they get. If they are incorrect, their opponent gets 10 points for each mistake they find. The score sheet itself is an assessment tool that the teacher can use to evaluate the students' understanding of factors.

Factor Game:
In this lesson students will start off in pairs, playing a factorfinding game for two players (it also has a player vs. computer mode). Students must find winning strategies, including finding numbers with the fewest factors and knowing which factors were already discounted from the previous rounds. Afterwards, the students will analyze game strategies, and be introduced to prime and composite numbers.
(from NCTM Illuminations)  Who Wants to be a Mathionaire?: This online game replicates the popular "Who Wants to be a Millionaire?" format, using multiplication facts. It's a fun way for kids to practice multiplication!

All About Multiplication:
This fourlesson unit from NCTM's Illuminations has students explore several meaning and representations of multiplication (number line, equal sets, arrays, and balanced equations). Other multiplication topics covered include: the commutative (order) property, the results of multiplying by 1 and 0, and the inverse property. Students will write and solve multiplication story problems, and convert word problems into equations. A bibliography of children's literature with a focus on multiplication is provided, which can be integrated so that students can connect through literature.
Individual Lessons
 Lesson 1: Hopping on the Number Line
 Students are shown how to generate products on a number line, highlighting the measurement aspect of addition. After some practice, they are asked to predict and then verify sums, and are also introduced to the commutative property.
 Lesson 2: Exploring Equal Sets
 Students explore the set model for multiplication. They find products using equal sets and present their results in a table. Next students apply their knowledge about multiplication in the creation of pictographs.
 Lesson 3: Modeling Multiplication With Streets and Avenues
 The array model of multiplication lays the foundation for the later study of area, and in this lesson, students will create arrays of the factors of 4 and 6 using toothpicks as representations of streets and avenues. The teacher will ask how many stoplights would be needed to fill each intersection. Zero and one will also be used as factors. Finally they will write their results as multiplication problems.
 Lesson 4: Balance Beam Discoveries
 The balance beam model of multiplication is examined in this lesson, along with another important relationship, the inverse of multiplication. This will lead naturally into the representation of multiplication facts as equations. Students will also practice multiplication facts by playing the Product Game.
 Jeopardy Labs: This is a math review using interactive jeopardy games.
 Math Vocabulary Bingo: This lesson activity is an opportunity to assess and reinforce students' understanding of mathematical vocabulary. Students will play Bingo, except instead of calling numbers out, the teacher will call out clues that utilize math vocabulary to indicate numbers, such as: "the number of inches in one yard." This will make students listen for details, perform calculations mentally, associate math vocabulary to specific examples, and to communicate their mathematical thinking. Bingo cards will need to be revised to be aligned with the grade level's Standards.
 Estimator Quiz: In this activity, students are quizzed on their ability to estimate sums, products, and percentages. The student can adjust the difficulty of the problems and how close they have to be to the actual answer. This activity allows students to practice estimating addition, multiplication, and percentages of large numbers. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.
 Estimator Four: In this activity, students play a game of connect four, but to place a piece on the board they have to correctly estimate an addition, multiplication, or percentage problem. Students can adjust the difficulty of the problems as well as how close the estimate has to be to the actual result. This activity allows students to practice estimating addition, multiplication, and percentages of large numbers (100s). This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.
 Estimate: This interactive applet helps develop number sense. The user mentally estimates a number that is represented by an arrow on a number line and then checks the estimate by clicking to have the exact number revealed. Users can choose a number range for whole numbers (between 0 and 10, 100, 1000 or 10,000) or decimals (tenths or hundredths between 0 and 1). An optional scale of tick marks provides guidance. The activity does not provide a scoring component.
 Perfect Patterns: This unit begins with nonnumeric patterns. The students use linkage cubes, pattern blocks and hundred square paper on day 1 to create visual (nonnumeric) patterns. Days 2 and 3 of the unit include increasing and decreasing numeric skip patterns. Students begin by identifying the pattern, extending it, then creating their own patterns in multiple ways using the same rule.
 Guess My Rule: Function Machine card game is used to reinforce student understanding of functions using an in/out table.
 All Things Being Equal: In this game, learners strategize to win the most cards by building number equations. Learners practice operations (addition, subtraction, multiplication, and division) to construct their equations. This activity guide contains sample questions to ask, literary connections, extensions, and alignment to local and national standards.
 Equivalent Fractions: "This lesson helps students discover how to obtain equal fractions by using both fraction strips and playing a fraction matching game. Students will learn that to obtain equal fractions they may multiply the numerator and denominator by the same number." (from ALEX  Alabama Learning Exchange)
 Factortris (FactorTris): This game is an adaptation of Tetris, in which the student is tasked with creating an arrays product and using the arrays to play Tetris. The game increases in difficulty as the student progresses, but unfortunately, the teacher can not adjust the difficulty of the game at the onset.
 Fun with Estimation: This activity is a timed game that allows teams of students to practice their estimation skills while shopping for groceries (empty containers acquired for classroom use).

Fractional Clothesline:
This lesson has the teacher stretch a lengthy string across the room, and then ask 5 students to equally distribute cards with the numbers 0, 1, 2, 3, and 4 written on them. Once this number line is done, students will use it to place cards with mixed numbers, proper fractions and improper fractions, which will help them realize the difference between these kinds of rational numbers. They will also discuss their strategies for placing the cards. Finally, students will play an estimation game practicing combining numbers from two cards. Exposure to looking at fractions in different ways will help their developing conceptual fraction sense.
 Reasonable Estimates: Students are taught to estimate, to identify appropriate situations using estimation, to calculate estimated sums and differences, and to evaluate answers for reasonableness.

Ordering numeric expressions :
Ordering decimals, fractions and percents.
 The MegaPenny Project: "Visualizing huge numbers can be very difficult. People regularly talk about millions of miles, billions of bytes, or trillions of dollars, yet it's still hard to grasp just how much a "billion" really is. The MegaPenny Project aims to help by taking one small everyday item, the U.S. penny, and building on that to answer the question: "What would a million, billion (or a trillion) pennies look like?" " from The Mega Penny Project.

Ordering numeric expressions :
Ordering decimals, fractions and percents.
 Clipart ETC Fractions: Illustrations that can be used for teaching and demonstrating fractions. Fractional representations are modeled in wedges of circles ("pieces of pie") and parts of polygons. There are also clipart images of numerical fractions, both proper and improper, from halves to twelfths. Fraction charts and fraction strips found in this collection can be used as manipulatives and are ready to print for classroom use.
 Clipart: Geometric Shapes: In this lesson, you will find clip art and various illustrations of polygons, circles, ellipses, star polygons, and inscribed shapes.
 All About Multiplication: Bibliography: List of five children's books with a multiplication focus (found on NCTM Illuminations site under "All About Multiplication").
 More Bang for your Buck!: In this openended MEA, students will work in teams to determine a procedure for ranking recycling companies. Students will need to calculate their return on recycling, make decisions based on a table of data, and write a letter to the client providing evidence for their decisions.
 True, False, and Open Sentences: "Students first explore arithmetic sentences to decide whether they are true or false. The lesson then introduces students to sentences that are neither true nor false but are algebraic equations, also called open sentences, such as x + 3 = 7 or 2 x = 12." from Math Solutions.
 The MegaPenny Project: "Visualizing huge numbers can be very difficult. People regularly talk about millions of miles, billions of bytes, or trillions of dollars, yet it's still hard to grasp just how much a "billion" really is. The MegaPenny Project aims to help by taking one small everyday item, the U.S. penny, and building on that to answer the question: "What would a million, billion (or a trillion) pennies look like?" " from The Mega Penny Project.
 Hand Spans, A Lesson for Third and Fourth Graders (Division and Measurement): A measurement activity to give students experience with the grouping model of division and practice with rulers and tape measures.

Ordering numeric expressions :
Ordering decimals, fractions and percents.
 Expanded Notation: With this stepbystep, interactive video from StudyJams students can earn more about expanded notation. Expanded notation, also called expanded form, is a handy way to write numbers that shows the place value of each digit. It looks like an addition problem and will help you break numbers down and understand their place value. A short, selfchecking quiz is also included with this link.
 Who Wants to be a Mathionaire?: This online game replicates the popular "Who Wants to be a Millionaire?" format, using multiplication facts. It's a fun way for kids to practice multiplication!
 Estimate: This interactive applet helps develop number sense. The user mentally estimates a number that is represented by an arrow on a number line and then checks the estimate by clicking to have the exact number revealed. Users can choose a number range for whole numbers (between 0 and 10, 100, 1000 or 10,000) or decimals (tenths or hundredths between 0 and 1). An optional scale of tick marks provides guidance. The activity does not provide a scoring component.
 The MegaPenny Project: "Visualizing huge numbers can be very difficult. People regularly talk about millions of miles, billions of bytes, or trillions of dollars, yet it's still hard to grasp just how much a "billion" really is. The MegaPenny Project aims to help by taking one small everyday item, the U.S. penny, and building on that to answer the question: "What would a million, billion (or a trillion) pennies look like?" " from The Mega Penny Project.
 Jeopardy Labs: This is a math review using interactive jeopardy games.

"Bar Model Math"  "Twice" as Nice:
In this lesson students will solve real world problems that have multiplicative comparisons in them. They will use the strategy of bar models to solve the problems.
 Math Vocabulary Bingo: This lesson activity is an opportunity to assess and reinforce students' understanding of mathematical vocabulary. Students will play Bingo, except instead of calling numbers out, the teacher will call out clues that utilize math vocabulary to indicate numbers, such as: "the number of inches in one yard." This will make students listen for details, perform calculations mentally, associate math vocabulary to specific examples, and to communicate their mathematical thinking. Bingo cards will need to be revised to be aligned with the grade level's Standards.
 Estimate: This interactive applet helps develop number sense. The user mentally estimates a number that is represented by an arrow on a number line and then checks the estimate by clicking to have the exact number revealed. Users can choose a number range for whole numbers (between 0 and 10, 100, 1000 or 10,000) or decimals (tenths or hundredths between 0 and 1). An optional scale of tick marks provides guidance. The activity does not provide a scoring component.
 Secret NumberPlace Value Game: Students will be able to use decimals through the thousandths place to name numbers between whole numbers through the use of number generators and critically thinking about where to place numbers on a place value chart.
 Fraction/Decimal War: This lesson plan introduces a game that will reinforce fractions and decimals.

Ordering numeric expressions :
Ordering decimals, fractions and percents.
 The Area of Tangram Pieces: The student uses tangrams to compute the area of polygons without formulas.
 True, False, and Open Sentences: "Students first explore arithmetic sentences to decide whether they are true or false. The lesson then introduces students to sentences that are neither true nor false but are algebraic equations, also called open sentences, such as x + 3 = 7 or 2 x = 12." from Math Solutions.
 Using Cubes and Isometric Drawings: Building Using the FrontRightTop View: Students explore drawing the frontrighttop view when given a threedimensional figure built from cubes, and explore building a threedimensional figure when given the frontrighttop view.
 Hand Spans, A Lesson for Third and Fourth Graders (Division and Measurement): A measurement activity to give students experience with the grouping model of division and practice with rulers and tape measures.
 Learning and Teaching Ratio and Proportion: Research Implications: This is a chapter describing the theory and lesson planning for teaching ratio and rate problems. The authors describe how students use reasoning about multiplication and division to solve ratio and rate problems.
 Factortris (FactorTris): This game is an adaptation of Tetris, in which the student is tasked with creating an arrays product and using the arrays to play Tetris. The game increases in difficulty as the student progresses, but unfortunately, the teacher can not adjust the difficulty of the game at the onset.
 Fun with Estimation: This activity is a timed game that allows teams of students to practice their estimation skills while shopping for groceries (empty containers acquired for classroom use).
 Jeopardy Labs: This is a math review using interactive jeopardy games.
 Math Vocabulary Bingo: This lesson activity is an opportunity to assess and reinforce students' understanding of mathematical vocabulary. Students will play Bingo, except instead of calling numbers out, the teacher will call out clues that utilize math vocabulary to indicate numbers, such as: "the number of inches in one yard." This will make students listen for details, perform calculations mentally, associate math vocabulary to specific examples, and to communicate their mathematical thinking. Bingo cards will need to be revised to be aligned with the grade level's Standards.
 Estimate: This interactive applet helps develop number sense. The user mentally estimates a number that is represented by an arrow on a number line and then checks the estimate by clicking to have the exact number revealed. Users can choose a number range for whole numbers (between 0 and 10, 100, 1000 or 10,000) or decimals (tenths or hundredths between 0 and 1). An optional scale of tick marks provides guidance. The activity does not provide a scoring component.
 Dynamic Decimals, Fractions and Money!: In this lesson, students will realize the connection between fractions, decimals and money through the use of a 100 grid.
 Learning and Teaching Ratio and Proportion: Research Implications: This is a chapter describing the theory and lesson planning for teaching ratio and rate problems. The authors describe how students use reasoning about multiplication and division to solve ratio and rate problems.

Are You Ready for a Hurricane?:
This activity allows students to determine the types of items that should be in a hurricane survival kit, use a budget and calculations to determine the items to include in the kit and gain an understanding of hurricanes and the need to prepare for them.
 More Bang for your Buck!: In this openended MEA, students will work in teams to determine a procedure for ranking recycling companies. Students will need to calculate their return on recycling, make decisions based on a table of data, and write a letter to the client providing evidence for their decisions.
 Cookies and Treats: Fourth graders will help Cookies and Treats find ecofriendly packaging for its cookies. Students will work with decimals and data in order to develop a procedure for ranking and choosing packaging for cookies.
 Celebrity Floor Plan Frenzy: Students will help an architect find the area of each room in a celebrity home and then determine the best location to build the home based on qualitative data about the locations.

Banana County Public SchoolPainters MEA:
This Model Eliciting Activity (MEA) is written at a 4th grade level.
This activity allows students to think critically using information provided. Students will write a procedure on how they determined which painting company would be suitable for the client.  Multiply and Conquer: In this lesson, students will segue from decomposing twodigit by onedigit into twodigit by twodigit problems. First, the problem 14 x 6 is written out using an array model (14 boxes by 6 boxes). Next, the 14 box side is decomposed into a 10x6 section and a 4x6 section, and then multiplication is carried out using the distributive property. After several of these types of problems, a twodigit by twodigit array is modeled for students. This time, both sides are decomposed into four grids. This lesson will provide conceptual knowledge for what happens in a 2 twodigit multiplication problem.
 Secret NumberPlace Value Game: Students will be able to use decimals through the thousandths place to name numbers between whole numbers through the use of number generators and critically thinking about where to place numbers on a place value chart.
 Dynamic Decimals, Fractions and Money!: In this lesson, students will realize the connection between fractions, decimals and money through the use of a 100 grid.

Banana County Public SchoolPainters MEA:
This Model Eliciting Activity (MEA) is written at a 4th grade level.
This activity allows students to think critically using information provided. Students will write a procedure on how they determined which painting company would be suitable for the client.  Fun with Fractions: Making and Investigating Fraction Strips: This lesson has students discover relationships between different fractions. From equally cut pieces of paper (a whole), fraction strips are made into halves, thirds, fourths, sixths, and eighths. Students then compare them to one another, discovering their relationships, including equivalent fractions.
15 Student Resources

Number Line Bars: A versatile tool that can be used to illustrate the operations of addition, subtraction, multiplication, and division. Virtual Manipulative (Primary Type) 
Arithmetic: The students will be given mutiplication and division problems which they must answer. They also have the option of being given a number then stating the factors of how that number was attained using either multiplication or division. Virtual Manipulative (Primary Type) 
Estimator Quiz: In this activity, students are quizzed on their ability to estimate sums, products, and percentages. The student can adjust the difficulty of the problems and how close they have to be to the actual answer. This activity allows students to practice estimating addition, multiplication, and percentages of large numbers. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet. AssessmentEducational Game (Primary Type) 
Estimator Four: In this activity, students play a game of connect four, but to place a piece on the board they have to correctly estimate an addition, multiplication, or percentage problem. Students can adjust the difficulty of the problems as well as how close the estimate has to be to the actual result. This activity allows students to practice estimating addition, multiplication, and percentages of large numbers (100s). This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet. Virtual ManipulativeEducational Game (Primary Type) 
Estimate: This interactive applet helps develop number sense. The user mentally estimates a number that is represented by an arrow on a number line and then checks the estimate by clicking to have the exact number revealed. Users can choose a number range for whole numbers (between 0 and 10, 100, 1000 or 10,000) or decimals (tenths or hundredths between 0 and 1). An optional scale of tick marks provides guidance. The activity does not provide a scoring component. Virtual Manipulative (Primary Type)Educational GameEducational Software / ToolTeaching IdeaInstructional Technique 
Ordering numeric expressions : Ordering decimals, fractions and percents.
Presentation/SlideshowVideo/Audio/AnimationTutorialTeaching Idea (Primary Type) 
Junior Architects: Finding Perimeter and Area: In this worksheet, students are directed to find the perimeter and area for a clubhouse in the form of rectangles, composite figures, and other polygons. The second sheet urged them to make their own designs for a clubhouse and find the perimeter and area. This resource is recommended as an introduction or review of perimeter and area.
(Found under "Finding Perimeter and Area" on NCTM's Illuminations)Worksheet (Primary Type) 
The Area of Tangram Pieces: The student uses tangrams to compute the area of polygons without formulas. Teaching Idea (Primary Type) 
Fractal Tool: Students investigate shapes that grow and change using an iterative process. Fractals are characterized by selfsimilarity, smaller sections that resemble the larger figure. From NCTM's Illuminations. Virtual Manipulative (Primary Type) 
Transformations  Translation: The user may drag or rotate a shape created with pattern block pieces, using or not using a coordinate axes and lattice points background.
(source: NLVM grade 68 "Transformations  Translation")Virtual Manipulative (Primary Type) 
The MegaPenny Project: "Visualizing huge numbers can be very difficult. People regularly talk about millions of miles, billions of bytes, or trillions of dollars, yet it's still hard to grasp just how much a "billion" really is. The MegaPenny Project aims to help by taking one small everyday item, the U.S. penny, and building on that to answer the question: "What would a million, billion (or a trillion) pennies look like?" " from The Mega Penny Project. Presentation/Slideshow (Primary Type)Text ResourceEducational Software / Tool 
Estimation: The students will be presented with two shapes and must estimate how many times the smaller will fit in the larger. They will be surprised at some of the results but will quickly learn and make adjustments. Virtual Manipulative (Primary Type) 
Expanded Notation: With this stepbystep, interactive video from StudyJams students can earn more about expanded notation. Expanded notation, also called expanded form, is a handy way to write numbers that shows the place value of each digit. It looks like an addition problem and will help you break numbers down and understand their place value. A short, selfchecking quiz is also included with this link. Virtual Manipulative (Primary Type)Tutorial 
Factortris (FactorTris): This game is an adaptation of Tetris, in which the student is tasked with creating an arrays product and using the arrays to play Tetris. The game increases in difficulty as the student progresses, but unfortunately, the teacher can not adjust the difficulty of the game at the onset. Educational Game (Primary Type)Teaching Idea 
Factor Tree: This virtual manipulative will help the students in exploring the prime factorization of numbers and see how to use the factorization of a pair of numbers to find the greatest common factor (GCF) and the least common factor (LCM). In the manipulative the number pairs are presented randomly, so that a student returning to the factor tree will most likely begin with a pair of numbers not seen before. Virtual Manipulative (Primary Type)
7 Parent Resources

Number Line Bars: A versatile tool that can be used to illustrate the operations of addition, subtraction, multiplication, and division. Virtual Manipulative (Primary Type) 
Clipart ETC Fractions: Illustrations that can be used for teaching and demonstrating fractions. Fractional representations are modeled in wedges of circles ("pieces of pie") and parts of polygons. There are also clipart images of numerical fractions, both proper and improper, from halves to twelfths. Fraction charts and fraction strips found in this collection can be used as manipulatives and are ready to print for classroom use. Image/Photograph (Primary Type) 
Ordering numeric expressions : Ordering decimals, fractions and percents.
Presentation/SlideshowVideo/Audio/AnimationTutorialTeaching Idea (Primary Type) 
All Things Being Equal: In this game, learners strategize to win the most cards by building number equations. Learners practice operations (addition, subtraction, multiplication, and division) to construct their equations. This activity guide contains sample questions to ask, literary connections, extensions, and alignment to local and national standards. Educational Game (Primary Type) 
Clipart: Geometric Shapes: In this lesson, you will find clip art and various illustrations of polygons, circles, ellipses, star polygons, and inscribed shapes. Image/Photograph (Primary Type) 
Transformations  Translation: The user may drag or rotate a shape created with pattern block pieces, using or not using a coordinate axes and lattice points background.
(source: NLVM grade 68 "Transformations  Translation")Virtual Manipulative (Primary Type) 
Factor Tree: This virtual manipulative will help the students in exploring the prime factorization of numbers and see how to use the factorization of a pair of numbers to find the greatest common factor (GCF) and the least common factor (LCM). In the manipulative the number pairs are presented randomly, so that a student returning to the factor tree will most likely begin with a pair of numbers not seen before. Virtual Manipulative (Primary Type)
Course Number: 5012060 
Course Path:
Section: Grades PreK to 12 Education Courses > Grade Group: Grades PreK to 5 Education Courses > Subject: Mathematics > SubSubject: General Mathematics >

Course Section: Grades PreK to 12 Education Courses  Abbreviated Title: MATH GRADE FOUR 
Course Length: Year (Y)  
Course Type: Core  
Course Status : Course Approved  
Keywords: Grades PreK To 5 Education Courses, PreK5, PreK To 5, Grades Pre K To 5 Education Courses, Pre K5, Pre K To 5, Elementary, Mathematics, General Mathematics, General, Math, Mathematics  Grade Four, MATH GRADE FOUR  
Grade Level(s): 4  
GENERAL NOTES
MAFS.4
In Grade 4, instructional time should focus on three critical areas: (1) developing understanding and fluency with multidigit multiplication, and developing understanding of dividing to find quotients involving multidigit dividends; (2) developing an understanding of fraction equivalence, addition and subtraction of fractions with like denominators, and multiplication of fractions by whole numbers; (3) understanding that geometric figures can be analyzed and classified based on their properties, such as having parallel sides, perpendicular sides, particular angle measures, and symmetry.
(1) Students generalize their understanding of place value to 1,000,000, understanding the relative sizes of numbers in each place. They apply their understanding of models for multiplication (equalsized groups, arrays, area models), place value, and properties of operations, in particular the distributive property, as they develop, discuss, and use efficient, accurate, and generalization methods to compute products of multidigit whole numbers. Depending on the numbers and the context, they select and accurately apply appropriate methods to estimate or mentally calculate products. They develop fluency with efficient procedures for multiplying whole numbers; understand and explain why the procedures work based on place value and properties of operations; and use them to solve problems. Students apply their understanding of models for division, place value, properties of operations, and the relationship of division to multiplication as they develop, discuss, and use efficient, accurate, and generalization procedures to find quotients involving multidigit dividends. They select and accurately apply appropriate methods to estimate and mentally calculate quotients, and interpret remainders based upon the context.
(2) Students develop understanding of fraction equivalence and operations with fractions. They recognize that two different fractions can be equal (e.g., 15/9 = 5/3), and they develop methods for generating and recognizing equivalent fractions. Students extend previous understandings about how fractions are built from unit fractions, composing fractions from unit fractions, decomposing fractions into unit fractions, and using the meaning of fractions and the meaning of multiplication to multiply a fraction by a whole number.
(3) Students describe, analyze, compare, and classify twodimensional shapes. Through building, drawing, and analyzing twodimensional shapes, students deepen their understanding of properties of twodimensional objects and the use of them to solve problems involving symmetry.