# Mathematics - Grade Four   (#5012060)

## General Course Information and Notes

### General Notes

MAFS.4

In Grade 4, instructional time should focus on three critical areas: (1) developing understanding and fluency with multi-digit multiplication, and developing understanding of dividing to find quotients involving multi-digit 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 (equal-sized 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 multi-digit 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 multi-digit 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 two-dimensional shapes. Through building, drawing, and analyzing two-dimensional shapes, students deepen their understanding of properties of two-dimensional objects and the use of them to solve problems involving symmetry.

English Language Development ELD Standards Special Notes Section:
Teachers are required to provide listening, speaking, reading and writing instruction that allows English language learners (ELL) to communicate information, ideas and concepts for academic success in the content area of Mathematics. For the given level of English language proficiency and with visual, graphic, or interactive support, students will interact with grade level words, expressions, sentences and discourse to process or produce language necessary for academic success. The ELD standard should specify a relevant content area concept or topic of study chosen by curriculum developers and teachers which maximizes an ELL’s need for communication and social skills. To access an ELL supporting document which delineates performance definitions and descriptors, please click on the following link:

Florida Standards Implementation Guide Focus Section:

The Mathematics Florida Standards Implementation Guide was created to support the teaching and learning of the Mathematics Florida Standards. The guide is compartmentalized into three components: focus, coherence, and rigor.Focus means narrowing the scope of content in each grade or course, so students achieve higher levels of understanding and experience math concepts more deeply. The Mathematics standards allow for the teaching and learning of mathematical concepts focused around major clusters at each grade level, enhanced by supporting and additional clusters. The major, supporting and additional clusters are identified, in relation to each grade or course. The cluster designations for this course are below.

Major Clusters

MAFS.4.OA.1 Use the four operations with whole numbers to solve problems.

MAFS.4.NBT.1 Generalize place value understanding for multi-digit whole numbers.

MAFS.4.NBT.2 Use place value understanding and properties of operations to perform multi-digit arithmetic.

MAFS.4.NF.1 Extend understanding of fraction equivalence and ordering.

MAFS.4.NF.2 Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.

MAFS.4.NF.3 Understand decimal notation for fractions, and compare decimal fractions.

Supporting Clusters

MAFS.4.OA.2 Gain familiarity with factors and multiples.

MAFS.4.MD.1 Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.

MAFS.4.MD.2 Represent and interpret data.

MAFS.4.OA.3 Generate and analyze patterns.

MAFS.4.MD.3 Geometric measurement: understand concepts of angle and measure angles.

MAFS.4.G.1 Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

Note: Clusters should not be sorted from major to supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting and additional clusters.

### General Information

Course Number: 5012060
Course Path:
Course Length: Year (Y)
Course Attributes:
• Class Size Core Required
Course Status: Course Approved

## Educator Certifications

One of these educator certification options is required to teach this course.

## Student Resources

Vetted resources students can use to learn the concepts and skills in this course.

## Original Student Tutorials

CPALMS Aquarium: Part 3 Division with Larger Numbers:

This is part 3 of in a three-part series. Click below to learn different strategies to help you become more efficient with division.

Type: Original Student Tutorial

Fractions at the Fair: Equivalent Tenths and Hundredths:

Learn about equivalent 10ths and 100ths and how to calculate these equivalent fractions at the fair in this interactive tutorial.

Type: Original Student Tutorial

Rounding Larger Numbers with Endangered Animals:

Learn how to round larger whole numbers to any place value while exploring endangered species.

Type: Original Student Tutorial

Use tile designs to explore how angles can be decomposed into smaller angles and how those parts can be shown as addends in equations in this interactive tutorial.

Type: Original Student Tutorial

Space: Multiplication as Comparison:

Launch into solving word problems that use multiplicative comparisons, drawings, and symbols in this space-themed interactive tutorial.

Type: Original Student Tutorial

Game On Part Two: Measuring Distance with Customary Units:

Learn to convert a larger customary measurement unit into equivalent smaller units, including converting miles to yards and feet in this sports-themed interactive tutorial.

This is Part 2 of a two-part series. Click HERE to open Part 1: Measuring Length with Customary Units.

Type: Original Student Tutorial

Game On Part One: Measuring Length with Customary Units:

Learn to convert a larger customary measurement unit into equivalent smaller units, including converting yards to feet and inches, in this sports-themed interactive tutorial.

Type: Original Student Tutorial

Buffy's Bakery Part 4- Multiplying a Fraction by a Whole: Standard Algorithm:

Help Buffy multiply fractions by whole numbers using the standard algorithm in addition to visual fraction models in this bakery-themed, interactive tutorial.

This is part 4 of a 4-part series. Click below to open other tutorials in the series.

Type: Original Student Tutorial

Wubble Double Trouble:

Join Pete as he explores patterns within patterns with feisty Wubbles and Dipples in this interactive tutorial.

Type: Original Student Tutorial

Buffy’s Bakery Part 3: Using Models to Multiply a Fraction by a Whole Number:

Help Buffy the Baker multiply a fraction by a whole using models in this sweet interactive tutorial.

This is part 3 of a 4-part series. Click below to open other tutorials in the series.

Type: Original Student Tutorial

Multiplying Fractions with Bake Sale Mania:

Find the total amounts of repeated fraction quantities by multiplying a fraction by a whole number using visual models that represent real-world problems and cookies in this interactive tutorial.

Type: Original Student Tutorial

Healthy Distance: Social Distancing Measurement Conversions:

Learn why it's sometimes important to use social distancing to reduce the spread of germs and how to estimate and convert this customary distance with this interactive tutorial.

Type: Original Student Tutorial

Comparing Fractions with Square Foot Gardens Part 2:

Use equivalent fractions to compare fractions in this garden-themed, interactive tutorials

This is Part 2 in a two-part series. Click HERE to open Part 1,  “Mama’s Pizza, Butterflies, & Comparing Fractions.”

Type: Original Student Tutorial

Learn about decimals on a number line and comparing decimals to save the Decis from a wizard's spell in this interactive tutorial.

Type: Original Student Tutorial

Mama's Pizza, Butterflies, and Comparing Fractions Part 1:

Help a family settle an argument about who got the most pizza and which butterfly was longer by comparing fractions using benchmarks and area models, in this interactive tutorial.

Type: Original Student Tutorial

Dream House Dilemma, Part 2: Perimeter:

Learn to calculate the perimeter of rectangular and composite shapes to help April finish designing her dream home in this interactive tutorial.

This is the second in a three-part series. Click below to open the other tutorials in the series.

Type: Original Student Tutorial

CPALMS Aquarium Part 2: Division Strategies:

Learn to solve division challenges using the partial quotients strategy with this interactive tutorial.

This is the second tutorial is a series on division strategies.

Type: Original Student Tutorial

Dream House Dilemma Part 3: Perimeter and a Missing Side:

Help April design her dream home while learning how to calculate perimeter and find a missing side measurement for a shape given the perimeter, in this interactive tutorial.

Type: Original Student Tutorial

CPALMS Aquarium: Connecting Multiplication and Division: Part 1:

Learn how multiplication connects to division to help understand what division is in this aquarium-themed, interactive tutorial.

This is part 1 of a two-part series. Click HERE to open Part 2.

Type: Original Student Tutorial

Dream House Dilemma Part 1: Area:

Help April calculate area and missing measurements for items in her perfect dream home in this interactive tutorial.

Type: Original Student Tutorial

Prime and Composite Backyard Dilemma :

Discover what makes prime and composite numbers unique thanks to an interesting backyard problem in this interactive tutorial.

Type: Original Student Tutorial

Who's Top Dog, Now? Comparing Numbers:

Learn how to compare numbers using the greater than and less than symbols in this interactive tutorial that compares some pretty cool things!

Type: Original Student Tutorial

Candy Engineer: Place Value:

Read and write multi-digit whole numbers using base-ten numerals and number names using the Base 10 Place value system in this interactive tutorial.

Type: Original Student Tutorial

Candy Sales Are Booming! Expanded Notation:

Learn how to write numbers using place value in different forms like standard, word, and expanded notation in this interactive tutorial.

Type: Original Student Tutorial

Multi-Digit Multiplication Magic Part 3: Recording Partial Products:

Calculate the product of multi-digit factors by decomposing factors and recording partial products in this interactive tutorial.

This is the third tutorial in a 3-part series. Click below to open the other tutorials in the series.

Type: Original Student Tutorial

Multi-Digit Multiplication Area Models: Part 2:

See the magical power of area models when multiplying multi-digit numbers, in this interactive tutorial.

This is the second tutorial in a 3-part series. Click below to open the other tutorials in the series.

Type: Original Student Tutorial

Multi-Digit Multiplication Magic Part 1: Arrays:

Area models are efficient tools for multi-digit multiplication, see just how magical they are in this interactive tutorial!

This is the first tutorial in a 3-part series. Click below to open the other tutorials in the series.

Type: Original Student Tutorial

Logging Lengths with Line Plots:

Learn how to create a line plot and analyze data in the line plot in this interactive tutorial.  You will also see how to add and subtract using the line plot to solve problems based on the line plots.

Type: Original Student Tutorial

The Nightmare of Pouring Polygons: Quadrilateral Classification:

Overcome the nightmare of quadrilateral classification based on the presence of parallel, perpendicular, and congruent sides as you complete this interactive tutorial.

Type: Original Student Tutorial

Deci Land Escape:

Help Rich escape Deci Land by learning how to write decimals that are related to fractions with denominators of 10 and 100 in this interactive tutorial.

Type: Original Student Tutorial

The Treasure of Aryabhata: Place Value Comparison:

Help solve mysteries built on patterns of ten to discover the treasure of our number system in this interactive student tutorial.

Type: Original Student Tutorial

Shape Surfing: Finding Parallel and Perpendicular Lines:

Help a surfing crab learn how to find parallel and perpendicular sides in a variety of polygons as you complete this interactive tutorial!

Type: Original Student Tutorial

Map the ROC: Angle Measure:

Learn how to measure angles with a protractor to help get a robot through an obstacle course in this interactive tutorial.

Type: Original Student Tutorial

ROC Star: Angle Classification:

Classify and name angles in two-dimensional shapes to help a robot create a path using angles in this interactive tutorial.

Type: Original Student Tutorial

Equivalent Fractions Part 1: Sharing at the Shelter:

Learn how to create equivalent fractions and visually see how they are equivalent in this interactive tutorial. This is part 1 of a 2 part series.

Type: Original Student Tutorial

Field Trip Frenzy (Part 4):

Learn when to write the remainder of a multi-step division process as a fraction or decimal in this interactive tutorial.

This is the final tutorial in the Field Trip Frenzy Series about remainders. Click below to open the other tutorials in this series.

Type: Original Student Tutorial

Field Trip Frenzy (Part 3):

Learn how to interpret remainders in multi-step division problems in this interactive tutorial

This is the third tutorial in the Field Trip Frenzy Series about remainders. Click below to open the other tutorials in this series.

Type: Original Student Tutorial

Field Trip Frenzy (Part 2):

Learn how to interpret remainders in multi-step division problems in this second interactive tutorial in the Field Trip Frenzy Series.

This is the second tutorial in the Field Trip Frenzy Series about remainders. Click below to open the other tutorials in this series.

Type: Original Student Tutorial

Dash or Bash: Discovering Angles:

Discover what an angle is by helping to program a robot through an obstacle course in this interactive tutorial.

Type: Original Student Tutorial

Field Trip Frenzy (Part 1):

Take a field trip while learning how to interpret remainders in multi-step division word problems.

This is part 1 of a four-part series of interactive tutorials. Click below to open the other tutorials in this series.

Type: Original Student Tutorial

Multiplying Math Magic:

Learn how to write multiplication equations based on multiplication comparisons and story problems in this magical math online tutorial!

Type: Original Student Tutorial

Adding It All Up with the Standard Algorithm:

Learn to add multi-digit numbers using the standard algorithm in this interactive tutorial.

Type: Original Student Tutorial

The Symmetry Sisters Save the Day:

Help the Symmetry Sisters save the City of Symmetry Line and the State of Arithmetic from the Radical Rat!

Type: Original Student Tutorial

The Leftover Dessert Dilemma:

Learn how to be able to decompose a fraction into a sum of fractions with common denominators.

Type: Original Student Tutorial

These are the Right Triangles:

Identify right triangles and explain the properties shared by all right triangles.

Type: Original Student Tutorial

The Geometry Super Heroes Save Mathopolis:

Join Parallel Man and Perpendicular Man as they help Mayor Mathematics save Mathopolis by identifying parallel lines and line segments, as well as perpendicular lines and line segments in two-dimensional figures.

Type: Original Student Tutorial

Equivalent Fractions, Part 2: Patterns in the Multiplication Table:

Learn how to find equivalent fractions in a multiplication table in this interactive tutorial.

This is part 2 of a 2 part series. Click HERE to open Part 1.

Type: Original Student Tutorial

## Educational Games

Arithmetic Workout:

This tutorial will help you to brush up on your multiplication, division and factoring skills with this exciting game.

Type: Educational Game

Sigma Prime: A Prime Factorization Game:

This fun and engaging game will test your knowledge of whole numbers as prime or composite. As you shoot the asteroids with a particular factor, the asteroids will break down by that chosen factor. Keep shooting the correct factors to totally eliminate the asteroids. But be careful, shooting the wrong factor has consequences!

Type: Educational Game

The Factor Game:

Test your factors skills with this fun factor game. Take turns choosing numbers from the board and identifying its factors. Outscore your opponent by identifying factors and using strategy to limit their score. Play against the computer or a friend.

Type: Educational Game

Ice Ice Maybe: An Operations Estimation Game:

This fun and interactive game helps practice estimation skills, using various operations of choice, including addition, subtraction, multiplication, division, using decimals, fractions, and percents.

Various levels of difficulty make this game appropriate for multiple age and ability levels.

Multiplication/Division: The multiplication and addition of fractions and decimals.

Percentages: Identify the percentage of a whole number.

Fractions: Multiply and divde a whole number by a fraction, as well as apply properties of operations.

Type: Educational Game

Flower Power: An Ordering of Rational Numbers Game:

This is a fun and interactive game that helps students practice ordering rational numbers, including decimals, fractions, and percents. You are planting and harvesting flowers for cash. Allow the bee to pollinate, and you can multiply your crops and cash rewards!

Type: Educational Game

The Product Game:

This four-lesson unit develops students' fluency with multiplication facts and their understanding of the relationship between factors and multiples. While playing the Product Game and making their own game boards, students develop strategic thinking. They use Venn diagrams to represent the relationships between the factors or products of two numbers. In the fourth lesson they make connections and expand their learning from the first three lessons.

• Lesson 1: Playing the Product Game
• Students learn to play the Product Game to better understand factors and products, and the relationships between them.
• Lesson 2: Making Your Own Product Game
• Students make their own Product Game boards in this lesson, which they will find challenging. They will learn a lot as they experiment and make mistakes about what factors and products to include in their games.
• Lesson 3: Classifying Numbers
• Students will represent the relationships between factors or products of two numbers using Venn diagrams.
• Lesson 4: Connections and Extensions
• Students will make more connections and expand on what they have learned in the first three lessons, will explain the effects of different moves on the game board, and will "Guess My Number" using various clues.

Type: Educational Game

Fraction Quiz:

Test your fraction skills by answering questions on this site. This quiz asks you to simplify fractions, convert fractions to decimals and percentages, and answer algebra questions involving fractions. You can even choose difficulty level, question types, and time limit.

Type: Educational 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.

Type: Educational Game

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, or 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.

Type: Educational Game

Number Puzzles:

This virtual manipulative poses problems requiring the students to position numbers in a diagram, so all numbers in a line add up to a given value.

Type: Educational Game

Product Game (Factors and Multiples):

This interactive game for two players develops students' fluency with multiplication facts, their understanding of the relationship between factors and products, and their strategic thinking. On a board displaying all the factors of the numbers 1-9, players take turns moving markers on the factor list and claiming their products. The first player to get four in a row wins the game.

Type: Educational Game

Decimal and Fraction:

This interactive Flash applet has students match fractions with their equivalent one- or two-place decimals. Students have a chance to correct errors until all matches are made.

Type: Educational Game

## Educational Software / Tool

Arithmetic Quiz:

In this activity, students solve arithmetic problems involving whole numbers, integers, addition, subtraction, multiplication, and division. This activity allows students to track their progress in learning how to perform arithmetic on whole numbers and integers. 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.

Type: Educational Software / Tool

To regroup or not to regroup:

This task presents an incomplete problem and asks students to choose numbers to subtract (subtrahends) so that the resulting problem requires different types of regrouping. This way students have to recognize the pattern and not just follow a memorized algorithm--in other words, they have to think about what happens in the subtraction process when we regroup. This task is appropriate to use after students have learned the standard US algorithm.

Ordering 4-digit numbers:

It is common for students to compare multi-digit numbers just by comparing the first digit, then the second digit, and so on. This task includes three-digit numbers with large hundreds digits and four-digit numbers with small thousands digits so that students must infer the presence of a 0 in the thousands place in order to compare. It also includes numbers with strategically placed zeros and an unusual request to order them from greatest to least in addition to the more traditional least to greatest.

Lines of symmetry for triangles:

This activity provides students an opportunity to recognize these distinguishing features of the different types of triangles before the technical language has been introduced. For finding the lines of symmetry, cut-out models of the four triangles would be helpful so that the students can fold them to find the lines.

This task provides students a chance to experiment with reflections of the plane and their impact on specific types of quadrilaterals. It is both interesting and important that these types of quadrilaterals can be distinguished by their lines of symmetry.

Lines of symmetry for circles:

This is an instructional task that gives students a chance to reason about lines of symmetry and discover that a circle has an an infinite number of lines of symmetry. Even though the concept of an infinite number of lines is fairly abstract, fourth graders can understand infinity in an informal way.

Finding an unknown angle:

The purpose of this task is to give 4th grade students a problem involving an unknown quantity that has a clear visual representation. Students must understand that the four interior angles of a rectangle are all right angles (4.G.2) and that right angles have a measure of 90° and that angle measure is additive (4.MD.7).

Are these right?:

The purpose of this task is for students to measure angles and decide whether the triangles are right or not. Students should already understand concepts of angle measurement (4.MD.5) and know how to measure angles using a protractor (4.MD.6) before working on this task.

The purpose of this task is adding fractions being with a focus on tenths and hundredths. Each part of this task emphasizes a unique aspect of 4.NF.5.

Making 22 Seventeenths in Different Ways:

This task is a straightforward task related to adding fractions with the same denominator. The main purpose is to emphasize that there are many ways to decompose a fraction as a sum of fractions, similar to decompositions of whole numbers that students should have seen in earlier grades (see e.g. K.OA.3).

Listing fractions in increasing size:

The fractions for this task have been carefully chosen to encourage and reward different methods of comparison. The first solution judiciously uses each of the following strategies when appropriate: comparing to benchmark fractions, finding a common denominator, finding a common numerator. The second and third solution shown use only either common denominators or numerators. Teachers should encourage multiple approaches to solving the problem. This task is mostly intended for instructional purposes, although it has value as a formative assessment item as well.

How Many Tenths and Hundredths?:

The purpose of this task is for students to finish the equations to make true statements. Parts (a) and (b) have the same solution, which emphasizes that the order in which we add doesn't matter (because addition is commutative), while parts (c) and (d) emphasize that the position of a digit in a decimal number is critical. The student must really think to encode the quantity in positional notation. In parts (e), (f), and (g), the base-ten units in 14 hundredths are bundled in different ways. In part (e), "hundredths" are thought of as units: 14 things = 10 things + 4 things. Part (h) addresses the notion of equivalence between hundredths and tenths.

Fraction Equivalence:

Students may not articulate every detail, but the basic idea for a case like the one shown here is that when you have equivalent fractions, you have just cut the pieces that represent the fraction into more but smaller pieces. Explaining fraction equivalences at higher grades can be a bit more involved (e.g. 6/8=9/12), but it can always be framed as subdividing the same quantity in different ways.

Explaining Fraction Equivalence with Pictures:

The purpose of this task is to provide students with an opportunity to explain fraction equivalence through visual models in a particular example. Students will need more opportunities to think about fraction equivalence with different examples and models, but this task represents a good first step.

Expanded Fractions and Decimals:

The purpose of this task is for students to show they understand the connection between fraction and decimal notation by writing the same numbers both ways. Comparing and contrasting the two solutions shown below shows why decimal notation can be confusing. The first solution shows the briefest way to represent each number, and the second solution makes all the zeros explicit.

Dimes and Pennies:

The purpose of this task is to help students gain a better understanding of fractions through the use of dimes and pennies.

Comparing two different pizzas:

The focus of this task is on understanding that fractions, in an explicit context, are fractions of a specific whole. In this this problem there are three different wholes: the medium pizza, the large pizza, and the two pizzas taken together. This task is best suited for instruction. Students can practice explaining their reasoning to each other in pairs or as part of a whole group discussion.

Comparing Sums of Unit Fractions:

The purpose of this task is to help develop students' understanding of addition of fractions; it is intended as an instructional task. Notice that students are not asked to find the sum because in grade 4, students are limited to computing sums of fractions with the same denominator. Rather, they need to apply a firm understanding of unit fractions (fractions with one in the numerator) and reason about their relative size.

Writing a Mixed Number as an Equivalent Fraction:

The purpose of this task is to help students understand and articulate the reasons for the steps in the usual algorithm for converting a mixed number into an equivalent fraction. Step two shows that the algorithm is merely a shortcut for finding a common denominator between two fractions. This concept is an important precursor to adding mixed numbers and fractions with like denominators and as such, step two should be a point of emphasis. This task is appropriate for either instruction or formative assessment.

Using Place Value:

Each part of this task highlights a slightly different aspect of place value as it relates to decimal notation. More than simply being comfortable with decimal notation, the point is for students to be able to move fluidly between and among the different ways that a single value can be represented and to understand the relative size of the numbers in each place.

Using Benchmarks to Compare Fractions:

This task is intended primarily for instruction. The goal is to provide examples for comparing two fractions, 1/5 and 2/7 in this case, by finding a benchmark fraction which lies in between the two. In Melissa's example, she chooses 1/4 as being larger than 1/5 and smaller than 2/7.

Sugar in six cans of soda:

This task provides a familiar context allowing students to visualize multiplication of a fraction by a whole number. This task could form part of a very rich activity which includes studying soda can labels.

Peaches:

This task provides a context where it is appropriate for students to subtract fractions with a common denominator; it could be used for either assessment or instructional purposes. For this particular task, teachers should anticipate two types of solution approaches: one where students subtract the whole numbers and the fractions separately and one where students convert the mixed numbers to improper fractions and then proceed to subtract.

Money in the piggy bank:

This task is designed to help students focus on the whole that a fraction refers to. It provides a context where there are two natural ways to view the coins: As equal parts of the set of coins in the piggy bank, and As money so each coin is assigned its monetary value. The important thing to realize here is that two different fractions can describe the same situation depending on what you choose to be the whole.

Comparing Growth, Variation 2:

The purpose of this task is to assess students' understanding of multiplicative and additive reasoning. We would hope that students would be able to see identify that Student A is just looking at how many feet are being added on, while the Student B is comparing how much the snakes grew in comparison to how long they were to begin with.

Comparing Growth, Variation 1:

The purpose of this task is to foster a classroom discussion that will highlight the difference between multiplicative and additive reasoning. Some students will argue that they grew the same amount (an example of "additive thinking"). Students who are studying multiplicative comparison problems might argue that Jewel grew more since it grew more with respect to its original length (an example of "multiplicative thinking").

Carnival Tickets:

The purpose of this task is for students to solve multi-step problems in a context involving a concept that supports financial literacy, namely inflation. Inflation is a sustained increase in the average price level. In this task, students can see that if the price level increases and people's incomes do not increase, they aren't able to purchase as many goods and services; in other words, their purchasing power decreases.

Double Plus One:

The purpose of this task is to help students gain a better understanding of patterns. This task is meant to be used in an instructional setting.

Comparing Money Raised:

The purpose of this task is to give students a better understanding of using four operations to solve problems.

Karl's Garden:

The purpose of the task is for students to solve a multi-step multiplication problem in a context that involves area. In addition, the numbers were chosen to determine if students have a common misconception related to multiplication. Since addition is both commutative and associative, we can reorder or regroup addends any way we like.

Identifying Multiples:

The goal of this task is to work on finding multiples of some whole numbers. After shading in the multiples of 2, 3, and 4 on the table, students will see a key difference. In the fourth grade, the emphasis here should be on seeing that there is a visual difference in patterns and that this difference is related to whether and how numbers factor. This task could be used to introduce the notion of a prime number, or if students are already familiar with primes and composites, this is a good task to reinforce these ideas.

Plastic Building Blocks:

The purpose of this task is to have students add mixed numbers with like denominators. This task illustrates the different kinds of solution approaches students might take to such a task. Two general approaches should be anticipated: one where students calculate exactly how many buckets of blocks the boys have to determine an answer, and one where students compare the given numbers to benchmark numbers.

Running Laps:

The purpose of this task is for students to compare two fractions that arise in a context. Because the fractions are equal, students need to be able to explain how they know that. Some students might stop at the second-to-last picture and note that it looks like they ran the same distance, but the explanation is not yet complete at that point.

## Tutorials

Number Pattern Example: Growing Sequence:

In this Khan Academy tutorial video a table is used to track a growing sequence of design.

Type: Tutorial

Recognizing Prime and Composite Numbers:

This Khan Academy tutorial video reviews how to determine if a number is prime or composite.

Type: Tutorial

Regrouping Numbers: 4,500 = 3 thousands + ? hundreds:

In this tutorial, you will look at regrouping a number by different place values.

Type: Tutorial

Categorizing Geometric Shapes: Practice Examples:

This Khan Academy tutorial video presents examples and explanations for categorizations of perpendicular sides and right, obtuse, and acute triangles.

Type: Tutorial

Examples of Triangle Categories:

In this Khan Academy tutorial video triangles are categorized by angles or side lengths of a specified size.

Type: Tutorial

Adding Two Fractions with Denominators 10 and 100:

The Khan Academy tutorial video presents a visual fraction model for adding 3/10 + 7/100 .

Type: Tutorial

Angles and arcs:

in this tutorial, students will learn about central angles and arcs of a circle.

Type: Tutorial

Solving for a Part of a Decomposed Angle:

This Khan Academy tutorial video presents the strategy for finding the measure of one of two adjacent angles, when the sum of both and measure of one are known.

Type: Tutorial

Parallel and Perpendicular Lines Introduced:

This Khan Academy tutorial video defines and illustrates parallel and perpendicular lines.

Type: Tutorial

Recognizing Acute, Right, and Obtuse Angles:

This Khan Academy tutorial video identifies acute, right, and obtuse angles and justifies each identification.

Type: Tutorial

Angle Measurements and Circle Arcs:

This Khan Academy tutorial video demonstrates the relationship between the measurement of an angle and the arc of a circle.

Type: Tutorial

Angle Basics:

This Khan Academy tutorial video presents how an angle is formed and labeled.

Type: Tutorial

Units of Measurement Word Problem: Making Change:

This Khan Academy tutorial video presents a strategy for computing the amount of change to be received after making a purchase.

Type: Tutorial

Time Word Problem: When to Leave to Get Home on Time:

In this Khan Academy tutorial video Chris is told to be home by 6:15. You know the number of minutes it takes him to get home. What time should he leave?

Type: Tutorial

U.S. Customary Units: Distance:

This Khan Academy tutorial video presents conventional examples that use specific customary units

Type: Tutorial

How to convert kilograms to milligrams and tons to ounces:

In this tutorial video from Khan Academy, explore the differences and similarities involved when converting between measurements in the metric and customary systems.

Type: Tutorial

How to convert gallons to quarts, quarts to pints, pints to cups, and cups to ounces:

In this video tutorial from Khan Academy, explore converting between gallons, quarts, pints, cups, and fluid ounces.

Type: Tutorial

How to convert kilometers to meters and meters centimeters:

In this video tutorial from Khan Academy, explore conversion within metric units of length, such as: kilometers, meters and centimeters.

Type: Tutorial

How to convert hours to minutes and minutes to seconds:

In this video tutorial from Khan Academy, explore conversion of units of time between hours, minutes and seconds.

Type: Tutorial

U.S. customary units: fluid volume:

In this video tutorial from Khan Academy, explore U.S. customary units of fluid volume (teaspoon, tablespoon, fluid ounce, cup, pint, quart, and gallon).

Type: Tutorial

U.S. customary units: weight:

In this video tutorial from Khan Academy, explore pounds, ounces and tons.

Type: Tutorial

Intro to lines, line segments, and rays:

In this video tutorial from Khan Academy, let's get familiar with the difference between lines, line segments, and rays.

Type: Tutorial

Area and Perimeter Word Problem: Table Dimensions:

This Khan Academy tutorial video presents a step-by-step solution for finding the length and width of a table when given its area and perimeter.

Type: Tutorial

Comparing Two Decimals with a Visual Model:

In this Khan Academy tutorial video two decimals are compared using grid diagrams.

Type: Tutorial

Decimals as Words:

In this Khan Academy video decimals are written and spoken in words.

Type: Tutorial

Decimals and Fractions from Grid and Number-Line Representations:

The Khan Academy video uses grid diagrams and number-line representations to say and write equivalent decimals and fractions.

Type: Tutorial

Grid Representations of Decimals:

The Khan Academy video illustrates how to determine and write the decimal represented by shaded grids.

Type: Tutorial

Visually Converting from Tenths to Hundredths:

In this Khan Academy video a fraction is converted from tenths to hundredths using grid diagrams.

Type: Tutorial

Multiplying a Fraction by a Whole Number:

In this Khan Academy video visual fraction models are used to represent the expressions and the products.

Type: Tutorial

What Fraction of Spider Eyes are Looking at Me?:

This Khan Academy video uses authentic pictures to present addition of two fractions with common denominators.

Type: Tutorial

Figuring Out How Much of a Pizza is Left:

This Khan Academy video solves two word problems using visual fraction models.

Type: Tutorial

Equivalent Fractions: Visual Models:

This Khan Academy video illustrates that fraction a/b is equivalent to fraction (a x n)/(b x n).

Type: Tutorial

Division: The importance of place value:

In this video tutorial from Khan Academy, learn about the importance of place value when dividing. Being able to perform the standard algorithm is the end goal, but it helps to understand how and why this process works.

Type: Tutorial

Division: Intro to remainders:

In this video tutorial from Khan Academy, you will get an introduction to the meaning of remainders.

Type: Tutorial

Multiplying: how to use the area model:

In this video tutorial from Khan Academy, view a demonstration of how to set-up an area model for multiplying a two-digit number by a two-digit number on graph or grid paper.

Type: Tutorial

Multiplying: 2-digit number times a 2-digit number (area model):

In this tutorial video from Khan Academy, view an example of how to multiply a two-digit number by a two-digit number using the area model. Make a connection between the area model and what the standard algorithm represents.

Type: Tutorial

Multiplying: 2-digit number times 2-digit number (using distributive property):

In this video tutorial from Khan Academy, view an example and a description of how the distributive property can be used to multiply a two-digit number by a two-digit number.

Type: Tutorial

Multiplying: 4-digits times 1-digit (using expanded form):

In this Khan Academy video tutorial, view an example of multiplying a 4-digit number by a 1-digit number by expanding the 4-digit number and multiplying by each digit individually. This video will help to build an understanding of the standard algorithm.

Type: Tutorial

Multiplying: 2-digit number times a 2-digit number (standard algorithm):

In this tutorial video from Khan Academy, view an example of how to multiply a 2-digit number by another 2-digit number. Be sure to stick around for the second example! The key is understanding the value of each digit!

Type: Tutorial

Multiplying: 3 digits times 1 digit (standard algorithm):

In this video tutorial from Khan Academy, view an example of how to solve a problem in which a 3-digit number is being multiplied by a 1-digit number using the standard algorithm.

Type: Tutorial

Multiplying: 2 digits times 1 digit (standard algorithm):

In this video tutorial from Khan Academy, view an example of how to solve a multiplication problem with a two-digit number multiplied by a one-digit number using the standard algorithm.

Type: Tutorial

Subtracting: regrouping twice:

In this video tutorial from Khan Academy, learn how to subtract in situations that require regrouping twice using the expanded forms of numbers, as well as the standard algorithm.

Type: Tutorial

Comparing Areas of Plots of Land:

Find area of two rectangles to solve a word problem.

Type: Tutorial

Comparing areas and perimeters of rectangles:

In this tutorial video from Khan Academy, explore the relationship between area and perimeter. For example, if you know that area and the length, can you find the perimeter?

Type: Tutorial

Mental technique for subtraction without regrouping:

In this Khan Academy video tutorial, consider an alternate algorithm for subtracting multi-digit numbers mentally. This video is best for students that are already comfortable with using regrouping to subtract using the standard algorithm.

Type: Tutorial

How to use an abacus (to represent multi-digit numbers):

In this tutorial video from Khan Academy, learn to use an abacus to represent multi-digit numbers. This video will explain how the beads on an abacus can each represent ten times the value of the bead to its right.

Type: Tutorial

Angle Basics:

This video discusses the definition of an angle and how to name an angle.

Type: Tutorial

Lines, Line segments and Rays:

This video discusses about the difference between lines, line segments and rays.

Type: Tutorial

Introduction to Decimals:

This tutorial for student audiences will provide a basic introduction to decimals. The tutorial presents a decimal as another way to represent a fraction. Students will be able to navigate the teaching portion of the tutorial at their own pace and test their understanding after each step of the lesson with a "Try This" section. The "Try This" section will monitor students answers and self-check by a right answer gaining an orange circle and a wrong answer graying out. Some "Try This" sections will read the decimal to the students as well.

Type: Tutorial

Comparing Fractions:

This tutorial for student audiences will assist learners with a further understanding that fractions are a way of showing part of a whole. Yet some fractions are larger than others. So this tutorial will help to refresh the understanding for the comparison of fractions. Students will be able to navigate the teaching portion of the tutorial at their own pace and test their understanding after each step of the lesson with a "Try This" section. The "Try This" section will monitor students answers and self-check by a right answer turning orange and a wrong answer dissolving.

Type: Tutorial

## Virtual Manipulatives

Perimeter Explorer:

This activity allows the user to test his or her skill at calculating the perimeter of a random shape. The user is given a random shape and asked to enter a value for the perimeter. The applet then informs the user whether or not the value is correct. The user may continue trying until he or she gets the correct answer.

This activity would work well in mixed ability groups of two or three for about 25 minutes if you use the exploration questions, and 10-15 minutes otherwise.

Type: Virtual Manipulative

Fractions - Equivalent:

Students use this virtual manipulative to visualize and name equivalent fractions. The applet presents a shape divided into equal parts, with some parts shaded. Students change the number of divisions of the shape to visualize equivalent fractions, name the fractions, and check their answers. Instructions for using the applet and teaching ideas for parents/teachers are available through the links at the top of the page.

Type: Virtual Manipulative

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.

Type: Virtual Manipulative

This virtual manipulative will help the students in understanding parts in relation to a whole group. Students will also learn to distinguish between characteristics of shapes, create and describe patterns in shapes, and identify lines of symmetry and create symmetrical patterns.

Type: Virtual Manipulative

Time - What Time Will It Be?:

This clock manipulative allows the user control of the hands of the clock and tell the elapsed time on both digital and analog clocks.

Type: Virtual Manipulative

Geoboard:

This extremely versatile manipulative can be used by learners of different grades. At early grades, this manipulative will help the students recognize, name, build, draw, and compare two-dimensional shapes. As they progress students can classify and understand relationships among types of two-dimensional objects using their defining properties. The application computes perimeter and area allowing students to explore patterns as dimensions change.

Type: Virtual Manipulative

Exploring Fractions:

Match shapes and numbers to earn stars in this fractions game.

• Match fractions using numbers and pictures
• make the same fractions using different numbers
• Match fractions in different picture patterns
• Compare fractions using numbers and patterns

Type: Virtual Manipulative

Fraction Finder- Number Line:

In this activity, you will graphically determine the value of two given fractions represented as points on a number line. You will then graphically find a fraction whose value is between the two given fractions and determine its value.

Type: Virtual Manipulative

This interactive Java applet helps users develop place value concepts involving addition of numbers with 1, 2, 3, or 4 digits. The virtual blocks are manipulated to model regrouping in order to solve addition problems posed by the applet. Users also may create and solve their own problems with whole numbers or up to 3 decimal places. The default of base 10 may be changed to any of the bases 2, 3, 4, or 5.

Type: Virtual Manipulative

Place Value Number Line:

With this interactive applet students can develop their understanding of place value and the relative position and magnitude of numbers in the base-10 number system. Users drag given numbers to their position on a number line, zooming in and out to increase precision. They can select from six levels of difficulty (Tens, Hundreds, Thousands, Millions, Billions, Decimals) and from 1 to 3 dots. There are three modes: Explore, Practice, Test.

Type: Virtual Manipulative

Base Blocks:

This virtual manipulative provides base blocks that consist of individual "units," "longs," "flats," and "blocks" (ten of each set for base 10). They can be used to show place value for numbers and to increase understanding of addition and subtraction. Also allows for representation of decimal numbers.

Type: Virtual Manipulative

This virtual manipulative offers pattern blocks of different shapes and in different colors which can be used for a range of activities. In addition, the resource offers descriptions of six example activities in the "Activities" section, and a sample lesson plan in the section "Parent/Teacher."

Type: Virtual Manipulative

Fraction Game:

This virtual manipulative allows individual students to work with fraction relationships. (There is also a link to a two-player version.)

Type: Virtual Manipulative

Shape Tool:

This virtual manipulative allows you to create, color, enlarge, shrink, rotate, reflect, slice, and glue geometric shapes, such as: squares, triangles, rhombi, trapezoids and hexagons.

Type: Virtual Manipulative

Follow the Pattern. Which shapes go next?:

Students select the shape that goes next in the pattern and place it in the row, then identify the overall pattern.

Type: Virtual Manipulative

Circle 99:

This interactive Java applet develops logical reasoning skills and fluency with two-digit addition. The game board consists of a ring of seven overlapping circles, each with 3 regions. Numbers are provided in five regions. The player drags 9 other numbers to the empty regions so that the sum of the numbers within each circle is 99.

Type: Virtual Manipulative

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this course.