## Course Standards

## General Course Information and Notes

### Version Description

In this course, instructional time should focus on three critical areas: (1) developing fluency with addition and subtraction of fractions, and developing understanding of the multiplication of fractions and of division of fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit fractions); (2) extending division to 2-digit divisors, integrating decimal fractions into the place value system and developing understanding of operations with decimals to hundredths, and developing fluency with whole number and decimal operations; and (3) developing understanding of volume.

(1) 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. Students apply their understanding of fractions and fraction models to represent the addition and subtraction of fractions with unlike denominators as equivalent calculations with like denominators. They develop fluency in calculating sums and differences of fractions, and make reasonable estimates of them. Students also use the meaning of fractions, of multiplication and division, and the relationship between multiplication and division to understand and explain why the procedures for multiplying and dividing fractions make sense. (Note: this is limited to the case of dividing unit fractions by whole numbers and whole numbers by unit fractions.)

(2) Students generalize their understanding of place value to 1,000,000, understanding the relative sizes of numbers in each place. 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. Students develop understanding of why division procedures work based on the meaning of base-ten numerals and properties of operations. They finalize fluency with multi-digit addition, subtraction, multiplication, and division. They apply their understandings of models for decimals, decimal notation, and properties of operations to add and subtract decimals to hundredths. They develop fluency in these computations, and make reasonable estimates of their results. Students use the relationship between decimals and fractions, as well as the relationship between finite decimals and whole numbers (i.e., a finite decimal multiplied by an appropriate power of 10 is a whole number), to understand and explain why the procedures for multiplying and dividing finite decimals make sense. They compute products and quotients of decimals to hundredths efficiently and accurately.

(3) Students recognize volume as an attribute of three-dimensional space. They understand that volume can be measured by finding the total number of same-size units of volume required to fill the space without gaps or overlaps. They understand that a 1-unit by 1-unit by 1-unit cube is the standard unit for measuring volume. They select appropriate units, strategies, and tools for solving problems that involve estimating and measuring volume. They decompose three-dimensional shapes and find volumes of right rectangular prisms by viewing them as decomposed into layers of arrays of cubes. They measure necessary attributes of shapes in order to determine volumes to solve real world and mathematical problems.

**Honors and Advanced Level Course Note: **Advanced courses require a greater demand on students through increased academic rigor. Academic rigor is obtained through the application, analysis, evaluation, and creation of complex ideas that are often abstract and multi-faceted. Students are challenged to think and collaborate critically on the content they are learning. Honors level rigor will be achieved by increasing text complexity through text selection, focus on high-level qualitative measures, and complexity of task. Instruction will be structured to give students a deeper understanding of conceptual themes and organization within and across disciplines. Academic rigor is more than simply assigning to students a greater quantity of work.

**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: http://www.cpalms.org/uploads/docs/standards/eld/MA.pdf.

For additional information on the development and implementation of the ELD standards, please contact the Bureau of Student Achievement through Language Acquisition at sala@fldoe.org.

### General Information

**Course Number:**5012065

**Course Path:**

**Abbreviated Title:**ACCEL MATH GRADE 4

**Course Length:**Year (Y)

**Course Attributes:**

- Honors
- Class Size Core Required

**Course Type:**Core Academic Course

**Course Level:**3

**Course Status:**Course Approved

**Grade Level(s):**4

## Educator Certifications

## Student Resources

## Original Student Tutorials

Apply your understanding of the defining attributes of all 2-dimensional figures covered in this series to classify their relationships using Euler and Venn Diagrams.

This part 8 in a 8-part series. Click below to explore the other tutorials in the series.

Part 1: "Figuring Out" 2D Figures

Part 2: Exploring Relationships with Venn & Euler Diagrams

Part 3: Classifying Triangles by Angles Using Euler Diagrams

Part 4: Classifying Triangles by Sides & Angles Using Venn and Euler Diagrams

Type: Original Student Tutorial

By the end of this tutorial you should be able to identify examples of quadrilaterals and their defining attributes to classify them using diagrams. We will focus on kites and other quadrilaterals in this tutorial.

This part 7 in a 8-part series. Click below to explore the other tutorials in the series.

Part 1: "Figuring Out" 2D Figures

Part 2: Exploring Relationships with Venn & Euler Diagrams

Part 3: Classifying Triangles by Angles Using Euler Diagrams

Part 4: Classifying Triangles by Sides & Angles Using Venn and Euler Diagrams

Part 5: Quadrilaterals

Part 6: Trapezoids

Part 8: (Coming Soon)

Type: Original Student Tutorial

Read word problems and use number lines with benchmarks to solve multi-step problems involving addition and subtraction of fractions with unlike denominators. In this tutorial, you will help Daisy and Angie paint pictures using fractions.

Type: Original Student Tutorial

Explore the defining attributes of trapezoids--a special type of quadrilateral--and classify them using diagrams in this interactive tutorial. You'll also learn how two different definitions for a trapezoid can change affect classifications of quadrilaterals.

This part 6 in a 6-part series. Click below to explore the other tutorials in the series.

Part 1: "Figuring Out" 2D Figures

Part 2: Exploring Relationships with Venn & Euler Diagrams

Part 3: Classifying Triangles by Angles Using Euler Diagrams

Part 4: Classifying Triangles by Sides & Angles Using Venn and Euler Diagrams

Type: Original Student Tutorial

Learn how to classify quadrilaterals--including parallelograms, rectangles, rhombi, and squares--based on their defining attributes using diagrams in this interactive tutorial.

This part 5 in a 6-part series. Click below to explore the other tutorials in the series.

**Part 1: "Figuring Out" 2D Figures****Part 2: Exploring Relationships with Venn & Euler Diagrams****Part 3: Classifying Triangles by Angles Using Euler Diagrams****Part 4: Classifying Triangles by Sides & Angles Using Venn and Euler Diagrams**- Part 6: (Coming Soon)

Type: Original Student Tutorial

Explore rounding decimals through the thousandths place as you help Tyrese the Turtle train to race the hare in this interactive tutorial.

Type: Original Student Tutorial

Learn more about division with larger numbers in this aquarium-themed, interactive tutorial.

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

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

Type: Original Student Tutorial

Learn how to convert time from seconds to minutes, minutes to hours, and hours to days. In this interactive tutorial, you will also practice converting time to fractional amounts.

Type: Original Student Tutorial

Help out at the fishing tournament while comparing decimals through the thousandths place in this interactive tutorial.

Type: Original Student Tutorial

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.

**Part 1: Visual Models and Multiplying Fractions****Part 2: Multiplying Fractions****Part 3 Using Models to Multiply a Fraction by a Whole Number**

Type: Original Student Tutorial

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

Type: Original Student Tutorial

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.

**Part 1: Visual Models and Multiplying Fractions****Part 2: Multiplying Fractions****Part 4: Multiplying a Fraction by a Whole Number - Standard Algorithm**

Type: Original Student Tutorial

Help Buffy the Baker multiply fractions less than one by relating the standard algorithm to visual models as he runs his bakery in this interactive tutorial.

This is part 2 of a 4-part series. Click below to open other tutorials in the series.

Type: Original Student Tutorial

Explore how to add fractions less than one with unlike denominators in this magical, interactive tutorial.

Type: Original Student Tutorial

Help Buffy the Baker use visual models to multiply fractions less than one as he runs his bakery in this interactive tutorial.

This is part 1 of a 4-part series. Click below to open other tutorials in the series.

**Part 2: Multiplying Fractions****Part 3 Using Models to Multiply a Fraction by a Whole Number****Part 4: Multiplying a Fraction by a Whole Number - Standard Algorithm**

Type: Original Student Tutorial

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

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

Help build a Grasshopper Hut for Gus by creating line plots and answering questions about the line plots in this interactive tutorial.

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

Help solve the problem of shipping video games and accessories to customers by calculating the volume of the containers needed in this interactive tutorial.

Type: Original Student Tutorial

Build on your previous knowledge of area and learn how to calculate volume in cubic units with this interactive tutorial.

Type: Original Student Tutorial

Explore base 10 and exponents in this baseball-themed, interactive tutorial.

Type: Original Student Tutorial

Help your town build a dog park by multiplying whole numbers by decimals to the tenths place in this interactive tutorial.

Type: Original Student Tutorial

Learn to subtract decimals to the hundredths place using place-value models and written expressions as you fix the topsy-turvy playground in this interactive tutorial.

Type: Original Student Tutorial

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

Help these aliens clean up the Sweet Treats Factory by learning to add decimals in this interactive mathematics tutorial.

Type: Original Student Tutorial

Learn how to show relationships represented in Venn & Euler Diagrams as you complete this interactive geometry tutorial.

This is part two of four. Click below to open the other tutorials in the series.

**Part 1: "Figuring Out" 2D Figures - Part 1**- Part 2 Exploring Relationships with Venn & Euler Diagrams
**Part 3: Classifying Triangles by Angles using Euler Diagrams****Part 4: Classifying Triangles by Sides and Angles using Venn and Euler Diagrams**

Type: Original Student Tutorial

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

Learn how triangles can be sorted and classified using side lengths and angle measures in this interactive tutorial.

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

**Part 1: "Figuring Out" 2D Figures - Part 1****Part 2 Exploring Relationships with Venn & Euler Diagrams****Part 3: Classifying Triangles by Angles using Euler Diagrams**- Part 4: Classifying Triangles by Sides and Angles using Venn and Euler Diagrams

Type: Original Student Tutorial

Try to escape from this room using multiplication as scaling in this interactive tutorial.

Type: Original Student Tutorial

Learn to classify triangles and use Euler diagrams to show relationships, in this interactive tutorial.

This is part-three of four. Click below to open the other tutorials in the series.

**Part 1: "Figuring Out" 2D Figures - Part 1****Part 2 Exploring Relationships with Venn & Euler Diagrams**- Part 3: Classifying Triangles by Angles using Euler Diagrams
**Part 4: Classifying Triangles by Sides and Angles using Venn and Euler Diagrams**

Type: Original Student Tutorial

Explore 2D (two-dimensional) figures and see how every 2D figure possesses unique attributes in this interactive tutorial.

This is part one of four. Click below to open the other tutorials in the series.

- Part 1: "Figuring Out" 2D Figures - Part 1
**Part 2 Exploring Relationships with Venn & Euler Diagrams****Part 3: Classifying Triangles by Angles using Euler Diagrams****Part 4: Classifying Triangles by Sides and Angles using Venn and Euler Diagrams**

Type: Original Student Tutorial

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

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

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

Practice plotting coordinates, in Quadrant I, using ordered pairs in this interactive tutorial for students.

Type: Original Student Tutorial

Learn about the basics of the coordinate plane, focus on Quadrant I and see why the coordinate plane is useful in everyday life in this interactive tutorial.

Type: Original Student Tutorial

Learn how the standard algorithm for multiplying numbers works and practice your skills in this interactive tutorial.

Type: Original Student Tutorial

Learn how to use repeat loops in this interactive tutorial. Repeat loops iterate though a list of instructions based on a desired number of times. Combined with variables, condition statements, if statements, and repeat loops we practice using order of operations to code.

Click below to check out the other tutorials in the series.

Type: Original Student Tutorial

Learn how to perform instructions using an if statement and explore relational operators (less than, greater than, equal and not equal to) and how they are used to compare to values in this interactive tutorial.

Type: Original Student Tutorial

Construct efficient lines of code using condition- and if-statements to solve equations as you complete this interactive tutorial. You'll also review the order of operations in expressions. This is part 2 of a 4-part series on coding.

Type: Original Student Tutorial

Learn how to define, declare and initialize variables as you start the journey to "bee" a coder in this interactive tutorial. Variables are structures used by computer programs to store information. You'll use your math skills to represent a fraction as a decimal to be stored in a variable. This is part 1 of a series of 4 in learning how to code.

Type: Original Student Tutorial

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

Type: Original Student Tutorial

Learn how to write mathematical expressions while making faces in this interactive tutorial!

Type: Original Student Tutorial

Learn to interpret data presented on a line plot and use operations on fractions to solve problems involving information presented in line plots as you complete this beach-themed, interactive tutorial.

Type: Original Student Tutorial

By the end of this tutorial you’ll know how to convert among different-sized customary units of weight, length, capacity, and units of time.

Type: Original Student Tutorial

Learn to evaluate expressions that have all four operations (multiplication, division, addition, and subtraction) and parentheses as you settle debates in this interactive tutorial.

Type: Original Student Tutorial

By the end of this tutorial, you will be able to read and write decimals to the thousandths using base-ten numerals, number names, and expanded form.

Type: Original Student Tutorial

Take flight as you learn to recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right with this interactive tutorial.

Type: Original Student Tutorial

Learn to identify a fraction as division of the numerator by the denominator using fraction models in this interactive tutorial.

Type: Original Student Tutorial

Learn how to accurately plot coordinates on a plane.

Type: Original Student Tutorial

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

Type: Original Student Tutorial

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

Type: Original Student Tutorial

Demonstrate how a rectangular prism can be carefully filled without gaps or overlaps using the same size unit cubes and then use this model to determine its volume.

Type: Original Student Tutorial

## Educational Games

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.

*Addition/**Subtraction:* The addition and subtraction of whole numbers, the addition and subtraction of decimals.

*Multiplication/Division: *The multiplication and addition of whole numbers.

*Percentages: *Identify the percentage of a whole number.

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

Type: Educational 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

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

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

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

This interactive applet gives students practice in making change in U.S. dollars and in four other currencies. Students are presented with a purchase amount and the amount paid, and they must enter the quantity of each denomination that make up the correct change. Students are rewarded for correct answers and are shown the correct change if they err. There are four levels of difficulty, ranging from amounts less than a dollar to amounts over $100.

Type: Educational Game

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

In this activity, students enter coordinates to make a path to get to a target destination while avoiding mines. This activity allows students to explore Cartesian coordinates and the Cartesian coordinate plane. 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

## Educational Software / Tool

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

## Problem-Solving Tasks

Students are asked to determine the number of unit cubes needed to construct cubes with given dimensions.

Type: Problem-Solving Task

Students are asked to find the volume of water in a tank that is 3/4 of the way full.

Type: Problem-Solving Task

Students are asked to find the height of a rectangular prism when given the length, width and volume.

Type: Problem-Solving Task

Students are asked to apply knowledge of volume of rectangular prisms to find the volume of an irregularly shaped object using the principle of displacement.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving 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.

Type: Problem-Solving Task

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).

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

This task asks students to exercise both of these complementary skills, writing an expression in part (a) and interpreting a given expression in (b). The numbers given in the problem are deliberately large and "ugly" to discourage students from calculating Eric's and Leila's scores. The focus of this problem is not on numerical answers, but instead on building and interpreting expressions that could be entered in a calculator or communicated to another student unfamiliar with the context.

Type: Problem-Solving Task

This purpose of this task is to help students understand what happens when you scale the dimensions of a right rectangular solid. This task provides an opportunity to compare the relative volumes of boxes in order to calculate the mass of clay required to fill them. These relative volumes can be calculated geometrically, filling the larger box with smaller boxes, or arithmetically using the given dimensions.

Type: Problem-Solving Task

When a division problem involving whole numbers does not result in a whole number quotient, it is important for students to be able to decide whether the context requires the result to be reported as a whole number with remainder (as with Part (b)) or a mixed number/decimal (as with Part (c)). Part (a) presents two variations on a context that require these two different responses to highlight the distinction between them.

Type: Problem-Solving Task

The purpose of this task is to help students see the connection between a÷b and ab in a particular concrete example. The relationship between the division problem 3÷8 and the fraction 3/8 is actually very subtle. This task is probably best suited for instruction or formative assessment.

Type: Problem-Solving Task

This task provides a context for performing division of a whole number by a unit fraction. This problem is a "How many groups?'' example of division: the "groups'' in this case are the servings of oatmeal and the question is asking how many servings (or groups) there are in the package.

Type: Problem-Solving Task

The purpose of this task is to provide students with a situation in which it is natural for them to divide a unit fraction by a non-zero whole number. Determining the amount of paint that Kulani needs for each wall illustrates an understanding of the meaning of dividing a unit fraction by a non-zero whole number.

Type: Problem-Solving Task

The purpose of this task is for students to find the answer to a question in context that can be represented by fraction multiplication. This task is appropriate for either instruction or assessment depending on how it is used and where students are in their understanding of fraction multiplication.

Type: Problem-Solving Task

The purpose of this task is to present students with a situation in which they need to divide a whole number by a unit fraction in order to find a solution. Calculating the number of origami stars that Avery and Megan can make illustrates students' understanding of the process of dividing a whole number by a unit fraction.

Type: Problem-Solving Task

The purpose of this task is to help students realize there are different ways to add mixed numbers and is most appropriate for use in an instructional setting. The two primary ways one can expect students to add are converting the mixed numbers to fractions greater than 1 or adding the whole numbers and fractional parts separately. It is good for students to develop a sense of which approach would be better in a particular context.

Type: Problem-Solving Task

The purpose of this instructional task is to motivate a discussion about adding fractions and the meaning of the common denominator. The different parts of the task have students moving back and forth between the abstract representation of the fractions and the meaning of the fractions in the context.

Type: Problem-Solving Task

This tasks lends itself very well to multiple solution methods. Students may learn a lot by comparing different methods. Students who are already comfortable with fraction multiplication can go straight to the numeric solutions given below. Students who are still unsure of the meanings of these operations can draw pictures or diagrams.

Type: Problem-Solving Task

The purpose of this task is to present students with a situation where it is natural to add fractions with unlike denominators; it can be used for either assessment or instructional purposes. Teachers should anticipate two types of solutions: one where students calculate the distance Alex ran to determine an answer, and one where students compare the two parts of his run to benchmark fractions.

Type: Problem-Solving Task

The purpose of this task is to familiarize students with multiplying fractions with real-world questions.

Type: Problem-Solving Task

The purpose of this task is to help students see that 4×(9+2) is four times as big as (9+2). Though this task may seem very simple, it provides students and teachers with a very useful visual for interpreting an expression without evaluating it because they can see for themselves that 4×(9+2) is four times as big as (9+2).

Type: Problem-Solving Task

The purpose of this task is to have students add fractions with unlike denominators and divide a unit fraction by a whole number. This accessible real-life context provides students with an opportunity to apply their understanding of addition as joining two separate quantities.

Type: Problem-Solving Task

Since tasks such as this will be among the first that students see, solutions which involve (sub)dividing a quantity into equal parts in order to find a fraction of the quantity should be emphasized. In particular, such solutions should be introduced if students do not generate them on their own. Students benefit from reasoning through the solution to such word problems before they are told that they can be solved by multiplying the fractions; this helps them develop meaning for fraction multiplication.

Type: Problem-Solving Task

The two solutions reflect different competencies described in 5.NF.5. The first solution uses the idea that multiplying by a fraction less than 1 results in a smaller value. The second actually uses the meaning of multiplying by 89 to explain why multiplying by that fraction will result in a smaller value.

Type: Problem-Solving Task

This is a good task to work with kids to try to explain their thinking clearly and precisely, although teachers should be willing to work with many different ways of explaining the relationship between the magnitude of the factors and the magnitude of the product.

Type: Problem-Solving Task

The purpose of this task is to generate a classroom discussion that helps students synthesize what they have learned about multiplication in previous grades. It builds on 3.OA.5 Apply properties of operations as strategies to multiply and divide and 4.OA.1 Interpret a multiplication equation as a comparison.

Type: Problem-Solving Task

This problem allows student to see words that can describe the expression from part (c) of "5.OA Watch out for Parentheses." Additionally , the words (add, sum) and (product, multiply) are all strategically used so that the student can see that these words have related meanings.

Type: Problem-Solving Task

This problem asks the student to evaluate six numerical expressions that contain the same integers and operations yet have differing results due to placement of parentheses. This type of problem helps students to see structure in numerical expressions. In later grades they will be working with similar ideas in the context of seeing and using structure in algebraic expressions.

Type: Problem-Solving Task

This task requires division of multi-digit numbers in the context of changing units and so illustrates 5.NBT.6 and 5.MD.1. In addition, the conversion problem requires two steps since 2011 minutes needs to be converted first to hours and minutes and then to days, hours, and minutes.

Type: Problem-Solving Task

This is the third problem in a series of three tasks involving fraction multiplication that can be solved with pictures or number lines. The first, 5.NF Running to school, does not require that the unit fractions that comprise 3/4 be subdivided in order to find 1/3 of 3/4. The second task, 5.NF Drinking Juice, does require students to subdivide the unit fractions that comprise 1/2 in order to find 3/4 of 1/2. This task also requires subdivision and involves multiplying a fraction and a mixed number.

Type: Problem-Solving Task

The purpose of this task is to gain a better understanding of multiplying and dividing with fractions. Students should use the diagram provided to support their findings.

Type: Problem-Solving Task

This problem helps students gain a better understanding of dividing with fractions.

Type: Problem-Solving Task

The purpose of this task is to provide students with a concrete experience they can relate to fraction multiplication. Perhaps more importantly, the task also purposefully relates length and locations of points on a number line, a common trouble spot for students. This task is meant for instruction and would be a useful as part of an introductory unit on fraction multiplication.

Type: Problem-Solving Task

Part (a) of this task asks students to use two different denominators to subtract fractions. The purpose of this is to help students realize that any common denominator will work, not just the least common denominator. Part (b) does not ask students to do it in more than one way; the purpose is to give them an opportunity to choose a denominator and possibly compare with another student who chose a different denominator. The purpose of part (c) is to help students move away from a reliance on drawing pictures. Students can draw a picture if they want, but this subtraction problem is easier to do symbolically, which helps students appreciate the power of symbolic notation.

Type: Problem-Solving Task

Part (a) of this task asks students to find and use two different common denominators to add the given fractions. The purpose of this question is to help students realize that they can use any common denominator to find a solution, not just the least common denominator. Part (b) does not ask students to solve the given addition problem in more than one way. Instead, the purpose of this question is to give students an opportunity to choose a denominator and possibly to compare their solution method with another student who chose a different denominator. The purpose of part (c) is to give students who are ready to work symbolically a chance to work more efficiently.

Type: Problem-Solving Task

The purpose of this task is to help students gain a better understanding of fractions and the conversion of fractions into smaller units.

Type: Problem-Solving Task

This task is intended to complement "5.NF How many servings of oatmeal?" and "7.RP Molly's run.'' All three tasks address the division problem 4÷1/3 but from different points of view. This task provides a how many in each group version of 4÷1/3. This task should be done together with the "How many servings of oatmeal" task with specific attention paid to the very different pictures representing the two situations.

Type: Problem-Solving Task

One goal of this task is to help students develop comfort and ease with adding fractions with unlike denominators. Another goal is to help them develop fraction number sense by having students decompose fractions.

Type: Problem-Solving Task

This is the second problem in a series of three tasks involving fraction multiplication that can be solved with pictures or number lines. This task does require students to subdivide the unit fractions that comprise 1/2 in order to find 3/4 of 1/2.

Type: Problem-Solving Task

This task addresses common errors that students make when adding fractions. It is very important for students to recognize that they only add fractions when the fractions refer to the same whole, and also when the fractions of the whole being added do not overlap. This set of questions is designed to enhance a student's understanding of when it is and is not appropriate to add fractions.

Type: Problem-Solving Task

This task requires students to recognize both "number of groups unknown" (part (a)) and "group size unknown" (part (d)) division problems in the context of a whole number divided by a unit fraction. It also addresses a common misconception that students have where they confuse dividing by 2 or multiplying by 1/2 with dividing by 1/2.

Type: Problem-Solving Task

The purpose of this task is to have students think about the meaning of multiplying a number by a fraction, and use this burgeoning understanding of fraction multiplication to make sense of the commutative property of multiplication in the case of fractions.

Type: Problem-Solving Task

The purpose of this task is for students to compare a number and its product with other numbers that are greater than and less than one. As written, this task could be used in a summative assessment context, but it might be more useful in an instructional setting where students are asked to explain their answers either to a partner or in a whole class discussion.

Type: Problem-Solving Task

This particular problem deals with multiplication. Even though students can solve this problem by multiplying, it is unlikely they will. Here it is much easier to answer the question if you can think of multiplying a number by a factor as scaling the number.

Type: Problem-Solving Task

The purpose of this task is to provide students with a concrete situation they can model by dividing a whole number by a unit fraction. For students who are just beginning to think about the meaning of division by a unit fraction (or students who have never cooked), the teacher can bring in a 1/4 cup measuring cup so that students can act it out. If students can reason through parts (a) and (b) successfully, they will be well-situated to think about part (c) which could yield different solution methods.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

This site uses visual models to better understand what is actually happening when students multiply and divide fractions. Using area models -- one that superimposes squares that are partitioned into the appropriate number of regions, and shaded as needed -- students multiply, divide, and translate the processes to decimals. The lesson uses an interactive simulation that allows students to create their own area models and is embedded with problems throughout for students to solve.

Type: Problem-Solving Task

This resource introduces students to the aspects a builder must think about before constructing a building. Students will study the cabin blueprint of Henry David Thoreau and then will find the surface area of the walls and how much paint would be needed. Then, students will find the volume of the cabin to determine the home heating needs. Third, students will study the blueprint and will create a 1/10 scale of it on graph paper and then will use art supplies to create a model of the cabin. Last, students will design and create models of furniture to scale for the cabin.

Type: Problem-Solving Task

## Tutorials

This Khan Academy tutorial video explains patterns in the placement of the decimal point, when a decimal is multiplied by a power of 10.

Type: Tutorial

This Khan Academy tutorial video presents the methodology of understanding and using patterns in the number of zeros of products that have a factor that is a power of 10.

Type: Tutorial

This Khan Academy tutorial video presents the pattern, when multiplying tens, that develops when we compare the number of factors of tens with the number of zeros in the product. The vocabulary, *exponent *and* base*, are introduced.

Type: Tutorial

This Khan Academy tutorial video interprets written statements and writes them as mathematical expressions.

Type: Tutorial

This Khan Academy tutorial video presents the application of parentheses notation in an expression.

Type: Tutorial

This Khan Academy tutorial video demonstrates how to write a simple expression from a word problem.

Type: Tutorial

This Khan Academy tutorial video illustrates the conversion equivalence of liters, milliliters, and kiloliters.

Type: Tutorial

This Khan Academy tutorial video presentation represents a word problem's solution on a coordinate plane to determine the number of blocks walked from a home to a school.

Type: Tutorial

This Khan Academy tutorial video presents how to graph an ordered pair of positive numbers on the x- and y-axis of a coordinate plane.

Type: Tutorial

This Khan Academy tutorial video presents a strategy for solving the following problem: given a dot plot with different measurements of trail mix in bags, find the amount of trail mix each bag would contain, if the total amount in all the bags was equally redistributed.

Type: Tutorial

This Khan Academy tutorial video develops a visual diagram to use to solve a distance problem that requires converting feet to yards and other computations.

Type: Tutorial

This Khan Academy tutorial video demonstrates a strategy for ordering four different-sized metric units.

Type: Tutorial

This Khan Academy tutorial video illustrates how to find the volume of an irregular solid figure by dividing the figure into two rectangular prisms and finding the volume of each.

Type: Tutorial

This Khan Academy tutorial video illustrates finding the volume of an irregular figure made up of unit cubes by separating the figure into two rectangular prisms and finding the volume of each part.

Type: Tutorial

This Khan Academy tutorial video illustrates measuring volume by counting unit cubes.

Type: Tutorial

This Khan Academy tutorial video describes measurement in one, two, and three dimensions.

Type: Tutorial

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

Type: Tutorial

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

Type: Tutorial

This Khan Academy tutorial video introduces quadrilaterals. their categories, and subcategories.

Type: Tutorial

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

Type: Tutorial

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

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

Type: Tutorial

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

Type: Tutorial

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

Type: Tutorial

This Khan Academy tutorial video presents using place-value to compare two decimals expressed to thousandths.

Type: Tutorial

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

Type: Tutorial

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

Type: Tutorial

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

Type: Tutorial

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

Type: Tutorial

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

Type: Tutorial

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

Type: Tutorial

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

Type: Tutorial

In this video, you will work through an example to correctly use the order of operations.

Type: Tutorial

In this video, you will see why it is important to have one agreed upon order of operations.

Type: Tutorial

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

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

Type: Tutorial

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

In this tutorial, the four operations are applied to fractions with the visualization of the number line. This tutorial starts by adding fractions with the same denominators and explains the logic behind multiplication of fractions. This tutorial also highlights the application and extension of previous understandings of mulitplication to multiply a fraction or whole number by a fraction.

a. Interpret the product (* a*/

*) x*

**b***as*

**q***parts of a partition of*

**a***into*

**q***equal parts; equivalently, as the result of a sequence of operations*

**b***x*

**a***. In general, (*

**qb***/*

**a***) x (*

**b***/*

**c***) =*

**d***/*

**ac***.*

**bd**Type: Tutorial

This tutorial explores the addition and subtraction of fractions with unlike denominators. Performing these operations on fractions with unlike denominators requires the creation of a 'common' denominator. Using the number line, this mathematical process can be easily visualized and connected to the final strategy of multiplying the denominators (a/b + c/d = ad +bc/bd).

Type: Tutorial

In this tutorial, students will be exposed to the strategy of finding the least common denominator for certain cases. Sometimes when finding a common denominator, an unnecessarily large common denominator is created (** a/b x c/d** =

*+*

**ad***). This chapter explains how to find the smallest possible common denominator.*

**bc****/bd***For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12.*

Type: Tutorial

The Cartesian Coordinate system, formed from the Cartesian product of the real number line with itself, allows algebraic equations to be visualized as geometric shapes in two or three dimensions.

Type: Tutorial

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

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

This tutorial for student audiences will assist learners with a further understanding of the rules for adding and subtracting with decimals. 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

This tutorial for student audiences will assist learners with a further understanding of the rules for adding and subtracting 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

In this lesson, students will be viewing a Khan Academy video that will show how to convert ratios using speed units.

Type: Tutorial

The video describes how to multiply fractions and state the answer in lowest terms.

Type: Tutorial

## Virtual Manipulatives

Students investigate shapes that grow and change using an iterative process. Fractals are characterized by self-similarity, smaller sections that resemble the larger figure. From NCTM's Illuminations.

Type: Virtual Manipulative

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

Type: Virtual Manipulative

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

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

Type: Virtual Manipulative

In this activity, students practice solving algebraic expressions using order of operations. The applet records their score so the student can track their progress. This activity allows students to practice applying the order of operations when solving problems. 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: Virtual Manipulative

Section:Grades PreK to 12 Education Courses >Grade Group:Grades PreK to 5 Education Courses >Subject:Mathematics >SubSubject:General Mathematics >