# MiddleSchoolPortal/Math Focal Points: Grade 5

<keywords content="middle school mathematics focal points division fractions decimals surface area volume concepts skills interactive activities solids curriculum practice understanding" /> <metadescription content="This free, standards-based, online publication developed for teachers features digital resources that support the teaching of the NCTM Curriculum Focal Points for grade 5: whole number division, fractions and decimals, three-dimensional shapes." />

## Math Focal Points - Grade 5 - Introduction

The National Council of Teachers of Mathematics (NCTM), in an effort to highlight the most important mathematical topics at each grade level, has developed Curriculum Focal Points for Prekindergarten Through Grade 8 Mathematics. Rather than a discrete topic to be checked off a list, NCTM emphasizes that a “focal point” is a “cluster of related knowledge, skills, and concepts.” These focal points, three at each grade level, specify “the mathematical content that a student needs to understand deeply and thoroughly for future mathematics learning.”

With this publication, the Middle School Portal begins a series on resources that support the teaching of the Curriculum Focal Points for grades 5 through 8. We begin with grade 5.

NCTM foresees that students at the fifth-grade level would work on developing understanding and mastery of division of whole numbers. To support this area, we feature online interactive activities that model the connection between multiplication and division of whole numbers, plus lessons that give direct instruction on the process of division.

Students at this level would also be developing an understanding and mastery of addition and subtraction of fractions and decimals. Resources in the fractions and decimals section concentrate on basic concepts, such as equivalent fractions and decimal place value, as well as visual demonstrations of addition and subtraction and opportunities for practice.

## Contents

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In grade 5, students would also learn to describe three-dimensional shapes and analyze their properties, including volume and surface area. Through the activities online and offline in the three-dimensional section, students can explore the properties of solids and go on to consider the rules that determine surface area and volume.

In Background Information for Teachers, you will find professional learning resources. Finally, we discuss the focal points as they are related to the NCTM Principles and Standards for School Mathematics.

Future publications in this series will look at the focal points for grades 6, 7, and 8. We hope the resources in the Math Focal Points series open your classes to a wider view of mathematics!

For a complete statement of the NCTM Curriculum Focal Points for grade 5 see below. NCTM Curriculum Focal Points for Grade 5

Number and Operations and Algebra: Developing an understanding of and fluency with division of whole numbers. Students apply their understanding of models for division, place value, properties, and the relationship of division to multiplication as they develop, discuss, and use efficient, accurate, and generalizable procedures to find quotients involving multidigit dividends. They select appropriate methods and apply them accurately to estimate quotients or calculate them mentally, depending on the context and numbers involved. They develop fluency with efficient procedures, including the standard algorithm, for dividing whole numbers, understand why the procedures work (on the basis of place value and properties of operations), and use them to solve problems. They consider the context in which a problem is situated to select the most useful form of the quotient for the solution, and they interpret it appropriately.

Number and Operations: Developing an understanding of and fluency with addition and subtraction of fractions and decimals. Students apply their understandings of fractions and fraction models to represent the addition and subtraction of fractions with unlike denominators as equivalent calculations with like denominators. They apply their understandings of decimal models, place value, and properties to add and subtract decimals. They develop fluency with standard procedures for adding and subtracting fractions and decimals. They make reasonable estimates of fraction and decimal sums and differences. Students add and subtract fractions and decimals to solve problems, including problems involving measurement.

Geometry and Measurement and Algebra: Describing three-dimensional shapes and analyzing their properties, including volume and surface area. Students relate two-dimensional shapes to three-dimensional shapes and analyze properties of polyhedral solids, describing them by the number of edges, faces, or vertices as well as the types of faces. Students recognize volume as an attribute of three-dimensional space. They understand that they can quantify volume by finding the total number of same-sized units of volume that they need to fill the space without gaps or overlaps. They understand that a cube that is 1 unit on an edge is the standard unit for measuring volume. They select appropriate units, strategies, and tools for solving problems that involve estimating or measuring volume. They decompose three-dimensional shapes and find surface areas and volumes of prisms. As they work with surface area, they find and justify relationships among the formulas for the areas of different polygons. They measure necessary attributes of shapes to use area formulas to solve problems.

## Background Information for Teachers

The following resources offer insights into the arithmetic and geometry underlying the Curriculum Focal Points for fifth graders. Each is an online workshop session designed for K-8 teachers who are looking for deeper understanding of mathematical content. The workshops were developed by Learning Math as free, college-level courses for educators.

Meanings and Models for Operations In this session, you can explore some of the laws that govern the basic operations, including division. Interactive mathematical models aid in examining division as inverse multiplication, as do questions for teacher discussion.

Solids Through these activities, developed for teachers, you will build and and manipulate the Platonic solids, exploring their properties and some of the geometric relationships between them. Other investigations focus on nets and cross sections of solids.

Measurement Relationships This workshop begins with an examination of the relationships between area and perimeter but goes on to the proportional relationship between surface area and volume and some of its applications. For instance, you will construct boxes and use graphs as you try to find the dimensions of the rectangular prism that holds the maximum volume. Video segments of teachers working on this activity lead to further discussion in this session from Learning Math: Measurement.

## Division of Whole Numbers

Students apply their understanding of models for division, place value, properties, and the relationship of division to multiplication as they develop, discuss, and use efficient, accurate, and generalizable procedures to find quotients involving multidigit dividends. They select appropriate methods and apply them accurately to estimate quotients or calculate them mentally, depending on the context and numbers involved. They develop fluency with efficient procedures, including the standard algorithm, for dividing whole numbers, understand why the procedures work (on the basis of place value and properties of operations), and use them to solve problems. They consider the context in which a problem is situated to select the most useful form of the quotient for the solution, and they interpret it appropriately (NCTM, 2006, p.17).

This list of resources begins with interactive activities that emphasize the connection between multiplication and division. Other resources deal directly with long division and offer lessons in that skill.

Factorize 2 In this interactive activity, students learn about division through building rectangular arrays on a grid. Each array represents the factorization of a number (ranging from 1 through 50). Students represent the product of two factors of the number as the area of a rectangle on a grid, an important representation of multiplication and division. The challenge is to find all possible divisors of the number.

The Product Game An excellent online game, easily adapted to a paper-and-pencil format, exercises skills with factors and multiples. The activity offers practice with multiplication of numbers from 1 through 9 and with division of their products.

Rectangle Division With this virtual manipulative, students use rectangular models to explore and practice division. From picture and equation, the learner sees that the dividend equals the divisor times another number plus a remainder. In a “Test Me” option, users are given division problems to solve. They can create a picture representing the division problem, then fill in the text boxes to give the equation and check their answer.

Coloring remainders in Pascal's triangle Using this online activity, the student practices division by finding remainders for a given number and clicking the appropriate numbers in Pascal's triangle to form a colored number pattern. The applet guides the student in finding all possible remainders, starting with zero, and will not allow an incorrect number to be colored.

Division - Table of Contents Many, many division topics are addressed here through short lessons and practice problems. In particular, instructions on how to perform long division are presented through well-set out examples.

Written division Pages, called “factsheets,” emphasize the meaning of place value in division and the concept of division as repeated subtraction, with well-explained examples of both short and long division. A game as well as worksheets and quizzes are available for practice.

## Fractions and Decimals

Students apply their understandings of fractions and fraction models to represent the addition and subtraction of fractions with unlike denominators as equivalent calculations with like denominators. They apply their understandings of decimal models, place value, and properties to add and subtract decimals. They develop fluency with standard procedures for adding and subtracting fractions and decimals. They make reasonable estimates of fraction and decimal sums and differences. Students add and subtract fractions and decimals to solve problems, including problems involving measurement (NCTM, 2006, p.17).

The resources here offer support in explaining the concepts underpinning addition and subtraction of fractions and decimals — the concepts of equivalent fractions and decimal place values. The resources also include demonstrations of addition and subtraction and opportunities for practice. Whenever possible, we selected sites that visually or interactively engage the learner.

We begin with resources emphasizing fractions. Please scroll down the page to find decimal resources.

Fractions

Visual Fractions An exceptional tutorial on fractions, including step-by-step, illustrated explanations of addition and subtraction. Both circle and line models help students visualize the operations with like and unlike denominators. Interactive problems allow students to use these visual models as they figure the numerical answers.

Equivalent fractions finder To add or subtract fractions with unlike denominators, students must thoroughly understand how to find equivalent fractions. Here users are shown a fraction displayed in an area model and on a number line. They must visually represent two unique fractions that are equivalent to the given fraction. The fractional value is shown on a number line after the students check to see if their fraction is correct.

Adding Fractions Students must do the usual exercise of finding equivalent fractions with common denominators, but here the fractions are represented visually as portions of a square. Once the computer checks that the fractions are correct, the students can drag the representations into a third box and enter the sum of the fractions. This is a learning experience! There are other activities on fractions as well, all worth checking out.

The Fractionator Created by math teacher Jeff LeMieux, the Fractionator offers online and offline tools to help students understand fractions. The online tools use unit squares to model two fractions to be added (or subtracted) and then create equivalent fraction models; with this visual aid, students complete the operation. They can request a new problem for each exercise or enter the two fractions themselves. Also provided are links to printable materials, such as overhead transparencies and student worksheets.

Who Wants Pizza? A Fun Way to Learn about Fractions Here are six lessons on the definition of a fraction, equivalent fractions, addition of fractions, and multiplication of fractions. Although too brief for a first introduction to fractions, the lessons can engage students in review or in extra practice. Students can respond online to get immediate feedback, or they can work the examples on grid paper.

Soccer Shootout Practice time! Students can practice the addition and subtraction of fractions at levels of difficulty ranging from Easy to Super Brain. Students play against the computer and are provided with a full solution when a wrong answer is entered.

Decimals

Builder Ted This interactive game deals with place value in decimals, necessary to understanding addition and subtraction. In the game scenario, students help Builder Ted by placing numbered bricks on a ladder in numerical order. At the first level of difficulty, all numbers are positive, but the two higher levels include negative numbers as well. If a number is placed incorrectly, all the bricks immediately fall and the player begins again. Tips for students are available as well as an explanation of the key ideas underlying the game.

Place Value Here is another exercise preliminary to decimal addition and subtraction. The user can type in any number, such as 3601.076, or let the computer choose a number. As the student passes the mouse over each digit in the number, the place value is shown. Also, how to say the number is given, plus a short exercise asking the student to identify the digit in, say, the thousandth position.

Base blocks decimals With this virtual manipulative, students can explore the meaning of place value and grouping as they add and subtract decimals. Base blocks consist of individual "units," "longs," "flats," and "blocks" (ten of each set for base 10). The blocks can represent negative as well as positive numbers with one to four decimal places and in five different bases. Students exchange and group the blocks as needed to solve the problem. Problems can be presented to or created by the students. All material is available in Spanish and French as well as English, including instructions for using the manipulative, information about bases and place value, and suggested questions for classroom use. M

Decimals This site offers bare-bones explanations of decimal topics and interactive practice. In the long list of topics are adding and subtracting decimals as well as adding and subtracting money. The computer sets the problem and gives immediate feedback to the student’s response. The bottom of each lesson page contains timed exercises.

## 3-D Shapes

Students relate two-dimensional shapes to three-dimensional shapes and analyze properties of polyhedral solids, describing them by the number of edges, faces, or vertices as well as the types of faces. Students recognize volume as an attribute of three-dimensional space. They understand that they can quantify volume by finding the total number of same-sized units of volume that they need to fill the space without gaps or overlaps. They understand that a cube that is 1 unit on an edge is the standard unit for measuring volume. They select appropriate units, strategies, and tools for solving problems that involve estimating or measuring volume. They decompose three-dimensional shapes and find surface areas and volumes of prisms. As they work with surface area, they find and justify relationships among the formulas for the areas of different polygons. They measure necessary attributes of shapes to use area formulas to solve problems (NCTM, 2006, p.17).

Work on three-dimensional shapes begins with hands-on play, either virtual or with actual materials. Using these resources, students can investigate the properties of solids and go on to consider the rules that determine volume and surface area.

2D to 3D morphing : flat 2D shapes rise up to make 3D forms As students follow the directions on the printable pages, they construct a pyramid, a cube, and an octahedron. They can fold the flat, two-dimensional polygons and see them rise up to form three-dimensional polyhedra. Each page is decorated with colorful images of the Cyberchase team so that one image appears on each face of the constructed three-dimensional objects.

Geometric Solids and Their Properties Using an applet, students investigate several polyhedra. They can revolve each shape, color each face, and mark each edge or vertex. They can even see the figure without the faces colored in — a skeletal view of the "bones" forming the shape. The lesson leads to Euler’s formula, connecting the number of edges, vertices, and faces, and ends with creating nets to form polyhedra. An excellent introduction to three-dimensional figures!

Cubes Students fill a box with cubes. This can be done online or using actual materials, depending on what’s available in your classroom. The number of cubes needed to fill the entire box is defined as the “volume” of the box. Students are challenged to determine a rule for finding the volume of a box when they know its width, depth, and height.

How high? : measurement (grades 6-8) With this virtual manipulative, students pour a liquid from one container to a container of the same shape, but of a different size. There are four shapes to choose from: rectangular prism, cylinder, cone, and pyramid. The left container is partially filled with liquid and the base dimensions are given. The student uses a slider to estimate how high the liquid will rise when poured into the second container. After clicking a button that initiates pouring, the student can compare the estimate with the results. Opens up interesting discussion on volume!

Surface area and volume This applet enables students to form and rotate both rectangular and triangular prisms. They can set the dimensions (width, depth, and height), observing how each change in dimension affects the shape of the prism as well as its volume and surface area. This is a quick way to collect data for a discussion of the relationship between surface area and volume. Users can rotate the figure and call for its front, side, or back view — very interesting with a triangular prism!

Keeping cool : when should you buy block ice or crushed ice? Which would melt faster: a large block of ice or the same block cut into three cubes? The prime consideration is surface area. The solution demonstrates how to calculate the surface area of the cubes as well as the area of the large block of ice. Related problems involve finding surface area and volume for irregular shapes and examining the relationship between surface area and volume in various situations.

Drip drops : how much water do you waste? In this activity, students are given a situation in which a leaky faucet is dripping at the rate of one drop every two seconds. They are asked to decide if the water lost in one week would fill a drinking glass, a sink, or a bathtub. The answer page shows students how to convert the drops to gallons using an equation or a table. Related questions ask students to consider how much water is lost in one year by a single leaky faucet and by two million leaky faucets. Real applications of work on volume!

Popcorn : if you like popcorn, which one would you buy? This challenge directs the student to use popcorn to compare the volumes of tall and short cylinders formed with 8- by 11-inch sheets of paper. The importance of being able to make visual estimates and find volumes is pointed out.

Platonic Solids (Grades 6-8) Students examine in detail the five Platonic solids — their shapes, vertices, edges, and regular polygonal faces. With the virtual manipulative, they can rotate each solid, viewing it from every angle, change its size, then use the transparent mode to see only the skeletal structure of the polyhedron.

Scaling the pyramids These activities engage students through their fascination with the sheer size of the Great Pyramid. In one hands-on activity, students use a template to construct a scale model of the Great Pyramid. In another, students are given the actual dimensions for two other pyramids and challenged to create their own models.

## SMARTR: Virtual Learning Experiences for Students

Visit our student site SMARTR to find related math-focused virtual learning experiences for your students! The SMARTR learning experiences were designed both for and by middle school aged students. Students from around the country participated in every stage of SMARTR’s development and each of the learning experiences includes multimedia content including videos, simulations, games and virtual activities.

## NCTM Standards

A question that naturally arises is: What do the Curriculum Focal Points have to do with the Principles and Standards for School Mathematics? The National Council of Teachers of Mathematics answers that identifying areas of emphasis at each grade level is the next step in implementing the principles and standards. Curriculum Focal Points for Prekindergarten Through Grade 8 "provides one possible response to the question of how to organize curriculum standards within a coherent, focused curriculum, by showing how to build on important mathematical content and connections identified for each grade level, pre-K–8" (NCTM, 2006, p. 12).

The Curriculum Focal Points draw on the content standards described in the Principles and Standards, at times clustering several topics in one focal point. Also, the process standards are pivotal to well-grounded instruction, for, according to the Curriculum Focal Points for Grade 5, “it is essential that these focal points be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations."

This Middle School Portal publication offers resources that will support you in teaching the key mathematical areas identified for grade 5. In particular, we aimed to provide visual and interactive models for division and place value, for addition and subtraction of fractions and decimals, and for three-dimensional shapes. These resources also provide opportunities for developing understanding of concepts underpinning standard procedures in arithmetic and measurement as well as for developing fluency with those procedures.

A description of the Focal Points for grade 5 is found at http://www.nctm.org/standards/focalpoints.aspx?id=334&ekmensel=c580fa7b_10_52_334_7

Curriculum Focal Points for Prekindergarten Through Grade 8: A Quest for Coherence may be viewed in its entirety at http://www.nctm.org/standards/content.aspx?id=270

Reference NCTM. (2006). Curriculum Focal Points for Kindergarten Through Grade 8 Mathematics: A Quest for Coherence. Reston, VA: Author