Georgia Performance Standards Framework

Fifth Grade MathematicsUnit 5 1st Edition

Grade 5 Mathematics Frameworks
Unit 5
Geometry and Measurement –
Solid Figures

GeorgiaDepartment of Education

Kathy Cox, State Superintendent of Schools

MATHEMATICS GRADE5UNIT 5: SUPER SOLID FIGURES

August 2009 Page 1 of 63

Copyright 2009 © All Rights Reserved

Georgia Performance Standards Framework

Fifth Grade MathematicsUnit 5 1st Edition

Unit 5Organizer

Geometry and Measurement – Solid Figures

(6 weeks)

TABLE OF CONTENTS

Overview...... 3

Key Standards & Related Standards...... 4

Enduring Understandings ...... 5

Essential Questions...... 6

Concepts & Skills to Maintain...... 6

Selected Terms and Symbols...... 7

Classroom Routines...... 9

Strategies for Teaching and Learning...... 9

Evidence of Learning...... 10

Tasks...... 11

  • Capacity Line-up...... 12
  • Fill ‘R Up...... 18
  • More Punch, Please!...... 24
  • Water Balloon Fun!...... 33
  • Differentiating Area and Volume...... 39
  • How Many Ways?...... 46
  • Super Solids...... 52

Culminating Task

Boxing Boxes...... 56

OVERVIEW

In this unit students will:

  • convert capacity measurements within a single system of measurement (customary, metric)
  • estimate and measure capacity using milliliters, liters, fluid ounces, cups, pints, quarts, and gallons
  • describe three-dimensional figures by faces, edges, and vertices
  • determine the formula for finding the volume of cubes and rectangular prisms
  • estimate and determine the volume of cubes and rectangular prisms
  • distinguish between volume and capacity

Although the units in this instructional framework emphasize key standards and big ideas at specific times of the year, routine topics such as addition and subtraction of decimals and fractions with like denominators, whole number computation, angle measurement, length/area/weight, number sense, data usage and representations, characteristics of 2-D and 3-D shapes, and order of operations should be addressed on an ongoing basis. Ideas related to the five process standards: problem solving, reasoning, connections, communication, and representation, should be addressed continually as well. The first unit should establish these routines, allowing students to gradually enhance their understanding of the concept of number and to develop computational proficiency.

To assure that this unit is taught with the appropriate emphasis, depth, and rigor, it is important that the tasks listed under “Evidence of Learning” be reviewed early in the planning process. A variety of resources should be utilized to supplement, but not completely replace, the textbook. Textbooks not only provide much needed content information, but excellent learning activities as well. The tasks in these units illustrate the types of learning activities that should be utilized from a variety of sources.

STANDARDS ADDRESSED IN THIS UNIT

Mathematical standards are interwoven and should be addressed throughout the year in as many different units and activities as possible in order to emphasize the natural connections that exist among mathematical topics.

KEY STANDARDS

M5M3.Students will measure capacity with appropriately chosen units and tools.

  1. Use milliliters, liters, fluid ounces, cups, pints, quarts, and gallons to measure capacity.
  2. Compare one unit to another within a single system of measurement.

M5M4. Students will understand and compute the volume of a simple geometric solid.

  1. Understand a cubic unit (u3) is represented by a cube in which each edge has the length of 1 unit.
  2. Identify the units used in computing volume as cubic centimeters (cm3), cubic meters (m3),cubic inches (in3), cubic feet (ft3), and cubic yards (yd3).
  3. Derive the formula for finding the volume of a cube and a rectangular prism using manipulatives.
  4. Compute the volume of a cube and a rectangular prism using formulae.
  5. Estimate the volume of a simple geometric solid.
  6. Understand the similarities and differences between volume and capacity.

RELATED STANDARDS

M5A1. Students will represent and interpret the relationships between quantities algebraically.

  1. Use variables, such as n or x, for unknown quantities in algebraic expressions.
  2. Investigate simple algebraic expressions by substituting numbers for the unknown.
  3. Determine that a formula will be reliable regardless of the type of number (whole numbers or decimals) substituted for the variable.

M5P1. Students will solve problems (using appropriate technology).

  1. Build new mathematical knowledge through problem solving.
  2. Solve problems that arise in mathematics and in other contexts.
  3. Apply and adapt a variety of appropriate strategies to solve problems.
  4. Monitor and reflect on the process of mathematical problem solving.

M5P2. Students will reason and evaluate mathematical arguments.

  1. Recognize reasoning and proof as fundamental aspects of mathematics.
  2. Make and investigate mathematical conjectures.
  3. Develop and evaluate mathematical arguments and proofs.
  4. Select and use various types of reasoning and methods of proof.

M5P3. Students will communicate mathematically.

a. Organize and consolidate their mathematical thinking through communication.

b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.

c. Analyze and evaluate the mathematical thinking and strategies of others.

d. Use the language of mathematics to express mathematical ideas precisely.

M5P4. Students will make connections among mathematical ideas and to other disciplines.

a. Recognize and use connections among mathematical ideas.

b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.

c. Recognize and apply mathematics in contexts outside of mathematics.

M5P5. Students will represent mathematics in multiple ways.

a. Create and use representations to organize, record, and communicate mathematical ideas.

b. Select, apply, and translate among mathematical representations to solve problems.

c. Use representations to model and interpret physical, social, and mathematical phenomena.

ENDURING UNDERSTANDINGS

  • Three-dimensional (3-D) figures are described by their faces (surfaces), edges, and vertices (singular is “vertex”).
  • Volume can be expressed in both customary and metric units.
  • Volume is represented in cubic units – cubic inches, cubic centimeters, cubic feet, etc.
  • Capacity is measured in fluid ounces, cups, pints, quarts, gallons, milliliters, and liters.
  • Volume refers to the space taken up by an object itself, while capacity refers to the amount of a liquid or other pourable substance a container holds.

ESSENTIAL QUESTIONS

  • Can different size containers have the same capacity?
  • How can we estimate and measure capacity?
  • What material is the best to use when measuring capacity?
  • What material is the best to use when measuring volume?
  • What connection can you make between the volumes of geometric solids?
  • Does volume change when you change the measurement material? Why or why not?
  • How do we measure volume?
  • How are fluid ounces, cups, pints, quarts, and gallons related?
  • How can fluid ounces, cups, pints, quarts, and gallons be used to measure capacity?
  • Why do we need to be able to convert between capacity units of measurement?
  • How do we compare metric measures of milliliters and liters?
  • How do we compare customary measures of fluid ounces, cups, pints, quarts, and gallons?
  • Why is volume represented with cubic units and area represented with square units?
  • How are area and volume alike and different?
  • Why is volume represented with cubic units?
  • How do we measure volume?
  • How can you find the volume of cubes and rectangular prisms?
  • How can you find the volume of cubes and rectangular prisms?
  • Why is volume represented with cubic units?
  • What connection can you make between the volumes of geometric solids?
  • How do we measure volume?
  • Can different size containers have the same capacity?
  • How can we measure capacity?
  • How can we measure volume?
  • How are volume and capacity the same? How are volume and capacity different?

CONCEPTS/SKILLS TO MAINTAIN

It is expected that students will have prior knowledge/experience related to the concepts and skills identified below. It may be necessary to pre-assess in order to determine if time needs to be spent on conceptual activities that help students develop a deeper understanding of these ideas.

  • number sense
  • computation with whole numbers and decimals, including application of order of operations
  • addition and subtraction of common fractions with like denominators
  • angle measurement
  • measuring length and finding perimeter and area of rectangles and squares
  • characteristics of 2-D and 3-D shapes
  • data usage and representations

SELECTED TERMS AND SYMBOLS

The following terms and symbols are often misunderstood.These concepts are not an inclusive list and should not be taught in isolation. However, due toevidence of frequent difficulty and misunderstanding associated with these concepts, instructors should pay particular attention to them and how their students are able to explain and apply them.

The definitions below are for teacher reference only and are not to be memorized by the students.Students should explore these concepts usingmodels and real life examples.Students should understand the concepts involved and be able to recognize and/or demonstrate them with words, models, pictures, or numbers.

The websites below are interactive and include a math glossary suitable for elementary children. Note – At the elementary level, different sources use different definitions. Please preview any website for alignment to the definitions given in the frameworks.

This web site has activities to help students more fully understand and retain new vocabulary (i.e. The definition page for dice actually generates rolls of the dice and gives students an opportunity to add them.)

Definitions and activities for these and other terms can be found on the Intermath website. Because Intermath is geared towards middle and high school, grade 3-5 students should be directed to specific information and activities.

Capacity: The greatest volume that a container can hold.

Cube: A regular polyhedron whose six faces are congruent squares.

Cubic Centimeter (cm3): Standard metric unit for measuring volume, each dimension is measured in centimeters.

Cubic Foot (ft3): Standard customary unit for measuring volume, each dimension is measured in feet.

Cubic Inch (in3): Standard customary unit for measuring volume, each dimension is measured in inches.

Cubic Meter (m3): Standard metric unit for measuring volume, each dimension is measured in meters.

Cubic Yard (yd3): Standard customary unit for measuring volume, each dimension is measured in yards.

Cup (c.): Standard customary unit for measuring capacity.

2 cups = 1 pint

Edge: The intersection of two surfaces in a three-dimensional figure.

Face: One of the flat surfaces that makes up a three-dimensional figure.

Fluid Ounce (fl. oz.): Standard customary unit for measuring capacity.

8 fl. oz. = 1 pint

Gallon (gal.): Customary unit for measuring capacity.

4 quarts = 1 gallon

Liter (L): Standard metric unit for measuring capacity.

Milliliter (mL): Standard metric unit for measuring capacity.

1 pint is about 500 mL

Pint (pt.): Standard customary unit for measuring capacity.

2 cups = 1 pint

2 pints = 1 quart

Quart (qt.): Standard customary unit for measuring capacity.

2 pints = 1quart

4 quarts = 1 gallon

Rectangular Prism: A prism that has a rectangle as its base.

Vertex of a 3-D object: The points where the edges of a 3-D object intersect. In early grades, referred to as “corners.”

Volume: Amount of space occupied by an object, usually measured in cubic units.

CLASSROOM ROUTINES

The importance of continuing the established classroom routines cannot be overstated. Daily routines must include obvious activities such as estimating, analyzing data, describing patterns, and answering daily questions. They should also include less obvious routines, such as how to select materials, how to use materials in a productive manner, how to put materials away, and how to access classroom technology such as computers and calculators. An additional routine is to allow plenty of time for children to explore new materials before attempting any directed activity with these new materials. The regular use of routines is important to the development of students’ number sense, flexibility, fluency, collaborative skills, and communication. These routines contribute to a rich, hands-on standards-based classroom and will support students’ performances on the tasks in this unit and throughout the school year.

STRATEGIES FOR TEACHING AND LEARNING

  • Students should be actively engaged by developing their own understanding.
  • Mathematics should be represented in as many ways as possible by using graphs, tables, pictures, symbols and words.
  • Interdisciplinary and cross curricular strategies should be used to reinforce and extend the learning activities.
  • Appropriate manipulatives and technology should be used to enhance student learning.
  • Students should be given opportunities to revise their work based on teacher feedback, peer feedback, and metacognition which includes self-assessment and reflection.
  • Students should write about the mathematical ideas and concepts they are learning.
  • Consideration of all students should be made during the planning and instruction of this unit. Teachers need to consider the following:

‐What level of support do my struggling students need in order to be successful with this unit?

‐In what way can I deepen the understanding of those students who are competent in this unit?

‐What real life connections can I make that will help my students utilize the skills practiced in this unit?

The following web-based games can be used as extension activities. The web sites contain logic games where students are required to fill containers or measure liquid. Please note, some of the web sites use advertising.

EVIDENCE OF LEARNING

By the conclusion of this unit, students should be able to demonstrate the following competencies:

  • Identify faces, edges, and vertices of cubes and rectangular prisms.
  • Understand volume can be by finding the product of the area of the base times the heightV = Bh.
  • Estimate and determine the volume of cubes and rectangular prisms.
  • Compare the capacity and volume of different objects with and without formulae.
  • Distinguishbetween capacity and volume.
  • Convert capacity measurements within a single system of measurement (customary, metric).
  • Measure liquids using standard customary and metric measures.

PERFORMANCE TASKS

The following tasks represent the level of depth, rigor, and complexity expected of all fifth grade students. These tasks or a task of similar depth and rigor should be used to demonstrate evidence of learning. It is important that all elements of a task be addressed throughout the learning process so that students understand what is expected of them. While some tasks are identified as a performance task, they also may be used for teaching and learning (learning task).

Task Name / Task Type
Grouping Strategy / Content Addressed
Capacity Line-up / LearningTask
Partner/Small Group Task / Estimate and measure capacity
Fill ‘RUp / Learning Task
Partner/Small Group Task / Estimate and measure the volume of geometric figures
More Punch, Please! / Performance Task
Individual/PartnerTask / Convert liquid measures within the Customary system
Water Balloon Fun! / Performance Task
Individual/PartnerTask / Compare capacity units of measure
Differentiating Area and Volume / Learning Task
Partner/Small Group Task / Compare/contrast the measures of area and volume
How ManyWays? / Learning Task
Individual/Partner Task / Develop a formula for determining the volume of cubes and rectangular prisms
Super Solids / Learning Task
Individual/Partner Task / Estimate and calculate the volume of rectangular prisms
Culminating Task:
Boxing Boxes / Performance Task
Individual/Partner Task / Consider volume and capacity to determine guidelines for packing boxes

LEARNING TASK:Capacity Line-Up

STANDARDS ADDRESSED

M5M3.Students will measure capacity with appropriately chosen units and tools.

  1. Use milliliters, liters, fluid ounces, cups, pints, quarts, and gallons to measure capacity.

M5M4. Students will understand and compute the volume of a simple geometric solid.

  1. Estimate the volume of a simple geometric solid.
  2. Understand the similarities and differences between volume and capacity.

M5P1. Students will solve problems (using appropriate technology).

  1. Build new mathematical knowledge through problem solving.
  2. Solve problems that arise in mathematics and in other contexts.

M5P2. Students will reason and evaluate mathematical arguments.

  1. Make and investigate mathematical conjectures.
  2. Develop and evaluate mathematical arguments and proofs.

M5P3. Students will communicate mathematically.

a. Organize and consolidate their mathematical thinking through communication.

b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.

c. Analyze and evaluate the mathematical thinking and strategies of others.

d. Use the language of mathematics to express mathematical ideas precisely.

M5P4. Students will make connections among mathematical ideas and to other disciplines.

a. Recognize and use connections among mathematical ideas.

b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.

c. Recognize and apply mathematics in contexts outside of mathematics.

ESSENTIAL QUESTIONS

  • Can different size containers have the same capacity?
  • How can we estimate and measure capacity?

MATERIALS

For class

  • A House for Birdie by Stuart J. Murphy

For each student

  • “Capacity Line-up, Measuring with Graduated Cylinder” student recording sheet

For each group

  • 6 containers of different size and shape, labeled A-F (i.e small jars, cans, plastic containers, and bottles)
  • Large bottleof water
  • Pan or tray for spillage
  • Set of graduated cylinders – be sure graduated cylinders are large enough to measure the capacity of the containers used for this task (be sure the graduated cylinders measure in milliliters (mL)
  • 2 sheets construction paper
  • Filler, such as packing peanuts, lima beans, rice, etc.
  • Tennis ball
  • Apple

GROUPING