Unit 1 Organizer:
|Grade 5 Mathematics Frameworks |
|Unit 5 |
|Geometry and Measurement – |
|Solid Figures |
Unit 5 Organizer
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.
a. Use milliliters, liters, fluid ounces, cups, pints, quarts, and gallons to measure capacity.
b. Compare one unit to another within a single system of measurement.
M5M4. Students will understand and compute the volume of a simple geometric solid.
a. Understand a cubic unit (u3) is represented by a cube in which each edge has the length of 1 unit.
b. Identify the units used in computing volume as cubic centimeters (cm3), cubic meters (m3), cubic inches (in3), cubic feet (ft3), and cubic yards (yd3).
c. Derive the formula for finding the volume of a cube and a rectangular prism using manipulatives.
d. Compute the volume of a cube and a rectangular prism using formulae.
e. Estimate the volume of a simple geometric solid.
f. Understand the similarities and differences between volume and capacity.
RELATED STANDARDS
M5A1. Students will represent and interpret the relationships between quantities algebraically.
a. Use variables, such as n or x, for unknown quantities in algebraic expressions.
b. Investigate simple algebraic expressions by substituting numbers for the unknown.
c. 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).
a. Build new mathematical knowledge through problem solving.
b. Solve problems that arise in mathematics and in other contexts.
c. Apply and adapt a variety of appropriate strategies to solve problems.
d. Monitor and reflect on the process of mathematical problem solving.
M5P2. Students will reason and evaluate mathematical arguments.
a. Recognize reasoning and proof as fundamental aspects of mathematics.
b. Make and investigate mathematical conjectures.
c. Develop and evaluate mathematical arguments and proofs.
d. 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 to evidence 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 using models 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.
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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 height V = B ( h.
• Estimate and determine the volume of cubes and rectangular prisms.
• Compare the capacity and volume of different objects with and without formulae.
• Distinguish between 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 |Content Addressed |
| |Grouping Strategy | |
|Capacity Line-up |Learning Task |Estimate and measure capacity |
| |Partner/Small Group Task | |
|Fill ‘R Up |Learning Task |Estimate and measure the volume of geometric figures |
| |Partner/Small Group Task | |
|More Punch, Please! |Performance Task |Convert liquid measures within the Customary system |
| |Individual/Partner Task | |
|Water Balloon Fun! |Performance Task |Compare capacity units of measure |
| |Individual/Partner Task | |
|Differentiating Area and Volume |Learning Task |Compare/contrast the measures of area and volume |
| |Partner/Small Group Task | |
|How Many Ways? |Learning Task |Develop a formula for determining the volume of cubes and |
| |Individual/Partner Task |rectangular prisms |
|Super Solids |Learning Task |Estimate and calculate the volume of rectangular prisms |
| |Individual/Partner Task | |
|Culminating Task: |Performance Task |Consider volume and capacity to determine guidelines for |
|Boxing Boxes |Individual/Partner Task |packing boxes |
LEARNING TASK: Capacity Line-Up
STANDARDS ADDRESSED
M5M3. Students will measure capacity with appropriately chosen units and tools.
a. 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.
e. Estimate the volume of a simple geometric solid.
f. Understand the similarities and differences between volume and capacity.
M5P1. Students will solve problems (using appropriate technology).
a. Build new mathematical knowledge through problem solving.
b. Solve problems that arise in mathematics and in other contexts.
M5P2. Students will reason and evaluate mathematical arguments.
b. Make and investigate mathematical conjectures.
c. 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 bottle of 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
Partner/Small Group Task
TASK DESCRIPTION, DEVELOPMENT, AND DISCUSSION
In this task, students will explore estimation and measurement of capacity and volume with real-world tools. Students will participate in exploratory activities to compare the capacity of different containers.
Comments
This task can be introduced by reading A House for Birdie by Stuart J. Murphy. The humorous story relates real-life objects to volume and capacity.
Give each group a set of 6 containers labeled A-F and ask the group to come to a consensus in ordering them from least to greatest capacity. Ask for explanations from the group. Even adults have difficulty judging how much different size containers hold in relation to each other, so don’t be surprised with a variety of answers. To illustrate this concept further, have each group take 2 pieces of construction paper. Make 2 tube shapes – one by taping the two long edges together and the other by taping the two short edges together. Ask students which holds more. Most groups roughly split in thirds for the answer – short and fat, tall and skinny, same. Have students place the skinny cylinder inside the fat one and fill the skinny one with filler to the top edge. Then lift it up allowing the filler to empty into the fat cylinder. Students will see that the capacity is the same because they hold the same amount. Challenge groups to revisit their original line-up and give explanations for changes and/or staying the same.
While comparing capacity is difficult, comparing volume is sometimes even harder. One method of comparing volume is displacement. To do this, use an apple and a tennis ball. Choose one container bigger than both items. Fill it completely with filler and then pour that into an empty holding container. Place the first object in the original container and fill it to the top with the filler. The remaining filler is equal to the volume of the object. Make a mark on the side of the holding container to show this amount. Repeat this with the second object. Compare the two marks to determine which object has a greater volume.
Students will use graduated cylinders for this task. To introduce the graduated cylinders, share how to use and read this tool. Encourage students to make sure they keep the graduated cylinder straight and still, read it at eye level, and look at the uppermost number to determine which cylinder to use.
To give students a first experience with graduated cylinders, pour 225 mL of water into a 500-mL cylinder. Ask for volunteers to state the measure. Point out that the measure cannot be accurately determined because the water level is between the lines. Ask if there is another cylinder that is more appropriate to use. This guides students to use the correct measurement tools to get the most accurate measurements. Each student then selects a container, fills it with water, estimates its capacity in mL, and records the estimate. Students use the estimate to select a cylinder and record its name. Pour the contents into the cylinder to check to see if the right tool was used. When the right tool is selected (the water level is readable at a line on the cylinder), students record the measure and cylinder used. Remind students that this task involves estimating and measuring capacity in metric units (mL) and challenge students to locate at home items that are measured in this same unit.
Background Knowledge
Students should have experience with basic capacity and conservation. Students will also need to be familiar with using liquid measuring tools (e.g. graduated cylinders). Milliliter can be abbreviated as mL or ml. In this unit, mL is used to highlight the liter, but students should be aware that both are acceptable and should be able to recognize the use of ml.
Task Directions
Students will follow the directions below from the “Capacity Line-up, Measuring with Graduated Cylinder” student recording sheet.
Record your estimate of the capacity of each container in the “Estimate” column below. Next, find the best cylinder to use to measure the capacity and record its capacity in the “Cylinder Used” column. Finally, record the capacity of each container in the “Actual Measure” Column.
Questions/Prompts for Formative Student Assessment
• Does size and shape always affect capacity? Why or why not?
• How can you compare the volume of 2 similar items?
• How did you decide which cylinder to use?
• What must you do to get the most accurate measure?
Questions for Teacher Reflection
• How well do students understand how size and shape affect capacity?
• Which students are able to compare the volume of two similar items?
• Which students are able to select the appropriate cylinder to use?
• Which students are able to accurately measure using the cylinder?
DIFFERENTIATION
Extension
• Have groups fill the largest container with water and then pour it into the second largest and then the third largest, etc. to see if their progression was correct.
Intervention
• Ask students to complete each row of their table before moving on to another container. This will allow them to develop some experience with capacity before making their next estimate.
TECHNOLOGY CONNECTION
• “Can You Fill It?” asks students to fill a container using the different sized measuring cups with the fewest number of pours.
• Asks students to compare the amount of water in two different-shaped containers.
• Student estimate how high a container will be filled with liquid from a container with a different size or a different shape.
Name______________________________________ Date ______________________________
Capacity Line-up
Measuring with Graduated Cylinders
Record your estimate for the capacity of each container in the “Estimate” column below. Next, find the best cylinder to use to measure
the capacity and record its capacity in the “Cylinder Used” column. Finally, record the capacity of each container in the “Actual Measure” Column.
|Container |Estimate |Cylinder Used |Actual Measure |
| |_____mL |_____mL |_____mL |
|A | | | |
| |_____mL |_____mL |_____mL |
|B | | | |
| |_____mL |_____mL |_____mL |
|C | | | |
| |_____mL |_____mL |_____mL |
|D | | | |
| |_____mL |_____mL |_____mL |
|E | | | |
| |_____mL |_____mL |_____mL |
|F | | | |
LEARNING TASK: Fill ‘R Up
STANDARDS ADDRESSED
M5M3. Students will measure capacity with appropriately chosen units and tools.
a. 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.
a. Estimate the volume of a simple geometric solid.
b. Understand the similarities and differences between volume and capacity.
M5P1. Students will solve problems (using appropriate technology).
a. Build new mathematical knowledge through problem solving.
b. Solve problems that arise in mathematics and in other contexts.
M5P2. Students will reason and evaluate mathematical arguments.
a. Recognize reasoning and proof as fundamental aspects of mathematics.
b. Make and investigate mathematical conjectures.
c. Develop and evaluate mathematical arguments and proofs.
d. 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.
ESSENTIAL QUESTIONS
• 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?
MATERIALS
For each student
• “Fill ‘R Up” student recording sheet
• “Fill ‘R Up, Measuring Stations” student recording sheet
For class
• Choose three: Rice, Sand, Small dry beans (such as black beans), Orecchiette pasta (tiny pasta disks “little ears”, elbow macaroni could be substituted)
• Water
• Pan or tray for spillage
For each group
• 1 set of geometric solids that can be filled (such as Power Solids, Relational Geometric Solids, or Volumetric Solids)
• funnel
• graduated cylinder or other measuring tool with mL
GROUPING
Small Group Task
TASK DESCRIPTION, DEVELOPMENT, AND DISCUSSION
In this task, students compare the volume of geometric figures. The volume of each figure is approximated by measuring its capacity.
Comments
This task can be introduced with a review of the basic solid figures. Provide a set of geometric solids to each group of students. Ask students to name the figures and then line them up according to an estimate of their volume. Record this on the “Fill ‘R Up” student recording sheet or in a math journal. Share ideas, thoughts, and methods for determining this estimated order.
This task has four stations. At each station students will have a graduated cylinder, one material for filling (rice, beans, water, pasta), and one set of clear geometric solids. If there is only one set of geometric solids, they may be divided using sets of 2 at each station. Using the funnel, have groups fill and measure each solid with the material at the station and record results. Then, they move to the next station and repeat until all stations are complete. If time permits, have students repeat the process again to compare measurements and to hold a discussion on the importance of accuracy but the expectation of human error.
Once students have completed all of the stations, ask students to order the solids according to the measurements, record the order on the “Fill ‘R Up” student recording sheet, and compare the order to their estimations. Ask students to share their findings and to justify their findings by describing the process they followed.
Background Knowledge
According to Van de Walle (2006) “volume typically refers to the amount of space that an object takes up” whereas “capacity is generally used to refer to the amount that container will hold” (p. 265). To distinguish further between the two terms, consider how the two are measured. Volume is measured using linear measures for each dimension (ft, cm, in, m, etc) while capacity is measured using liquid measures (L, mL, qt, pt, g, etc). However, Van de Walle reminds educators, “having made these distinctions [between volume and capacity], they are not ones to worry about. The term volume can also be used to refer to the capacity of a container” (p. 266).
Van de Walle, J. A. & Lovin, L. H. (2006). Teaching students-centered mathematics: Grades 3-5. Boston: Pearson Education, Inc.
Students should have experience with basic capacity and conservation. Students will also need to be familiar with using liquid measuring tools (e.g. graduated cylinders). Milliliter can be abbreviated as mL or ml. In this unit, mL is used to highlight the liter, but students should be aware that both are acceptable and should be able to recognize the use of ml.
Task Directions
Students will follow the directions below from the “Fill ‘R Up” student recording sheet and the “Fill ‘R Up, Measuring Stations” student recording sheet.
“Fill ‘R Up” student recording sheet
• Record each geometric solid according to an estimate of their volume. The first will be the smallest; the last will be the largest.
• Once you have completed the “Fill ‘R Up, Measuring Stations” student recording sheet, record each geometric solid from smallest to largest, according to the actual measures.
“Fill ‘R Up, Measuring Stations” student recording sheet
• Fill each geometric solid with the filling material at each station. Then measure the filling material using the graduated cylinder provided.
Filling Materials: Water, Rice, Sand, Beans, or Pasta
• Return to each station and measure the amount of filling for each geometric solid a second time.
• Based on your measurements, order the volume of each solid in the table on the “Fill ‘R Up” student recording sheet.
Questions/Prompts for Formative Student Assessment
• Is height or width more important in volume measurement?
• Which material provides the most accurate measurement? Why?
• Why are your results different when you measured the second time?
• Are you surprised by the volume measure of any of the figures? Why?
Questions for Teacher Reflection
• Which students are able to accurately measure with the various materials?
• Which students are able to order volumes of solids accurately?
DIFFERENTIATION
Extension
• Use the conversion of mL to liters and compare to cubic centimeters using centimeter cubes and water.
• Ask students to compare the relationships between the materials used for filling the solids and between the different solids. Several things may come to light in this discussion.
‐ The amount of filler used to fill the container is the same even though the size of the filler may be different.
‐ The volume of the cube is three times the volume of the square pyramid (with the same base area and height). The volume of the square pyramid is 1/3 the volume of the prism with the same base area and height.
‐ The volume of the cylinder (with the same base area and height) is three times the volume of a cone. The volume of a cone is 1/3 the volume of the cylinder with the same base area and height.
‐ The volume of a sphere is 2/3 the volume of the cylinder (if the radius of each figure is the same and the height of the cylinder is 2r).
Intervention
• Have an adult work with a small group of students who need support using a graduated cylinder.
Name______________________________________ Date ______________________________
Fill ‘R Up
• In the first column, order the geometric solids according to your estimate of their volume. The first in the list will be the smallest; the last will be the largest.
• Now complete the “Fill ‘R Up, Measuring Stations” student recording sheet.
• In the second column, order the geometric solids from smallest to largest, according to the actual measures.
|Geometric Solids |
|(In order from least to greatest by volume) |
| |
|Estimation |Actual after Measurement |
|1. |1. |
|2. |2. |
|3. |3. |
|4. |4. |
|5. |5. |
|6. |6. |
|7. |7. |
|8. |8. |
Name _____________________________________ Date_______________________________
Fill ‘R Up
Measuring Stations
1. At each station fill the geometric solids with the filling material. Then measure the filling material using the graduated cylinder provided.
Filling Materials: Water, Rice, Sand, Beans, or Pasta
2. Return to each station and measure the amount of filling for each geometric solid a second time.
3. Based on your measurements, order the solids by volume and record your results in the table on the “Fill ‘R Up” student recording sheet.
|Geometric Solid |Filling Material: WATER |Geometric Solid |Filling Material: |
| |Volume |Volume | |Volume |Volume |
| |Measurement 1 |Measurement 2 | |Measurement 1 |Measurement 2 |
|Square pyramid | | |Square pyramid | | |
|Rectangular prism | | |Rectangular prism | | |
|Triangular prism | | |Triangular prism | | |
|Triangular Pyramid | | |Triangular Pyramid | | |
|Cylinder | | |Cylinder | | |
|Cone | | |Cone | | |
|Sphere | | |Sphere | | |
| |
|Geometric Solid |Filling Material: |Geometric Solid |Filling Material: |
| |Volume Measurement 1 |Volume Measurement 2 | |Volume Measurement 1 |Volume Measurement 2 |
|Square pyramid | | |Square pyramid | | |
|Rectangular prism | | |Rectangular prism | | |
|Triangular prism | | |Triangular prism | | |
|Triangular Pyramid | | |Triangular Pyramid | | |
|Cylinder | | |Cylinder | | |
|Cone | | |Cone | | |
|Sphere | | |Sphere | | |
PERFORMANCE TASK: More Punch, Please!
M5M3. Students will measure capacity with appropriately chosen units and tools.
a. Use milliliters, liters, fluid ounces, cups, pints, quarts, and gallons to measure capacity.
b. Compare one unit to another within a single system of measurement.
M5P1. Students will solve problems (using appropriate technology).
c. Build new mathematical knowledge through problem solving.
d. Solve problems that arise in mathematics and in other contexts.
e. Apply and adapt a variety of appropriate strategies to solve problems.
f. Monitor and reflect on the process of mathematical problem solving.
M5P2. Students will reason and evaluate mathematical arguments.
a. Recognize reasoning and proof as fundamental aspects of mathematics.
b. Make and investigate mathematical conjectures.
c. Develop and evaluate mathematical arguments and proofs.
d. 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.
ESSENTIAL QUESTIONS
• 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?
MATERIALS
“More Punch, Please!” student recording sheet
GROUPING
Individual/Partner Task
TASK DESCRIPTION, DEVELOPMENT, AND DISCUSSION
In this task students work with liquid measures to determine the amount of punch needed for a class party.
Comments
Before this task is introduced (or as an opening to this task), students could be asked to create a graphic representing the relationship between customary measurements of capacity. Using rice, sand, or water, they can find the relationships between the different units of measure (cups, pints, quarts, and gallons). Once students know how the different units of measure are related, they can create a graphic representation of these relationships. Allow students to create a graphic representation that makes sense to them. Then allow students to share their graphic representations with students in their small group or choose two or three students who created different representations to share their graphic representation with the class. One possible representation is shown to the right.
In the graphic representation shown, each “C” represents a cup, each “P” represents a pint, each “Q” represents a quart, and the “G” represents a gallon. This model shows there are 16 cups in a gallon, 8 pints in a gallon, and 4 quarts in a gallon. Students can then be asked to convert between different customary measurements using their model as a reference. For example, ask students questions such as:
• If I have 2 quarts of punch, how many cups do I have?
• How many quarts would 12 cups fill?
• How many pints would be needed to fill 3 quarts?
• How many cups are in 6 pints?
• If one cup is 8 fluid ounces, how many fluid ounces are in a pint? Quart? Gallon?
The quantities used in the recipe in this task can be adjusted for the number of students in fifth grade at your school. Also, the context of the task can be adapted to better suit a particular school’s traditions (e.g. fifth grade dance rather than a fifth grade party).
Task Directions
Students will follow the directions below from the “More Punch, Please!” student recording sheet.
We are making punch for a fifth grade party. A little more than 100 students will attend the party. The recipe below will serve 16 students.
Party Punch
Serves 16 (serving size: 8 fluid ounces)
Ingredients:
2 Pints Strawberry Sherbet
2 Quarts Fruit Punch, chilled
32 Fluid Ounces Lemon-Lime flavored carbonated beverage, chilled
Directions:
Place sherbet in punch bowl. Pour in fruit punch and lemon-lime soda.
Answer the following questions about the punch for the party. Show all work and explain how you know your answers are accurate.
1. How much of each ingredient needs to be purchased to serve punch at the party? Rewrite the recipe to serve over 100 students.
2. How many total gallons of punch can be made with the ingredients purchased?
3. If each serving is 12 fluid ounces, how many servings can be made with the ingredients purchased?
Background Knowledge
Before students are given this task, they will need to be familiar with customary units of measure (see “Comments” section above for more information).
According to Van de Walle (2006) “volume typically refers to the amount of space that an object takes up” whereas “capacity is generally used to refer to the amount that container will hold” (p. 265). To distinguish further between the two terms, consider how the two are measured. Volume is measured using linear measures (ft, cm, in, m, etc) while capacity is measured using liquid measures (L, mL, qt, pt, g, etc). However, Van de Walle reminds educators, “having made these distinctions [between volume and capacity], they are not ones to worry about. The term volume can also be used to refer to the capacity of a container” (p. 266).
Van de Walle, J. A. & Lovin, L. H. (2006). Teaching students-centered mathematics: Grades 3-5. Boston: Pearson Education, Inc.
The answers to the questions presented in this task are given below.
1. How much of each ingredient needs to be purchased to serve punch at the party? Rewrite the recipe to serve over 100 students.
16 ( 7 = 112 therefore, multiplying the recipe by 7 will allow for more than 100 students.
The following ingredients will be needed:
Strawberry Sherbet
2 pints of Strawberry Sherbet ( 7 = 14 pints of Strawberry Sherbet
There are 8 pints in one gallon, 14 pints ÷ 8 pints per gallon = 1 gallon and 6 pints.
There are 2 pints in one quart, 6 pints ÷ 2 pints per quart = 3 quarts.
Therefore, 1 gallon and 3 quarts of Strawberry Sherbet will be needed.
Fruit Punch
2 quarts of Fruit Punch ( 7 = 14 quarts of Fruit Punch
There are 4 quarts in one gallon, 14 quarts ÷ 4 quarts per gallon = 3 gallons and 2 quarts.
So, 3 gallons and 2 quarts of Fruit Punch will be needed.
Lemon-Lime Soda
32 fluid ounces of Lemon-Lime Soda ( 7 = 224 fluid ounces
There are 8 fluid ounces in one cup, 224 fluid ounces ÷ 8 fluid ounces per cup = 28 cups.
There are 16 cups in one gallon, 28 cups ÷ 16 cups per gallon = 1 gallon and 12 cups.
There are 4 cups in one quart, 12 cups ÷ 4 cups per quart = 3 quarts.
Therefore, 1 gallon and 3 quarts of Lemon-Lime Soda will be needed.
2. How many total gallons of punch can be made with the ingredients purchased?
Add 1 gallon and 3 quarts, 3 gallons and 2 quarts, and 1 gallons and 3 quarts. There is a total of 5 gallons and 8 quarts. But there are 4 quarts in a gallon, so 8 quarts ÷ 4 quarts per gallon = 2 gallons. Adding the 2 gallons + 5 gallons, means there will be a total of 7 gallons of punch.
3. If each serving is 12 fluid ounces, how many servings can be made with the ingredients purchased?
There is a total of 128 fluid ounces in one gallon (8 fluid ounces per cup ( 16 cups per gallon = 128 fluid ounces per gallon). 7 gallons ( 128 fluid ounces per gallon = 896 fluid ounces total.
Divide the total number of fluid ounces by the number of fluid ounces per serving, 896 ÷ 12 = 74 twelve fluid-ounce servings and 8 fluid ounces left over.
Questions/Prompts for Formative Student Assessment
• How many batches of the recipe will you need? How do you know?
• How much sherbet will you need to buy? How many pints do you need? How many pints in a gallon? How many gallons is that? How do you know?
• How many quarts of Fruit Punch do you need? How many quarts in a gallon? How many gallons of Fruit Punch do you need? How do you know?
• How much Lemon-Lime soda do you need? How many fluid ounces in a gallon? How many gallons of Lemon-lime soda do you need? How do you know?
• How many fluid ounces in a gallon? How many fluid ounces of punch will you make? How many 8 fluid ounce servings is that? How do you know?
Questions for Teacher Reflection
• Which students are able to show how fluid ounces, cups, pints, quarts, and gallons can be used to measure capacity?
• Which students are able to move flexibly from one unit of measure to another within the same system?
• Which students are able to show how fluid ounces, cups, pints, quarts, and gallons are related?
DIFFERENTIATION
Extension
• Encourage students to find a different punch recipe and to rewrite the recipe to serve over 100 students.
• Ask students to determine what size drink is typical (they can consider the type of cup being used, whether ice will be available, and other factors that may influence the amount of punch served to each student). Once students have collected data, they can display the data, choosing the most effective data display.
Intervention
• Some students may need opportunities to develop an understanding of how different measures are related by filling cup, pint, quart, and gallon containers with rice, sand, or water to determine the relationships between these liquid measures.
• Some students may benefit from using a chart to help them organize their thinking and their work. See sample below, “More Punch, Please!, Version 2” student recording sheet.
Name______________________________________ Date ______________________________
More Punch, Please!
Version 2
We are making punch for a fifth grade party. A little more than 100 students will attend the party. The recipe below will serve 16 students.
Answer the following questions about the punch for the party. Show all of your work and explain how you know your answers are accurate.
1. How much of each ingredient needs to be purchased to serve punch at the party? Rewrite the recipe to serve over 100 students.
|Party Punch |
|Serves 16 |Serves ______ |
|(serving size: 8 fluid ounces) |(serving size: 8 fluid ounces) |
|2 Pints Sherbet |_____ Pints Sherbet |
|2 Quarts Punch |_____ Quarts Punch |
|32 Fluid Ounces Lemon-Lime |_____ Fluid Ounces Lemon-Lime |
| |
2. How many total gallons of punch can be made with the ingredients purchased?
|Party Punch |
| |
3. If each serving is 12 fluid ounces, how many servings can be made with the ingredients purchased?
|Party Punch |
| |
|_______ Total Gallons of Punch 1 gallon = 128 fl. oz. |
| |
|_______ Total fluid ounces of Punch |
| |
|_______ Total 12 ounce servings |
Name______________________________________ Date ______________________________
More Punch, Please!
We are making punch for a fifth grade party. A little more than 100 students will attend the party. The recipe below will serve 16 students.
Party Punch
Serves 16 (serving size: 8 fluid ounces)
Ingredients:
2 Pints Strawberry Sherbet
2 Quarts Fruit Punch, chilled
32 Fluid Ounces Lemon-Lime flavored carbonated beverage, chilled
Directions:
Place sherbet in punch bowl. Pour in fruit punch and lemon-lime soda.
Answer the following questions about the punch for the party. Show all work and explain how you know your answers are accurate.
1. How much of each ingredient needs to be purchased to serve punch at the party? Rewrite the recipe to serve over 100 students.
2. How many total gallons of punch can be made with the ingredients purchased?
3. If each serving is 12 fluid ounces, how many servings can be made with the ingredients purchased?
PERFORMANCE TASK: Water Balloon Fun!
M5M3. Students will measure capacity with appropriately chosen units and tools.
c. Use milliliters, liters, fluid ounces, cups, pints, quarts, and gallons to measure capacity.
d. Compare one unit to another within a single system of measurement.
M5P1. Students will solve problems (using appropriate technology).
g. Build new mathematical knowledge through problem solving.
h. Solve problems that arise in mathematics and in other contexts.
i. Apply and adapt a variety of appropriate strategies to solve problems.
j. Monitor and reflect on the process of mathematical problem solving.
M5P2. Students will reason and evaluate mathematical arguments.
e. Recognize reasoning and proof as fundamental aspects of mathematics.
f. Make and investigate mathematical conjectures.
g. Develop and evaluate mathematical arguments and proofs.
h. 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.
ESSENTIAL QUESTIONS
• How do we compare metric measures of milliliters and liters?
• How do we compare customary measures of fluid ounces, cups, pints, quarts, and gallons?
MATERIALS
• “Water Balloon Fun!” student recording sheet
• Pastry School in Paris: An Adventure in Capacity by Cindy Neuschwander or similar book about liquid measure
GROUPING
Individual/Partner Task
TASK DESCRIPTION, DEVELOPMENT, AND DISCUSSION
In this task students compare liquid measures using milliliters, liters, fluid ounces, cups, pints, quarts, and gallons.
Comments
This task can be introduced by reading Pastry School in Paris: An Adventure in Capacity by Cindy Neuschwander or similar book about liquid measure. While reading the story, discuss the concepts and relationships that are used during the story.
As students are working, observe the strategies they use to solve the given problems. Consider strategies that would be helpful for other members of the class to see and understand. While students are working, ask selected students if they would be willing to share their work with the class. During the lesson summary, have students share the strategies they used with their classmates.
Task Directions
Students will follow the directions below from the “Water Balloon Fun!” student recording sheet.
Use what you know about the relationship between metric measures of capacity (milliliters, liters) or customary measures (fluid ounces, cups, pints, quarts, gallons) of capacity to solve the following problems. Show all of your work and explain your thinking.
1. The package for Matt’s water balloons says that each balloon holds 300 mL of water. How many balloons can he fill if he has 2 liters of water?
2. Beverly has 1.5 liters of water to fill six water balloons. Each balloon holds 0.35 liter of water. Does she have enough water to fill all six balloons? If not, how many balloons can she fill?
3. Camille has 6 water balloons. Each is filled with 4 fluid ounces of water. Bibi has 5 balloons. Each is filled with 1 cup of water. Whose balloons contain the most water?
4. Charlie filled all of his balloons with 2 quarts of water. Warren filled each of his 6 balloons with 1[pic] cups of water. Whose balloons contain the most water?
Background Knowledge
Before students are given this task, they will need to be familiar with customary and metric units of measure. Encourage students to refer to the graphic they created during the “More Punch, Please!” task (see page 24 of this unit).
Also, students should have had experiences that demonstrated the relationship between milliliters and liters (e.g. how the relationship between a millimeter and a meter are like the relationship between milliliters and liters – it takes 1,000 millimeters to make a meter and it takes 1,000 milliliters to make a liter; how a graduated cylinder that holds 100 mL would need to be filled 10 times in order to fill a 1 liter bottle).
The answers to the questions presented in this task are given below.
1. The package for Matt’s water balloons says that each balloon holds 300 mL of water. How many balloons can he fill if he has 2 liters of water?
Students could make a table to help them solve this problem.
|Number of Balloons|Amount of Water |
| |(in mL) |
|1 |300 |
|2 |600 |
|3 |900 |
|4 |1,200 |
|5 |1,500 |
|6 |1,800 |
|7 |2,100 |
So, Matt can fill 6 balloons with 2 liters of water.
2. Beverly has 1.5 liters of water to fill six water balloons. Each balloon holds 0.35 liter of water. Does she have enough water to fill all six balloons? If not, how many balloons can she fill?
Students could use a picture to solve this problem. To fill six balloons, Beverly would need 0.35 ( 6 liters of water.
She would need a total of 0.35 liter ( 6 = 2.1 liters of water. She only has 1.5 liters of water. She only has enough to fill 1.5 liters ÷ 0.35 liters or approximately 4.29 balloons.
So, Beverly can only fill 4 balloons with 1.5 liters of water.
3. Camille has 6 water balloons. Each is filled with 4 fluid ounces of water. Bibi has 5 balloons. Each is filled with 1 cup of water. Whose balloons contain the most water?
Students would need to use the following relationships to solve this problem.
8 fluid ounces = 1 cup
Camille has 4 fluid ounces ( 6 water balloons = 24 fluid ounces. Knowing that 8 ounces equals 1 cup, we know that Camille has 24 fluid ounces ÷ 8 fluid ounces = 3 cups of water.
Bibi has 1 cup ( 5 balloons = 5 cups of water. 5 cups is more than 3 cups. So, Bibi’s balloons contain 2 cups more water.
4. Charlie filled all of his balloons with 2 quarts of water. Warren filled each of his 6 balloons with 1[pic] cups of water. Whose balloons contain the most water?
Students could make a table to help them solve this problem.
|Warren’s Balloons |
|Number of Balloons|Amount of Water |
| |(in Cups) |
|1 |1[pic] |
|2 |3 |
|3 |4[pic] |
|4 |6 |
|5 |7[pic] |
|6 |9 |
Questions/Prompts for Formative Student Assessment
• How can you compare those amounts of water?
• What are the relationships you need to know in order to solve this problem? How do you know?
• How will you use those relationships to solve this problem?
• How much water will he (or she) need? How do you know?
• Who has more water in their balloons? How do you know?
• How can you model this problem?
• Why did you choose to model the problem this way?
Questions for Teacher Reflection
• Which students are able to compare liquid measures in the customary system?
• Which students are able to compare liquid measures in the metric system?
• Which students are able to move flexibly from one unit of measure to another within the same system?
• Which students are able to show how fluid ounces, cups, pints, quarts, and gallons are related?
DIFFERENTIATION
Extension
• Encourage students to create story problems that require the comparison of liquid measures and to solve the problems. Ask a partner to solve the problems.
Intervention
• Some students may need opportunities to solve the problems using measuring cups and water, rice, or sand.
• Provide students with a table that displays the relationships between different units of measure. This will allow students to focus on what the problem is asking.
• Similarly, some students may benefit by using a calculator to solve these problems. That would allow them to concentrate on the problem, not the operations required using decimal numbers.
Name______________________________________ Date ______________________________
Water Balloon Fun!
Use what you know about the relationship between metric measures of capacity (milliliters, liters) or customary measures (fluid ounces, cups, pints, quarts, gallons) of capacity to solve the following problems. Show all of your work and explain your thinking.
|The package for Matt’s water balloons says | |
|that each balloon holds 300 mL of water. How| |
|many balloons can he fill if he has 2 liters| |
|of water? | |
|Beverly has 1.5 liters of water to fill six | |
|water balloons. Each balloon holds 0.35 | |
|liter of water. Does she have enough water | |
|to fill all six balloons? If not, how many | |
|balloons can she fill? | |
|Camille has 6 water balloons. Each is filled| |
|with 4 fluid ounces of water. Bibi has 5 | |
|balloons. Each is filled with 1 cup of | |
|water. Whose balloons contain the most | |
|water? | |
|Charlie filled all of his balloons with 2 | |
|quarts of water. Warren filled each of his 6| |
|balloons with 1[pic] cups of water. Whose | |
|balloons contain the most water? | |
LEARNING TASK: Differentiating Area and Volume
STANDARDS ADDRESSED
M5M4. Students will understand and compute the volume of a simple geometric solid.
a. Understand a cubic unit (u3) is represented by a cube in which each edge has the length of 1 unit.
b. Identify the units used in computing volume as cubic centimeters (cm3), cubic meters (m3), cubic inches (in3), cubic feet (ft3), and cubic yards (yd3).
M5P1. Students will solve problems (using appropriate technology).
a. Build new mathematical knowledge through problem solving.
b. Solve problems that arise in mathematics and in other contexts.
c. Apply and adapt a variety of appropriate strategies to solve problems.
d. Monitor and reflect on the process of mathematical problem solving.
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.
ESSENTIAL QUESTIONS
• Why is volume represented with cubic units and area represented with square units?
• How are area and volume alike and different?
MATERIALS
• “Differentiating Area and Volume” student recording sheet
• Newspaper
• Construction paper
• Copy paper
• Grid paper (cm, in)
• Scissors
• Masking tape
• Rulers
• Meter sticks
• Measuring tape
• Cardstock or poster board
• Markers
GROUPING
Small Group
TASK DESCRIPTION, DEVELOPMENT, AND DISCUSSION
Students create a display of square and cubic units in order to compare/contrast the measures of area and volume.
Comments
This is a cooperative learning activity in problem solving. Students are provided with materials, but no initial instruction is given on how to build the models. This task will help give students a tangible model of square units and cubic units.
To open this task, students can discuss in their small groups what they know about area and volume. Key points of a class discussion can be recorded on chart paper.
Students will work in small groups to build models to represent units of area and units of volume. When the groups have completed their projects they will share with the class what they built, what each is called, and how each compares to some of the other models built by other groups.
Background Knowledge
Students should realize that the square units represent 2-dimensional objects and have both length and width. If students are having difficulty determining how to create these, have a class discussion about the word “square.” What comes to mind? How do you think this word might be related to area?
Note: The figures above are not drawn to scale.
Students should also realize that the cubic units represent 3-dimensional objects and have length, width, and height. If students are having difficulty determining how to create these, have a class discussion about the words “cube” and “cubic.” What comes to mind? How do you think these words might be related to volume?
Note: The figures above are not drawn to scale.
Task Directions
Students will follow the directions below from the “Differentiating Area and Volume” student recording sheet.
Create a display for area and volume by creating the following models. Use newspaper, construction paper, copy paper, grid paper, scissors, masking tape, meter sticks, markers and/or cardboard to build the models.
• Area models – 1 cm2, 4 cm2, 1 in2, 4 in2, 1 ft2, 1 m2
• Volume models – 1 cm3, 8 cm3, 1 in3, 8 in3, 1 ft3, 1 m3
At the end of the work period, each group will share their completed models and explain what has been built, what each is called, and how your models compare with some of the other models built by the other groups.
Individually, answer the following questions:
• How are area and volume alike?
• How are area and volume different?
• Why is area labeled with square units?
• Why is volume labeled with cubic units?
• Think about your home – bedroom, kitchen, bathroom, living room.
‐ What would you measure in square units? Why?
‐ What would you measure in cubic units? Why?
Questions/Prompts for Formative Student Assessment
• What does cm2 mean? cm3? How do you know?
• What does in2 mean? in3? How do you know?
• What does ft2 mean? ft3? How do you know?
• What does m2 mean? m3? How do you know?
• What shape is used to represent cm2? cm3? in2? in3? ft2? ft3? m2? m3?
• How can you create a shape that represents 4 cm2? What length would you use? How do you know?
• How can you create a shape that represents 8 cm3? What length would you use? How do you know?
Questions for Teacher Reflection
• Which students are able to distinguish between square units and cubic units?
• Which students are able to describe how area and volume are alike and different?
DIFFERENTIATION
Extension
• Ask students to describe the relationship between 4 cm2 and 8 cm3 as well as 9 cm2 and 27 cm3. Then have students generate other pairs of numbers that have the same relationship. What do they notice? (Students may use 1 cm cubes placed on a 4 cm2 or 9 cm2 square to determine the dimensions of a cube built on the square.)
Intervention
• Allow students to create at least some of the figures using a word processing or a drawing computer program. This will allow students to easily create right angles, equal side lengths, and cubes with equal edge lengths.
• Students may benefit from using I” square tiles, 1” cubes, and similar 1 cm materials to create some of these models, especially 4 cm2, 4 in2, 8 cm3, and 8 in3.
TECHNOLOGY CONNECTION
• The activity is called “Cubes.” Students are able to select the dimensions of a cube and then fill it with cubic units.
Name______________________________________ Date ______________________________
Differentiating Area and Volume
Create a display for area and volume by creating the following models. Use newspaper, construction paper, copy paper, grid paper, scissors, masking tape, meter sticks, markers and/or cardboard to build the models.
• Area models – 1 cm 2, 4 cm2, 1 in2, 4 in2, 1 ft2, 1 m2
• Volume models – 1 cm3, 8 cm3, 1 in3, 8 in3, 1 ft3, 1 m3
At the end of the work period, each group will share their completed models and explain what has been built, what each is called, and how your models compare with some of the other models built by the other groups.
Individually, answer the following questions:
1. How are area and volume alike?
__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
______________________________________________________________________________
2. How are area and volume different? __________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
______________________________________________________________________________
3. Why is area labeled with square units?
__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
______________________________________________________________________________
4. Why is volume labeled with cubic units?
__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
______________________________________________________________________________
5. Think about your home – bedroom, kitchen, bathroom, living room.
What would you measure in square units? Why?
_________________________________________________________________________________________________________________________________________________________________________________________________________________________________
___________________________________________________________________________
What would you measure in cubic units? Why?
________________________________________________________________________________________________________________________________________________________________________________________________
________________________________________________________________
LEARNING TASK: How Many Ways?
STANDARDS ADDRESSED
M5M4. Students will understand and compute the volume of a simple geometric solid.
a. Understand a cubic unit (u3) is represented by a cube in which each edge has the length of 1 unit.
b. Identify the units used in computing volume as cubic centimeters (cm3), cubic meters (m3), cubic inches (in3), cubic feet (ft3), and cubic yards (yd3).
c. Derive the formula for finding the volume of a cube and a rectangular prism using manipulatives.
M5A1. Students will represent and interpret the relationships between quantities algebraically.
a. Use variables, such as n or x, for unknown quantities in algebraic expressions.
b. Investigate simple algebraic expressions by substituting numbers for the unknown.
M5P1. Students will solve problems (using appropriate technology).
a. Build new mathematical knowledge through problem solving.
b. Solve problems that arise in mathematics and in other contexts.
c. Apply and adapt a variety of appropriate strategies to solve problems.
M5P2. Students will reason and evaluate mathematical arguments.
a. Recognize reasoning and proof as fundamental aspects of mathematics.
b. Make and investigate mathematical conjectures.
c. Develop and evaluate mathematical arguments and proofs.
d. 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.
ESSENTIAL QUESTIONS
• Why is volume represented with cubic units?
• How do we measure volume?
• How can you find the volume of cubes and rectangular prisms?
MATERIALS
• “How Many Ways?” student recording sheet
• Snap cubes
GROUPING
Partner/Small Group Task
TASK DESCRIPTION, DEVELOPMENT, AND DISCUSSION
In this task, students will use 24 snap cubes to build cubes and rectangular prisms in order to generalize a formula for the volume of rectangular prisms.
Comments
To introduce this task ask students to make a cube and a rectangular prism using snap cubes. Discuss the attributes of cubes and rectangular prisms – faces, edges, and vertices. Initiate a conversation about the figures:
• What is the shape of the cube’s base?
• What is the shape of the rectangular prism’s base? The base of each is a rectangle (remember a square is a rectangle!).
Students should notice that the cube and rectangular prism are made up of repeated layers of the base. Describe the base of the figure as the first floor of a rectangular-prism-shaped building. Ask students, “What is the area of the base? Next, discuss the height of the figure. Ask students, “How many layers high is the cube?” or “How many layers high is the prism?” The number of layers will represent the height. DO NOT LEAD THE DISCUSSION TO THE VOLUME FORMULA. Students will use the results of this task to determine the volume formula for rectangular prisms on their own.
While working on the task, students do not need to fill in all ten rows of the “How Many Ways?” student recording sheet. Some students may recognize that there are only six different ways to create a rectangular prism using 24 snap cubes. For students who have found four or five ways to build a rectangular prism, tell them they have not found all of the possible ways without telling them exactly how many ways are possible. It is important for students to recognize when they have found all possible ways and to prove that they have found all of the possible rectangular prisms.
Once students have completed the task, lead a class discussion about the similarities and differences between the rectangular prisms they created using 24 snap cubes. Allow students to explain what they think about finding the volume of each prism they created. Also, allow students to share their conjectures about an efficient method to find the volume of any rectangular prism. Finally, as a class, come to a consensus regarding an efficient method for finding the volume of a rectangular prism.
Background Knowledge
Students should have had experiences with the attributes of rectangular prisms, such as faces, edges, and vertices, in fourth grade. This task will build upon this understanding.
The “How Many Ways?” student recording sheet asks students to determine the area of the base of each prism using the measurements of base and height of the solid’s BASE. The general formula for the area of a parallelogram (as discussed in the task on deriving formulas) is A = bh. Knowing the general formula for the area of a parallelogram enables students to memorize ONE formula for the area of rectangles, squares, and parallelograms since each of these shapes is a parallelogram.
The general formula for the volume of a prism is V = Bh, where B is the area of the BASE of the prism and h is the height of the prism. Knowing the general formula for the volume of a prism prevents students from having to memorize different formulas for each of the types of prisms they encounter.
There are six possible rectangular prisms that can be made from 24 snap cubes.
1 ( 1 ( 24
1 ( 2 ( 12
1 ( 3 ( 8
1 ( 4 ( 6
2 ( 2 ( 6
2 ( 3 ( 4
Students may identify rectangular prisms with the same dimensions in a different order, for example, 1 ( 4 ( 6, 1 ( 6 ( 4, 6 ( 1 ( 4, 6 ( 4 ( 1, 4 ( 1 ( 6, 4 ( 6 ( 1. All of these are the same rectangular prism, just orientated differently. It is okay for students to include these different orientations on their recording sheet. However, some students may need to be encouraged to find different rectangular prisms.
Task Directions
Students will follow the directions below from the “How Many Ways?” student recording sheet.
1. Count out 24 cubes.
2. Build all the rectangular prisms that can be made with the 24 cubes. For each rectangular prism, record the dimensions and volume in the table below.
3. What do you notice about the rectangular prisms you created?
4. How can you find the volume without building and counting the cubes?
[pic]
Questions/Prompts for Formative Student Assessment
• What is the shape of the rectangular prism’s base? How can you find the area of the base?
• What is the height of the rectangular prism? How do you know? (How many layers or “floors” does it have?)
• What is the volume of the rectangular prism? How do you know? (How many snap cubes did you use to make the rectangular prism? How do you know?)
Questions for Teacher Reflection
• Which students are able to discuss how cubes are also rectangular prisms?
• Which students are able to build different rectangular prisms using 24 snap cubes?
• Which students are able to determine the formula for the volume of rectangular prisms?
DIFFERENTIATION
Extension
• Ask students to suggest possible dimensions for a rectangular prism that has a volume of 42 cm3 without using snap cubes.
• Ask students to explore the similarities and differences of a rectangular prism with dimensions 3 cm x 4 cm x 5 cm and a rectangular prism with dimensions 5 cm x 3 cm x 4 cm. Students can consider the attributes and volumes of each of the prisms.
• Students can calculate the area of each surface of the solid and determine the total surface area.
Intervention
• Some students may need organizational support from a peer or by working in a small group with an adult. This person may help students recognize duplications in their table as well as help them recognize patterns that become evident in the table.
• Some students may benefit from using the “Cubes” applet on the Illuminations web site (see link in “Technology Connection” below). It allows students to easily manipulate the size of the rectangular prism and then build the rectangular prism using unit cubes.
TECHNOLOGY CONNECTION
• The activity is called “Cubes.” Students are able to select the dimensions of a cube and then fill it with cubic units.
Name______________________________________ Date ______________________________
How Many Ways?
1. Count out 24 cubes.
2. Build all the rectangular prisms that can be made with the 24 cubes. For each rectangular prism, record the dimensions and volume in the table below.
3. What do you notice about the rectangular prisms you created?
4. How can you find the volume without building and counting the cubes?
|Shape # |Area of the BASE of the Solid | Number of Layers of the Base |Volume |
| |A = bh |(Height of Solid) | |
| |base |height | | |
|1 | | | | |
|2 | | | | |
|3 | | | | |
|4 | | | | |
|5 | | | | |
|6 | | | | |
|7 | | | | |
|8 | | | | |
|9 | | | | |
|10 | | | | |
LEARNING TASK: Super Solids
STANDARDS ADDRESSED
M5M4. Students will understand and compute the volume of a simple geometric solid.
d. Compute the volume of a cube and a rectangular prism using formulae.
e. Estimate the volume of a simple geometric solid.
M5A1. Students will represent and interpret the relationships between quantities algebraically.
a. 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).
a. Build new mathematical knowledge through problem solving.
b. Solve problems that arise in mathematics and in other contexts.
c. Apply and adapt a variety of appropriate strategies to solve problems.
d. Monitor and reflect on the process of mathematical problem solving.
M5P2. Students will reason and evaluate mathematical arguments.
a. Recognize reasoning and proof as fundamental aspects of mathematics.
b. Make and investigate mathematical conjectures.
c. Develop and evaluate mathematical arguments and proofs.
d. 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.
ESSENTIAL QUESTIONS
• 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?
MATERIALS
• Empty boxes (such as shoe, cereal, cracker, etc.)
• Centimeter cubes
• Rulers or measuring tapes
• “Super Solids” task sheet
GROUPING
Partner/Small Group Task
TASK DESCRIPTION, DEVELOPMENT, AND DISCUSSION
In this task, students will estimate and find the volume of real-world objects.
Comments
For each object, students will estimate the number of centimeter cubes that will be needed completely fill the box. (They should NOT fill the box with centimeter cubes to estimate.) After all estimates have been recorded, students will use their measurement tools to determine the volume of each box. All measurements should be to the nearest tenth of a centimeter.
After students have found the volume of each box, compare results. Discuss any discrepancies. Allow pairs of students to share their strategies for making their estimate and determining the volume.
Background Knowledge
Students should realize that square units represent 2-dimensional objects and have both length and width, while cubic units represent 3-dimensional objects and have length, width, and height.
Students should have had experiences with the attributes of rectangular prisms, such as faces, edges, and vertices, in fourth grade. This task will build upon this understanding.
The general formula for the area of a parallelogram is A = bh. Knowing the general formula for the area of a parallelogram enables students to memorize ONE formula for the area of rectangles, squares, and parallelograms since each of these shapes is a parallelogram.
The general formula for the volume of a prism is V = Bh, where B is the area of the BASE of the prism and h is the height of the prism. Knowing the general formula for the volume of a prism prevents students from having to memorize different formulas for each of the types of prisms they encounter.
Task Directions
Students will follow the directions below from the “Super Solids” student recording sheet.
For each object you choose, estimate the number of centimeter cubes that will be needed to completely fill the box. Once you have recorded your estimate, measure the object to determine the volume of each box.
(All measurements should be recorded to the nearest tenth of a centimeter.
[pic]
Questions/Prompts for Formative Student Assessment
• How did you find your estimate for the volume of your rectangular prism?
• How did you find the area of the base of your prism?
• How did you find the volume of your prism?
• What is [pic] ( [pic]? What is 0.1 ( 0.1? Where should you place your decimal in your answer? How do you know? (Students should recognize that [pic] ( [pic] = [pic] and that [pic] ( [pic] = [pic]. Therefore, 0.1 ( 0.1 = 0.01 and 0.01 ( 0.1 = 0.001.
Questions for Teacher Reflection
• Which students are able to make reasonable estimates?
• Which students are able to accurately measure to the nearest tenth of a centimeter?
• Which students are able to fluently compute using decimals?
• Which students are able to accurately compute the volume of each box?
DIFFERENTIATION
Extension
• Students can calculate the area of each surface of the solid and determine the total surface area.
Intervention
• Encourage students to fill their boxes with centimeter cubes. This allows students to use models when determining volume.
Name______________________________________ Date ______________________________
Super Solids
For each object you choose, estimate the number of centimeter cubes
that will be needed to completely fill the box. Once you have recorded your estimate, measure the object to determine the volume of each box.
(All measurements should be recorded to the nearest tenth of a centimeter.
|Object |Estimate |Area of Base |Height |Volume |
| |in cm3 |A = b ( h |of Prism |of Prism in cm3 |
| | | | |A = B ( h |
| | | | | |
| | | | | |
| | | | | |
| | | | | |
Unit 5 Culminating Task
PERFORMANCE TASK: Boxing Boxes
STANDARDS ADDRESSED
M5M4. Students will understand and compute the volume of a simple geometric solid.
a. Understand a cubic unit (u3) is represented by a cube in which each edge has the length of 1 unit.
b. Identify the units used in computing volume as cubic centimeters (cm3), cubic meters (m3), cubic inches (in3), cubic feet (ft3), and cubic yards (yd3).
c. Derive the formula for finding the volume of a cube and a rectangular prism using manipulatives.
d. Compute the volume of a cube and a rectangular prism using formulae.
e. Estimate the volume of a simple geometric solid.
f. Understand the similarities and differences between volume and capacity.
M5P1. Students will solve problems (using appropriate technology).
a. Build new mathematical knowledge through problem solving.
b. Solve problems that arise in mathematics and in other contexts.
c. Apply and adapt a variety of appropriate strategies to solve problems.
d. Monitor and reflect on the process of mathematical problem solving.
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.
ESSENTIAL QUESTIONS
• 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?
MATERIALS
• “Boxing Boxes” student recording sheet
• Snap cubes and/or 1” grid paper (several sheets per student), scissors, and clear tape
• “Boxing Boxes, Part II” student recording sheet (optional)
GROUPING
Individual/Partner Task
TASK DESCRIPTION, DEVELOPMENT, AND DISCUSSION
In this task, students explore volume while packing shipping boxes with various-sized merchandise boxes.
Comments
This task can be introduced by asking small groups of students to create the different sized merchandise boxes using grid paper or snap cubes. If using grid paper, students will need to sketch the nets for the boxes described on 1” grid paper and then cut the nets out and fold them to create the rectangular prisms. If using snap cubes, students can create the required rectangular prisms with snap cubes using the dimensions required. Students can then use these models while working on the task.
Allow students to create their own chart for the “Boxing Boxes” task that makes sense to them. Then allow students to share their chart with students in their small group and choose two or three students who created different charts to share their work with the class.
Background Knowledge
According to Van de Walle (2006) “volume typically refers to the amount of space that an object takes up” whereas “capacity is generally used to refer to the amount that container will hold” (p. 265). To distinguish further between the two terms, consider how the two are typically measured. Volume is measured using linear measures (ft, cm, in, m, etc) while capacity is measured using liquid measures (L, mL, qt, pt, g, etc). However, Van de Walle reminds educators, “having made these distinctions [between volume and capacity], they are not ones to worry about. The term volume can also be used to refer to the capacity of a container” (p. 266).
Van de Walle, J. A. & Lovin, L. H. (2006). Teaching students-centered mathematics: Grades 3-5. Boston: Pearson Education, Inc.
Notice that the capacity of the standard shipping box is 12 ft3. Therefore, the sum of the volumes of the merchandise boxes packed must equal 12 ft3 for each packing plan (see table below).
[pic]
A sample solution to the task is given below.
Additionally, students will need to write a letter to their boss explaining how to use the chart they created.
Task Directions
Students will follow the directions below from the “Boxing Boxes” student recording sheet.
You have been hired by Boxes Unlimited to help determine the best way to package merchandise for shipping.
Boxes Unlimited has a standard shipping box which will hold merchandise measuring 2 ft by 3 ft by 2 ft.
Boxes Unlimited needs to pack merchandise they receive into the standard shipping box. The merchandise arrives in four different box sizes.
Merchandise Box W is 1 ft. x 3 ft. x 2 ft.
Merchandise Box X is 1 ft. x 2 ft. x 2 ft.
Merchandise Box Y is 2 ft. x 2 ft. x 2 ft.
Merchandise Box Z is 1 ft. x 1 ft. x 1 ft.
Your task is to create a chart for employees to use as a reference when they are packing boxes for shipment. Be sure to include the volume of each merchandise box and the capacity of the standard shipping box on your chart.
Write a report to your boss explaining how to read your chart.
Questions/Prompts for Formative Student Assessment
• Have you found all of the possible ways to fill the standard shipping box? How do you know?
• What is the total capacity of the standard shipping box? Will the merchandise completely fill the standard shipping box? How do you know?
• How are you organizing your packing chart? Why did you choose this type of organizational chart?
• Can you explain how your chart could be used by the employees who pack boxes?
Questions for Teacher Reflection
• Which students are able to visualize filling the shipping boxes?
• Which students used the models to find different ways to fill the shipping boxes?
• Which students are able to accurately determine various packing options?
DIFFERENTIATION
Extension
• Ask students to consider a large shipping box with dimensions of 3 ft ( 3 ft ( 3 ft. What are the ways that this packing box could be filled with the given merchandise boxes? Students could work the task with this large shipping box rather than the regular shipping box. Next, students who worked with the large shipping box could be paired with students who worked on the standard shipping box. Partners could then be asked to determine which size box would be a better choice and justify their thinking.
Intervention
• Encourage students to use snap-cubes to create models of the merchandise boxes.
• Students who would benefit from a chart in which to record their work should be provided one. A sample is given below. See “Boxing Boxes, Part II” student recording sheet.
TECHNOLOGY CONNECTION
• Students can create models of the boxes using 1” grid paper. Each inch can represent 1 ft when creating models.
Name______________________________________ Date ______________________________
Boxing Boxes
You have been hired by Boxes Unlimited to help determine the best way to package merchandise for shipping.
Boxes Unlimited has a standard shipping box which will hold merchandise measuring 2 ft by 3 ft by 2 ft.
Boxes Unlimited needs to pack merchandise they receive into the standard shipping box. The merchandise arrives in four different box sizes.
Merchandise Box W is 1 ft. x 3 ft. x 2 ft.
Merchandise Box X is 1 ft. x 2 ft. x 2 ft.
Merchandise Box Y is 2 ft. x 2 ft. x 2 ft.
Merchandise Box Z is 1 ft. x 1 ft. x 1 ft.
Your task is to create a chart for employees to use as a reference when they are packing boxes for shipment. Be sure to include the volume of each merchandise box and the capacity of the standard shipping box on your chart.
Write a report to your boss explaining how to read your chart.
Name______________________________________ Date ______________________________
Boxing Boxes
Part II
The volume of the merchandise boxes are as follows:
Merchandise Box W: _________________________________
Merchandise Box X: _________________________________
Merchandise Box Y: _________________________________
Merchandise Box Z: _________________________________
The capacity of the standard shipping box is _________________________________ .
[pic]
-----------------------
2 cups = 1 pint
2 pints = 1 quart
So, 4 cups = 1 quart
Using the relationship above and the chart to the left, we know that Warren would need 9 cups of water to fill 6 balloons. If 4 cups = 1 quart, then 8 cups = 2 quarts. Therefore Warren used 2 quarts ( 1 cup of water. That is 1 cup more than the amount of water Charlie used. So, Warren’s balloons contain more water.
0.35 liters
0.35 liters
0.35 liters
0.35 liters
0.35 liters
0.35 liters
1,000 mL = 1 L
Using the relationship above and the chart to the left, we know that Matt could fill 6 balloons with 2 liters of water (with 200 mL left over). In order to fill 7 balloons, he would need 2,100 mL or 2 liters 100 mL. So, he would need an additional 100 mL to fill 7 balloons.
1 gallon = 4 quarts
1 gallon = 8 pints
1 gallon = 16 cups
1 gallon = 128 fluid ounces
1 quart = 2 pints
1 pint = 2 cups
1 cup = 8 fluid ounces
1 ft
1 ft
1 ft
1 in
1 in
1 in
1 cm
1 cm
1 cm
1 m
1 m
1 m
1 m
1 cm
1 in
1 ft
1 cm
MATHEMATICS
Party Punch Directions:
Serves 16 (serving size: 8 fluid ounces) Place sherbet in punch bowl. Pour in fruit punch and lemon-lime soda.
Ingredients:
2 Pints Strawberry Sherbet
2 Quarts Fruit Punch, chilled
32 Fluid Ounces Lemon-Lime flavored carbonated beverage, chilled
1 in
1 ft
Merchandise Packing Guide
[pic]
The volume of the merchandise boxes are as follows:
Merchandise Box W: 1 ft x 3 ft x 2 ft = 6 ft3
Merchandise Box X: 1 ft x 2 ft x 2 ft = 4 ft3
Merchandise Box Y: 2 ft x 2 ft x 2 ft = 8 ft3
Merchandise Box Z: 1 ft x 1 ft x 1 ft = 1 ft3
The capacity of the standard shipping box is 2 ft ( 3 ft ( 2 ft = 12 ft3
1 m
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