Tab 9: Volume and Capacity Table of Contents

Transcription

1 Tab 9: Volume and Capacity Table of Contents Master Materials List 9-iii Foundations for Capacity and Volume 9-1 Handout 1-Grade Level Expectations for Development of Attributes of Capacity and Volume 9-10 Handout 2-Grade Level Expectations for Development of Attributes of Capacity and Volume Answer Key* 9-11 Transparency 1/Handout 3 -Clarifications from TEA Concerning Capacity and Volume 9-12 The following activities are optional for trainings with only grades K-2 teachers. Trainers should make a decision about whether to include these activities based on the experiences and understanding of their audience. Skeletons in the Closet 9-14 Handout 1-Recording Sheet 9-22 Fill Er Up! 9-23 Handout 1-Recording Sheet 9-29 Transparency 1-Class Recording Chart 9-30 Full to Capacity 9-31 Handout 1-Participant Recording Sheet 9-43 Race to a Gallon 9-44 Handout 1-Spinner Sheets 9-50 Handout 2-Recording Sheet 9-54 Handout 3-Summary Sheet 9-55 Handout 4-Summary Sheet Answer Key* 9-56 * This document was developed as a resource for trainers, but it may be used with participants at the trainer's discretion. Tab 9: Volume and Capacity: Table of Contents 9-i

2 My Tub Runneth Over! 9-58 Handout 1-My Tub Runneth Over! Skit Script 9-65 Handout 2-Recording Sheet # Handout 3-Recording Sheet # Handout 4-Recording Sheet # Handout 5-Recording Sheet #3 Answer Key* 9-71 Handout 6-Recording Sheet # Sizing Up the Cube 9-73 Handout 1-Recording Sheet 9-79 * This document was developed as a resource for trainers, but it may be used with participants at the trainer's discretion. Tab 9: Volume and Capacity: Table of Contents 9-ii

3 Tab 9: Volume and Capacity Master Materials List Dishpans Funnel or 2-liter bottle with end cut out Variety of empty assorted containers Variety of scoops/spoons with no markings Ziploc bag of dried beans, peas, rice, or other material Foundations for Capacity and Volume Handouts and Transparency The following materials are not within this tab of the notebook, but they can be accessed by clicking on the link below. K-5 Mathematics TEKS The following materials are needed for the optional activities: Skeletons in the Closet, Fill Er Up, Full to Capacity, Race to a Gallon, My Tub Runneth Over, and Sizing Up the Cube inch (or other small diameter) dowel rods precut into 1 foot lengths 1 liter water bottle 1-cup measuring cup (scoop type) 1-pint plastic jar 1-quart plastic jar Base ten blocks Beach towels Centimeter cubes (make sure that you have Rainbow cubes or the plastic centimeter cubes that are approximately 1 gram each so that the cubes will sink) Centimeter and inch grid paper on cardstock Centimeter grid paper on plain paper Clear spinner or paper clip to use as a spinner Collection of small objects such as marbles, small toys such as soldiers or dinosaurs, and so on (waterproof, non-floating, able to fit into the graduated cylinder) Corrugated cardboard cut into 12 inch by 12 inch squares Dishpan or mop bucket Empty gallon jug with 1-gallon level marked Eyedroppers Family size boxes of cereal Funnel Graduated cylinder calibrated in milliliters either 25 milliliter or 50 milliliter capacity Graduated cylinders with larger capacities Large bowl - large enough to hold the contents of an entire box of cereal Markers Masking tape Meter sticks Modeling clay, Play-Doh or adhesive putty (Handi-tak) Tab 9: Volume and Capacity: Master Materials List 9-iii

4 One-inch 90-degree elbow (PVC pipe) with one-inch inlet Paper towels Plastic decimeter cube model with masking tape over the labels Plastic, paper, or styrofoam bowls - cereal bowl size Rulers with inch and centimeter markings Scissors Tape Variety of clean, small, empty boxes Variety of scoops/spoons with no markings Water in a container that allows participants to pour easily Yardsticks Ziploc bag of dried beans, peas, rice, or other material Skeletons in the Closet Handout Fill Er Up! Handout and Transparency Full to Capacity Handout Race to a Gallon Handouts My Tub Runneth Over Handouts Sizing Up the Cube Handout The following materials are not within this tab of the notebook, but they can be accessed by clicking on the links below. Centimeter grid paper Inch grid paper Full to Capacity Summary PowerPoint K-5 Mathematics TEKS Be cautious when copying graph paper on the copy machine. The copier can distort the size of the grid. Printing grid paper referenced in the Materials List above works well; however, be sure to turn off any scaling before printing. Tab 9: Volume and Capacity: Master Materials List 9-iv

5 Activity: Foundations for Capacity and Volume TEKS: Overview: See Appendix for student lessons with correlated TEKS for capacity and volume. This lesson is designed to give teachers an opportunity to reflect on the vertical articulation of capacity and volume concepts in the TEKS from Kindergarten through Grade 5. A strong foundation in the early grades is essential for students to progress to the desired level of understanding at the end of Grade 5. It is helpful for teachers of Grades 3-5 to understand what concepts should already be in place when the students come to them. It is also beneficial for teachers of Grades K-2 to realize where the curriculum is headed so that they can help their students build a strong foundation that will carry them into the next phase of their learning. In order to look at the big picture, this lesson begins with a focus on understanding the attribute to be measured which is typically developed by direct comparisons (comparing and ordering), followed by the use of tools for non-standard units to assign a numerical value to the measurement. Typically, these concepts are addressed in the early grades, so participants will have opportunity to engage in several handson activities that deal with comparing, ordering, and the use of nonstandard units. Finally, the use of tools for standard units for capacity and volume are addressed. This activity should be included in all trainings for Grades K-2 teachers. Trainers should consider experiences and understanding of their audience in deciding whether to include this activity for trainings with Grades 3-5 teachers. Materials: Grades K-5 TEKS Handout 1-Grade Level Expectations for Development of Attributes of Capacity and Volume (page 9-10) Handout 2-Grade Level Expectations for Development of Attributes of Capacity and Volume Answer Key (page 9-11) Handout 3/Transparency 1 & 2-Clarifications from TEA concerning Capacity and Volume (pages ) Variety of empty assorted containers, 5 or 6 containers per group Dishpan, 1 per group Ziploc bag of dried beans, peas, rice, or other material, 1 per group Funnel or 2-liter bottle with the end cut out, 1 per group Variety of scoops with no markings, 2 or 3 sizes for each group Foundations for Capacity and Volume 9-1

6 Grouping: Time: 4 participants per group minutes Lesson: 1. Remind participants of the activity during the Measurement Overview (pages ) where they were asked to take two identical transparencies and roll each into a cylinder. On the first, the cylinder was formed by taping the two longer edges of the transparency together. On the second, the cylinder was formed by taping the two shorter edges of the transparency together. The participants were asked to predict what would happen if the contents of the tall, more narrow cylinder were poured into the shorter, wider cylinder. The activity from the Introduction to Measurement session demonstrates direct comparison of capacities which helps students form a strong foundation in the early grades in order for them to progress in their knowledge of capacity and volume. In this lesson, we will investigate several ideas that can be used in the early grades to develop this measurement sense for capacity and volume. 2. As stated by Van de Walle (2006), measurement involves comparing an attribute of an object with a unit that possesses that same attribute. So, one of the most critical aspects of teaching measurement in the early grades is to help students understand what it means to measure a given attribute. After understanding the attribute to be measured, students need to choose a unit of measure, then compare the attribute of the object and the same attribute of the unit chosen for the measurement. The measure of the attribute is a count of how many units are needed to fill, cover, or match the attribute of the object Foundations for Capacity and Volume 9-2

7 being measured (Van de Walle, 2006, p. 224). Part I: Direct Comparisons Comparing and Ordering 3. So, let s look at what types of activities can build a strong foundation for capacity and volume concepts. Let s begin with making comparisons. When possible, have your students use direct comparisons. 4. Ask participants to take a look at the Kindergarten TEKS for Measurement (K.10). Notice that the TEKS statement says directly compares. Kindergarten students should have many opportunities to directly compare two containers according to capacity (holds more, holds less, or holds the same). (K.10C) Also note that the TEKS statement refers to comparative language: holds more, holds less, holds the same. Ask participants to summarize the capacity/volume TEKS regarding capacity and volume on Handout 1-Grade Level Expectations for Development of Attributes of Capacity and Volume (page 9-10) as they go through the lesson. Handout 2 (page 9-11) is a completed version of the Grade Level Expectations for Development of Attributes of Capacity and Volume. If participants have done the activity Of Course We Have Standards (and Nonstandards)! (pages ), they will have already looked at the capacity and volume TEKS for Kindergarten through 5 th grade, along with the TEKS dealing with the attributes of length, area, and weight/mass. In that activity, the participants were focusing solely on the use of nonstandard units and standard Foundations for Capacity and Volume 9-3

8 5. Provide a wide variety of assorted containers for the group. Ask a representative from each group to come forward; then, give the representative two containers that are somewhat close in their capacities. When the representatives go back to their groups, each group should predict if one of the containers will hold more, or if the containers will hold the same. units. This lesson provides a more in-depth look at what is expected and appropriate for volume and capacity activities at each grade level. You ll need a large assortment of containers for the group. Go for lots of variety. Remind the participants that for their own classrooms, they can ask their students to bring a container or containers from home to donate to the class set to be used for capacity investigations. In addition, the teachers might consider placing this item on the school supply list for the start of the school year. 6. After each group has made its prediction, the group should verify which container holds more, holds less, or if the two containers hold the same. Give each group a dishpan (to catch any spills) and a Ziploc bag of dried beans, peas, rice, or the like. Each group will also need a funnel. Have the group members directly compare the capacities by filling one container, then pouring the contents into the other container. Make sure that the funnel has a wide enough mouth for the material to flow freely. You can also use an empty 2-liter bottle with the bottom cut out for a funnel. Remind participants that they should make sure that their students know how to interpret the results when they transfer the contents of one container into the other container. Asking the students for a verbal description of what happen and for their conclusion is a good way to assess this activity so that the teacher can follow up with additional questions if students are not comprehending the Foundations for Capacity and Volume 9-4

9 7. Now let s look at the capacity TEKS for 1 st grade: How do the TEKS for capacity in 1 st grade compare and contrast with the Kindergarten TEKS for capacity? The student is still doing direct comparisons and using comparative language (holds more, holds less, holds the same). Now the student is asked to "compare and order two or more containers according to capacity (from holds the most to holds the least)" (1.7E). 8. Ask a representative from each group to come back to get more containers. Each group now should have the two containers from the previous capacity direct comparison plus another 3 or 4 containers of varying capacities. Ask each group to order the containers from holds the most to holds the least. Each group should make a prediction first. Then the groups should use the dried beans, peas, rice, or other material to directly compare the capacities as they check to see if their predictions were accurate. Part II: Using Tools (Non-Standard Units) 9. Progressing to the 2 nd grade TEKS, let s look at the capacity concepts in the Measurement Strand. What things are similar to the Kindergarten and 1 st grade TEKS for the concept of capacity? Students are still making direct comparisons when needed, and they are still using comparative language (holds more, holds less, holds the same). What things are extending the scope of the concept of capacity within the 2 nd grade concept. Foundations for Capacity and Volume 9-5

10 TEKS? The students are now selecting and using non-standard units to describe capacity. In addition, the students are to recognize and use models that approximate standard units (from both the metric and customary systems) for length, weight/mass, capacity, and time. However, in the Student Expectations, length is the attribute where standard units are specifically addressed. Non-standard units are the emphasis for capacity measurements in 2 nd grade. 10. So, the next step in providing instruction designed to build a strong foundation for capacity concepts (and measurement concepts in general) is to move the students into the use of non-standard units. Now the students begin to see measurement as a way to assign a numerical value to the attribute being measured. 11. Have the participants mix up the order of their containers from the previous capacity investigation using direct comparisons, then trade the set of containers with another group. Each group should still have 5 or 6 containers of various shapes and sizes. The task for each group is to sort the containers by capacity again from holds the most to holds the least, but this time the participants will use a non-standard unit for capacity to compare the containers (2 nd Grade TEKS) instead of pouring the contents from one container to the other as they did in the previous part of the lesson (1 st Grade TEKS). 12. Give each group a scoop with no markings to use as a non-standard unit. The participants will scoop the dried beans, peas, rice, or other material into the container and record how many units it took to fill the container. The group should record the measurement on a sheet of paper, then proceed to scoop Make sure that the groups are filling the scoop to a consistent level each time. The scoop should be full to capacity but not overflowing. Be aware that some containers Foundations for Capacity and Volume 9-6

11 materials into another container to measure the capacity in terms of the number of scoops it takes to fill the container. The containers can be placed in order based on the measurement values themselves instead of by direct comparison. may not hold a whole number of scoops of the material. The participants may need to record a fraction of a scoop at times. This fraction talk can begin at this level in conjunction with 2.2 (C) where students are expected to use concrete models to determine if a fractional part of a whole is closer to 0, 1 2, or Now give each group a different, smaller scoop and have them measure the capacities of their containers using this new non-standard unit. 14. Ask: What happened to your measurements when you used a smaller scoop? The numerical value for the measurement was larger since the scoop was smaller, it took more scoops to fill the container. Ask: Did the amount of material that the container held change? No.the container still held the same amount. The number we used to describe the capacity changed, though, because of the unit we were using. 15. Ask: If I were to give you a larger scoop, what would happen to your capacity measurements? The numerical value for the measurement would be smaller. Since the scoop is larger, it will take fewer scoops to fill the container. The amount of material didn t change just the number we use to describe the capacity changes because of the unit we use. Part III: Using Tools (Standard Units) 16. Now let s look at the 3 rd grade TEKS for capacity in the measurement strand. Foundations for Capacity and Volume 9-7

12 What aspects are the same? Students are still making direct comparisons when needed and are still using comparative language (holds more, holds less, holds the same). What aspects are extended? The student is now selecting and using standard units to describe capacity/volume. Note that this TEKS is the first time that the word volume is introduced. In the student expectations, the students are expected to (E) identify concrete models that approximate standard units for capacity and then use them to measure capacity; and (F) use concrete models that approximate cubic units to determine the volume of a given container or other three-dimensional geometric figure. 17. So, we now need our instruction to guide the students to develop an understanding of standard units. Activities for developing this understanding of standard units could look like the things we have done throughout this series of lessons. 18. Now let s look at the 4 th grade TEKS capacity and volume concepts in the Measurement Strand. What is different? Notice that direct comparisons and uses comparative language do not appear in the 4 th grade TEKS. What does this imply? It implies that students should have mastered these concepts by now. They may still perform direct comparisons, and they will still use comparative language, but these things should be in place in the student s knowledge base by now. Foundations for Capacity and Volume 9-8

13 Also note that the students are using tools and standard units in both the metric system and the customary system. The other big concept here is performing simple conversions between different units of capacity within the customary measurement system. For volume, the students are estimating volume in cubic units. 19. Finally, let s look at the 5 th grade TEKS for capacity and volume in the Measurement Strand. Note that the Knowledge and Skills statement uses capacity/volume, but the student expectations refer to volume, but not capacity. The use of formulas begins at this level, and from this point forward, capacity is not mentioned in the TEKS. Remember that capacity is the same as the maximum volume. 20. Examples of activities to address the capacity/volume ideas in the 4 th grade and 5 th grade TEKS would look very similar to the lessons we have done throughout this series. 21. As participants reflect on the lesson, have them review their completed Grade Level Expectations for Development of Attributes of Capacity and Volume Recording Sheet as you share with them the information contained on the sheet labeled Handout 3/Transparency 1-Clarifications from TEA Concerning Capacity and Volume (pages ). You might wish to use the Clarifications from TEA Concerning Capacity and Volume just as a handout, or you might wish to make a transparency of the page to project as you share the information with the participants. Assessment: Resources: Assessment is done informally throughout the lesson during the discussions. Van de Walle, J. A. (2006). Teaching student-centered mathematics: Grades K-3. Boston: Pearson Education, Inc. Foundations for Capacity and Volume 9-9

14 Grade Level Expectations for Development of Attributes of Capacity and Volume Kindergarten Fifth Grade As you investigate the TEKS dealing with capacity and volume for each grade, summarize your findings below. Attributes Kindergarten 1 st Grade 2 nd Grade 3 rd Grade 4 th Grade 5 th Grade Capacity and Volume Handout 1 Foundations for Capacity and Volume 9-10

15 Grade Level Expectations for Development of Attributes of Capacity and Volume Kindergarten Fifth Grade Sample Answer Key As you investigate the TEKS dealing with capacity and volume for each grade, summarize your findings below. Attributes Kindergarten 1 st Grade 2 nd Grade 3 rd Grade 4 th Grade 5 th Grade Capacity and Volume Perform direct comparisons according to capacity two containers Use comparative language Perform direct comparisons according to capacity two or more containers Use comparative language Perform direct comparisons according to capacity Use comparative language Select and use nonstandard units to describe capacity Recognize and use models that approximate standard units for capacity (SI and customary) Perform direct comparisons according to capacity Use comparative language Select and use standard units to describe capacity/volume Identify concrete models that approximate standard units for capacity and use them to measure capacity Use concrete models that approximate cubic units to determine the volume of a given container or other threedimensional geometric figure Use measurement tools to measure capacity/volume Estimate and use measurement tools to determine capacity Perform simple conversions between different units of capacity Use concrete models of standard cubic units to measure volume Estimate volume in cubic units Apply measurement concepts involving capacity/volume Perform simple conversions within the same measurement system Connect models for volume with the volume formula (rectangular prisms) Select and use appropriate units and formulas to measure volume Handout 2 Foundations for Capacity and Volume 9-11

16 Clarifications from TEA Concerning Capacity and Volume The following information was received via from TEA to clarify some questions from the MTR writing team concerning the capacity and volume TEKS. Definitions Capacity is the description of the maximum amount of material a container or receptacle can hold/contain. This is the maximum volume of the container. Volume is how much 3-dimensional space a given material occupies. Volume is a property of the material. We (math) typically talk about volume with cubic units. The concept of volume involves many developmental milestones; the idea of conservation is one example. If I have 36 blocks, the volume of the figure is 36 blocks regardless of how I rearrange the 36 blocks. Another example is subdivision, breaking a larger object into smaller congruent sub-pieces. In kindergarten and first grade, children are asked to compare the capacity of different containers. They do not have to subdivide the containers into smaller units either standard or nonstandard units. In second grade, children determine the capacity of a given container using a nonstandard unit. In essence the students are measuring the volume of the container using a nonstandard unit. We are just not using the word volume. Developmentally we are subdividing but not worrying about conservation because we are restricted to the structure of our containers. In third grade, children identify concrete models that approximate standard units for capacity. Here is where misconceptions come into play. These are also units for determining volume. For instance the volume of a tablespoon is 3 teaspoons. The capacity of the tablespoon is also 3 teaspoons because the maximum volume (in teaspoons) a tablespoon will hold is 3 teaspoons. We also begin to use concrete cubic units to determine the volume of a container or other figure. Since we are finding the maximum number of these units we are finding the capacity of the container. In 4 th grade we are still talking about estimating the capacity (maximum volume) but using standard and SI (metric) units. We also convert between different units of capacity which are the same units as volume. We use concrete models of standard cubic units to measure volume. We do not have to deal just with containers now. At 5 th grade there is a discrepancy between the knowledge statement and the student expectations. We say capacity/volume in the knowledge statement but only use the term volume in the student expectations. We begin the use of formulas at this grade level. From this point on capacity is not mentioned in the TEKS. Transparency1/Handout 3-1 Foundations for Capacity and Volume 9-12

17 An implied message in the TEKS appears to be that capacity can be measured concretely and volume must be calculated. This is incorrect. When you determine the capacity of a container you have determined its maximum volume no matter what methodology is used. The units are the same for capacity or volume. Science does not use the term capacity only the term volume. The SI (metric) unit of volume is the cubic centimeter or 1 milliliter. A one liter pitcher also has a volume of 1 cubic decimeter. Its capacity or maximum volume is the same one liter or 1 cubic decimeter. Capacity is not a subset, it is a description. Volume is a property. Volume can be converted within customary units such as quarts and cubic inches. It is just much simpler with SI (metric) units. Transparency1/Handout 3-2 Foundations for Capacity and Volume 9-13

18 Activity: TEKS: Skeletons in the Closet (Optional activity for Grades K-2 Training) See Appendix for student version of this activity with correlated TEKS. Overview: In this activity, participants will reinforce their knowledge about the attributes of models for the following standard units: cubic centimeter, cubic meter, cubic inch, cubic foot, and cubic yard. Concrete models for the standard units will be constructed in order to help participants visualize how much space the standard units would each occupy. In addition, selecting an appropriate unit for measuring the volume of various objects will be addressed. This activity should be included in all trainings for Grades 3-5 teachers. Materials: Handout 1-Recording Sheet (page 9-22) Centimeter cubes, 1 per participant 1-centimeter grid paper, 1 sheet per participant 1-inch grid paper on cardstock, 1 sheet per participant Markers Scissors Tape Rulers with inch and centimeter markings Corrugated cardboard cut into 12 inch by 12 inch squares, 2 squares per group Modeling clay, Play-Doh or adhesive putty (Handi-tak) 1 4 -inch (or other small diameter) dowel rods precut into 1 foot lengths, 4 per group One-inch 90-degree elbow (PVC pipe) with one-inch inlet, 16 total for the large group Meter stick, 1 per group You will also need 12 meter sticks total for the large group. If you have fewer than 12 groups, have some extra metersticks on hand. Yardstick, 1 per group You will also need 12 yardsticks total for the large group. If you have fewer than 12 groups, have some extra yardsticks on hand. Grouping: Time: Groups of 4 participants minutes Skeletons in the Closet 9-14

19 Lesson: 1. Have each participant select a centimeter cube. Ask: What attributes of the centimeter cube can be measured with the centimeter ruler? length, width, and/or height Even though participants are working in small groups, it is beneficial for each participant to have his or her own centimeter cube and centimeter ruler to use during this activity. 2. Have each participant measure the length and the width of one face of the centimeter cube using the centimeter ruler. Ask: What did you find for the measures of the length and width of one face? The length and width of one face are each 1 centimeter in length. 3. Have participants use their centimeter rulers to measure the length and width of one small square on a sheet of centimeter grid paper. (See Materials List for link to centimeter grid paper.) Ask: What did you find for the measures of the length and width of one small square on the grid paper? The length and width of one small square on the grid paper are each 1 centimeter in length. Be cautious when copying graph paper on the copy machine. The copier can distort the size of the grid. Printing grid paper directly from the URLs referenced in the Materials List works well; however, be sure to turn off any scaling before printing. Remind participants that one small square on the grid paper represents 1 square centimeter since the length and width are each 1 centimeter in length. 4. Ask: How many squares on the grid paper would it take to cover one face of the centimeter cube? It would take one square of the grid paper or 1 square centimeter. Ask: What attribute of the centimeter cube are we measuring when we say that it would take one square centimeter to cover one face of the centimeter cube? area of one face Skeletons in the Closet 9-15

20 5. Volume is typically measured in cubic units. Since this centimeter cube is already in the shape of a cube, what can we say about its volume? Its volume is 1 cubic centimeter. 6. Have each group record a list of attributes and their measures for the centimeter cube on Handout 1-Recording Sheet (page 9-22). Although lists may vary, some important attributes of the centimeter cube necessary for our discussion of volume in the remainder of this training are listed below: 12 edges that measure 1 centimeter in length 6 faces that measure 1 square centimeter in area; each face is a square with a length and width of 1 centimeter in length 1 cube that measures 1 cubic centimeter in volume 7. Tell participants that they have just described a model for the standard unit of a cubic centimeter. 8. In this lesson, we will build models for other standard cubic units so that we might develop a feel for how much space would be occupied by each. What attributes do you think a cubic inch would have? 12 edges that measure 1 inch in length 6 faces that measure 1 square inch in area; each face is a square with a length and width of 1 inch in length 1 cube that measures 1 cubic inch in volume 9. Give each participant a sheet of 1-inch grid paper on cardstock (See Materials List for grid paper link). Guide them to shade in 6 square inches in an arrangement that can be For each of the subsequent models built during this lesson, we will follow the same basic procedure: predict, build, verify, record. You might choose to let participants find their own nets for a cube. Another challenge is to have participants find all possible Skeletons in the Closet 9-16

21 folded to make a cube (a net for a cube) such as the following: nets for a cube. Have each participant cut out the net, then fold into a cube and tape closed. Have participants verify that the attributes they predicted are indeed true for the inchcube, then record these attributes on Recording Sheet # So, now we have a model for a cubic centimeter (metric system) and for a cubic inch (customary). 11. Let s name some objects for which the cubic centimeter would be an appropriate choice for measuring volume. Metric units are sometimes called SI units. SI stands for System Internationale. Responses will vary, but the objects should be fairly small. Now let s name some objects for which the cubic inch would be an appropriate choice for measuring volume. 12. What if I need to measure the volume of this room? Would cubic centimeters be appropriate? What about cubic inches? No. Even though the volume of the room could be measured in cubic centimeters or in cubic inches, those cubic units are very small when compared to the volume of the room. We need a larger cubic unit. 13. Let s develop a model for a cubic foot. Ask: What attributes do you think will be present on a cubic foot? 12 edges that measure 1 foot in length 6 faces that measure 1 square foot in area; each face is a square with a Skeletons in the Closet 9-17

22 length and width of 1 foot each 1 cube with a volume of 1 cubic foot 14. Have each group build a model for a cubic foot. Give each group two pre-cut squares of corrugated cardboard that are 12 inches by 12 inches. Using a ruler, have each group verify that the squares have an area of 1 square foot. Place a small piece of modeling clay, Play- Doh or adhesive putty in each corner of one of the squares. Then take 4 dowel rods (precut to 1 foot each in length) and stand one dowel rod up in each corner of the square. Next, place a small piece of modeling clay, Play-Doh or adhesive putty on top of each dowel rod, then lay the other square on the top of the cube. The dowel rods form the vertical edges of the cube. A print-shop can cut squares to these dimensions very quickly and easily. By constructing the cubic foot in this manner, the model is easy to disassemble for storage. Instead of building the cubic foot in this manner, you might consider using a manufactured box for this model. Office supply stores like Office Depot or Staples sell cardboard boxes which measure 12" by 12" by 12". The boxes store flat for convenience. 15. Now we have a skeleton model for a cubic foot. Our model is see-through, but a solid cube this size or a box this size would have the same volume 1 cubic foot. 16. Have participants verify that the attributes that they listed for a cubic foot are true, then remind them to record these attributes on the Recording Sheet. 17. Have the groups stack their cubic foot models close together to give the participants a visual reference for spaces that measure more than 1 cubic foot in volume. 18. Let s name some objects for which the cubic foot would be an appropriate unit for measuring volume. Skeletons in the Closet 9-18

23 19. Now let s consider a cubic yard. What attributes do you think will be present on a cubic yard? 12 edges that measure 1 yard in length 6 faces that measure 1 square yard in area; each face is a square with a length and width of 1 yard 1 cube with a volume of 1 cubic yard What about a cubic meter? What attributes do you think will be present on a cubic meter? 12 edges that measure 1 meter in length 6 faces that measure 1 square meter in area; each face is a square with a length and width of 1 yard 1 cube with a volume of 1 cubic meter 20. As a large group, build a model for a cubic yard and for the cubic meter. Ask for three or four volunteers to help build the next two models for the large group to see. Use the PVC elbows with the 1-inch inlet as the vertices for the cubic unit. Place the yardsticks into the PVC with the yardsticks serving as edges. Ask participants to verify that the attributes listed above for the cubic yard are correct. Next, use the metersticks to build a cubic meter in the same manner. Ask participants to verify that the attributes listed above for the cubic meter are correct. Since these units are larger, building one cubic yard and one cubic meter per the entire large group will be sufficient. You may choose to have the volunteers remain with the model to hold it steady. You might also place a ball of modeling clay or Play-Doh into the PVC pipe to help hold the edges in place. By doing so, the participants who help build the model could then step away (at least one at a time, depending on how sturdy the model seems to be) to see the full effect. Models to build the cubic meter are available commercially if you wish to use those. These models will disassemble easily for storage (hence the name Skeletons in the Closet) so that the teacher can bring them out when needed. Skeletons in the Closet 9-19

24 Remember, our models are skeleton models so that we can disassemble them and store them easily. A solid cube or a box with these dimensions would take up the same amount of space that our skeleton models occupy. Building the cubic meter and the cubic yard side by side is a very powerful visualization. You may also be able to build the cubic yard inside of the cubic meter to help participants get a better feel for how much larger the cubic meter is compared to the cubic yard. 21. Let s name some objects for which the cubic yard would be an appropriate unit for measuring volume. What about the cubic meter? Let s name some objects for which the cubic meter would be an appropriate unit for measuring volume. 22. These units are some of the standard units for measuring volume: Cubic centimeter and cubic meter in the metric system; Cubic inch, cubic foot, and cubic yard in the customary system 23. Leave some of the models on display in the classroom so that participants can refer to them as needed. Assessment: Identify a real-world object with the attribute of volume. Ask the participants to identify an appropriate standard unit with which to measure the object s volume. Extensions: Given a specific volume in either cubic centimeters or cubic inches (because the models for these units are less cumbersome), ask participants to find all of the possible rectangular prisms with that volume. Participants can build the prisms with centimeter cubes or with 1-inch blocks, recording the dimensions for the length, width, and height of each prism. Skeletons in the Closet 9-20

25 Another extension is to ask participants to identify relationships between units. For example, how many cubic inches are in 1 cubic foot? How many cubic feet are in 1 cubic yard? How many cubic centimeters are in 1 cubic meter? Participants can use the models they have built to help them see the connections. For example, participants can bring their cubic foot models to the front of the room to compare to the cubic yard model. Stacking three cubic feet for the length, width, and height of the cubic yard can help participants see a visual for the conversion that there are 27 cubic feet in one cubic yard. Another extension would be to investigate how much more volume the cubic meter contains compared to the cubic yard. Skeletons in the Closet 9-21

26 Skeletons in the Closet Recording Sheet Attributes of the Cubic Centimeter Attributes of the Cubic Meter Attributes of the Cubic Inch Attributes of the Cubic Foot Attributes of the Cubic Yard Handout 1 Skeletons in the Closet 9-22

27 Activity: Fill Er Up! (Optional activity for Grades K-2 Training) TEKS: See Appendix for student version of this activity with correlated TEKS. Overview: Volume is a term used to refer to the size of three-dimensional regions. Volume can be measured in solid and liquid units. According to Van de Walle (2004), standard units for solid volume are expressed in terms of length units, such as cubic meters, cubic inches, cubic centimeters, and so on. In this activity, participants will have opportunity to refine their knowledge of solid volume as they fill a three-dimensional region (rectangular prism) with cubic centimeters. Throughout the activity, participants will strengthen their measurement skills for linear measurement and for area. Finally, the participants will investigate how to guide their students to develop the formula for the volume of a rectangular prism and to connect the formula to the model. This activity should be included in all trainings for Grades 3-5 teachers. Materials: Centimeter grid paper, at least one sheet per participant Base ten blocks Ruler with centimeter markings, one for each participant Meter sticks, one per group of participants Variety of clean, small, empty boxes, 1 per group or pair of participants Scissors, one pair of scissors for every 2 participants Handout 1-Recording Sheet, one per participant (page 9-29) Transparency 1-Class Recording Chart (page 9-30) Grouping: Small groups (3-4 participants per group) or in pairs Time: 30 minutes Lesson: 1. Have each participant select a unit from a set of base ten blocks. Reminisce about quantities of units in each type of block: 1 rod = 10 units, 1 flat = 100 units, 1 cube = 1000 units. Even though the participants are working in small groups, it is beneficial for each person to have his or her own base-ten block unit and centimeter ruler to use during this activity. Fill Er Up! 9-23

28 2. Ask: What attributes of the cubic centimeter do you recall? (See previous lesson Skeletons in the Closet, Pages ) 12 edges that measure 1 centimeter in length 6 faces that measure 1 square centimeter in area; each face is a square with a length and width of 1 centimeter in length 1 cube that measures 1 cubic centimeter in volume The unit in the base-ten blocks is a model for a cubic centimeter. 3. Have participants select an empty box. Have participants cover the bottom of a box in a single layer of base-ten block unit cubes and record the number of cubes it took to cover the bottom of the box on Handout 1- Recording Sheet (page 9-29). Have participants sketch a diagram of the bottom layer they have constructed from the top view of the box (looking down over the layer of base-ten block unit cubes). Provide a variety of box sizes, but make sure that the bottom of the box will fit easily onto a sheet of centimeter grid paper. Also, make sure that the boxes are small enough so that it will be feasible for the participant to fill the box with cubic centimeter units later in the activity and not lose sight of the mathematics illustrated by the activity. 4. Have participants use their centimeter rulers to measure the length and width of the bottom of the box. Each person should record these measurements on the Recording Sheet. 5. Direct participants to set the empty box on the grid paper and to carefully trace an outline of the bottom of the box onto the grid paper. (See Materials List for link to centimeter grid paper.) Each participant should then cut out the traced model of the bottom of the box from the centimeter grid paper. Ask: What is the area of the bottom of the box? answers will vary Make sure that the participants line up one corner of the box with a corner of the grid paper so that the length and width of the box are traced along the grid lines. Remind participants that area can be found by A = l w, and that area is the number of squares it takes to cover a two-dimensional region. Fill Er Up! 9-24

29 Have participants record the area of the bottom of the box on the Recording Sheet. 6. Discuss/compare/contrast the following: a. the sketch of the cubes (on the recording sheet) that cover the bottom of the box; b. the centimeter grid paper cut-out model for the bottom of the box; and c. the bottom layer of cubes in the box. 7. Replace the bottom layer of unit cubes in the box if the cubes have come out of position. Then, have the participant construct more layers to put into the box, one layer at a time, until it is completely full of unit cubes, making sure to notate the total number of layers used on the Recording Sheet. Participants should notice that the centimeter grid paper cut-out model and the sketch from the recording sheet are very similar while the layer of unit cubes is different because of the thickness, or height, of each unit cube. Depending on the size of the box, participants may wish to use larger base ten block pieces to fill their boxes. If you decide to allow participants to use rods or flats, make sure to discuss the quantitative value of each as it is used. Make sure that the participants understand that if they use 1 rod, they are really using 10 cubic centimeters and so on. 8. Have participants use their centimeter rulers to measure and record the height of the box and record the height on the Recording Sheet. Discuss/compare the number of layers used with the measurement of the height of the box. 9. Have participants predict the total number of centimeter cubes used to fill the box completely and record their predictions on the Recording Sheet using the area of the base and the number of layers used (height of the box). 10. Have participants empty cubes from their boxes, then count and record actual number of centimeter cubes used to fill the box. Discuss with participants that they have just found a model for the volume of the box. Volume is measured in cubic units because we are finding how many cubes it takes to fill a space. In this activity, we are finding the maximum volume because we are filling the box completely (maximum volume = capacity) with cubes. Fill Er Up! 9-25

30 11. Discuss with participants possible number sentences that can be used to find the total number of centimeter cubes used to fill the box using the information recorded on the Recording Sheet. These number sentences show what it looks like mathematically. Sample number sentences: 5 layers of 88 centimeter cubes = 88 cm cm cm cm cm 3 = 440 cm 3 or 5 layers of 88 centimeter cubes = 5 88 cm Collect data (width, length, and height of box; area of bottom of box; volume of box) from each group to create a class chart. See Transparency 1-Class Recording Chart (page 9-30) as a guide. 13. Direct participants attention to a large box in the training room. If you don t have a large box readily available, then use the room itself (if it is a rectangular prism) or have the participants visualize a large box such as a refrigerator box. Ask: Would you want to fill this box with centimeter cubes? No! While we could fill the box with centimeter cubes, it would take too long to physically do it. So we need a more efficient method than physically filling the box with cubes and counting the cubes. Ask: Can you find a way to determine the number of cubes it would take to fill a box (volume) without having to physically fill the box with cubes and count them? Use the data collected on the class chart to help you look for patterns. Yes. We could find the area of the bottom of the box, then multiply that value by the height. Or, we could multiply the length of the box, the width of the box, and the height Reinforce the fact that we are finding the maximum volume because we are filling the box completely (maximum volume = capacity) with cubes. Fill Er Up! 9-26

31 of the box. Can we write that using symbols? Yes. We can write that V=Area of the Base x height or we can write that V=l x w x h. 14. As the participants reflect on the lesson, remind them of the following: It is important to connect the formula to the model, so be prepared to help your students see that the area of the bottom of the box is the area of the base of the rectangular prism. In addition, the l w portion of the volume formula is the same as the area formula for a rectangle and represents finding the area of the bottom of the box. The h portion then represents the number of layers of cubes it would take to fill the region. Assessment: Extensions: Have each group find a box that is not empty either in the classroom, around the school, or at home. Since the box is not empty, tell them that this would be an excellent opportunity to test their strategies for finding volume without actually filling the box with cubes. Have each group calculate the volume of the box, then present their findings to the class as they justify their answer. Have participants use estimation to identify items in the classroom, around the school, or at home, that have a volume less than a given value. Then have them find items that have a volume more than a given value. After making their estimates, have the participants measure the volume of their selected items to determine if their selections truly fit the criteria. Have each group of participants build a model of one square meter using a meter stick and newsprint paper. Using their model, have each group find the area of the classroom floor in square meters and the volume of the classroom in cubic meters. Have each group devise a strategy for finding these measurements, then explain in writing and through drawings how they determined the measurements. Fill Er Up! 9-27

32 Given a constant volume (such as 18 cm 3, or 24 cm 3, or 48 cm 3, etc.), have pairs of participants use centimeter cubes to build all the possible rectangular prisms and record the various dimensions. Is it possible to build a rectangular prism with a volume of 23 cubic units? 37 cubic units, etc.? If so, what could be the dimensions? Have participants do mat-planning activities recording different views (top view, front view, side view, etc.) of rectangular prisms built using cubes and use the recordings to determine the volume or the total number of cubes used to build the prism. Have participants repeat the lesson activity using cubic inches and/or other cubic units. Reinforce the idea that it takes more of the smaller units than the larger units to measure the same volume. Resources: Lovin, L.H., & Van de Walle, J. A. (2006). Teaching student-centered mathematics: Grades 3-5. Boston: Pearson Education, Inc. Van de Walle, J. A. (2004). Elementary and middle school mathematics: Teaching developmentally. Boston: Pearson Education, Inc. Fill Er Up! 9-28

33 Fill Er Up! Recording Sheet Sketch of Bottom Layer (looking down from the top) Number of Centimeter Cubes to Make Bottom Layer Number of Layers Used to Fill the Box Prediction of Total Number of Centimeter Cubes Used to Fill the Box Actual Number of Centimeter Cubes Used to Fill the Box Width of Box in Centimeters Length of Box in Centimeters Number Sentence Describing the Area of the Bottom of the Box Height of Box in Centimeters How is the number of centimeters in the width and the length related to the number of centimeter cubes used to make the bottom layer? How is the number of centimeters in the height related to the number of layers of centimeters cubes it took to fill the box? Explain the method you used to predict the total number of centimeter cubes used to fill the box. How can you use your sketch of the bottom layer and the number of layers used to fill the box to write a number sentence that gives the total number of centimeter cubes used to fill the box? Handout 1 Fill Er Up! 9-29

34 Fill Er Up! Class Recording Chart Group Width of Box in Centimeters Length of Box in Centimeters Height of Box in Centimeters Area of Bottom of the Box in Square Centimeters Volume of the Box in Cubic Centimeters Look for patterns in the data collected. Can you find a way to determine the number of cubes it would take to fill a box (volume) without having to physically fill the box with cubes and count them? Transparency 1 Fill Er Up! 9-30