Enhanced Instructional Transition Guide

Transcription

1 Enhanced Instructional Transition Guide / Unit 08: Suggested Duration: 12 days Unit 08: Measurement: Three-Dimensional (12 days) Possible Lesson 01 (12 days) POSSIBLE LESSON 01 (12 days) This lesson is one approach to teaching the State Standards associated with this unit. Districts are encouraged to customize this lesson by supplementing with districtapproved resources, materials, and activities to best meet the needs of learners. The duration for this lesson is only a recommendation, and districts may modify the time frame to meet students needs. To better understand how your district is implementing CSCOPE lessons, please contact your child s teacher. (For your convenience, please find linked the TEA Commissioner s List of State Board of Education Approved Instructional Resources and Midcycle State Adopted Instructional Materials.) Lesson Synopsis: Students investigate geometric properties, nets to classify three-dimensional figures, estimation, measurement, and formulas for volume. TEKS: The Texas Essential Knowledge and Skills (TEKS) listed below are the standards adopted by the State Board of Education, which are required by Texas law. Any standard that has a strike-through (e.g. sample phrase) indicates that portion of the standard is taught in a previous or subsequent unit. The TEKS are available on the Texas Education Agency website at Number, operation, and quantitative reasoning.. The student adds, subtracts, multiplies, or divides to solve problems and justify solutions. The student is expected to: 7.2B Use addition, subtraction, multiplication, and division to solve problems involving fractions and decimals. Readiness Standard 7.2F Select and use appropriate operations to solve problems and justify the selections. Readiness Standard 7.2G Determine the reasonableness of a solution to a problem. Supporting Standard 7.6 Geometry and spatial reasoning.. The student compares and classifies two- and three-dimensional figures using geometric vocabulary and properties. The student is expected to: 7.6C Use properties to classify three-dimensional figures, including pyramids, cones, prisms, and cylinders. Supporting Standard 7.8 Geometry and spatial reasoning.. The student uses geometry to model and describe the physical world. The student is expected to: page 1 of 65

2 Enhanced Instructional Transition Guide / Unit 08: Suggested Duration: 12 days 7.8B Make a net (two-dimensional model) of the surface area of a three-dimensional figure. Supporting Standard 7.8C Use geometric concepts and properties to solve problems in fields such as art and architecture. Supporting Standard 7.9 Measurement.. The student solves application problems involving estimation and measurement. The student is expected to: 7.9B Connect models for volume of prisms (triangular and rectangular) and cylinders to formulas of prisms (triangular and rectangular) and cylinders. Supporting Standard 7.9C Estimate measurements and solve application problems involving volume of prisms (rectangular and triangular) and cylinders. Readiness Standard Underlying Processes and Mathematical Tools TEKS: 7.13 Underlying processes and mathematical tools.. The student applies mathematics to solve problems connected to everyday experiences, investigations in other disciplines, and activities in and outside of school. The student is expected to: 7.13A Identify and apply mathematics to everyday experiences, to activities in and outside of school, with other disciplines, and with other mathematical topics. 7.13B Use a problem-solving model that incorporates understanding the problem, making a plan, carrying out the plan, and evaluating the solution for reasonableness. 7.13C Select or develop an appropriate problem-solving strategy from a variety of different types, including drawing a picture, looking for a pattern, systematic guessing and checking, acting it out, making a table, working a simpler problem, or working backwards to solve a problem. 7.13D Select tools such as real objects, manipulatives, paper/pencil, and technology or techniques such as mental math, estimation, and number sense to solve problems Underlying processes and mathematical tools.. The student communicates about mathematics through informal and mathematical language, representations, and models. The student is expected to: 7.14A Communicate mathematical ideas using language, efficient tools, appropriate units, and graphical, numerical, physical, or algebraic mathematical models Underlying processes and mathematical tools.. The student uses logical reasoning to make conjectures and verify conclusions. The page 2 of 65

3 Enhanced Instructional Transition Guide / Unit 08: Suggested Duration: 12 days student is expected to: 7.15A Make conjectures from patterns or sets of examples and nonexamples. 7.15B Validate his/her conclusions using mathematical properties and relationships. Performance Indicator(s): Grade 07 Unit 08 PI 01 Create a presentation (e.g., Prezi, poster, etc.) to compare and contrast the volume of two three-dimensional figures (e.g., cylinder, triangular prism, rectangular prism, etc.) to solve a real-life problem situation. Include the classification of each three-dimensional figure, the formula needed to calculate the volume of each three-dimensional figure, and a written justification of the solution process used to determine a reasonable estimate of each three-dimensional figure. Validate the solution to the real-life problem situation with mathematical properties and relationships. Sample Performance Indicator: Maverick operates a jam company and needs to decide the best container to ship his product to stores. The Container Shop has sent Maverick a net and a verbal description of the two containers they are able to provide. Maverick must decide which container can hold the most jam. page 3 of 65

4 Enhanced Instructional Transition Guide / Unit 08: Suggested Duration: 12 days Create a presentation comparing and contrasting the two containers that includes a sketch of the two-dimensional net of each container, the three-dimensional classification of the containers, the equation needed to calculate the volume of each container, and a written justification of the solution process used to determine a reasonable estimate of the volume of each container. Use mathematical properties and relationships to validate which container Maverick should purchase to ship his jam. Standard(s): 7.2B, 7.2F, 7.2G, 7.6C, 7.8B, 7.8C, 7.9B, 7.9C, 7.13A, 7.13B, 7.13C, 7.13D, 7.14A, 7.15A, 7.15B ELPS ELPS.c.1H, ELPS.c.3J, ELPS.c.5F, ELPS.c.5G Key Understanding(s): Conjectures of the classification of three-dimensional figures can be validated by the mathematical properties, including the relationships of angle measurements, shape of the base, number of edges, and vertices. The surface area of a three-dimensional figure or real-life example may be represented with a two-dimensional model called a net. Real-life problems may be modeled and measurement application problems may be solved using three-dimensional models built from twodimensional models called nets. The formula for volume of prisms (triangular and rectangular) and cylinders can be generated from concrete models and applied to solve real-life problem situations by multiplying the area of the base times the height of the figure. Formulas for volume of prisms (triangular and rectangular) and cylinders may be applied to solve problem situations and communicate the appropriate unit of measure. page 4 of 65

5 Enhanced Instructional Transition Guide / Unit 08: Suggested Duration: 12 days The process of evaluating a formula that must be rewritten to solve for another variable involves using a plan or strategy to keep the values on both sides of the formula equally balanced and validating the solution for reasonableness. Misconception(s): Some students may think the lateral surface of a cylinder is a rectangle instead of a curved surface because of the net model. Some students may think that B is synonymous with b, the length of the base, instead of B, which represents the area of the base of a threedimensional figure. Some students may think that the figure which rests on the bottom of a prism is the base Underdeveloped Concept(s): Students may think they are measuring in inches, when they are actually using the centimeter side of the ruler. Vocabulary of Instruction: cone cylinder net prism pyramid three-dimensional figure unit of measure volume Materials List: cardstock (optional) (6 sheets per 3 students) centimeter cubes (36 per 2 students) index card (1 per 2 students) math journal (1 per student) paper (plain) (6 sheets per 3 students, 1 sheet per teacher) ruler (standard) (1 per student) scissors (1 per 3 students, 1 per teacher) STAAR Reference Materials (1 per student) tape (clear) (optional) (1 per teacher) three-dimensional models (rectangular prism, triangular prism, cylinder, cone, triangular pyramid, square pyramid) (1 set per 3 students) page 5 of 65

6 Enhanced Instructional Transition Guide / Unit 08: Suggested Duration: 12 days Attachments: All attachments associated with this lesson are referenced in the body of the lesson. Due to considerations for grading or student assessment, attachments that are connected with Performance Indicators or serve as answer keys are available in the district site and are not accessible on the public website. Three-Dimensional Match KEY Three-Dimensional Match Three-Dimensional Models Three-Dimensional Figures KEY Three-Dimensional Figures Frayer Diagram for Three-Dimensional Figures SAMPLE KEY Frayer Diagram for Three-Dimensional Figures Attributes of Three-Dimensional Figures KEY Attributes of Three-Dimensional Figures Notes and Examples of Three-Dimensional Figures KEY Notes and Examples of Three-Dimensional Figures Nets Activity Recording Sheet KEY Nets Activity Recording Sheet Three-Dimensional Decisions KEY page 6 of 65

7 Enhanced Instructional Transition Guide / Unit 08: Suggested Duration: 12 days Three-Dimensional Decisions Building Cube Models KEY Building Cube Models Volume Formulas Prisms and Pyramids Volume Formula Cylinders Applications for Volume KEY Applications for Volume GETTING READY FOR INSTRUCTION Teachers are encouraged to supplement and substitute resources, materials, and activities to meet the needs of learners. These lessons are one approach to teaching the TEKS/Specificity as well as addressing the Performance Indicators associated with each unit. District personnel may create original lessons using the Content Creator in the Tools Tab. All originally authored lessons can be saved in the My CSCOPE Tab within the My Content area. Suggested Day Suggested Instructional Procedures Notes for Teacher 1 2 Topics: Three-dimensional figures Engage 1 Students use logic and reasoning skills to identify and categorize real-life three-dimensional prisms, pyramids, and curved surfaces. Students justify categorizations by describing the attributes of each figure. Spiraling Review ATTACHMENTS Teacher Resource: Three-Dimensional Match KEY (1 per teacher) Teacher Resource: Three-Dimensional Match (1 per teacher) page 7 of 65

8 Enhanced Instructional Transition Guide / Unit 08: Suggested Duration: 12 days Suggested Day Suggested Instructional Procedures Notes for Teacher Instructional Procedures: 1. Display teacher resource: Three-Dimensional Match. Instruct students to match each figure to the prism, pyramid, or curved surface category and record each match and justification in their math journal. Allow approximately 1 2 minutes for students to complete the activity. 2. Place students in pairs. Instruct student pairs to compare their matches and justifications. Allow time for students to complete the activity. Monitor and assess student pairs to check for understanding. Facilitate a class discussion to debrief student solutions. Ask: What do you know about a right prism? (A right prism is a solid formed by polygons that are connected at their edges. They have two bases that are congruent and parallel polygons and have side (lateral) faces that are rectangles formed by segments connecting the corresponding vertices of the bases.) What do you know about a pyramid? (A pyramid is a solid formed by polygons connected at their edges with one base that is a polygon. The side (lateral) faces are triangles formed by segments connecting the vertices of the base to the common vertex of the side faces.) What do you know about a cylinder? (A cylinder is a solid with two congruent MATERIALS math journal (1 per student) TEACHER NOTE In elementary math, vertex is defined in two-dimensional figures as the point where two sides meet. In threedimensional figures, vertex is defined as the point where three or more edges meet. Vertex is not defined for the three-dimensional curved surface figure, the cone, but may be referred to as a point. As students' progress though the mathematics curriculum, the definition for vertex is extended to include the three-dimensional curved surface figure, the cone. Vertex for the threedimensional curved surface figure, the cone, is defined as the point where infinite sides meet. This lesson transitions from using "point" to using "vertex" or "apex" of a cone. page 8 of 65

9 Enhanced Instructional Transition Guide / Unit 08: Suggested Duration: 12 days Suggested Day Suggested Instructional Procedures circular bases that are parallel and has a curved side surface.) What do you know about a cone? (A cone is a solid with one circular base and a curved side surface that meets at a vertex or apex point not on the base.) What is the specific name for Figure A? Figure B? Answers may vary. Cone; etc. Notes for Teacher TEACHER NOTE According to TEA, a face of a three-dimensional figure is defined as a flat surface in the shape of a twodimensional figure. Since the circular bases of a cylinder and cone are flat surfaces in the shape of a twodimensional figure, they could also be considered "faces." However, for elementary, TEA only lists curved surface and circular base(s) as the attributes of a cylinder and cone. So, CSCOPE uses these attributes in alignment with TEA. Topics: ATTACHMENTS Three-dimensional figures Explore/Explain 1 Students explore the similarities and differences of prisms, cylinders, and cones. Students investigate the attributes of three-dimensional figures and formalize the geometric vocabulary associated with these figures. Instructional Procedures: 1. Prior to instruction, if three-dimensional models are not available, use class resource: Three-Dimensional Models for every 3 students, by copying on cardstock, cutting apart, and assembling each three-dimensional figure as described. 2. Place students in groups of 3 and distribute a set of three-dimensional models Class Resource (optional): Three- Dimensional Models (1 per 3 students) Teacher Resource: Three-Dimensional Figures KEY (1 per teacher) Handout: Three-Dimensional Figures (1 per student) Teacher Resource: Frayer Diagram for Three-Dimensional Figures SAMPLE KEY (1 per teacher) Teacher Resource: Frayer Diagram for Three-Dimensional Figures (1 per teacher) page 9 of 65

10 Enhanced Instructional Transition Guide / Unit 08: Suggested Duration: 12 days Suggested Day Suggested Instructional Procedures (rectangular prism, a triangular prism, a cylinder, a cone, a triangular pyramid, and a rectangular pyramid) to each group. Instruct student groups to sort their set of threedimensional models into similar groupings and record a list of the similarities and differences among the groupings of the three-dimensional models in their math journal. Allow time for students to complete the activity. Monitor and assess student groups to check for understanding. Facilitate a class discussion to debrief student solutions. Ask: What geometric figures make up this three-dimensional figure? Answers may vary. Possible answers: Circle; Rectangles; Triangles; etc. How does a circle differ from a rectangle? From a triangle? (A circle has no vertices. A circle is not a polygon. A rectangle and a triangle are polygons.) What are some similarities among the three-dimensional figures in this group? Answers may vary. Possible answers: The bases are circles. All the faces and bases are rectangles. The side (lateral) faces are rectangles and the two bases are triangles; etc. What part of the three-dimensional figure represents the faces, vertices, and edges? (Faces: flat surface in the shape of a 2-D figure; vertices: intersection of line segment endpoints; edges: line segments where two polygons intersect.) How many vertices are on this three-dimensional figure? Answers may vary. 1 vertex. No vertices. Eight vertices. Five vertices. Six vertices. Four vertices; etc. Notes for Teacher MATERIALS three-dimensional models (rectangular prism, triangular prism, cylinder, cone, triangular pyramid, square pyramid) (1 set per 3 students) cardstock (optional) (6 sheets per 3 students) scissors (optional) (1 per teacher) tape (clear) (optional) (1 per teacher) math journal (1 per student) TEACHER NOTE Students investigate the attributes of three-dimensional figures such as number of vertices, edges, types of bases, and lateral faces. Students may sort the figures in multiple ways. They should be able to justify their method of sorting. 3. Distribute handout: Three-Dimensional Figures to each student. Facilitate a class discussion to formalize different geometric terms and how to categorize threedimensional figures based on bases and lateral faces while reinforcing the vocabulary page 10 of 65

11 Enhanced Instructional Transition Guide / Unit 08: Suggested Duration: 12 days Suggested Day Suggested Instructional Procedures from the handout. Instruct students to complete their handout throughout the discussion. 4. Display teacher resource: Frayer Diagram for Three-Dimensional Figures. Model completing the diagram for a rectangular prism. 5. Instruct students to complete a Frayer Diagram for a triangular prism, a cylinder, a cone, a triangular pyramid, and a rectangular pyramid in their math journal. Allow time for students to complete the activity. Monitor and assess students to check for understanding. Facilitate a class discussion to debrief student solutions, as needed. Notes for Teacher 3 Topics: Attributes of three-dimensional figures Explore/Explain 2 Students formalize attributes of three-dimensional figures and summarize findings and patterns in a table. Instructional Procedures: 1. Place students in groups of 3. Distribute handout: Attributes of Three-Dimensional Figures to each student and a set of three-dimensional models (rectangular prism, a triangular prism, a cylinder, a cone, a triangular pyramid, and a rectangular pyramid) to each group. Instruct student groups to identify the attribute of each three-dimensional figure. Allow time for students to complete the activity. Monitor and assess student groups to check for understanding. Facilitate a class discussion to debrief student solutions. Spiraling Review ATTACHMENTS Teacher Resource: Attributes of Three- Dimensional Figures KEY (1 per teacher) Handout: Attributes of Three- Dimensional Figures (1 per student) Teacher Resource: Notes and Examples of Three-Dimensional Figures KEY (1 per teacher) Handout: Notes and Examples of Three-Dimensional Figures (1 per student) page 11 of 65

12 Enhanced Instructional Transition Guide / Unit 08: Suggested Duration: 12 days Suggested Day Suggested Instructional Procedures Notes for Teacher Ask: What types of bases are there for this prism? Answers may vary. Triangular; etc. What types of lateral faces are there for this prism? Answers may vary. Rectangle; etc. How many vertices are there for this prism? Answers may vary. 6; etc. How many faces are there for this prism? Answers may vary. 5; etc. How many edges are there for this prism? Answers may vary. 9; etc. What patterns do you notice in the table? Answers may vary. The number of vertices is two times the number of edges on one base; etc. How can you use the patterns in the table to determine the requested information? Answers may vary. To determine the number of vertices, you can multiply 2 times the 9 edges on the base and determine there will be 18 total vertices; etc. What type of base is there for this pyramid? Answers may vary. Triangular; etc. What types of lateral faces are there for this pyramid? Answers may vary. Triangles; etc. How many vertices are there for this pyramid? Answers may vary. 4; etc. How many faces are there for this pyramid? Answers may vary. 4; etc. How many edges are there for this pyramid? Answers may vary. 6; etc. What patterns do you notice in the table? Answers may vary. The number of vertices is one plus the number of edges on the base; etc. How can you use the patterns in the table to determine the requested information? Answers may vary. The number of vertices is equal to one plus the 9 MATERIALS three-dimensional models (rectangular prism, triangular prism, cylinder, cone, triangular pyramid, square pyramid) (1 set per 3 students) page 12 of 65

13 Enhanced Instructional Transition Guide / Unit 08: Suggested Duration: 12 days Suggested Day Suggested Instructional Procedures edges on the base; etc. How is the shape of the base used to determine the name of the threedimensional figure? (The type of polygon that forms the base determines part of the name of the three-dimensional figure.) Notes for Teacher 2. Distribute handout: Notes and Examples of Three-Dimensional Figures to each student. Facilitate a class discussion to formalize the attributes of three-dimensional figures with and without curved surfaces. Instruct students to complete the handout as independent practice and/or homework. 4 Topics: Three-dimensional figures Two-dimensional nets Explore/Explain 3 Students create a two-dimensional net of a three-dimensional figure and begin identifying relationships between the shape of the base and the classifications of a three-dimensional figure. Instructional Procedures: 1. Place students in groups of 3. Distribute handout: Nets Activity Recording Sheet to each student and a set of three-dimensional models (rectangular prism, a triangular prism, a cylinder, a cone, a triangular pyramid, and a rectangular pyramid) to each group. Explain to students that they can create a two-dimensional net of each threedimensional figure by placing the model on a sheet of paper, tracing the base, and Spiraling Review ATTACHMENTS Teacher Resource: Nets Activity Recording Sheet KEY (1 per teacher) Handout: Nets Activity Recording Sheet (1 per student) MATERIALS three-dimensional models (rectangular prism, triangular prism, cylinder, cone, triangular pyramid, square pyramid) (1 set page 13 of 65

14 Enhanced Instructional Transition Guide / Unit 08: Suggested Duration: 12 days Suggested Day Suggested Instructional Procedures then rotating the model to trace the faces. Demonstrate the process of creating a net of a triangular prism. 2. Distribute 6 sheets of paper and a pair of scissors to each group. Instruct student groups to sketch each three-dimensional figure listed on their handout: Nets Activity Recording Sheet, identify the type of base, and sketch the net for each threedimensional model. Allow time for students to complete the activity. Monitor and assess student groups to check for understanding. Facilitate a class discussion to debrief student solutions. Ask: What type of three-dimensional figure do you have? Answers may vary. Rectangular prism, triangular prism, triangular pyramid, rectangular pyramid, cylinder, cone; etc. What type of base(s) does the three-dimensional figure have? How many? Answers may vary. Rectangle. Triangle. Circle. One base pyramid or cone. Two bases prism or cylinder; etc. Does the three-dimensional figure have any lateral faces? What type? Answers may vary. Yes, Rectangle. Yes, Triangle. No, a cylinder or cone does not have lateral faces; etc. How can you verify your two-dimensional model is a net for your threedimensional figure? Answers may vary. Wrap the two-dimensional model around the three-dimensional model; etc. Is there a different net you can draw for this same three-dimensional figure? (yes) Notes for Teacher per 3 students) paper (plain) (6 sheets per 3 students, 1 sheet per teacher) scissors (1 per 3 students, 1 per teacher) State Resources MTR 6 8: Semantic Feature Analysis Charts Attributes of 2-D and 3-D Figures; Envelope Tetrahedrons page 14 of 65

15 Enhanced Instructional Transition Guide / Unit 08: Suggested Duration: 12 days Suggested Day Suggested Instructional Procedures Notes for Teacher 5 Topics: Three-dimensional figures Elaborate 1 Students extend concepts of the attributes, classification, net, and perspective views of threedimensional figures. Instructional Procedures: Spiraling Review ATTACHMENTS Teacher Resource: Three-Dimensional Decisions KEY (1 per teacher) Handout: Three-Dimensional Decisions (1 per student) 1. Place students in pairs and distribute handout: Three-Dimensional Decisions to each student. Instruct pairs to identify the attributes and name of each threedimensional figure. Allow time for students to complete the activity. Monitor and assess student pairs to check for understanding. Facilitate a class discussion to debrief student solutions. Ask: How many bases does the three-dimensional figure have? Answers may vary. 1; etc. What type of base(s) does the three-dimensional figure have? Identify the base(s) in the sketch. Answers may vary. Square; etc. How many faces does the three-dimensional figure have? Answers may vary. 3; etc. Why type of faces does the three-dimensional figure have? Answers may vary. Triangles; etc. How many vertices does the three-dimensional figure have? Identify the page 15 of 65

16 Enhanced Instructional Transition Guide / Unit 08: Suggested Duration: 12 days Suggested Day Suggested Instructional Procedures vertices in the sketch. Answers may vary. 4 vertices; etc. How many edges does the three-dimensional figure have? Identify the edges in the sketch. Answers may vary. 6 edges; etc. What name can you give the three-dimensional figure based on the attributes you have listed? Answers may vary. Triangular Pyramid; etc. Notes for Teacher 6 Topics: Volume Engage 2 Student use logic and reasoning skills to build rectangular prisms and investigate volume. Instructional Procedures: 1. Place students in pairs. Distribute handout: Building Cube Models and the STAAR Reference Materials to each student and 36 centimeter cubes to each student pair. Explain to students that they will investigate how the base of the model and the height may be used to calculate the number of cubes needed to build the rectangular prism. Instruct student pairs to build the model with their centimeter cubes to represent the front, top, and side views of a rectangular prism. Allow time for students to complete the activity. Monitor and assess student groups to check for understanding. Facilitate a class discussion to debrief student solutions. Ask: Spiraling Review ATTACHMENTS Teacher Resource: Building Cube Models KEY (1 per teacher) Handout: Building Cube Models (1 per student) MATERIALS STAAR Reference Materials (1 per student) centimeter cubes (36 per 2 students) What is the area of the base of the rectangular prism? Answers may vary. 6 page 16 of 65

17 Enhanced Instructional Transition Guide / Unit 08: Suggested Duration: 12 days Suggested Day Suggested Instructional Procedures cubes; etc. How many cubes high is the rectangular prism? Answers may vary. 2 cubes; etc. How can the area of the base of the rectangular prism and the height of the rectangular prism be used to calculate the total number of cubes used to build the rectangular prism? (Calculate the area of the base = base height and then multiply this product by the height: Area of Base height.) What formula on the STAAR Reference Materials is related to how you can calculate the volume of this rectangular prism? (V = Bh) Notes for Teacher 7 8 Topics: Volume Rectangular and triangular prisms Explore/Explain 4 Students connect models of a rectangular prism to the formula for the volume of a rectangular prism. Students apply the formulas for volume of rectangular and triangular prisms. Instructional Procedures: 1. Distribute the STAAR Reference Materials and handout: Volume Formulas Prisms and Pyramids to each student. Spiraling Review ATTACHMENTS Handout: Volume Formulas Prisms and Pyramids (1 per student) Teacher Resource: Volume Formulas Prisms and Pyramids (1 per teacher) MATERIALS page 17 of 65

18 Enhanced Instructional Transition Guide / Unit 08: Suggested Duration: 12 days Suggested Day Suggested Instructional Procedures 2. Display teacher resource: Volume Formulas Prisms and Pyramids. Facilitate a class discussion about how the models are connected to the volume formulas for a rectangular prism and triangular prism on the STAAR Reference Materials. Notes for Teacher STAAR Reference Materials (1 per student) State Resources MTR 6 8: Volume It all Stacks Up Topics: Volume of prisms Elaborate 2 Students apply concepts of measurement to find the volume of prisms in real-life problem situations. Instructional Procedures: 1. Place students in pairs and distribute an index card to each pair. Instruct students to create and record a question that connects volume to a real-life experience on the front of the card, then record the solutions process that could be used to solve the problem on the back of the card. Allow time for students to complete the activity. Monitor and assess student pairs to check for understanding. 2. Instruct student pairs to exchange index cards with another student pair, solve and record the solution process for the new question in their math journal, and compare their answer with the solution on the back of the index card. Repeat this activity until each pair has solved every volume problem situation created by the class. Allow time Spiraling Review MATERIALS index card (1 per 2 students) math journal (1 per student) page 18 of 65

19 Enhanced Instructional Transition Guide / Unit 08: Suggested Duration: 12 days Suggested Day Suggested Instructional Procedures for students to complete the activity. Monitor and assess student pairs to check for understanding. Facilitate a class discussion to debrief student solutions, as needed. 3. Instruct students to bring a cylinder shaped container to the next class meeting. Notes for Teacher 9 10 Topics: Volume Cylinder Explore/Explain 5 Students connect models of cylinders to the formula for volume of a cylinder. Instructional Procedures: 1. Distribute the STAAR Reference Materials and handout: Volume Formula Cylinders to each student. 2. Display teacher resource: Volume Formula Cylinders. Facilitate a class discussion about how the models are connected to the volume formula for a cylinder on their STAAR Reference Materials. Spiraling Review ATTACHMENTS Handout: Volume Formula Cylinders (1 per student) Teacher Resource: Volume Formula Cylinders (1 per teacher) MATERIALS STAAR Reference Materials (1 per student) State Resources MTR 6 8: Volume It all Stacks Up Topics: MATERIALS page 19 of 65

20 Enhanced Instructional Transition Guide / Unit 08: Suggested Duration: 12 days Suggested Day Suggested Instructional Procedures Notes for Teacher Volume of cylinders Elaborate 3 Students apply concepts of measurement to find the volume of cylinders in real-life problem situations. STAAR Reference Materials (1 per student) ruler (standard) (1 per student) math journal (1 per student) Instructional Procedures: 1. Prior to instruction, invite students to display their cylinders, brought from home, around the classroom. 2. Place students in pairs. Distribute the STAAR Reference Materials and ruler to each student. Instruct students to rotate around the room and use their ruler to measure the dimensions of each cylinder and record the solution process to calculate the volume of each cylinder, along with a written justification of the importance of calculating the volume of the cylinder in relationship to real-life application, in their math journal. Allow time for students to complete the activity. Monitor and assess student pairs to check for understanding. Facilitate a class discussion to debrief student solutions Topics: Volume Prisms Cylinders Elaborate 4 Spiraling Review ATTACHMENTS Teacher Resource: Applications for Volume KEY (1 per teacher) page 20 of 65

21 Enhanced Instructional Transition Guide / Unit 08: Suggested Duration: 12 days Suggested Day Suggested Instructional Procedures Students compare and contrast the formulas for volume of prisms and cylinders, determining the differences and similarities. Students use appropriate formulas to solve real-life problem situations. Notes for Teacher Handout: Applications for Volume (1 per student) Instructional Procedures: 1. Place students in pairs. Distribute the STAAR Reference Materials and handout: Applications for Volume to each student. Instruct students to use their STAAR Reference Materials to solve each problem. Allow time for students to complete the activity. Monitor and assess student pairs to check for understanding. Facilitate a class discussion to debrief student solutions. Ask: MATERIALS STAAR Reference Materials (1 per student) What shape is the area of the base of the figure? Answers may vary. Rectangular, triangular, or circular; etc. How do you calculate the area of the base? Answers may vary. If the base is a square, you can use A = bh; etc. What is the height of the figure? Answers may vary. 3 ft; etc. How do you use the area of the base and the height to calculate the volume? (Area of Base height = Volume) Evaluate 1 MATERIALS Instructional Procedures: 1. Assess student understanding of related concepts and processes by using the Performance Indicator(s) aligned to this lesson. STAAR Reference Materials (1 per student) page 21 of 65

22 Enhanced Instructional Transition Guide / Unit 08: Suggested Duration: 12 days Suggested Day Suggested Instructional Procedures Notes for Teacher Performance Indicator(s): Grade 07 Unit 08 PI 01 Create a presentation (e.g., Prezi, poster, etc.) to compare and contrast the volume of two threedimensional figures (e.g., cylinder, triangular prism, rectangular prism, etc.) to solve a real-life problem situation. Include the classification of each three-dimensional figure, the formula needed to calculate the volume of each three-dimensional figure, and a written justification of the solution process used to determine a reasonable estimate of each three-dimensional figure. Validate the solution to the real-life problem situation with mathematical properties and relationships. Sample Performance Indicator: Maverick operates a jam company and needs to decide the best container to ship his product to stores. The Container Shop has sent Maverick a net and a verbal description of the two containers they are able to provide. Maverick must decide which container can hold the most jam. page 22 of 65

23 Enhanced Instructional Transition Guide / Unit 08: Suggested Duration: 12 days Suggested Day Suggested Instructional Procedures Notes for Teacher Create a presentation comparing and contrasting the two containers that includes a sketch of the two-dimensional net of each container, the threedimensional classification of the containers, the equation needed to calculate the volume of each container, and a written justification of the solution process used to determine a reasonable estimate of the volume of each container. Use mathematical properties and relationships to validate which container Maverick should purchase to ship his jam. Standard(s): 7.2B, 7.2F, 7.2G, 7.6C, 7.8B, 7.8C, 7.9B, 7.9C, 7.13A, 7.13B, 7.13C, 7.13D, 7.14A, 7.15A, 7.15B ELPS ELPS.c.1H, ELPS.c.3J, ELPS.c.5F, ELPS.c.5G 05/08/13 page 23 of 65

24 Enhanced Instructional Transition Guide / Unit 08: Suggested Duration: 12 days page 24 of 65

25 Three-Dimensional Match KEY 1. Match each figure with one of the categories listed below. 2. Be able to justify your response. Figure A Figure B Figure C Figure D Figure E Figure F Prism Category (Rectangular Prism) Figure B Figure C (container) All the faces are rectangles Pyramid Category (Rectangular Pyramid) Figure F The side (lateral) faces are triangles Curved Surface Category (Cone) Figure A (cone part) (Cylinder) Figure D (can) Figure E Figure C (contents) There are no edges. The bases are circular. 2012, TESCCC 10/12/12 page 1 of 1

26 Three-Dimensional Match HS 1. Match each figure with one of the categories listed below. 2. Be able to justify your response. Figure A Figure B Figure C Figure D Figure E Figure F Prism Category Pyramid Category Curved Surface Category 2012, TESCCC 10/12/12 page 1 of 1

27 Three-Dimensional Models HS Copy on cardstock. Cut out along solid lines. Fold along dotted lines. Tape together to form a threedimensional figure. 2012, TESCCC 10/12/12 page 1 of 6

28 Three-Dimensional Models HS Copy on cardstock. Cut out along solid lines. Fold along dotted lines. Tape together to form a threedimensional figure. 2012, TESCCC 10/12/12 page 2 of 6

29 Three-Dimensional Models HS Copy on cardstock. Cut out along solid lines. Fold along dotted lines. Tape together to form a threedimensional figure. 2012, TESCCC 10/12/12 page 3 of 6

30 Three-Dimensional Models HS Copy on cardstock. Cut out along solid lines. Fold along dotted lines. Tape together to form a threedimensional figure. 2012, TESCCC 10/12/12 page 4 of 6

31 Three-Dimensional Models HS Copy on cardstock. Cut out along solid lines. Fold along dotted lines. Tape together to form a threedimensional figure. 2012, TESCCC 10/12/12 page 5 of 6

32 Three-Dimensional Models HS Copy on cardstock. Cut out along solid lines. Fold along dotted lines. Tape together to form a threedimensional figure. 2012, TESCCC 10/12/12 page 6 of 6

33 Three-Dimensional Figures KEY Types of Three-Dimensional Figures and Their Characteristics Prisms Prisms contain two bases that are parallel, congruent polygons. Lateral faces are polygons that make up the sides of the figure. If the prism is a right prism, the non-base faces will be rectangles. Prisms are named by the type of polygon used for the bases. Rectangular Triangular Prism Prism 1. Describe a base according to the diagram above. Congruent and parallel polygons. 2. Describe a face of the right prism according to the diagram above. Rectangles formed by segments that connect the corresponding vertices of the bases. 3. Describe an edge according to the diagram above. A segment where two faces intersect. 4. Describe a vertex according to the diagram above. The point of intersection of three or more faces. Pyramids Pyramids contain one base that is a polygon. Lateral faces are triangular and fold in to share a common vertex. Pyramids are named by the type of polygon used for the base. Pyramids explored at this level will have triangular or rectangular bases. Figure A Figure B 5. Find and label the bases, faces, edges, and vertices of the pyramids. 6. Which one, Figure A or Figure B, is the rectangular pyramid? How do you know? Figure B since it has a rectangular base. 7. Which one, Figure A or Figure B, is the triangular pyramid? How do you know? Figure A since it has a triangular base. 2012, TESCCC 10/12/12 page 1 of 2

34 Three-Dimensional Figures KEY Types of Three-Dimensional Figures and Their Characteristics Cylinders Cylinders contain parallel, congruent, and circular bases. A curved surface forms the sides. Cylinders do not contain vertices or faces. 8. Why is a cylinder not a prism or a pyramid? Prisms and pyramids have bases and faces that are polygons. The circular bases of the cylinder are not polygons and the curved surface of the side of the cylinder is not a polygon. Cones Cones contain one circular base and a curved surface. The curved surface meets at a vertex or apex point that is not on the base. 9. Give two characteristics a cone has that a pyramid does not have. A cone has a circular base. A cone has a curved surface for its side. 2012, TESCCC 10/12/12 page 2 of 2

35 Three-Dimensional Figures Types of Three-Dimensional Figures and Their Characteristics Prisms Prisms contain two bases that are parallel, congruent polygons. Lateral faces are polygons that make up the sides of the figure. If the prism is a right prism, the non-base faces will be rectangles. Prisms are named by the type of polygon used for the bases. Rectangular Prism Triangular Prism 1. Describe a base according to the diagram above. 2. Describe a face of the right prism according to the diagram above. 3. Describe an edge according to the diagram above. 4. Describe a vertex according to the diagram above. Pyramids Pyramids contain one base that is a polygon. Lateral faces are triangular and fold in to share a common vertex. Pyramids are named by the type of polygon used for the base. Pyramids explored at this level will have triangular or rectangular bases. Figure A Figure B 5. Find and label the bases, faces, edges, and vertices of the pyramids. 6. Which one, Figure A or Figure B, is the rectangular pyramid? How do you know? 7. Which one, Figure A or Figure B, is the triangular pyramid? How do you know? 2012, TESCCC 10/12/12 page 1 of 2

36 Three-Dimensional Figures Types of Three-Dimensional Figures and Their Characteristics Cylinders Cylinders contain parallel, congruent, and circular bases. A curved surface forms the sides. Cylinders do not contain vertices or faces. 8. Why is a cylinder not a prism or a pyramid? Cones Cones contain one circular base and a curved surface. The curved surface meets at a vertex or apex point that is not on the base. 9. Give two characteristics a cone has that a pyramid does not have. 2012, TESCCC 10/12/12 page 2 of 2

37 Frayer Diagram for Three-Dimensional Figures SAMPLE KEY Name of Three-Dimensional Figure Rectangular Prism Curved or Non-Curved Figure Non-curved Model Sketch of Three-Dimensional Figure Rectangle Shape that Forms Base(s) of the Figure 6 Faces: 4 Lateral Faces + 2 Bases 12 Edges 8 Vertices 2012, TESCCC 10/12/12 page 1 of 1

38 Frayer Diagram for Three-Dimensional Figures Name of Three-Dimensional Figure Curved or Non-Curved Figure Sketch of Three-Dimensional Figure Shape that Forms Base(s) of the Figure Number of Faces, Bases, Edges, and Vertices 2012, TESCCC 10/12/12 page 1 of 1

39 Attributes of Three-Dimensional Figures KEY Use the three-dimensional models for the triangular prism and the rectangular prism to complete the first two rows of the table. Look for patterns in the table to complete the remaining rows and answer the questions below. Type of Prism Triangular Prism Rectangular Prism Pentagonal Prism Hexagonal Prism Name of Faces 2 triangle bases 3 rectangle (lateral faces) 2 rectangle bases 4 rectangle (lateral faces) 2 pentagon bases 5 rectangle (lateral faces) 2 hexagon bases 6 rectangle (lateral faces) A Pattern for Prisms Number of Vertices Number of Faces Number of Edges Describe any number patterns in your chart. Number of Vertices = 2 (number of edges on 1 base) Number of Faces = 2 bases + (number of edges on 1 base) Number of Edges = 3 (number of edges on 1 base) Suppose you made a prism that had a base with nine edges. Without making the prism, describe how you know the number of vertices, edges, and faces it had. Number of Vertices = 2 (number of edges on 1 base) Number of Vertices = 2 9 = 18 Number of Faces = 2 bases + (number of edges on 1 base) Number of Faces = = 11 Number of Edges = 3 (number of edges on 1 base) Number of Edges = 3 9 = Describe a prism with 24 edges. Number of Edges = 3 (Number of Edges on 1 base) 24 edges = 3 x x = 8 x = number of edges on 1 base 8 edges the base is an octagon The bases of the prisms are octagons so we have an octagonal prism with 16 vertices and 10 faces Describe a prism with 21 edges. Number of Edges = 3 (Number of Edges on 1 base) 21 edges = 3 x x = 7 x = number of edges on 1 base 7 edges the base is a heptagon. The bases of the prisms are heptagons so we have a heptagonal prism with 14 vertices and 9 faces. 2012, TESCCC 10/12/12 page 1 of 2

40 Attributes of Three-Dimensional Figures KEY Use the three-dimensional models for the triangular pyramid and the square pyramid to complete the first two rows of the table. Look for patterns in the table to complete the remaining rows and answer the questions below. Type of Pyramid Triangular Pyramid Square Pyramid Pentagonal Pyramid Hexagonal Pyramid Name of Faces triangle (1 base) 3 triangles (lateral faces) square (1 base) 4 triangles (lateral faces) pentagon (base) 5 triangles (lateral faces) hexagon (base) 6 triangles (lateral faces) A Pattern for Pyramids Number of Vertices Number of Faces Number of Edges Describe any number patterns in your chart. Number of Vertices = 1 + (number of edges on base) Number of Faces = 1 base + (number of edges on base) Number of Edges = 2 (number of edges on base) Suppose you made a pyramid with a base that had nine edges. Without making the pyramid, describe how you know the number of vertices, edges, and faces it had. Number of Vertices = 1 + (number of edges on base) Number of Vertices = = 10 Number of Faces = 1 base + (number of edges on base) Number of Faces = = 10 Number of Edges = 2 (number of edges on base) Number of Edges = 2 9 = Describe a pyramid with 20 edges. Number of Edges = 2 (number of edges on base) 20 edges = 2 x x = 10 x = number of edges on base 10 edges the base is a decagon The base of the pyramid is a decagon so we have a decagonal pyramid with 11 vertices and 11 faces Describe a pyramid with 13 edges. Number of Edges = 2 (number of edges on base) 13 edges = 2 x x = 6.5 x = number of edges on base 6.5 edges not possible since we must have an even number of edges 2 (number of edges on base). This will not make a pyramid. 2012, TESCCC 10/12/12 page 2 of 2

41 Attributes of Three-Dimensional Figures Use the three-dimensional models for the triangular prism and the rectangular prism to complete the first two rows of the table. Look for patterns in the table to complete the remaining rows and answer the questions below. A Pattern for Prisms Type of Prism Name of Faces Number of Vertices Number of Faces Number of Edges Triangular Prism Rectangular Prism Pentagonal Prism Hexagonal Prism 1. Describe any number patterns in your chart Suppose you made a prism that had a base with nine edges. Without making the prism, describe how you know the number of vertices, edges, and faces it had Describe a prism with 24 edges Describe a prism with 21 edges. 2012, TESCCC 10/12/12 page 1 of 2

42 Attributes of Three-Dimensional Figures Use the three-dimensional models for the triangular pyramid and the square pyramid to complete the first two rows of the table. Look for patterns in the table to complete the remaining rows and answer the questions below. A Pattern for Pyramids Type of Pyramid Name of Faces Number of Vertices Number of Faces Number of Edges Triangular Pyramid Square Pyramid Pentagonal Pyramid Hexagonal Pyramid 1. Describe any number patterns in your chart Suppose you made a pyramid with a base that had nine edges. Without making the pyramid, describe how you know the number of vertices, edges, and faces it had Describe a pyramid with 20 edges Describe the pyramid with 13 edges. 2012, TESCCC 10/12/12 page 2 of 2

43 Notes and Examples of Three-Dimensional Figures KEY List the attributes of three-dimensional figures without curved surfaces. Prisms H G D C E F A B Rectangular Prism Triangular Prism Attributes Vertices Edges Lateral faces Bases Definition Vertices are the points where three or more line segments intersect. Edges are line segments where polygon surfaces meet. Lateral faces are polygons that make up the sides of the figure. Parallel, congruent polygons that define the type of figure. Rectangular Prism A, B, C, D, E, F, G, H AB, BC, CD, AD, AE, BF, DH, CG, EF, FG, GH, HE ABCD, EFGH, BCGF, AEHD, ABFE, DCGH Rectangles ABFE, DCGH In this rectangular prism, any of the faces that are parallel and congruent could be the bases. Triangular Prism P, Q, R, S, T, U PQ, QR, RP, PS, UT, US, QT, RU, TS PQR, STU, PQTS, QRUT, PSUR Triangles PQR, STU 2012, TESCCC 10/12/12 page 1 of 3

44 Notes and Examples of Three-Dimensional Figures KEY Pyramids E Q D C M N A B P Identify the attributes of the pyramids above by completing the table. Attribute Vertices Definition Vertices are the points where three or more line segments meet. Pyramid ABCDE Pyramid NMPQ A, B, C, D, E M, N, P, Q Edges Edges are line segments where polygon surfaces meet. AB, BC, CD, AD, AE, BE, CE, DE MN, MP, NP, MQ, NQ, PQ Lateral faces Lateral faces are polygons that make up the sides of the figure. ABE, BCE, CDE, ADE ABCD MPQ, NPQ, MNQ, MNP Base Polygon opposite the vertex that defines the type of figure. Rectangle ABCD Triangle MNP Name Rectangular Pyramid Triangular Pyramid 2012, TESCCC 10/12/12 page 2 of 3

45 Notes and Examples of Three-Dimensional Figures KEY List the attributes of three-dimensional figures with curved surfaces. Cylinders and Cones A Q P B Attribute Definition Cylinder Cone The vertex is the point at Vertex which the curved surface Point P None meets that is not on the 1 vertex plane of the circular base. Base(s) Bases are the circular regions and their interiors. 2 congruent circular bases Circle A Circle B 1 circular base Circle Q 2012, TESCCC 10/12/12 page 3 of 3

46 Notes and Examples of Three-Dimensional Figures List the attributes of three-dimensional figures without curved surfaces. Prisms Rectangular Prism Triangular Prism Attribute Definition Rectangular Prism Triangular Prism Vertices Edges Lateral faces Bases 2012, TESCCC 10/12/12 page 1 of 3

47 Notes and Examples of Three-Dimensional Figures Pyramids E Q D C M N A B Identify the attributes of the pyramids above by completing the table. P Attribute Definition Pyramid ABCDE Pyramid NMPQ Vertices Edges Lateral faces Base Name 2012, TESCCC 10/12/12 page 2 of 3

48 Notes and Examples of Three-Dimensional Figures List the attributes of three-dimensional figures with curved surfaces. Cylinders and Cones A Q P B Attribute Definition Cylinder Cone Vertex Base(s) 2012, TESCCC 10/12/12 page 3 of 3

49 Nets Activity Recording Sheet KEY Name of Three-Dimensional Figure Sketch of Three-Dimensional Figure Samples Type of Base Sketch of Net Samples Triangular Pyramid triangle (1) Rectangular Pyramid rectangle (1) Rectangular Prism rectangle (2) Triangular Prism triangle (2) Cylinder circle (2) Cone circle (1) 2012, TESCCC 10/12/12 page 1 of 1

50 Nets Activity Recording Sheet Name of Three-Dimensional Figure Sketch of Three-Dimensional Figure Type of Base Sketch of Net Triangular Pyramid Rectangular Pyramid Rectangular Prism Triangular Prism Cylinder Cone 2012, TESCCC 10/12/12 page 1 of 1

51 Three-Dimensional Decisions KEY 1. Identify the attributes: vertices, edges, bases, and lateral faces of each three-dimensional figure. Classify the figure using the attributes. a. Figure Attributes Classification (Name) 4 Vertices: A, B, C, D 6 Edges: AB, BC, AC, AD, BD, CD 3 lateral faces: ABD, BCD, ACD Triangular Pyramid 1 base: ABC b. c. D A H E C B G F 8 Vertices: A, B, C, D, E, F, G, H 12 Edges: AB, BC, AE, AD, BF, CD, CG, HD, EF, EH, FG, GH 4 lateral faces: ABCD, BCGF, EFGH, ADHE 2 bases: ABFE, CDHG The other faces could also be the bases. Rectangular Prism No vertices 1 lateral curved surface Cylinder 2 circular bases Circle A and circle B 2012, TESCCC 05/01/13 page 1 of 4

52 Three-Dimensional Decisions KEY 2. Identify the attributes: vertices, edges, bases, and lateral faces of each three-dimensional figure. Classify the figure using the attributes. a. Figure Attributes Classification (Name) 5 Vertices: A, B, C, D, E 8 Edges: AB, BC, AE, AD, BE, CD, CE, DE 4 lateral faces: ABE, BCE, CDE, ADE Rectangular Pyramid 1 base: ABCD b. 6 Vertices: A, B, C, D, E, F 9 Edges: AB, BC, BE, AD, CA, CF, DF, EF, ED 3 lateral faces: ACFD, BCFE, ABED Triangular Prism c. 2 bases: ABC, DEF 1 vertex point A 1 lateral curved surface Cone 1 circular base: B 2012, TESCCC 05/01/13 page 2 of 4

53 Three-Dimensional Decisions KEY 3. Draw a two-dimensional net for each three-dimensional figure. Classify the figure. a. Three-Dimensional Figure Classification (Name) Two-Dimensional Net Rectangular Pyramid b. Triangular Prism c. Cone 2012, TESCCC 05/01/13 page 3 of 4

54 Three-Dimensional Decisions KEY 4. Draw a two-dimensional net for each three-dimensional figure. Classify the figure. a. Three-Dimensional Figure Classification (Name) Two-Dimensional Net Triangular Pyramid b. Rectangular Prism c. Cylinder 2012, TESCCC 05/01/13 page 4 of 4

55 Three-Dimensional Decisions 1. Identify the attributes: vertices, edges, bases, and lateral faces of each three-dimensional figure. Classify the figure using the attributes. a. Figure Attributes Classification (Name) b. c. 2012, TESCCC 10/12/12 page 1 of 4

56 Three-Dimensional Decisions 2. Identify the attributes: vertices, edges, bases, and lateral faces of each three-dimensional figure. Classify the figure using the attributes. a. Figure Attributes Classification (Name) b. c. 2012, TESCCC 10/12/12 page 2 of 4

57 Three-Dimensional Decisions 3. Draw a two-dimensional net for each three-dimensional figure. Classify the figure. a. Three-Dimensional Figure Classification (Name) Two-Dimensional Net b. c. 2012, TESCCC 10/12/12 page 3 of 4

58 Three-Dimensional Decisions 4. Draw a two-dimensional net for each three-dimensional figure. Classify the figure. a. Three-Dimensional Figure Classification (Name) Two-Dimensional Net b. c. 2012, TESCCC 10/12/12 page 4 of 4

59 Building Cube Models KEY Use the front, top, and side views to build a rectangular prism. Indicate how many cubes were used to build each rectangular prism. Front View Top View Side View Total Number of Cubic Units How many cubes did you use to build the bottom layer? 6 cubes 2. How many cubes high is the rectangular prism? 2 cubes 3. How is the area of the base of the rectangular prism and the height related to the total number of cubes needed to build the rectangular prism? Area of base height = total number of cubes Front View Top View Side View Total Number of Cubic Units 6 4. How many cubes did you use to build the bottom layer? 2 cubes 5. How many cubes high is the rectangular prism? 3 cubes 6. How is the area of the base of the rectangular prism and the height related to the total number of cubes needed to build the rectangular prism? Area of base height = total number of cubes 2012, TESCCC 10/12/12 page 1 of 1

60 Building Cube Models Use the front, top, and side views to build a rectangular prism. Indicate how many cubes were used to build each rectangular prism. Front View Top View Side View Total Number of Cubic Units 1. How many cubes did you use to build the bottom layer? 2. How many cubes high is the rectangular prism? 3. How is the area of the base of the rectangular prism and the height related to the total number of cubes needed to build the rectangular prism? Front View Top View Side View Total Number of Cubic Units 4. How many cubes did you use to build the bottom layer? 5. How many cubes high is the rectangular prism? 6. How is the area of the base of the rectangular prism and the height related to the total number of cubes needed to build the rectangular prism? 2012, TESCCC 10/12/12 page 1 of 1

61 Volume Formulas Prisms and Pyramids Volume of a Rectangular Prism Math Notes Stage 1 Stage 2 The volume of a rectangular prism is the measure of the amount of space occupied in the rectangular prism. The volume is measured using cubic units. Stage 3 The attributes of a rectangular prism: 6 faces 4 lateral rectangular faces and 2 rectangular bases 8 vertices 12 edges Stage 4 A rectangular prism is similar to layers of stacked rectangular bases. Each layer is covered with cubic units. To determine the number of cubic units occupying the space in a rectangular prism, we can count the cubic units individually or by multiples of 2 s, multiples of 3 s, etc. or we can calculate the number of cubic units that cover one rectangular layer and multiply by the number of rectangular layers which represents the height of the rectangular prism. Stage 5 If a rectangular base has a layer of 15 square units and the rectangular prism is 3 units high, then the number of cubic units that will fill the rectangular prism is: # of cubic units per layer times # of layers. To determine how many cubic units will cover the rectangular base above, we can count each individual cubic unit or we can use the formula: A = b h to calculate the number of cubic units to cover the area of the rectangular base. It is important to note that only the square face of the cubic unit will cover the area of the rectangular base. Calculate the number of square units that will cover the area of the rectangular base above: Area = 5 3 = 15 u 2 Stage 6 Write a formula in terms of the area of the rectangular base of a rectangular prism and the height of a rectangular prism to calculate the volume of a rectangular prism. Use this formula to calculate the volume of the rectangular prism in Stage 5. Formula for Volume of a Rectangular Prism: Area of Rectangular Base times height: V = B h Volume of Rectangular Prism in Stage 5: V = (5 3) 3 = 45 cubic units 2012, TESCCC 10/12/12 page 1 of 2

62 Volume Formulas Prisms and Pyramids Volume of a Triangular Prism Math Notes Stage 1 Stage 2 The volume of a triangular prism is the measure of the amount of space occupied in the triangular prism. The volume is measured using cubic units. Stage 3 The attributes of a triangular prism: 5 faces 3 lateral rectangular faces and 2 triangular bases 6 vertices 9 edges Stage 4 A triangular prism is similar to layers of stacked triangular bases. Each layer is covered with cubic units. To determine the number of cubic units occupying the space in a triangular prism, we can count the cubic units individually or by multiples of 2 s, multiples of 3 s, etc. or we can calculate the number of cubic units that cover one triangular layer and multiply by the number of triangular layers which represents the height of the triangular prism. To determine how many cubic units will cover the triangular base above, we can count each individual cubic unit or we can use the formula: A = 1 b h to calculate 2 the number of cubic units to cover the area of the triangular base. It is important to note that only the square face of the cubic unit will cover the area of the triangular base. Calculate the number of square units that will cover the area of the triangular base above: Area = 1 (4 3) = 6 u2 2 Stage 5 If a triangular base has a layer of 6 square units and the triangular prism is 3 units high, then the number of cubic units that will fill the triangular prism is: # of cubic units per layer times # of layers. Stage 6 Write a formula in terms of the area of the triangular base of a triangular prism and the height of a triangular prism to calculate the volume of a triangular prism. Use this formula to calculate the volume of the triangular prism in Stage 5. Formula for Volume of a Triangular Prism: Area of Triangular Base times height: V = B h Volume of Triangular Prism in Stage 5: V = (4 3 2) 3 = 18 cubic units 2012, TESCCC 10/12/12 page 2 of 2

63 Volume Formulas Cylinders Volume of a Cylinder Math Notes Stage 1 Stage 2 The attributes of a cylinder: 2 circular bases 1 curved lateral surface Stage 3 The volume of a cylinder is the measure of the amount of space occupied in the cylinder. The volume is measured using cubic units. Stage 4 A cylinder is similar to layers of stacked circular bases. Each layer is covered with cubic units. To determine the number of cubic units occupying the space in a cylinder, we can count the cubic units individually or by multiples of 2 s, multiples of 3 s, etc. or we can calculate the number of cubic units that cover one circular layer and multiply by the number of circular layers which represents the height of the cylinder. Stage 5 If each circular layer has square units and the cylinder is 3 units high, then the number of cubic units that will fill the cylinder is: # of cubic units per layer times # of layers. To determine how many cubic units will cover the circular base above, we can count each individual cubic unit or we can use the formula: A = π r 2 to calculate the number of cubic units to cover the area of the circular base. It is important to note that only the square face of the cubic unit will cover the area of the circular base. Calculate the number of square units that will cover the area of the circular base above: Area = 3.14(6) 2 = 3.14(36) = u 2 Stage 6 Write a formula in terms of the radius of the circular base of a cylinder and the height of a cylinder to calculate the volume of a cylinder. Use this formula to calculate the volume of the cylinder in Stage 5. Formula for Volume of a Cylinder: Area of Circular Base times height: V = B h = π r 2 h Volume of Cylinder in Stage 5: V = 3.14(6) 2 3 = cubic units 2012, TESCCC 10/12/12 page 1 of 1