Volume Measurement Core Mathematics Partnership Building Mathematical Knowledge and High-Leverage Instruction for Student Success Core Math July 25, 2016 1:00-2:30

Learning Intention We are learning to help children understand the process of volume measurement. We will be successful when we can: Explain measurement of volume as an example of the general measurement process; Explain why measuring the volume of a rectangular prism can be reduced to (calculated by) multiplication; Use the Moving and Combining Principles to find volumes of simple geometric shapes.

What is “Volume”? Answer this question (silently!) on your own. Compare your answer with those of others at your table. Come to consensus on an answer to report out to the whole group. Add grid to poster or white board for groups to record their lengths Group Measurement

CCSSM Volume Measurement Standards Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition. 5.MD.3 Recognize volume as an attribute of solid figures and understand concepts of volume measurement a. A cube with side length 1 unit, called “a unit cube,” is said to have “one cubic unit” of volume, and can be used to measure volume. b. A plane figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units. 5.MD.4 Measure volumes by counting unit cubes. Note the exact parallel with 3.MD.5 and 3.MD.6 for area!

Sound Familiar?? Turn to page 5 in the Geometric Measurement Progressions. Find the comparison of Grade 3 and Grade 5 Standards. Study each set of standards and note the differences between the them. What is the message the CCSSM authors are sending us about finding area and volume measurements? What are the implications for teachers as a school team? Should participants read paragraph on page 4?

Justify the Volume of a Rectangular Prism? Use snap cubes to build a rectangular prism with side lengths 2 by 3 by 4. How many unit cubes are in this prism? How many ways can you find this number? Write expressions that summarize each of your methods. Change to picture of cube.

Seeing the Prism Terms of Layers

Connecting Nets to Volume Find the nets you constructed from the session earlier today. With a partner… Discuss how you can see the volume of the 2 x 3 x 4 prism by viewing the net. Justify your predictions by using snap cubes.

Connecting Nets to Volume Discuss how you can see the volume of the 1 x 3 x 4 prism. Viewing the net. Relating it to the volume of the 2 by 3 by 4.

Connecting Nets to Volume Discuss how you can see the volume of the ½ x 3 x 4 prism. Viewing the net. Relating it to the volume of the 1 by 3 by 4. In what ways would students be engaged in the following standards… MP6, MP7, MP8?

What is the Volume of a Rectangular Prism? What is the volume of a 1/2 x 1/3 x 4 rectangular prism? What about a 1/2 x 2/3 x 4 rectangular prism? What can you say about the volume of any rectangular prism? The goal here is to see that the volume of a rectangular prism will be given by length x width x height (or area of base x height) even when the dimensions are fractions of a unit. Participants should be reasoning with the combining principle, and the analogy with their earlier work on areas of rectangles should be made explicit. (See next slide.)

CCSSM Volume Measurement Standards Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition. 5.MD.5 Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume. a. Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication. c. Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems. Part b. of this standard is to “apply the formulas…”

CCSSM Volume Measurement Standards Geometry: Solve real-world and mathematical problems involving area, surface area, and volume. 6.G.2 Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = lwh and V = bh to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. Part b. of this standard is to “apply the formulas…”

Properties of Volume “Moving principle”: the volume of a solid figure is not changed under a rigid motion. (Congruent figures have equal volumes.) “Combining principle”: the total volume of two (or more) non-overlapping solid figures is equal to the sum of their volumes.

Seeing the Prism Terms of Layers

What is the Student Misconception? Seeing Arrays as Unstructured sets. Unstructured Sets Seeing Arrays in terms of sides or faces.

Comparison with Area Think back to your work on areas of rectangles on Day 3. What similarities and differences are there between that work and today’s work with volume?

Mulching the garden Beth has to mulch her garden. Her garden measures 20 ft by 24 ft. She wants to put 6 inches of mulch on her garden. How many cubic feet of mulch does she need to purchase? How many cubic yards of mulch would there be?

Learning Intention We are learning to help children understand the process of volume measurement. We will be successful when we can: Explain measurement of volume as an example of the general measurement process; Explain why measuring the volume of a rectangular prism can be reduced to (calculated by) multiplication; Use the Moving and Combining Principles to find volumes of simple geometric shapes.

Session Ends Here! Complete log

Core Mathematics Partnership Project Disclaimer Core Mathematics Partnership Project University of Wisconsin-Milwaukee, 2013-2016   This material was developed for the Core Mathematics Partnership project through the University of Wisconsin-Milwaukee, Center for Mathematics and Science Education Research (CMSER). This material may be used by schools to support learning of teachers and staff provided appropriate attribution and acknowledgement of its source. Other use of this work without prior written permission is prohibited—including reproduction, modification, distribution, or re-publication and use by non-profit organizations and commercial vendors. This project was supported through a grant from the Wisconsin ESEA Title II, Part B, Mathematics and Science Partnerships.