Unit 5

Area and Geometric Measurement:  Volume

Math

Unit Length and Description:

10 days

Students’ prior work with the area model prepares students for work with area and volume in Unit 5.  Students will recognize volume as an attribute to solid figures as they learn that a 1 unit by 1 unit by 1 unit cube (a cubic unit) is the standard unit for measuring volume.  Students will find the volume of a rectangular prism by counting unit cubes.  Students will practice measuring volume using cubic centimeters, cubic inches, cubic feet, and improvised units.  Work will also involve applying the formulas V = l x w x h and V = b x h.  In Unit 5, students will select appropriate units, strategies, and tools for solving real word problems involving estimating and measuring volume. This work will include situations where students find the volume of figures composed of two right rectangular prisms.  Work with area and volume also reinforces recently learned fraction concepts and scaling.  Questions about how the area changes when a rectangle is scaled by a whole or fractional scale factor may be asked and missing fractional sides may be found.

Standards:

 Major Clusters NF – Number and Operations - Fractions Apply and extend previous understandings of multiplication and division to multiply and divide fractions. 5.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. Interpret the product of (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b.  For example, use a visual fraction model to show (2/3 × 4 = 8/3, and create a story context for this equation.  Do the same with (2/3) × (4/5) = 8/15.  (In general, (a/b) × (c/d) = ac/bd.) Major Clusters Measurement and Data 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 solid 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, using cubic cm, cubic in, cubic ft, and improvised units. 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. b.   Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems. 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. Foundational Standards 3.MD.C.5 Recognize area as an attribute of plane figures and understand concepts of area measurement. a.   A square with side length 1 unit, called “a unit square,” is said to have “one square unit” of area, and can be used to measure area. b.    A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units. 3.OA.B.5 Apply properties of operations as strategies to multiply and divide.2 Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.) 4.MD.A.3 Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor. Standards for Mathematical Practices 1.   Make sense of problems and persevere in solving them. 2.   Reason abstractly and quantitatively. 3.   Construct viable arguments and critique the reasoning of others. 4.   Model with mathematics. 5.   Use appropriate tools strategically. 6.   Attend to precision. 7.   Look for and make use of structure. 8.   Look for and express regularity in repeated reasoning. Instructional Outcomes ·        5.MD.3: o   I can recognize that volume is measured as cubic units and is found by packing unit cubes in solid figures. o   I can identify volume as an attribute of a solid figure. o   I can recognize that a cube with 1 unit side length is “one cubic unit” of volume. ·        5.MD.4: o   I can measure volume by counting unit cubes. o   I can measure volume in cubic centimeters, cubic inches, cubic feet, and improvised units. ·        5.MD.5a: o   I can multiply the three dimensions in any order to calculate volume. o   I can idenitfy that the base (b) can be determined by multiplying length times width. o   I can explain that mulipltying length, width, and height of a right rectangular prism is the same as determining the area of the base and multiplying by its height. §  Volume = length x width x height (V = l x w x h) §  Volume = area of base x height (V = b x h) o   I can pack the area of the base of a right rectangular prism and determine the height then multiply or use repeated addition to find the volume. o   I can prove multiplying length by width by height is the same as packing a rectangular prism with cubic units. ·        5.MD.5b: o   I can find volume of right rectangular prisms to solve real world problems. ·        5.MD.5c o   I can add the volume of two right rectangular prisms to find the total volume. o   I can find the total volume of two right rectangular prisms to solve real world problems. ·        5.NF.4b o   I can multiply fractional side lengths to find the volume of a rectangular prism.

Enduring Understandings:

·        Volume is represented in cubic units.

·        Volume can be expressed in both customary and metric units.

·        A square unit could have fractional lengths.  As long as the lengths of a square unit are the same, it is still considered a square unit.

Essential Questions:

·        How do I use the language of math to make sense of/solve a problem?

·        How can the volume of cubes and rectangular prisms be found?

·        What is the relationship among the volumes of geometric solids?

·        Why is volume represented with cubic units and area represented with square units?

·        How do you find volume using fractional lengths?