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Addition and Multiplication with Area and Volume
Addition and Multiplication with Area and Volume
Drawing, Analysis, and Classification of Two-Dimensional Figures
Students construct, analyze, and classify trapezoids, kites, parallelograms, rectangles, rhombuses, and squares. They identify properties of quadrilaterals that involve pairs of parallel sides, angle measures, side lengths, diagonals, and lines of symmetry and use these properties to create a hierarchy of quadrilaterals. They use the hierarchy to determine the most specific name of any quadrilateral and all names for the quadrilateral.
Areas of Rectangular Figures with Fraction Side Lengths
Students find areas of rectangles with fraction side lengths, first by tiling with tiles that are squares with unit-fraction side lengths and then by tiling with rectangles with fraction side lengths. Through reasoning, students determine that the area of any rectangle, including one with fraction side lengths, can be found by multiplying the rectangle’s length by its width. Students use an area model to multiply mixed numbers, and then they solve real-world and mathematical problems involving multiplication of mixed numbers.
Volume Concepts
In their first formal study of volume, students count the number of unit cubes that pack right rectangular prisms. Then they build right rectangular prisms with improvised units to find volume by using something other than unit cubes. They compose and decompose right rectangular prisms into layers in different ways, finding the volume of each layer and multiplying the number of layers by the volume of each layer to find the volume of the right rectangular prism. Students explore conceptual ideas about volume and capacity by differentiating between packing with cubes and filling with a liquid.
Volume and the Operations of Multiplication and Addition
Students synthesize the work of topic C by determining that the volume of any right rectangular prism is calculated either by multiplying the area of the base by the height, V = B × h, or by multiplying the three dimensions of the prism, V = l × w × h. They use these two formulas to find volumes and unknown dimensions of right rectangular prisms in both mathematical and real-world problems. Students find the volume of a figure composed of right rectangular prisms by decomposing the figure into right rectangular prisms, finding the volume of each prism, and adding the volumes together.
Common Core State Standards covered in Module 5
5.NF.B.4: Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
5.NF.B.4.a:Interpret the product (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.)
5.NF.B.4.b: Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.
5.NF.B.6: Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
5.MD.C.3: Recognize volume as an attribute of solid figures and understand concepts of volume measurement.
5.MD.C.3.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.
5.MD.C.3.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.C.4: Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.
5.MD.C.5: Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.
5.MD.C.5.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.
5.MD.C.5.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.
5.MD.C.5.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.
5.G.B.3: Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.
5.G.B.4: Classify two-dimensional figures in a hierarchy based on properties.
Mathematical Practices
Mathematical Practices
Students can:
1. Solve problems without giving up.
2. Think about numbers in many ways
3. explain their thinking and try to understand others.
4. Show their work in many ways.
5. Use math tools and explain why they used them.
6. Work carefully and check their work.
7. Use what they know to solve new problems.
8. Solve problems by looking for rules and patterns.
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