In module 5, students use their understanding of the areas of rectangles to develop formulas for the area of a parallelogram and the area of a triangle. Students apply their prior knowledge of area, equivalent numerical expressions, the properties of operations, and coordinate graphing as they find the areas of composite polygons and trapezoids. They identify attributes of the faces of right prisms and pyramids and use the net of a solid to determine its surface area. By packing right rectangular prisms with cubes of fractional edge lengths, students determine that the formulas
V = lwh and V = Bh
can be applied to find the volume of any right rectangular prism with positive, rational number edge lengths. Students apply these formulas to solve real-world and mathematical problems, and they write and solve single-variable equations to determine unknown measurements of a prism.
In this topic, students extend their understanding of areas of polygons to develop the formulas for the area of a parallelogram and the area of a triangle. They compose parallelograms into rectangles to develop the formula A = bh, recognizing that the area of a parallelogram is equal to the area of a rectangle with the same base and the same height as the parallelogram. They compose two identical triangles into rectangles or parallelograms to develop the formula A = 1/2bh, recognizing that the area of a triangle is exactly half the area of a parallelogram with the same base and the same height as the triangle. Finally, students solve real-world and mathematical problems by finding the areas of triangles.
In topic B, students apply their understanding of the areas of rectangles and triangles to find the areas of composite polygons and trapezoids. First, they solve problems involving figures graphed in the coordinate plane, using their understanding of rational number coordinates to determine side lengths and find the perimeters and areas of figures. Then students transition to working with trapezoids and composite figures not in the coordinate plane. They use known measurements to determine unknown measurements in a figure and explore multiple strategies to find the areas of polygons, including composition, decomposition, and subtraction.
Students begin topic C by exploring the properties of right prisms and pyramids and identifying attributes of the faces of right prisms and pyramids. Then students represent right prisms and pyramids with nets and determine whether a given figure is a net of a solid. Students find the surface areas of solids by using a net and then generalize a formula for the surface area of a right rectangular prism. They apply their understanding of surface area in a modeling task that involves designing boxes for a pack of 12 ice cream sandwiches to meet given criteria.
In topic D, students find the volumes of right rectangular prisms that have fractional edge lengths. Building from prior knowledge about packing prisms with unit cubes, students explore packing right rectangular prisms with cubes of fractional edge lengths. Students determine that the formulas
V = lwh and V = Bh
can be applied to find the volume of any right rectangular prism. They apply these formulas to solve real-world and mathematical problems, including problems about solids composed of right rectangular prisms.