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.
6.C.6: Apply the order of operations and properties of operations (identity, inverse, commutative properties of addition and multiplication, associative properties of addition and multiplication, and distributive property) to evaluate numerical expressions with nonnegative rational numbers, including those using grouping symbols, such as parentheses, and involving whole number exponents.
6.AF.1: Evaluate expressions for specific values of their variables, including expressions with whole-number exponents and those that arise from formulas used in geometry and other real-world problems.
6.AF.2: Apply the properties of operations (e.g., identity, inverse, commutative, associative, distributive properties) to create equivalent linear expressions and to justify whether two linear expressions are equivalent when the two expressions name the same number regardless of which value is substituted into them.
6.AF.5: Solve equations of the form x + p = q, x − p = q, px = q, and x/p = q fluently for cases in which p, q and x are all nonnegative rational numbers. Represent real-world problems using equations of these forms and solve such problems.
6.GM.4: Find the area of complex shapes composed of polygons by composing or decomposing into simple shapes; apply this technique to solve real-world and other mathematical problems.
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Lesson at a Glance
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