Dear Parents and Caregivers, Thank you for supporting your child to achieve success in school. We value your input and active participation in your child’s education. These letters are designed to help you understand the work your child brings home and the academic expectations of Arizona’s College and Career Ready Standards. Your child is developing the necessary skills and knowledge to help them compute, think, and reason mathematically. This letter is about area, volume, and surface area in sixth grade. End-­‐of-­‐year goals In sixth grade, students build upon the foundations from fourth and fifth grades of reasoning about relationships among shapes to determine area, surface area, and volume. They find areas of right triangles, other triangles, and special quadrilaterals by decomposing these shapes, rearranging or removing pieces, and relating shapes to rectangles. They also use nets to model and find the surface area of 3-­‐dimensional figures. They apply their understanding of the concept of volume that they discovered in fifth grade to find the volume of rectangular prisms with fractional side lengths using formulas. Vocabulary •
3-­‐dimensional figure: figures that have length, width and height, such as rectangular prisms •
Area: the measure, in square units, of the interior region of a 2-­‐dimensional figure (u2) •
Surface area: the total area of the faces, including bases, and curved surfaces of a solid figure •
Volume: the number of cubic units it takes to fill a solid (u3) •
Net: a 2-­‐dimensional shape that can be folded into a 3-­‐dimensional figure Finding the area of a 2-­‐dimensional figure Students have practiced finding the area of rectangles and triangles in prior grades. Sixth grade students use that understanding to find the area of special quadrilaterals, including rectangles, parallelograms, trapezoids, rhombi, and kites. For example, to find the area of the trapezoid below, the student might see that it is made up of two triangles and a rectangle. By applying the formulas for finding the area of a triangle and the area of a rectangle, they could find the area of the trapezoid. Area is always written in square units. (u2) 4 12 6 Area of a triangle: A = Area of a rectangle: A = Decompose the trapezoid into two triangles and one rectangle. To find the area of each triangle solve • 3 • 4 = 6u2. To find area of both triangles add 6 + 6 = 12u2. This leaves a rectangle with the length of 4 and the width of 6. Solve using the area of a rectangle formula: 4 • 6 = 24u2. Then add the areas of the triangles to the area of the rectangle to find the total area of the trapezoid. 12u2 + 24u2 = 36u2. Mesa Public Schools/Grade 6/Area, Volume, & Surface Area/2013 Authorization to reprint or disseminate must be granted by Mesa Public Schools (February-­‐2014). Models for surface area of 3-­‐dimensional figures Students will use nets, composed of rectangles and triangles, to represent the surface area of rectangular prisms. It may be helpful to think of surface area as the amount of paper one might need to cover the outside of a box, without overlaps. To determine surface area, students can draw 2-­‐
dimensional figures that represent each side of the box. For example, they could draw six squares, all of the same size to represent the six faces of a cube. Since a cube has 6 faces of equal area, the student could find the area of each face by squaring the side (s) length (A = s2). Then add those areas to find the total for the cube. This is the same as using the formula to find the surface area of a cube. (SA = 6s2) To find the surface area of 3-­‐dimensional figures with faces that are triangles and rectangles, such as square pyramids or prisms, a student would create the net for each figure, find the areas of each face represented in the net, and add those areas together. A square pyramid has four triangular faces and one square base. Find the area of each triangular face and the square base. Add the areas together to find the surface area. 3-­‐D shape net Finding volume of a right rectangular prism Fifth grade students learned to find the volume of right rectangular prisms by using cubes or blocks to fill the prism. In sixth grade, students use this understanding to find the volume of right rectangular prisms with fractional edge lengths. They will discover the relationship between the total volume and the area of the base. This leads them to the formulas for volume: • V = Bh volume equals B (area of the base) multiplied by h (height) OR • V = lwh volume equals l (length) multiplied by w (width) multiplied by h (height) How to help at home •
Encourage your child to use a variety of strategies to solve problems and to explain the reasoning behind the strategy they chose. •
Contact your child’s teacher with questions. •
Watch these videos on using nets, finding volume and surface area from Learn Zillion http://learnzillion.com/lessonsets/278-­‐use-­‐nets-­‐to-­‐represent-­‐threedimensional-­‐figures-­‐and-­‐find-­‐surface-­‐area •
Remember, mistakes are a part of learning. Mesa Public Schools/Grade 6/Area, Volume, & Surface Area/2013 Authorization to reprint or disseminate must be granted by Mesa Public Schools (February-­‐2014).