Strand: GEOMETRY (6.G)

Solve real-world and mathematical problems involving area, surface area, and volume (Standards 6.G.1–4).
• 24 Unit Squares
The purpose of this activity is to help students think a little more flexibly about the concept of area before studying, generally, the areas of triangles and special quadrilaterals.
• 2D Nets and 3D Decorative Boxes
Calculating the surface area of cardboard boxes is the focus of this interactive activity. The classroom activity takes this knowledge and asks the students for figure out how many square inches of wrapping paper is needed to wrap a gift. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• A Place in Space
In this lesson the student is asked to describe a point in space using coordinates.
• Area
This lesson is designed to help students be able to calculate the area of a random shape on a grid, explain the correlation between the size of the perimeter and the number of different possible areas that can be contained within that perimeter.
• Area Explorations
In this lesson, students will explore the area of irregular shapes to find multiple different methods for calculating area
• Areas of Irregular Shapes: Building Sailboats
Learn how wooden boat builders use a variety of mathematical concepts when custom designing their vessels. This video focuses on how area, volume, and measurements of irregular shapes are used in the engineering process, taking math out of the classroom and into real world problem solving.
• Areas of Right Triangles
This task is intended to help build understanding as students work toward deriving a general formula for the area of any triangle. The purpose of this task is for students to use what they know about area and express regularity in repeated reasoning to generate a formula for area of a right triangle.
The purpose of this task is for students to use what they know about area to find the areas of special quadrilaterals. Depending on previous instruction, methods may include decomposing the figures into right triangles and rectangles, or drawing a rectangle to encircle the figure and subtracting areas of right triangles that are not part of the original figure.
The purpose of this task is two-fold. One is to provide students with a multi-step problem involving volume. The other is to give them a chance to discuss the difference between exact calculations and their meaning in a context.
• Base and Height
In this scenario a teacher has given students a task to label the base and height of a triangle and shows 3 students' solutions. Students must then identify which, if any, of the solutions are correct and explain why.
• Boxed In and Wrapped Up
This lesson asks students to find the volume and surface area of a rectangular box and then convert it into a cubical box with the same volume.
• Cartesian Coordinate System
This lesson is designed to help students understand the Cartesian plane, specifically how to plot points, read coordinates and find the ratio of the rise over run for slope.
• Chapter 5 - Mathematical Foundations (UMSMP)
This is Chapter 5 of the Utah Middle School Math: Grade 6 textbook. It provides a Mathematical Foundation for Geometry.
• Chapter 5 - Student Workbook (UMSMP)
This is Chapter 5 of the Utah Middle School Math: Grade 6 student workbook. It covers the following topics: Geometry.
• Christo's Building
This task is primarily about volume and surface area, although it also gives students an early look at converting between measurements in scale models and the real objects they correspond to.
• Computing Volume Progression 1
This is the first in a series of four tasks that gradually build in complexity. The purpose of this series of tasks is to build in a natural way from accessible, concrete problems involving volume to a more abstract understanding of volume. The purpose of this first task is to see the relationship between the side-lengths of a cube and its volume.
• Computing Volume Progression 2
This is the second in a series of four tasks that gradually build in complexity. The purpose of this series of tasks is to build in a natural way from accessible, concrete problems involving volume to a more abstract understanding of volume. In this iteration, we do away with the lines that delineate individual unit cubes (which makes it more abstract) and generalize from cubes to rectangular prisms. However, the calculations are the same as in 6.G Computing Volume Progression 1.
• Computing Volume Progression 3
This is the third in a series of four tasks that gradually build in complexity. The purpose of this series of tasks is to build in a natural way from accessible, concrete problems involving volume to a more abstract understanding of volume. Here, we are given the volume and are asked to find the height.
• Computing Volume Progression 4
This is the last in a series of four tasks that gradually build in complexity. The purpose of this series of tasks is to build in a natural way from accessible, concrete problems involving volume to a more abstract understanding of volume. This problem is based on Archimedes Principle that the volume of an immersed object is equivalent to the volume of the displaced water.
• Determining Surface Area with Unit Blocks, Rulers, and Nets
In this video students are shown how to calculate the surface area of a prism. The classroom activity in the lesson requires that students apply this knowledge and measure the surface areas of real 3-Dl objects. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Finding Areas of Polygons (6th grade)
This task asks students to find the area of polygons that are best suited for increasingly abstract methods.
• Fruit Boxes
A grocer wants to sell boxes of fruit. The student's task is to find out the largest volume box he can make using a 36 inch square of card.
• Geometry (6.G) - 6th Grade Core Guide
The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for Mathematics Grade 6 - Geometry.
• Grade 6 Math Module 5: Area, Surface Area, and Volume Problems (EngageNY)
In this module, students utilize their previous experiences in order to understand and develop formulas for area, volume, and surface area. Students use composition and decomposition to determine the area of triangles, quadrilaterals, and other polygons. Extending skills from Module 3 where they used coordinates and absolute value to find distances between points on a coordinate plane, students determine distance, perimeter, and area on the coordinate plane in real-world contexts
In order to assist educators with the implementation of the Common Core, the New York State Education Department provides curricular modules in Pre-K-Grade 12 English Language Arts and Mathematics that schools and districts can adopt or adapt for local purposes.
• Grade 6 Unit 5: Area and Volume (Georgia Standards)
In this unit students will: Find areas of right, equilateral, isosceles, and scalene triangles, and special quadrilaterals. Find areas of composite figures and polygons by composing into rectangles and decomposing into triangles and other shapes. Solve real-world and mathematical problems involving area. Decipher and draw views of rectangular and triangular prisms from a variety of perspectives. Recognize and construct nets for rectangular and triangular prisms. Find the surface area of rectangular and triangular prisms by using manipulatives and by constructing nets.
• Grade 6 Unit 7: Rational Explorations: Numbers and their Opposites (Georgia Standards)
In this unit students will understand that positive and negative numbers are used together to describe quantities having opposite directions or values, understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line. Recognize that the opposite of the opposite of a number is the number itself.
• Great Modeling Tasks in Three Acts - File Cabinet
This surface area activity has students answer the question: How many stickies cover the cabinet?
• Horizontal and Vertical Distances on the Cartesian Graph
In this activity students place marine animals on a Cartesian graph and then determine the horizontal and vertical distance between them. The classroom activity builds on the student's understanding of distances between points on a Cartesian graph. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Nets for Pyramids and Prisms
The goal of this task is to work with nets for three-dimensional shapes and use them to calculate surface area.
• Painting a Barn
Given the dimensions of a barn, the square footage covered by a gallon of paint, and the price of the paint, students must find the cost of painting the barn and explain their work.
• Pentagon Puzzles
By deconstructing pentagons into triangles, students in this activity learn how to calculate the area of pentagons.
• Polygons in the Coordinate Plane
The purpose of this task is for students to practice plotting points in the coordinate plane and finding the areas of polygons. This task assumes that students already understand how to find areas of polygons by decomposing them into rectangles and triangles.
• Same Base and Height, Variation 1
This is the first version of a task asking students to find the areas of triangles that have the same base and height, and is the most concrete.
• Same Base and Height, Variation 2
This is the second version of a task asking students to find the areas of triangles that have the same base and height. This presentation is more abstract as students are not using physical models.
• Scale Models and Three-Dimensional Scaling in Practice
Students can use this interactive to explore how an object changes when enlarged by a factor of 10. They put this understanding to use in the activity when they compare two cubes of different sizes by volume and surface area. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Sierpinski's Carpet
The purpose of this task is to help motivate the usefulness of exponential notation in a geometric context and to give students an opportunity to see that sometimes it is easier to write a number as a numeric expression rather than evaluating the expression.
In this task, students must figure out how to make a cardboard box just big enough for 12 bottles.
• Surface Area , Area and Volume: The Big Sleep
In this video, Bianca is planning a sleepover for friends. She has to figure out how many people she can invite because the floor will only hold so many sleeping bags. She must calculate both the surface area of the floor and the surface area of a sleeping bag. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Surface Area and Volume
An online activity is the focus of this lesson plan to help students understand the concepts of surface area and volume.
• Surface Area of a Rectangular Prisms
This lesson will help students understand surface area and solve problems using the surface area of a rectangular prism.
• Surface Area of Prisms
In this lesson students will understand surface area and how solve for the surface area of triangular prisms.
• Table for 22: A Real-World Geometry Project
This Teaching Channel video has students apply knowledge of area and perimeter to solve real-world problems. This site provides a lesson plan and student handouts. (13 minutes)
• The Largest Container: Problems Using Volume and Shape
By using a single sheet of paper this interactive leads students to construct shapes, calculate volume, and think about the relationships between different shapes. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Triangle Area
This interactive lesson plan will help students understand how to find the area of a right triangle.
• Triangle Explorer
The applet in this lesson allows students to draw triangles and calculate their area.
• Volume of Prisms
This is a lesson designed to help students understand how to solve problems for the volume of triangular prisms.
• Volume of Rectangular Prisms
This lesson is designed to help students understand how to solve for the volume of rectangular prisms.
• Volumes with Fractional Edge Lengths
The purpose of this task is to introduce students to fractional units for volume.
• Walking the Block
The purpose of this task is for students to apply the calculation of distances on a coordinate plane to a real life context. Though explicit coordinates are not given in the problem, the reasoning behind finding the side lengths of the rectangles in the plane is present and this activity could prepare for formalizing of this with the Cartesian coordinate plane later on.
• Wallpaper Decomposition
The purpose of this task is for students to experiment with composition and decomposition of polygons to examine shapes in a real world context. To find the area of the wall, students will decompose a pentagon into simpler shapes (for example, a rectangle and a triangle).
• What's Fun About Surface Area?
In this Teaching Channel video an educator helps students construct an understanding of surface area. (7 minutes)

http://www.uen.org - in partnership with Utah State Board of Education (USBE) and Utah System of Higher Education (USHE).  Send questions or comments to USBE Specialist - Lindsey Henderson and see the Mathematics - Secondary website. For general questions about Utah's Core Standards contact the Director - Jennifer Throndsen .

These materials have been produced by and for the teachers of the State of Utah. Copies of these materials may be freely reproduced for teacher and classroom use. When distributing these materials, credit should be given to Utah State Board of Education. These materials may not be published, in whole or part, or in any other format, without the written permission of the Utah State Board of Education, 250 East 500 South, PO Box 144200, Salt Lake City, Utah 84114-4200.