Surface Area and Volume

Type of Unit: Conceptual

Prior Knowledge

Students should be able to:

Identify rectangles, parallelograms, trapezoids, and triangles and their bases and heights.
Identify cubes, rectangular prisms, and pyramids and their faces, edges, and vertices.
Understand that area of a 2-D figure is a measure of the figure's surface and that it is measured in square units.
Understand volume of a 3-D figure is a measure of the space the figure occupies and is measured in cubic units.

Lesson Flow

The unit begins with an exploratory lesson about the volumes of containers. Then in Lessons 2–5, students investigate areas of 2-D figures. To find the area of a parallelogram, students consider how it can be rearranged to form a rectangle. To find the area of a trapezoid, students think about how two copies of the trapezoid can be put together to form a parallelogram. To find the area of a triangle, students consider how two copies of the triangle can be put together to form a parallelogram. By sketching and analyzing several parallelograms, trapezoids, and triangles, students develop area formulas for these figures. Students then find areas of composite figures by decomposing them into familiar figures. In the last lesson on area, students estimate the area of an irregular figure by overlaying it with a grid. In Lesson 6, the focus shifts to 3-D figures. Students build rectangular prisms from unit cubes and develop a formula for finding the volume of any rectangular prism. In Lesson 7, students analyze and create nets for prisms. In Lesson 8, students compare a cube to a square pyramid with the same base and height as the cube. They consider the number of faces, edges, and vertices, as well as the surface area and volume. In Lesson 9, students use their knowledge of volume, area, and linear measurements to solve a packing problem.

Four full-year digital course, built from the ground up and fully-aligned to the Common Core State Standards, for 7th grade Mathematics. Created using research-based approaches to teaching and learning, the Open Access Common Core Course for Mathematics is designed with student-centered learning in mind, including activities for students to develop valuable 21st century skills and academic mindset.

Zooming In On Figures

Unit Overview

Type of Unit: Concept; Project

Length of Unit: 18 days and 5 days for project

Prior Knowledge

Students should be able to:

Find the area of triangles and special quadrilaterals.
Use nets composed of triangles and rectangles in order to find the surface area of solids.
Find the volume of right rectangular prisms.
Solve proportions.

Lesson Flow

After an initial exploratory lesson that gets students thinking in general about geometry and its application in real-world contexts, the unit is divided into two concept development sections: the first focuses on two-dimensional (2-D) figures and measures, and the second looks at three-dimensional (3-D) figures and measures.
The first set of conceptual lessons looks at 2-D figures and area and length calculations. Students explore finding the area of polygons by deconstructing them into known figures. This exploration will lead to looking at regular polygons and deriving a general formula. The general formula for polygons leads to the formula for the area of a circle. Students will also investigate the ratio of circumference to diameter ( pi ). All of this will be applied toward looking at scale and the way that length and area are affected. All the lessons noted above will feature examples of real-world contexts.
The second set of conceptual development lessons focuses on 3-D figures and surface area and volume calculations. Students will revisit nets to arrive at a general formula for finding the surface area of any right prism. Students will extend their knowledge of area of polygons to surface area calculations as well as a general formula for the volume of any right prism. Students will explore the 3-D surface that results from a plane slicing through a rectangular prism or pyramid. Students will also explore 3-D figures composed of cubes, finding the surface area and volume by looking at 3-D views.
The unit ends with a unit examination and project presentations.

Students will critique their work from the Self Check in the previous lesson and redo the task after receiving feedback. Students will then take a quiz to review the goals of the unit.Key ConceptsStudents understand how to find the surface area (using nets) and volume of rectangular prisms. They have extended that knowledge to all right prisms and were able to generalize rules for both measurements. Students also found the surface area (and volume) of figures made up of cubes by looking at the 2-D views.GoalsCritique and revise student work.Apply skills learned in the unit.Understand 3-D measurements:Surface area and volume of right prismsArea and circumference of circlesSurface area and volume of figures composed of cubesSWD: Consider the prerequisite skills for this Putting it Together lesson. Students with disabilities may need direct instruction and/or guided practice with the skills needed to complete the tasks in this lesson. It may be helpful to pull individual students or a small group for direct instruction or guided practice with the skills they have learned thus far in this unit. While students have had multiple exposures to the domain-specific terms, students with disabilities will benefit from repetition and review of these terms. As students move through the lesson, check to ensure they understand the meaning of included domain-specific vocabulary. Use every opportunity to review and reinforce the meaning of domain-specific terms to promote comprehension and recall.

Students will extend their knowledge of surface area and nets of rectangular prisms to generalize a formula for the surface area of any prism.Key ConceptsStudents know how to find the surface area of a rectangular prism using a net and adding the areas for pairs of congruent faces. Students have not seen that the lateral surface forms one long rectangle whose length is the perimeter of the base and whose width is the height of the prism.Using this idea, the surface area of any right prism can be found using the formula:SA = 2B + (perimeter of the base)hGoalsFind a general formula for surface area of prisms.Find the surface area of different prisms.SWD: Generalization of skills can be particularly challenging for some students with disabilities. Students may need direct instruction on the connection between what they already understand and a general formula.Some students with disabilities may have difficulty recalling formulas when it comes time to apply them. Once students discover the formula SA = 2B + (perimeter of the base)h, consider posting the formula in the classroom and encouraging students to add the formula(s) to the resources they have available when completing classwork and homework.

Lesson OverviewStudents will work on the final portion of their project which includes creating the nets for the sides, making a slice in one of their buildings, and putting their buildings together. Once their two model buildings are complete, they will find the surface area and volume for their models and the full-size buildings their models represent.Key ConceptsThe second part of the project is essentially a review of the second half of the unit, while still using scale drawings. Students will find the surface area of a prism as well as the surface area of a truncated prism. The second prism will require estimating and problem solving to figure out the net and find the surface area. Students will also be drawing the figure using scale to find actual surface area.GoalsRedraw a scale drawing at a different scale.Find measurements using a scale drawing.Find the surface area of a prism.SWD: Students with disabilities may have a more challenging time identifying areas of improvement to target in their projects. It may be helpful to model explicitly for students (using an example project or student sample) how to review a project using the rubric to assess and plan for revisions based on that assessment.Students with fine motor difficulties may require grid paper with a larger scale. Whenever motor tasks are required, consider adaptive tools or supplementary materials that may benefit students with disabilities.Students with disabilities may struggle to recall prerequisite skills as they move through the project. It may be necessary to check in with students to review and reinforce estimation skills.

This lesson explores the drag force on airplanes. The students will be introduced to the concept of conservation of energy and how it relates to drag. Students will explore the relationship between drag and the shape, speed and size of an object.

Driving Question: How can we as 7th grade math students use surface area to make a mug that retains heat for the longest amount of time possible?

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Students build scale models of objects of their choice. In class they measure the original object and pick a scale, deciding either to scale it up or scale it down. Then they create the models at home. Students give two presentations along the way, one after their calculations are done, and another after the models are completed. They learn how engineers use scale models in their designs of structures, products and systems. Two student worksheets as well as rubrics for project and presentation expectations and grading are provided.

Learners explore three-dimensional geometric frames including cubes and tetrahedrons, as they create bubble wands with pipe cleaners and drinking straws. Then they investigate how soap film flows into a state of minimum energy when they lift the wand up from the bubble solution. Learners also see how light reflection and interference create shimmering colors in the bubbles.

In this task, students will design a fish tank that meets the volume and surface area criteria indicated in the task.

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