This is an instructional task that gives students a chance to reason about lines of symmetry and discover that a circle has an an infinite number of lines of symmetry. Even though the concept of an infinite number of lines is fairly abstract, fourth graders can understand infinity in an informal way.

This task provides students a chance to experiment with reflections of the plane and their impact on specific types of quadrilaterals. It is both interesting and important that these types of quadrilaterals can be distinguished by their lines of symmetry.

This task is intended for instruction, providing the students with a chance to experiment with physical models of triangles, gaining spatial intuition by executing reflections.

This task can be implemented in a variety of ways. For a class with previous exposure to the incenter or angle bisectors, part (a) could be a quick exercise in geometric constructions,. Alternatively, this could be part of a full introduction to angle bisectors, culminating in a full proof that the three angle bisectors are concurrent, an essentially complete proof of which is found in the solution below.

This instructional task gives students a hands-on experience with composing and decomposing geometric figures.

In this problem-based learning module, students will work collaboratively to improve the accessibility or safety of their school or community. For example, students could identify that accessibility ramps need to be added to the school property or additional sidewalks need to be created/repaired to increase the safety of students as they walk to school. Students would work together to create models of these improvements and create a communications plan that informs the stakeholders of the materials needed to create these improvements (i.e. using volume to determine the amount of concrete, using angles to determine measurements for ramps, etc..).

A Google Drive folder created by Kevin Fink of materials for Math 2231 - Calculus with Analytical Geometry at the College of DuPage. This is intended to be a foundation for anyone wanting to teach Math 2231 where students would need to spend little to no money for materials during the semester. Includes links to a textbook on openstax, a myopenmath shell template, links to lecture videos, supplemental homework assignments and answers, and the materials for an Area project about approximating area under curves.This resource will be updated as needed. For the google drive folder and the most recent version, visit: https://drive.google.com/drive/folders/1KhhTIXOT2a0oEX1hqMIhDsNi4DaO6NSk?usp=sharing

This interactive textbook includes practice problems, examples, and chapters that range from calculating angles and probability to annuities and applying geometry. The content in this e-text was developed using a variety of open educational resources, that were remixed to fit a math applications course in higher education.

Students use a rectangular area model to understand the distributive property. They watch a video to find how to express the area of a rectangle in two different ways. Then they find the area of rectangular garden plots in two ways.Key ConceptsThe distributive property can be used to rewrite an expression as an equivalent expression that is easier to work with. The distributive property states that multiplication distributes over addition.Applying multiplication to quantities that have been combined by addition: a(b + c)Applying multiplication to each quantity individually, and then adding the products together: ab + acThe distributive property can be represented with a geometric model. The area of this rectangle can be found in two ways: a(b + c) or ab + ac. The equality of these two expressions, a(b + c) = ab + ac, is the distributive property.Goals and Learning ObjectivesUse a geometric model to understand the distributive property.Write equivalent expressions using the distributive property.

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.

Lesson OverviewStudents find the area of a parallelogram by rearranging it to form a rectangle. They find the area of a trapezoid by putting together two copies of it to form a parallelogram. By doing these activities and by analyzing the dimensions and areas of several examples of each figure, students develop and understand area formulas for parallelograms and trapezoids.Key ConceptsA parallelogram is a quadrilateral with two pairs of parallel sides. The base of a parallelogram can be any of the four sides. The height is the perpendicular distance from the base to the opposite side.A trapezoid is a quadrilateral with exactly one pair of parallel sides. The bases of a trapezoid are the parallel sides. The height is the perpendicular distance between the bases.You can cut a parallelogram into two pieces and reassemble them to form a rectangle. Because the area does not change, the area of the rectangle is the same as the area of the parallelogram. This gives the parallelogram area formula A = bh.You can put two identical trapezoids together to form a parallelogram with the same height as the trapezoid and a base length equal to the sum of the base lengths of the trapezoid. The area of the parallelogram is (b1 + b2)h, so the area of the trapezoid is one-half of this area. Thus, the trapezoid area formula is A = 12(b1 + b2)h.Goals and Learning ObjectivesDevelop and explore the formula for the area of a parallelogram.Develop and explore the formula for the area of a trapezoid.

Lesson OverviewStudents find the area of a triangle by putting together a triangle and a copy of the triangle to form a parallelogram with the same base and height as the triangle. Students also create several examples of triangles and look for relationships among the base, height, and area measures. These activities lead students to develop and understand a formula for the area of a triangle.Key ConceptsTo find the area of a triangle, you must know the length of a base and the corresponding height. The base of a triangle can be any of the three sides. The height is the perpendicular distance from the vertex opposite the base to the line containing the base. The height can be found inside or outside the triangle, or it can be the length of one of the sides.You can put together a triangle and a copy of the triangle to form a parallelogram with the same base and height as the triangle. The area of the original triangle is half of the area of the parallelogram. Because the area formula for a parallelogram is A = bh, the area formula for a triangle is A = 12bh.Goals and Learning ObjectivesDevelop and explore the formula for the area of a triangle.

Lesson OverviewStudents use what they know about finding the areas of basic figures to find areas of composite figures.Key ConceptsA composite figure is a figure that can be divided into two or more basic figures.The area of a composite figure can be found by dividing it into basic figures whose areas can be calculated easily.For some figures, the area can also be found by surrounding the figure with a basic figure, creating other basic figures “between” the original figure and the surrounding figure. The area of the original figure can then be found by subtracting the basic figure.Goals and Learning ObjectivesFind the area of composite figures by decomposing and composing them into more basic figures. 

Lesson OverviewStudents make two different rectangular prisms by folding two 812 in. by 11 in. sheets of paper in different ways. Then students use reasoning to compare the total areas of the faces of the two prisms (i.e., their surface areas). Students also predict how the amounts of space inside the prisms (i.e., their volumes) compare. They will check their predictions in Lesson 6.Key ConceptsStudents compare the total area of the faces (i.e., surface area) of one rectangular prism to the total area of the faces of another prism. Students make predictions about which prism has the greater amount of space inside (i.e., the greater volume). Students do not compute actual surface areas or volumes. This exploration helps pave the way for a more formal study of volume in Lesson 6 and a more formal study of surface area in Lesson 7.Goals and Learning ObjectivesExplore how the surface areas and volumes of two different prisms made from the same-sized sheet of paper compare.

Lesson OverviewStudents revise their packing plans based on teacher feedback and then take a quiz.Students will use their knowledge of volume, area, and linear measurements to solve problems. They will draw diagrams to help them solve a problem and track and review their choice of problem-solving strategies.Key ConceptsConcepts from previous lessons are integrated into this assessment task: finding the volume of rectangular prisms. Students apply their knowledge, review their work, and make revisions based on feedback from the teacher and their peers. This process creates a deeper understanding of the concepts.Goals and Learning ObjectivesApply your knowledge of the volume of rectangular prisms.Track and review your choice of strategy when problem-solving.

Lesson OverviewStudents explore nets—2-D patterns that can be folded to form 3-D figures. They start by examining several patterns and determining which nets form a cube. Then, they sketch nets for rectangular prisms. They also find the surface area of the rectangular prisms.ELL: Remind students of the units used to measure area and volume. Use this opportunity to reinforce why square units are used for area (2-D) and cubed units are used for volume (3-D).MathematicsA net is a 2-D pattern that can be folded to form a 3-D figure. In this lesson, the focus is on nets for rectangular prisms. There are many possible nets for any given prism. For example, there are 11 different nets for a cube, as shown below.The surface area of a prism is the area of its net.Goals and Learning ObjectivesIdentify nets for cubes.Sketch the net of a rectangular prism.Find the surface area of a rectangular prism.

Lesson OverviewStudents use scissors to transform a net for a unit cube into a net for a square pyramid. They then investigate how changing a figure from a cube to a square pyramid affects the number of faces, edges, and vertices and how it changes the surface area and volume.Key ConceptsA square pyramid is a 3-D figure with a square base and four triangular faces.In this lesson, the net for a cube is transformed into a net for a square pyramid. This requires cutting off one square completely and changing four others into isosceles triangles.It is easy to see that the surface area of the pyramid is less than the surface area of the cube, because part of the cube's surface is cut off to create the pyramid. Specifically, the surface area of the pyramid is 3 square units, and the surface area of the cube is 6 square units. Students will be able to see visually that the volume of the pyramid is less than that of the cube.Students consider the number of faces, vertices, and edges of the two figures. A face is a flat side of a figure. An edge is a segment where 2 faces meet. A vertex is the point where three or more faces meet. A cube has 6 faces, 8 vertices, and 12 edges. A square pyramid has 5 faces, 5 vertices, and 8 edges.Goals and Learning ObjectivesChange the net of a cube into the net of a pyramid.Find the surface area of the pyramid. 

Lesson OverviewStudents estimate the area of Lake Chad by overlaying a grid on the map of the lake.Key ConceptThe area of an irregular figure can be found by overlaying a grid on the figure. By estimating the number of grid squares the figure covers and multiplying by the area of each square, you can find the approximate area of the figure. The accuracy of the estimate depends on the size of the grid squares. Using a smaller grid leads to a more accurate estimate because more whole grid squares are completely filled. However, using a smaller grid also requires more counting and more combining of partially-filled squares and is, therefore, more time-consuming. Using a larger grid gives a quicker, but rougher, estimate of the area.Goals and Learning ObjectivesUse a grid to find the area of an irregular figure.MaterialsMap of Lake Chad handout (one for each pair of students)Rulers, optional (one for each pair of students)

Lesson OverviewStudents build prisms with fractional side lengths by using unit-fraction cubes (i.e., cubes with side lengths that are unit fractions, such as 13 unit or 14 unit). Students verify that the volume formula for rectangular prisms, V = lwh or V = bh, applies to prisms with side lengths that are not whole numbers.Key ConceptsIn fifth grade, students found volumes of prisms with whole-number dimensions by finding the number of unit cubes that fit inside the prisms. They found that the total number of unit cubes required is the number of unit cubes in one layer (which is the same as the area of the base) times the number of layers (which is the same as the height). This idea was generalized as V = lwh, where l, w, and h are the length, width, and height of the prism, or as V = Bh, where B is the area of the base of the prism and h is the height.Unit cubes in each layer = 3 × 4Number of layers = 5Total number of unit cubes = 3 × 4 × 5 = 60Volume = 60 cubic unitsIn this lesson, students extend this idea to prisms with fractional side lengths. They build prisms using unit-fraction cubes. The volume is the number of unit-fraction cubes in the prism times the volume of each unit-fraction cube. Students show that this result is the same as the volume found by using the formula.For example, you can build a 45-unit by 35-unit by 25-unit prism using 15-unit cubes. This requires 4 × 3  × 2, or 24, 15-unit cubes. Each 15-unit cube has a volume of 1125 cubic unit, so the total volume is 24125 cubic units. This is the same volume obtained by using the formula V = lwh:V=lwh=45×35×25=24125.15-unit cubes in each layer = 3 × 4Number of layers = 2Total number of 15-unit cubes = 3 × 4 × 2 =  24Volume = 24 × 1125 = 24125 cubic units Goals and Learning ObjectivesVerify that the volume formula for rectangular prisms, V = lwh or V = Bh, applies to prisms with side lengths that are not whole numbers.