Geometry

CAD Crafts teach geometry through play, exploration, creative problem solving, and challenges.
Students will understand geometric principles intuitively by constructing original toys with 2D shapes and 3D forms using 21st Century technology.

CAD Craft projects are meant to build upon what a student already understands from formal education and to elaborate upon and reinforce these concepts. Additionally, for students in primary grades, projects will introduce rudimentary concepts that they will encounter in future grades. CAD Craft projects are meant to challenge student's thinking. CAD Craft projects are also meant to allow students the freedom to explore mathematical and geometric concepts in greater depth by applying math to play, "real world" problem solving, design thinking, and graphic arts.

The standards outlined here are guidelines for and examples of CAD Craft's reinforcement of Common Core Standards for Geometry and is not exhaustive of the topics students will intuit from designing their own toy.

Grade 1

Grade 1 students will design toys and explore principles of geometry utilizing software and mathematical tools. Students will make connections between geometry concepts and real objects through play, design, and problem solving.

Reason with shapes and their attributes.

CCSS.MATH.CONTENT.1.G.A.1

"Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes."

Students will explore basic shapes by planning designs and creating original toys.
Students will recognize that their toys and everyday objects are made of basic shapes and 3D forms.
Students will describe their toy in terms of the shapes and forms used to create it.

CCSS.MATH.CONTENT.1.G.A.2

"Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape."

Students will create a toy design utilizing basic shapes in a computer graphics program.
Students will be challenged to adapt new shapes and forms to create a similar toy.

e.g. Can you make a toy design using only circles?
e.g. If you made a car using boxes, could you make a car using spheres? What would it look like?

Students will define their toy in terms of component parts and be able to create and recognize unfolded 3D objects such as boxes and pyramids

e.g. What does a box look like when you flatten all of its sides? Could you refold it to be a box again?

Students will recognize that 3D objects can be described in terms of faces and that each face makes up a part of the whole.

e.g. The sides of a box are rectangles.

CCSS.MATH.CONTENT.1.G.A.3

Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

Students will describe their toy in terms of component parts and learn to visualize objects in terms of parts of a whole
Students will create designs using basic shapes in computer graphics software using additive methods (see video)

Grade 2

Reason with shapes and their attributes.

Grade 2 students will demonstrate competency in Grade 1 standards and create toy projects that explore geometry concepts in greater depth

CCSS.MATH.CONTENT.2.G.A.1

"Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes."

Students will be challenged to create toys following certain parameters.

e.g. Make a teddy bear using only heart shapes. Design a folded paper car using only hexagons and circles.

Students will define their toy in terms of component parts and recognize the 2D "unfolding" of 3D objects such as truncated pyramids and prisms with polygon bases (see example)

(preliminary introduction to Grade 6 standard 6.G.A.4: "Represent three-dimensional figures using nets made up of rectangles and triangles...")
e.g. How can you make a doll cup with angled, rather than straight, sides? What would this cup look like unfolded from the base?
e.g. What are the different ways I could unfold and construct my paper toy?
See Expanding Thinking—"Unfolding: Polygons & Nets"

Students will recognize that 3D objects can be described in terms of faces and that each face makes up a part of the whole.

e.g. A box has 6 rectangular faces. A pyramid has 5 faces: 4 triangles and a square base.
e.g. Challenge Question—What does a house look like from the left side, right side, top, back, and front? How can you use this information to make a toy house? How can we use this method to make any toy?

SLIDE SHOW OF HOUSE FACES

CCSS.MATH.CONTENT.2.G.A.2

"Partition a rectangle into rows and columns of same-size squares and count to find the total number of them."

Students can use grids as a method of scaling by drawing a design on a small grid, then redrawing the design on a larger grid, using what they see in each small grid square as a reference to draw in the larger grid squares.
Students will become familiar using grid guidelines in computer graphics software

e.g. A student might use grids to help them design a box or plan out a design for a doll quilt.
(preliminary introduction to Grade 5 standards 5.G.A.1 & 5.G.A.2: "Graph points on the coordinate plane to solve real-world and mathematical problems.")

CCSS.MATH.CONTENT.2.G.A.3

"Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape."

Students will describe their toy in terms of component parts and learn to visualize objects in terms of parts of a whole
Students will create designs using basic shapes in computer graphics software using both additive and subtractive methods (see video below)

e.g. A student might discover that a heart shape can be used to make the sides and arms of a doll chair by removing the bottom third of the heart and top quarter of the shape (see example)

Grade 3

Grade 3 students will demonstrate competency in Grades 1 & 2 standards. Students will create toy projects that explore geometry concepts in greater depth by applying what they know about properties of objects and forms and by defining their toy designs using broader geometry vocabulary.

Reason with shapes and their attributes.

CCSS.MATH.CONTENT.3.G.A.1

Understand that shapes in different categories...may share attributes...and that the shared attributes can define a larger category...

Students will recognize that most of their toy designs will follow a generalized method of construction
Students will be able to generalize the process of toy modeling in 2D and 3D using geometry vocabulary

In relation to designing a 3D paper toy with computer software, students will be able to describe that they start with a 2D shape as the base. Then, they "extrude" the shape into a 3D form. Students will be able to define what their 2D and 3D objects are (for example, a rectangle is extruded to make a rectangular prism)

Students can classify their design forms as fitting into specific categories of shapes and forms

e.g. Students will recognize that forms such as boxes have sides that are quadrilaterals.
(preliminary introduction to Grade 5 standard 5.G.B.3: "Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category...")

Students will recognize that they can make many different and more complex 3D forms to create a toy by using irregular shapes

e.g. A student might decide to make a set of irregular and funky paper tea cups using prisms with irregular bases rather than strictly rectangular prisms. Or a student might make a series of "concept" cars using the same idea.

Students will recognize and describe variations of their design that can be created by adapting the basic forms of 3D and 2D models

Students will recognize and describe that categories of toys (like tea cups) can be made of 3D forms such as prisms. Variations can be made by changing the base of the prism. Students will recognize, for example, that they can make different styles of tea cups depending on if they use a rectangular prism for the cup or if they make a cup with 6 or 8 faces using hexagonal or octagonal prisms.

Students will be able to describe the relationship of their toy designs to others' work in terms of the shapes and forms used in the designs' creation

e.g. A toy car and a doll closet are both made of prisms. The doll furniture is made of a prism that is taller than it is wide or deep. A car is made from a prism that is deeper than it is tall or wide.

Grade 4

Grade 4 students will demonstrate competency in Grades 1, 2, & 3 standards and create toy projects that explore geometry concepts in greater depth by formally recognizing principles of symmetry in design construction of paper and/or fabric toys.

CCSS.MATH.CONTENT.4.G.A.3

Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.

Students will utilize symmetry to construct original toys
Students will learn to use symmetry tools in graphic design and 3D modeling software

e.g. Students will understand that they can make symmetric shapes in computer graphics software by drawing one half of the image then duplicating and reflecting this image with software tools. Or, depending on the program they use, students might also explore automated symmetry tools. The symmetric shapes they generate can be used for any design!

Students will create double-sided toys using reflection and translation tools in computer graphics software

e.g. A student may make a doll dresser with lined drawers by creating a drawer pattern that can be folded in half before being folded into a box, so that the folded paper has designs on both sides. The student would be able to define their unfolded pattern as being"reflectional symmetric." Once folded into a final product, their drawer pattern would create a box with a decorated inside and outside. (See example, below)

Students will be able to recognize and define where and how they used symmetry in making their toy
If their toy is symmetric, students will be able to informally classify which axis their toy is symmetric about

e.g. A student might describe a toy car as being symmetric if it were cut in half from the middle of the hood to the middle of the trunk, longways. An advanced student could describe this toy as being symmetric about the x or y axis when using 3D modeling software.
e.g. A student might describe a handmade doll as being symmetric from front to back if it were unsewn, or symmetric from left to right. An advanced student would describe this doll as being "mirror symmetric."

(See Extending Thinking—"Symmetry")

Grade 5

Grade 5 students will demonstrate competency in Grades 1, 2, 3, & 4 standards and create toy projects that explore geometry concepts in greater depth by using features of 3D modeling software to experiment with varying the angles of polygons and modifying their basic toy designs. Students will generalize about 3D forms and describe relationships among all 3D paper toys using advanced geometry vocabulary.

Classify two-dimensional figures into categories based on their properties.

CCSS.MATH.CONTENT.5.G.B.3

Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category.

Students will classify their toy design components as fitting into general categories of shapes and forms

Students will recognize that forms such as prisms belong to a series of objects called polyhedrons. 2D shapes such as rhombuses, rectangles, and pentagons are classified as polygons.
e.g. A student may explain that paper craft toys like cars and tea sets could be classified as being made of polyhedrons. The sides of paper craft models are called faces, and each face is a polygon.

Students will recognize and describe variations of their design that can be created by adapting the basic forms of 3D and 2D models

Students will recognize and describe that categories of toys (like tea cups) can be made of polyhedrons such as prisms. Variations can be made by changing the polygon base of the prism. Students will recognize, for example, that they can make different styles of tea cups depending on if they use a rectangular prism for the cup or if they make a cup with 6 or 8 faces using hexagonal or octagonal prisms

Students will be able to describe the similarities among their toy designs and others' toys in broad and specific terms of the shapes and forms used in the designs' creation

e.g. All 3D paper toys are made from polygons. The toys differ in the number of polygons and type of polygons used to create them. In general, toys like cars are made from polyhedrons that have quadrilateral faces, like rectangular prisms. However, it is possible to make a car with other prisms like hexagonal prisms. These polyhedrons might be used to make paper headlights or a grill for the car.

CCSS.MATH.CONTENT.5.G.B.4

Classify two-dimensional figures in a hierarchy based on properties.

Students will recognize that most of their toy designs will follow a generalized method of construction
Students will be able to generalize the process of toy modeling in 2D and 3D using advanced geometry vocabulary

In relation to designing a 3D paper toy with computer software, students will be able to describe that they start with a polygon and "extrude" this polygon to make a polyhedron. The polyhedron form is modified to create a recognizable toy object. The process of "unfolding" their toy for printing onto paper creates the 2D net of their polyhedron form.
e.g. For example, a student may make a toy drum with computer drafting software and describe the model on screen as an octagonal prism. They would describe the method they used to make a paper model of this toy as unfolding each face of the object into a series of connected polygons called a net. The net can is used to reconstruct the drum by folding each face so that the edges of each polygon are joined.
(preliminary introduction to Grade 6 standard 6.G.A.4: "Represent three-dimensional figures using nets made up of rectangles and triangles...")

(See Extending Thinking—"Polygons & Nets")

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