# What are Penrose Tiles?

Create many interesting designs that tessellate by fitting together Penrose tile shapes!

**Grades covered:**through 12

**Curriculum Topics:**Identifying Geometric Shapes, Tessellations, Penrose Tiling

### Standards

Idea Sheets are cross-referenced to subjects listed in the Common Core, Next Generation Science Standards, and California Content Standards.

Mathematical Practices: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. ||Common Core Mathematics||Kindergarten||Mathematical Practices|||K.G.1. Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as above, below, beside, in front of, behind, and next to.||Common Core Mathematics||Kindergarten||Geometry||Identify And Describe Shapes (Squares, Circles, Triangles, Rectangles, Hexagons, Cubes, Cones, Cylinders, And Spheres)|||K.G.2. Correctly name shapes regardless of their orientations or overall size.||Common Core Mathematics||Kindergarten||Geometry||Identify And Describe Shapes (Squares, Circles, Triangles, Rectangles, Hexagons, Cubes, Cones, Cylinders, And Spheres)|||K.G.3. Identify shapes as two-dimensional (lying in a plane, “”flat””) or three- dimensional (“”solid””).||Common Core Mathematics||Kindergarten||Geometry||Identify And Describe Shapes (Squares, Circles, Triangles, Rectangles, Hexagons, Cubes, Cones, Cylinders, And Spheres)|||K.G.4. Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts (e.g., number of sides and vertices/””corners””) and other attributes (e.g., having sides of equal length).||Common Core Mathematics||Kindergarten||Geometry||Analyze, Compare, Create, And Compose Shapes|||K.MD.1. Describe measurable attributes of objects, such as length or weight. Describe several measurable attributes of a single object.||Common Core Mathematics||Kindergarten||Measurement And Data||Describe And Compare Measurable Attributes|||K.MD.2. Directly compare two objects with a measurable attribute in common, to see which object has “”more of””/””less of”” the attribute, and describe the difference. For example, directly compare the heights of two children and describe one child as taller/shorter.||Common Core Mathematics||Kindergarten||Measurement And Data||Describe And Compare Measurable Attributes|||K.MD.3. Classify objects into given categories; count the numbers of objects in each category and sort the categories by count.3||Common Core Mathematics||Kindergarten||Measurement And Data||Classify Objects And Count The Number Of Objects In Each Category|||Mathematical Practices: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. ||Common Core Mathematics||Grade 1||Mathematical Practices|||1.G.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.||Common Core Mathematics||Grade 1||Geometry||Reason With Shapes And Their Attributes|||1.G.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.4||Common Core Mathematics||Grade 1||Geometry||Reason With Shapes And Their Attributes|||Mathematical Practices: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. ||Common Core Mathematics||Grade 2||Mathematical Practices|||Reason With Shapes And Their Attributes||Common Core Mathematics||Grade 2||Geometry|||Mathematical Practices: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. ||Common Core Mathematics||Grade 3||Mathematical Practices|||Reason With Shapes And Their Attributes||Common Core Mathematics||Grade 3||Geometry|||Mathematical Practices: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. ||Common Core Mathematics||Grade 4||Mathematical Practices|||Draw And Identify Lines And Angles, And Classify Shapes By Properties Of Their Lines And Angles||Common Core Mathematics||Grade 4||Geometry|||4.OA.5. Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule “”Add 3″” and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.||Common Core Mathematics||Grade 4||Operations And Algebraic Thinking||Generate And Analyze Patterns|||Mathematical Practices: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. ||Common Core Mathematics||Grade 5||Mathematical Practices|||5.G.3. Understand that attributes belonging to a category of two- dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.||Common Core Mathematics||Grade 5||Geometry||Classify Two-Dimensional Figures Into Categories Based On Their Properties|||5.G.4. Classify two-dimensional figures in a hierarchy based on properties.||Common Core Mathematics||Grade 5||Geometry||Classify Two-Dimensional Figures Into Categories Based On Their Properties|||Solve Real-World And Mathematical Problems Involving Area, Surface Area, And Volume||Common Core Mathematics||Grade 6||Geometry|||Mathematical Practices: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. ||Common Core Mathematics||Grade 6||Mathematical Practices|||Draw, Construct, And Describe Geometrical Figures And Describe The Relationships Between Them||Common Core Mathematics||Grade 7||Geometry|||Mathematical Practices: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. ||Common Core Mathematics||Grade 7||Mathematical Practices|||Solve Real-Life And Mathematical Problems Involving Angle Measure, Area, Surface Area, And Volume||Common Core Mathematics||Grade 7||Geometry|||Understand Congruence And Similarity Using Physical Models, Trans- Parencies, Or Geometry Software||Common Core Mathematics||Grade 8||Geometry|||Mathematical Practices: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. ||Common Core Mathematics||Grade 8||Mathematical Practices|||Mathematical Practices: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. ||Common Core Mathematics||High School||Mathematical Practices

2.1 Identify and describe common geometric objects (e.g., circle, triangle, square, rectangle, cube, sphere, cone).||CA Mathematics||Grade Kindergarten||03. Measurement and Geometry||2.0 Students identify common objects in their environment and describe the geometric features.|||2.2 Compare familiar plane and solid objects by common attributes (e.g., position, shape, size, roundness, number of corners).||CA Mathematics||Grade Kindergarten||03. Measurement and Geometry||2.0 Students identify common objects in their environment and describe the geometric features.|||2.0 Students identify common geometric figures, classify them by common attributes, and describe their relative position or their location in space.||CA Mathematics||Grade 1||03. Measurement and Geometry|||2.1 Describe and classify plane and solid geometric shapes (e.g., circle, triangle, square, rectangle, sphere, pyramid, cube, rectangular prism) according to the number and shape of faces, edges, and vertices.||CA Mathematics||Grade 2||03. Measurement and Geometry||2.0 Students identify and describe the attributes of common figures in the plane and of common objects in space.|||2.2 Put shapes together and take them apart to form other shapes (e.g.,two congruent right triangles can be arranged to form a rectangle).||CA Mathematics||Grade 2||03. Measurement and Geometry||2.0 Students identify and describe the attributes of common figures in the plane and of common objects in space.|||2.0 Students describe and compare the attributes of plane and solid geometric figures and use their understanding to show relationships and solve problems.||CA Mathematics||Grade 3||03. Measurement and Geometry|||3.4 Identify figures that have bilateral and rotational symmetry.||CA Mathematics||Grade 4||03. Measurement and Geometry||3.0 Students demonstrate an understanding of plane and solid geometric objects and use this knowledge to show relationships and solve problems.|||3.7 Know the definitions of different triangles (e.g., equilateral, isosceles, scalene) and identify their attributes.||CA Mathematics||Grade 4||03. Measurement and Geometry||3.0 Students demonstrate an understanding of plane and solid geometric objects and use this knowledge to show relationships and solve problems.|||3.8 Know the definition of different quadrilaterals (e.g., rhombus, square, rectangle, parallelogram, trapezoid).||CA Mathematics||Grade 4||03. Measurement and Geometry||3.0 Students demonstrate an understanding of plane and solid geometric objects and use this knowledge to show relationships and solve problems.|||2.0 Students identify and describe the properties of two-dimensional figures.||CA Mathematics||Grade 6||03. Measurement and Geometry|||2.2 Estimate and compute the area of more complex or irregular two- and three-dimensional figures by breaking the figures down into more basic geometric objects.||CA Mathematics||Grade 7||03. Measurement and Geometry||2.0 Students compute the perimeter, area and volume of common geometric objects and use the results to find measures of less common objects. They know how perimeter, area, and volume are affected by changes of scale.|||10.0 Students compute areas of polygons, including rectangles, scalene triangles, equilateral triangles, rhombi, parallelograms, and trapezoids.||CA Mathematics||Grade 8 – 12||02. Geometry|||12.0 Students find and use measures of sides and of interior and exterior angles of triangles and polygons to classify figures and solve problems.||CA Mathematics||Grade 8 – 12||02. Geometry|||22.0 Students know the effect of rigid motions on figures in the coordinate plane and space, including rotations, translations, and reflections.||CA Mathematics||Grade 8 – 12||02. Geometry|||08.0 Students know, derive, and solve problems involving the perimeter, circumference, area, volume, lateral area, and surface area of common geometric figures.||CA Mathematics||Grade 8 – 12||02. Geometry

**Categories include:**

- Grades K-2
- Grades 3-5
- Grades 6-8
- Math

**Tags:**Measurement & Data, Patterns, Geometry

**Subjects covered:**["Math"]