Pentomino Puzzler (Geometry)

Time Required: One-two 45 minute sessions
Skill Level: Intermediate
Recommended Grades: 3rd to 5th

In this activity, students will work in pairs to create a set of 3Doodled pentominoes and then assemble them to solve the 6x10 rectangle. Pentominoes are tetris-like shapes that your students will enjoy designing and working with, as they're problem solving. The focus of this lesson is on the "Standards for Mathematical Practice," which compose the critical components of thinking and reasoning.

Note: Any links outside of the3doodler.com are optional resources. We can’t ensure their upkeep or accuracy.

Knowledge

Students have
  • understand the structure of dominoes as a shape made up of two squares attached.

  • understand the structure of trominoes as a shape made up of three squares attached edge to edge. *See Appendix.

  • understand the structure of polyominoes as a shape made up of any number of squares attached along their edges. *See Appendix.

  • had practice in rotating, sliding and flipping shapes.

  • had plenty of practice in using the 3Doodler. *This activity requires quite a bit of control.

Objectives

Students will
  • learn about the 12-pentomino shapes.

  • be able to recognize and identify the 12-pentomino shapes.

  • Verify experimentally the properties of rotations, reflections, and translations.

  • 3Doodle a set of pentominoes.

  • use pentominoes to attempt to solve the 6 x10 rectangle puzzle.

Materials

Students will need
  • 3Doodler (1 per pair)

  • scissors (1 per pair)

  • sandpaper or nail file (1 per pair)

  • Pentomino worksheet (1 per pair)

Lesson Plan

Instructions

Step 1

Assign 5-equal sized squares to each pair or students. Ask them to work with a partner to see how many different composite shapes they can create from these 5 squares connected edge to edge.

Step 2

Display your tablet, laptop or computer screen on the board with the Pentomino worksheet to introduce the 12-pentomino shapes. Note that each pentomino is composed of 5 equal sized squares attached edge to edge. Practice recognizing and identifying each pentomino by name.
*See Pentomino worksheet in Resources.

Step 3

State the goal: To create a 3Doodled set of pentominoes and use them to solve the 6 x10 rectangle puzzle. Note that there are 2,339 solutions to this puzzle, but that finding even one is extremely difficult.
Option: Display the 2,339 solution page. Don't worry, your students will not remember the solutions.
Solution Page

Step 4

Optional: Work together as a group to try and solve a 6 x10 pentomino puzzle online at this page.

Note that pieces can be flipped (reflection), slid (translation) or turned (rotation). These are called "transformations."

Step 5

Optional: Assign one tablet, laptop or computer per pair. Give students time to practice solving the 6 x 10 puzzle online Here

Step 6

Model how to use the Pentomino worksheet stencils and 3Doodler to create a set of pentominoes. Then use 3Doodled pentominoes to solve the 6x10 puzzle on the Pentomino worksheet mat. Note that it is important to trace stencils as accurately as possible, so that pentominoes fit together neatly. Stray strands or uneven edges can be trimmed with scissors or filed down with sandpaper.

Step 7

Hand out Pentomino worksheet and 3Doodlers. Circle the room to assist and assess as students work.

Wrap Up

Assemble students to discuss what they found most challenging about this assignment and how they solved problems. Students share their pentomino puzzle pieces and solutions on Twitter #3DoodlerEDU

Assessment

Review student’s pentomino pieces, attempts at solving the 6x10 puzzle, and feedback during discussion.

Possible Extensions

  • For students in grades 4 and 5 discuss the area and perimeter of pentomino shapes. Compare the area and perimeter of different pentomino pieces. Note observations.

  • Challenge students to create different sized pentomino rectangles with all 12 or some of the pentominoes.

  • *There are 4 different rectangles that can be created with all 12 pieces: 3x20, 4x15, 5x12 and 6x10.

  • Create animals using all or some of the pentomino pieces. Create a puzzle mat based on animal.

Vocabulary

  • flip - a transformation creating a mirror image of a figure on the opposite side of a line. A flip is also called a reflection.

  • slide - a transformation that slides a figure a given distance in a given direction. A slide is also called a translation.

  • transformation - a general term for four specific ways to manipulate the orientation of a point, a line, or shape.

  • turn - the transformation that occurs when a figure is turned a certain angle and direction around a point. A turn is also called a rotation.

Educational Standards

Common Core
CCSS.MATH.PRACTICE.MP1

Make sense of problems and persevere in solving them.

In This Lesson

Students will understand the meaning of the 6x10 pentomino problem and then attempt various configurations of pentomino pieces to attempt to solve it.

Common Core
CCSS.MATH.PRACTICE.MP4

Model with mathematics.

In This Lesson

Students will create a set of 3Doodled pentominoes to attempt to solve the 6x10 puzzle in the real world rather than online.

Common Core
CCSS.MATH.PRACTICE.MP5

Use appropriate tools strategically.

In This Lesson

Students will use 3Doodlers to explore and deepen their understanding of pentomino concepts.

Common Core
CCSS.MATH.PRACTICE.MP6

Attend to precision.

In This Lesson

Students will use precision in the design of their petomino pieces, as well as their communication about pentomino solutions.

Common Core
CCSS.MATH.PRACTICE.MP7

Look for and make use of structure.

In This Lesson

Students will look closely to discern a pattern or structure within pentomino solutions.

Next Gen Science
3-5-ETS1-1

Engineering Design Define a simple design problem reflecting a need or a want that includes specified criteria for success and constraints on materials, time, or cost.

In This Lesson

Students will create a series of jumping circles based on a set of criteria.

Next Gen Science
3-5-ETS1-2

Engineering Design Generate and compare multiple possible solutions to a problem based on how well each is likely to meet the criteria and constraints of the problem.

In This Lesson

Students will make observations about mathematical relationships based on solutions to multiple Jumping Circles.

CS Teachers
1A-A-3-5

Decompose (break down) a larger problem into smaller sub-problems with teacher guidance or independently.

In This Lesson

Students will break down the larger problem of solving a 6x10 pentomino puzzle into smaller tasks: identifying and recognizing pentominoes, practicing the 6x10 puzzle online, constructing a 3Doodled set of pentominoes and using them to attempt to solve the 6 x10 puzzle on a mat.

ISTE
1C

Use technology to seek feedback that informs and improves their practice and to demonstrate their learning in a variety of ways.

In This Lesson

Students will use the 3Doodler to create a concrete model of a pentomino set. Students will use these pentominoes to attempt various solutions to the 6x10 puzzle.

ISTE
4D

Exhibit a tolerance for ambiguity, perseverance and the capacity to work with open-ended problems.

In This Lesson

Students will demonstrate willingness and competency within an open-ended task with more than one possible outcome.

ISTE
6B

Create original works or responsibly repurpose or remix digital resources into new creations.

In This Lesson

Students will use a 3Doodler to design a set of pentominoes.

ISTE
7A

Use collaborative technologies to work with others, including peers, experts or community members, to examine issues and problems from multiple viewpoints.

In This Lesson

Students will use a 3Doodler with a partner and work together to attempt to solve the 610 pentomino puzzle.

Back to Lessons