Why I love this problem:
It’s open to so much creative play (especially if you have pentomino tiles). Students can try their luck at taking a tile and scaling it up and then trying to cover it with the original size tiles.
You can give students some grid paper and ask them to make scale versions of some of the shapes first. You’ll get a much better understanding of whether they truly understand scale because these shapes are more complicated than rectangles that they would typically scale. In addition, students may not intuitively realized that when you scale the dimensions by a factor the area scales by that factor squared! When you ask how many tiles are required to cover a shape that was scaled up by 2, they’ll be surprised to find out the answer if 4 no matter which shape we are talking about!
Finally, this activity is open to lots of mistakes and trial and error. It’s a reminder for students that when you are on the edge of your learning / ability, you WILL make mistakes.
Tip: Start with the simpler shapes like he square and the line tetromino. Then try the T-shape or the S-shape. Then build up to the pentominoes.
Grade Band: 3rd – 8th
Math Content: Geometry, area, and scaling.
- 3.MD – Geometric measurement: understand concepts of area and relate area to multiplication and to addition.
- 3.G – Reason with shapes and their attributes.
- 4.G – Draw and identify lines and angles, and classify shapes by properties of their lines and angles.
- 5.OA – Analyze patterns and relationships.
- 5.G – Classify two-dimensional figures into categories based on their properties.
- 6.RP – Understand ratio concepts and use ratio reasoning to solve problems.
- 6.G – Solve real-world and mathematical problems involving area, surface area, and volume.
- 7.RP – Analyze proportional relationships and use them to solve real-world and mathematical problems.
- 7.G – Draw, construct, and describe geometrical figures and describe the relationships between them. (Solve problems involving scale drawings.)
Standards of Mathematical Practice:
- Make sense of problems and persevere in solving them.
- Reason abstractly and quantitatively.
- Construct viable arguments and critique the reasoning of others.
- Model with mathematics.
- Use appropriate tools strategicailly
- Attend to precision.
- Look for and make use of structure.
Strategies to try:
As you explore this scenario, you might want to try one or more of these strategies. Sometimes using more than one is best.
- Teacher: Take a tetromino and draw some scale models. Make some correct and other incorrect. Challenges students to identify the mistakes.
- This mostly requires a lot of trial and error. However, if you can tile a polyomino into a square, that is a good way to make a scale model of ANY polyomino.
- For students that find this complicated, allow them to use simpler shapes include pattern blocks.
Questions to explore:
- Is there more than one way to tile a scale model?
- Are there tetrominoes or pentominoes that can NEVER be “rep-tiled”. If so, which ones? How do you know for sure?
- How many polyominoes are need to make a 2x version? 3x? 4x?
Students can Uuse mathigon.org/polypad as a tool to explore this treat as well. My students loved trying things in polypad. They also enjoyed watching me use polypad and using Zoom annotation tools to guide me to find Rep-Tile solutions. Note: polypad is free and no account is required. However, if anyone who creates an account can save and share their polypad drawings.