Mathematics
Indices
Class 8
Intermediate
About 15 min
Premium
Laws of indices: folding a sheet of paper
Fold, unfold and repeat folds of a sheet of paper to discover the laws of indices, and find out how thick the stack would get.
Core concept
2^m × 2^n = 2^(m + n), 2^m ÷ 2^n = 2^(m − n) and (2^m)^n = 2^(mn); 2^0 = 1 and 2^−n = 1/2^n
Topic: Indices
Fold a sheet of paper in half and it has 2 layers; fold again and it has 4; every fold doubles the layers. After m folds there are 2 × 2 × … × 2 = 2^m layers: 2 is the base and m is the index. Folding n more times multiplies by 2^n again, unfolding n times divides by 2^n, and doing the same m folds n times over raises 2^m to the power n. Watching the layers shows the laws of indices, and what 2^0 and 2^−1 must mean.
What you will learn
- Use the product, quotient and power laws of indices.
- Explain why 2^0 = 1 and 2^−n = 1/2^n.
- Appreciate how quickly repeated doubling grows.
Before you start
- Repeated multiplication and powers such as 2³ = 8.
Materials
- A sheet of newspaper
Equipment
- A ruler
Procedure
- Tap − or + to choose m, the number of folds to start with, and n.
- Tap the chip to fold n more times, unfold n times, or repeat the m folds n times.
- Read the law, the number of layers and how thick the stack would be.
- Record 2³ × 2², an unfolding that leaves exactly one sheet, an unfolding that goes past one sheet, and a repeated fold with 2⁶ or more layers.
How this simulation is simplified
- Each sheet is 0.1 mm thick, and the paper can be folded as often as we like.