Can you think of pictorial ways to visualise the sequence of Powers of 2? Powers of 3?
Powers of 2 can be visualized pictorially as:
(1) Doubling squares - start with 1 dot, then 2 dots in a line, then 4 dots in a square, then 8 dots, then 16 dots in a square, and so on.
(2) Branching tree structure where each branch splits into two.
(3) Binary tree representation where each node splits into two children.
Powers of 3 can be visualized as:
(1) Tripling triangles - each triangle splits into 3 smaller triangles.
(2) Ternary tree where each node branches into three.
(3) Cubes that triple in one dimension - 1 cube, then 3 cubes in a row, then 9 cubes in a arrangement, then 27 cubes in a ×3 cube.
The chapter shows one way of thinking about Powers of 2 using doubling patterns.
Explanation
The chapter mentions in section 1.3 that there are pictorial ways to visualize powers of 2, and shows one representation. Students are encouraged to think of their own visual representations. Powers of 2 naturally relate to doubling patterns and binary branching, while powers of 3 relate to tripling patterns and ternary structures.
Solution Steps
Powers of 2 - Method 1: Doubling squares - start with 1 dot, then 2 dots in a line, then 4 dots in a square, then 8 dots, then 16 dots in a square
Powers of 2 - Method 2: Branching tree structure where each branch splits into two
Powers of 2 - Method 3: Binary tree representation where each node splits into two children
Powers of 3 - Method 1: Tripling triangles - each triangle splits into 3 smaller triangles
Powers of 3 - Method 2: Ternary tree where each node branches into three
Powers of 3 - Method 3: Cubes that triple in one dimension - 1 cube, 3 cubes in a row, 9 cubes in arrangement, 27 cubes in ×3 cube