1. Equivalents

2. Piaget Concrete Semi-concrete A manipulative
Something I can move around and investigate things with
Semi-concrete
A picture of the things I have been moving around
If I have never moved it around before this becomes semi-abstract for me

3. Piaget Semi-abstract Abstract
Symbols or alternate representations for my semi-concrete models
Good time to use algebraic thinking with students
Abstract
Using numbers to represent all parts of the original concrete model

4. Equivalent Representations
Percents
Decimals
Ratios
Fractions
25% = 0.25 = 25/100 = ¼
Need to be able to fill in missing equivalent values on a table

5. Circle Graph Representations
5/3
12.5%
1.25
All circles contain exactly 100%