# Seeing Structure in Expressions

### Symbols Come to Life: Meaningful Mathematics with A-SSE

Common Core standard A-SSE is (page 64) Algebra - Seeing Structure in Expressions. Excerpts:

1. Interpret expressions that represent a quantity in terms of its context.★ a. Interpret parts of an expression, such as terms, factors, and coefficients.

3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.★

1. Handshake Problem - (one of my all-time favorite problems) If there are 12 people in a room and each person shakes hands with every other person once, how many handshakes are there? (This problem can be solved in many ways and is connected to many other problems and math concepts.) One solution is given below, along with a general solution for n people. Explain why these expressions work.

1. Delivery Trucks (Illustrative Mathematics) - Expressions are written using up to 4 variables. The question is what does each expression mean in terms of the problem's context. (PDF)
2. Frosted Cake Investigation - A cube of cake is dropped into a vat of frosting....This investigation links number patterns, 3D geometry, and algebra. (Hint/spoiler alert: Four functions are generated, from degree 0 through degree 3. The degree of the polynomial function corresponds to the degree of the geometrical object it is representing.)
3. Profit of a Company (Illustrative Mathematics) - Three equivalent (quadratic) expressions for profit are given (in terms of price, x). Students are to select the expression which gives specific information (such as profit 0 or price 0). (PDF)
4. Horseshoes in Flight (NCTM) - Similar in nature to the last activity. The idea is that the various forms that a quadratic expression can be written provide different pieces of information (about the scenario and graph), x-intercepts, y-intercept, and maximum/minimum point. (A student handout is provided.)
5. Spotting Numbers - Online game. Investigate various dot patterns. There are about 10 patterns in each of 3 difficulty levels. This makes the connection between number patterns, algebraic expressions (functions), and geometry. A useful key is to understand one-dimensional growth (which will be associated with a linear expression), 2D growth (quadratic expression), 3D growth (cubic expression), and exponential. Examples to print. Worksheet to print.
6. Seeing Dots (Illustrative Mathematics) - Two expressions are given. A dot pattern (similar to Spotting Numbers above) is given. Students are to explain why each expression represents the dot pattern (which makes the expressions equivalent). (PDF)
7. Interpreting Algebraic Expressions (MAP/MARS) - This is a lesson. This is a card sorting/matching activity. This helps students translate between words, algebraic expressions, tables (of numbers), and areas. (more info)

By Jim Olsen, JR-Olsen@wiu.edu, Western Illinois University. This webpage created for my ICTM presentation on October 18, 2014 (session #99)