Exploring Mathematical Tasks Using the Representation Star RAMP 2013

Your first REAL test: Question: What is Algebra? Answer: The intensive study of the last three letters of the alphabet.

A Typical Algebra Experience 1.Here is an equation: y = 3x Make a table of x and y values using whole number values of x and then find the y values, 3.Plot the points on a Cartesian coordinate system. 4.Connect the points with a line.

Consider... What if the equation came last ?

Let’s Play!

Equations Arise From Physical Situations How many tiles are needed for Pile 5? ? Pile 1 Pile 2 Pile 3 Pile 4 Pile 5

Piles of Tiles A table can help communicate the number of tiles that must be added to form each successive pile? (the recursive rule) ? Pile Tiles Pile 1 Pile 2 Pile 3 Pile 4 Pile 5

Piles of Tiles How many tiles in pile 457? ? Pile Tiles Pile 1 Pile 2 Pile 3 Pile 4 Pile 5

Piles of Tiles A table can help communicate the number of tiles that must be added to form each successive pile? (the recursive rule) ? Pile Tiles Pile 1 Pile 2 Pile 3 Pile 4 Pile 5

Piles of Tiles Physical objects can help find the explicit rule to determine the number of tiles in Pile N? Pile 1 Pile 2 Pile 3 Pile

Piles of Tiles Tiles = 3n + 1 For pile N = 457 Tiles = 3x Tiles = 1372 Pile Tiles

Piles of Tiles Graphing the Information. Pile Tiles Tiles = 3n + 1 n = pile number

Piles of Tiles The information can be visually analyzed. PileTiles

Piles of Tiles How is the change, add 3 tiles, from one pile to the next (recursive form) reflected in the graph? Explain. How is the term 3n and the value 1 (explicit form) reflected in the graph? Explain. Y = 3n + 1

Piles of Tiles The recursive rule “Add 3 tiles” reflects the constant rate of change of the linear function. The 3n term of the explicit formula is the “repeated addition of ‘add 3’” Y = 3n + 1

Representation Star

Piles of Tiles Pile Tiles What rule will tell the number of tiles needed for Pile N? Tiles = 3n + 1

Your first REAL test (revisited): Question: What is Algebra? Answer: Algebra is a way of thinking and a set of concepts and skills that enable students to generalize, model, and analyze mathematical situations. Algebra provides a systematic way to investigate relationships, helping to describe, organize, and understand the world... Algebra is more than a set of procedures for manipulating symbols. (NCTM Position Statement, September 2008)

Let’s Play Some More!

The Mirror Problem Parts Corner Edge Center A company makes “bordered” square mirrors. Each mirror is constructed of 1 foot by 1 foot square mirror “tiles.” The mirror is constructed from the “stock” parts. How many “tiles” of each of the following stock tiles are needed to construct a “bordered” mirror of the given dimensions?

The Mirror Problem

Mirror Size Number of 2 borders tiles Number of 1 border tiles Number of No border tiles 2 ft x 2 ft400 3 ft x 3 ft 4 ft x 4 ft 5 ft x 5 ft 6 ft x 6 ft 7 ft by 7 ft 8 ft by 8 ft 9 ft by 9 ft 10 ft x 10 ft

The Mirror Problem Mirror Size Number of “Tiles” (2 borders) Number of “Tiles” (1 border) Number of “Tiles” (No borders) Total Number of “Tiles” 2 ft x 2 ft ft x 3 ft ft x 4 ft ft x 5 ft ft x 6 ft ft by 7 ft ft by 8 ft ft by 9 ft ft x 10 ft

The Mirror Problem

The Mirror Problem Mirror Size Number of “Tiles” (2 borders) Number of “Tiles” (1 border) Number of “Tiles” (No borders) Total Number of “Tiles” 2 ft x 2 ft ft x 3 ft ft x 4 ft ft x 5 ft ft x 6 ft ft by 7 ft ft by 8 ft ft by 9 ft ft by 8 ft ::::: n ft by n ft 44(n-2)(n-2) 2 n2n2

The Mirror Problem Mirror Size # of 2 borders tiles # of 1 border tiles # of No border tiles 2 ft x 2 ft 3 ft x 3 ft 4 ft x 4 ft 5 ft x 5 ft All squares have 4 corners 1 B ord. Tiles = 4(n-2)

Extending the Problem What if we extended the problem to 3D?

Painted Cube Problem A four-inch cube is painted blue on all sides. It is then cut into one- inch-cubes. What fraction of all the one-inch cubes are painted on exactly one side?

Painted Cube Problem Suppose you consider a set of painted cubes, each of which is made up of several smaller cubes. Use patterns to fill in the blanks in the table that follows. The last entries (for a cube with length of edge 10 in) have been filled in so that you can check the patterns you obtain. Explain thoroughly why the patterns arise and can be extended.

Painted Cube Problem Length of Edge (n) Total Cubes # of small cubes with the indicated # of painted faces

CREDIT Both the “Piles of Tiles Task” and “Mirror Task” were borrowed from presentations made by Mr. Jim Rubillos, Executive Director NCTM ( ) at 2012 Annual PAMTE Symposium Link to NCTM Algebra Position PaperNCTM Algebra Position Paper