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 + 1

2. 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 TilesA table can help communicate the number of tiles that must be added to form each successive pile? (the recursive rule)

?

Pile 1 2 3 4 5 6 7 8 ..

Tiles

Pile 1 Pile 2 Pile 3 Pile 4 Pile 5

Piles of Tiles

How many tiles in pile 457?

?

Pile 1 2 3 4 5 6 7 8 ..

Tiles 4 7 10 13 16 19 22 25 ..

Pile 1 Pile 2 Pile 3 Pile 4 Pile 5

Piles of TilesA table can help communicate the number of tiles that must be added to form each successive pile? (the recursive rule)

?

Pile 1 2 3 4 5 6 7 8 ..

Tiles 4 7 10 13 16 19 22 25 ..

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 4

3+1 3+3+1 3+3+3+1 3+3+3+3+1

Piles of Tiles

Tiles = 3n + 1

For pile N = 457Tiles = 3x457 + 1 Tiles = 1372

Pile 1 2 3 4 ..Tiles 3+1 3+3+1 3+3+3+1 ..

Piles of Tiles

Graphing the

Information.

0

5

10

15

20

25

30

35

40

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Pile

Til

es

Pile 1 2 3 4 5 6 7 8Tiles 4 7 10 13 16 19 22 25

Tiles = 3n + 1

n = pile number

Piles of Tiles

0

5

10

15

20

25

30

35

40

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Pile

The information can be visually analyzed.

Pile Tiles

0 1

1 4

2 7

3 10

4 13

5 16

6 19

7 22

8 25

9 28

10 31

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.

0

5

10

15

20

25

30

35

40

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Pile

Til

es

Y = 3n + 1

Piles of Tiles

0

5

10

15

20

25

30

35

40

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Pile

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 TilesPile 0 1 2 3 4 5 6

Tiles 1 4 7 10 13 16 19

What rule will tell the number of tiles needed for Pile N?

Tiles = 3n + 1

0

5

10

15

20

25

30

35

40

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Pile

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

The Mirror ProblemMirror Size

Number of2 borders tiles

Number of1 border tiles

Number ofNo border tiles

2 ft x 2 ft 4 0 0

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)

TotalNumber

of “Tiles”

2 ft x 2 ft 4 0 0 4

3 ft x 3 ft 4 4 1 9

4 ft x 4 ft 4 8 4 16

5 ft x 5 ft 4 12 9 25

6 ft x 6 ft 4 16 16 36

7 ft by 7 ft 4 20 25 49

8 ft by 8 ft 4 24 36 64

9 ft by 9 ft 4 28 49 81

10 ft x 10 ft 4 32 64 100

The Mirror Problem

1 2 3 4 5 6 7 8 9 10 11

The Mirror Problem Mirror Size

Number of“Tiles”

(2 borders)

Number of“Tiles”

(1 border)

Number of“Tiles”

(No borders)

TotalNumber

of “Tiles”

2 ft x 2 ft 4 0 0 4

3 ft x 3 ft 4 4 1 9

4 ft x 4 ft 4 8 4 16

5 ft x 5 ft 4 12 9 25

6 ft x 6 ft 4 16 16 36

7 ft by 7 ft 4 20 25 49

8 ft by 8 ft 4 24 36 64

9 ft by 9 ft 4 28 49 81

8 ft by 8 ft 4 32 64 100

: : : : :

n ft by n ft 4 4(n-2) (n-2)2 n2

The Mirror ProblemMirror Size

# of2 borders

tiles

# of1 border

tiles

# ofNo

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 3 2 1 0

23456789

10

# 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 (2001-2009) at 2012 Annual PAMTE Symposium

• Link to NCTM Algebra Position Paper