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
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