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Exploring Mathematical Tasks Using the Representation Star
RAMP 2013
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Your first REAL test:
• Question: What is Algebra?• Answer: The intensive study of the last three
letters of the alphabet.
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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.
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Consider . . .
• What if the equation came last ?
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Let’s Play!
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Equations Arise From Physical Situations
How many tiles are needed for Pile 5?
?
Pile 1 Pile 2 Pile 3 Pile 4 Pile 5
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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
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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
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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
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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
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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 ..
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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
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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
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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
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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
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Representation Star
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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
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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)
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Let’s Play Some More!
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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?
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The Mirror Problem
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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
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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
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The Mirror Problem
1 2 3 4 5 6 7 8 9 10 11
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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
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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)
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Extending the Problem
• What if we extended the problem to 3D?
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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?
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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.
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Painted Cube Problem
Length of Edge(n)
Total Cubes 3 2 1 0
23456789
10
# of small cubes with the indicated # of painted faces
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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