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Where Does that Algebraic Equation Come From?Moving From Concrete Experience to Symbolic Form"
Jim RubilloJim [email protected]@verizon.net
What is Algebra?
The intensive study of the last three letters of
the alphabet.
The Policy Dilemma
Algebra in Grade 7/8 or
Algebra When Ready?
Algebra When Ready
Only when students exhibit demonstrable success with prerequisite skills—not at a prescribed grade level—should they focus explicitly and extensively on algebra, whether in a course titled Algebra 1 or within an integrated mathematics curriculum.
Exposing students to such coursework before they are ready often leads to frustration, failure, and negative attitudes toward mathematics and learning.
NCTM Position : Algebra: What, When, and for Whom (September 2011)
Major Themes that Start in PreK and Go all the Way through Grade 12
• Exploring and extending patterns
• Representing mathematical ideas with symbols and objects
• Using mathematical models to represent quantitative relationships
• Analyzing change in various contexts
What We Know About Student Difficulties in Algebra
• Lack of proficiency in proportionality (fractions, decimals, ratios, percent)
• Understanding the equal sign (do something vs. equality, balance)
• Using Variables (placeholder vs. changing value)
• Making the transition from words (verbal or written) to symbols.
• Understanding of function concept (rule or formula)
• Lack of exposure to multiple representations (numbers, graphs, tables, symbols, etc.)
What We Know About Student Difficulties in AlgebraUnderstanding the equal sign
(do something vs. equality, balance)
3 + 5 = or (3x + 5) – (x -3) =versus
4 + 6 = 6 + 4 or y + (-y) = 0
Using Variables (placeholder vs. changing value)
2x + 3 = 17versus
y = 3x2 – 19x - 14
The Far Too Typical Experience!
1. Here is an equation: y = 3x + 4
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.
Opinion: In a student’s first experience, the equation should come last, not first.
SHOW ME!
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 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 Tiles
A 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
Piles of Tiles
Pile 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
Playing with the Four Basic Operations
1 + 1 =1 – 1 =1 × 1 =1 ÷ 1 =_______ Total =
20114
2 + 2 =2 – 2 =2 × 2 =2 ÷ 2 =_______ Total =
40419
Playing with the Four Basic Operations
3 + 3 =3 – 3 =3 × 3 =3 ÷ 3 =_______ Total =
609116
Playing with the Four Basic Operations
4 + 4 =4 – 4 =4 × 4 =4 ÷ 4 =_______ Total =
8 016 125
Playing with the Four Basic Operations
n 1 2 3 4 5 6 7 8 9
Total 4 9 16 25
Willing to Predict?
Playing with the Four Basic Operations
n 1 2 3 4 5 6 7 8 9
Total 4 9 16 25 36 49 64 81 100
Willing to Predict?
Playing with the Four Basic Operations
Playing with the Four Basic Operations
n 1 2 3 4 5 6 7 .. n
Total 4 9 16 25 36 49 64
Willing to Generalize?
n + n =n – n =n × n =n ÷ n =_______ Total =
2n 0 n2
1 _______________n2 + 2n + 1 = (n + 1)2
Equations Arise From Physical Situations
What is the perimeter of shape 6?
Find the Perimeter
Shape 1 2 3 4 5 6 7 8 ..
Perimeter
A table can help communicate the length of the sides that must be added to a shape to find the perimeter of the next shape? (the recursive rule)
Find the Perimeter
A table can help communicate the length of the sides that must be added to a shape to find the perimeter of the next shape? (the recursive rule)
Shape 1 2 3 4 5 6 7 8 ..
Perimeter 4 6 8 10 12 14 16 18
Find the Perimeter
Can we find the perimeter of shape N without using the recursive rule? (the explicit rule)
Shape 1 2 3 4 5 6 .. N
Perimeter 4 6 8 10 12 14 .. 2N + 2
Equations Arise From Physical Situations
What is the perimeter of shape 6?
Find the Perimeter
Shape 1 2 3 4 5 6 7 8 ..
Perimeter
A table can help communicate the length of the sides that must be added to a shape to find the perimeter of the next shape? (the recursive rule)
Shape 1 2 3 4 5 6 7 8 ..
Perimeter 6 11 16 21 26 31 36
A table can help communicate the length of the sides that must be added to a shape to find the perimeter of the next shape? (the recursive rule)
Find the Perimeter
Can we find the perimeter of shape N without using the recursive rule? (the explicit rule)
Shape 1 2 3 4 5 6 .. N
Perimeter 5 11 16 21 26 31 .. 5N + 1
Find the Perimeter
How many beams are needed to build a bridge of length n?
Bridge of length 6
Bridge of length n
B = n + 2n + (n - 1)
B = 3n + (n - 1)
B = 4 + 3(n – 1) + (n – 2)
B = 4n – 1
where n is the length of the bridge and B is the number of beams needed
How many beams are needed to build a bridge of length n?
Follow the Fold(s)
Folds 0 1 2 3 4 5 6 7 8 9 10
Sides
Folds 44 -- 49 -- 67 -- 82 -- 876 --
Sides
Follow the Fold(s)
What’s the Graph?
Folds 0 1 2 3 4 5 6 7 8 9 10
Sides 4 5 6 5 4 5 6 5 4 5 6
7
6
5
4
3
2
1 7
0 3
0 1 2 3 4 5 6 7 8
PENCILS
ERASERS
Total Cost Table: Example 1
7
6
5
4 28
3
2 24
1
0
0 1 2 3 4 5 6 7 8
PENCILS
ERASERS
Total Cost Table Example 2
Multiple Representations
The understanding of mathematics is advanced when
concepts are explored in a variety of forms including symbols,
graphs, tables, physical models, as well as spoken and
written words.