Where Does that Algebraic Equation Come From?Moving From Concrete Experience to Symbolic Form"

Jim RubilloJim RubilloJRubillo@verizon.netJRubillo@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.