Algebraic Thinking and Algebra

Algebraic Thinking involves:•Recognizing patterns•Modeling situations with objects, pictures or symbols•Analyzing the effects of change

Understanding variables (letters that stand for an unknown) is key to algebraic thinking.

The evolution of algebraic thinking (an example)

3

3+ =

1 + 2 = 3

+ 2 = 3

x + 2 = 3

MAP released item, 2006, grade 3

DESE released item 2004 MAP test, grade 4

MAP released item, 2006, 5th grade

MAP released item, 2006, grade 7

What does an equal sign mean?

1 + 2 = 3

1 + 2 =

ETA Cuisenaire

Model the equation using a balance scale.

+ 2 = 3

Model the equation using a balance scale.

+ 2 = 3

2 1

+ 2 = 3

-2 -2

= 1

x

x

http://illuminations.nctm.org/ActivityDetail.aspx?ID=33

http://illuminations.nctm.org/ActivityDetail.aspx?ID=26

Properties of Equations

Addition Property of Equality

If a = b, then a + c = b + c

4 = 44 + 2 = 4 + 2

Multiplication Property of Equality

If a = b, then a · c = b · c

4 = 44 · 2 = 4 · 2

Distributive Property of Multiplication over Addition

a(b + c) = a · b + a · c

2(3 + 4) = 2 · 3 + 2 · 4

5(14) = 5(10 + 4)= 5 · 10 + 5 · 4= 50 + 20= 70

Model the equation 2x + 1 = 7 using a balance scale.

+ 1 = 7

1 6

2xx + x

33

If you triple a number and add 3, the result is 36. Find the number.

Algebra Algebraic methods

3x + 3 = 36- 3 -3

3x = 333 3

x = 11

3 3633 311 11 11

Kate bought a TV on sale for $160. If the sale was a “1/3 off” sale, what was the original price of the TV?

AlgebraLet x = the original price

original price - 1/3 of the original price = sale price

160x3

1-x

160x3

2

2

3

2

3

x = 240

Kate bought a TV on sale for $160. If the sale was a “1/3 off” sale, what was the original price of the TV?

Algebraic methods

off3

1$160 Sale price

$80 $80 $80

3(80) = $240

Write down any three consecutive numbers.Multiply the first and third numbers.Square the middle number.What do you notice?

5 6 73635

10 11 12121120

First times last is one less than square of middle

n – 1 n n + 1n2

(n – 1)(n + 1)

(n – 1)(n + 1) = n2 – 1 difference of squares

Functions

A function from set A to set B is a correspondence from set A to set B in which every element of set A is paired with exactly one element of set B.

Input(domain)

Output(range)

DESE sample MAP item, 3rd grade

Representations of FunctionsTable

Input1234

Output48

1216

Ordered pairs{(1, 4), (2, 8), (3, 12), (4, 16)}

Domain: {1, 2, 3, 4}Range: {4, 8, 12, 16}

Arrow Diagram1234

48

1216

Function Notation

f(x) = 4x

Representations of FunctionsGraph

Input1234

Output48

1216

input

output

1 2 3 4

16

12

8

4

Use function notation to write a rule for the function{(1, 2), (2, 5), (3, 8), (4, 11)}

Input1234

Output258

11

a1 =a2 =a3 =a4 =

Arithmetic sequencean = a1 + (n – 1)d

a1 = 2d = 3

an = 2 + (n – 1)3an = 2 + 3n – 3an = 3n – 1

f(x) = 3x - 1

Composite Functions

x

f gf(x) g(f(x))

Notation:

(g o f)(x) means g(f(x))

If f(x) = 2x – 1 and g(x) = 3x + 1, what is (f o g)(3) ?

(f o g)(3) = f(g(3))= f(10)

= 19

3

g f10 19

g(3) = 3(3) + 1= 9 + 1= 10

f(10) = 2(10) – 1 = 20 – 1 = 19