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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.
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
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
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)
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
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