Algebraic thinking involves recognizing patterns, modeling situations with symbols, and analyzing change. It relies on understanding variables to represent unknown quantities. The document traces the evolution of algebraic thinking from simple equations to more complex concepts like functions, composite functions, and properties of equations. It provides examples of how algebraic reasoning and symbols can be used to represent and solve real-world problems.
2. 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.
3.
4.
5. The evolution of algebraic thinking (an example)
3
+ =3
1 + 2 = 3
+ 2 = 3
x + 2 = 3
17. 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
18. Model the equation 2x + 1 = 7 using a balance scale.
x +2x + 1
x = 7
1 6
3 3
19. If you triple a number and add 3, the result is 36. Find
the number.
Algebra Algebraic methods
3x + 3 = 36
- 3 -3
3x = 33 3 11 36 11
33 11 3
3 3
x = 11
20. 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?
Algebra
Let x = the original price
original price - 1/3 of the original price = sale price
1
x - x 160
3
3 2 3
x 160
2 3 2
x = 240
21. 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
$80 $80 $80
1
$160 Sale price off
3
3(80) = $240
22. Write down any three consecutive numbers.
Multiply the first and third numbers.
Square the middle number.
What do you notice?
First times last is one less than square of middle
5 6 7
36
n–1 n n+1
35
n2
(n – 1)(n + 1)
10 11 12
121
(n – 1)(n + 1) = n2 – 1
120
difference of squares
23. 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)