Anthony Chan September, 2018 1 2018 Georgia Adult Education Conference
Attendees will be able to: Make difficult math concepts simple and help their students discover math principles on their own. This will help dispel their fear of mathematics. 2
I hear and I forget. I see and I remember. I do and I understand. --Chinese proverb If you tell me, I may forget. If I discover, I will remember--for it is mine. 3 --Anthony Chan
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Information overload = Limited learning (Copyrighted) Alpha Educational Products and Services (2009)/ www.alphaeducational.com
Mathematics was discovered and not invented. When someone discovers something, he owns it. 7
8 The key to every door in the math world.
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What s so scary about math? Symbols Terminology Wording of the problems The formulas The rules The procedure The reasoning process 10
A major difference between arithmetic and algebra In arithmetic we say: In algebra we say: 11 There are 10 pigs. There are 10 animals. There are 8 little pigs and 2 big pigs. There are 8 brown pigs and 2 black pigs. (Copyrighted) Alpha Educational Products and Services)/ www.alphaeducational.com
Arithmetic which is based entirely on known number values. Algebra is based on the concept of unknown values called variables. Variables are the basis for algebra. 12
What do notice? 2 x 3= 6 2(3)= 6 2 3= 6 y = 3 x = 2 y = 3 2y= 6 xy= 6
3 f ( x) dx + 2 f ( x) dx = 3 + 2 = 5 = 3p + 2p = 5p 3 f ( x) dx + 2 f ( x) dx = 5 f ( x) dx (Copyrighted) Alpha Educational Products and Services)/ www.alphaeducational.com
ones tens Can we add these numbers in any order we wish? 57 + 12 15
(Copyrighted) Alpha Educational Products and Services / www.alphaeducational.com a 3 + a = 5 3 2 = 3 2 = 5a x 3 2 + x 2 = 5 3 2 + 2 2 = 5x 2 + = 5 5 2
9 + 5 = 9 + 5 9d 9x 5p + = 9 d + 5p 5y + = 9 x + 5y 9x 4 + 5 y 2 = 9 x 4 + 5y 2 5 6 + = 5 + 6 5 x 4 + 6 x 2 = 5 x 4 + 6x 2 17 (Copyrighted) Alpha Educational Products and Services)/ www.alphaeducational.com
Clarifying like terms 4 dogs + 3 dogs = 4 dogs + 3 cats = From 4 dogs subtract 2 cats. 18
What can we teach with these? (Copyrighted) Alpha Educational Products and Services/ www.alphaeducational.com
1 8 4 5 + = 8 8 Use errors to benefit 2 5 2 4 + = 5 5 your students Have THEM correct you. 1 4 2 3 + = 4 4 Remember, whatever your students discover, they own. 1 x 3 7 4 5 + = x x 2 4 + = 7 8 5 7 Also, let your students observe some things on their own so they can discover principles for themselves. 20 (Copyrighted) Alpha Educational Products and Services / www.alphaeducational.com
Rectangles 1 2 3 3 1 2 3 4 5 5 (Copyrighted) Alpha Educational Products and Services (2009)/ www.alphaeducational.com
How much of the chocolate was separated? 1 or 4 5 20 (Copyrighted) Alpha Educational Products and Services (2009)/ www.alphaeducational.com
You can have a half of a quarter of this pie or a quarter of a half of the pie. Prove with a diagram. 23
Expressions and Equations X+1 3y 1 = 4xy 3 1 = What s the difference? X+1 = 5 3y 1 = 8 4xy = -8 3 1 = 2 24
Simplify this equation 2x + 4 = 6 2 2 2 x + 2 = 3 25 (Copyrighted) Alpha Educational Products and Services)/ www.alphaeducational.com
Simplify this equation 1 + 1 4 2 (4) (4) (4) = 4 3 1 + 2 = 3 26 (Copyrighted) Alpha Educational Products and Services)/ www.alphaeducational.com
How do we solve these? 1 + x 4 2 (12) (12) (12) (12) = 3 2 + 2 1 3 + 6x = 4 + 6 27 (Copyrighted) Alpha Educational Products and Services)/ www.alphaeducational.com
How do we add these? 2 + 3 4 8 x 2 + y 3 (Copyrighted) Alpha Educational Products and Services)/ www.alphaeducational.com
Expo what? 29 (Copyrighted) Alpha Educational Products and Services)/ www.alphaeducational.com
Be a math detective. Always ask the right questions. a) 3 x 3 = 9 3 x 3 = 3 2 = 9 b) 4 x 4 = 4 2 = 16 c) 7 x 7 x 7 = 7 3 = 343 2 4 d) 2 x 2 x 2 x 2 = = 16 e) 2 x 2 x 2 x 2 x 2 x2 = 2 6 = 64 What do you notice? Why? What could have been done differently in exercise a? What does the 4 in d represent? Can you come up with a general rule? f) a x a x a x a x a x a = a 6 Does your rule apply to letters?
How do you feel about the following? a m a n = a m+n (a m ) n = a mn
How do our students feel about the following? Simplify 7 2 7 3 = a 2 a 2 = x 4 x 2 = y 3 y 5 = m 2 m 4 = Simplify (7 3 ) 2 = (5 2 ) 4 = (a 3 ) 2 = (x 5 ) 2 = (y 4 ) 3 =
Try to figure out the pattern 7 2 7 3 = 7 5 a 2 a 2 = a 4 x 4 x 2 = x 6 y 3 y 5 = y 8 a m a n = a m+n What math skills are needed to teach this algebraic concept? m 5 m 2 = m 7 33
Try to figure out the pattern (7 3 ) 2 = 7 6 (5 2 ) 4 = 5 8 (a 3 ) 2 = a 6 (x 5 ) 2 = x 10 (a m ) n = a mn What math skills are needed to teach this algebraic concept? (y 4 ) 3 = y 12 34
Try to figure out the pattern 7 5 7 2 = 7 3 5 6 5 4 = 5 2 x 7 x 4 = x 3 What math skills are needed to teach this algebraic concept? y 5 y 3 = y 2 m 7 m 3 = mn 34 (Copyrighted) Alpha Educational Products and Services (www.alphaeducational.com)
What s the difference? (+2) 2-2 2 (-2)2 (+2) 2 =(+2) (+2) = +4-2 2 = - (2) (2) = -4 (-2) 2 =(-2) (-2) = +4 36
Do you want your students to eat the whole hamburger in one bite? Give them a bite at a time only enough that they can handle. a b c PYTHAGOREAN THEOREM Help list the concepts that students should know prior to using the Pythagorean theorem: Recognizing right triangles (includes characteristic of a right triangle) Recognizing the legs and hypotenuse of a right triangle Squaring a number Solving square roots Solving simple quadratic equations
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m t 2 1 3 4 n 6 8 5 7 40
Ratio St. Equation St. Geometry St. Chan St. Second St. Graph St. Y St. Please find my car. Tell me first the street and then the road. Young Rd. (X Y) Fourth Rd. Algebra Rd. (2 4 ) (Second St., Fourth Rd. ) X Rd. Function Rd. 41 (Copyrighted) Alpha Educational Products and Services / www.alphaeducational.com
Please find my car. Tell me first the street and then the road. (X Y) (2 ) 4 42 (Copyrighted) Alpha Educational Products and Services / www.alphaeducational.com
Positive slope Negative slope (Copyrighted) Alpha Educational Products and Services / www.alphaeducational.com
Types of Slope Zero Positive Undefined 44 Negative
a f Compare these two angles. With permission from Alpha Educational Products and Services
a f b c n a a p b m b w m n h (Copyrighted) Alpha Educational Products and Services (2009)/ www.alphaeducational.com
t t y 90 p 90 All right angles measure 90. g p 1 90 y a 45 g r 90 α m With permission from Alpha Educational Products and Services
f y p r p 3 f y right angles Compare these two sets of angles? t a y t p y acute angles g α r m With permission from Alpha Educational Products and Services
f y p 3 r f y right angles Compare these two sets of angles? t y t p p y g a obtuse angles α r m With permission from Alpha Educational Products and Services
y f r t y t p 1 2 3 p a r f straight angles Compare these two sets of angles? t y t p p y g a obtuse angles y α r m With permission from Alpha Educational Products and Services p
Find the right angles y r d m p c
Which two houses are next door to each other? p y g r 52 (Copyrighted) Alpha Educational Products and Services / www.alphaeducational.com
Which two houses are opposite each other? y p g r 53 (Copyrighted) Alpha Educational Products and Services / www.alphaeducational.com
54 Geometry symbols Symbol Symbol Name Meaning / definition parallel parallel lines congruent to equivalence of geometric shapes and size ~ similarity same shapes, not same size Δ triangle triangle shape right angle = 90 AB line infinite line AB ray a line that start from point A to infinity AB line segment a piece of line from point A to point B
55 Math Detective
How do our students feel about the following? Simplify the following polynomials: 2)(7x 4 + 2x 3 +3x) + (2x 4 + 4x 3 +2x) = 3)(3x 3 + 3x 2 4x + 5) + (x 3 2x 2 + x 4) = 4) (x 3 + 5x 2 2x) + (x 3 + 3x 6) + ( 2x 2 + x 2) = 5) (6x 4 + x 3 2x) + (2x 2 + x 2) = (Copyrighted) Alpha Educational Products and Services (2009)/ www.alphaeducational.com
( 7 + 5 + 4 ) (Copyrighted) Algebra I Can do!(2009) Anthony R. Chan and Paul Candelaria (Copyrighted) Alpha Educational Products and Services (2009)/ www.alphaeducational.com
+ (8 + 2 + 5 ) + (6 + 6 + 4 ) = ( 14 + 8 + 9 ) (Copyrighted) Alpha Educational Products and Services (2009)/ www.alphaeducational.com
(8 x 4 + 2 x 3 + 5 x ) + (4 + 6 + 4 x ) = x 4 x 3 (12 x 4 + 8 x 3 + 9 x ) Do you know the name of the algebraic concept that was just taught? The addition of Polynomials We have the tendency to scare our students when we tell them the name of the lesson at the beginning of the lesson. (Copyrighted) Alpha Educational Products and Services (2009)/ www.alphaeducational.com
Use parentheses to explain what was done in this basket. ( 4 x 3 + 3 x 2 + 4 x + 7) (Copyrighted) Alpha Educational Products and Services / www.alphaeducational.com
+ Let s combine these two baskets. ( 4 x 3 + 3 x 2 + 4 x + 7) + ( 2 x 3 + 2 x 2 + 3 x + 2) ( 6 x 3 + 5 x 2 + 7 x + 9) (Copyrighted) Alpha Educational Products and Services / www.alphaeducational.com
Can you add these?: 1) (7 + 2 + 3 ) + (2 + 4 + 2 ) = Simplify the following polynomials: 2)(7x 4 + 2x 3 +3x) + (2x 4 + 4x 3 +2x) = 3)(3x 3 + 3x 2 4x + 5) + (x 3 2x 2 + x 4) = 4) (x 3 + 5x 2 2x) + (x 3 + 3x 6) + ( 2x 2 + x 2) = 5) (6x 4 + x 3 2x) + (2x 2 + x 2) = (Copyrighted) Alpha Educational Products and Services (2009)/ www.alphaeducational.com
2 2 ( 7 + 5 + 4 ) ( 14 + 10 + 8 ) (Copyrighted) Alpha Educational Products and Services (2009)/ www.alphaeducational.com
x 4 x 3 x 2 (5 +3 + 2 ) x 4 x 3 x (10 + 6 + 4 ) Do you know the name of the algebraic concept that was just taught? The Distributive Property What are some other examples of the daily use of the distributive property? (Copyrighted) Alpha Educational Products and Services (2009)/ www.alphaeducational.com
Math wise, what can you teach with a piece of paper and a pair of scissors? 65 GEDtestingservice.com GED.com
Math wise, what can you teach with a paper airplane? 66 GEDtestingservice.com GED.com
Math wise, what can you teach with beans? 67 GEDtestingservice.com GED.com
5 x = 0 x 5 = 0 (2+3) x = 0 68 (m+3) x = 0
Possible ways to get zero when multiplying two numbers ( # 0 ) x ( 0# 0 ) = 0 69
Factoring Trinomial of the form x 2 + bx + c 70
Which Two Numbers Game Always respond with the bigger number first Which two numbers when multiplied will give 6 and when added will give 7? 6 and 1 Which two number, when multiplied will give 6 and when subtracted will give 5? 6 and 1 Which two numbers, when multiplied will give 10 and when subtracted will give 3? 5 and 2 Which two numbers when multiplied will give 24 and when added will give 14? 12 and 2 Which two numbers when multiplied will give 20 and when added will give 9? 5 and 4 How can this game be used? (Copyrighted) Alpha Educational Products and Services / www.alphaeducational.com
Factoring Quadratic Trinomial Expressions x 2 + bx + c x 2 + 7x + 10 ( ) ( ) x 2-5x --14 ( ) ( ) Multiply the signs + Multiply the signs + Two numbers when multiplied will give and when (added) will give. 5 and 2 Two numbers when multiplied will give and when (subtracted) will give. 7 and 2 (Copyrighted) Alpha Educational Products and Services / www.alphaeducational.com
Factoring Quadratic Trinomial Expressions x 2 + bx + c x 2-5x 5 + 6 ( ) ( ) x 2 + 5x - 6 ( ) ( ) Multiply the signs - Multiply the signs - Two numbers when multiplied will give and when (added) will give. 3 and 2 Two numbers when multiplied will give and when (subtracted) will give. 6 and 1 (Copyrighted) Alpha Educational Products and Services / www.alphaeducational.com
Factoring Trinomials of the form 1) p 2) k 3) x 4) p 5) x 2 2 2 2 2 + 7 p 11k + 30 18x y + 12 (p + 4) (p + 3) (k - 6) (k -5) + 80 (x - 10) (x - 8) + 15p + 56 (p + 8) (p + 7) 2 + 11xy + 18 (xy + 9) (xy + 2) 2 x + bx + c 6) x 7) y 8) x 9) z 2 2 2 2 10) x + + 3y 10 14x + 9z 2 x (x + 3) (x - 2) (y + 5) (y - 2) 5x 6 (x 15) (x+1) 15 10 (z + 10) (z - 1) (x - 9) (x - 4) 36 74
x 2 + bx + c Make up a 3-term quadratic equation with coefficient 1 in the x 2 term. Check if it is factorable. 75
Quadratic formula ax 2 + bx + c = 0 x = -b + 2a b2 4ac (Copyrighted) Alpha Educational Products and Services / www.alphaeducational.com
Quadratic formula ax 2 + bx + c = 0 b x = -( ) + ( b )2 4( a)( c) 2( a) (Copyrighted) Alpha Educational Products and Services / www.alphaeducational.com
Quadratic formula ax 2 + bx + c = 0 a 2x 2 + 5x + 3 = 0 b x = -( ) + ( b )2 4( a)( c) 2( a) (Copyrighted) Alpha Educational Products and Services / www.alphaeducational.com
Quadratic formula ax 2 + bx + c = 0 a -2x 2 - -5x + 3 = 0 b ) x = -( ) + ( b )2 4( a ( c ) 2( a ( (Copyrighted) Alpha Educational Products and Services / www.alphaeducational.com
Thank you!!!!! Anthony Chan achan@alphaeducational.com www.alphaeducational.com