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The Bracket Method: there is a simple way to learn algebra
No More Double Signs
2009-2010
Page 1
2009-2010
The Bracket Method: there is a simple way to learn algebra
No more double signs
A “+”, or “-” sign can mean many things; create a list of words for both signs:
“+”: positive, plus, add, _________________________________________________________________
_____________________________________________________________________________________
“-”: negative, subtract, minus, ____________________________________________________________
_____________________________________________________________________________________
Circle all double signs and replace with one sign, then punch the problem in your calculator exactly as you see it!
TI-83
The colored buttons on your calculator are math operations (add and subtract). Use these
buttons only! Do not touch the negative sign if you want to add or subtract!
1: 51 + -36 – 46 + 71 + -26
2: 15 + 62 - +18 - -12 - 6
Add
Subtract
What happens when the first number is “-“? You must tell the calculator the starting point is negative! The only
time you use the “negative sign” when adding or subtracting is to tell the calculator the first number is negative.
TI-83
The negative (-) button is only used to tell the calculator the starting point of an addition or
subtraction problem is negative. If the first number is negative, use this button.
3: -32 - +65 – 90 - -61 + 17
4: -28 – 18 + 92 + -28 + -65
5: -120 – 85 + -62 - +24 - 82
6: 18 – 9 + -68 + 25 – 98 - -8 - +89
-
-
-
+
7: 87 - 68 + 54 + 8 - 12 - 37
-
-
+
8: -61 + 72 - 6 - 35 - 74
Negative
The Big Ugly (TBU): -125.89 + -95.32 + 58.6 - -71.6 + 36.9 - 17.68 - +85.13 + 98.2 - -84.2
No More Double Signs
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The Bracket Method: there is a simple way to learn algebra
2009-2010
No More Double Signs
Everything in math is either positive or negative; the sign of the number tells you both: (1) if it is positive or
negative; and (2) to add or subtract when combining. No more “5 + -10” or “6 - +7”. A number is either + or –!
Write the following real life scenarios as a mathematical expression with one sign in front of each number
1: balance of your checking account is $134
2: write a check for $103
3: the temperature drops by 16 degrees, then falls another 6 degrees.
4: you sell $350 at a garage sale
5: you lose 5 points on an exam for not writing your name, but you got the extra credit right for 2 extra points
6: 60 students left the concert early. 23 students came in late.
7: the fence has to be 7 feet longer
A little tougher…remember one sign for each number!
9: Starting temperature is 67 degrees; the temperature rises 15 degrees at noon, then goes up 8 more degrees
by dinner time; the temperature then drops 19 degrees at sundown, and decreases another 5 during the night.
b: What is the final temperature?
10: The business account had a starting balance of $1340. You write a check for $245; then you take $160 out
at an ATM; then you make a deposit for $378; finally, you write another check for $29.
b) What is the final balance?
Speak Geek
Expression: numbers and/or variables put together in a mathematical sentence with + or – signs (no equal sign).
No More Double Signs
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2009-2010
The Bracket Method: there is a simple way to learn algebra
No more double signs
Find the Mistakes:
1: 51 + -36 – 46 + 71 + -26
2: 15 + 62 - +18 - -12 – 6
51 - 36 - 46 – 71 – 26
15 + 62 – 18 – 12 – 6
3: 18 - 52+ -53 - -12 – 97 + -31
18 – 52 + 53 + 12 – 97 + 31
Speak Geek
Commutative Property: Do you see the “Co” in the word Commutative? Remember Co -“change order”. It says
you can change the order of an addition (or subtraction) problem, without changing the order.
Prove the Commutative property by going through the following examples.
a & b are the same problem, just a different order. Put both in the calculator; see if you get the same answer.
4a: -32 - 65 – 90 + 61 + 17
5a: -120 + 85 - 62 + 24 – 82
6a: -14 + 75 - 12 - 34 + 45
4b: 61 – 32 + 17 – 65 – 90
5b: 85 – 62 - 82 + 24 – 120
6b: 45 – 12 - 34 + 75 - 14
Now that you know the Commutative Property works: find the mistakes; why don’t these problems work?
7a: -120 – 85 + 62 - 24 + 82
8a: -14 + 75 - 12 - 34 + 45
9a: -15 – 5 + 82 - 24 + 59
7b: -85 + 62 + 82 + 24 -120
8b: 45 – 12 + 34 – 75 + 14
9b: -24 - 15 + 59 + 5 + 82
Fraction Time: Get rid of double signs; Use your calculator to find the answer; put the answer as a fraction.
10:
3
5
+
7
9
3
11: − 5 −
3
TBU: 5 +
−1
4
−
+1
8
−7
9
−
−
4
7
+
−3
4
+1
4
−6
7
+
+
+
4
7
−5
6
−
TI-83
+
−6
7
3
7
A fraction is a division problem in disguise. To enter a fraction,
simply type the top number, “÷”, and the bottom number. To be
safe, put all fractions inside (); do not put the operation sign inside
the ()
example: (3÷5) – (5÷7)
1: [Math]
Don’t worry about simplifying or reducing your answers, the
calculator can do that: Press [Math], [1], [Enter]. You should be
able to change any answer from decimal to fraction with the press
of 3 buttons: “Math, one, enter”.
2: [1]
3: [Enter]
No More Double Signs
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The Bracket Method: there is a simple way to learn algebra
How to Bracket
2009-2010
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The Bracket Method: there is a simple way to learn algebra
2009-2010
How to Bracket
Bracketing Terms: the first step to the bracket method is to be able to “bracket the terms” in an expression
What is the rule for bracketing:_____________________________________________________________
______________________________________________________________________________________
Bracket the following expressions
1: 5x2 – 6(3x) + 5y(8x) - 17
2: -6x(4y) – 7x(2y-7t) + 6z – 8(5) + 19(-t)
3: -2(5x) + 6(2x-7+8y) + 9(3)(-7)
4: 6x2(4x) – 2(5y)(-6)(-9y) – 6t – 7t3 + 2
5: -5 + 6y – (5)(-8x)(6y) - 8t(-6x)(3y4) – 4(-7x) – (-x)
Don’t panic, stick with your rules!
6: 5 6 + 7 − 3𝑥 − 8 7𝑥 − (2!)(6𝑥 − 5𝑦)
7: 5𝑥 7𝑦 − 6𝑥 2 + sin 6𝑥 − cos(3𝑥 − 9)
TBU: -6(5x) + 6x(6y)(-6) – (4x +5-7y) – 8(4)(-6) + 5x – 6(-x)(-y) + 4 – 6(-8) – 2 + 9x(-3x)(-6x)
How well have you trained your eyes? Without bracketing, write the number terms in each expression?
8: 5x(-40) – 7(-3) – (3x-4) – (3-6y)
9: 4(3x)(4y)(-5z) – 2(3) – (53y) – 2(-5)
10: -6(3x)(-6y) – 6x – 6(4x-7) - 6(7)
11: -5x(7y)(-6) – 2x2 + 5(-3x-2y2) – x(5y) + (-t)
Speak Geek
Term: a part of a mathematical expression or equation separated by + or – signs.
How to Bracket
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The Bracket Method: there is a simple way to learn algebra
How to Mash
2009-2010
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2009-2010
The Bracket Method: there is a simple way to learn algebra
Multiplication = Mash potatoes
In algebra, the easiest operation is multiplication. Just put it all the signs, numbers, letters in one pile (like your
uncle’s mash potato plate at thanksgiving: potato’s, butter, peas, carrots, etc.),
Multiply each term
1: 4(3x)(6y)
3: 7x(7y)(5t)(2)
4: 6(2x)(5)(2t)(10)
5: x(y)(t)(z)
6: 2(x)(5)(2y)
Don’t forget about signs? (In multiplication and division, every 2 negative signs is equal to a positive!)
Even # of – signs = _____
Odd # of – signs = _____
Multiply each term: (1) signs; (2) numbers; (3) letters
7: -3(4x)(7y)(-5t)
8: 4t(6y)(-5)(-8z)
9: -6(-4x)(-7y)(5t)(10z)
10: 4(-3x)(12y)(-4)(-9z)
11: -(2)(3x)(-7y)(-10)(-8t)
12: -x(y)(-z)(t)
13): -6.8x(2.3y)(-7.1z)
14: 2.5(-6.8x)(6.25y)(-6.5t)
15: -(3.4x)(8.4y)(-9.2t)(-4)
TBU:
3
5
7𝑥
3
−4
2
8
𝑦
−5𝑡 −𝑡 (−8𝑧)
Math Geeks Only:
16: Is 5(4)(-6)(-10) the same as -6(5)(-10)(4)?
17: What property allows you to change the order and keep the same answer? (ahem…change order)
How to Mash
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2009-2010
The Bracket Method: there is a simple way to learn algebra
Bracket and Mash
Each term (bracket) is a potential multiplication (mash) group. After you bracket, mash each term.
Bracket and then mash each term separately (signs, numbers, letters)
1: 4(3x) - 4(-5x)(2y) - 3(4y)
2: -5x(4y)(2) – 5(-2x) + 7(3t)(-5x)
3: 3(-5) + 3x(-7y) - 3(-x)(y)(-t)
4: -(4)(-5) – 5(-2x)(-7) + 4(-6x)(y)
5: -(-4x) + (-6)(-5t)(7z) – x(-y)(-z)
6: 7(3) – 4(-8x) + 4x(2y) – 6(-8t)
7: x – 4t(6)(3) – 6 + 2x(-6)
8: -6(-4)(-5)(7) – (2x)(3y)(-6y)3
9: -(-1)(-1)(-5) + 3x(-3z) + 5(-y)
10: 3.2(-6.5) + 4x(-5.7)(-2.1) – 3.6z(-4.5t)
11: -12(7x)(-15y) – 5(-19)(6t) – (-12x)(-25)
12: -t – (-x)(-t)(-y) – (v)(-z)(-t) – x(z)
TBU:
3
5
𝑥 7𝑦 − 5
How to Mash
2
3
𝑥
4
− 7 𝑡 + 5.6𝑡 −5𝑥
2
5
𝑦 − −7.2𝑡 + 5.8
7
9
𝑦
6
5
9
𝑡 − 𝑥 −𝑦 −8 + 3
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2009-2010
The Bracket Method: there is a simple way to learn algebra
Mixed Review
Part I: No more double signs – you know what to do!
1: -32 - +65 – 90 - -61 + 17
2: -28 – 18 + 92 + -28 + -65
3: -120 – 85 + -62 - +24 - 82
4: 18 – 9 + -68 + 25 – 98 - -8 - +89
Part II: Bracket the following terms – don’t do the math, just see how many terms there are!
5: 5(2x) – 7(-3) + x(y)(-z)
7:
3
4
4x-7 + 3 49-5x +
6: sin32 – 14(x) + 4cos(3x) – 2(3)(4)
2
5
8: 4 – (7 -2x) – (x)(2y) + 17(4) – 2x3 + 16(-x2)(x)3
x (y)
Part III: Mash the following terms
9: 2(-3x)(-6y)
10: -(-3x)(4y)(-7z)
12: 7(-6y)(5x)(-t)
13:
3
5
5𝑦
2
11: -2(-x)(-7)(-y)
3
𝑡 (− 8 𝑥)
3
14: -(-x)(-y)(-t)(-z)
Part IV: Bracket terms; mash each term.
15: 2(-4) + 4x(-6y) - 3(-x)(5)(-t)
17: -(-4x) + (-3)(-2t)(7z) – x(-2)(-z)
How to Mash
16: -(3)(-5) – 2(-3x)(-5) + 3(-6x)(y)
18: 5(4) – (-8z) + 4x(2y) – 3(-7t)
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The Bracket Method: there is a simple way to learn algebra
How to Distribute (Pizza Delivery)
2009-2010
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The Bracket Method: there is a simple way to learn algebra
How to Distribute (Pizza Delivery)
2009-2010
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The Bracket Method: there is a simple way to learn algebra
2009-2010
Distribute: the second type of multiplication
Write M (mash) or D (distribute) above each term…then do it!
1: 6(3x)(-5)
2: 7(5x + 7)
3: 7(x + 9)
4: -5(6 – x + 7y)
5: -(6x)(+7)
6: -(7 – 5x)
7: 5(4x)(7y)(6z)
8: 3(5x-8y+9)
Distribute each problem (draw those arrows if you are not sure!)
9: 6( 4x + 8)
10: -5(7 + 3t)
11: -7(6 – 8x)
12: 2y( -7 – 4x)
13: - ( 5x – 8)
14: 5(t – x)
15: -5(2x – y + 5)
16: (6 – 5x + 7y)
TBU1: -2xy(6t – 9 + 4z – u)
TBU2:
How to Distribute (Pizza Delivery)
3
1
5
(5𝑥 − 3 𝑦 + 9 𝑧)
4
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The Bracket Method: there is a simple way to learn algebra
2009-2010
Mixed Review
Part I: No more double signs – you know what to do!
1: -22 - +61 – 23 - -16 + 37
2: -18 – 38 + 52 + -8 + -25
3: -12 – 65 + -75 - +44 - 62
4: 8 – 19 + -38 + 27 – 81 - -15 - +64
Part II: Bracket the following terms – don’t do the math, just see how many terms there are!
5: 3(-2x) – 5(-2) + x(3y)(-4)
3
7: -3 49 + 4 4x + 𝑥 2 4 (𝑦)3
6: 32(7!) – (2x) + tan(3x) – 2!(3)(4)
8: – (y -2x) – (-x2)(-18x)3 - (5x)(3y) + 17(4) – 2x3(5x)
Part III: Bracket terms; mash each term
9: 2(5x) - 4(-x)(2y) - 8(5y)
10: – 6(-2x) - 5x(4y)(2) + (4t)(-7x)
11: -(-7) - 2(-x)(y)(-7) + 3x(-7y)
12: -(3)(-2)(7)(v) – 5(-2x)(-7t)(-w) + 4(-6x)(y)+(2)
Part IV:. Mash or Distribute the following terms
13: 6(3x)(-5)
16:
2 1
1
( 𝑥 − 3)
3 2
How to Distribute (Pizza Delivery)
14: 7(5x + 7)
15: -(4x2 – 5x – 8)
17: -(-4x)(-7)(-6y)
18: -5(6 – x + 7y)
Page 14
The Bracket Method: there is a simple way to learn algebra
PEMD: The Bracket Way
2009-2010
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2009-2010
The Bracket Method: there is a simple way to learn algebra
Bracket and Identify
Find Terms (Bracket), then write M (mash) or D (distribute) or S (solo) above the bracket; do not do the math!
1: 4(2x) – 3(2 +6y)
2: -2(5y) + 4(2)(5) – 3(2x)
3: -3(4 -5x) – 2(4y) + 2(3t)(-z)
4: 5(3x-6y) – 4(-5)(-3) – 4(t)
5: -(4x)(-5) – (5y – 7)
6: -3(2x) – 4(5 – 3y) – 4w + 2(3xy – 5t)
Put it all together now: bracket, decide and label each term (M, D or S); do the math one term at a time!
7: 4y + 7(3x – 8)
8: 5x + 6y(3x) – 7y(-3)
9: 7 – 3y(-4)(-7) + 4(5x)
10: 5x + 3(5y – 6t) - 8
11: -(-3x)(-8) – (5y – 10)
12: -3t(2z) – (4x - 8) + (-8y)
13: -(x)(-y)(-t) + cos(90) – 3x(2t)
14:
15: 3 2𝑥 − 3𝑦 − 7𝑥
2
TBU: 3𝑥 5 − 7𝑦 +
1 2
5
𝑦 − (−𝑡)
3 7
PEMD: The Bracket Way
3
4
2
5𝑥 − 3 𝑥
4
7
5
𝑦 + 3𝑥(6 𝑡)
16: − 3𝑥 − 𝑦 − 17𝑧 − −4𝑥
2
5
3
1
4
−6𝑦 + 5 𝑥(−9𝑡)
3
1
4
3
3
𝑦 − 6𝑡 + 3𝑡 −7𝑦 + 3 𝑧 4𝑥 − 8 𝑦 + 8 − 5 𝑡 𝑥 − 2 𝑧 + 7 − 8 (− 4)
Page 16
The Bracket Method: there is a simple way to learn algebra
2009-2010
Algebra Multiplication…how good are you?
Part I: Bracket terms; don’t do the math; just label each term M, D or S.
1: 5x(-5) – (-3)(-5) – 7y(3x – 10)
3: 7 – 4(3x – 12) + 8 – 7y
2: 4!(-7y) - .387(2x) – 4cosx
4: 6 28 + 5𝑥 2 (2𝑦 3 − 8) − 2 (4)sin 𝑥
Part II: Mash or Distribute each term
5: -(3x – 5)
7: 2(4x)(+7)(-8)
6: -(4x)(-5)
8: -2(7 – x)
Part III: Find the mistakes!
9: 5(3x) – 7y – (4t – 8)
15x 7y -4t - 8
11: 5x – 7x(3t) + 4 – (2)(-x)
2x + 3t + 8x
10: 6 + 4(4x – 8y)
10 + 4x – 8y
12: -(3x – 7y) – 3 + (3t – 8z)
-3x -7y -9y +24z
Part IV: Put it all together now: bracket, decide and label each term with (M or D or S), and do the math one
term at a time.
13: 6(3x) + 5(3y – 17) – (-6)(-3v)
14: -(-6)(-5x) – 7 + 3(4y – 8z)
15: -5(2y) – 9x(-7y)(-2) – 8(-3)
16: 6(4 – 7x) – 2y – (7t – 9z) – 3(-2x)(-z)(-t)
PEMD: The Bracket Way
Page 17
The Bracket Method: there is a simple way to learn algebra
Basic Exponents
2009-2010
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2009-2010
The Bracket Method: there is a simple way to learn algebra
What About Exponents?
Speak Geek
3
Exponent: math notation used to show repeat multiplication (mashing) of numbers or variables (ie. xxx=x )
3 2
Expanded Notation: writing terms without the use of exponents (ie. xxxyy instead of x y )
Alphabetical Order: it doesn’t really matter, but mathematicians do prefer you write variables in alphabetical order
Mash and write the answers in expanded notation (the long way)
1: 2x(3x)(-5x)
2: -4x(2y)(-3x)
3: -x(4x)(2t)(-5x)
4: -2x(3t)(4x)(-10t)(-t)
Part b: Now, in the box below, rewrite your answer using exponents (exponential notation)
Distribute and write the answers in expanded notation
5: 3x(2x + 5y)
6: -2y(6x – 5xy)
7: -xy(2x + 4y – 2xy) 8) –x(x – y + 2xy)
Part b: Now below your expanded answer, write the answer using exponents
Put it all together: find terms, mash or distribute; use exponents if necessary
9: 2x(-3x) - (2y)(3y)(-4y) + 3x(2x)(-5x)
10: -3(2x)(-x) + 3(-5y)(-6y) – (-2y)(3x)
11: 3x(2 – 4x) + 3(2y)(-3y) – x(xy)
12: 2xy(3y – 4x) – 5y(y)(-y)
13: -(2x)(-5x)(-y) + 5y(3x)(-2y)
14: 3(4x)(5x) – 5y(2y)(-10y) + 6(2)(-10)
TBU:
2
3
𝑥 4𝑥 −
Basic Exponents
5
4
3𝑥 −
5
6
+ 4𝑥
7
8
𝑥
8
9
2
𝑦 + 2𝑦 − 5 𝑦 + 4𝑥𝑦 −
4
3
(6𝑥𝑦)(8𝑡)
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The Bracket Method: there is a simple way to learn algebra
2009-2010
What About Exponents?
Write the following expanded answers in exponential form
1: 7xx2x3
2: -42y3xy2x5
3: -x4x2txt3y2
4: -2xy3t4xt3t2y
Math Geeks Only: What property allows you to re-arrange 5x3yxy2 into 5xx3yy2? (hint: change order?!)
Mash and write the answers in expanded notation (the long way)
1: 2x(3x)(-5x2)
2: -4x3(2y)(-3x2)
3: -x(4x3)(2t)(-5x)
4: -2x(3t2)(4x2)(-10t)(-t3)
Part b: Now, below your expanded answer, write the answer using exponents
Distribute and write the answers in expanded notation
5: 3x(2x3 + 5y)
6: -3y2(6x – 5xy)
7: -xy2(2x + 4y – 2x3y)
8: –x(x2 – y + 2x3y)
Part b: Now below your expanded answer, write the answer using exponents
Put it all together: bracket, mash or distribute; use exponents if necessary
9: 2x(-3x) - (2y)(3y)(-4y) + 3x(2x)(-5x)
11: 3x(2x – 4y) + 3(2y)(-3y) – x(2xy)
13: -(2x)(-5x)(-y) + 5(3y)(-2y) – 3(2x)(-5)
Basic Exponents
10: -3(2x)(-x) + 3x(-5y)(-6y) – x(-2y)(3x)
12: 2x(3 – 4x) – 5y(2y)(-y)
14: 3(4x)(5x) – 5y(2y)(-10y) + 6(2)(-10)
Page 20
2009-2010
The Bracket Method: there is a simple way to learn algebra
The Challenge
The problems below are considered tough! If you can handle these, you “officially” have algebra skills!
1: 5x(3x) – 4(x – 7) – (-8)(-y)
2: 4 – 5(4x – 8y) – (x)(3x)(-x)
3: 5y(3x – 4y) – 4xy(5x) + 6y(-y)(2y)
4: 5x3(-3x2) – 2x(5x2 – 7y2) + y2(4y3)
5: -x(-x2)(-x3) + x2(y2) – (x2 – y2)
6: -2xy(4x2) + 2xy(3x – 4y) – 4x2(5x3)
7: 2x3(3x) – 4y2(3y – 4x) - 5x3
8: -(-3x2)(x4) – 5(4x – 7y) – 6 + (3x)(-2x2)(3y)
9: 3x2y3(5x2 – 7y3) + x3y(5xy2 – 7xy)
10: 2x(-2y) – 4x2(x2 - 5x) – xy(-x3)(-y2)
2
1
11: 5 𝑥 −15𝑥 2 − 3 𝑦 2 9𝑦 − 21𝑦 3
12:
1
2
3
𝑥 4𝑥
5
TBU: −(−3𝑥) −5𝑥 − 3𝑥 2 −5𝑥 4 + 3 𝑥 6𝑥 2 − 7𝑦 3 + 6 𝑥 2 𝑦 3
Basic Exponents
3
5
1
6
3
𝑥 2 − 3 𝑦 2 (7 − 4 𝑦 3 )
9
10
3
5
𝑥𝑦 2 − 4 𝑥(8 𝑥 3 − 9𝑦)
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The Bracket Method: there is a simple way to learn algebra
Clean-up: Combine Like Terms
2009-2010
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The Bracket Method: there is a simple way to learn algebra
Clean-up: Combine Like Terms
2009-2010
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The Bracket Method: there is a simple way to learn algebra
2009-2010
Combine = “Clean-up”
Learn how to identify like terms: use underlines, double underlines…, and circle constants (regular numbers).
1: 5t – 7y + 4 - 8t – 12t + 8x
2: 3x + 6x2 -7xy - 18x2 + 10x – 17xy
3: 4xy – 7x – xy2 + 8xy – x + 3x2y
4: 3x – 5y – 3x2 – 6x3 + x – 7x4 + 51 - 51x3
5: 3 2 + 6 3 − 2 2 + 2 − 7 3
6: 5t – 7t – t2 + 10t3 – 5 – 6t2 + 9t3 – 6
7: 6x – 3x2 – 10xy +8x2y – 6xy2 + 6x2 – 8x3 + x2 – 7xy + 15x2y -3x2y2 – 15 + 4xy – x +2x3
Clean these up...combine the following expressions
8: 7x – 12x + 8x – 20x + 11x
9: 3y + 11y + 8y + 20y – 40y
10: 3t2 + 9t2 - 4t2 – 21t2 + 6t2
11: -8x5 + 9x5 – 3x5 – x5 +8x5
12: -101x + 56x – 68x -71x -211x + 57x
13: 15y + 9y -12y + y – 5y
Cleaning up (combining) is all about adding or subtracting; the sign tells you what to do.
14: 5x - 6x + 8x + 9y + 18y – 7y
15: -9y + 7x – 13x + 18y + 9x -21y – 5x
16: 8x – 5x2 – 11x2 + 7x - 17x2 + 8x
17: 12x – 15y + 9 + 4y – 17 – 9y + 10x - 5
Speak Geek
Coefficient: a number in front of a variable
Constant: a number by itself (no variables): a regular number (all constants/regular numbers are like terms)
Like Terms: terms where the non-coefficient parts (the “stuff” after the front number) are the same
Clean-up: Combine Like Terms
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The Bracket Method: there is a simple way to learn algebra
2009-2010
Combine = “Clean-up”
Clean-up the mess: Identify a term, find all of them, and combine (add or subtract depending on the sign)
1:
Answer:
2:
Answer:
3:
Answer:
Clean-up: Combine Like Terms
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The Bracket Method: there is a simple way to learn algebra
2009-2010
Combine = “Clean-up”
Clean-up the mess: Identify a term, find all of them, and combine (add or subtract depending on the sign)
1:
Answer:
2:
Answer:
Now that you have seen the worst, try some basic problems: do the same thing!
3: -5x + 17y + 12 + 13xy - 3y – 11x + 8xy - 28
4: 4x2 – 5x3 + 9xy – 10x3 + 13xy2 – 2x2y + 25xy – 14x2y – xy2
5: 2 3 − 8 2 + 6 3 − 2 − 3 3 + 15 2
Clean-up: Combine Like Terms
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The Bracket Method: there is a simple way to learn algebra
Combining Like Terms (Clean-up)
Math Geeks Only: If xx = x2, then why is x+x not x2?
Find the Mistakes
1: 4x – 5x2 + 8 – 6x2 – 11x + 3x2 - 7
2: 6x + 18y - 4 + 17y – y + 8x - 9
-4x - 11x2 + 1
14x + 35y + 13
Identify and combine (clean-up)
3: 3x – 5y +6x – 18 +29y – 84x -17
4: 5x2 – 7x – 8x2 -18x + 34x2 – 17x2
5: 5x + x2 + 17x – 4x2 + 10 – x + 5x2 - 34
6: 16x + 14 xy – 5y – 8xy – 7y2 + 12 – 9y2 - xy
7: −5 2 + 7 3 − 8 3 + 11 2 − 2 3 + 3 2
8: 6x3 – 4x2 + 28x – 15 + 7x3 + 14x2 + 8x - 18
9: x + xy – y + 4xy – 4y + x + xy – 3x – 6y
10: 3x2 – 4xy2 + 7y2 – 7x2y – 9y2 – 2 +17xy2 + 11
TBU1:
3
4
1
2
𝑥 − 2 𝑥 2 − 6 + 7𝑥 − 3 𝑥 2 + 15
Clean-up: Combine Like Terms
2
1
1
1
5
TBU2: −11 − 5 𝑦 − 4 𝑦 2 − 6 𝑦 + 𝑦 − 3 𝑦 2 + 7 + 8 𝑦 2
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The Bracket Method: there is a simple way to learn algebra
Simplify: The Bracket Way
2009-2010
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The Bracket Method: there is a simple way to learn algebra
2009-2010
Simplify = Bracket
Find the Mistakes
1: 5x – 3(5 – 8x) + 3(-4)
5x - 15 + 24x - 12
29x + 27
2: 7 + 3(6x – 2) - 5x
10(6x – 2) - 5
60x – 20 - 5
60x - 25
Simplify: (1) Bracket terms, (2) Mash, Distribute (or Solo) each term, (3) Clean-up
3: 17 – 5(3x – 8)
4: 5x + 2(-4) – 7x(-5) - 20
5: 20(-2x) – 7(-8) + 2x – 3(-7)
6: 5(2x - 8) – 4(9x - 5)
7: -3(4) – (3x – 6) + 5 + 7(-4x)
8: 5x – 7(-2x)(-5) – 2(x – 4)
9: 3(4x2 – 5x -8) – 2x2 – 4(3x) - (-2)
10: -(-4)(-3y) + 3x -2(-4x -7y) + 5(-4y)
TBU: 7(3x)(-7) – 4x – 6(-3)(-1) - (4x – 8y + 7) + 14y – x(-5) – 8(-3y + 8x) + 5 + 2(- 7)(-6) – x – y
Simplify: The Bracket Way
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The Bracket Method: there is a simple way to learn algebra
The Challenge (Part II)
If you can simplify these…you can simplify anything!
Remember: (1) find terms, (2) Mash or Distribute each term, (3) Clean-up (combine)
1: 5x(2x) – 5(3x2 – 7x - 8) + (-7)
2: -2(-3x3) – 2x(-5x) – 6x2 + 5x(-2x)(-7x)
3: 2(x3 – 18x2 + 9x – 34) – (x3 – 8x2 – 14x - 21)
4: 5(2x – 6y) – (4x)(-3y) – 7(-2x) + x(3y) – (-x)
5: x(x) – 2(3x2 – 7x – 9) – 3(2x) - 18
6: -(-6x3) + 18x2 – 2x2(-5x) – x(x – 5)
7: 5(-2y) – 7y(4y – 8) – 9 +2(7y) – (-4y2)
8: 2xy(3x – 8y) – 3x(-4y2) – x(9x)(-3y) + 7y(-2xy)
9: x(x3 – 9x2 + 4x – 21) – 2x(x2 – 4x) – 2(-11)
10:
3
8
2
3
𝑥 4𝑥 − 5𝑥 2 + 3 𝑥 2 − 3𝑥 − 8 − 4 (−7)
TBU: -4x2(3x2) – 7x3(-5) – x(x3 – 6x2 – 10) – x(-4x)(-7x) + 18x2(-x) + 5x3(-7x) – 6x(18 – 7x – x2 + 5x3)
Simplify: The Bracket Way
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The Bracket Method: there is a simple way to learn algebra
Solving: Junk & Divide
2009-2010
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The Bracket Method: there is a simple way to learn algebra
Solving: Junk & Divide
2009-2010
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The Bracket Method: there is a simple way to learn algebra
Solving: Junk & Divide
2009-2010
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The Bracket Method: there is a simple way to learn algebra
Solving Equations: Junk & Divide
Speak Geek
Equation: when one mathematical expression is set equal to another.
Solving Basic Rules:
1. Draw the ___________
2. Work on the _____________________________________
3. Whatever you do to one side of the wall, _______________________________________________
Draw the wall; put an * above the junk (do not solve!)
1: 7x – 27 = 76
2: 35 = -5 + 7x
3: 16 – 8x = 123
5: 4x - 7π = 237
6: 71 = 2 3 − 7𝑥
7: 9𝑥 −
4
7
= 78
4: -352 = 21x + 7
8:
6
5
+ 7𝑥 = 89
Now go back (1-8) and show how you would “get rid of the junk” (do not dive yet, that is next)
Solve: Put it all together now…Junk & Divide
9: 17 – 9x = 71
10: -5x – 35 = -72
11: 89 = 4x - 12
12: -57 = -9 + 17x
13: 12 – 7x = -81
14: 13x + 78 = -114
15: 171 = 5 – 5.2x
16: 17 – x = 58
Solving: Junk & Divide
7
17: 5𝑥 + = 16
3
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The Bracket Method: there is a simple way to learn algebra
Solve: Junk & Divide
1: -17x + 167.4 = -431.5
2: -89 + 63.2x = 178.9
3: 54.5 = -76.2 – 18.4x
4: 1235 = 123x – 86
5: 1 – x = 0
6: 67 – 12x = -782
These are so easy, they are tough (sometimes you only do junk, and sometimes you skip right to divide)!?
Speak Geek
One Step Equations: equations which can be solved in one mathematical step.
7: 5x = 876
8: -15 + x = 91
9: -17 = -4x
10: -346 = 89 + x
11: - x = -86
12: x – 13 = -21
Don’t panic, they’re only fractions (remember to use parenthesis)…junk and divide!
13: 5𝑥 −
16: 6 =
3
8
2
3
= 28
𝑥−
15
3
Solving: Junk & Divide
14: 17 +
17:
5
9
𝑥−
5
6
𝑥 = − 34
27
3
= 34
15: 174 = 115 −
18:
46
7
=
1
4
𝑥+
2
5
𝑥
21
5
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The Bracket Method: there is a simple way to learn algebra
2009-2010
Simplify and Solve
Bracket, Junk & Divide
1: 7 + 2(3x – 8) = 123
2: 5x – 2(3x) – 18(-4) = 231
3: 4(-2) – 6(3x – 8) + 8(-3x) = -500
3: -5x – 4(x – 8) = -324
5: 1056 = 4(-2x) – 18(3) – 7
6: -78 = 7(2) – 3(4x – 8) + 2(-5x)
7: -(3x – 8) – x + 17 – 3(-7x) = -378
8: 78 = 17 – 2(x - 8) + 5(-7x) – (-5)(-8)
TBU: (5x)(-17)(-4) – (12x2 – 6) – 5x – 4(5 – x) – (-2)(-8)(-3x) + 2(3x)(-7) + 4(3x2 – 7x +9) = -1578
Solving: Junk & Divide
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The Bracket Method: there is a simple way to learn algebra
X’s on Both Sides
“make one disappear”, junk & divide
1: 5x – 19 = 54 + 8x
2: -15 + 6x = 25 + 8x
3: 24 – 8x = -57 - 11x
4: 34 – 18x = -71 + 9x
5: 8x + 19 = -6x – 23
6: -16x + 31 = -7x + 23
7: 34.5 - 83.1x = 25.4x + 46.1
8: 5.26x + 8.12 = -8.95 + 9.74
9: -23 + 8x = 40 – 8x
These are strange…think about which x you want to disappear!?
10: 13 + 9x = 6x
11: -78 – 7x = -12x
12: 15x = 74 + 6x
13: -19x = -7x + 23
14: – x = 75 + 8x
15: 40 + 9x = 19x
Hmmm?16:
6x – 8 = 15 + 6x
Solving: Junk & Divide
Hmmm?17:
21 – 7x = -7x + 21
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The Bracket Method: there is a simple way to learn algebra
The Challenge (Part III)
If you can solve these…you can solve anything!
1: 6 + 2(3x – 8) = 7x -9
2: -3(x – 8) = -4(5x – 7)
3: -4(-3) – (6x – 8) + 7 = 5x – 18
4: 7x – 8(-5x) + 7 = 17 – 5x
5: 5(2x) – 17(-2) = 2(7x – 8)
6: -(-5) – 4(2x – 8) = 3x – 7(-2) + 8x
7: 4x – 2(-3x) + 2(5) = 7 – 3(2x – 8) – 9x
9: 5(-2)(3) – (4x – 9) – 4(-7) = 3(-2) -3(4)(-5x) – 9
11:
3
4
2𝑥 − 8 =
2
3
6𝑥 − 14(−3)
Solving: Junk & Divide
8: 4(3x + 21) = 5x – 7x + 15x
10: 2(-4) – x – 2(3) = 8 + 8x – 19 + 4x
12:
2
3
4 −
4
5
3
3𝑥 − 8 =
1
2
7
2
4𝑥 − 8 −2 (− 21 )
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2009-2010
The Bracket Method: there is a simple way to learn algebra
Solving for Letters
Speak Geek
Implicit Formulas: solving a formula for another variable
When solving for formulas, nothing changes (except the answers are ugly looking). Identify what you are
trying to solve for; get rid of the junk; divide.
1: solve for x: 5x – t = 4g
2: solve for t: 4g – 7t = cy
3: solve for v: 3tv = 35g
4: solve for g: 16t2 – 5g = F
5: Solve for r: 6πrh = 156
6: solve for h: fm – gh = 38t
7: solve for v: 6t2 – 7t – 5vh = 324
8: solve for g: 7y – 6t – 8g = 56
9: solve for r: 7d – 3πrh = -72g
One of the most common things you will do in algebra is to solve for y (get y alone!).
10: y + 4x = 7
11: 5x - y = 18
12: 5x + 8y = 21
13: 2x – 7y = -21
14: 7y – 4x – 8 = 18
15: 15 = 5x – 3y
Solving: Junk & Divide
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The Bracket Method: there is a simple way to learn algebra
2009-2010
Adding to your repertoire
Junk & Divide can be used to solve most any type of linear algebraic equation. However, once you get good at
this method, there are certain methods you should learn to make solving faster or easier!
1) Cross Multiplication: used anytime you have fractions set equal to each other.
2) “Dot” Method, or IHF (I hate fractions), or GCF (this is the official “geek” name: it stands for greatest
common factor): used when you have a lot of fractions in an equation, and you just want to get rid of
them!
3) Graph the equation and find the intersect: every equation can be solved this way.
4) Two Sticks, Two Equations: used to solve absolute value problems.
5) The Inverse of Square is Square Root: Used to solve x2 or 𝑥 problems
Solving: Junk & Divide
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