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UNIT 1– Math 621
Simplifying Expressions
Description:
This unit focuses on using quantities to model and analyze situations, interpret expressions, and create expressions to describe situations. Students will work in one-variable and multivariable settings, attain fluency with such algebraic manipulations as use of the distributive property, simple factoring, and connecting the structure of expressions to the contextual meanings of those expressions. Broadly, the unit will help students with the transition into using the tools of algebra to model and explore scenarios.
***Date Assigned/ Due- Next Class
RESOURCES
1.1 PEMDAS
Khan Academy PEMDAS:
https://www.khanacademy.org/math/pre-algebra/order-of-operations/order_of_operations/v/introduction-to-order-of-operations
1.2: Linear Combination and Writing Math Expressions
How to Write Expressions from Variables: https://www.khanacademy.org/math/algebra/introduction-to-algebra/writing-expressions-tutorial/v/writing-expressions-1
1.3: Properties (associative, commutative, distributive, additive/mult identities, additive/mult inverses)
Arithmetic Properties: https://www.khanacademy.org/math/pre-algebra/order-of-operations/arithmetic_properties/v/commutative-law-of-addition
1.4: GCF and simple factoring
GCF: https://www.khanacademy.org/math/pre-algebra/factors-multiples/greatest_common_divisor/v/greatest-common-divisor-factor-exercise
1.5: Evaluating expressions (substitution) - (more practice on PEMDAS in the context of substitution)
Evaluating Expressions with Multiple Variables: https://www.khanacademy.org/math/cc-sixth-grade-math/cc-6th-expressions-and-variables/cc-6th-substitution/v/evaluating-expressions-in-two-variables
1.6: Simplifying expressions & combining like terms
Simplifying using Distributive Property and Combining Like Terms: https://www.khanacademy.org/math/cc-seventh-grade-math/cc-7th-variables-expressions/cc-7th-manipulating-expressions/v/combining-like-terms-and-the-distributive-property
page 2
Section 1.1 Notes
The Order of Operations
A set of guidelines used to simplify mathematical expressions. When simplifying an expression, the order is perform all operations inside any parenthesis first, followed by evaluating all exponents. Third, do all multiplication and division at the same time, from left to right. Lastly, do all addition and subtraction at the same time, from left to right.
Some people refer to the order of operations as PEMDAS. The order is as follows:
Parenthesis Do anything in parenthesis first.
Exponents Next, all powers (exponents) need to be evaluated.
Multiplication/Division Multiplication and division must be done at the same time, from left to right, because they are inverses of each other.
Addition/Subtraction Addition and subtraction are also done together, from left to right.
Example A
Simplify
Solution: Parenthesis:
Exponents:
Multiplication:
Add/Subtract:
Example B
Simplify
Solution: Think of everything in the numerator as if it were in its own set of parenthesis as well as everything in the denominator. The problem can be rewritten as
When there are multiple operations in a set of parenthesis, use the Order of Operations within each set.
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Class Notes:Step 1: Do as much as you can to simplify everything inside the parenthesis first
Step 2: Simplify every exponential number in the numerical expression
Step 3: Multiply and divide whichever comes first, from left to right
Step 4: Add and subtract whichever comes first, from left to right
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Section 1.1PEMDAS classwork Name _________________________________
Evaluate these expressions using correct order of operations. Show your steps. No calculators.
1. 4 + 32 2. 5 + (2 + 3) 2
3. 30 – 3(8 – 3) 2 ÷ 5 4. 18 – 6 * 2
5. 2 * 9 – 3(6 – 1) + 1 6. 36 ÷ 4(5 – 2) + 6
7. 2 + 3[5 + (4 – 1) 2] 8. 2 + [-1(-2 – 1)] 2
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9. 16 ÷ 2[8 – 3(4 – 2)] + 1
10. 14x + 5[6 – (2x + 3)]
Make the following expressions equal to 35 by placing parenthesis.
11.
8 – 3 • 9 – 2 = 3512.
15 + 10 • 8 4 = 35
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Section 1.1Homework Name _______________________________________
1. 2.
3. 4.
5.
6.
7. 8.
9.
10. Using #9 (above), insert parenthesis to make the expression equal 1. You may need to use more than one set of parentheses.
= 1
11. Using #9 (above), insert parenthesis to make the expression equal 23. You may need to use more than one set of parentheses.
= 23page 9
Section 1.2Algebraic Expressions- Intro Name ________________________________________
Match each English phrase or sentence with its algebraic translation.
1. Five less than twice a number.
2. Five more than twice a number.
3. Five is less than twice a number.
4. Five more than twice a number is less than ten.
5. Five less than twice a number is greater than ten times the number.
6. Five less than twice times a number is less than the number.
(a) 2n 5 (b) 5 > 2n (c) 5 < 2n (d) 5 2n(e) 2n 5 (f) 2n 5 > 10 (g) 2n 5 10 (h) 2n 5 > 10n(i) 2n 5 < n (j) 2n 5 < 10 (k) 2n 5 > 10 (l) 2n 5 < 10
Translate into words.
7. > 8. ≠
9. ≤ 10.
Translate into words as illustrated in the example.example: n + 4 Four more than a number.
11. x 3
12. w 7
13. 5n
14. 5n 1
15. y < 8
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Section 1.2 Class Notes – Algebraic Expressions
A linear combination is an expression in the form _______________________________________________________.
Example 1:
In a professional hockey game, a win is worth two points, a tie is worth one point and a loss is worth zero points. Write an expression to express the relationship among the number of wins (W), the number of ties (T) and the number of losses (L).
Example 2:
Ivan bought P pounds of peaches at $2.29 per pound and G pounds of grapes at $3.79 per pound.
a. Write an expression that gives the amount Ivan paid for the peaches and the grapes.
b. Suppose he bought 2 pounds of peaches and 3 pounds of grapes. How much did he spend in total?
c. Suppose he spent $15.24 in total. If he only bought one pound of grapes, how many pounds of peaches did he buy?
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Section 1.2 Class Practice
1. The expression 37P + 32R is called a __________________________________________________ of P and R.
2. Suppose you bought D DVD’s and B Books. If each DVD cost $14.95 and each book you bought off the ‘specials’ table cost $6.95, write an expression that tells how much you spent.
3. On a quiz show, 20 points are given for correct answers to regular questions and 50 points are given for correct answers in the bonus round. Let R represent the number of regular questions answered correctly and B represent the number of bonus questions answered correctly.
a. Write an expression that gives the total number of points earned.
b. Suppose that a contestant earned 650 points. Write an equation relating R, B, and the number of points earned.
c. Give three different possible solutions to the equation you wrote in part b.
Review: Translate into an algebraic expression or sentence.
1. p less than y
2. p is less than y
3. Five more than three times a number
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Choose the correct answer for each question.
(a) x y (b) x y (c) xy (d) (e)
4. You give a friend y dollars. You had x dollars. How much do you have left?
5. You drove x miles in y hours. What was your rate?
6. You buy x pencils at y cents each. What is the total cost?
7. Sara is y years old. Her sister is x years older. How old is her sister?
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Section 1.2Homework Name _________________________________________
Write algebraic expressions for each of these situations.
1. Suppose a collector now owns D dolls and is buying 6 new dolls per year. How many dolls will the collector have after y years?
_______________________________________
2. You give a friend d dollars. You had y dollars before. What do you have left? _______________________________________
3. Mrs. Bell is y years old. Her daughter is d years younger. How old is her daughter? _______________________________________
4. You drove m miles in h hours. What was your rate? _______________________________________
5. You buy g granola bars at c cents per bar. What is the total cost? ___________________________________
6. Suppose you buy p pizzas at $7.99 each and d drinks at $1.25 each. Write an expression that tells how much you spend in total.
7. Suppose zucchini sells for $1.29 per pound and tomatoes sell for $2.99 per pound.
a. What will be the cost of 2.5 pounds of zucchini and 3.4 pounds of tomatoes combined?
b. What will be the cost of Z pounds of zucchini and T pounds of tomatoes?
8. William spent last weekend mowing lawns. He charged $25 for small lawns, $50 for large lawns and earned a total of $275. Let S be the number of small lawns and L be the number of large lawns that he mowed.
a. Write an equation relating S, L, and the amount of money earned.
b. List all possible numbers of large and small lawns that William could have mowed.
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Section 1.3 Class NotesArithmetic Properties
The properties of arithmetic enable us to solve mathematical equations. Notice that these properties hold for addition and multiplication.
Name Form Example
Commutative Property of Addition a + b = b + a
Commutative Propertyof Multiplication ab = ba
Associative Propertyof Addition a + (b + c) = (a + b) + c
Associative Propertyof Multiplication a(bc) = (ab)c
Additive Identity a + 0 = a
Multiplicative Identity a·1 = a
Additive Inverse
Multiplicative Inverse
Distributive Property of Multiplication over Addition a(b + c) = ab + ac
From the Identity Property, we can say that 0 is the additive identity and 1 is the multiplicative identity.
Similarly, from the Inverse Property, is the additive inverse of a and is the multiplicative inverse
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of a because they both equal the identity for their respective operations.
Example
Identify the property used in the equations below.
a) ___________________________________________________
b) ___________________________________________________
c) ___________________________________________________
d) ___________________________________________________
e) ___________________________________________________
f) ___________________________________________________
g) ___________________________________________________
h) ___________________________________________________
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Section 1.3 Arithmetic PropertiesPractice Sheet #1 Name _________________________________________
Match each example with the property that is illustrated.
(a) Additive Identity
(b) Additive Inverse
(c) Associative Property of Addition
(d) Associative Property of Multiplication
(e) Commutative Property of Addition
(f) Commutative Property of Multiplication
(g) Distributive Property of Mult over Addn
(h) Multiplicative Identity
(i) Multiplicative Inverse
__________ 1. 2(3 + 4x) = 6 + 8x
__________ 2. 2 + (3 + 4) = (2 + 3) + 4
__________ 3. 3 + (-3) = 0
__________ 4.
__________ 5.
__________ 6.
__________ 7. 2 + 3 + 4 = 4 + 2 + 3
__________ 8.
__________ 9. (3 + 7) + (2 + 1) = (2 + 1) + (3 + 7)
__________ 10.
__________ 11. 8 + 0 = 8
__________ 12.
13. Which statement illustrates the associative property of multiplication?
(a)
(b)
(c)
(d)
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14. Which statement illustrates the additive identity?
(a) 2 – 2 = 0
(b) 2 + 0 =2
(c)
(d)
15. Use the distributive property to rewrite without parentheses.
(a) 16x - 20
(b) 16x -
(c) 16x - 16
(d) 16x - 4
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1.3 Properties of Real NumbersPractice Sheet #2 Name _________________________________________
Match each example with the property that is illustrated.
(a) Additive Identity
(b) Additive Inverse
(c) Associative Property of Addition
(d) Associative Property of Multiplication
(e) Commutative Property of Addition
(f) Commutative Property of Multiplication
(g) Distributive Property of Mult over Addn
(h) Multiplicative Identity
(i) Multiplicative Inverse
__________ 1. m + 3 = 3 + m
__________ 2.
__________ 3. a + (b + c) = (a + b) + c
__________ 4.
__________ 5.
__________ 6.
__________ 7.
__________ 8.
__________ 9.
__________ 10.
__________ 11.
__________ 12. ab + xy = ba + yx
__________ 13.
__________ 14.
__________ 15.
__________ 16.
__________ 17.
__________ 18.
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1.3 Homework Name ________________________________
Multiple Choice. Choose the one best answer for each problem.
________ 1. What is the additive inverse of x – y?
(a) x + y (b) -x + y (c) x – y (d) -x – y
________ 2. Which of the following illustrates the associative property of addition?
(a) 8 + (3 + 9) = (8 + 3) + 9 (b) 8 + 12 = 12 + 8
(c) (d) 8 + (-8) = 0
________ 3. Use the distributive property to rewrite the expression -3(5 – 2x) without parentheses.
(a) -15 – 6x (b) 6x – 15 (c) 6x + 15 (d) -8 – 5x
________ 4. Which equation below shows the correct use of the distributive property?
(a) (b)
(c) (d)
________ 5. Which statement illustrates the associative property of multiplication?
(a) (b)
(c) (d)
________ 6. Which statement illustrates the commutative property of addition?
(a) (5 + 3) + 6 = 5 + (3 + 6) (b) (5 + 3) + 6 = (3 + 5) + 6
(c) (5 + 3) + 6 = 8 + 6 (d) (5 + 3) + 6 = 14
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Match each example with the property that is illustrated.
(a) Additive Identity
(b) Additive Inverse
(c) Associative Property of Addition
(d) Associative Property of Multiplication
(e) Commutative Property of Addition
(f) Commutative Property of Multiplication
(g) Distributive Property of Mult over Addn
(h) Multiplicative Identity
(i) Multiplicative Inverse
__________ 7. m(a + b)= ma + mb
__________ 8.
__________ 9.
__________ 10.
__________ 11.
__________ 12.
__________ 13.
__________ 14.
__________ 15.
__________ 16.
__________ 17.
__________ 18.
__________ 19.
__________ 20.
Use the numbers and the property to create an equation.
Example: 2, 3 and 4 using the associative property of multiplication:
21. 4, 6, and 9 using the Associative property of addition: _____________________________________________
22. 11 and 3 using the commutative property of multiplication: ______________________________________
23. 13 and 2 using the commutative property of addition: _______________________1.3 Properties of Real Numbers
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Practice Sheet #3- Review Name ______________________________
Notice that these properties hold for addition and multiplication.
Commutative Property of Addition/MultiplicationAssociative Property of Addition/MultiplicationDistributive Property of Multiplication over AdditionAdditive InverseMultiplicative InverseAdditive IdentityMultiplicative Identity
Problems. All variables represent real numbers.
1. Use the Commutative Property of Addition to rewrite the expression 5x + 4. ______________________
2. Use the Commutative Property of Multiplication to rewrite the expression . ______________________
3. Use the Associative Property of Addition to rewrite the expression (7x + 4) + 11. ______________________
4. Use the Associative Property of Multiplication to rewrite the expression ______________________
5. Use the Distributive Property to rewrite the expression 6(x + 4). ______________________
6. Use the Distributive Property to rewrite the expression 5x+10. ______________________
Identify which property is illustrated in each statement below.
7. x(yz) = x(zy) ______________________________________
8. 2(x + y) = 2x + 2y ______________________________________
9. (x + y) + z = (y + x) + z ______________________________________
10. (x + y) + z = x + (y + z) ______________________________________
11. 2[x (a + d)] = (2x) (a + d) ______________________________________
12. ______________________________________
13. 3a + 3b = 3(a + b) ______________________________________
14. x + (-x) = 0 ______________________________________
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15. Show using examples that Subtraction and Division are NOT commutative or associative operations.
16. Does ? What property is being used here?
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Section 1.4GCF and Simple Factoring Notes Name:
GCF: _______________________________________ ________________________________ _________________________________
The Greatest Common Factor of a set of numbers is
Method for finding the GCF: Example: Find the GCF of 20 and 45
a. List the factors of each number. 20:
b. Identify the common factors.45:
c. Choose the greatest of those.
Sometimes the GCF includes a variable. For example, let’s look at 4x2 and 16x5.
4x2:
16x5:
Try it! Find the GCF of the following sets of numbers.
1. 26, 52 2. 4, 20, 15
3. 25a3, 15a6, 10a 4. 3k4, 8k, 5k5
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How to Factor Out the GCF from an expression:
Now we are going to rewrite expressions so that they are a multiplication of the GCF and the rest of the expression.
Example: 4x+20
a. Identify the GCF.
b. Divide each term in the expression by the GCF.
How can you check your final expression is equivalent to the original?
Note: If there is no GCF, that means the expression is already as simplified as possible and is called prime.
Let’s try it together! Factor out the GCF from each expression. Write PRIME if there is no GCF.
1. 2.
3. 4.
5. 6.
page 26
Now try it on your own. Factor out the GCF from each expression. Write PRIME if there is no GCF.
7. 8.
9. 10.
11. 12.
CHALLENGE!!! Push yourself!
13. 14.
15. 16.
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1.4 Homework Name: GCF and More Properties Practice
Match the property with the given equation.
1. 4x + 20 = 4(x + 5)
A) Identity Property of AdditionB) Distributive PropertyC) Commutative Property of AdditionD) Associative Property of Addition
2. 3 + x = x + 3
A) Associative Property of AdditionB) Commutative Property of MultiplicationC) Identity Property of AdditionD) Commutative Property of Addition
3. 8(9) = 9(8)
A) Commutative Property of MultiplicationB) Commutative Property of AdditionC) Associative Property of MultiplicationD) Identity Property of Multiplication
4. 8y - 16 = 4(2y - 4)
A) Associative Property of AdditionB) Distributive PropertyC) Identity Property of AdditionD) Commutative Property of Addition
5. 3(4y + 1) = 12y + 3
A) Associative Property of AdditionB) Commutative Property of AdditionC) Identity Property of AdditionD) Distributive Property
6. (2 + 3) + 4 = 2 + (3 + 4)
A) Distributive PropertyB) Associative Property of AdditionC) Identity Property of AdditionD) Commutative Property of Addition
7. (8 x 3) x 4 = 8 x (3 x 4)
A) Distributive PropertyB) Identity Property of MultiplicationC) Associative Property of MultiplicationD) Commutative Property of Multiplication
8. b + (m + n) = (m + n) + b
A) Identity Property of AdditionB) Commutative Property of AdditionC) Distributive PropertyD) Associative Property of Addition
9. 3(1)=3
A) Identity Property of MultiplicationB) Inverse Property of MultiplicationC) Inverse Property of AdditionD) Identity Property of Addition
10. 2-2=0
A) Identity Property of MultiplicationB) Inverse Property of MultiplicationC) Inverse Property of AdditionD) Identity Property of Addition
11. 4 + 0 = 4
A) Identity Property of MultiplicationB) Inverse Property of MultiplicationC) Inverse Property of AdditionD) Identity Property of Addition
12. 2(1/2)=1
A) Identity Property of MultiplicationB) Inverse Property of MultiplicationC) Inverse Property of AdditionD) Identity Property of Addition
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Factor out the GCF from each expression. Write prime if the expression is already simplified.
13) 36x4 + 8x3 + 20x 14) 7x5 − 7x3 + 7x
15) 54x 4 + 63x − 81 16) 4a7+ 3a2 + 7a
17) 20x3 − 40x + 12 18) 60x5 + 10x4 + 20x3
19) 5x 12 + 3x4 + 5x3 20) −45k3 − 18k2 + 9k
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Section 1.5 NotesEvaluating Expressions Name:
You have probably seen letters in a mathematical expression, such as 3x+8.
These letters, also called variables, represent an unknown number. One of the goals of algebra is to solve various equations for a variable.
To evaluate an expression or equation, we would need to substitute in a given value for the variable and test it. In order for the given value to be true for an equation, the two sides of the equation must simplify to the same number.
Example A: Evaluate 2x2 – 9 for x = 3.
Example B: Determine if x = 5 is a solution to 3x – 11 = 14.
Example C: Sometimes you are given a formula to use. For example, the formula for the area of
a circle is A=πr 2
, where r is the radius in feet. If the radius is 4ft, find the area.
Vocabulary Check
Variable- A letter used to represent an unknown value.
Expression- A group of variables, numbers, and operators.
Equation- Two expressions joined by an equal sign.
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Solution- A numeric value that makes an equation true
1.5 Classwork
Below are some formulas that we will use for practice today. You do NOT need to memorize these.
Area of a Circle with radius = r.
A=πr 2
Surface area of a rectangular prism with length = l,
width = w and height = h.
SA=2 wl+2wh+2 lh
Volume of a cone withradius = r and height = h.
V=1
3πr2 h
Children’s Dose of some Medications giventhe adult dose = A
and the child’s age = g (years).
C=( gg+12 )⋅A
Volume of a Sphere withradius = r
V= 4
3πr3
Surface Area of a Cylinder with radius = r and height = h.
SA=2πr2+2 π rh
Use the appropriate formula from the list above to find the values requested below.1. Find the surface area of a rectangular prism with length = 10 in, width = 4 in, & height = 5 in.
2. Find the volume of a cone with radius = 3 cm and height = 12 cm.
3. Find the Area of a circle with radius = 5 in. 4. Find the appropriate dose of a medication for a 6 year old child if the Adult dose is 120 mg.
page
r
31
5. Find the surface area of a cylinder with radius = 5 cm and height = 20 cm.
6. Find the volume of a sphere with radius = 6 in.
7. Here’s another useful formula. Suppose you stand at the top of a tall building and you drop something (like a penny) off the roof. The distance the object has fallen depends upon how long it has been falling. Let t stand for “time” (in seconds). If we measure the distance
in feet, the formula is: .
If we measure in meters, it is: d=1
2(9 .8 ) t2
----------------------
Note: The general formula is d= 1
2gt 2
where “g” stands for “gravitational constant.” The gravitation constant for feet = 32 ft/sec and the gravitation constant for meters = 9.8 m/sec. for objects near the surface of the Earth. The numbers are different near the surface of the moon or planets other than Earth.
8. Suppose you stand at the top of a very tall building and drop a penny off the top. How many feet will it have traveled after 2 second?
Suppose you drop a penny off the same building. How many meters will it have travelled in 2.5 seconds?
Evaluate each variable expression when a = 4 and b = 5
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9. 3(a + b) 10.
11.
1.5 Homework Evaluating expressions Name _______________________________________
Evaluate the following expressions for the given value.
1. Given x = 5, evaluate 4x – 1 2. Given a = -1, evaluate -2a + 7
3. Given t = 4, evaluate 4. Given c = -9, evaluate
5. Given p = -2, evaluate 6. Given m = 1, evaluate
Determine if the given values are solutions to the equations below.
7. Given x = 4, evaluate 8. Given x = 1, evaluate
9. Given z = -2, evaluate 10. Given b = 6, evaluate
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Find the value of each expression, given that a = -1, b = 2, c = -4, and d = 0.
7. 8.
9. 10.
For #11-16, use the equation
11. Is y = 4 a solution to this equation? 12. Is y = -4 a solution to this equation?
13. Is y = 3 a solution to this equation? 14. Is y = -3 a solution to this equation?
15. Do you think there are any other solutions to this equation other than those found above?
16. Using the solutions found in #11-14, find the sum of those solutions and their product. What do you notice?
Sum = ______________
Product = ______________
Look back at the original equation and at the sum & product of the solutions you found. What do you notice?
page 34
Section 1.6 NotesSimplifying Expressions
Corey has a bowl of fruit that consists of 5 apples, 4 oranges, and 3 limes. Katelyn went to the farmer’s market and picked up 2 apples, 5 limes, and an orange. How many apples, oranges, and limes do Corey and Katelyn have combined?
Combining like terms is much like grouping together different fruits, like apples and oranges.
Every algebraic expression has at least one term. A term is a number or is the product of a number and a variable. Terms are separated by addition and subtraction signs. A constant is a term that has no variable.
Given: 3x + 7 3x and 7 are terms (7 is also called a constant)
Algebraic expressions often have like terms that can be combined. Like terms are terms that have the same variable. All constants are like terms. The coefficient, the number multiplied by a variable in a term, does not need to be the same in order for the terms to be like terms.
Example 1: Match pairs of like terms: 6x 2y 3m x 9 3m 5 14y
In order to simplify an algebraic expression you must combine all like terms. When combining like terms you must remember that the operation in front of the term (addition or subtraction) must stay attached to the term. Rewrite the expression by grouping like terms together before adding or subtracting the coefficients to simplify.
Example 2: Simplify each algebraic expression by combining like terms.
a. 4x + 3x + 5x ____________________________________
b. 8y + 6 + 3 + 4y + 1 ____________________________________
c. 9m + 10p − 3p − 2m + 4m ____________________________________
d. 4d + 12 + d – 12 ____________________________________
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Simplify each algebraic expression by combining like terms.
1. 4y + 2y + 3y 4. 5x + 9y + 3x − 2y 7. 22u − 6u + 4t + 4t − 8u
2. 8x + x − 5x 5. 14p + 8 + 1− 14p 8. x + y + x + x + 2y − x
3. 10m − 4m + 2m − 3m 6. 11 + 5d − 3d – 4 9. 15 + 8n + 3n − 2 − 13
1.6 ClassworkCombining Like Terms Name _________________________________________
Combine like terms within each expression
1. 8x – 10 – 5x + 7 2. 4a – 7b + 5a – 20b + 3
3. 4.
5. 6.
Find the GCF of the following expressions and use the Distributive Property to simplify each one.
7. 5x – 10 8. 12a + 3
9. 10.
11. 12.
We can also use the Distributive Property and GCF to pull out common variables from an expression. Find the GCF and use the Distributive Property to simplify the following expressions.
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13. 14.
15. 16.
17. 18.
Section 1.6 HomeworkSimplifying Algebraic Expressions Name ________________________________
Simplify the following expressions as much as possible. If the expression cannot be simplified, write “cannot be simplified.”
1. = 2. =
3. = 4. =
5. = 6. =
7. = 8. =
Find the GCF of the following expressions and use the Distributive Property to simplify each one.
9. = 10. =
11. = 12. =
page 37
We can also use the Distributive Property and GCF to pull out common variables from an expression. Find the GCF and use the Distributive Property to simplify the following expressions.
13. = 14. =
15. =
Unit 1 Review & Study Guide Name ________________________________________
Lesson 1.1: You should know the order of arithmetic operations (PEMDAS) and be able to use them to simplify arithmetic expressions.
Simplify the following expressions using the correct order of operations.
1. 2.
3. 4.
5. 6.
Lesson 1.2: You should be able to translate English sentences into mathematics sentences and to use mathematical expressions to represent real-life situations.
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Translate these mathematical symbols into words
7. > 8. ≠
9. ≤ 10.
Write an algebraic expression or equation for each situation.
11. A number is 6 less than twice another number.
12. Four greater than a number.
13. A number is greater than 4 times another number.
14. Liz bought food for a party. Each pizza cost $7.99, each bottle of soda was $2.99 and each bag of chips was $1.49. How much did she spend in total? ________________________________________
15. Hannah bought G bags of Gold Fish at $2.29 per bag and C bags of Chex Mix at $1.79 per bag.
a. Write an expression that gives the amount Hannah paid for all of the food.
b. Suppose she bought 5 bags of Gold Fish and 3 bags of Chex Mix. How much did she spend in total?
c. Suppose she spent $13.03 in total. If he only bought one bag of Gold Fish, how many bags of Chex Mix did she buy?
Lesson 1.3: You should know the properties of arithmetic and be able to identify which property is illustrated given a mathematical sentence. You should also be able to give examples of properties.
Match the example column with the proper property given on the left column
a. Commutative Property of Addition
b. Associative Property of Multiplication
c. Distributive Property of Multiplication over Addition
d. Additive Inverse
__________ 16.
__________ 17. a + 0 = a
__________ 18.
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e. Multiplicative Inverse
f. Additive Identity
g. Multiplicative Identity
h. Commutative Property of Multiplication
i. Associative Property of Addition
__________ 19. x(m + n) = xm + xn
__________ 20. p + -p = 0
__________ 21.
__________ 22. xy + 2z = 2z + xy
__________ 23.
__________ 24. (m + p) + s = m + (p + s)
__________ 25.
__________ 26.
__________ 27.
Lesson 1.4: You should be able to factor linear expressions using the GCF
Find the GCF and use the Distributive Property to simplify the following expressions.
28. 29.
30. 31.
32. 33.
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34. 35.
Lesson 1.5 You should be able to evaluate expressions & equations
36. If x = 3: 37. Solve for V given that and p = 3.14, r = 4 and h = 7
Evaluate the following expressions given x = 3, y = 5, and z = -2
38. 39.
40. The formula gives us the area A for a regular polygon. The variable a represents the length of the apothem and P represents the perimeter. If the perimeter is 36 cm and the
apothem is about cm, find the area for the polygon.
Lesson 1.6 You should be able to simplify algebraic equations by combining like terms
Simplify the following expressions as much as possible.
41. 42.
page 41
43. 44.
45. 46.
47. 48.
49. 50.
Unit 1 Review & Study Guide Answers
1.
2.
3.
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4.
5.
6.
7. Greater than 8. not equal 9. less than or equal 10. approximately
11. n = 2x – 6 12. n + 4 13. n > 4x 14. 7.99p + 2.99s + 1.49 c
15a. 2.29G + 1.79C 15b, 5(2.29) + 3(1.79) = $16.82
15c. 2.29 + 1.79C = 13.03; 1.79C = 10.74; C = 6 answer: 6 bags of Chex Mix
16. h 17. f 18. e 19. c 20. d 21. b22. a 23. g 24. i 25. h 26. c 27. a
28. 3(x + y) 29. 30. 7(c – 2) 31. 2(x – y + 5z0
32. x(x – 9) 33. 34. 4(3x + 2y – 15 z) 35. 3x(x + 2)
36. 37. V = 3.14 (16) (7) = 351.68 38.
39. 40.
41. 5p 42. -2 – 5x 43. 44. -5x
45. 6x + 17 46. 8x – 6 47. 48. 14x – 10
49. 6x – 1 50. -1.6x – 6.6
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Unit 1 Review #2 Name: _______________________________________________
Evaluate these expressions using the correct order of operations. Show your steps.
1. (6+3 )÷(9−22) 2.
3. 100−(6−8 )2+7 4.
5. 6. 2 + [-3(-8 – 2)] 2
Add parenthesis to make each equation true.7.
8.
Insert the proper operation signs (+,–, x, ÷) and grouping symbols, when needed, to make each sentence true.9.
5________2________3________3 = -2
10.
6________7________8________2 = 9
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Match each example with the property that is illustrated.
(a) Additive Identity
(b) Additive Inverse
(c) Associative Property of Addition
(d) Associative Property of Multiplication
(e) Commutative Property of Addition
(f) Commutative Property of Multiplication
(g) Distributive Property of Mult over Addn
(h) Multiplicative Identity
(i) Multiplicative Inverse
11. __________ 17. 4 3 = 3 4 __________
12. __________ 18. 4x 1 = 4x __________
13. __________ 19. __________
14. __________ 20. __________
15. __________ 21.
12+(− 1
2 )=0__________
16. __________ 22. __________
23.Use the Commutative Property of Multiplication to rewrite the given expression.
4 5 6= ______________________________
24.Use the Distributive Property of Multiplication over Addition.
8x - 24= _______________________________
25. Is a solution to the equation Show work and explain in words.
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Area of a Circle with radius = r.
A=πr 2
Surface area of a rectangular prism with length = l,
width = w and height = h.
SA=2 wl+2wh+2 lh
Volume of a cone withradius = r and height = h.
V=1
3πr2 h
Children’s Dose of some Medications giventhe adult dose = A
and the child’s age = g (years).
C=( gg+12 )⋅A
Volume of a Sphere withradius = r
V= 4
3πr3
Surface Area of a Cylinder with radius = r and height = h.
SA=2πr2+2 π rh
Use the appropriate formula from the list above. Show work. Include UNITS in your answers. 26. Find the volume of a sphere with radius
6cm.27. Find the surface area of a cylinder with
radius 2in. and height 5in.
28. Find the surface area of a rectangular prism with length 2ft, width 9ft, and height 4ft.
29. Find the dose of medication for a 4 years old child if the adult dose is 12mg.
page
r
46
30. Evaluate the following expressions when a = -2, b = 6, and c = -1.
a.
a−bc2 b+c
b. 5 b−2c+a(2 b+c )
Simplify the following as much as possible. If an expression is already simplified, say so.31) 32)
33) 34)
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35) 36)
37) 38)
39.
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Answers:
1. 2. 10 3. 103 4. 36
5. 6 6. 902 7. 8. 9. 10.
11. e 12. a 13. g 14. i 15. d 16. b
17. f 18. h 19. e 20. c 21. b 22. g
23.
24. 8(x – 3)
25. If were a solution to the equation
then Let’s see if it does:
-20 – 12 = -32 and -2(-5+21) = -2(16) = -32 Since both sides equal -32, x = -5 is a solution to the equation
4x – 12 = -2(x + 21).
26. cm3
27. = in2
28.
= 36 + 72 + 16 124 ft2
29.
mg.
30a.
30b.
31. 4x + 5 32. –x + 19 33. 5x + 7 34. 14w2 – 4w
35. simplified already 36. 37. 38. 12x + 3y
39. -123x
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