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Summer Math Packet
Number Sense & Pre-ΒβAlgebra Skills For Students Entering Algebra
No Calculators!! Within the first few days of your Algebra course you will be assessed on the prerequisite skills outlined in this packet. The packet will not be graded; however, you are responsible for the material. The assessment will count as a full test grade in your first quarter average.
Topics Covered
1. Central Tendencies
2. Greatest Common Factor/Least Common Multiple
3. Fractions
4. Order of Operations
5. Working with Integers
6. Evaluation Algebraic Expressions and Formulas
7. Properties of Operations
8. Solving Multi-ΒβStep Equations
9. Solving Multi-ΒβStep Inequalities
10. Linear Functions
11. Polynomials
Central Tendencies (Mean, Median, Mode, and Range)
Mean is the sum of the values in a set of data divided by the number of values.
Median is the middle value of a set of data written in ascending order. If there are two middle values, the median is the mean of those values.
Mode is the most frequent value in a set of data.
Range is the difference between the greatest and least value in a set of data.
Exercises:
Find the mean, median, mode, and range of each set of data.
1. 108, 93, 426, 766, 518, 210
2. 21.5, 35.5, 49.5, 16.3, 35.5
GCF & LCM
Example:
Find the Greatest Common Factor (GCF) and Least Common Multiple (LCM) of 24 and 32.
Exercises:
Find the GCF.
1. 42, 60
3. 27π₯!π¦!, 45π₯!
2. 24π₯π¦!, 42π₯π¦
4. 11, 21
Find the LCM
5. 27, 18
7. 9π₯!π¦, 15π₯π¦!
6. 15x, 18π₯π¦
8. 64, 48
Fractions (Addition, Subtraction, Multiplication, and Division)
Miscellaneous
Write the fractions in lowest terms.
1. !!"
3. !"!!!!"!"
2. !"!"
4. !"!"!!
!"!!!!!
Solve for π.
5. !"!"= !
!"
7. !"!"= !
!"
6. !"!"= !
!
8. !!= !"
!
Write as improper fractions.
9. 2 !!
10. β4 !!
Write as mixed numbers.
11. β !!
12. !"!
Addition and Subtraction
Find each sum or difference. Write your answer in simplest form.
13. β !!+ !
!
15. !!"β !
!
17. 5 !!!β 2 !
!
14. 3 !!+ 2 !
!
16. 6 !!"+ β1 !
!
18. 2 !!"β 9 !
!
Multiplication and Division
Find each product or quotient. Write your answer in simplest form.
19. β !!β !!"
21. 2 !!β β3 !
!
23. !!β !!"β !!
20. β !!Γ· β !
!"
22. β3 !!Γ· 4 !
!
24. 6 !!Γ· 4
Order of Operations
When several operations are indicated in a numerical expression, proceed in the following order: work within the parentheses, expand each power, multiply and divide (whichever comes first), and finally, add or subtract (whichever comes first).
PEMDAS (βPlease Excuse My Dear Aunt Sallyβ) is an acronym that provides a good way to remember your order of operation.
P: Parentheses E: Exponents MD: Multiply or Divide, whichever comes first AS: Add or Subtract, whichever comes first
Simplify.
1. 2! β 3 3! β 8
3. 4! β 4 5! β 32 Γ· 8 β 4
5. 5! β 3 6 + β2 20 + β15
7. 15 β 3 4! β 10 + 25 Γ· 5 β 15
9. β32 + 32
2. 4! + 10 4 β 10 5! β 20
4. 8 β 5 Γ· 10 + 2 2! β 8! Γ· 2
6. 4! + (β10)(30 β 8 β 5)
8. 10 β 5 20 β 2 3! + 1
10. !"!!"Γ·!!
!!!β!
Working with Integers
Adding and Subtracting:
1st: Rewrite all subtraction as addition then⦠! If the integers have the same signs, add their absolute values. The sum willhave the same sign of the addends.
! If the integers have different signs, subtract their absolute values. The sumhas the sign of the addend with the greater absolute value.
Multiplying and Dividing:
! The product or quotient of two integers having the same sign is positive.! The product or quotient of two integers having different signs is negative.
Find each sum, difference, product, or quotient.
1. β13 + 19
4. β27 β 93
7. β45 Γ· 9
10. 8(β17)
2. 37 + (β13)
5. β46 β (β32)
8. β84 Γ· β12
11. β24 β β6
3. β18 + (β29)
6. 9 β 83
9. !"#!!!
12. β62 8
13. There is a 6Β° drop in temperature over the past hour. If it is 55Β° now, whatwas the temperature an hour ago?
14. It is β9Β° now. The temperature will drop 5Β° in two hours. What will thetemperature be in two hours?
Evaluating Expressions and Formulas
To evaluate an expression, first replace the variable by a given value. Then simplify the resulting numerical expression.
Evaluate the expression when π₯ = β2 and π¦ = 5.
1. π₯ + π¦
3. 2π₯ β π¦
5. !!!!!!
2. π₯! + π¦!
4. β2(π¦ β 2π₯)
6. !!!!
Properties of Operations
Commutative Property of Addition: π + π = π + π
Associative Property of Addition:
(π + π) + π = π + (π + π)
Identity Property of Addition: π + 0 = π
Commutative Property of Multiplication: π Γ π = π Γ π
Associative Property of Multiplication:
π Γ π Γ π = π Γ (π Γ π)
Identity Property of Multiplication: π Γ 1 = π
Name the property illustrated by each expression.
1. 8 Γ 12 = 12 Γ 8
3. 2 + 5 + 12 = 5 + 2 + 12
5. 1π₯ = π₯
7. 3 + 4 + 5 = 3 + (5 + 4)
9. (4 + 8) + 5 = 4 + (8 + 5)
2. 3 Γ 2 Γ 5 = 3 Γ 2 Γ 5
4. π₯π¦ + 0 = π₯π¦
6. 5 + 7 = 7 + 5
8. 3π₯π¦ = 3π₯π¦(1)
10. 5 Γ 6 Γ 8 = 8 Γ 5 Γ 6
Solving Multi-ΒβStep Equations
Procedure: To solve multi-Ββstep equationsβ¦ 1. Fully simplify both sides of the equation2. Get all variables to one side of the equation.3. Use inverse operations to isolate the variable
**undo addition and subtraction first**
Ex. Ex.
Exercises
Solve and check each equation.
1. β2π₯ + 7 = 25
3. 15 β 2 π€ + 5 = 11
5. β4(π + 5) = β32
7. 3 β 2π₯ = 15
9. 17 + 3π₯ = 4π₯ β 9
2. 3 β 8π₯ = β141
4. 12 β 4π = 6π + 2
6. 12 β 2π₯ + 5 = β1
8. !!β 7=12
10. β3 6π β 12 = 36 β 18π
Solving Multi-ΒβStep Inequalities
Note: Solve a multi-Ββstep inequality just like you would solve a multi-Ββstep equation. However, if you multiply or divide both sides of an inequality by a negative number, then the inequality sign reverses.
Ex. Ex.
Exercises
Find and graph the solution set of each inequality.
1. 3π₯ + 8 > 17
3. 2π£ + 7 β₯ 11
5. !!!!β€ 4
7. 2π§ β 5 < β21 β 2π§
9. 3π₯ β 5 > 6π₯ + 13
2. β6π¦ + 3 > 9 β 7π¦
4. 7 > 3 + !!
6. 4π + 4 < 4(5 β 3π)
8. 8π β 10 β₯ 6(3 β π)
1. 7 π¦ + 5 β 10 β€ 2π¦
Linear Functions
Exercises
Tell whether each ordered pair is a solution of the equation.
1. 3π₯ + π¦ = β11, (β4, 1) 2. 2π₯ β π¦ = 4, (3,β2)
Find the intercepts of the equations graph.
3. 3π₯ β 4π¦ = β12 4. π¦ = β2π₯ β 8
Find the slope through the given points.
5. (4, 7), (β3, 6) 6. (β5, 7), (β5,β14)
Identify the slope and y-Ββintercept of the line with the given equation.
7. π¦ = 2π₯ β 12 8. 2π₯ β 3π¦ = β6
Write an equation of the line that is parallel to the given line and passes through the given point.
9. π¦ = β2π₯ β 6, (0,β4) 10. β2π₯ + 3π¦ = 12, (3, 2)
Graph the equation using any method.
11. π¦ = 2π₯ β 3 12. β2π₯ β 3π¦ = 12
Polynomials Examples
A polynomial is in Standard Form if it is simplified and the terms are arraigned so the degree of each term increases (or stays the same) from left to right.
Find the difference: 6π₯! β 5π₯ + 2 β β3π₯! β 8π₯ + 3
First: Turn the expression into an addition problem by distributing the negative to the second expression. 6π₯! β 5π₯ + 2 + 3π₯! + 8π₯ β 3
Then: Combine like terms 6π₯! + 3π₯! + β5π₯ + 8π₯ + 2+β3 = 9π₯! + 3π₯ β 1
Find the product: 3π₯ 2π₯! β 5 = 3π₯ 2π₯! + 3π₯ β5 = 6π₯! β 15π₯
Find the quotient: 8π! + 4π! β 6π
2π =8π!
2π +4π!
2π +β6π2π = 4π! + 2π β 3
Find the product using the F.O.I.L. method (F: first, O: outer, I: inner, L: last): 2π₯ β 3 π₯ + 5 = 2π₯ π₯ + 2π₯ 5 β 3 π₯ β 3 5 = 2π₯! + 7π₯ β 15
Exercises
Write the expression in standard form.
1. 13 β 4π₯ + 3π₯! 2. 4π¦! β 2 2π¦ β 3 + π¦
Find the sum or difference.
3. 3π₯! β 5π₯ + 2 + 5π₯! + 9π₯ β 5 4. 8π¦! + 2π¦ β 6 β 3π¦! β 5π¦ + 2
Find the product or quotient.
5. 3π₯(π₯! β 5)
7. !!!!!!!!!!!!
9. (π₯ + 2)(π₯ + 3)
6. 4π§! β 5π§ + 2 6π§
8. !"!!!!!!!!!!
!!!!
10. π¦ β 3 π¦ + 10