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LN Summer Math Requirement – Algebra Review For students entering Algebra II

The purpose of this packet is to ensure that students are prepared for Algebra II. The Topics

contained in this packet are the core Algebra I concepts that students must understand to be

successful in Algebra II. There are 9 concepts addressed in this packet:

Topics

A. Simplifying Polynomial Expressions – Practice Set 1

B. Solving Equations - Practice Sets 2 & 3

C. Rules of Exponents - Practice Set 4

D. Binomial Multiplication - Practice Set 5

E. Factoring – Practice Set 6

F. Radicals – Practice Set 7

G. Lines – Practice Sets 8, 9, 10 & 11

H. Solving Systems of Equations – Practice Set 12

I. Absolute Values – Practice Sets 13

For each concept listed, there is an explanation with examples, a problem set, and a listing of

websites that deals with that particular topic. The websites include tutorials, videos, and extra

practice problems. An answer key is provided at the end of the packet.

Below is a list of websites that can help you with Algebra II concepts over the course of this

next year. http://www.hippocampus.org/HippoCampus/Algebra%20%26%20Geometry;jsessionid=1304A407D3C4842D1BF2266960BE6EB3 http://www.purplemath.com/modules/index.htm http://www.wtamu.edu/academic/anns/mps/math/mathlab/beg_algebra/ http://www.khanacademy.org/#library-section http://www.regentsprep.org/Regents/math/ALGEBRA/math-ALGEBRA.htm http://patrickjmt.com/

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Tutorials and Problems in this packet were compiled from the following sources:

Math teachers at Lawrence North High School

Prentice Hall Algebra 1 textbook http://frankumstein.com/worksheets.htm http://www.hcpss.org/parents/summer_enteringalgebra2_2010.pdf http://www.docstoc.com/docs/124240773/ALGEBRA-II-%ef%bf%bd-SUMMER-PACKET---DOC http://www.algebrafunsheets.com/algebratutorials/tutorials.php?name=SolvingAbsValueBasic.html http://teacherspace.swindsor.k12.ct.us/staff/smazzonna/documents/Summeralg1review_000.pdf

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Online Tutorials & Additional Practice – simplifying expressions http://www.purplemath.com/modules/polyadd.htm

http://www.regentsprep.org/Regents/math/ALGEBRA/AV2/indexAV2.htm

http://www.purplemath.com/modules/polymult.htm

http://www.regentsprep.org/Regents/math/ALGEBRA/AV3/indexAV3.htm

http://www.wtamu.edu/academic/anns/mps/math/mathlab/beg_algebra/beg_alg_tut27_addpoly.htm

http://www.wtamu.edu/academic/anns/mps/math/mathlab/beg_algebra/beg_alg_tut28_multpoly.htm

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Online tutorials & additional practice – solving equations http://www.khanacademy.org/math/algebra/solving-linear-equations-and-inequalities/basic-

equation-practice/v/equations-3

http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut7_lineq.htm

http://www.purplemath.com/modules/solvelin4.htm

http://www.montereyinstitute.org/courses/Algebra1/U02L1T2_RESOURCE/index.html

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Online tutorials and extra practice – exponents http://www.purplemath.com/modules/exponent.htm http://www.algebralab.org/lessons/lesson.aspx?file=Algebra_ExponentsRules.xml http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut23_exppart1.htm http://www.khanacademy.org/math/cc-sixth-grade-math/cc-6th-arithmetic-operations/cc-6th-exponents/v/introduction-to-exponents

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Online tutorials and extra practice – multiplying polynomials:

http://www.khanacademy.org/math/algebra/multiplying-factoring-expression/multiplying-binomials/v/multiplication-of-polynomials

http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut26_multpoly.htm

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___* ___ = -63 and ___ + ___ = -2

𝑥2 − 2𝑥 − 63

Factoring with a leading coefficient

2𝑥2 + 5𝑥 − 3

Factoring the Difference of Two

Squares

𝑥2 − 63

1. Multiply and rewrite

2. Factor

3. Give Back

4. Simplify

5. Swing back

1. Signs are ALWAYS opposite

2. Take the square root of both terms

( - ) ( + )

1. Group common terms

2. Pull out the GCF

3. Factor the groups

4. Rewrite

Factoring by GROUPING

2𝑥³ + 𝑥² − 12𝑥 − 6

Signs + + ( + ) ( + )

- + ( - ) ( - )

- - ( - ) ( + )

+ - ( + ) ( - )

FACTORING BASICS

Factoring without a leading coefficient

Use Borrow-Giveback

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11. 2x2 + 7x – 4 12. 3x2 + 19x + 6 13. 6x2 – 5x – 4 14. 4x2 – 16x + 7

Online tutorials and extra practice - factoring Factoring Trinomials (skip substitution method): http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut28_facttri.htm

Factoring a Trinomial: http://www.algebrahelp.com/lessons/factoring/trinomial/

Video: http://www.khanacademy.org/math/algebra2/polynomial_and_rational/quad_factoring/v/factoring-quadratic-expressions

Practice Set 6

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Practice Set 7

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Online Tutorials & Additional Practice – Radicals http://www.freemathhelp.com/Lessons/Algebra_1_Simplifying_Radicals_BB.htm http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut37_radical.htm http://www.khanacademy.org/math/arithmetic/exponents-radicals/radical-radicals/v/simplifying-radicals

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Practice Set 8 Find the slope of the line between that contains these points

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Practice Set 10

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Linear Equations in Two Variables

Examples:

a) Find the slope of the line passing through the points (-1, 2) and (3, 5).

slope = m = y

2- y

1

x2- x

1

m = 5-2

3 - (-1)

3

4

b) Graph y = 2/3 x - 4 with slope-intercept method.

Reminder: y = mx + b is slope-intercept form where m =. slope and b = y-intercept.

Therefore, slope is 2/3 and the y-intercept is – 4.

Graph accordingly.

c) Graph 3x - 2y - 8 = 0 with slope-intercept method.

Put in Slope-Intercept form: y = -3/2 x + 4

m = 3/2 b = -4

d) Write the equation of the line with a slope of 3 and passing through the point (2, -1)

y = mx + b

-1 = 3(2) + b

-7 = b Equation: y = 3x – 7

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Practice Set 11

Write an equation, in slope-intercept form using the given information.

1) (5, 4) m = 2

3

2) (-2, 4) m = -3

3) (-6, -3) (-2, -5)

Online tutorials and extra practice Using the slope and y-intercept to graph lines: http://www.purplemath.com/modules/slopgrph.htm Straight-line equations (slope-intercept form): http://www.purplemath.com/modules/strtlneq.htm Slopes and Equations of Lines: http://www.regentsprep.org/Regents/math/ALGEBRA/AC1/indexAC1.htm

List of videos to watch: http://www.khanacademy.org/search?page_search_query=lines

Video: http://www.montereyinstitute.org/courses/Algebra1/U06L1T1_RESOURCE/index.html

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H. Solving Systems of Equations

Solve for x and y:

x = 2y + 5 3x + 7y = 2

Using substitution method:

3(2y + 5) + 7y = 2

6y + 15 + 7y = 2

13y = -13

y = -1

x = 2(-1) + 5

x=3

Solution: (3, -1)

Solve for x and y:

3x + 5y = 1 2x + 3y = 0

Using linear combination (addition/

subtraction) method:

3(3x + 5y = 1)

-5(2x + 3y = 0)

9x + 15y = 3

-l0x - 15y = 0

-1x = 3

x = -3

2(-3) + 3y = 0

y=2

Solution: (-3, 2)

Solve each system of equations by either the substitution method or the linear combination (addition/

subtraction) method. Write your answer as an ordered pair.

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Practice Set 12

1. y = 2x + 4

-3x + y = -9

2. 2x + 3y = 6

-3x + 2y = 17

3. x – 2y = 5

3x – 5y = 8

4. 3x + 7y = -1

6x + 7y = 0

Online tutorials and extra practice – solving systems of equations http://www.purplemath.com/modules/systlin1.htm http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut19_systwo.htm http://www.khanacademy.org/math/algebra2/systems_eq_ineq/systems_tutorial_precalc/v/trolls--tolls--and-systems-of-equations http://www.montereyinstitute.org/courses/Algebra1/U06L1T2_RESOURCE/index.html http://www.montereyinstitute.org/courses/Algebra1/U06L1T3_RESOURCE/index.html

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J. Absolute Values

Absolute Value Equations

Example 1:

|x - 3| = 9 The absolute value term is already by itself on the left side.

x - 3 = 9 and x - 3 = -9

We need to create 2 equations. The first equation is exactly like

the original problem, but without the absolute value. The second

equation is created by keeping everything the same on the variable

side, but just changing the sign of the expression on the other side

(in this problem, 9 becomes -9).

x - 3 = 9 and x - 3 = -9

+3 +3 +3 +3

x = 12 and x = -6

Solve each equation separately. Since we have 2 equations, most of

the time there will be 2 answers (you won't always have 2 answers).

Answer: x = 12 and x = -6

Example 2:

|x + 4| - 2 = 11

Our first goal in absolute value equations is to isolate the absolute

value. In this case, on the absolute value side of the equation, we

have the terms |x + 4| and -2. In other words, we have to get rid

of that -2.

|x + 4| - 2 = 11

+2 +2

|x + 4| = 13

Adding 2 to both sides gets the absolute value by itself.

Now we have to create our 2 equations to solve.

x + 4 = 13 and x + 4 = -13

The first equation is the same as our isolated equation in the

previous step, but without the absolute value. For the second

equation, we'll just change the sign of the other side of the

expression so that 13 becomes -13.

x + 4 = 13 and x + 4 = -13

-4 -4 -4 -4

x = 9 and x = -17

Solve each equation separately.

Answer: x = 9 and x = -17

TIP

Remember, absolute value means distance from zero on a number line, and distance is always

positive. So, the absolute value of something can NEVER be negative. (I'm not saying your

solutions can't be negative, we had lots of negative solutions in the above problems). Just that

the answer to the question "what is the distance from zero" is always positive. Look at the

examples below.

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Example 3:

|x| = -3

The absolute value is already isolated. This problem is asking "what

number is -3 away from zero on a number line?" We know the absolute

value of anything HAS to be positive, so this problem has no solutions

because nothing is -3 away from zero.

Answer: No Solution

-3|x + 2| = 9

To isolate our absolute value, we have to divide both sides by -3, which

transforms our problem to: |x + 2| = -3. Now that the absolute value is

isolated, we can clearly see that this problem has no solutions.

Answer: No Solution

-2|x + 5| = -10

TIP Just because an equation has negative numbers in it does not mean

it is automatically a No Solution problem. You cannot make this

determination until the absolute value has been isolated! To isolate this

equation, you have to divide both sides by -2. This gives us the equation:

|x + 5| = 5. I won't finish out this problem, but you can see now that

this problem will have solutions.

(which are -10 and 0, in case you are curious).

Problem Set 13

Solve.

1. |𝑥| = 3 2. |𝑥 − 4| = −3

3. |𝑥| − 10 = −3 4. −3|𝑥| = −6

5. 12 = −4|𝑥| 6. |𝑥 + 2| = 6

Online tutorials and extra practice: http://www.algebrafunsheets.com/algebratutorials/tutorials.php?name=SolvingAbsValueBasic.html http://www.purplemath.com/modules/solveabs.htm http://www.khanacademy.org/math/algebra/solving-linear-equations-and-inequalities/absolute-value-equations/v/absolute-value-equations

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ANSWERS TO PRACTICE SETS

PRACTICE SET 1

1. 24𝑥 + 3𝑦 2. −15𝑦2 + 37𝑦 + 22 3. 9𝑛 − 3

4. −22𝑏 + 6 5. 160𝑞𝑥 + 110𝑞 6. −5𝑥 + 6

7. 74𝑧 − 24𝑤 8. 56𝑐 − 117 9. −27𝑥2 + 54𝑥 − 9

10. 31𝑥 − 𝑦 + 42

PRACTICE SET 2

1. 7 2. 26 3. 9.5 4. 13

5. 3.5 6. 16 7. 19 8. -2.8

9. 9.5 10. 0

PRACTICE SET 3

1. V = W - Y 2. 𝑤 =81

9𝑟 3. 𝑓 = −3 +

2

3𝑑

4. 𝑥 =10−𝑡

𝑑 5. 𝑔 =

𝑃+1620

180 6. 𝑥 =

9𝑦+5ℎ+𝑢

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PRACTICE SET 4

1. 𝑐8 2. 𝑚12 3. 𝑘20 4. 1

5. 𝑝11𝑞7 6. 9𝑧9 7. −𝑡21 8. 3𝑓3

9. 60ℎ8𝑘5

10. 𝑎3𝑏4

3𝑐

11. 81𝑚8𝑛4

12. 1

13. 30𝑎3𝑏4𝑐

14. 4𝑥

15. 24𝑥4𝑦7

PRACTICE SET 5

1. x2 + x - 90 2. x

2 - 5x - 84 3. x

2 - 12x + 20

4. x2 + 73x - 648 5. 8x

2 + 2x - 3 6. 18x

2 - 100x + 50

7. -6x2 - 20x - 16 8. x

2 + 20x + 100 9. x

2 - 10x + 25

10. 4x2 - 12x + 9

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PRACTICE SET 6

1. 3x ( x + 2) 2. 4ab2(a – 4b + 2c) 3. (x – 5)(x + 5)

4. (n + 5)(n + 3) 5. (g – 5)(g – 4) 6. (d + 7)(d – 4)

7. (z – 10)(z + 3) 8. (m + 9)2

9. 4y(y – 3)(y + 3)

10. 5(x + 9)(x – 3) 11. (x + 4)(2x – 1) 12. (x + 6)(3x + 1)

13. (3x – 4)(2x + 1) 14. (2x – 7)(2x – 1)

PRACTICE SET 7

1. 11 2. 103 3. 75 4. 212

5. 69 6. 8 7. 560

8. 321 9. 1940 10. 3

55 or 5

3

5

PRACTICE SET 8

1. -3 2.

3

2

3. 2

1

4. 0

5. 1 6. Undefined

PRACTICE SET 9

1. Slope: 2; y-intercept: (0,5) 2. Slope: 1

2; y-intercept: (0, -3)

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3. Slope: -2

5; y-intercept: (0, 4) 4. Slope: -3; y-intercept: (0, 0)

5. Slope: -1; y-intercept: (0, 2) 6. Slope: 1; y-intercept: (0, 0)

PRACTICE SET 10

1. 3x + y = 3 2. 5x + 2y = 10

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3. y = 4 4. 4x – 3y = 9

5. -2x + 6y = 12 6. x = -3

PRACTICE SET 11

1. y =-2

3x +

22

3

2. y = -3x-2 3. y =

-1

2x - 6

PRACTICE SET 12

1. (13, 30) 2. (– 3, 4)

3. (– 9, – 7) 4.

7

2,

3

1

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PRACTICE SET 13

1. -3, 3 2. 1, 7 3. -7, 7

4. -2, 2 5. No solution 6. -8, 4