Arithmetic and Algebra Worksheets, Study notes of Algebra

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Essentials to Mathematics

Arithmetic and Algebra Worksheets

Shirleen Luttrell

circle.adventist.org

Contents

7i – Solving Radical Equations.............................................................................................................................. 110 Chapter 7 Test........................................................................................................................................................ 111 Chapter 8 Expanding & Solving Polynomials ........................................................................................................... 113 8a – Multipl y ing Pol y nomials ................................................................................................................................ 114 8b – Factoring ........................................................................................................................................................ 115 8c – Factoring ........................................................................................................................................................ 116 8d – Factoring ........................................................................................................................................................ 117 8e – Solve b y Factoring ......................................................................................................................................... 118 8f – Difference of Squares ..................................................................................................................................... 119 8g – More Practice Factoring ................................................................................................................................ 120 8h –Quadratic Formula .......................................................................................................................................... 121 8i – more Quadratic Formula ................................................................................................................................. 122 8j – Graphing Quadratics ....................................................................................................................................... 123 8k – Graphing Quadratics ...................................................................................................................................... 124 8L – Quadratic Word Problems ............................................................................................................................. 125 8m –Comple x Numbers ......................................................................................................................................... 126 8n –Comple x Numbers continued.......................................................................................................................... 127 Chapter 8 Test........................................................................................................................................................ 128

Algebra Cumulative Review...................................................................................................................................... 130

Appendix ................................................................................................................................................................... 131 1 Sets and Operations of Numbers ........................................................................................................................ 132 2 Functions and Relations ...................................................................................................................................... 133 3 - Linear Functions ............................................................................................................................................... 134 4- Systems of Linear Functions ............................................................................................................................. 135 5 - Quadratic Functions ......................................................................................................................................... 136 6a - Exponentials ................................................................................................................................................... 137 6b - Logarithmic Functions ................................................................................................................................... 138 6c - Solving Logarithmic Functions ...................................................................................................................... 139 7 - Rational Functions ............................................................................................................................................ 140 8 - Irrational Functions .......................................................................................................................................... 141 9 - Polynomials of any Degree .............................................................................................................................. 142

Final Exam................................................................................................................................................................. 144

Answer Key ............................................................................................................................................................... 155

Acknowledgements

I want to acknowledge that this booklet does not contain all the worksheets needed to cover the entire algebra curriculum. This book began ten years ago when I assisted a colleague, Dr. Keith Calkins, remediate high school students entering a rigorous advanced mathematics program. The worksheets I developed then focused on common weak areas my students needed to strengthen. Since that time I worked a couple of years with Dr. Lynelle Weldon who directed the task to remediate university students before placing them into university mathematics courses. A few of her study guides became a blue print for a few of mine and those got inserted into this book as well. Then I spent three years developing a two-year pre-algebra course for a combined seventh and eighth grade class. Since there was always an influx of new students each year, the curriculum was the same each year with the difference only in the activities and worksheets. The worksheets I developed were for certain days when I could find no resources on hand for what I wanted the students to master. These worksheets found their way into this book as well. So you can conclude that this booklet you are perusing is a compilation of ten years of supplemental writing. Hopefully you will find it useful.

I want to thank Dr Calkins and Dr Weldon for their inspiration and their examples! Pun intended.

Chapter 1 Number System

Prior Skills:

• Convert fractions to decimal for sheet 1c
• Time measurements for sheet 1c
• Basic understanding of decimal and fractions for sheet 1d

Name: ______________________ Date: _____

1b – Number Systems

Complex Numbers - All numbers are complex. Their form is a + bi. These numbers will be taught later!

Real Numbers – numbers found on the “number line”. If written as a complex number, they would look like a+0i.

Imaginary numbers - points not on the standard number line. If written as complex, they would have form 0+b i.

Zero - It is both real and imaginary.

Rational Numbers – Real numbers that can be expressed as a ratio of two integers. If written as a decimal, they would be terminating or repeating.

Irrational Numbers - reals that CANNOT be expressed as a ratio of integers. If written as a decimal, they would be nonterminating and nonrepeating decimals.

Transcendental Numbers - irrational numbers that can NOT be solved by algebraic methods

Integers - whole numbers and their opposites

Non-integers - another name for a reduced fraction where 1 is NOT in the denominator.

Whole numbers - 0, 1, 2, 3…

Natural Numbers (counting numbers) - 1, 2, 3…

Digits - whole numbers from 0 to 9, those numbers which make up our numerals

Even - integers divisible by 2

Odd - integers that are NOT divisible by 2

Positive - reals greater than 0

Negative - reals less than 0

Answer the following about numbers:

1. On a separate piece of paper, create a hierarchy for the number systems above.

For each branch, list three examples of the number system.

2. Which of the following is not a rational number?

3. Which of the following is not a rational number?

4. Which is not an integer? 2 -2 0 ½ 4 2

5. What type of number is this: (rational, irrational, integer, real...)

A. -3.4 B. 5 C. 12 D. 0

6. Explain which decimals are rational numbers? How can you tell them from an irrational number?

Name: ______________________ Date: _____

1c - Number Systems Mathematicians use short-hand notation when referring to number systems:

N - natural, Z - integer, Q - rational, R - real, C - complex.

1. Check off which number systems the following numbers are:

N Z Q R C

π

4. 81

27

0

2. How many minutes are there in two and one half days?
3. How many seconds are there in a day?
4. Some rational numbers can be expressed in decimal form. Express the following in decimal,

showing all work:

a. ¼ b.

1

6

c.

1

9

d.

5

12

e.

7

100

5. Explain which decimals are rational and which are irrational. For example: π ≈ 3.141592... is irrational.
6. Write the following decimals as a ratio of two integers:

a. 0.315 b. 3.151515...

Name: ______________________ Date: _____

1e-Adding & Subtracting Integers

Perform the operations without a calculator. Show work by plotting the operations on a number line.

1. There are several ways to add or subtract integers. Some think of money debts, others think

of protons versus electrons. The following example is showing how addition is about gaining

and subtraction about losing in terms of the real number line.

-4 + 9 7 - 5 -3 - 3 1 - 7 1 + 3 = 5 = 2 = -6 = -6 = 4

-4 0 1 2 3 4 5 -6 -5 -4 -3 1 2 3 4

Simplify the following by doing the indicated operation:

2. • 4 - 9 3. -7 – 5 4. -3 + 3
3. -1 – 7 6. 1 - 3 7. -5 – 9
4. 8 – 3 9. 53 – 42 10. 31 – 82
5. • 44 + 53 12. - 35 + 35 13. 23 – 17
6. 2 – 4 – 6 15. 2 + (- 4) – 6 16. 0 – 2 + 6

Name: ______________________ Date: _____

1f- more Adding & Subtracting Integers

You may use your calculator only to check your answers. Simplify the expressions.

Find the result.

1. 7 + 3 2. -6 – 3 3. -8 – 6 4. 6 + (-3)
2. (+6) + (-3) 6. -7 + (-8) 7. - 4 + (+2) 8. 4 + (-2)
3. 5 – 8 10. -78 – 21 11. -32 – 21 12. -55 – 44
4. 34 – 43 14. -34 + 68 15. 54 – 59 16. -90 + 90
5. 3 – 6 18. –4 + 5 19. 4 – 5 20. 6 – 5
6. 7 – 17 22. 10 – 15 23. 0 – 1 24. -3 + 4
7. -14 + 25 26. -5 + 10 27. -1 + 8 28. -7 + 23
8. 8 – 3 30. 3 – 6 31. 10 – 6 32. 4 – 7
9. -1 + 3 34. -10 + 6 35. -7 + 4 36. -7 + 8

Translate the following expression and find the integer that represents the overall change.

37. The temperature starts at -15°C, drops 10°C, rises 5°C and rises 8°C.
38. A person starts with $50, earns $12, spends $15, earns $18, and spends $22.
39. A submarine starts at sea level, dives down 125 m, dives another 72 m, and rises 42 m.
40. An elevator starts on the seventh floor, descends 5 floors and ascends 9 floors.

Name: ______________________ Date: _____

1h-Expanding Numbers

You may use your calculator only to check your answers.

State the place value of the ‘5’ in each number below:

1. 78,513 2. 960,500 3. 5,000,
2. 85,723 5. 23,985 6. 234,

Write the following in expanded form:

7. 34
8. 5345
9. 4,000,
10. 203,
11. 432
12. 865,342,

Write each of the following in standard form:

13. (4×100,000) + (5×10,000) + (3×1000) + (8×100)
14. (9×1,000,000) + (7×1)
15. (6×1) + (7×10) + (8×100) + (6×1000) + (7×10,000)
16. (3×100) + (4×1000) + (7×1) + (9×10) + (4×1,000,000)

Name: ______________________ Date: _____

Test REVIEW: Integers

SHOW WORK. A calculator is NOT allowed on this test. You must work alone. Questions regarding interpreting the directions are allowed. Simplify your answers!

1. Solve the following for x:

a. x – 12 = 28 b. x + 17 = 0 c. 6 – x = 2

2. Evaluate the following:

a. 13 - 12 b. -13 + 12 c. -13 - 12

3. Evaluate the following: a. -2(- 4) b. 4(-8) c. -(-7) d. -5 × 4
4. Evaluate the following:

a. 25 ÷ (-5) b. -32 ÷ 8 c. -18 ÷ (-2)

5. Write the following number in standard form:

4(100,000) + 6(1,000) + 5(100) + 3(10)

6. Write in expanded form: 43,507.

Chapter 2 Fractions

Prior skills:

• For sheet 2a, know perimeter
• For sheet 2n, know area

An asterisk (*) next to a question, such as question 17 & 18 on sheet 2i implies that the student may find the question challenging. The questions may have come from an activity we did in class prior to the worksheet. If you using the worksheets without other resources, just beware

that the students may have difficulty with asterisk questions.

Name: ______________________ Date: _____

2a-Finding Fractions

For each question, translate the equation and then solve by mental math.

1. Darcy decides to eat only ⅓ of a candy bar. Draw a candy bar and shade in what was eaten.
2. Students at SLA walked 20 laps to help the Terry Fox Foundation. Some walked only ¾ the

laps. Make 20 squares to represent the laps and shade in the amount some only walked.

3. The perimeter of the park is about 4 miles. Someone walked only ¼ of it. Draw a circle and

color the fraction of the circle walked.

4. Nicole bought four apples. One was eaten this morning. What fraction of apples are left?
5. There are 150 days of school. If students have been in school for 15 days, what fraction of

the school year is left?

6. The perimeter of the building is 400 feet and is getting a new coat of paint, what fraction of

the building is left to paint if only 100 feet got painted? Draw a picture of the outline of the

building and where it’s painted. Does your drawing look like others?

Name: ______________________ Date: _____

2c-Adding & Subtracting Fractions of Different Denominators

Simplify the following without the use of a calculator. Leave as a proper fraction.

1. 23 + 125 2. 3 15 − 1 38 3. 5 6

5

• 12

4. 2 4 1 3

4

• 5 5. 8 9

3

• 4 6. 1 7

3

• 4

7. 2 1

1 2

3 − (^4) 8. 10 5 3 6

1 − 9 9. 4 2

5 9

13 − 15

10. 1 1 3

14 − 15 11. ⅞ + ⅔ 12. ⅓ - ⅜

Name: ______________________ Date: _____

2d-Multiplying & Dividing Fractions

Use your calculator only to check your answer. Leave answer as simplified proper fractions.

1. 43 ⋅ 89 2. 4 5

10 × (^17) 3. 5 6

3 × 10

6 7

14 × 15 5.

8 9

27 × 28 6. 3 7

21 ( 10 )

7. 2 43 × 425 8. − 4 5

15 ( 24 ) (^) 9. 3 4 ×^3

4 5 (^ −^15 ) 11.^3

2 ( − 3 ) 12. 2 3 1 4

1 × 3

13. 3 4

2 3

1 × 11 14. 2 3

1 17

2 × 5 15. 5 7

1 3

1 × 2

16. 2 1 2

5 ÷ 6 17. 4 11 3 5

1 ÷ 2 18. 5 7

10 ÷ 21

8 9

1 ÷ 39 20. 4 ½ ∙ 8 ⅔^ 21. (8⅔) / (4½)

Question 21 is an example why there are many types of parentheses and division symbols. Some

symbols make the question cluttered and hard to read.