Description
In This collection, we will go deep into math.
This collection will help all math and high school students.
Study Set Content:
Essentials to Mathematics
Arithmetic and Algebra Worksheets
Shirleen Luttrell
2012
circle.adventist.org
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Contents
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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.
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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
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Name: ______________________
Date: _____
1a- Translating Mathematical Symbols
For each question, translate the equation and then solve by mental math. No calculator!
Example: 3x = 21.
Translation: Three times a number is 21.
Answer: x = 7
1.
x - 4 = 13
2.
x + 5 = 8
3.
8 - x = 5
4.
4x = 12
5.
2x = 6
6.
T + 7 = 10
7.
14 - t = 5
8.
21 - x = 13
9.
Y ÷ 3 = 6
10.
9 ÷ P = 1
11.
8×P = 32
12.
6 × R = 54
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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?
3.1
3.01
3.111...
3.1234322344523...
3½
3. Which of the following is not a rational number?
3.4
-3.4
3.444...
-3.444
3.040040004...
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?
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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
3.4
0 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...
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Name: ______________________
Date: _____
1d - Which Number is Bigger?
You may use your calculator to convert into decimal if necessary to answer the questions.
1. Plot the numbers on a real number line: -
1⅓, 0, 2 ⅔, 1¾, -⅝, -⅔,
1
3
5
,
2
5
7
, -3, 2 ½
2. Plot the numbers on a real number line: -2.3, -3.3, 4.3, 4, 2, -2, -3.8
3. Plot on a real number line: -
1 ½, 3⅛, -2⅞, 1.5, 1.05, -1¾, 1.7
6. Which is bigger? Fill in the blank with <, >, or =.
a. -3 ___ 3
b. - 4 ___ -5
c. -10.4 ___ -10.3
d. 4.5 ___ 4.55
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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
5. -1 – 7
6. 1 - 3
7. -5 – 9
8. 8 – 3
9. 53 – 42
10. 31 – 82
11. - 44 + 53
12. - 35 + 35
13. 23 – 17
14. 2 – 4 – 6
15. 2 + (- 4) – 6
16. 0 – 2 + 6
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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)
5. (+6) + (-3)
6. -7 + (-8)
7. - 4 + (+2)
8. 4 + (-2)
9. 5 – 8
10. -78 – 21
11. -32 – 21
12. -55 – 44
13. 34 – 43
14. -34 + 68
15. 54 – 59
16. -90 + 90
17. 3 – 6
18. –4 + 5
19. 4 – 5
20. 6 – 5
21. 7 – 17
22. 10 – 15
23. 0 – 1
24. -3 + 4
25. -14 + 25
26. -5 + 10
27. -1 + 8
28. -7 + 23
29. 8 – 3
30. 3 – 6
31. 10 – 6
32. 4 – 7
33. -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.
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Name: ______________________
Date: _____
1g-Multiplying & Dividing Integers
You may use your calculator only to check your answers.
1. (-8)(-3)
2. -6(- 4)
3. 5(-9)
4. 10(-3)
5. -6(-3)
6. -2(5)
7. 15(- 4)
8. 16(-3)
9. 17×(-5)
10. (-8)(-9)
11. (-7)(31)
12. 90(100)
13. 8 × (-3)
14. -3 × (-2)
15. -5 × -14
16. 12×12
17. -9 ÷ -3
18. -18 ÷ (-9)
19. 20 ÷ (-5)
20. -72 ÷ (-8)
21. -100 ÷ (-10)
22. -35 ÷ 7
23. 36 ÷ 4
24. 81 ÷ (-3)
25. 95 ÷ (-19)
26. -32 ÷ 4
27. 64 ÷ 8
28. 42 ÷ (-6)
29. -(-9)
30. -0
31. (-1)(-1)(-1)(-1)(-1)
32. 2(-3)(4)(-5)(6)
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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. 5,000,732
4. 85,723
5. 23,985
6. 234,951
Write the following in expanded form:
7. 34
8. 5345
9. 4,000,001
10. 203,432
11. 432
12. 865,342,422
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)
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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.
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Chapter 1 Test
Name: _____________________
Date: _________
SHOW WORK
. A calculator is allowed on this test, but work still must be shown for full
credit. Only questions regarding interpreting the directions are allowed – no talking to anyone
but the teacher until all have finished and have submitted their test.
1. Put the following numbers in order from smallest to biggest:
-9, 9, -1, 1, 7, -7, 0, 2, -3, -5, 6
2. Write each number in standard form:
a. 500,000 + 1,000 + 70 + 2
b. 20,000 + 300 + 4
c. 4,000,000 + 800,000 + 2,000 + 900
3. Rewrite using algebraic symbols:
a. three less than a number B. Five more than twice a number is thirteen.
4. Fill in the blanks with either standard form or expanded form.
a. 62,723 =
(
) _ _ _ _ _ (
) _ _ _ _ _ _ (
)
6 10
7 10
3 10
4
2
0
×
+
+ ×
+
+ ×
b. ________________= (
)
(
)
3 10
9 10
2
5
×
+ ×
5. Solve:
a. x - 2 = 5
b. x -(-3) = 7
6. Circle each of the following types of numbers that best describes 4.5.
a. Real
b. Rational
c. Complex d. Transcendental
e. Integer
Bonus: Fill in the next three numbers that continues the sequence: 1, 3, 6, 10,__, __, __
5
5
5
5
5
5+0
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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.
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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?
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Name: ______________________
Date: _____
2b-Proper and Improper Fractions
Write as a mixed number:
1.
11
3
2.
24
5
3.
43
12
4.
31
8
5.
33
5
6.
56
15
7.
23
16
8.
7
3
Write as an improper fraction:
9. 2
7
8
10. 5
6
11
11. 10
2
7
12.
6
9
11
13.
2
5
7
14.
3
11
12
15.
5
3
10
16.
13
1
5
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Name: ______________________
Date: _____
2c-Adding & Subtracting Fractions of Different Denominators
Simplify the following without the use of a calculator. Leave as a proper fraction.
1.
2
3
5
12
+
2.
3
1
1
5
3
8
−
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
3
5
6
1
9
−
9.
4
2
5
9
13
15
−
10.
1
1
3
14
15
−
11. ⅞ + ⅔
12. ⅓ - ⅜
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Name: ______________________
Date: _____
2d-Multiplying & Dividing Fractions
Use your calculator only to check your answer. Leave answer as simplified proper fractions.
1.
3
4
8
9
⋅
2.
4
5
10
17
×
3.
5
6
3
10
×
4.
6
7
14
15
×
5.
8
9
27
28
×
6.
3
7
21
10
( )
7.
2
4
3
4
2
5
×
8.
−
4
5
15
24
( )
9.
3
4
3
×
10.
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
÷
19.
8
9
1
9
3
÷
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.
