CONTENTS David Hilbert, a Biographical Sketch of a Great Mathematician ........... 2-4 The Math Key ........................................................................................... 5-8 Math Vocabulary/Problem Examples .................................................... 9-13 Math Challenge/Crossword Mix Puzzle ................................................... 14 Careers Involving Math ....................................................................... 15-17 Math Challenge Answer ............................................................................ 19 Crossword Mix Answers……………………………………………………….20
Author & Editor, Justice Adams American Way Middle School Mrs. Brown-Jones/Math-7th grade
~1~
David Hilbert: A Biographical Sketch of a Great Mathematician David Hilbert (January 23, 1862 – February 14, 1943) was a German mathematician and is recognized as one of the most influential mathematicians of the 19th and 20th centuries. Hilbert was born the first of two children to Otto and Maria Therese (Erdtmann) Hilbert in the Province of Prussia, either in Königsberg, per Hilbert, or Wehlau (Znamensk since 1946), near Königsberg, where his father worked
at the time. Hilbert entered
the Friedrichskolleg
Gymnasium (Collegium
fridericianum) in the fall
of 1872. Philosopher,
Immanuel Kant, had
attended the school 140 years
earlier. Hilbert transferred
to Wilhelm Gymnasium in
the fall of 1879. He
graduated in the spring of
1880, and enrolled into
the University of Könisberg
(the “Albertina”) in the
fall of 1880. Hilbert obtained
his doctorate degree in
1885 with a dissertation
entitled Über invariante
Eigenschaften spezieller
binärer Formen,
insbesondere der
Kugelfunktionen ("On the
invariant properties of special
binary forms, in particular
the spherical harmonic
functions"), which was
written under Ferdinand von
Lindemann. Hilbert
taught and maintained
professorship at the University of Könisberg from 1886 to 1895. Hilbert’s presentation of a collection of problems in 1900 led the way for much of the 20th century’s mathematical research. Hilbert and his students significantly contributed to developing important tools that are employed in modern mathematical physics. Hilbert is known for being among the first to distinguish between mathematics and metamathematics, the study of mathematics itself utilizing mathematical methods. Metatheories, mathematical theories about other mathematical theories, are produced from this study. Hilbert met and married Käthe Jerosch (1864-1945), the daughter of a Könisberg merchant, in 1892; they had their first and only child, Franz Hilbert (1893-1969), the following year. In 1895 he obtained the position of Chairman of Mathematics at the University of Göttingen, the best
~2~
research center for mathematics in the world at the time, through the intervention of Christian Felix Klein (April 25, 1849- June 22, 1925), another German mathematician who is known for his work in complex analysis, group theory, non-Euclidean geometry, and the connections between group theory and geometry. Hilbert remained at the University of Göttingen for the remnant of his life. His social circle there entailed Amalie Emmy Noether (March 23, 1882-April 14, 1935), another influential German mathematician known for her contributions to theoretical physics and abstract algebra, and Alonzo Church (June 14, 1903-August 11, 1995), an American mathematician and logician who contributed significantly to the foundations of theoretical computer science as well as mathematical logic. Several of Hilbert’s 69 Ph.D. students went on to become famous mathematicians, including Richard Courant, Felix Berstein, Otto Blumenthal, Erich Hecke, Wilhelm Ackermann, and Hugo Steinhaus. Hilbert set forth what is recognized as the best contemplated and most successful list of 23 unsolved problems ever presented by a single mathematician at the International Congress of Mathematicians in Paris in 1900. Some of these problems have been solved, while some continue to this day to remain a challenge for mathematicians, and a few are now taken to be open-ended or to have no fixed answers that would restrict future changes. In 1899 Hilbert published Grundlagen der Geometrie (Foundations of Geometry), in which he proposes a formal set, Hilbert’s axioms, substituting the traditional axioms of Euclid. In 1931, Kurt Gödel showed, or it was accepted, as with his incompleteness theorem, that Hilbert’s notion of axiomated mathematics with definitive principles that would quench uncertainties was unfounded. Hilbert’s subsequent achievements of proof theory were established as consistent with theories of centripetal concern to mathematics, however. Insomuch as Hilbert’s work initiated the course of clarification of logic Gödel’s work led to development of recursion or computability theory, and thus mathematical logic as a distinct discipline by the 1930s. In 1934, Hilbert published Grundlagen der Mathematik, a 2-volume work on his views of the foundations of mathematics. In terms of religion, Hilbert was agnostic, and urged that mathematical truth was independent of the existence of God, though, amusingly, Paul Gordon referred to Hilbert’s resolution of Gordon’s basis theorem of the finiteness of generators for binary forms (Gordan’s Problem) as
~3~
“theology,” which is the study of God and His relationship to the universe, initially rejecting, but later accepting it. And, amusingly, many of Hilbert’s colleagues were Jewish, and he lived to see the Nazis purge many of the prominent faculty members at the University of Göttingen in 1933, including Emund Landau (February 14, 1877- February 19, 1938), a German Jewish mathematician who worked in the fields of complex analysis and number theory, as well as Emmy Noether and Hermann Weyl. Hilbert has perhaps best and unknowingly established that it is inevitable to stumble upon God, the Beginning and the End, in the ultimate quest for truth, and that perhaps the universe, however vast, is indeed finite, though constant in its growth and immeasurable by any mathematical instrument known to mankind. Perhaps, indeed, it is not impossible to learn, and, quoting Hilbert, Wir müssen wissen. Wir werden wissen. We must know. We will know.
~4~
If your assignment is not derived from your math
The Math Key
textbook, but is rather a hand-out from your instructor, you may also utilize the index and/or
The important thing to remember about math is that, like any other subject, it requires reading. In most instances it is not possible to merely look at a math problem and solve it, unless you’ve been previously taught and retain the knowledge of that skill. After exposure and even mastery of some mathematical concepts you may need to refer back to instructional material when solving similar problems after especially extended periods of time.
glossary of your math text book to locate the instructions, unless, of course, the assignment is a closed-book test or quiz. There is also a Quick Reference guide in the back of your text book that lists formulas as well as metric conversion and other charts. Additionally, there is an answer key or a Selected Answers and Solutions section in the back of your text book that gives the answers to selected math problems throughout your textbook; these should only be used to
Knowing Your Math Book
verify whether or not you have accurately solved The instructions for solving math problems usually precede or can be found by flipping back a page or a few pages from the assignments in your math text book. Some instructions can be found on the same
a problem. It also helps with solving other similar problems. You should never utilize the answer key or Selected Answers and Solutions section of your math book to simply cheat.
page as the assignment. If the instructions for a math assignment do not appear on the same page or a few pages preceding an assignment you may find them by referring to the alphabetized index and/or the glossary in the back of your math book. Both of these usually refer to page numbers where a
Utilizing Math Tools/Resources Take full advantage of any opportunities with which you may be presented to utilize math resources, like afterschool tutoring, if possible, and/or online math tutorials, preferably those that are school-based.
particular mathematical concept or operation can be found (although the glossary for the 7th grade math
Use a calculator to check and double
textbook that is utilized at American Way does not
check your answers to assigned math problems. It is
refer to any page numbers for any defined
also acceptable, for the purpose of saving time, to
mathematical terms). The glossary also exhibits
utilize a calculator to derive answers and/or perform
examples of certain mathematical concepts or
certain mathematical operations that you are
operations. The index lists subcategories of a
capable of manually solving.
particular concept or term, making them easier to find.
~5~
This acceptability is subject to instructor authorization, however, where any classroom work is concerned. Instructors often have calculators that are available for student use. It is ideal to purchase or have your parents/guardians to purchase a personal calculator for you, however, as it is not necessarily a given that your math instructor or future math instructors will have them available for use. Additionally, you can utilize the calculator on your computer by clicking the start button, “All Programs,” “Accessories,” and then “Calculator.” You may also access either of numerous online calculators via search engines like Google; to perform a Google search go to www.Google.com and type “online calculators,” in this instance. You can also utilize the calculator on your cellular phone if you have one or have access to one. Do Your Homework Simply put. Do your homework. While it may account for a smaller percentage of your grade, homework is actually where you’ll achieve or should achieve the bulk of your learning. The classroom lecture is designed to pinpoint major concepts from your reading/homework assignments, and that’s in any class. Even when you are not assigned homework, take it upon yourself to study and/or correct missed problems on your graded papers. This may require research, contacting a homework hotline, or carrying those graded papers with missed problems to tutoring sessions. Whether your instructor honors your corrections or not, you will have advanced a step in your learning process by gaining understanding of previously un-mastered material.
Study Groups Form a pattern of good study habits as well as a study group for maximum benefit. A group of your classmates and/or peers who are studying the same or similar material can be an excellent resource for solving complex problems and retaining vital information. You may arrange to meet at a local library, at either group member’s home (with parental permission and discretion, of course) and/or computer conference. Exchange telephone numbers and/or email addresses with each member of your study group and be prepared to participate as needed. Learning from Errors It is very important to heed missed problems on your graded papers. This is true of all subjects. If your instructor does not return your graded papers, ask for them, or for at least copies of them. Feedback from
~6~
your instructor is a vital aspect of your learning. Very often missed problems are resultant of carelessness or not writing problems as they appear in original text. Of course if you have not first written a problem correctly your answer won’t be accurate regardless of whether or not you have performed the necessary operations accurately. So take the time to assure that you write all problems correctly; and employ the same manner of precaution as you work through problems. Speak Up If you don’t understand something, ask questions, whether it’s in class, during a tutoring session, or during a study group meeting or conference. Never sit in silence and pretend to understand something you don’t—your instructor will never be shy of asking you questions, noting further that your lack of understanding will be reflected in your grades. So by all means ask away.
Project Tips Microsoft Excel may be utilized to create coordinate planes and crossword puzzles from scratch. Click “Insert” and “Shapes” to insert arrows and lines for arrows into coordinate planes. You can also search for templates for math crossword puzzles via the Google search engine at www.google.com. To create a cartoon using Windows Movie Maker, first either manually or electronically draw your cartoon in the numerous stages of motion or what will appear as motion in the final presentation. Your drawings should be as large as possible, as they will appear smaller in the video. Scan and save any manual drawings to your media device, i.e. a Universal Serial Bus (USB) flash or jump drive and/or the hard drive of your computer. You can use Paint to electronically draw your cartoon or to copy and paste images drawn in Microsoft Word utilizing shapes. To use Paint, click “Start” and then “Accessories” and scroll down to and click on “Paint.” To draw in Microsoft Word click “Insert” and then “Shapes” and select and maneuver the shapes to create the desired object. You will need a microphone to speak for or record voices for your cartoon(s). If you do not have a microphone, you may elect to use callouts (dialogue boxes) by clicking “Insert” and “Shapes” and scrolling down to and selecting the style of callout that you wish to use. The next step will be uploading the pictures into Windows Movie Maker. Click “Start” and “All Programs” and scroll to and click on Windows Movie Maker. Upload each picture in the order that they should appear
~7~
in the storyboard by clicking on the “pictures” link and chronologically selecting each file. Next, drag each picture into the appropriate box in the storyboard at the bottom of the page. Once you have finished adding all of the synonymous pictures, save and/or publish the project by clicking on “File” in the top left hand corner of the page and then selecting “Save As” (or “Save,” if the project has not been previously saved) and/or “Publish Project.” Select the location to which you’d like to save the file and click “Next” and then “Finish.” Finally, to share your movie, upload it to a site like YouTube at www.youtube.com or via Google at www.google.com (Note: Registering for and utilizing a Google email account privileges you to automatic YouTube access and various other Google features under the same username and password).
S-P-E-A-K Speak up and be counted. Ask questions so that you’ll learn.
Click here to see my animated cheer. http://www.youtube.com/watch?v=8ETuWyHHp5U
~8~
Math Vocabulary 1. Numerical expressions – A combination of numbers and operations. *Examples: 4 + (10 – 7) + 1
12 – 8 + (5-2)
26 ÷ 2 + (9 x 3)
4+3+1=8
4+3=7
13 + 27 = 40
2. Order of operations – The rules to follow when more than one operation is used in a numerical expression. 1. Evaluate the expressions inside grouping symbols. 2. Evaluate all powers 3. Multiply and divide in order from left to right. 4. Add and subtract in order from left to right. Note: Multiplication comes before addition or subtraction, regardless of where it appears in the equation. * Examples: 12 + (12 – 4)
18 – 5+ 43
17 + 4 · 1 X 1 = 21
12 + 8 = 20
13 + 4 x 4 (16) x 4 = 77
21 x 1 = 21
3. Variable – A symbol, usually a letter, used to represent a number in mathematical expressions or sentences. 4. Algebra (pgs. 33, 40, 43, 59, 96, 97, 106-107, 110-112, 119, 149, 184, 212, 234, 238, 296, 350, 403, 409, 465, 562, 567, 586, 676-678, 681, 684-685, 696, 705, 708, 735-736 & 740) – A branch of mathematics that involves expressions with variables. 5. Algebraic expression (pgs. 34-37) – A combination of variables, numbers, expressions, and at least one operation. Follow these steps to write an algebraic expression. 1. Words: Describe the situation. Use only the most important words. 2. Variable: Choose a variable that represents the unknown quantity. 3. Expression: Write an algebraic expression that represents your verbal description.
~9~
*Examples: Evaluate x – 3 if x = 21. x – 3 = 21 – 3 = 18
Evaluate 60 ÷ d if d is 3. 60 ÷ d = 60 ÷ 3
Evaluate 13x3 – 5 if x is 2 13 x3 – 5 = 13 (2x2x2) – 5
= 20
13 x 8 – 5 104 – 5 99
6. Coefficient (pgs. 34 & 215) – The numerical factor of a term that contains a variable. 8. Rational numbers (pgs. 124-197) – Numbers that can be written as the ratio of two integers in which the denominator is not zero. All integers, fractions, mixed numbers, and percents are rational numbers. *Examples:
9. Common denominator (pg. 134) – A common multiple of the denominators of two or more fractions. 24 is a common denominator for , , and because 24 is the LCM of 3, 8, and 4. 10. Least common denominator (LCD) (pgs. 134 & 147) – The least common multiple of the denominators of two or more fractions. You can use the LCD to compare fractions. 11. Two-step equation (pgs. 228-234) – An equation having two different operations. 12. Percent proportion (pgs. 332-336) – One ratio or fraction that compares part of a quantity to the whole quantity. The other ratio is the equivalent percent written as a fraction with a denominator of 100.
13. Direct variation? (pgs. 405-410 & 834) – The relationship between two variable quantities that have a constant ratio. 14. Non-linear function? (pg. 395) – A function for which the graph is not a straight line.
~ 10 ~
(Direct/Inverse proportions & Non-linear relationships info not found) 15. Decimals – (a.) based on the number 10 (n.) a fraction with a denominator of ten or a power of ten, shown by a point (decimal point) before the numerator.
16. Integers – any numbers from the set {…, –4, –3, –2, –1, 0, 1, 2, 3, 4, …}, where … means continues without end. A deposit of $40 = $40
35°C below zero = ˗35
A profit of $12 = $12
17. Squares – A square is the product of a number and itself. 26 x 26 676
8 x 8 = 64
18. Square roots – the factors multiplied to form perfect squares, which are numbers with square roots that are whole numbers. 100 is a perfect square because 10 is its square root. =9
= 21
19. Proportions functions – A proportion is an equation stating that two ratios or rates are equivalent.
4x = 3 x 8
1.7a = 12 x 1
1w = 4 x 10
4x = 24
1.7a = 12
1w = 40
x = 24 ÷ 4
a = 12 ÷ 1.7
w = 40 ÷ 1
x=6
a = 7.06
w = 40
~ 11 ~
20. Proportional Relationships – when a change in one of two variables always accompanies a change in the other, and if the changes are always related by use of a constant.
Cost $ of an Item Quantity
4 1
8 2
12 3
16 4
Ingredient Additive 1 Additive 2
1 2 3
2 4 6
3 6 9
4 8 12
Time (per hrs) Pay
1 12
2 3 4 24 36 48
21. Equations - An equation is a mathematical sentence that contains an equals sign, =, stating that two quantities are equal. 18 + y = 21
x–5=3
4+h=–1
y = 21 – 18
x=3+5
h = – 1+ – 4
y=3
x=8
h=–5
22. Inequalities – open sentences that use <, >, ≠, ≤, or ≥ to compare two quantities. x+2<8 –2–2 x<6
w – 5 ≥ 2.2 +5+5 w ≥ 7.2
y+3 > 2 4 –3 – 3 4 4 y >
23. Sequences – ordered lists of numbers, such as 0, 1, 2, 3 or 2, 4, 6, 8. Each number in a sequence is called a term. In an arithmetic sequence, each term is found by adding the same number to the previous term. 7, 14, 21, 28,… 35, 42, 49 (Added number 7)
21, 24, 27, 30,… 33, 36, 39 (Added number 3)
0.4, 0.6, 0.8, 1,… 1.2, 1.4, 1.6 (Added number 0.2)
24. Coordinate plane – a plane in which a horizontal* number line and a vertical number line intersect at their zero points. Also called a coordinate grid (*horizontal means parallel to the horizon; level or across, and vertical means of, relating to, or located at the vertex; upright). The origin is the point of intersection of the two number lines. Quadrants are the four sections of the coordinate plane.
~ 12 ~
The x-coordinate corresponds to a number on the x-axis, the horizontal number line. The y-coordinate corresponds to a number on the y-axis, the vertical number line. An ordered pair is a pair of numbers, i.e. (3, −5), used to locate a point in the coordinate plane.
* Coordinate Plane Example
y
B
7 6 5 4
A
3 2 1
−7 −6 −5 −4 −3 −2 −1
0
1
2 3
4
5
6
7
x
−1 −2 −3 −4
C
−5 −6 −7
A (−5, 3)
B (4, 7)
~ 13 ~
C (7, −3)
MATH CHALLENGE Tiffany purchases a dozen sharp pencils for $1.74. How much would she need to purchase three and a half dozen pencils?
CROSSWORD MIX PUZZLE 1
2
3
4
5
7
6
15
16
8
9
11
10
12
13
14
Across 1. The solution for 7−b = 4 3. The value of 44 + 6 ÷ 2 – 3 4. The exponential form of 10x10x10 5. The solution for 4a = 12 8. Evaluate 124 12.18 + 6 (12–3) – 7 x 2 13. 8 · 8 – 6 · 7
Down 2. The next three terms for the sequence 3, 6, 9 6. The next three terms for the sequence 7, 14, 21 7. x + 21 = 23 9. Solve 2x + x = 18 10. Solve the proportion 12 = 3 y 1 11. Evaluate 43 − (10−2) 14. 12 + (6–2) 15. 538 a – 28a = 1530 16. 12 = 18 8 x
~ 14 ~
Careers Involving Math The average middle school student may question the relevance of math or how, beyond measuring or counting, it may be applicable to the world outside of the classroom. Ultimately, we should think of all math as the numerical manifestation of real problems, or those math word problems that we’re taught and periodically asked to solve. School, where we’re introduced to a variety of material, however irrelevant it may seem to us now, places us in the best position to discover ourselves. It compels us to ponder our likes and dislikes, and what we’d gravitate to in terms of prospective careers. Any bachelor’s degree (and some associates or technical diplomas) could more than double one’s income prospects. There are a number of rewarding math-related careers; and, due to the lack of individuals with advanced math skills, these professions are inclined to pay very well. The National Council of Teachers of Mathematics provides information on math careers on its website. http://www.nctm.org/. There are essentially two traditional divisions of mathematics and math careers; they are applied mathematics and pure mathematics. Applied mathematicians work in all fields of engineering, science, and industry, while pure mathematicians fixate on the study of math itself. The following are some popular math-related careers: 1. Mathematicians utilize advance theories to derive and propose solutions to various issues in engineering, economics, and business, generally. A mathematician is considered to be the best of all professions, according to the Wall Street Journal. A mathematician can earn a low-end salary of $55,680, a median salary of $101,040, and a high-end salary of $152,140. Many universities require completion of math-related courses in engineering, computer science, and economics to earn a math degree. A Ph.D. is usually required to gain employment as a mathematician in the private sector, while entry-level positions with the federal government are obtainable with a bachelor’s degree in mathematics. 2. Teachers — Math teachers plan lessons, write math assignments and tests, and administer those assignments, of course. Teachers for k-12 mathematics are in high demand, and the
~ 15 ~
demand is expected to continue in growth. College math professors are hired at universities and state and technical colleges and may work as few as three hours per week at graduate universities or twelve to sixteen hours weekly at undergraduate colleges or universities. A bachelor’s degree and completion of a teacher preparation program is required to become a math teacher. Math teachers are qualified to teach in any secondary school after completing a degree. A low end salary for a math teacher is $34,600, while a median salary is $52,200, and a high end salary is $82,000. 3. Engineers use applied math and science skills to develop solutions to technical issues. Engineers might specialize in electronics, biomedicine, civil, or aerospace engineering, or other top-paying fields, like chemical, petroleum, mechanical, or computer engineering. Continuing education is very important to the field due to rapid technological advancement. A minimum of a bachelor’s degree is required for most entry-level engineering positions. Salaries for engineers vary according to the field, but are generally high paying. A median annual salary for a consulting software engineer, for example, is $123,000. 4. Economists study how society distributes resources, like land, raw materials, machinery, and labor for the production of goods and services. Economists research issues like inflation, taxes, energy costs, employment levels, etc. to predict economic trends. For many private sector jobs and advancement to more responsible positions a master’s or Ph.D degree in economics is required. Economists earn a low-end salary of $48,250, a median salary of $89,450, and a high-end salary of $155,490. 5. Computer scientists use their mathematical expertise and research to develop new computer technologies for business-related and other use. They often work alongside engineers and other specialists. Computer Scientists usually need a Ph.D. for employment, though some entry level positions with the government are open to those with bachelor’s degrees. Computer scientists earn a low-end salary of $57,630, a median salary of $100,660, and a high-end salary of $153,120. 6. Air Traffic Controllers coordinate the movement of air traffic, assuring that planes stay a safe distance apart from one another and that delays are minimized. An air traffic controller uses math to direct an aircraft on the speed and altitude that they should fly. Additionally, they must learn and work with automated instruments and special computer programs. Most air traffic controllers are employed by the Federal Aviation Administration (FAA). They work in
~ 16 ~
towers and flight service stations at airports, and in route traffic control centers. A small number work for the United States Department of Defense. An air traffic controllerâ&#x20AC;&#x2122;s salary can range from $54,480 yearly at low end to $108,040 yearly at median, or $165,000 yearly at high end. 7. Urban Planners utilize math to design the arrangement, appearance, and functionality of cities and towns. Nearly 70% of Urban Planners are employed by local governments. An increasing proportion of planners are employed in the private sector at architectural, engineering, and other related firms/companies; they are also significantly employed with management, scientific, and technical consulting agencies. Others are employed in state government in housing, transportation, and environmental protection agencies, while a few work for the federal government. A bachelorâ&#x20AC;&#x2122;s degree in mathematics, geography, political science, economics, or environmental design is particularly good preparation for the field. Most entry level jobs in local, state, and federal government require a masterâ&#x20AC;&#x2122;s degree in urban or regional planning or a similar field from an accredited university. An urban planner can earn a low-end salary of $41,040, a median salary of $64,100, and a high end salary of $98,060.
~ 17 ~
WORKS CITED “Areas of Mathematics.” Wikipedia. 29 April 2013 <http://en.wikipedia.org/wiki/Areas_of_mathematics>. “Careers.” Weusemath.org (BYU Mathematics Department). 2010-2013 <http://weusemath.org/?page_id=800>. “David Hilbert.” Wikipedia. 2013 <http://en.wikipedia.org/wiki/David_Hilbert> “Mathematics Careers.” The Mathematical Association of America. 2013 <http://www.maa.org/careers/>. Newman, Rick. “The 10 Most Overpaid Jobs: Even in this tough economy, some workers earn boffo pay for relatively easy work.” US News & World Report LP. 21 March 2013 <http://money.usnews.com/money/careers/articles/2013/03/21/the-10-most-overpaid-jobs>. “What are Popular Math Related Careers.” Degree Directory.org. 2003-2013 <http://degreedirectory.org/articles/What_are_Popular_Math_Related_Careers.html>. “Why Choose a Mathematics-Related Profession?” Department of Mathematics, UC Regents, Davis campus. 2013 <http://www.math.ucdavis.edu/~kouba/MathJobs.html>.
~ 18 ~
MATH CHALLENGE ANSWER Tiffany purchases a dozen sharp pencils for $1.74. How much would she need to purchase three and a half dozen pencils?
12x = $1.74 x 42 12 x =$73.08
Get a Jump Start on Higher Education & Your Career Attention middle schoolers!!!! Did you know that there are approximately seven colleges/universities offering dual enrollment programs that include college math courses to Memphis City high school students (the University of Memphis, LeMoyne-Owen College, Christian Brothers University, Victory University, the Tennessee Technology Center of Memphis, Tennessee State University, and Southwest Tennessee Community College http://www.mcsk12.net/de/? Now is the time for planning.
~ 19 ~
CROSSWORD MIX ANSWERS
1
3
2
3
1
4
2
5
6
5
103 3
7
2
2 6
15
1
2
5
8
16
3
12
8
1
3
8
5
2
0
7
3
6
0
2
4 9
2
6
11
35
10
12
4
2
13
9
2 14
1 6
Down
Across
2. 12, 15, 18 6. 28, 35, 42 7. 2 9. 6 10. 4 11. 35 14. 16 15. 3 16. 12
1. b = 3 3. 256 4. 103 5. 3 8. 20,736 12. 202 13. 22
~ 20 ~
2
~ 21 ~
~ 22 ~