ࡱ> km5fghijm~\H L g pbjbjVV r<r<S,Z+Z+Z+4+++h+ 2+od\67nnno===ccccccc$f}ic9Z+=|===cno])d3U3U3U=nZ+oc3U=c3U3Uah$v cou/)>6bc?d0odFbjQj cjZ+ c==3U=====ccT===od====j========= :  Natural Numbers, Whole Numbers, Integers, Rational and Irrational Numbers, Real Numbers, and Imaginary Numbers Levels 1-3, Numbers: Concepts & Properties Natural Numbers______________________________________ Whole Numbers ______________________________________ Integers_____________________________________________ Rational Numbers_____________________________________ Irrational Numbers____________________________________ Real Numbers _______________________________________ Imaginary Numbers ___________________________________      Properties of Rational Numbers Non Repeating Decimals are Rational Numbers. Why? Examples: a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c. .52 = _____ Is .743521 rational? Explain. 2. Decimals are Rational Numbers. Why? Example: Write  EMBED Equation.DSMT4 as a quotient of two integers. Let  EMBED Equation.DSMT4  Notice that the decimal repeats after the 10ths place. So we want to multiply both sides by 10.  EMBED Equation.DSMT4  Now subtract equal amounts from both sides. Remember  EMBED Equation.DSMT4    EMBED Equation.DSMT4   EMBED Equation.DSMT4  Next, solve for x.  EMBED Equation.DSMT4  Example: Write  EMBED Equation.DSMT4 as a quotient of two integers. Let  EMBED Equation.DSMT4  The decimal repeats after the 100ths place. So we want to multiply both sides by 100.  EMBED Equation.DSMT4  Subtract equal amounts from both sides. Remember  EMBED Equation.DSMT4    EMBED Equation.DSMT4   EMBED Equation.DSMT4  Next, solve for x.  EMBED Equation.DSMT4  Example: Write  EMBED Equation.DSMT4 as a quotient of two integers. Let __________ The decimal repeats after the _________ place, so we want to multiply both sides by _________. _______________ ____ EMBED Equation.DSMT4  3. What do we want to subtract from both sides? ______________ Remember  EMBED Equation.DSMT4 _____  EMBED Equation.DSMT4   EMBED Equation.DSMT4  4. Try this: Write EMBED Equation.DSMT4 as a quotient of two integers. 5. What do you think . EMBED Equation.DSMT4  is equal to? Try it! Properties of Irrationals Radicals that cannot be written as a fraction of integers. Examples: a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  6. Write your own ___________ Other Irrational numbers: Examples: a. .21211211121111 b. 723.723372333 c.  EMBED Equation.DSMT4  d.  EMBED Equation.DSMT4  EMBED Equation.DSMT4  _______________ 7. Write your own irrational number ___________________ Imaginary Numbers Notation:  EMBED Equation.DSMT4  Examples: a. 3i b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  8. Write two of your own examples A._______ B._________ 9. Which of the following is not a real number? a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  d.  EMBED Equation.DSMT4  e.  EMBED Equation.DSMT4  Homework 1 Name __________________________________ Period______________ 1. Write the following decimals as fractions of integers. a. .62 = _______ b. .931 = _______ c. .8 = _______ d. . EMBED Equation.DSMT4  = _______ e. .03 = _______ f. . EMBED Equation.DSMT4 = _______ 2. State whether each of the following is Rational or Irrational. Explain your decision. a.  EMBED Equation.DSMT4  ____________ Explain:__________________________ b.  EMBED Equation.DSMT4  _____________ Explain: __________________________ c.  EMBED Equation.DSMT4 _____________ Explain: __________________________ d. . EMBED Equation.DSMT4  _____________ Explain: __________________________ e.  EMBED Equation.DSMT4  _____________ Explain: __________________________ f. .54554555455554 _____________ Explain: _______________________ 3. State whether each of the following is Real or Imaginary. a.  EMBED Equation.DSMT4  __________________ b.  EMBED Equation.DSMT4 __________________ c.  EMBED Equation.DSMT4  EMBED Equation.DSMT4  _________________ d.  EMBED Equation.DSMT4  ________________ e. . EMBED Equation.DSMT4  __________________ f. 17i __________________ 4. Which of the following is not a rational number? A. 3.14 B.  EMBED Equation.DSMT4  C. 0 D.  EMBED Equation.DSMT4  E.  EMBED Equation.DSMT4  Properties of the Real Numbers (Basic Operations and Number Properties) Name_____________________________ Period____________ There are some basic rules that allow us to solve algebra problems. They are: The Commutative Properties The Associative Properties The Distributive Property The Identity Properties The Inverse Properties The Commutative Properties The Commutative Property of Addition tells us that the order of adding two values doesnt matter. In symbols: For any real numbers  EMBED Equation.DSMT4   EMBED Equation.DSMT4  1. Draw a diagram that shows this property. The Commutative Property of Multiplication lets us know that the order doesnt matter when we multiply two values. For any real numbers  EMBED Equation.DSMT4   EMBED Equation.DSMT4    2. The array shows a representation of the product  EMBED Equation.DSMT4 . How might this array also represent the product  EMBED Equation.DSMT4 ? 3. Explain why is there no Commutative Property of Subtraction or Commutative Property of Division? Illustrate with specific examples. The Associative Properties The Associative Property of Addition lets us know that grouping addition problems with three or more numbers in different ways does not change the sum. Formally, for any real numbers  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Draw a diagram that illustrates this property. The Associative Property of Multiplication shows that grouping factors differently does not affect the value of a product of 3 or more factors. Distributive Property In the morning Angela picked 3 bundles of 5 flowers, and in the afternoon she picked 7 bundles of 5 flowers. How many did she pick in all? Look at the diagram below to think of another way to solve the flower problem. ***** ***** Morning ***** ***** ***** ***** ***** Afternoon ***** ***** ***** You can see from the diagram that Angela picked 10 bundles during the day, so 50 flowers in all. This means that  EMBED Equation.DSMT4 . The flower example illustrates the Distributive Property. In symbols: For any real numbers  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Use the distributive property to write an equivalent expression. 6.  EMBED Equation.DSMT4  7.  EMBED Equation.DSMT4  8.  EMBED Equation.DSMT4  9.  EMBED Equation.DSMT4  10. Substitute some values for a, b, and c for  EMBED Equation.DSMT4 . Do you think this equation is true for all values of a, b, and c? Because of the relationship  EMBED Equation.DSMT4  division problems may be written as multiplication problems, and visa versa. Can you see why this would make the equation in #10 for all values? Notice that the division problem  EMBED Equation.DSMT4  can be written as the fraction  EMBED Equation.DSMT4 . With this in mind you dont have to do long division to evaluate  EMBED Equation.DSMT4  if you dont want to.  EMBED Equation.DSMT4  You can divide 124 in any convenient way you want! Try these. Remember there is no specific way to split up the numerator. 11.  EMBED Equation.DSMT4  12.  EMBED Equation.DSMT4  Another way to use the distributive property is to break up multiplication problems. For example  EMBED Equation.DSMT4  may be written as  EMBED Equation.DSMT4 , which equals  EMBED Equation.DSMT4 . Does this make the problem easier to do in your head? Try these. 13.  EMBED Equation.DSMT4  14.  EMBED Equation.DSMT4  Inverse Properties The inverse property of addition lets us know that every number has an opposite, and that when you add a number with its opposite you get zero. Formally, for any real number EMBED Equation.DSMT4 , there exist a number  EMBED Equation.DSMT4  such that  EMBED Equation.DSMT4  The inverse property of multiplication is the rule that relates multiplication and division. It tells us that for any real number  EMBED Equation.DSMT4  there is a number  EMBED Equation.DSMT4  such that  EMBED Equation.DSMT4   EMBED Equation.DSMT4  are called multiplicative inverses or reciprocals of each other. Identity Properties The identity property of addition tells us something very important about the number zero. Adding zero to any number does not change the value of the number. In symbols: For any real number  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Believe it or not, we can use this idea to make addition easier sometimes. For example:  EMBED Equation.DSMT4  can be written as  EMBED Equation.DSMT4 . (I know, it seems silly, but wait!)  EMBED Equation.DSMT4  15. Why is  EMBED Equation.DSMT4 the same as  EMBED Equation.DSMT4 ? Try these addition problems. 16.  EMBED Equation.DSMT4  18.  EMBED Equation.DSMT4  The relationship between multiplication and division  EMBED Equation.DSMT4  The Identity Property of Multiplication gives us a very important tool for solving problems. It states that you can multiply any number be 1, and the value of the number remains the same. In symbols: For any real number  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Again, these may not seem like a big deal, but it is! This property allows us to do all kinds of mathematics. Simplify:  EMBED Equation.DSMT4  (Show two ways) Now try  EMBED Equation.DSMT4  (show two ways) Simplify 19.  EMBED Equation.DSMT4  20.  EMBED Equation.DSMT4  Discuss the equation  EMBED Equation.DSMT4 ? 21. Which expression would be appropriate to complete the following equation in order for the equation to illustrate the identity property of addition:  EMBED Equation.DSMT4  F.  EMBED Equation.DSMT4  G.  EMBED Equation.DSMT4  H.  EMBED Equation.DSMT4  J. 5 + 7 K. 12 Homework Name___________________________ Period____________ Match each equation with the property it illustrates? 1.  EMBED Equation.DSMT4  ________ A. Commutative Property of Multiplication 2. 2 + 5 + (-5) = 2 + 0 ________ B. Identity Property of Multiplication 3.  EMBED Equation.DSMT4  ________ C. Associative Property of Addition 4.  EMBED Equation.DSMT4  ________ D. Commutative Property of Addition 5. 3 + (2 + 8) = 3 + (8 + 2) ________ E. Identity Property of Addition 6. 7 + 4 + 0 = 11 ________ F. Associative Property of Multiplication 7. 5(2) + 7(2) = ( 12)(2) ________ G. Inverse Property of Multiplication 8.  EMBED Equation.DSMT4  ________ H. Inverse Property of Addition Use the Properties of the real numbers to simplify each expression. Please NO calculators. 9.  EMBED Equation.DSMT4  10.  EMBED Equation.DSMT4  11.  EMBED Equation.DSMT4  12.  EMBED Equation.DSMT4  13.  EMBED Equation.DSMT4  14.  EMBED Equation.DSMT4  Order of Operations & Evaluating Expressions Level 4, Numbers: Concepts & Properties Exhibit knowledge of elementary number concepts including absolute value Level 2, Expressions, Equations, & Inequalities Order of Operations Mathematical operations should always be performed in the following order 1. Parentheses and absolute values 2. Exponents / Radicals 3. Multiplication and Division (from left to right as you encounter them) 4. Addition and Subtraction (from left to right as you encounter them) Parentheses Notation: ( ) , [ ], grouping symbols (implied parentheses) Examples: a.  EMBED Equation.DSMT4 = ________ b.  EMBED Equation.DSMT4 = _______ c.  EMBED Equation.DSMT4 _________ d.  EMBED Equation.DSMT4 = _______ 1. Try example (b) above without the parentheses. Your equation should be  EMBED Equation.DSMT4 = _____________. Do you get the same answer as the example? ________ Find the value of the following expressions: 2.  EMBED Equation.DSMT4 = ___________ 3.  EMBED Equation.DSMT4 = _________ Absolute Values Examples: a. |-8| = ______ b. |5| = ______ c. |12-18| = ______ d.  EMBED Equation.DSMT4 ________ Find the value of the following expressions: 3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  Exponents / Radicals Notation:  EMBED Equation.DSMT4 , etc. Remember to use implied parentheses with radicals. So  EMBED Equation.DSMT4 =  EMBED Equation.DSMT4  Examples: a.  EMBED Equation.DSMT4 _________ b.  EMBED Equation.DSMT4 = _________ c.  EMBED Equation.DSMT4  = ________ d.  EMBED Equation.DSMT4 = _______ Evaluate the following expressions: 4.  EMBED Equation.DSMT4 _________ 5.  EMBED Equation.DSMT4 = __________ Examples: a.  EMBED Equation.DSMT4 = ________ b.  EMBED Equation.DSMT4 = _______ c.  EMBED Equation.DSMT4  = _______ d.  EMBED Equation.DSMT4 = _______ 6. What is the value of  EMBED Equation.DSMT4 ? a. 9 b. 940 c. 27 d. -8 e. -9 Homework Name ________________________________ Period _______________ Evaluate the following expressions:  EMBED Equation.DSMT4  15.  EMBED Equation.DSMT4  A. -12 B. 6 C. 16 D. -18 E. 8 Prime Numbers (levels 3 & 4 Numbers concepts and properties) Prime Number __________________________________________________________ 1. Is 1 a prime number? Why or why not? ______________________________ 2. Is 2 a prime number? Why or why not? ______________________________ 3. List the prime numbers that are less than 30: __________________________ Composite Number ______________________________________________________ 4. List the composite numbers less than 20: _____________________________ 5. A number is divisible by 2 if _______________________________ 6. A number is divisible by 3 if _______________________________ 7. A number is divisible by 4 if _______________________________ 8. A number is divisible by 5 if _______________________________ 9. A number is divisible by 6 if _______________________________ 10. A number is divisible by 8 if _______________________________ 11. A number is divisible by 10 if ______________________________ 12. State whether each number is prime or composite a. 12 _____________ b. 59 _____________ c. 129 ____________ d. 31 ____________ Prime Factorization 13. Factoring a number means to write it as ______________________________ Examples: a. Write the prime factors of 35  35  7 * 5 So  EMBED Equation.DSMT4  b. Write the prime factors of 48 48 So 48 = __________ Write the prime factors of the following numbers: 14. 72 15. 693 16. Which of the following numbers is NOT prime? A. 43 B. 51 C. 73 D. 97 E. 101 Homework Name________________________________ Period______________ NO CALCULATOR! State whether each number is prime or composite. Justify your answer. 1. 19 2. 99 3. 52 4. 3125 5. Is 9046 divisible by 8? Explain. 6. Is 1974345 divisible by 3? Is it divisible by 5? Is it divisible by 10? Explain. Write each number in prime factored form. 7. 165 8. 124 9. 67 10. 1852 11. Which of the following numbers is prime? F. 51 G. 52 H. 53 I. 54 J. 55 Multiplication & Division Level 2, Expressions, Equations, & Inequalities - Solve simple equations using integers Multiplicative Inverse _________________________________ Example: a. What is the multiplicative inverse of 3?  EMBED Equation.DSMT4  is the multiplicative inverse because  EMBED Equation.DSMT4  b. What is the multiplicative inverse of  EMBED Equation.DSMT4 ? ____ is the multiplicative inverse because _____________ Try these: Find the multiplicative inverse of the following: 1.  EMBED Equation.DSMT4  ______ is the multiplicative inverse because ____________ 2.  EMBED Equation.DSMT4  _______ is the multiplicative inverse because ___________ Multiplication and division Recall:  EMBED Equation.DSMT4  Example:  EMBED Equation.DSMT4  Notice that the denominator did not change! 1.  EMBED Equation.DSMT4 ___________ 1. How can we use this concept to help us to solve a division problem? ____________________________________________________________ Example:  EMBED Equation.DSMT4 =  EMBED Equation.DSMT4  Try these: Solve without a calculator and leave as a mixed fraction, if necessary. Show all of your steps. 1.  EMBED Equation.DSMT4 = 2.  EMBED Equation.DSMT4 = Fractions Examples: a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4   EMBED Equation.DSMT4  EMBED Equation.DSMT4  commutative property and factoring 9 =  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  Simplify c.  EMBED Equation.DSMT4 = 1. What should be our first step? ______________________ __________________ __________________ __________________ __________________ __________________ 2. What did we do in the above examples that helped us to avoid the term canceling? ____________________________________ Try these: 3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  = 5.  EMBED Equation.DSMT4 = 6.  EMBED Equation.DSMT4  7.  EMBED Equation.DSMT4 = 8. How many curtains can be made from 20 meters of cloth if each curtain requires 2 EMBED Equation.DSMT4 meters? A. 50 B. 20 C. 12 D. 8 E. 4 Homework Name ____________________________________ Period __________________ Change the following fractions to mixed fractions without using a calculator. 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  = 3.  EMBED Equation.DSMT4 = 4.  EMBED Equation.DSMT4 = Simplify without canceling: 5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4 = 7.  EMBED Equation.DSMT4 = 8.  EMBED Equation.DSMT4  9.  EMBED Equation.DSMT4 = 10.  EMBED Equation.DSMT4  11.  EMBED Equation.DSMT4  12.  EMBED Equation.DSMT4  13.  EMBED Equation.DSMT4  A.  EMBED Equation.DSMT4  B.  EMBED Equation.DSMT4  C.  EMBED Equation.DSMT4  D.  EMBED Equation.DSMT4  E. None of these. Operations with Fractions (levels 3 & 4 Numbers: Concept & Properties) 1. What is a fraction?__________________________________ 2. Numerator ______________________ 3. Denominator_____________________ 4. Words that mean add _________________________________ 5. Words that mean subtract ______________________________ 6. Words that mean multiply ______________________________ 7. Words that mean divide ________________________________ Addition and Subtraction 8. What must fractions have in common before they can be added or subtracted?_______________________________________ Example:  EMBED Equation.DSMT4 ________ Finding a Least Common Denominator (LCD): 1. Find the prime factors of each denominator. 2. The LCD is the product of the highest occurring powers of each factor for the two numbers. Why do we want to use the Least Common Denominator? ____________________________________________________________ Example:  EMBED Equation.DSMT4  Prime factors of 16 _____________ Prime factors of 20 _____________ So now  EMBED Equation.DSMT4 __________________ = __________________ = __________________ = __________________ Example:  EMBED Equation.DSMT4  9. What is the first step? _________________________ 10. Rewrite the equation: 11. The least common denominator is __________________ 12. Solve the equation showing all of your steps. Multiplication Examples: a.  EMBED Equation.DSMT4 = b.  EMBED Equation.DSMT4  _______________ = ________________ commutative property =________________ =________________ =________________ c.  EMBED Equation.DSMT4 = 13. What is our first step? _____________________ =__________________ =__________________ =__________________ =__________________ Division Examples: a.  EMBED Equation.DSMT4 =__________________ =____________________ =____________________ =____________________ b.  EMBED Equation.DSMT4 = ____________________ = ____________________ = ____________________ = ____________________ c.  EMBED Equation.DSMT4 ____________________ = ____________________ = ____________________ = ____________________ Combinations of Operations Order of Operations 1. ______________________________________________ 2. ______________________________________________ 3. ______________________________________________ 4. ______________________________________________ Examples: a.  EMBED Equation.DSMT4  __________________ = __________________ = __________________ = __________________ = __________________ b.  EMBED Equation.DSMT4 __________________ = __________________ = __________________ = __________________ = __________________ 14. What is the sum of the fractions  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4  A.  EMBED Equation.DSMT4  B.  EMBED Equation.DSMT4  C.  EMBED Equation.DSMT4  D.  EMBED Equation.DSMT4  E.  EMBED Equation.DSMT4  15. When completely simplified,  EMBED Equation.DSMT4  F.  EMBED Equation.DSMT4  G.  EMBED Equation.DSMT4  H.  EMBED Equation.DSMT4  J.  EMBED Equation.DSMT4  K.  EMBED Equation.DSMT4  16.  EMBED Equation.DSMT4  A.  EMBED Equation.DSMT4  B.  EMBED Equation.DSMT4  C.  EMBED Equation.DSMT4  D.  EMBED Equation.DSMT4  E.  EMBED Equation.DSMT4  Homework Name ________________________________ Period _______________ 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4 = 3.  EMBED Equation.DSMT4 = 4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4 = 7.  EMBED Equation.DSMT4  A. 8 B.  EMBED Equation.DSMT4  C.  EMBED Equation.DSMT4  D.  EMBED Equation.DSMT4  E.  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Ratios, Proportions, and Percentages Level 2, Numbers: Concepts & Properties Recognize equivalent fractions and fractions in lowest terms Level 4, Basic Operations & Applications Solve routine two-step or three-step arithmetic problems involving concepts such as rate and proportion, tax added, and percentage off. Ratio __________________________ Notation : (colon) or  EMBED Equation.DSMT4  means to 6:1 is read 6 to 1 This can also be written as  EMBED Equation.DSMT4 . Notice that we do not simplify this to equal 6. We leave the 1 in the denominator!  EMBED Equation.DSMT4  is read as 2 to 5 This can also be written as __________ Example: In a classroom there are 25 students total 15 girls 10 boys The ratio of girls to boys is 15:10 or 3:2 when reduced We could also write this ratio as  EMBED Equation.DSMT4  Write the ratios of the following in two ways. Remember to reduce. 1. Girls to Total Students __________ ____________ 2. Total students to Boys __________ ____________ Proportions _____________________________________ Example  EMBED Equation.DSMT4  To solve this we can use __________________.  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  simplify 3. Can you think of another way to solve this problem without using cross multiplication? Show your work below. Example  EMBED Equation.DSMT4  _________________ _____________________ __________________ _____________________ __________________ _____________________ __________________ _____________________  EMBED Equation.DSMT4 _______ Try these: Solve the following proportions 4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  For each of the following, state whether or not the two fractions are equivalent. Show your work! 6.  EMBED Equation.DSMT4  7.  EMBED Equation.DSMT4  Applications of Proportions Examples: a. Find the length of x.  Set up a proportion: Solve the proportion: b. Jessica found a recipe for her favorite Toll House Chocolate Chip cookies but the recipe makes too many cookies. Jessica wants to make only  EMBED Equation.DSMT4 of the recipe. Find the amount of each ingredient that Jessica needs to use. (This recipe really works!!) Original recipe: Reduced recipe: 6 cups all-purpose flour 3 teaspoons baking soda 3 teaspoons salt 3 cups (6 sticks) butter, softened 2 cups granulated sugar 2 cups packed brown sugar 3 teaspoons vanilla extract 6 large eggs 6 cups chocolate chips 3 cups chopped nuts Percentages Percentage problems can be written as a comparison of equal proportions. Formula:  EMBED Equation.DSMT4  or  EMBED Equation.DSMT4  Where base is the total quantity, amount is the portion of the base, and part is the part of 100. Example: 15 is 50% of 30 Set up the proportion using the formula  EMBED Equation.DSMT4  8. Are  EMBED Equation.DSMT4  equivalent fractions? Show why or why not below. 9. Can you think of another way to set up this problem without using proportions? Be prepared to share with your classmates. Example: Find 15% of 600. 10. What are we going to put in place of our unknown quantity?________  EMBED Equation.DSMT4  _________ x = _____ Example: A newspaper ad offered a set of tires at a sale price of $258. The regular price was $300. What percent of the regular price was the savings? What is our unknown quantity now? _______  EMBED Equation.DSMT4  _________ x = ______ Try these: 11. What is 48.6% of 19? 12. 12% of what number is 3600? 13. The interest in 1 year on deposits of $11,000 was $682. What percent interest was paid? 14. What Percent of 24 is 18? A. 75% B. 150% C. 25% D. 33 EMBED Equation.DSMT4 % E. 133 EMBED Equation.DSMT4 % Homework Name ________________________________ Period__________________ 1. Sovle :  EMBED Equation.DSMT4  2. Solve:  EMBED Equation.DSMT4  3. What percent of 48 is 96? 4. 25% of what number is 150? 5. What is 26% of 480? 6. 35% of 430 is what number? 7. If 6 gallons of premium unleaded gasoline cost $11.34, how much would it cost to completely fill a 15-gallon tank? 8. If sales tax on a $16.00 compact disc is $1.32, how much would the sales tax be on a $120.00 compact disc player? 9. If the sales tax rate is 6.5% and I have collected $3400 in sales tax, how much were my sales? 10. A used automobile dealership recently reduced the price of a used compact car from $18,500 to $17,020. What is the percentage decrease from the old price to the new price? 11. Corey received 10 toys for his birthday and 12 toys for Christmas. By what percent did the number of toys increase? Show your work!! A. 10% B. 12% C. 20% D. 2% E. 16 EMBED Equation.DSMT4 % Simplifying Expressions Level 3, Expressions, Equations, & Inequalities Level 4, Numbers: Concepts & Properties - Exhibit knowledge of elementary number concepts including greatest common factor Order of Operations Mathematical operations should always be performed in the following order 1. Parentheses and absolute value 2. Exponents / Radicals 3. Multiplication and Division (from left to right as you encounter them) 4. Addition and Subtraction (from left to right as you encounter them) Parentheses Notation: ( ) , [ ], { } grouping symbols (implied parentheses) 1. What do we do when parentheses are nested within other parentheses or brackets? __________________________________________________ Ex.  EMBED Equation.DSMT4 = ________________ ________________ ________________ ________________ Try this: 2.  EMBED Equation.DSMT4  = Exponents / Radicals Notation:  EMBED Equation.DSMT4 , etc. Distributive Property ___________________________________________ Example:  EMBED Equation.DSMT4 = ___________________ 3.  EMBED Equation.DSMT4 = 4.  EMBED Equation.DSMT4 = Factoring Greatest Common Factor ____________________________ Find the greatest common factor for each set of numbers: 5. 30, 45 6. 72, 120, 432 7.  EMBED Equation.DSMT4  Example: Factor  EMBED Equation.DSMT4  _____________ _____________ Distributive property Factor the following: 8.  EMBED Equation.DSMT4  9.  EMBED Equation.DSMT4  10.  EMBED Equation.DSMT4  Combining Like Terms Remember  EMBED Equation.DSMT4  11. What are like terms? _______________________________ 12. Which of the following are like terms:  EMBED Equation.DSMT4 . Group the like terms together. __________________________________ 13. What can we do with like terms when we are simplifying expressions? ________________________________ Example:  EMBED Equation.DSMT4 = _________________ = _________________ 14.  EMBED Equation.DSMT4  15.  EMBED Equation.DSMT4  = 16.  EMBED Equation.DSMT4  17.  EMBED Equation.DSMT4 = 18. Can #17 be simplified in two different ways? Explain. __________________________________ 19. What is the simplified form of  EMBED Equation.DSMT4  A.  EMBED Equation.DSMT4  B.  EMBED Equation.DSMT4  C.  EMBED Equation.DSMT4  D.  EMBED Equation.DSMT4  E.  EMBED Equation.DSMT4  Homework Name ______________________________________ Period ____________________ 1. List the order of operations (in order of first to last): _______________________________________________________________ Find the value of each expression. 2.  EMBED Equation.DSMT4 = 3.  EMBED Equation.DSMT4 = Factor the following: 4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  7.  EMBED Equation.DSMT4  8. Group like terms together:  EMBED Equation.DSMT4  Simplify 9.  EMBED Equation.DSMT4  10.  EMBED Equation.DSMT4  11.  EMBED Equation.DSMT4  12.  EMBED Equation.DSMT4  13.  EMBED Equation.DSMT4  14.  EMBED Equation.DSMT4  15.  EMBED Equation.DSMT4  A. -12 B. 6 C. 16 D. -18 E. 8 Evaluating Expressions Name_______________ level 3 Period__________ Variables as place holders Write a verbal expression that is equivalent to the algebraic expression. Example: Algebraic Expression:  EMBED Equation.DSMT4  Verbal expression: Five more than the product of 2 and some value x. 1.  EMBED Equation.DSMT4 , ________________________________________________________________________ 2.  EMBED Equation.DSMT4  ________________________________________________________________________ 3.  EMBED Equation.DSMT4  Why do we use two different variables? 4. 5xy 5.  EMBED Equation.DSMT4  _______________________________________________________________________ 6.  EMBED Equation.DSMT4  7.  EMBED Equation.DSMT4  8.  EMBED Equation.DSMT4  Substitution property: If a = b then:  EMBED Equation.DSMT4  example:  EMBED Equation.DSMT4  10. Your example_______________  EMBED Equation.DSMT4  example:  EMBED Equation.DSMT4  11. Your example________________ 12. Can you use other operations to illustrate the substitution property? If so, show some examples Using the substitution property to evaluate expressions If x = 3 then: 13. What is the value of  EMBED Equation.DSMT4 ? _________________________ 14. What is the value of  EMBED Equation.DSMT4 ?_________________________ 15. Evaluate  EMBED Equation.DSMT4  ________________________________ Given x = -3 and y = 2, evaluate the following expressions. 16.  EMBED Equation.DSMT4  _________________ 17.  EMBED Equation.DSMT4  _________________ 18.  EMBED Equation.DSMT4  _________________ 19.  EMBED Equation.DSMT4  _________________ 20.  EMBED Equation.DSMT4  _________________ ACT problem: If x = -2 and y = 3, then  EMBED Equation.DSMT4  A. 16 B. -34 C. -38 D. 20 E. 144 Homework Name________________________ Period______ Write verbal expressions for each of the algebraic expressions. 1.  EMBED Equation.DSMT4  ________________________________________________________________________ 2.  EMBED Equation.DSMT4  ________________________________________________________________________ 3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  _______________________________________________________________________ 6.  EMBED Equation.DSMT4  7.  EMBED Equation.DSMT4  8.  EMBED Equation.DSMT4  Given  EMBED Equation.DSMT4  and EMBED Equation.DSMT4 , evaluate the following expressions. 9.  EMBED Equation.DSMT4  ________________________ 10.  EMBED Equation.DSMT4  ________________________ 11.  EMBED Equation.DSMT4  ________________________ 12. If x = 3 and y = -4, then  EMBED Equation.DSMT4  A. 52 B. -44 C. -52 D. 148 E. 44 Solving Single Variable Equations Level 4, Expressions, Equations, & Inequalities - Solve routine first-degree equations Additive Inverse ____________________________________ Examples: Find the additive inverse of the following a.  EMBED Equation.DSMT4   EMBED Equation.DSMT4  b. 7  EMBED Equation.DSMT4  Try these find the additive inverse of each: 1. 30 the additive inverse is________ because ____________ 2.  EMBED Equation.DSMT4  the additive inverse is _______ because ____________ Multiplicative Inverse _______________________________________ Examples: Find the multiplicative inverse of the following a. 4  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4   EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Try these find the multiplicative inverse for each: 3. 13 the multiplicative inverse is ______ because ____________ 4.  EMBED Equation.DSMT4  the multiplicative inverse is ______ because ____________ 5.  EMBED Equation.DSMT4  the multiplicative inverse is ______ because ____________ Solution Set ___________________________________ Notation: { } Single Step Equations Remember to use additive inverses or multiplicative inverses as needed. Example: a.  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  Try these. Find the solution set for each problem. 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  Distributive Property ___________________________________________ Example a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4 = ____________ = _________ Try these. Use the distributive property. 4.  EMBED Equation.DSMT4  ____________ = _________ 5.  EMBED Equation.DSMT4 = ____________ = _________ Combining Like Terms Remember  EMBED Equation.DSMT4  Examples: Simplify the following expressions a.  EMBED Equation.DSMT4   EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  ______________ because ________________ ___________ Simplify Multi-step Equations Example: A.  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  The solution set is {4} B.  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  What is the next step? _____________________ _________________ _____________________ _________________ _____________________ _________________ _____________________ _________________ _____________________  EMBED Equation.DSMT4  The solution set is {____} Try these. Find the solution set for each equation. 6.  EMBED Equation.DSMT4  7.  EMBED Equation.DSMT4  8. What is the solution set of  EMBED Equation.DSMT4  A.  EMBED Equation.DSMT4  B.  EMBED Equation.DSMT4  C.  EMBED Equation.DSMT4  D.  EMBED Equation.DSMT4  E.  EMBED Equation.DSMT4  Homework Name _____________________________ Period ________________ 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  7.  EMBED Equation.DSMT4  8.  EMBED Equation.DSMT4  9.  EMBED Equation.DSMT4  10.  EMBED Equation.DSMT4  11. Find the solution set:  EMBED Equation.DSMT4  Show your work!! A.  EMBED Equation.DSMT4  B.  EMBED Equation.DSMT4  C.  EMBED Equation.DSMT4  D.  EMBED Equation.DSMT4  E. { } Variables, Expressions, Word Problems (level 3) Name_______________ Period__________ Unknowns Identify the unknowns in the following sentences: 1. What is five more than 6? __________________ 2. She is known for her math skills. __________________ 3. Eight more than some value is 48. __________________ 4. Where on the number line is EMBED Equation.DSMT4 ? __________________ 5. Find the time between 1:00 and 2:00 where the minute and hour hands of a clock are in the same position. __________________ Variables Definition: ____________________________________________________________ Purposes: _____________________________________________________________ _______________________________________________________________ _______________________________________________________________ For each of the following relationships a) write a verbal description b) write a formula. 1. The area of a triangle. a) b) 2. The perimeter of a rectangle a) b) Find the price a shirt before tax if it costs $43 including 6% sales tax. Write the following expressions algebraically. 1. 5 less than some number 2. 3 more than a value 3. The product of two different numbers. 4. 7 times the sum of and number and 4 5. Six less than the product of 3 and some number. 6. The quotient of eight and a number is subtracted from the product of five and the same number. Equal Words: Ex: equals is the same as Can you think of some others? Equations: What number when divided by two is 7 less than that same number? When the product of a number and 5 is decreased by 4, the number is tripled. Find the number. What happens to the area of a circle when its diameter is multiplied by 6? ACT problem: If 2 less than five times a certain number is 1 more than twice the same number, which equation can be used to find the number? A.  EMBED Equation.DSMT4  B.  EMBED Equation.DSMT4  C.  EMBED Equation.DSMT4  D.  EMBED Equation.DSMT4  E.  EMBED Equation.DSMT4  Checking Solutions (level 3) Name________________________ Period__________ A value or values that make an equation true when substituted for unknowns are called solutions. It is common to say that solutions satisfy an equation. Since 2 is the solution to 3x + 7 = 13, 2 satisfies the equation 3x + 7 = 13. 1. What value makes x + 3 = 5 true? _____ Are there any other values?_____________ 2. Is 2 a solution to  EMBED Equation.DSMT4 ________Explain______________________________ 3. Is 3 a solution to EMBED Equation.DSMT4 _______Explain_______________________________ 4. Does -2 satisfy  EMBED Equation.DSMT4  ______Explain________________________________ 5. Is the ordered pair (5, -1) a solution the linear equation x + y = 4? 6. Find three other ordered pairs that satisfy x + y = 4? 7. Plot the points from #5 & 6 on the coordinate axis using an appropriate scale.   8. What appears to be happening? 9. How many points satisfy x + y = 4? 10. A graph is the mapping of all of the points on a coordinate system that satisfy an equation. Sketch the graph of x + y = 4 above where you plotted the points. 11. Find a point that does not satisfy x + y = 4, then plot it on the coordinate system. Describe the position of this point. _______________________________________ 12. Which of the following ordered pairs satisfies the equation 3x - 2y = 5? F.  EMBED Equation.DSMT4  G.  EMBED Equation.DSMT4  H.  EMBED Equation.DSMT4  J.  EMBED Equation.DSMT4  K.  EMBED Equation.DSMT4  Homework Name___________________________ Period_______ While taking the mathematics section of the ACT, Doug solved several equations. The following are some of them. Check his work to see if his answers were correct. Show the work that leads you to your conclusion. 1. What is the solution set of the following system of equations?  EMBED Equation.DSMT4  Dougs solution: H. {(4, -1)} Correct ____________ Not Correct___________ 2. What is the solution set for  EMBED Equation.DSMT4 ? Dougs solution: J.  EMBED Equation.DSMT4  Correct ____________ Not Correct___________ 3. What is the solution set of the equation  EMBED Equation.DSMT4 ? Dougs answer: K.  EMBED Equation.DSMT4  Correct ____________ Not Correct___________ 4. What is the solution set of the equation  EMBED Equation.DSMT4 ? Dougs answer: B.  EMBED Equation.DSMT4  Correct ____________ Not Correct___________ 5. Which of the following points lie on the graph of circle whose equation is EMBED Equation.DSMT4 ? Dougs answer: D.  EMBED Equation.DSMT4  Correct ____________ Not Correct___________ Absolute Value Inequalities (level 4) Name_____________________________ Period___________ Write the meanings of each symbol > _______________________ < _______________________  EMBED Equation.DSMT4  _______________________  EMBED Equation.DSMT4  _______________________ Graph the following inequalities on the number lines provided.  x > -3  x  EMBED Equation.DSMT4   x < 5   EMBED Equation.DSMT4  What happens when you multiply or divide both sides of an inequality by a negative value? How do you adjust the inequality to retain the same meaning? Additive Inverses What is the additive inverse of 3? ________________________ What do additive inverses have in common with each other? How do additive inverses relate to the number 0?  EMBED Equation.DSMT4  means the distance a is from 0. Why is the value of  EMBED Equation.DSMT4  always a positive number? Evaluate 1.  EMBED Equation.DSMT4  = 2.  EMBED Equation.DSMT4  3. Now consider  EMBED Equation.DSMT4 . What makes  EMBED Equation.DSMT4  different from the absolute value of a known value? (Hint: Try some values out for x) 4. Can you make a general statement about  EMBED Equation.DSMT4 ? 5. What value(s) of x will make the equation  EMBED Equation.DSMT4  true? 6. What is a number that satisfies the inequality  EMBED Equation.DSMT4 ? 7. How about a number that makes  EMBED Equation.DSMT4  true? 8. Try your solutions from questions 6 and 7 in the inequality  EMBED Equation.DSMT4 . Do they satisfy this inequality also? Explain? Show the solution of  EMBED Equation.DSMT4  on a number line. Since a number cannot be both greater than 17 and less than -17, the two different solution sets must be joined with the word or. Write the solution set of  EMBED Equation.DSMT4 . 9. To contrast the difference of the solution of  EMBED Equation.DSMT4  to that of  EMBED Equation.DSMT4  lets reverse the inequality symbols on the inequalities shown in problems 6 and 7. Change  EMBED Equation.DSMT4  to  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  to  EMBED Equation.DSMT4 . a) Find a solution to  EMBED Equation.DSMT4 . b) Find a solution to  EMBED Equation.DSMT4 . Now substitute the solutions from a) and b) in for x for the inequality  EMBED Equation.DSMT4 . c) What did you find out? d) Draw a number line to show the solution of  EMBED Equation.DSMT4 . 10. Is it possible for a number to satisfy  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 ? 11. What word would use to join the solution sets of  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  to form the solution set of  EMBED Equation.DSMT4 ? 12. Can you think of another way to write  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4  without the word and? Note:  EMBED Equation.DSMT4 means  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4  or in words all of the values between -3 and 3. Why is the notation  EMBED Equation.DSMT4  an improper way to write  EMBED Equation.DSMT4 or EMBED Equation.DSMT4 ? Solve the following inequalities: 13.  EMBED Equation.DSMT4  14.  EMBED Equation.DSMT4  Think about the definition of absolute value then consider the solutions to the following inequalities: 15.  EMBED Equation.DSMT4  16.  EMBED Equation.DSMT4  17. Comment on problems 15 and 16. 18. Use the solution from problem 13 to find the solution set of  EMBED Equation.DSMT4 . 19. Use the solution from problem 14 to find the solution set of  EMBED Equation.DSMT4 . Solve and graph the absolute value inequalities. 20.  EMBED Equation.DSMT4  21.  EMBED Equation.DSMT4  22.  EMBED Equation.DSMT4  23.  EMBED Equation.DSMT4  Homework Name_________________________ Period_____________ Find and graph the solution set of each of the following absolute value inequalities. 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4   3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4   5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4   7.  EMBED Equation.DSMT4  8.  EMBED Equation.DSMT4  Products and Powers of Polynomial Expressions (level 4) Name__________________________ Period________________ Arrays: One way to represent the product of two numbers is to draw a rectangular array. For example the product  EMBED Equation.DSMT4  can be illustrated with the following array:    This array shows that 3 groups of 2 are the same as 2 groups of 3. The Product is 6. Some people do two-digit multiplication using a similar method. For example, consider the product:  EMBED Equation.DSMT4    SHAPE \* MERGEFORMAT  Fill in the chart with the appropriate products. Use the diagram to find the product  EMBED Equation.DSMT4 . Likewise the product of two binomials can be obtained by using arrays. For example, the product  EMBED Equation.DSMT4  can be found with the following array.    Fill each blank with the appropriate expression or value, then combine like terms to obtain the product  EMBED Equation.DSMT4 .  Now lets find the product of  EMBED Equation.DSMT4      EMBED Equation.DSMT4  Now try  EMBED Equation.DSMT4  using the array provided below.    EMBED Equation.DSMT4 = What is the simplified form of the expression EMBED Equation.DSMT4 ? A.  EMBED Equation.DSMT4  B.  EMBED Equation.DSMT4  C.  EMBED Equation.DSMT4  D.  EMBED Equation.DSMT4  E.  EMBED Equation.DSMT4   Homework Name___________________________ Period__________________ Use the provided arrays to find the following products or powers. 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4   3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  Find the following products or powers. 7.  EMBED Equation.DSMT4  8.  EMBED Equation.DSMT4  9. A similar technique can be applied to this product EMBED Equation.DSMT4  10.  EMBED Equation.DSMT4  11.  EMBED Equation.DSMT4  12. Compare and comment on the results of problems 10 and 11. Factoring Polynomials (level 5) Quadratics of the form  EMBED Equation.DSMT4  NOTE: The first thing you need to look for is common factors. Example:  EMBED Equation.DSMT4  ______________ prime factors of 1 and -4 ______________ the two factors that add up to bx ______________ substituting for bx ______________ factor by grouping ______________ factored form of  EMBED Equation.DSMT4  Example:  EMBED Equation.DSMT4  ______________ ______________ ______________ ______________ ______________ Try these: 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  Difference of Squares  EMBED Equation.DSMT4  EMBED Equation.DSMT4  Example:  EMBED Equation.DSMT4  _______________ _______________ _______________ _______________ Example:  EMBED Equation.DSMT4  _______________ _______________ _______________ _______________ Try these: 3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5. Factor completely over integers:  EMBED Equation.DSMT4  A. (2x 5)(2x 3) B. (4x + 5)(x 3) C. (2x + 3)(2x 5) D. (2x 15)(2x + 1) E. (2x 3)(2x + 5) Homework Name ___________________________________ Period ____________ 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  7.  EMBED Equation.DSMT4  8.  EMBED Equation.DSMT4  9. What is the completely factored form of  EMBED Equation.DSMT4 . A. (3m 3n)(m + n) B. (3m n)(m + 3n) C. (3m + n)(m 3n) D. 3(m n)(m + n) E. (m + 3n)(3m n) Rectangular Coordinate System (levels 3,4,& 5) Name______________________ Period_____________ Who is Ren Descartes? Graph the coordinate of 5 on the horizontal number line.    Graph the coordinate of 3 on the vertical number line.  Ordered Pair (5, 3) is called an ordered pair. The name ordered pair is appropriate in that the order of the numbers matters! The graph of the ordered pair (5,3) is a particular point in a defined plane which we call The Cartesian Coordinate System. The graph can be obtained by using the two graphs we made above when we intersect them at 0. This point of intersection is called the origin. It is the graph of the ordered pair (0,0).  Are (5, 3) and (3, 5) the same point? Explain. Use an appropriate scale to graph the following points: A(3,-2), B(-4,1), C(-1,-5), D(3,4), E(0,4), F(3,0),G(0,-2), and H(-3,0). Quadrants:  Characterize a point in quadrant I._________________ Characterize a point in quadrant II_________________ Characterize a point in quadrant III________________ Characterize a point in quadrant IV________________ The points where graphs cross the y-axis are called y-intercepts. What is common to all y-intercepts?________________________________________ The points on the x-axis where graphs cross are called by three names: x-intercepts, roots, or zeros. The fact that there are three names for these points suggests their importance in mathematics. What is common to all x-intercepts?_________________________________________ Find the x-intercept and y-intercept of the linear equation  EMBED Equation.DSMT4 . x-intercept__________ y-intercept__________ The numbering of the quadrants also suggests a direction of rotation? Can you give another name for a positive rotation?________________________________________ What is another description of a negative rotation?_____________________________ What is the number of degrees in one complete revolution?______________________ A transformation maps an initial image called a preimage onto a final image called an image. Transformations include rotations, translations, and reflections. The reflection of (3, 4) over the x axis is (3, -4). Which point is the image?________________, the preimage?______________ What is the image of the reflection of (3, 4) over the y axis?_____________________ When (3, 4) is rotated clockwise  EMBED Equation.DSMT4  what is the image?_____________________ Another name for a translation is a slide. A translation is a mapping in a straight direction. It can be vertical, horizontal, or a combination of both. What is the image of the transformation of (3,4) after a translation of 5 units up, and 2 units left? _________________ If a translation of (2, 3) results in an image of (7, -2), what would be the image of (6, -1) be under the same translation?____________________________ Describe the translation of (2, 3) to (7, -2)______________________________________ ________________________________________________________________________ In the rectangular coordinate system, the point associated with the ordered pair (-4, 0) is located in which quadrant? A. I B. II C. III D. IV E. None of these Homework Name_______________________ Period__________ Match each point with the word that describes its position on the coordinate system. 1. (0, 5) ________ A. quadrant I 2. (-3, -4) ________ B. quadrant IV 3. (2, 0) ________ C. quadrant II 4. (7, -3) ________ D. x-intercept 5. (-4, 17) ________ E. quadrant III 6. (25, .01) ________ F. y-intercept 7. Find the image of a  EMBED Equation.DSMT4 counterclockwise rotation of (-4, 5). ____________ 8. Find the image of the reflection of (-4, -7) across the x-axis. ____________ 9. Find the x-intercept of the graph of the linear equation EMBED Equation.DSMT4 . ____________ 10. The image of a  EMBED Equation.DSMT4 rotation of (-2, 5) lies in which quadrant? ____________ 11. Find the image of the reflection (a, b) across the y-axis. ____________ 12. Find the y-intercept of the graph of the linear equation 3x + y = 4. ____________ 13. The ordered pair (-4, 0) lies in which quadrant: A. IV B. III C. II D. I E. none of these Types of Equations (level 4) Name___________________________ Period____________ Polynomial Equations Monomial terms in one variable Linear Equations Example Standard Form ______________________ ____________ Slope-Intercept form ______________________ ____________ Double intercept form ______________________ ____________ Point-Slope form ______________________ ____________ Graph  EMBED Equation.DSMT4   EMBED Equation.DSMT4   Quadratic Equations Polynomial of degree 2 ______________________ ____________ Standard form ______________________ ____________ Vertex form ______________________ ____________ Characteristics of a quadratic equation: Sketch the shape of a quadratic equation. Higher Order Polynomials Sketch the possible shapes of each: 2nd degree 3rd degree 4th degree 5th degree   Conic Sections Ellipses Circles Hyperbolas Form: __________________ ____________________ ________________ Example:________________ ____________________ ________________ Shapes: (sketch) Identify the type of equation as linear, quadratic, a circle, an ellipse, or a hyperbola. 1.  EMBED Equation.DSMT4  _____________________ 2.  EMBED Equation.DSMT4  _____________________ 3.  EMBED Equation.DSMT4  _____________________ 4.  EMBED Equation.DSMT4  _____________________ 5.  EMBED Equation.DSMT4  _____________________ 6.  EMBED Equation.DSMT4  _____________________ 7.  EMBED Equation.DSMT4  _____________________ 8.  EMBED Equation.DSMT4  _____________________ Identify the type of equation indicated by the follow sketches. A._____________ B. _______________ C. _______________  D. ________________ E. ______________________ F.________________  Which of the following is NOT a quadratic equation in one variable? F.  EMBED Equation.DSMT4  G.  EMBED Equation.DSMT4  H.  EMBED Equation.DSMT4  J.  EMBED Equation.DSMT4  K.  EMBED Equation.DSMT4  Homework Name_______________________ Period________ Identify each of the equations as linear, quadratic, a circle, an ellipse, or a hyperbola. 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  7.  EMBED Equation.DSMT4  8.  EMBED Equation.DSMT4  9.  EMBED Equation.DSMT4  10. Which of the following is a non-linear equation? A.  EMBED Equation.DSMT4  B.  EMBED Equation.DSMT4  C.  EMBED Equation.DSMT4  D.  EMBED Equation.DSMT4  E.  EMBED Equation.DSMT4  Systems of Linear Equations in Two Variables (level 6) Name__________________ Period________ Graphical Solution:__________________________________________ Solve:  EMBED Equation.DSMT4 graphically.  Solving systems of equations graphically is usually not practical, but the process illustrates an important fact: The solution is common to both equations. In other words, the x and y values must satisfy both equations so that  EMBED Equation.DSMT4  is the solution. This idea leads us to two other ways to solve systems of equations, substitution and elimination. Substitution: Lets do the same problem again using the idea that the solution is common for both equations. Since the (x, y) is the same, we can solve for either variable in one equation, then substitute the equivalent expression into the other equation like this: Since  EMBED Equation.DSMT4  in the first equation, we can replace the y in the second equation with EMBED Equation.DSMT4 .  EMBED Equation.DSMT4    EMBED Equation.DSMT4  We get an equation on one variable.  EMBED Equation.DSMT4  Distribute to get:  EMBED Equation.DSMT4  Combine like-terms:  EMBED Equation.DSMT4  Isolate x:  EMBED Equation.DSMT4  Since 3 is the solution for x in both equations, we can substitute 3 back into the other equation.  EMBED Equation.DSMT4  The solution:  EMBED Equation.DSMT4  Lets try this one:  EMBED Equation.DSMT4  We can substitute for either variable, but lets all substitute for x in  EMBED Equation.DSMT4  so that we are all doing this problem basically the same way. 1. Is there any advantage for this substitution? 2. Why should we put  EMBED Equation.DSMT4  in parenthesis when we make the substitution? 3. Re-write  EMBED Equation.DSMT4  with the substitution for x then solve for x. 4. How do you get the y value for the solution? 5. How can you check your answer? Elimination: Since one side of a true equation must be equal to its other side, equivalent operations performed to both sides results in an equivalent true equation. The principle in the above box enables us to solve systems of equations another way, by elimination. Note that our original system  EMBED Equation.DSMT4  may be written as  EMBED Equation.DSMT4 . 6. What changed? What was done to the original equation? 7. Why if we add  EMBED Equation.DSMT4 would it have to be equal to  EMBED Equation.DSMT4 ? 8. When the left side of the equations,  EMBED Equation.DSMT4  are added, and the right side values are added,  EMBED Equation.DSMT4 , we get another equation.  EMBED Equation.DSMT4  Write the resulting equation then solve it. 9. Why did the expressions in x go away? 10. Now that we know the y value for the solution, what can we do to find the x? 11. Write the solution. ___________________. Now lets solve another system of equations.  EMBED Equation.DSMT4  12. How is this problem different from  EMBED Equation.DSMT4 ? 13. Consider the system  EMBED Equation.DSMT4 . What could we multiply both sides of equation  EMBED Equation.DSMT4  by so that the y variables would be eliminated when the left sides are added together? 14. If we multiply both sides of  EMBED Equation.DSMT4 by 4 we get  EMBED Equation.DSMT4 . The equivalent system is now  EMBED Equation.DSMT4 . 15. Add the left and right sides then solve the system for x. 16. How could you find the y value? 17. Back to the original problem EMBED Equation.DSMT4 . What could be done to eliminate the x expressions? There are many ways to get opposites that add to zero, but one easy way is shown below:  EMBED Equation.DSMT4  The equivalent system is:  EMBED Equation.DSMT4  18. Solve for y. We have looked at three ways to solve systems of equations. Which would you use for each of the systems shown below? 19.  EMBED Equation.DSMT4  20.  EMBED Equation.DSMT4  21. ________________ _______________ __________________ There are three possible scenarios for systems of equations: ONE SOLUTION INFINITLY MANY SOLUTIONS NO SOLUTION   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  Homework Name______________________ Period_____________ Solve by the method of your choice. 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  Simplifying Radical Expressions (level 5) Name_____________ Period____  Perfect Squares Perfect Cubes Perfect fourth powers  EMBED Equation.DSMT4  Product Property of Radicals If  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  are real numbers, and  EMBED Equation.DSMT4  is an integer, then  EMBED Equation.DSMT4 6 Simplifying a Radical Expression Ex 1:  EMBED Equation.DSMT4  Ex 2:  EMBED Equation.DSMT4  Ex 3:  EMBED Equation.DSMT4  you try:  EMBED Equation.DSMT4  Step 1: Write each factor of the radicand as the product of two factors, one of which is a perfect power of the index. Ex 1:  EMBED Equation.DSMT4  Ex 2:  EMBED Equation.DSMT4  Ex 3:  EMBED Equation.DSMT4  ___________ Step 2: Write the radicand as a product of the two radicals, one of which contains the perfect power factors, the other contain the rest of the factors. Ex 1:  EMBED Equation.DSMT4  Ex 2:  EMBED Equation.DSMT4  Ex 3:  EMBED Equation.DSMT4  ___________ Step 3: Take the root of the perfect power factors. Ex 1:  EMBED Equation.DSMT4  Ex 2:  EMBED Equation.DSMT4  Ex 3:  EMBED Equation.DSMT4  ____________ Homework Name____________________________ Period____________ Multiply 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  Simplify 4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  7.  EMBED Equation.DSMT4  8.  EMBED Equation.DSMT4  9.  EMBED Equation.DSMT4  Multiply and simplify 10.  EMBED Equation.DSMT4  11.  EMBED Equation.DSMT4  12.  EMBED Equation.DSMT4  What is the equivalent expression, in simplest radical form to  EMBED Equation.DSMT4 ? A.  EMBED Equation.DSMT4  B.  EMBED Equation.DSMT4  C.  EMBED Equation.DSMT4  D.  EMBED Equation.DSMT4  E. None of these Triangles (levels 1-4) Types of Triangles Acute triangle _______________________________________ Sketch: Obtuse triangle ______________________________________ Sketch: Right triangle ______________________________________ Sketch: Scalene triangle ____________________________________ Sketch: Isosceles triangle ___________________________________ Sketch: Equilateral triangle _________________________________ Sketch: State whether the following triangles are Acute, Obtuse, or Right. 1. ___________________ 2. ______________________   3. ____________________ 4. _______________________   State whether the following triangles are Scalene, Isosceles, or Equilateral 5. _____________________ 6. _______________________   7. ______________________ 8. _______________________   Other Properties of Triangles Angle bisector _______________________________________ Sketch: Median of a triangle ___________________________________ Sketch: What occurs when we draw all three median lines of a triangle? Perpendicular bisector of a triangle ________________________________ Sketch: What occurs when we draw all three perpendicular bisectors on a triangle? Altitude of a triangle ________________________________________ Sketch: How is the altitude of the triangle related to the triangles area? Area of a triangle _______________________ Sum of the measures of the angles in a triangle = __________ Homework Name _____________________________________ Period _____________________ 1. Draw the angle bisector for  EMBED Equation.DSMT4 . 2. Draw the median of  EMBED Equation.DSMT4  Label the intersecting point as D, and Label the intersecting point as G, label the measurements of  EMBED Equation.DSMT4 , and label the lengths of segments and  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 .   3. For the following triangle,  a. What can you say about the measurements of  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 ? Justify your answer. b. What can you say about the measurements of  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 ? Justify your answer. c. List all the possible names for  EMBED Equation.DSMT4 . 4. Find the missing angle. 5. Find the area of the triangle if  EMBED Equation.DSMT4 =6 and  EMBED Equation.DSMT4 = 12   6. If, in  EMBED Equation.DSMT4  is drawn so that AD = DC, then what is  EMBED Equation.DSMT4 ?  A. An angle bisector B. An altitude C. A median D. A perpendicular bisector of  EMBED Equation.DSMT4  E. A transversal Parallel Lines Cut by a Transversal (level 4) Name____________________________ Period___________ Define: Parallel lines ________________________________________________________________________ Supplementary Angles ________________________________________________________________________ Adjacent Angles ________________________________________________________________________ Vertical Angles ________________________________________________________________________ Transversal ________________________________________________________________________     Corresponding Angles ________________________________________________________________________ Alternate Interior Angles ________________________________________________________________________ Alternate Exterior Angles ________________________________________________________________________ Consecutive Interior Angles (Same Side Interior) ________________________________________________________________________ Consecutive Exterior Angles ________________________________________________________________________ When parallel lines are cut by a transversal: 1. Corresponding angles are__________________ 2. Alternate interior angles are________________ 3. Alternate exterior angles are________________ 4. Consecutive interior angles are _________________ 5. Consecutive exterior angles are _________________ Conversely, lines are parallel if when cut by a transversal any of 1-5 are true. In the diagram, lines m and n in a plane are cut by transversal l. Which statement would allow the conclusion that m||n?  A.  EMBED Equation.DSMT4  B.  EMBED Equation.DSMT4  C.  EMBED Equation.DSMT4  D.  EMBED Equation.DSMT4  E.  EMBED Equation.DSMT4  Which pair of lines is parallel? A)  SHAPE \* MERGEFORMAT  B)  SHAPE \* MERGEFORMAT  C)  SHAPE \* MERGEFORMAT  D)  SHAPE \* MERGEFORMAT  Homework Name__________________ Period_________ 1. The figure shows line  EMBED Equation.DSMT4  intersecting line  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 .  SHAPE \* MERGEFORMAT  In the figure,  EMBED Equation.3 and  EMBED Equation.3  are ______________________. A) alternate interior angles B) alternate exterior angles C) corresponding angles D) consecutive interior angles 2. Line  EMBED Equation.DSMT4  and line  EMBED Equation.DSMT4  are parallel lines. Line  EMBED Equation.DSMT4  is a transversal. What kind of angles are  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 ?  SHAPE \* MERGEFORMAT  A) alternate interior angles B) alternate exterior angles C) consecutive interior angles D) corresponding angles 3. Two parallel sections of pipe are joined with a connecting pipe as shown. What is the value of x?  SHAPE \* MERGEFORMAT  A)  EMBED Equation.DSMT4  B)  EMBED Equation.DSMT4  C)  EMBED Equation.DSMT4  D)  EMBED Equation.DSMT4  4. In the accompanying diagram, parallel lines  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4  are cut by transversal  EMBED Equation.DSMT4 .  SHAPE \* MERGEFORMAT  A)  EMBED Equation.DSMT4  B)  EMBED Equation.DSMT4  is the complement of  EMBED Equation.DSMT4  C)  EMBED Equation.DSMT4  is the supplement of  EMBED Equation.DSMT4  D)  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  are right angles 5. Given:  EMBED Equation.DSMT4   SHAPE \* MERGEFORMAT  Which must be true if  EMBED Equation.DSMT4  ?  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  D)  EMBED Equation.DSMT4  6. Line n intersects lines EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , and  EMBED Equation.DSMT4 , forming the indicated  angles.        Which two lines are parallel? A) p and q B) p and r C) q and r D) r and s 7. Which statement would be sufficient to prove that line  EMBED Equation.DSMT4 is parallel to line  EMBED Equation.DSMT4 ?  SHAPE \* MERGEFORMAT  A)  EMBED Equation.DSMT4  B)  EMBED Equation.DSMT4  C)  EMBED Equation.DSMT4  D)  EMBED Equation.DSMT4  8. In the diagram below,  EMBED Equation.DSMT4   SHAPE \* MERGEFORMAT  Which of the following conclusions does not have to be true? A)  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  are supplementary angles. B) Line  EMBED Equation.DSMT4  is parallel to line  EMBED Equation.DSMT4  C)  EMBED Equation.DSMT4  D)  EMBED Equation.DSMT4  9. Line  EMBED Equation.DSMT4  and line  EMBED Equation.DSMT4  are parallel lines. Line  EMBED Equation.DSMT4  is a transversal. Which equation would you use to find the value of x? Explain your reasoning.  SHAPE \* MERGEFORMAT  A)  EMBED Equation.DSMT4 , because if parallel lines are cut by a transversal, then alternate interior angles are supplementary. B)  EMBED Equation.DSMT4 , because if parallel lines are cut by a transversal, then alternate interior angles are congruent. C)  EMBED Equation.DSMT4 , because if parallel lines are cut by a transversal, then consecutive interior angles are congruent. D)  EMBED Equation.DSMT4 , because if parallel lines are cut by a transversal, then consecutive interior angles are supplementary. 10. In the diagram, lines  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4  in a plane are cut by a transversal  EMBED Equation.DSMT4 . Which statement would allow the conclusion that  EMBED Equation.DSMT4 ?   SHAPE \* MERGEFORMAT  A)  EMBED Equation.DSMT4  B)  EMBED Equation.DSMT4  C)  EMBED Equation.DSMT4  D)  EMBED Equation.DSMT4  11. In the figure below,  EMBED Equation.DSMT4   SHAPE \* MERGEFORMAT  What is the value of x? A) 40 B) 50 C) 80 D) 90 12. In the diagram below, line n is parallel to line  EMBED Equation.DSMT4  and line  EMBED Equation.DSMT4  is parallel to line  EMBED Equation.DSMT4 . If  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 what is the measure of  EMBED Equation.DSMT4   SHAPE \* MERGEFORMAT  A)  EMBED Equation.DSMT4  B)  EMBED Equation.DSMT4  C)  EMBED Equation.DSMT4  D)  EMBED Equation.DSMT4  Circles (level 4) 1. What is a circle? _____________________________________________________ 2. How do we name circles? _____________________________________________  Parts of Circles Radius _____________________________ Notation:_________________________  Diameter __________________________ Notation: ______________________ 3. How are the radius and diameter related? __________________________ 4. How is are the radius and diameter related to the circumference of the circle? _____________________________________________________________  Chord ____________________________ Notation ______________________  Secant ____________________________ Notation ______________________  Tangent __________________________ Notation ______________________  Using the picture above: 5. Name the circle. 6. List all radii of the circle. 7. List all of the chords in the circle. 8. Name all diameters drawn on the circle. 9. Name all secant lines of the circle. 10. Name all the tangent lines of the circle. Angles in Circles Central Angle _____________________________ Notation __________________________  Inscribed Angle ____________________________ Notation ________________________ For circle P below, name the Central Angle and the Inscribed Angle.  11. Central angle _______________ 12. Inscribed angle _____________ 13. Which of the following is a secant line?  F. Segment  EMBED Equation.DSMT4  G. Segment  EMBED Equation.DSMT4  H. Line  EMBED Equation.DSMT4  I. Line  EMBED Equation.DSMT4  J. Segment  EMBED Equation.DSMT4  Homework Name ___________________________________ Period ___________________ Refer to the circle below for problems 1 8.  1. Name the circle. 2. Name all possible radii. 3. Name all possible diameters. 4. Name all chords. 5. Name all secant lines. 6. Name all tangent lines. 7. Suppose VX = 16 cm. Find the length of  EMBED Equation.DSMT4 . 8. Suppose XZ = 5 inches. Find the circumference of the circle. 9. On the diagram, which of the following is a chord?  F.  EMBED Equation.DSMT4  G.  EMBED Equation.DSMT4  H.  EMBED Equation.DSMT4  J.  EMBED Equation.DSMT4  K.  EMBED Equation.DSMT4  Triangles, Circles, and Angles (level 5) Name__________________________ Period______ Type of triangle Sketch _________________________ has exactly two congruent sides. _________________________ has three congruent sides. ________________________ has one angle that measures more than EMBED Equation.DSMT4 . ________________________ has no angles that measure more than EMBED Equation.DSMT4 . ________________________ has one angle that measures  EMBED Equation.DSMT4  ________________________ has no congruent sides Triangle parts label Isosceles triangles _______________________ are the two congruent sides. The _______________________ is the non-congruent side. ______________________ are the congruent angles. The _____________________ is the non-congruent angle.  Triangle parts label Right triangles The_____________________ is the side opposite the right angle and is the longest side. The __________________ are the other two sides. Diameter Central Angle  Inscribed angle  Inscribed angles that share the same intersected arc  Draw inscribed angles from points A, B, and C which intercept the semicircle  EMBED Equation.DSMT4   What are the measures of each the angles formed? The diameter of a circle is one side of a triangle, and the vertex is on the circle. What kind of triangle is formed? A. Isosceles B. Right C. Acute D. Scalene E. Equilateral Homework Name_____________________________ Period_________ 1. Solve for x x=_______________  2. Find the measure of  EMBED Equation.DSMT4   EMBED Equation.DSMT4 =_______________  Proportional secant and tangent segments (level 5) Name______________________ Period__________ Central Angle Segment Addition  Inscribed angle  Inscribed angles that share the same intersected arc  Similar triangles   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  CPSTP  Proportional secant segments  Proportional secant and tangent segments  In the circle shown,  EMBED Equation.DSMT4  is a tangent and  EMBED Equation.DSMT4  is a secant. If the length of  EMBED Equation.DSMT4  is 6 and the length of  EMBED Equation.DSMT4  is 4, what is the length of EMBED Equation.DSMT4 ?  F.  EMBED Equation.DSMT4  G. 8 H.  EMBED Equation.DSMT4  J.  EMBED Equation.DSMT4  K. 10 Homework Name___________________________ Period___ Solve for x. Write answers in simplest radical form. 1. 1.________________  2. 2.________________   3. 3_________________   4.________________ 4.  SHAPE \* MERGEFORMAT  Right Triangles (levels 4-6) Find and label the length of the diagonal of the square. Find the lengths of all segments shown in the equilateral triangle whose altitude is shown.    3-4-5 triangles 5-12-13 Period______ Find the unknown side. 1. 2.  3.   4.  EMBED Equation.DSMT4   5. 6.  7. 8. Find the diagonal of a square whose perimeter is 28 inches. 9. 10.  EMBED Equation.DSMT4  11. 12.   EMBED Equation.DSMT4   13. 14.   EMBED Equation.DSMT4  5 15. Find the altitude of an equilateral triangle whose perimeter is 18 inches. 16. Find the length of a side of a square that has a diagonal measuring 10 feet. Triangle Inequality (level 4) Name_______________ Period__________ Two sides of the following figure are fixed, and one side is not. Identify the lengths of the fixed sides.  Could it be drawn a different way, but still retain its given characteristics? Why do you think the word hinge is an appropriate name for the angle? Given the way it is drawn, what would you call the figure? If you called it a triangle, does it always have to be a triangle? What else could it be? For the figure to be a triangle, describe the smallest value for the unknown side. Describe the largest value. What is the rule for finding the smallest possible value for the third side of a triangle given the lengths of the other sides? What is the rule for finding the longest possible value for the third side of a triangle given the lengths of the other sides? ACT problem: Given a triangle with one side measuring 4 centimeters and another side measuring 6 centimeters, all of the following can be the length, in centimeters, of the third side EXCEPT: A. 3 B. 5 C. 7 D. 9 E. 1 Homework Name__________________________________ Period____________ Given the length of its sides, classify each triangle as scalene, right, isosceles, equilateral, 30-60-90, or no triangle. 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