1 Page1 Grade 9 Mathematics Unit #1 Number Sense Sub-Unit #1 Rational Numbers Lesson Topic I Can 1 Ordering & Adding Create a number line to order integers Integers Identify integers Add integers 2 Subtracting Integers Subtract Integers 3 Multiplication & Division Multiply Integers with Integers Divide Integers 4 Order of Operations with Solve problems involving order of operations Integers 5 Identifying and Identify Rational Numbers Representing the Identify Irrational Numbers Number Systems Relate Whole Numbers, Rational Numbers and Integers Describe a strategy that can create an irrational number from a rational number 6 Ordering, Adding and Create a number line to place decimals in order Subtracting Decimals Determine a decimal number between two given numbers Add decimals Subtract decimals 7 Multiplication and Multiply decimals Division of Decimals Divide decimals 8 Ordering Fractions Create a number line to order fractions Determine a fraction between two given numbers 9 Addition and Subtraction Add Fractions of Fractions Subtract Fractions 10 Multiplication and Multiply Fractions Division of Fractions Divide Fractions 11 Order of Operations with Perform the correct order of operations for decimals. Rational Numbers Perform the correct order of operations for fractions. Quiz #1 Rational Numbers Demonstrate your understanding of above topics
2 Page2 Lesson #1: Ordering & Adding Integers Integers are used in everyday life, when we deal with weather, finances, sports, geography and science. Key Vocabulary: Define the following using the glossary in Math Makes Sense 9 Integer Whole Number Natural Number Positive Integer Negative Integer Zero Pair Opposite Integer Zero Property Distributive Property Commutative Property Product Quotient Order of Operations
3 Page3 Ordering Integers Create a Number Line to Order the Following Integers: -3, 4, -9, 5, -6, 0. Step 1: Determine the lowest integer and the highest integer. Step 2: Draw a straight line for your number line. Use a straight edge and be sure to put arrows on both ends. Step 3: Plot your lowest number to the far left hand side of your line; plot your highest number to the far right hand side of your line. Mark equal intervals between to indicate each integer between the two. Step 4: Label each interval. Step 5: Place the numbers onto the number line by indicating its place with a dot or a line and writing the number ABOVE the number line. Possibilities when adding Integers: Money Method: Think about the numbers in the addition statement as money positive numbers is money you have, while negative numbers is money you owe. This would change the equation 3 + (-4) to You have $3 and owe $4. What would you have? Use this to add the following: a) b) c) d) e) f)
4 Page4 g) h) Ordering and Adding Integers Assignment: Order the following integers on a the number line below: Steps: 1. Sketch a number line and include markings for every integer 2. Place dots and label the numbers you are ordering on the number line a) -3, 5, 6, -1, -7, 0 b) 2, -3, 4, -5, 6, -7 Assignment: Add the following integers: (-7) + (+2) = (-2) + (-4) = (-1) + (+8) = (+5) + (-7) = (+9) + (+3) = (+2) + (-7) = (-2) + (+9) = (-2) + (+1) = (+6) + (-8) = (+2) + (-6) = (-3) + (-8) = (-6) + (-8) = (0) + (-2) = (-3) + (+6) = (-8) + (+4) = (0) + (-7) = (-8) + (-1) = (+5) + (+1) = (+6) + (+48) = (+90) + (-14) = (-57) + (-89) =
5 Page5 (+73) + (-55) = (+72) + (-92) = (-71) + (+47) = (-47) + (-61) = (-75) + (+78) = (-53) + (+57) = (-25) + (-31) = (-46) + (-3) = (+95) + (-58) =
6 Page6 Lesson #2: Subtracting Integers Subtracting Integers Money Method, continued, with Same, Change, Change Subtracting Integers requires the changing of signs within the mathematical statement. One way to remember to do this properly is Same, Change, Change. What this means is to leave the first numbers sign the same, no matter what it is, and change the second two. Ex) would mean to leave the first number, -5, as itself. Change the subtraction sign to an addition, then change the positive three to negative 3. This means: becomes. You would then apply the money method to solve, meaning you d think I owe $5 and I also owe $3. Ex) Subtract the following, using the method of Same, Change, Change. a) b) c) d) e) f)
7 Page7 Subtracting Integers Assignment: Subtract the following integers: (-6) - (0) = (-7) - (+8) = (-5) - (+3) = (+2) - (-6) = (+2) - (+8) = (-8) - (-8) = (-7) - (+1) = (+1) - (+9) = (0) - (+8) = (+9) - (-6) = (-2) - (-6) = (+5) - (+3) = (-1) - (+3) = (+2) - (+2) = (+7) - (-3) = (-9) - (-9) = (+2) - (0) = (-6) - (+8) = (-3) - (-7) = (+5) - (+1) = (-8) - (+7) = (-1) - (-1) = (+8) - (+2) = (-4) - (+8) = (+2) - (+3) = (+3) - (-1) = (+7) - (+7) = (-4) - (+4) = (-5) - (-8) = (0) - (+4) = (-9) - (+7) = (+3) - (-5) = (+9) - (-2) = (-4) - (-9) = (-8) - (+6) = (-7) - (0) = (+28) - (-51) = (-94) - (-73) = (-65) - (-41) = (+52) - (+47) = (+35) - (+68) = (-74) - (-61) = (+65) - (-59) = (+2) - (+78) = (-10) - (-12) = (+33) - (-53) = (+49) - (-52) = (-61) - (+92) = (-9) - (-91) = (-47) - (+3) = (+69) - (+39) =
8 Page8 Mixing Adding and Subtracting Integers: Find the sum or difference for each question. (+14) - (+9) = (+6) + (-13) = (-4) + (+1) = (-6) + (+10) = (+14) - (-4) = (0) + (+15) = (0) + (-9) = (+1) + (+24) = (+3) + (-14) = (-24) + (+14) = (+17) + (+14) = (+23) + (+14) = (+14) + (+9) = (+21) + (+21) = (-3) - (+4) = (+5) - (+3) = (-22) + (+20) = (+18) - (+21) = (+6) + (-25) = (+17) + (+6) = (+24) + (+11) = (+21) - (+3) = (+35) - (+24) = (-5) + (+8) = (-4) - (-9) = (+25) - (+2) = (-22) + (-22) = (-44) - (-19) = (+3) - (+10) = (+22) - (+4) = (+14) - (+9) = (+5) (+6) + (-13) = (-7) (-4) + (+1)
9 Page9 Lesson #3: Multiplication & Division with Integers When multiplying or dividing integers you treat multiply or divide the numbers as though they are natural numbers, then use the properties below to determine the sign of the product or a quotient. Properties for Multiplication of two Integers: Solve for the following products: a) c) b) d) When two or more numbers are multiplied the result is called the product. Properties for Division of two Integers: This Symbolically Means the same as this: Solve for the following quotients: a) c) b) d)
10 Page10 e) g) f) h) Multiplication & Division of Integers Assignment: Find each product. (-6)x0 = 7x3 = 6x(-10) = (-3)x(-5) = 8x(-2) = (-4)x(-10) = 10x(-3) = 3x5 = 9x(-4) = 10x4 = 10x(-4) = 5x9 = 0x(-10) = 11x11 = 2x3 = (-4)x(-12) = (-4)x(-6) = (-10)x(-2) = 3x12 = 4x7 = 3x(-3) = Find each quotient. 96 (-12) = (-40) (-10) = (-55) 5 = 63 9 = (-63) (-9) = (-8) 2 = (-90) (-10) = 36 (-6) = =
11 Page = 49 7 = (-100) 10 = 35 5 = (-25) 5 = 48 (-4) = = (-24) (-12) = (-96) 8 = 60 (-5) = (-30) 5 = 14 2 = (-14) (-7) = (-16) 2 = (-110) 10 = (-66) 11 = (-63) 9 = 80 (-10) = (-36) (-12) = 18 9 = 18 (-2) = Perform the indicated operation and simplify. 7 + (-8) = 16 (-4) = 9 - (-11) = = 8 (-10) = 4 - (-6) = (-3) + 11 = (-2) + 5 = 20 (-2) = (-8) 4 = (-9) 6 = (-1) (-10) = 3 + (-11) = 1 (-7) = 1 - (-9) =
12 Page12 Lesson #4: Order of Operations of Integers In lessons #1 and #2 you learned how to add, subtract, multiply and divide integers. Now you will learn to extend your ability to evaluate expressions containing positive and negative integers. Before learning to evaluate using the order of operations you must know how to apply an exponent to an integer. An exponent is a base (integer) raised to a specific power. For example is an exponent. The base is the number 3 and the power is the number 2. This means to multiply the base by itself two times. Evaluate the following: Exponential Form Expanded Form Simplified When applying the order of operations the acronym BEDMAS can be used to remember the order in which to perform the operations. 1. First any expression within Brackets must be performed. Brackets may be inside of brackets, in which case the innermost brackets must be calculated first. 2. Secondly any Exponents must be calculated. 3. Thirdly any Division or Multiplication operations must be performed. These should be done as they appear from left to right. 4. Lastly and Addition or Subtraction operations must be performed. These also should be done as they appear from left to right. Evaluate the expression Apply BEDMAS You must begin by solving within the brackets. Use subtraction of integer rules to subtract these. Find the solution by adding the integers. Example 1: Evaluate the expression.
13 Page13 Evaluate the expression. Apply BEDMAS. Insert all invisible 1 s to make the negatives simpler. Since there are no calculations within brackets you get to move on to evaluating the exponent. Multiply and Divide from left to right. DO NOT CONFUSE THE SUBTRACTION SIGN BETWEEN AS A MULTIPLICATION! Use rules of subtracting integers to make this into a simpler expression. Evaluate. Example 2: Evaluate the expression a) b) Evaluate the expression [ ] [ ] [ ] [ ]. Apply BEDMAS. Beware of Invisible Brackets that exist around the entire numerator and entire denominator. Insert Invisible Brackets and Invisible 1 s Perform any multiplications within Brackets (once inside the brackets BEDMAS must be applied again) Add and subtract within the top brackets. Divide the integers. Example 3: Evaluate the expression a) b)
14 Page14 Order of Operations of Integers Assignment: Perform the operations in the correct order x x (3 + (-1)) (-5) (-9) 5. 5 (-1) (-2) 7. 1 x 1+(-9) (-9 + (-3)) 9. [(10 (-10)) 7 ] x (-6) ((-6) x 2) 11. (6 - (-10)) (4 (-2)) 12. (2+1) x 6 (-9) 13. (-2) 2 (1 (-1)) (10 (-2) - 5)
15 Page x (-9 + (-9) ) x (-2 + 3) x (-9 + 7) (-10 + (-1 - (-5)) x 5) 18. (10-8) x (-4) 2 x ( (-4) + (-6)) ( (1 - (-6))) + (-4)
16 Page16 Lesson #5: Identifying and Representing the Number Systems Numbers are grouped into systems called the Number Systems. The Natural Numbers: The Natural Numbers (N) are the simplest counting numbers we know of. They are never negative, nor are they decimals or fractions. They begin at the number 1 and increase as though you are counting objects. Ex) 1, 2, 3, 4, are Natural Numbers The Whole Numbers: The Whole Numbers (W) expand upon the natural numbers with the inclusion of the number zero. Whole numbers are also never negative, nor are they decimals or fractions. They begin at the number zero and increase as though you are counting. Ex) 0, 1, 2, 3, 4, are Whole Numbers The Integers: Integers (I) are whole numbers along with their negative opposites. When simplified Integers will never be fractions (with a denominator other than 1) or a decimal. Ex), -3, -2, -1, 0, 1, 2, 3, are Integers The Rational Numbers: Rational Numbers are any number that can be written in the form, where a and b are Integers and. Rational Numbers also include decimals that repeat or terminate (since they can be written as fractions) Powers and Square roots may be rational numbers if their standard form is a rational number. The Irrational Numbers: Irrational numbers are any numbers that are not rational. That is to say any non-terminating, non-repeating decimals, any imperfect square (etc.) roots. All of the above numbers can be grouped into the Real Number System. That is to say, numbers that exist and can be plotted on a number line. Non-real (or Imaginary) numbers do exist but will not be taught at this level.
17 Page17 Venn Diagram of the Number Systems: Writing Rational Numbers in different forms: Many forms of rational numbers look different but represent the same number. For example, the number 5 can be represented by each of the following:, etc. Any rational number that reduces to 5 is the same value.
18 Page18 Dealing with negative in Rational Numbers: By convention, the negative symbol on a fraction is written in one of two places: either attached to the numerator of the fraction:, or out front of the fraction:. This means these two fractions have the same value, even though they look different. Many times people will leave the negative sign with the denominator. This doesn t change the value, but doesn t meet criteria of what is considered appropriate. Thus:. When a rational number has the same value in the numerator and the same value in the denominator and the same overall sign as another they are equivalent. Ex) Write 3 equivalent fractions to each of the following: a) b) Assignment: Page 101 #5, 6; page 103 #26, 27:
19 Page19 Lesson #6: Ordering, Adding & Subtracting Decimals When ordering decimal numbers we can use a similar strategy to ordering integers but extend the strategy to the right side of the decimal. When we order integers: Positive integers are larger than negative integers Ie: 5 is larger than -5 When comparing positive integers: o The more place values to the left of the decimal the larger the number is Ie: 0.3 is larger than o If two numbers have the same number of places, the first digit from the left that is larger will determine which value is larger. Ie: is larger than When comparing negative integers: o Use opposite rules as to positive integers o The fewer place values to the left of the decimal, the larger the number Ie: is larger than 0.3 o If two numbers have the same number of places, the first digits from the left that is larger will determine which value is smaller. Ie: is larger than Remember! When a decimal repeats the number that repeat are designated with a line above them: Ie: = and = Ex) Compare the following decimals with <, >, = a) b) c) Adding and Subtracting Decimals: When adding or subtracting decimals it is very important that you are adding values together for the same place value, which means you must line up the decimal when adding numbers with decimals. Ex) First: Line up the decimals: Then add/subtract columns as they appear from right to left. Ex 2)
20 Page20 Assignment: Comparing Decimals 1. Compare each pair of decimals using a <, > or = sign :32 2. Order the following sets of decimals on a number line. a) 3.42, 4.3, 4.32, 3.44 b) -3.42, -4.3, -4.32, Order each set of decimals. 0.62, 0.38, 0.35, 0.49, 0.1, 0.21, 0.54, 0.6, 0.51, 0.28 Least to Greatest 0.23, 0.2, 0.77, 0.49, 0.74, 0.91, 0.65, 0.23, 0.03, 0.83 Greatest to Least
21 Page , 1.41, 1.18, 1.34, 1.44, 1.52, 1.61, 1.03, 1.26, 1.56 Least to Greatest 4. Add or subtract the following decimals.
23 5. Add or subtract the following as indicated. Use loose leaf to line up decimal places. Page23
24 Page24 Lesson #7: Multiplying and Dividing Decimals **Remember rules for Multiplication and Division of Integers: Positive x Positive = Positive Negative x Negative = Positive Positive x Negative = Negative Negative x Positive = Negative Positive Positive = Positive Negative Negative = Positive Positive Negative = Negative Negative Positive = Negative In other words: if the signs are the same the product or quotient will be positive; if the signs are different the product or quotient will be negative. Multiplication with Decimals: When multiplying decimals, multiply them as you would whole numbers (ignore the decimal place). Then count the total number of decimal places in both multipliers (the numbers you are multiplying); this number is the total number of decimal places in the product. Ex) a) Multiply First: Set it up vertically: Then perform multiplication. Count decimal places and place decimal. b) Dividing with Decimals: Set up the division as a fraction. Multiply numerator and denominator in your fraction (which is both divisor and dividend within the fraction) by multiples of 10 until there is no more decimal places. Then divide numerator by denominator. Ex) a) Divide Set up fraction:
25 Page25 Multiply by 10 s: Divide. b) c) Assignment: Part A:
26 Page26 Part B: Part C:
28 Page28 Lesson #8: Ordering Fractions Fractions Review: The denominator goes in da bottom. The denominator also determines the number of pieces one whole is divided or cut into. The numerator goes on the top and determines how many pieces there are. Fractions can be represented with a pie chart picture where a circle is cut into the number of pieces indicated by the denominator and the number indicated in the numerator is shaded in. Whole numbers are indicated by an entire filled in circle. Ex) Represent the following fractions with a picture: a) b) c) d) Equivalent Fractions are numbers which appear different but have the same value. They may have different numerators and denominators, be improper and proper, etc. You can determine if they are equivalent by reducing until the fraction is in lowest terms. Ex) Determine if the two fractions are equivalent. a) b) c) Ex 2) Write 2 fractions that are equivalent to each: a) b)
29 Page29 Lowest Common Denominator LCD) is the number which is the smallest multiple of two given different denominators. This can be found easily be writing multiples of both denominators out and finding the lowest one in common. Ex) Find the LCD for the two given fractions: a) b) Converting Decimals to Fractions: Method 1: The simplest way to convert a fraction is to divide the numerator by the denominator: Ex) ; Method 2: Make an equivalent fraction with a base of power 10 (10, 100, 1000, etc.); then move the decimal place appropriately: Ex) Since we are dividing by 100 we move the decimal place 2 times to the left. Converting Decimal to Fractions: Any decimal can be converted to a fraction by multiplying the number by a power of 10 to make the decimal a whole number and putting that power of 10 as the denominator of the fraction. Ex) 1.65 = The fraction then must be fully reduced by dividing by common factors. Ordering Fractions: Ex: Write the numbers in order from greatest to least:
30 Page30 Method: Use common denominators: (Do positive numbers first): (And negative): Assignment: (TO BE DONE IN ASSIGNMENT SECTION ON LOOSE LEAF!) Page 101 #10, 14, 15, 24, 25:
31 Page31 Lesson #9: Adding and Subtraction Fractions: Use common denominators: **REMEMBER: The denominator in a fraction determines the number of pieces one whole is divided (cut) into. The numerator in a fraction determines how many pieces there are. Since denominators determine the size of the piece, it is important to ensure the denominators are equivalent or common. Example 1: Add: Sequence: 1. Make all fractions improper. 2. Make determine common denominator; multiply numerator and denominator by the number required to get the denominator: 3. Smoosh them together on one denominator. (This step may be skipped when you have gotten 5 correct in a row in your assignment) 4. Add/Subtract the numerators. Ex 2) Assignment: page 111 # 11; page 119 #9, 12, 13abc
32 Page32 Multiplication of Fractions: Lesson #10: Multiplication and Division of Fraction: When multiplying fractions you simply multiply the numerator by the numerator and the denominator by the denominator. Fractions must be in improper form and must be fully reduced. You may choose to reduce before multiplying if you prefer. Ex) Multiply the following: a) b) c) d) Division of Fractions: When dividing fraction we multiply by the reciprocal. This is a fraction that has been inverted. Ex) The reciprocal of is. Note: when reciprocating a negative number the negative stays with the numerator out of convention. That is to say that the reciprocal of is. Fractions must still be fully reduced. Ex) Divide the following fractions: a) b) c) d) Assignment: Page 127 #4, 7; page 134 #4 and the questions on the following page:
35 Page35 Lesson #11: Order of Operations with Rational Numbers: Recall the correct Order of Operations for mathematics using the acronym BEDMAS: B E D M A S Use the order of operations to complete the following: Ex 1) Ex 2) [ ] Assignment: Page 140 #3ac, 4ac, 7ac, 8, 12acd Quiz Review: Page 144 #1-5, 7-12, 14, 16, 19, 20, 23