Grade 9 Mathematics Unit #1 Number Sense SubUnit #1 Rational Numbers. with Integers Divide Integers


 Calvin Hoover
 10 months ago
 Views:
Transcription
1 Page1 Grade 9 Mathematics Unit #1 Number Sense SubUnit #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. (108) 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 nonterminating, nonrepeating 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. Nonreal (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.
22 Page22
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:
27 Page27
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:
33 Page33
34 Page34
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 #15, 712, 14, 16, 19, 20, 23
MATH0910 Review Concepts (Haugen)
Unit 1 Whole Numbers and Fractions MATH0910 Review Concepts (Haugen) Exam 1 Sections 1.5, 1.6, 1.7, 1.8, 2.1, 2.2, 2.3, 2.4, and 2.5 Dividing Whole Numbers Equivalent ways of expressing division: a b,
More informationChapter 1: Order of Operations, Fractions & Percents
HOSP 1107 (Business Math) Learning Centre Chapter 1: Order of Operations, Fractions & Percents ORDER OF OPERATIONS When finding the value of an expression, the operations must be carried out in a certain
More informationPREPARATION FOR MATH TESTING at CityLab Academy
PREPARATION FOR MATH TESTING at CityLab Academy compiled by Gloria Vachino, M.S. Refresh your math skills with a MATH REVIEW and find out if you are ready for the math entrance test by taking a PRETEST
More information1.1. Basic Concepts. Write sets using set notation. Write sets using set notation. Write sets using set notation. Write sets using set notation.
1.1 Basic Concepts Write sets using set notation. Objectives A set is a collection of objects called the elements or members of the set. 1 2 3 4 5 6 7 Write sets using set notation. Use number lines. Know
More informationTYPES OF NUMBERS. Example 2. Example 1. Problems. Answers
TYPES OF NUMBERS When two or more integers are multiplied together, each number is a factor of the product. Nonnegative integers that have exactly two factors, namely, one and itself, are called prime
More informationChapter 4 Fractions and Mixed Numbers
Chapter 4 Fractions and Mixed Numbers 4.1 Introduction to Fractions and Mixed Numbers Parts of a Fraction Whole numbers are used to count whole things. To refer to a part of a whole, fractions are used.
More informationImproper Fractions and Mixed Numbers
This assignment includes practice problems covering a variety of mathematical concepts. Do NOT use a calculator in this assignment. The assignment will be collected on the first full day of class. All
More information2 is the BASE 5 is the EXPONENT. Power Repeated Standard Multiplication. To evaluate a power means to find the answer in standard form.
Grade 9 Mathematics Unit : Powers and Exponent Rules Sec.1 What is a Power 5 is the BASE 5 is the EXPONENT The entire 5 is called a POWER. 5 = written as repeated multiplication. 5 = 3 written in standard
More informationAlgebra 1A and 1B Summer Packet
Algebra 1A and 1B Summer Packet Name: Calculators are not allowed on the summer math packet. This packet is due the first week of school and will be counted as a grade. You will also be tested over the
More informationAccuplacer Arithmetic Study Guide
Testing Center Student Success Center Accuplacer Arithmetic Study Guide I. Terms Numerator: which tells how many parts you have (the number on top) Denominator: which tells how many parts in the whole
More informationCopy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.
Algebra 2  Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers  {1,2,3,4,...}
More information3.1. RATIONAL EXPRESSIONS
3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers
More informationHOSPITALITY Math Assessment Preparation Guide. Introduction Operations with Whole Numbers Operations with Integers 9
HOSPITALITY Math Assessment Preparation Guide Please note that the guide is for reference only and that it does not represent an exact match with the assessment content. The Assessment Centre at George
More informationCOMPASS Numerical Skills/PreAlgebra Preparation Guide. Introduction Operations with Integers Absolute Value of Numbers 13
COMPASS Numerical Skills/PreAlgebra Preparation Guide Please note that the guide is for reference only and that it does not represent an exact match with the assessment content. The Assessment Centre
More informationParamedic Program PreAdmission Mathematics Test Study Guide
Paramedic Program PreAdmission Mathematics Test Study Guide 05/13 1 Table of Contents Page 1 Page 2 Page 3 Page 4 Page 5 Page 6 Page 7 Page 8 Page 9 Page 10 Page 11 Page 12 Page 13 Page 14 Page 15 Page
More informationQuick Reference ebook
This file is distributed FREE OF CHARGE by the publisher Quick Reference Handbooks and the author. Quick Reference ebook Click on Contents or Index in the left panel to locate a topic. The math facts listed
More informationChapter 15 Radical Expressions and Equations Notes
Chapter 15 Radical Expressions and Equations Notes 15.1 Introduction to Radical Expressions The symbol is called the square root and is defined as follows: a = c only if c = a Sample Problem: Simplify
More informationWord Problems. Simplifying Word Problems
Word Problems This sheet is designed as a review aid. If you have not previously studied this concept, or if after reviewing the contents you still don t pass, you should enroll in the appropriate math
More informationNow that we have a handle on the integers, we will turn our attention to other types of numbers.
1.2 Rational Numbers Now that we have a handle on the integers, we will turn our attention to other types of numbers. We start with the following definitions. Definition: Rational Number any number that
More informationChapter 4  Decimals
Chapter 4  Decimals $34.99 decimal notation ex. The cost of an object. ex. The balance of your bank account ex The amount owed ex. The tax on a purchase. Just like Whole Numbers Place Value  1.23456789
More informationMAT Mathematical Concepts and Applications
MAT.1180  Mathematical Concepts and Applications Chapter (Aug, 7) Number Theory: Prime and Composite Numbers. The set of Natural numbers, aka, Counting numbers, denoted by N, is N = {1,,, 4,, 6,...} If
More informationExponents, Factors, and Fractions. Chapter 3
Exponents, Factors, and Fractions Chapter 3 Exponents and Order of Operations Lesson 31 Terms An exponent tells you how many times a number is used as a factor A base is the number that is multiplied
More informationOperations on Decimals
Operations on Decimals Addition and subtraction of decimals To add decimals, write the numbers so that the decimal points are on a vertical line. Add as you would with whole numbers. Then write the decimal
More informationAlgebra 1: Topic 1 Notes
Algebra 1: Topic 1 Notes Review: Order of Operations Please Parentheses Excuse Exponents My Multiplication Dear Division Aunt Addition Sally Subtraction Table of Contents 1. Order of Operations & Evaluating
More informationHow To Math Properties
CLOSURE a + b is a real number; when you add 2 real numbers, the result is also a real number. and 5 are both real numbers, + 5 8 and the sum, 8, is also a real number. a b is a real number; when you subtract
More informationIntegers, I, is a set of numbers that include positive and negative numbers and zero.
Grade 9 Math Unit 3: Rational Numbers Section 3.1: What is a Rational Number? Integers, I, is a set of numbers that include positive and negative numbers and zero. Imagine a number line These numbers are
More informationChanging a Mixed Number to an Improper Fraction
Example: Write 48 4 48 4 = 48 8 4 8 = 8 8 = 2 8 2 = 4 in lowest terms. Find a number that divides evenly into both the numerator and denominator of the fraction. For the fraction on the left, there are
More informationeday Lessons Mathematics Grade 8 Student Name:
eday Lessons Mathematics Grade 8 Student Name: Common Core State Standards Expressions and Equations Work with radicals and integer exponents. 3. Use numbers expressed in the form of a single digit times
More informationACCUPLACER Arithmetic Assessment Preparation Guide
ACCUPLACER Arithmetic Assessment Preparation Guide Please note that the guide is for reference only and that it does not represent an exact match with the assessment content. The Assessment Centre at George
More informationSection R.2. Fractions
Section R.2 Fractions Learning objectives Fraction properties of 0 and 1 Writing equivalent fractions Writing fractions in simplest form Multiplying and dividing fractions Adding and subtracting fractions
More informationQ N R. Sep 5 7:55 AM THE NUMBER SYSTEM
Q W I TITLE: Q N R Sep 5 7:55 AM THE NUMBER SYSTEM N NATURAL NUMBERS All positive non zero numbers; in other words, all positive numbers. This does not include zero. These are the numbers we use to count.
More informationSection P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities
Section P.9 Notes Page P.9 Linear Inequalities and Absolute Value Inequalities Sometimes the answer to certain math problems is not just a single answer. Sometimes a range of answers might be the answer.
More informationMath 016. Materials With Exercises
Math 06 Materials With Exercises June 00, nd version TABLE OF CONTENTS Lesson Natural numbers; Operations on natural numbers: Multiplication by powers of 0; Opposite operations; Commutative Property of
More information6th Grade Vocabulary Words
1. sum the answer when you add Ex: 3 + 9 = 12 12 is the sum 2. difference the answer when you subtract Ex: 179 = 8 difference 8 is the 3. the answer when you multiply Ex: 7 x 8 = 56 56 is the 4. quotient
More informationThe numbers that make up the set of Real Numbers can be classified as counting numbers whole numbers integers rational numbers irrational numbers
Section 1.8 The numbers that make up the set of Real Numbers can be classified as counting numbers whole numbers integers rational numbers irrational numbers Each is said to be a subset of the real numbers.
More information47 Numerator Denominator
JH WEEKLIES ISSUE #22 20122013 Mathematics Fractions Mathematicians often have to deal with numbers that are not whole numbers (1, 2, 3 etc.). The preferred way to represent these partial numbers (rational
More informationSometimes it is easier to leave a number written as an exponent. For example, it is much easier to write
4.0 Exponent Property Review First let s start with a review of what exponents are. Recall that 3 means taking four 3 s and multiplying them together. So we know that 3 3 3 3 381. You might also recall
More informationMath Help and Additional Practice Websites
Name: Math Help and Additional Practice Websites http://www.coolmath.com www.aplusmath.com/ http://www.mathplayground.com/games.html http://www.ixl.com/math/grade7 http://www.softschools.com/grades/6th_and_7th.jsp
More informationReteaching. Properties of Operations
 Properties of Operations The commutative properties state that changing the order of addends or factors in a multiplication or addition expression does not change the sum or the product. Examples: 5
More informationUNIT 1 VOCABULARY: RATIONAL AND IRRATIONAL NUMBERS
UNIT VOCABULARY: RATIONAL AND IRRATIONAL NUMBERS 0. How to read fractions? REMEMBER! TERMS OF A FRACTION Fractions are written in the form number b is not 0. The number a is called the numerator, and tells
More informationSelfDirected Course: Transitional Math Module 2: Fractions
Lesson #1: Comparing Fractions Comparing fractions means finding out which fraction is larger or smaller than the other. To compare fractions, use the following inequality and equal signs:  greater than
More informationExponents, Polynomials and Functions. Copyright Cengage Learning. All rights reserved.
Exponents, Polynomials and Functions 3 Copyright Cengage Learning. All rights reserved. 3.1 Rules for Exponents Copyright Cengage Learning. All rights reserved. Rules for Exponents The basic concept of
More informationThe integer is the base number and the exponent (or power). The exponent tells how many times the base number is multiplied by itself.
Exponents An integer is multiplied by itself one or more times. The integer is the base number and the exponent (or power). The exponent tells how many times the base number is multiplied by itself. Example:
More information28: Square Roots and Real Numbers. 28: Square Roots and Real Numbers
OBJECTIVE: You must be able to find a square root, classify numbers, and graph solution of inequalities on number lines. square root  one of two equal factors of a number A number that will multiply by
More informationBasic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704.
Basic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704. The purpose of this Basic Math Refresher is to review basic math concepts so that students enrolled in PUBP704:
More informationSTRAND B: Number Theory. UNIT B2 Number Classification and Bases: Text * * * * * Contents. Section. B2.1 Number Classification. B2.
STRAND B: Number Theory B2 Number Classification and Bases Text Contents * * * * * Section B2. Number Classification B2.2 Binary Numbers B2.3 Adding and Subtracting Binary Numbers B2.4 Multiplying Binary
More information1.1 THE REAL NUMBERS. section. The Integers. The Rational Numbers
2 (1 2) Chapter 1 Real Numbers and Their Properties 1.1 THE REAL NUMBERS In this section In arithmetic we use only positive numbers and zero, but in algebra we use negative numbers also. The numbers that
More informationClick on the links below to jump directly to the relevant section
Click on the links below to jump directly to the relevant section Basic review Writing fractions in simplest form Comparing fractions Converting between Improper fractions and whole/mixed numbers Operations
More informationEXPONENTS. To the applicant: KEY WORDS AND CONVERTING WORDS TO EQUATIONS
To the applicant: The following information will help you review math that is included in the Paraprofessional written examination for the Conejo Valley Unified School District. The Education Code requires
More informationChapter 1. Real Numbers Operations
www.ck1.org Chapter 1. Real Numbers Operations Review Answers 1 1. (a) 101 (b) 8 (c) 1 1 (d) 1 7 (e) xy z. (a) 10 (b) 14 (c) 5 66 (d) 1 (e) 7x 10 (f) y x (g) 5 (h) (i) 44 x. At 48 square feet per pint
More informationOrder of Operations  PEMDAS. Rules for Multiplying or Dividing Positive/Negative Numbers
Order of Operations  PEMDAS *When evaluating an expression, follow this order to complete the simplification: Parenthesis ( ) EX. (52)+3=6 (5 minus 2 must be done before adding 3 because it is in parenthesis.)
More informationAble Enrichment Centre  Prep Level Curriculum
Able Enrichment Centre  Prep Level Curriculum Unit 1: Number Systems Number Line Converting expanded form into standard form or vice versa. Define: Prime Number, Natural Number, Integer, Rational Number,
More informationAccuplacer Arithmetic Study Guide
Accuplacer Arithmetic Study Guide Section One: Terms Numerator: The number on top of a fraction which tells how many parts you have. Denominator: The number on the bottom of a fraction which tells how
More informationUse order of operations to simplify. Show all steps in the space provided below each problem. INTEGER OPERATIONS
ORDER OF OPERATIONS In the following order: 1) Work inside the grouping smbols such as parenthesis and brackets. ) Evaluate the powers. 3) Do the multiplication and/or division in order from left to right.
More informationAlgebra Revision Sheet Questions 2 and 3 of Paper 1
Algebra Revision Sheet Questions and of Paper Simple Equations Step Get rid of brackets or fractions Step Take the x s to one side of the equals sign and the numbers to the other (remember to change the
More informationSummer Math Packet. Number Sense & Math Skills For Students Entering PreAlgebra. No Calculators!!
Summer Math Packet Number Sense & Math Skills For Students Entering PreAlgebra No Calculators!! Within the first few days of your PreAlgebra course you will be assessed on the prerequisite skills outlined
More informationWelcome to Basic Math Skills!
Basic Math Skills Welcome to Basic Math Skills! Most students find the math sections to be the most difficult. Basic Math Skills was designed to give you a refresher on the basics of math. There are lots
More informationMathematics Success Grade 8
T68 Mathematics Success Grade 8 [OBJECTIVE] The student will determine the square roots of perfect squares, and categorize rational numbers in the real number system and apply this understanding to solve
More informationGrade 6 Math Circles. Algebra
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Grade 6 Math Circles October 8/9, 2013 Algebra Note: Some material and examples from the Tuesday lesson were changed for the Wednesday lesson. These notes
More informationA rational number is a number that can be written as where a and b are integers and b 0.
S E L S O N Rational Numbers Goal: Perform operations on rational numbers. Vocabulary Rational number: Additive inverse: A rational number is a number that can be a written as where a and b are integers
More informationIntroduction to Fractions
Introduction to Fractions Fractions represent parts of a whole. The top part of a fraction is called the numerator, while the bottom part of a fraction is called the denominator. The denominator states
More informationVocabulary Words and Definitions for Algebra
Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms
More informationAll you have to do is THINK!
Mathematician s Notes BE A MATHEMATICIAN! All you have to do is THINK! Imagination is more important than knowledge. Albert Einstein We only think when confronted with a problem. John Dewey It s not that
More informationWorkbook Pages Teaching Guides 2544
RNUM: Real Numbers Workbook Pages  Building Blocks.... Factor Pairings..... Match Up on Fractions.... Fractions Using a Calculator... Charting the Real Numbers.... 5 Venn Diagram of the Real Numbers.....
More informationGrade 7 Math The Number System Grade 7 Math Grade 7 Math Start Date: August 30, 2012 End Date : September 28, 2012
Unit Overview Students will be able to: describe real life situations for quantities that combine to make zero understand the distance between two on a number line show how opposites have a sum of zero
More informationFoundations for Middle School 6 th Grade Syllabus
Unit 1:, and Percents (Lessons 0 19) and Percents (Lessons 0 19) Foundations for Middle School 6 th Grade Syllabus Solving for every variable Foundations for Middle School 6 th Grade Syllabus 1 Unit 1:
More informationYear 1 Maths Expectations
Times Tables I can count in 2 s, 5 s and 10 s from zero. Year 1 Maths Expectations Addition I know my number facts to 20. I can add in tens and ones using a structured number line. Subtraction I know all
More informationSolution: There are TWO square roots of 196, a positive number and a negative number. So, since and 14 2
5.7 Introduction to Square Roots The Square of a Number The number x is called the square of the number x. EX) 9 9 9 81, the number 81 is the square of the number 9. 4 4 4 16, the number 16 is the square
More informationMath 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.
Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used
More informationSIXTH GRADE MATH. Quarter 1
Quarter 1 SIXTH GRADE MATH Numeration  Place value  Comparing and ordering whole numbers  Place value of exponents  place value of decimals  multiplication and division by 10, 100, 1,000  comparing
More information1.6 The Order of Operations
1.6 The Order of Operations Contents: Operations Grouping Symbols The Order of Operations Exponents and Negative Numbers Negative Square Roots Square Root of a Negative Number Order of Operations and Negative
More informationTable of Contents Sequence List
Table of Contents Sequence List 368102215 Level 1 Level 5 1 A1 Numbers 010 63 H1 Algebraic Expressions 2 A2 Comparing Numbers 010 64 H2 Operations and Properties 3 A3 Addition 010 65 H3 Evaluating
More informationG RADE 9 MATHEMATICS (10F)
G RADE 9 MATHEMATICS (10F) Midterm Practice Examination Answer Key G RADE 9 MATHEMATICS (10F) Midterm Practice Examination Answer Key Instructions The midterm examination will be weighted as follows Modules
More informationMATH 90 CHAPTER 1 Name:.
MATH 90 CHAPTER 1 Name:. 1.1 Introduction to Algebra Need To Know What are Algebraic Expressions? Translating Expressions Equations What is Algebra? They say the only thing that stays the same is change.
More informationSixth Grade Math Pacing Guide Page County Public Schools MATH 6/7 1st Nine Weeks: Days Unit: Decimals B
Sixth Grade Math Pacing Guide MATH 6/7 1 st Nine Weeks: Unit: Decimals 6.4 Compare and order whole numbers and decimals using concrete materials, drawings, pictures and mathematical symbols. 6.6B Find
More informationAdd and subtract 1digit and 2digit numbers to 20, including zero. Measure and begin to record length, mass, volume and time
Year 1 Maths  Key Objectives Count to and across 100 from any number Count, read and write numbers to 100 in numerals Read and write mathematical symbols: +,  and = Identify "one more" and "one less"
More informationUnit 1, Concept 1 Number Sense, Fractions, and Algebraic Thinking Instructional Resources: Carnegie Learning: Bridge to Algebra
Unit 1, 1 Number Sense, Fractions, and Algebraic Thinking 7NS 1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers
More informationMath Foundations IIB Grade Levels 912
Math Foundations IIB Grade Levels 912 Math Foundations IIB introduces students to the following concepts: integers coordinate graphing ratio and proportion multistep equations and inequalities points,
More informationA fraction is a noninteger quantity expressed in terms of a numerator and a denominator.
1 Fractions Adding & Subtracting A fraction is a noninteger quantity expressed in terms of a numerator and a denominator. 1. FRACTION DEFINITIONS 1) Proper fraction: numerator is less than the denominator.
More informationLESSON SUMMARY. Manipulation of Real Numbers
LESSON SUMMARY CXC CSEC MATHEMATICS UNIT TWO: COMPUTATION Lesson 2 Manipulation of Real Numbers Textbook: Mathematics, A Complete Course by Raymond Toolsie, Volume 1. (Some helpful exercises and page numbers
More informationMultiplication and Division with Rational Numbers
Multiplication and Division with Rational Numbers Kitty Hawk, North Carolina, is famous for being the place where the first airplane flight took place. The brothers who flew these first flights grew up
More informationAlgorithm set of steps used to solve a mathematical computation. Area The number of square units that covers a shape or figure
Fifth Grade CCSS Math Vocabulary Word List *Terms with an asterisk are meant for teacher knowledge only students need to learn the concept but not necessarily the term. Addend Any number being added Algorithm
More informationPreAlgebra Class 3  Fractions I
PreAlgebra Class 3  Fractions I Contents 1 What is a fraction? 1 1.1 Fractions as division............................... 2 2 Representations of fractions 3 2.1 Improper fractions................................
More informationCD 1 Real Numbers, Variables, and Algebraic Expressions
CD 1 Real Numbers, Variables, and Algebraic Expressions The Algebra I Interactive Series is designed to incorporate all modalities of learning into one easy to use learning tool; thereby reinforcing learning
More informationHFCC Math Lab Arithmetic  4. Addition, Subtraction, Multiplication and Division of Mixed Numbers
HFCC Math Lab Arithmetic  Addition, Subtraction, Multiplication and Division of Mixed Numbers Part I: Addition and Subtraction of Mixed Numbers There are two ways of adding and subtracting mixed numbers.
More informationUnit Essential Question: When do we need standard symbols, operations, and rules in mathematics? (CAIU)
Page 1 Whole Numbers Unit Essential : When do we need standard symbols, operations, and rules in mathematics? (CAIU) M6.A.3.2.1 Whole Number Operations Dividing with one digit (showing three forms of answers)
More informationSquare Roots. Learning Objectives. PreActivity
Section 1. PreActivity Preparation Square Roots Our number system has two important sets of numbers: rational and irrational. The most common irrational numbers result from taking the square root of nonperfect
More informationMultiplying Decimal Numbers by 10, by 100, and by 1000
Multiplying Decimal Numbers by 10, by 100, and by 1000 Lesson 111 111 To multiply by 10, shift the decimal to the right one place. To multiply by 100, shift the decimal to the right two places. To multiply
More informationExample 1 Example 2 Example 3. The set of the ages of the children in my family { 27, 24, 21, 19 } The set of Counting Numbers
Section 0 1A: The Real Number System We often look at a set as a collection of objects with a common connection. We use brackets like { } to show the set and we put the objects in the set inside the brackets
More informationPrime and Composite Numbers Prime Factorization
Prime and Composite Numbers Prime Factorization Reteaching Math Course, Lesson A prime number is a whole number greater than that has exactly two factors, the number itself and. Examples: Factors of are
More informationa) b) 4 + ( 5) c) d) 2  ( 2) f)
CLASS VII New Integers A 1. The integer consist of, and numbers 2. The numbers less than zero are called integer. 3. All the numbers which are less than zero have sign. 4. Zero is greater than integer.
More informationSimplifying Radical Expressions
In order to simplifying radical expression, it s important to understand a few essential properties. Product Property of Like Bases a a = a Multiplication of like bases is equal to the base raised to the
More informationDECIMALS are special fractions whose denominators are powers of 10.
DECIMALS DECIMALS are special fractions whose denominators are powers of 10. Since decimals are special fractions, then all the rules we have already learned for fractions should work for decimals. The
More informationDecimals Adding and Subtracting
1 Decimals Adding and Subtracting Decimals are a group of digits, which express numbers or measurements in units, tens, and multiples of 10. The digits for units and multiples of 10 are followed by a decimal
More informationHow do you compare numbers? On a number line, larger numbers are to the right and smaller numbers are to the left.
The verbal answers to all of the following questions should be memorized before completion of prealgebra. Answers that are not memorized will hinder your ability to succeed in algebra 1. Number Basics
More informationVocabulary Cards and Word Walls Revised: June 2, 2011
Vocabulary Cards and Word Walls Revised: June 2, 2011 Important Notes for Teachers: The vocabulary cards in this file match the Common Core, the math curriculum adopted by the Utah State Board of Education,
More informationFractions and Linear Equations
Fractions and Linear Equations Fraction Operations While you can perform operations on fractions using the calculator, for this worksheet you must perform the operations by hand. You must show all steps
More informationAlgebra I. Copyright 2014 Fuel Education LLC. All rights reserved.
Algebra I COURSE DESCRIPTION: The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical expressions, equations, graphs, and other topics, with an emphasis
More informationChapter 1.1 Rational and Irrational Numbers
Chapter 1.1 Rational and Irrational Numbers A rational number is a number that can be written as a ratio or the quotient of two integers a and b written a/b where b 0. Integers, fractions and mixed numbers,
More informationNumerator Denominator
Fractions A fraction is any part of a group, number or whole. Fractions are always written as Numerator Denominator A unitary fraction is one where the numerator is always 1 e.g 1 1 1 1 1...etc... 2 3
More information