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Name:________________________Chapter 1 Test Date:____ Chapter 1: Numbers, Expressions, and Equations Lesson 1.1 Place Value Warm-Up: 1. 7 x 100 = 2. 6 x 1,000 = 3. 9 x 10,000 = 4. 3 x 1,000,000 = Place Value: There are several ways to read and write a number Places: There are 3 places in each period ~ H T O. Periods: Trillions, Billions, Millions, Thousands, Ones (Units). Standard form: 67,240,000 (commas are used to show periods) Word form: sixty-seven million, two hundred forty thousand Short-word form: 67 million, 240 thousand Expanded form: 60,000,000 + 7,000,000 + 200,000 + 40,000 OR (6 x 10,000,000) + (7 x 1,000,000) + (2 x 100,000) + (4 x 10,000) ***place value chart*** Example 1: Write the place and value of each underlined digit: 4,536,021,985______________________________ _________________________________________ _________________________________________ Example 2: Write each number in short-word form and expanded form. 6,354,000,120,000__________________________ ________________________________________ 77,537,000,320_______________________________ ___________________________________________ Example 3: Write each number in standard form and word form. 20,000 + 5,000 + 60 + 2 __________________________ _____________________________________________ 15 trillion, 450 billion, 89 million, 34__________________ _____________________________________________ Example 4: What does the zero in 62,778,304,525 represent? _____________________________________________ Lesson 1.2 Exponents Warm-Up: 1. 3 x 3 x 3 x 3 = 2. 8 x 8 = 3. 10 x 10 x 10 = Exponents are another way to write repeated factors. base: is the repeated factor 34 exponent base exponent or power: tells how many times the base is used as a factor exponential form: 27 (a base with an exponent) expanded form: 2,324 = (2 x 1000) + (3 x 100) + (2 x 10) + (4 x 1) 2,324 = (2 x 103) + (3 x 102) + (2 x 101) + (4 x 100) 2,324 = (2 • 103) + (3 • 102) + (2 • 101) + (4 • 100) *expanded form using exponents- A number written in expanded form with the place values written in exponential form (using exponents): squared: a number squared is the second power of a number. 72 = 7 squared or 7 to the second power cubed: A number cubed is the third power of a number. 53 = 5 cubed or 5 to the third power. Example 1: What are two ways to read 6³?______________________ ______________________________________________________ Example 2: Write 6³as a product and then evaluate._______________ ______________________________________________________ Example 3: Write 15 x 15 x 15 x 15 x 15 x 15 in exponential form______ ______________________________________________________ Example 4: Write 35,402 in expanded form using exponents._________ ______________________________________________________ Example 5: Are 25 and 52 equal? Explain why or why not? Justify your answer._________________________________________________ _______________________________________________________ _______________________________________________________ Lesson 1.3 Comparing and Ordering Whole Numbers Warm-Up: Give the place of each underlined digit. 1. 43,005 =___________________________________________ 2. 578 = _____________________________________________ 3. 201,459 = __________________________________________ 4. 7,655 = ____________________________________________ > < You can compare two numbers by using “ greater than” or “ Numbers can also be = equal to. Example 1: Use less than.” < or > to compare. 783 ____ 2,001 201,053 ____ 200,855 4,619 ____ 4,618 1,211 ____ 1,121 569,120 ____596,120 5,856 ____5,860 Lesson 1.3 You can order numbers from greatest to least and least to greatest. Read directions carefully. Example 1: Order these numbers from greatest to least: 26,750; 26,810; 29,035; 28,208 ______________________ ______________________ ______________________ ______________________ Order these numbers from least to greatest: 26,750; 26,810; 29,035; 28,208 ______________________ ______________________ ______________________ ______________________ Example 2: Write 3 numbers that are between 23,455,700 and 23,455,789. 23,455,700 ______________________ ______________________ ______________________ 23,455,789 Lesson 1.4 Rounding Whole Numbers Warm-Up: Give the value of each underlined digit. 1. 28,709 _______ 2. 468 _______ 3. 833,406 _______ 4. 1,398 _______ There are times when an exact answer is not needed therefore allowing you to round. round: To give an approximation for a number to the nearest one, ten, hundred, thousand, and so on Use these four steps to round numbers: 1. Find the rounding place and underline that digit. 2. Look at the digit to the right of this place and circle that digit. 3. If the circled digit is less than 5 leave the underlined digit the same. If the circled digit is greater than or = to 5 round the underlined digit up one number. 4. Change all digits to the right of the underlined digit (rounding digit) to zeros. Example 1: Round 285,610,627 to the nearest ten million then nearest ten thousand. 285,610,627 ____________________________ 285,610,627 ____________________________ Example 2: Round each number to the underlined place. 6,808,386 ______________________ 52,799,335 ______________________ 3,101,922 ______________________ 45,699,825 ______________________ 895,774 ______________________ Example 3: Write three numbers that would round to 5,600 when rounded to the nearest hundred. _____________;_____________;_____________ Lesson 1.5 Estimating Sums and Differences Warm-Up: Round 634,998,050 to the indicated places. 1. hundreds ____________________________ 2. 10-thousands__________________________ 3. 100-thousands_________________________ 4. 100-millions___________________________ You can estimate the sums and differences of whole numbers in a variety of ways: 1. round: To give an approximation for a number to the nearest one, ten, hundred, thousand, and so on. Example: 668 + 4,239 2. front-end estimation: To find a number that is close to an exact answer. You use front-end estimation to get a rough estimate. Add only the first digits that have the same place value. Example: 7,924 + 3,168 + 209 3. front-end estimation with adjusting: A method of front-end estimation that adjusts the result based on the remaining digits of each addend. Use this method to get a better answer. Example: Add the front end digits. 6,829 + 3,401 Add the hundreds. Add the front end estimate by adding the estimate from the front end digits. So, 6,829 + 3,401 ≈ _______________ 4. clustering: An estimation method where numbers that are approximately equal are treated as if they were equal. *You can use clustering when to estimate when the numbers are close together. Example: 412 + 398 + 385 + 409 So, 412 + 398 + 385 + 409 ≈ _________. List four estimating methods for adding (sums) and subtracting differences. 1.___________________________ 2.___________________________ 3.___________________________ 4.___________________________ Lesson 1.6 Estimating Products and Quotients Warm-Up: Round each number to one nonzero digit. 1. 5,880 __________________________ 2. 30,998 __________________________ 3. 998 __________________________ 4. 4,500 __________________________ You can estimate products and quotients or whole numbers in a variety of ways: 1. round: To give an approximation for a number to the nearest one, ten, hundred, thousand, and so on. 6,192 x 11 = 6,000 x 10 ≈ 2. compatible numbers: Numbers that are easy to compute mentally. 24 x 41 25 x 40 = 3. range: The difference of the greatest and the lowest number in a set of data. underestimate: When two factors are rounded down, the product is an underestimate. Example: 342 x 687 300 x 600 = ____________ overestimate: When two factors are rounded up, the product is an overestimate. 342 x 687 400 x 700 = _____________ Example: Estimate each answer. Tell which method you used. 1. 562 x 9,031 ≈ ______________, ______________ 2. 638 ÷ 72 ≈ ______________, _______________ 3. 5,893 ÷ 301 ≈ ______________, ______________ 4. 489 x 2,970 ≈ ______________, ______________ Estimating products means __________________________________ ______________________________________________________. Estimating quotients means__________________________________ ______________________________________________________. Lesson 1.7 Read and Understand What steps can help you get started with solving a problem? Step 1: What do you know? Step 2: What are you trying to find? **Read and understand is the first phase of the problem-solving process. Lesson 1.8 Order of Operations Warm-Up: 1. 12 + 5 – 3 2. 72 ÷ 8 3. 4 x 5 x 8 4. 34 numerical expression: An expression, consisting of numbers and operations to be performed, that represents a particular value. order of operations: The order in which to perform operations when evaluating expressions with more than one operation. P E M D A S Lesson 1.8 Order of Operations Example 1: 18 – 3 x 2 Example 2: 40 ÷ (2 x 4) Example 3: 9 + 23 Example 4: 4 x (5 + 5) ÷ 20 + 6 Example 5: (3 x 2) + 32 Example 6: 10 – 6 + 4 x 1 A plan for order of operations Lesson 1.8 A plan would be to first look for parenthesis, then look for exponents, multiply or divide which ever comes first going from left to right, next look for subtraction or addition which ever comes first from left to right. After you perform each step you bring down whatever is left. It should look like an upside down triangle. Tell which operation you would perform first to evaluate 14 – 9 + 2 – 3. ______________________________________________________. Explain why 5 x 3 + 42 does not equal 95.________________________ ______________________________________________________ How would you generate a plan to solve an order of operations problem? _______________________________________________________ _______________________________________________________ Lesson 1.9 Properties of Operations Warm-Up: Decide if statement if each statement if true or false. 1. 28 + 678 = 678 + 28 __________ 2. 24 ÷ 4 = 4 ÷ 24 __________ 3. (18 – 6) + 12 = 18 – (6 + 12) __________ 4. 14 + (6 + 37) = (14 + 6) + 37 __________ Properties of Addition and Multiplication Commutative Properties: The properties that state the order of the addends or factors does not affect the sum (+) or product (x). 9 + 15 = 15 + 9 4 x 12 = 12 x 4 Associative Properties: Properties that state the way in which addends or factors are grouped does not affect the sum (+) or product (x). 4 + (5 + 6) = (4 + 5) + 6 (3 x 2) x 4 = 3 x (2 x 4) Identity Properties: The properties that state the sum of any number and zero is that number and the product of any number and 1 is that number. 567 + 0 = 567 422 x 1 = 422 Multiplication Properties of Zero: The property that states the product of any number and zero is zero. 389 x 0 = 0 Find the missing number. Then tell what property or properties are shown. Example 1: 15 + (48 + 5) = 15 + ( __ + 48) ____________________ Example 2: (78 + 29) + __ = 78 + 29 ____________________ Example 3: 84 ÷ 12 = 84 ÷ 12 x __ _____________________ Example 4: (26 x __) x 30 = 30 x (26 x 4) _____________________ Show two different ways to compute 5 x 6 x 3. ____________________ ____________________ Do you get the same answer each time? Why or why not? _________________________________________________________ _________________________________________________________ Lesson 1.10 Mental Math: Using the Distributive Property Warm-Up: Tell which property is shown. 1. 47 + 68 = 68 + 47 2. 568 = 568 x 1 3. (39 x 5) x 6 = 39 x (5 x 6) _____________________ _____________________ _____________________ Distributive Property: The Distributive Property states that multiplying a sum by a number produces the same result as multiplying each addend by the number and adding the products. 8 x (20 + 4) (8 x 20) + (8 x 4) 160 + 32 192 **another way to solve using pencil and paper: (add first then multiply) 8 x (20 + 4) 8x 24 192 break apart: using the Distributive property to compute mentally Example 1: Find 47 x 6 using mental math. 47 x 6 (40 + 7)6 Break apart 47. 40(6) + 7(6) Use the Distributive Property. 240 + 42 282 Example 2: Find (9)45 + (9)5 using mental math. (9)45 + 9(5) Each product has a 9 as one of its factors. 9(45 + 5) Use the Distributive Property. (9)50 450 Find each missing number. 1. 8(40 + 2) = 8( __ ) + 8(2) 2. 5(8 + 12) = 5(8) + __ (12) 3. 9( __ + 7) = 9(80) + 9(7) Use the Distributive Property to multiply mentally. 1. (35 + 7)2 ____________________ 2. 7(150) 15(40 + 4) Does the Distributive Property work with subtraction? Try it with 9(50 – 1) and (20 – 4)5. Lesson 1.11 Mental Math Strategies Warm-Up: Solve mentally 1. 25 + 89 + 275 = _______ 2. 20 x (40 + 8) = _______ 3. 2 x 137 x 5 = _______ 4. 350 ÷ 10 = __________ You can use the properties of operations to compute mentally. Here are three different strategies you can use: Strategy A: Break apart the numbers. What You Think: Find 38 + 77 using mental math. Break apart the numbers into tens and ones. Add the tens: 30 + 70 = 100 Add the ones: 8 + 7 = 15 Add the sums: 100 + 15 = 115 So, 38 + 77 = 115 Why It Works: 38 + 77 (30 + 8) + (70 + 7) Commutative and Associative (30 + 70) + (8 + 7) Properties of Addition 100 + 15 115 Strategy B: Look for multiples of 10 or 100. Find 20 x 6 x 5 using mental math Multiply number pairs having a product of 10 or 100. Multiply the other number. 20 x 5 = 100 100 x 6= 600 So, 20 x 5 x 6 = 600. Strategy C: Use compensation. compensation: Choosing numbers close to the numbers in a problem, and then adjusting the answer to compensate for the numbers chosen. Add an amount to one number and subtract the same amount from the sum. 39 + 88 l 39 + 1 Add 9 to 31 l 40 + 88 = 128 128 – 1 = 127 Subtract 1 from the answer to compensate. equal additions: Adding the same number to two numbers in a subtraction problem does not affect the difference. 742 – 295 (742 + 5) - (295 + 5) Adding 5 to both numbers does not affect the difference. 747 – 300 447 Practice Problems: Compute mentally 1. 20 x 39 x 5 __________ 2. 973 – 645 __________ 3. 159 + 328 ___________ 4. 4 x 23 x 250________ 5. Explain the steps you can use to find 794 – 439 mentally. _________________________________________________________ _________________________________________________________ _________________________________________________________ Lesson 1.12 Plan and Solve What steps can help you find a solution to a problem? Step 1: Choose a strategy •Show what you know Draw a picture Make an organized list Make a Table Make a Graph Act It Out Use Objects •Look for a Pattern •Try, Check, and Revise •Write an Equation •Use Logical Reasoning •Solve a Simpler Problem •Work Backward Step 2: Stuck: Don’t give up. Try these suggestions: •Reread the problem •Tell what you know •Identify key facts and details •Tell the problem in your own words •Show the main idea •Try a different strategy •Retrace your steps Step 3: Answer the question in the problem. Lesson 1.13 Variables and Expressions Warm-Up: Use symbols to write the expression. 1. sum of 16 and 29 2. difference of 216 and 89 variable: A quantity that can change or vary, often represented with a letter algebraic expression: A mathematical phrase containing variables, numbers and operational symbols. evaluate: To find the number that an algebraic expression names by replacing a variable with a given number. (Solve) substitution: To put something in another’s place or replace the variable with a number. Writing Expressions The following common words and phrases indicate addition, subtraction, multiplication, and division. Addition plus the sum of increased by total more than added to Subtraction minus the difference decreased by fewer than less than subtracted Multiplication times the product of multiplied by of Operation Word Phrase Addition the sum of 9 and a number n a number m increased by 8 six more than a number t add eighteen to a number h seventy-seven plus number r the difference of 12 and a number n seven less than a number y ten decreased by a number p the product of 4 and a number k fifteen times a number t two multiplied by a number m the quotient of a number divided by five twenty-five divided by a number m Subtraction Multiplication Division Division divided by the quotient of per Algebraic Phrase 9+n m+8 t+6 h + 18 77 + r 12 – n y–7 10 – p 4k 15t 2m a/5 25/m Practice Problems: 1. 5 less than k ___________________________________________ 2. n increased by 7 ______________________________________ 3. twice b divided by 9 _____________________________________ 4. Evaluate each expression for x = 5. 45/x ____________ 20x – 1 ___________ 5. Write two different word phrases for 5 – d.___________________ ____________________________________________________ ____________________________________________________ x2 – 19 ________ Lesson 1.14 Properties of Equality Warm-Up: 1. 12 + 19 – 19 2. 15 – 8 + 8 3. 6 x 7 ÷ 7 4. 18 ÷ 3 x 3 ______________________ ______________________ ______________________ ______________________ Lesson 1.15 Solving Equations with Whole Numbers Warm-Up: Explain how to get the variable alone 1. 12x = 60 ______________________________ 2. d – 10 = 10 ______________________________ 3. 32 = 8 + a ______________________________ equation: A mathematical sentence stating that two expressions are equal properties of equality: Properties that state performing the same operation to both the same operation to both sides of an equation keeps the equation balanced inverse operations: Operations that “undo” each other, such as addition and subtraction, or multiplication or division Steps to Solving Equations with Integers: 1. Locate the side of the equation with the variable and determine the operation being used. 2. Use the inverse operation on the side with the variable. The inverse of addition is subtraction. The inverse of subtraction is addition. The inverse of multiplication is division. The inverse of division is multiplication. 3. Use the properties of equality to balance the equation. Whatever you do to one side of the equation you must do to the other side of the equation. Keep the balance! 4. Solve the equation to find out the answer to the variable. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Steps for Checking your Algebraic Equations: *** The check must be written as follows: Check: F 1. Write the original formula/equation. S 2. Substitute the value of the variable for the variable. Prove it by doing the work. S 3. Solve…check to make sure the equation is =. Lesson 1.16 Problem-Solving Skill: Look Back and Check workbook page 52