WHOLE NUMBERS REVIEW A set is a collection of objects. The set of natural numbers is {1,2,3,4,5,.} The set of whole numbers is {0,1,2,3,4,5, } Whole numbers are used for counting objects (such as money, but not cents!) However, they do not include fractions or decimals. The digits in a whole number have place value. Place Value 3 4 5 5 7 6 4 0 2 8 9 7 4 1 5 Three- Digit Groups (separated by commas) Billions group Thousands group Millions group Ones group Trillions group Verbal Form: 30,542 = Thirty thousand, five hundred forty-two. (Notice we don t use the word and.) Standard Notation uses only digits (0 through 9) and commas to state the number. 16 million = 16,000,000
Rounding Whole Numbers: Step 1: Locate the rounding digit, which is the digit at the place value you are rounding to. Step 2: Look at the digit directly to the right of the rounding digit. This is the test digit. If the test digit is < 5 (less than 5), keep the rounding digit the same and change all digits to the right of it to 0. If the test digit is 5 (greater than or equal to 5), then increase the rounding digit by 1 and change all the digits to the right of it to 0. On a number line, numbers are written so that ascending numbers are to the right. 10 is to the left of 20 on the number line, so 20 is greater than 10 (20 > 10). It can also be stated that descending numbers are to the left, so 10 is less than 20 (10 < 20) Ask "is 48 closer to 40 or 50?" It is closer to 50, it is about 50. Ask "is 81 is closer to 80 or 90?" It is closer to 80. It is about 80. Round off the same way for larger numbers. 0 1000 2000 Is 1888 closer to 1000 or 2000? 1888 is closer to 2000. Is 43,556 closer to 40,000, or 50,000? Compare the first two numbers. 43 is closer to 40 than 50. It is 40,000. "Tricky" numbers are those with 5 as the number that decides. Is 650 closer to 600 or 700? On a number line 650 is half way. Math has a rule for this. When the digit is 5 or greater, round up. When the digit is less than 5, round down. Example: Round 24 to the nearest ten. Round 36 to the nearest ten 24 35 Rounds down Rounds up You may need to estimate to a certain place. Look at the number in the place to the right. Then round. Round 684 to the tens place. The number 4 in the ones place to the right of the tens. It rounds down to 680. Round 6423: to the nearest 10 is 6420. to the nearest hundred is 6400. to the nearest thousand, it is 6000. Round 589,457: to the nearest 10, it is. to the nearest 100, it is. to the nearest 1000, it is. to the nearest 10,000 is. to the nearest 100,000 is. You can round it to the nearest million, it is 1,000,000. 48 1888
Adding and Subtracting Whole Numbers Mathematical properties are often used to simplify computation. Below are three addition properties stated in words, shown with a numeric example, and shown with an algebraic example. The Zero Property of Addition is also called the Identity Property of Addition. Associative Property of Addition When numbers or variables are added, for example (2 + 3) + 4 = 2 + (3 + 4) and (a + b) + c = a + (b + c) The addends can be grouped in different ways without changing the result. Commutative Property of Addition When numbers or variables are added, for example 2 + 3 = 3 + 2 and a + b = b + a, The order of the addends can be changed without changing the result. Zero Property of Addition When 0 is added to a number or variable, for example, 2 + 0 = 2 and a + 0 = a, the result is the same number or variable. These properties can be used when adding numbers in your head. Example: 337 + 18 = (300 + 30 + 7) + (10 + 8) = 300 + (30 + 10) + (7+ 8) = 300 + 40 + 15 = 355 Estimating Sums and Differences When an exact answer is not necessary, an estimate can be used. The most common method of estimating sums and differences is called front-end rounding, which is to round each number to its largest place value, so that all but the first digit of the number is 0. Examples: Estimate 4,894 + 429 Round 4,894 to the nearest thousand. 4,894 5,000 Round 429 to the nearest hundred 429 400 Add the rounded numbers. 5,000 + 400 = 5,400 The actual answer is 4,894 + 429 = 5353, which is close to 5,400. If both addends are rounded up, the estimated sum will be greater than the actual sum, and if both addends are rounded down, the estimated sum will be less than the actual sum. Such generalizations are not possible with subtraction. Estimate 6,209 383. What is the largest place value of 6,209? Round to the nearest. 6,209 -> What is the largest place value of 383? Round to the nearest. 383 -> Subtract the rounded numbers. - = The actual answer is 6,209 383 = 5826 How close was your estimate?
Addition Terms: The numbers being added are called addends. Subtraction Terms: The number you are taking away is the subtrahend, the number you are subtracting from is the minuend, and the answer is the difference. Multiplication Terms: The numbers being multiplied together are the factors. The result of the multiplication is the product. Division Terms: The number you are dividing by is the divisor. The number you are dividing into is the dividend. The answer to the division problem is the quotient. The amount over if the dividend is not exactly divisible by the divisor is the remainder. 1. Subtraction is the inverse of addition: If a - b = c, then c + b = a So if 10-3 = 7, then 7 + 3 = 10 Check subtraction by adding your difference answer to the subtrahend. Your new answer should be the minuend. 2. Division is the inverse of multiplication: If a b=c, then c x b = a So if 8 2=4, then 4 x 2 = 8 Check division like this: Quotient X Divisor + Remainder = Dividend. Multiply your answer to the division problem by the number you are dividing by, then add the remainder. Your answer should equal the number you divided into. 3. Addition and multiplication follow the commutative properties: a + b = b + a, so 8 + 3 = 3 + 8 a x b = b x a, so 5 x 2 = 2 x 5 Check division like this: Quotient X Divisor + Remainder = Dividend. Multiply your answer to the division problem by the number you are dividing by, then add the remainder. Your answer should equal the number you divided into. The perimeter of a rectangle or a square is the distance around it. This is found by adding up the lengths of all its sides. P = length + length + width + width The area of a rectangle or square is is the amount of material needed to "cover" it completely. This is found by the product of its length and width: A = l x w. The average (or mean) of a set of objects is the sum of the objects values divided by the number of objects.
ORDER OF OPERATIONS Please: do all operations within Parentheses and other grouping symbols (such as [ ], or operations in numerators and denominators of fractions) from innermost outward. Excuse: calculate Exponents My,Dear: do all Multiplication and Division as they occur from left to right Aunt,Sally: do all Addition and Subtraction as they occur from left to right. Example: 20 2 + 3(8-6) 2 Expression in parentheses gets calculated first = 20 2 + 3(2) 2 Next comes all items with exponents = 20 2 + 3(4) Next in order comes multiplication. Multiplication and Division always come before addition or division, even if to the right. = 20 2 + 12 Now when choosing between when to do addition and when to do subtraction, always go from left to right, so do 20-2 first, because the subtraction is to the left of the addition. = 18 + 12 Now finally we can do the addition. = 30 SECTION 1.7 Solving Application Problems Problem-Solving Strategy: 1. Analyze the problem. What are you trying to find? What s the given info? 2. Work out a plan before starting. Draw a sketch if possible. Look for indicator words (e.g. gained, lost, times, per) to know which operations (+,-, x, ) to use. 3. Estimate a reasonable answer. 4. Solve the problem. 5. Check your work. If the answer is not reasonable, start over. Example: Each home in a subdivision requires 180 ft of fencing. Find the number of homes that can be fenced with 5760 ft of fencing material. 1) What are we trying to find? Total number of homes that can be fenced. Given info: 5760 ft of material, 180 ft of fencing per home. 2) Work out a plan. We want to know how many homes can be fenced with 5760 ft of fencing. If we divide 180 feet per home into 5760 feet we will get the total number of homes ( The feet units cancel out). 3) Estimate a reasonable answer. Round 5760 to 6000 and 180 to 200. 6000 200 = 30 4) Solve the problem. 5) Check your work. Check division. Does 32x18=5760? Yes. Is 32 close to our estimate, 30? Yes. So our answer is 32 homes. 1home 5760 ft 180 ft 5760 180 homes 32 180 5760-540 360-360 0