Arithmetic with Whole Numbers and Money Variables and Evaluation

Transcription

1 rithmetic with Whole Numbers and Money Variables and Evaluation Reteaching 1 Math Course 2, Lesson 1 ddition addend + addend = sum $ $81.5 Subtraction minuend subtrahend = difference efore adding or subtracting, write the numbers in a column and align the digits in the ones place. To add or subtract money, write zeros as needed to include two digits to the right of each decimal point Multiplication factor factor = product $ Find each partial product. Then add to find the final product. When multiplying money, place a decimal point in the product to denote cents. 901 $9.9 Division dividend divisor = quotient $7.60 When dividing money, place the decimal point in the 8 ) $60.80 quotient above the decimal point in the dividend. mathematical expression uses numbers, operations, and variables to represent value. variable is a letter that stands for any unknown number. To evaluate an expression with variables, substitute the given values for the variables and perform the calculations. 1. Which is greater, the quotient of 16 divided by or the difference when the Solve 2. minuend is 16 and the subtrahend is 12? =. $2.5 6 =. 1(19) = Evaluate each expression for x = 20 and y = x + y = 6. x y = 7. xy = 8. x y = Saxon Math Course 2 Harcourt chieve Inc. and Stephen Hake. ll rights reserved. 1

2 Properties of Operations Reteaching 2 Math Course 2, Lesson 2 Property Definition Example Commutative Property of ddition Commutative Property of Multiplication Identity Property of ddition Identity Property of Multiplication Property of Zero for Multiplication ssociative Property of ddition ssociative Property of Multiplication Changing the order of the addends does not change the sum. Changing the order of the factors does not change the product. When zero is added to a number, the sum is equal to the given number. Zero is the additive identity. When a number is multiplied by one, the product is equal to the given number. One is the multiplicative identity. When a number is multiplied by zero, the product is zero. How the addends are grouped does not affect the sum. How the factors are grouped does not affect the product. 2 + = = 5 5 = 20 5 = 20 a + 0 = a a 1 = a a 0 = 0 (a + b) + c = a + (b + c) (a b) c = a (b c) Inverse operations undo each other. To undo addition, subtract. 6 + = = 10 = 6 To undo multiplication, divide. 6 = 2 2 = = Name the property that justifies each statement. 1. x + y = y + x 2. xy = yx Use the numbers 2, 5, and 10 to illustrate each property.. ssociative Property of ddition. ssociative Property of Multiplication Simplify each expression. 5. (20 5) (5 28) 0 2 Harcourt chieve Inc. and Stephen Hake. ll rights reserved. Saxon Math Course 2

3 Unknown Numbers in ddition, Subtraction, Multiplication, and Division Reteaching Math Course 2, Lesson n equation is a statement that two quantities are equal. + = 7 variable is a letter that stands for any unknown number. 5 + n = 9 Unknown numbers in problems can be found by using opposite operations. ddition: To find the unknown addend subtract n + 1 = = n Subtraction: To find the unknown minuend add n = = n To find the unknown subtrahend subtract 5 n = = n Multiplication: To find the unknown factor divide Examples: n = 6 6 = n n = 2 2 ( ) = n Division: To find the unknown dividend n 2 = 8 Division: To find the unknown divisor 8 n = 2 multiply 8 2 = n divide 8 2 = n Find the value of each unknown number z = 7 2. x + 25 = w = 12 z = x = w =. 16 m = c = f = 75 m = c = f = 7. d = t = 6 9. y 7 = 9 d = t = y = Saxon Math Course 2 Harcourt chieve Inc. and Stephen Hake. ll rights reserved.

4 Number Line Sequences less opposites origin Reteaching Math Course 2, Lesson greater negative numbers positive numbers Integers are the numbers {...,, 2, 1, 0, 1, 2,,...}. Opposites are numbers the same distance to the right and left of the origin. Comparison symbols show equals (=), less than (<), and greater than (>). Examples: + 2 = 5 5 < 5 > 6 plus 2 equals 5 5 is less than 5 is greater than 6 Use a number line to subtract integers. Simplify lways start at the origin. 2. Go to the right (+).. Go to the left ( ) 5.. The answer is 2. When a larger number is to be subtracted from a smaller one, reverse the order of the numbers and write the difference as a negative number = = 6 sequence is an ordered list of numbers that follow a pattern. Find the next number in this sequence: 1,, 5, 7, 9, Find the pattern. 1,, 5, 7, 9 The pattern is = 11 Find the next three numbers in each sequence ,, 9, 27, ,, 7, 1,... Replace each circle with the proper comparison symbol in problems Simplify: Use digits and comparison symbols to write the statement. 6. The sum of 2 and 8 is greater than the quotient of 8 divided by = 2 Harcourt chieve Inc. and Stephen Hake. ll rights reserved. Saxon Math Course 2

5 Place Value through Hundred Trillions Reading and Writing Whole Numbers Reteaching 5 Math Course 2, Lesson 5 Whole Number Place Values hundred trillions ten trillions trillions hundred billions ten billions billions hundred millions ten millions millions hundred thousands ten thousands thousands hundreds tens ones decimal point,,,,. trillions billions millions thousands units To change from standard numbers to expanded notation: Name the place value of each nonzero digit. 256 = ( 1000) + (2 100) + (5 10) + (6 1) To change from expanded notation to standard numbers: 1. Count the places in the first parentheses. 2. Draw digit lines for each place.. Fill in the digit lines. ( 1000) + (6 10) + (2 1),, To write numbers with words: 1. Put a comma after the words trillion, billion, million, and thousand. 2. lways put three digits after a comma. Write each number in expanded notation. 1. 7, ,090 Use digits to write each number.. Seven hundred thirty-one billion forty. Twelve thousand six hundred one 5. Subtract thirty-nine thousand from fifty million. Write the difference in words. Saxon Math Course 2 Harcourt chieve Inc. and Stephen Hake. ll rights reserved. 5

6 Factors Divisibility Reteaching 6 Math Course 2, Lesson 6 factor is a whole number that divides another number without a remainder. To list the factors of a whole number: 1. Start with the number End with the number given.. Then find all the other factors of the given number.. List the factors in order. Write each factor only once. Examples: Factors of 9: 1,, 9 Factors of 12: 1, 2,,, 6, 12 To find the greatest common factor (GCF) of two numbers: 1. List (in order) the factors of the smaller number. 2. Starting with the greatest factor, cross off factors that are NOT factors of the larger number.. The greatest remaining factor is the GCF. Find the greatest common factor of 18 and Factors of 18: 1, 2,, 6, 9, Cross off 18 and 9 because they are not factors of 0: 1, 2,, 6, 9, 18. GCF is 6. number is able to be divided by... 2 if the last digit is even. if the last two digits can be divided by. 8 if the last three digits can be divided by 8. 5 if the last digit is 0 or if the last digit is 0. Tests for Divisibility if the sum of the digits can be divided by. 6 if the number can be divided by 2 and by. 9 if the sum of the digits can be divided by 9. List the factors of each number Which numbers are factors of both 2 and 6?. What is the greatest common factor of 2 and 6? 6 Harcourt chieve Inc. and Stephen Hake. ll rights reserved. Saxon Math Course 2

7 Lines, ngles, and Planes Lines Reteaching 7 Math Course 2, Lesson 7 plane is a flat surface without end. Parallel Not Perpendicular Perpendicular Lines in a plane either intersect at one point or never intersect. Intersecting Horizontal Oblique Vertical Segment Line Ray Line or Line ;, Ray ; Types of ngles Segment or Segment ;, Obtuse cute Right Straight Use figure CD for Which angle is obtuse? D 2. Which angle is acute?. Which side of the figure is perpendicular to side DC? C. Which side of the figure is NOT parallel or perpendicular to any other side? Use line D for 5 and 6. C D 5. Name a ray on line D that intersects ray C in exactly one point. 6. What type of angle is formed by line D? Saxon Math Course 2 Harcourt chieve Inc. and Stephen Hake. ll rights reserved. 7

8 Fractions and Percents Inch Ruler Reteaching 8 Math Course 2, Lesson 8 Fractions and percents are used to name parts of a whole. fraction is written with two numbers and a division bar. numerator 1 division bar denominator The denominator of a percent is always percent means mixed number is a whole number plus a fraction. Point represents what mixed number? is shaded. parts equal 100%. 1 = 100% = 25% = 25% = 75% The whole number is 8.To find the fraction count the spaces (not the marks) between 8 and 9. This is the denominator (5). Count the spaces to point. This is the numerator (2). Point represents the mixed number Here is a magnified view of an inch ruler with divisions to 1 16 of an inch inch Three quarters is what fraction of a dollar? 2. What fraction of the circle on the right is shaded?. What percent of the circle is NOT shaded?. Point represents what mixed number? 5. Measure to the nearest sixteenth of an inch. 8 Harcourt chieve Inc. and Stephen Hake. ll rights reserved. Saxon Math Course 2

9 Reteaching 9 Math Course 2, Lesson 9 dding, Subtracting, and Multiplying Fractions Reciprocals To add fractions that have the same denominators, add the numerators. The denominator does not change = 5 5 = 1 To subtract fractions that have the same denominators, subtract the numerators. The denominator does not change = 9 To multiply fractions, multiply across both numerators and denominators. 1 2 = 8 To find the reciprocal, flip (reverse the terms of) the fraction. a 1 1 a The product of a fraction and its reciprocal is 1. = = 1 The Inverse Property of Multiplication says that a 1 a = 1 if a is not 0. Simplify Write the reciprocal of each fraction Find the value of each unknown number a = b = c = 1 5 Saxon Math Course 2 Harcourt chieve Inc. and Stephen Hake. ll rights reserved. 9

10 Writing Division nswers as Mixed Numbers Improper Fractions Reteaching 10 Math Course 2, Lesson 10 To write an answer as a mixed number, show the remainder as the numerator and the divisor as the denominator of the fraction. 6 R 1 ) To change an improper fraction to a mixed number, divide the numerator by the denominator. 1 R 2 5 ) To change a mixed number to an improper fraction: 1. Multiply the whole number and the denominator. 2. Then add that product and the numerator.. Keep the same denominator ( ) + 1 = = = 1 1 = 1 Notice each circle has been divided into equal parts, and 1 parts have been shaded. Write each quotient as a mixed number Write each improper fraction as either a whole number or a mixed number Write each number as an improper fraction Harcourt chieve Inc. and Stephen Hake. ll rights reserved. Saxon Math Course 2