Prime and Composite Numbers Prime Factorization

Transcription

1 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 and. Factors of are and. A composite number has more than two factors. Examples: 4 Factors of 4 are,, and 4. 6 Factors of 6 are,,, and 6. Prime factorization expresses a composite number as a product of its prime factors. Examples: 4 = 6 = 9 = = 40 To factor a number using a factor tree: Example:. Write any two factors of the given number Continue this process until each factor is a 6 7 prime number.. Circle the prime factors. 4. Write the prime factors in order. 40 = 7 Notice that some factors will occur more than once. To factor a number using division by primes:. Write the given number in a division box.. Begin dividing by a prime number that is a factor.. Divide the answer by a prime number that is a factor. 4. Repeat this process until the quotient is.. The divisors are the prime factors of the given number. Write the prime factors in order.. List the composite numbers from to 0? Write the prime factorization of the following numbers Write the prime factorization of 64 and 64, using exponents. Example: 7 ) 7 ) ) 0 ) 0 ) = 7 Saxon Math Course Harcourt Achieve Inc. and Stephen Hake. All rights reserved.

2 Problems About a Fraction of a Group Reteaching Math Course, Lesson To solve problems about a fraction of a group use diagrams. Example: Debbie scored of her team s 6 points. How many points did she score?. Draw a rectangle. This stands for the total. 6 points. Divide the rectangle into the same number of parts as the denominator. The denominator of is, so divide the rectangle into equal parts. 6 points. Divide the total by the denominator. 6 points 6 = points Write that answer in each part. points points 4. Bracket the parts into fractions. 6 points Debbie scored 4 points. points points points Debbie scored Team scored Draw a diagram of each statement. Then answer the questions that follow. Fifty people saw the movie. Two fifths were children.. How many children saw the movie?. How many adults saw the movie? 0 people 0 people 0 people 0 people 0 people 0 people children adults Twenty-five percent of the books were mysteries.. What fraction of the books were not mysteries? 4. How many books were mysteries? books books books books books 4 4 mystery books non-mystery books 4 Harcourt Achieve Inc. and Stephen Hake. All rights reserved. Saxon Math Course

3 Subtracting Mixed Numbers with Regrouping Reteaching Math Course, Lesson To subtract mixed numbers that require regrouping:. Borrow from the whole number and rename as a fraction.. Combine the top fraction with the renamed.. Then subtract. 4. Reduce if possible. Example: Simplify % % 4. 6 % 6 % % 7 4 % Saxon Math Course Harcourt Achieve Inc. and Stephen Hake. All rights reserved.

4 Reducing Fractions, Part Reteaching 4 Math Course, Lesson 4 Reduce large numbers by prime factorization. Cancel matching factors. Example: 4 0 = = Cancel means reduce before multiplying. 8 Pair any numerator 8 with any denominator. 4 = 4 Cancel. (Reduce the pairs.) Multiply the reduced terms. Another way to find the greatest common factor (GCF) of two numbers:. Write the prime factorization of each number.. Circle the common factors.. Multiply these common factors to find the GCF. Example: Find the GCF of 4 and = 64 = GCF = = 8 Reduce Simplify 4 and Harcourt Achieve Inc. and Stephen Hake. All rights reserved. Saxon Math Course

5 Dividing Fractions Reteaching Math Course, Lesson The reciprocal of any whole number or fraction is the reverse of terms. The product of a term and its reciprocal is. Examples: The reciprocal of is. The reciprocal of is, or. The reciprocal of 6 is 6. To divide whole numbers sometimes multiply by the reciprocal of the divisor. Example: 4 = 4 = 8 4 = 8 To divide fractions, multiply by the reciprocal of the divisor. Example: = =. Copy the first number (fraction).. Change to.. Write the reciprocal of the second fraction. 4. Cancel (reduce pairs) if possible.. Multiply. = Simplify Saxon Math Course Harcourt Achieve Inc. and Stephen Hake. All rights reserved. 7

6 Multiplying and Dividing Mixed Numbers Reteaching 6 Math Course, Lesson 6 To multiply and divide mixed numbers:. Change mixed numbers to improper ( top heavy ) fractions.. Then multiply or divide.. Simplify (reduce and/or convert) as necessary. Examples: Multiply Divide Change mixed numbers to improper fractions. = 6 Multiply. 0 0 = 4 Change mixed numbers to improper fractions. Multiply by reciprocal of the divisor. = 4 6 Simplify. = Simplify. Simplify 6: Harcourt Achieve Inc. and Stephen Hake. All rights reserved. Saxon Math Course

7 Multiples Least Common Multiple Equivalent Division Problems Reteaching 7 Math Course, Lesson 7 Use times table in Reference Guide to find the least common multiple (LCM) of small numbers. Example: Find the LCM of 6 and 8. Look down the 6 s and the 8 s columns. Find the first number that is the same in both columns. LCM = 4 Use prime factorization to find the LCM of larger numbers. Example: Find the LCM of 4 and 60.. Factor each number into prime factors. 4 = 60 =. Write each prime factor the greatest number of times it was used to form either number.. Multiply these factors to find the LCM. LCM = = 40 Use equivalent division to make complicated problems easier. Cancel matching zeros. Multiply or divide the dividend and divisor by the same number to form a new problem that is easier to calculate. Example: 0 Multiply both numbers by to form an equivalent problem. 0 = 44 0 = 44 0 Find the least common multiple (LCM) of each pair or group of numbers.. 4 and 0. and., 7, and 0 4., 6, and 8. 4 and and 6 Saxon Math Course Harcourt Achieve Inc. and Stephen Hake. All rights reserved. 9

8 Two-Step Word Problems Average, Part Reteaching 8 Math Course, Lesson 8 Two-step word problems can be written as two-step computation problems. Example: 0 (6 + ) Parentheses can make the problem easier. Example: Julie went to the store with $0. She bought 8 cans of dog food for 67 per can. How much money did she have left? $0 (0.67 8). Find out how much she spent = $.6. Then find out how much money she had left. $0 $.6 = $4.64 Calculating an average is often a two-step process.. Add the items.. Divide by the number of items. The answer must be between the smallest and the largest numbers. Another name for average is mean. Example: There were people in the first row, 7 in the second row, and 0 in the third row. What was the average number of people in each of the rows? people 7 people 0 people per row + 0 people rows ) 0 people 0 people The average of two numbers is the number halfway between the given numbers.. Hilda s scores on five games were the following: 8, 89, 94, 99, and 00. What was her average score?. Myrna bought 7 pounds of meat for a barbecue. She paid $.49 per pound and gave the clerk a $0 bill. How much change should she receive?. Jim drove 7 miles in hours 0 minutes. How many miles per hour did Jim drive? 4. What is the average (mean) of 0 and 40? 0 Harcourt Achieve Inc. and Stephen Hake. All rights reserved. Saxon Math Course

9 Rounding Whole Numbers Rounding Mixed Numbers Estimating Answers Reteaching 9 Math Course, Lesson 9 Before working a problem, estimate by rounding the numbers first. To round whole numbers:. Circle the place value you are rounding to.. Underline the digit to its right.. Ask Is the underlined number or more? Yes Add to the circled number. No Circled number stays the same. 4. Replace the underlined number (and any numbers after it) with zero. Examples: To round mixed numbers, compare each fraction to. If the fraction is equal to or greater than If the fraction is less than Examples: Round, to the nearest thousand., round up to the next whole number., round down. 7 ( because > ) 6 ( because 8 < ). Round 7,90 to the nearest hundred.. Estimate the quotient when 7,80 is divided by Estimate the sum of 4 and 0. Saxon Math Course Harcourt Achieve Inc. and Stephen Hake. All rights reserved.

10 Reteaching 0 Math Course, Lesson 0 Common Denominators Adding and Subtracting Fractions with Different Denominators To add or subtract fractions with different denominators:. Find the least common multiple (LCM) of the denominators. If the denominators are small numbers, use the times table. Look down the columns of each denominator. The first number that is the same in both columns is the LCM. If the denominators are large numbers, use prime factorization. Factor each denominator into prime factors. Write each prime factor the greatest number of times it was used to form either number. Multiply these factors to find the LCM.. Rename the fractions using the LCM as the new common denominator.. Add or subtract. 4. Simplify. Example: = = = 4. Rewrite 4 7 and so they have common denominators. Then find the sum and simplify. Simplify Compare: Harcourt Achieve Inc. and Stephen Hake. All rights reserved. Saxon Math Course