Lesson 1.2.4 Concept: Products, Factors, and Factor Pairs Vocabulary: Factors – numbers that create new numbers when they are multiplied. ( 3 and 4 are factors of 12) Product – a number resulting from a multiplication or two or more numbers. ( 25 is a product of 5 x 5) Prime Factor – a factor that is also a prime number. (2 x 3 = 6, 2 and 3 are prime factors of 6)

In mathematics, factors are numbers that create new numbers when they are multiplied.  A number resulting from multiplication is called a product.  In other words, since 2(3) = 6, 2 and 3 are factors of 6, while 6 is the product of 2 and 3.  Also, 1(6) = 6, so 1 and 6 are two more factors of 6. T The number 6 has four factors; 1, 2, 3, and 6.  In this lesson, you will use an extended multiplication table to discover some interesting patterns of numbers and their factors. 

73. Have you ever noticed how many patterns exist in a simple multiplication table?  Such a table is a great tool for exploring products and their factors. Get a Multiplication Chart from your teacher, or use the 1-73 Student eTool (CPM). Fill in the missing products to complete the table. With your team, describe at least three ways you could use the table to figure out the missing numbers (besides simply multiplying the row and column numbers) .

74. Gloria was looking at the multiplication table and noticed an interesting pattern. “Look,” she said to her team.  “All of the prime numbers show up only two times as products in the table, and they are always on the edges.” Discuss Gloria’s observation with your team.  Then choose one color to mark all of the prime numbers on your multiplication chart.  Explain why the placement of the prime numbers makes sense? Multiplication Chart link

Composite numbers, even numbers, odd numbers, any single number. 75. See how many patterns you can discover in the multiplication table. Gloria’s observation in problem concept problem #74 related to prime numbers.  What other kinds of numbers do you know about?  In your group write a list of the kinds of numbers you have discussed.  (You may want to look back at Lesson 1.2.3 to refresh your memory.)  Show where these different kinds of numbers appear on the table using different colors. Make a key/legend. b. What patterns can you find in the locations of the numbers of each type?  Explain your observations. Composite numbers, even numbers, odd numbers, any single number.

76. Consider the number 36 Choose a color or design (circle or x the number) and mark every 36 that appears in the table. Imagine that more rows and columns are added to the multiplication table until it is as big as your classroom floor.  Would 36 appear more times in this larger table?  If so, how many more times and where?  If not, how can you be sure? List all of the factor pairs of 36.  (A factor pair is a pair of numbers that multiply to give a particular product.  For example, 2 and 10 make up a factor pair of 20, because 2 · 10 = 20.)  How do the factor pairs of 36 relate to where it is found in the table?  What does each factor pair tell you about the possible rectangular arrays for 36?  How many factors does 36 have?

77. Frequency is the number of times an item appears in a set of data 77. Frequency is the number of times an item appears in a set of data.  What does the frequency of a number in the multiplication table tell you about the rectangular arrays that are possible for that number? Gloria noticed that the number 12 appears, as a product, 6 times in the table.  She wonders, “Shouldn’t there be 6 different rectangular arrays for 12?”  What do you think?  Work with your team to draw all of the different rectangular arrays for 12 and explain how they relate to the table. How many different rectangular arrays can be drawn to represent the number 48?  How many times would 48 appear as a product in a table as big as the classroom?  Is there a relationship between these answers?

200 = 2 x 2 x 2 x 5 x 5 200 = 23 x 52 78. PRIME FACTORIZATION What are all the factors of 200? A prime factor is a factor that is also a prime number.  What are the prime factors of 200? Sometimes it is useful to represent a number as the product of prime factors.  How could you write 200 as a product using only prime factors?  Writing a number as a product of only prime numbers is called prime factorization How could you write 200 as a product using only prime factors?  Writing a number as a product of only prime numbers is called prime factorization 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, and 200  200 4 · 50 2 · 2   10 · 5 5 · 2 200 = 2 x 2 x 2 x 5 x 5 or 200 = 23 x 52 c. What if you used 10 · 20 instead of 4 · 50?  Finish this prime factorization using a prime factor tree.

Learning Log #2 Title this entry “Characteristics of Numbers and Prime Factorization” and label it with today’s date. Reflect about the number characteristics and categories that you have investigated. You may look at your lesson notes in your Concept Notebook and refer to your toolkit notebook. In this learning log entry… Summarize the 7 characteristics of numbers you explored in Lesson 1.2.3 (such as prime and composite). Describe how you use these properties to write the prime factorization of a number. #93 Checkpoint Quiz Round the decimals to the specified place in parts (a) through (c).  Place the correct inequality sign (<  or  >) in parts (d) through (f). 17.1936  (hundredths) 0.2302  (thousandths) 8.256  (tenths) 47.2_____47.197 1.0032 _____1.00032 0.0089 _____0.03

Tonight’s homework is… 1.2.4 Review & Preview, problems #85 to #92, #94. Label your assignment with your name and Lesson number in the upper right hand corner of a piece of notebook paper. (Lesson 1.2.4) Show all work and justify your answers for full credit.