Lesson 1.2.3 Concept: Characteristics of Numbers Vocabulary: Rectangular Array – objects arranged in rows and columns filling a rectangular shape.

Turn to page 8 in your CPM toolkit Turn to page 8 in your CPM toolkit. Record definitions for the following terms in your Concept Notebook… Natural Numbers: Even: Odd: Whole Numbers: Factors of a Number: Prime Number: Composite Number: **Class Practice: Place your initials and birth day (1-31) on a small dot. Be ready to add the day of your birthday to the class Venn diagram.

Can all numbers be represented the same way? In Lesson 1.2.1, you worked with your team to find different ways of showing different numbers of pennies.  One arrangement that can be used to represent any whole number is a rectangular array.  An example is shown at right.  The horizontal lines of pennies are called rows, while the vertical lines of pennies are called columns. Factors of 15 are… 1 x 15 3 x 5 In this lesson, you will use rectangular arrays to investigate some properties of numbers.  As you work on the problems in this lesson, use the following questions to help focus your team’s discussion. Can all numbers be represented the same way? What can we learn about a number from its representations?

What can we learn about a number from its representations? Focus Questions: Can all numbers be represented the same way? What can we learn about a number from its representations? 62. HOW MANY PENNIES?  Jenny, Ann, and Gigi have different numbers of pennies.  Each girl has between 10 and 40 pennies.  You must figure out all the possible numbers of pennies that each girl could have.  Use the clues given below.  Be ready to explain your thinking to the class. Jenny can arrange all of her pennies into a rectangular array that looks like a square.  Looking like a square means it has the same number of rows as columns.  Ann can arrange all of her pennies into five different rectangular arrays. Whenever Gigi arranges her pennies into a rectangular array with more than one row or column, she has a remainder (some leftover pennies).

What can we learn about a number from its representations? Focus Questions: Can all numbers be represented the same way? What can we learn about a number from its representations? 63.  What can you learn about a number from its rectangular arrays?  Consider this question as you complete parts (a) and (b) below. A number that can be arranged into more than one rectangular array, such as Ann’s in part (b) of problem 1-62, is called a composite number.  List all composite numbers less than 15, and show the factor pairs that equal that number.   Consider the number 17, which could be Gigi’s number.  Seventeen pennies can be arranged into only one rectangular array: 1 penny by 17 pennies.  Any number, like 17, that can form only one rectangular array is called a prime number.  Work with your team to find all prime numbers less than 25. 4 1 * 4 2 * 2

What can we learn about a number from its representations? Focus Questions: Can all numbers be represented the same way? What can we learn about a number from its representations? 64. Jenny, Ann, and Gigi were thinking about odd and even numbers.  (When even numbers are divided by two, there is no remainder.  When odd numbers are divided by two, there is a remainder of one.)   Jenny said, “Odd numbers cannot be formed into a rectangle with two rows.  Does that mean they are prime?”   Consider Jenny’s question with your team.  Are all odd numbers prime?  List all odd numbers less that 25. Write a factor pair for any number possible without using the number itself and 1. Write a counterexample.  A counterexample is an example that can be used to show a statement is false (in this case, finding a number that is odd but not prime). 

What can we learn about a number from its representations? Focus Questions: Can all numbers be represented the same way? What can we learn about a number from its representations? 65. HOW MANY PENNIES?  Work with your team to figure out how many pennies (between 15 and 35) each person could have. You may want to use diagrams or expressions to help you determine your answers. Xander arranges his pennies into a rectangle with more than one row, he always has some leftover pennies.  When he uses two equal rows or three equal rows, he has one leftover penny.  When he arranges them into a rectangle with four equal rows, he has three leftover pennies. Jorge arranges his pennies into a rectangle with two equal rows, three equal rows, or five equal rows, he has one leftover penny.  When he arranges his pennies into a rectangle with four equal rows, he has three leftover pennies.  How many pennies could Jorge have?

Tonight’s homework is… 1.2.3 Review & Preview, problems #68 to #72. Label your assignment with your name and Lesson number in the upper right hand corner of a piece of notebook paper. (Lesson 1.2.3) Show all work and justify your answers for full credit.

Daily Closure: Return Group folder to the math box. Return your individual Concept Notebook to the math section of your binder. Return the group supply box to the cart after making sure all supplies have been stored in the box and the lid is secured. Record Review/Preview for lesson 1.1.5 #36-40 in your homework planner.