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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 above. The horizontal lines of pennies are called rows, while the vertical lines of pennies are called columns.
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?
· Have each team member create rectangular array with different numbers of rows and columns, and then share with each other.
1-62. HOW MANY PENNIES? Part One
Jenny, Ann, and Gigi have different numbers of pennies. Each girl has between 10 and 40 pennies. Work with your team to 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.
1. 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.
2. Ann can arrange all of her pennies into five different rectangular arrays.
3. Whenever Gigi arranges her pennies into a rectangular array with more than one row or column, she has a remainder (some leftover pennies).
· 1-63. What can you learn about a number from its rectangular arrays? Consider this question as you complete parts (a) and (b) below.
1. 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.
2. Consider the number 17, which could be Gigi’s number. Seventeenpennies can be arranged into only one rectangular array: 1penny by 17pennies. 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.
· 1-64.Jenny, Ann, and Gigi were thinking about odd and even numbers. (When even numbers are divided by two, there is no remainder. Whenodd 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? If so, explain how you know. If not, find 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).
· 1-65. HOW MANY PENNIES? Part Two
· Work with your team to figure out how many pennies (between 10 and 40) each person could have. You may want to use diagrams or expressions to help you determine your answers. Can you find more than one possible answer?
1. When 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.
2. When 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?
3. When Louisa arranges her pennies into a rectangle with two equal rows, three equal rows, four equal rows, or six equal rows, she has one leftover penny. When she arranges her pennies into a rectangle with five equal rows, she has two leftover pennies. How many pennies could Louisa have?
· 1-66. Follow your teacher’s directions to add the day of your birthday to the class Venn diagram like the one at right. For example, if you were born on August 16, you would place the number 16 in the diagram in section A.
1. For each of the sections labeled A through D in the diagram, choose a number and explain why it belongs in its section.
2. If you have not already done so, talk with your team about where the numbers 0,1,and2 belong in the diagram. Be ready to share your thinking with the class.
· 1-67. Additional Challenge: What if, instead of using pennies, you use small cubes to represent numbers? You could make a flat rectangular array, similar to what you can make with pennies, but you can also make a three-dimensional rectangular prism.
· For instance, you could represent the number 12 as a rectangular prism in different ways, as shown below.
· How many different rectangular prisms can be formed with the following numbers of cubes? Try to draw some of them.
1. 12
2. 16
3. 30
4. 120
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· Natural Numbers
· The numbers {1, 2, 3, 4, 5, 6, …} are called natural numbers or counting numbers. A natural number is even if it is divisible by two with no remainder. Otherwise the natural number is odd. The whole numbers include the natural numbers and zero.
· If one natural number divides evenlyinto another, the first one is called a factor of the second. For example, the factors of 12 are 1, 2, 3, 4, 6, and12. If a number has exactly two factors (1 and itself), it is called a prime number. If a number has more than two factors, it is called a composite number. The number 1 has only one factor, so it is neither prime nor composite.
· The prime numbers less than 40 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 31, and 37.
· 1-68. Harry had a pile of 48 pennies. He organized them into a rectangular array with exactly four rows with 12 pennies in each row.
· Draw diagrams to represent at least two other rectangular arrays he could use. Do you think there are more? Explain your thinking.
· 1-69. For each number of pennies below, arrange them first into a complete rectangular array and then into a different rectangular array that has a remainder of one (so there is one extra penny). Write an expression for each arrangement.
1. 10 pennies
2. 15 pennies
3. 25 pennies
· 1-70. How many pennies are represented by each expression below?
1. 3 + (4 · 5)
2. (4 · 3) + 7
3. (2 · 3) + 5 + (4 · 2)
· 1-71. Using whole numbers, fractions, and decimals, write at least eight addition equations that have a sum of 10. Write more if you can.
· 1-72. Use your knowledge of place value and decimals place the correct inequality sign (<, > ) between each pair of numbers.
1. 5.207 ___ 5.27
2. 3.006 ___ 3.06
3. 2.408 ___ 2.40
4. Round each number in part (b) to the nearest tenth.