NAME DATE. b) Then do the same for Jett s pennies (6 sets of 9 pennies with 4 leftover pennies).


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1 NAME DATE 1.2.2/1.2.3 NOTES Cody and Jett each have a handful of pennies. Cody has arranged his pennies into 3 sets of 16, and has 9 leftover pennies. Jett has 6 sets of 9 pennies, and 4 leftover pennies. Each student thinks he has the most pennies. a) Work on your own to draw a diagram and write an expression with numbers and symbols. Both your diagram and expression should represent the way Cody could have arranged his pennies (3 sets of 16 with 9 leftover pennies). Is there a different way you could have represented this same number of pennies with a diagram and numerical expression? b) Then do the same for Jett s pennies (6 sets of 9 pennies with 4 leftover pennies). c) Compare your results with your team. d) Copy the different numerical expressions for each student from your team to your paper. How many different ways did your team find to represent the number of pennies with diagrams and numerical expressions? e) With your team, decide which arrangements best represent the groups of pennies held by Cody and Jett. a. Which student has more pennies? How did you figure this out? b. Jett decided to rearrange all of his pennies into groups of 10, even though one group will not be complete. How many groups can he make? How can he represent his new grouping with a number expression?
2 154. Allen and Dwayne would like help comparing two piles of pennies. The pennies are arranged and represented in the diagrams at right. a) Which pile has more pennies? How do you know? b) With your team, write expressions that represent each pile. Then write a comparison of these expressions showing if one is greater than the other or if they are the same Doreen and Delilah were comparing pennies from two teams. They wrote the comparison statement 6(16) + 3 > 4(9) + 4(9) + 2. What arrangements could this represent? Can you find more than one possibility? Work with your team to draw diagrams of the arrangements of pennies that Doreen and Delilah could have been comparing The figure below is reprinted from problem Using different colors within the pattern may help your team find various patterns. a) Working with a partner, write as many numerical expressions as you can to describe the number and organization of dots in this figure. How many different ways can you see the pattern? b) Now compare your numerical expressions with those from the rest of your team. Are some easier to match to the diagram than others? As a team, choose two numerical expressions that represent the dots in the figure in very different ways. Be sure that everyone has these two expressions written on their own papers. c) Find the value of both expressions. How do they compare?
3 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. 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? 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. (Be sure to draw the arrays on your paper to use later.) a. 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. b. Whenever Gigi arranges her pennies into a rectangular array with more than one row or column, she has a remainder (some leftover pennies). c. Ann can arrange all of her pennies into five different rectangular arrays.
4 163. What can you learn about a number from its rectangular arrays? Consider this question as you complete parts (a) and (b) below. a. A number that can be arranged into more than one rectangular array, such as Ann s in part (b) of problem 162, is called a composite number. List all composite numbers less than 15. b. 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 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? 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).
5 NAME DATE 1.2.2/1.2.3 HOMEWORK Rules Review: (Remember to check examples in the Rules Section of your binder for help!) Write in words and expanded form, what each numerical expression represents and then solve. 1.) 4 x (7 + 3) 2.) 3 (9 + 6) Words: Expanded: Solution: Words: Expanded: Solution: Which is greater: three sets of (5 2) or two sets of (2 + 3)? Draw diagrams to support your conclusion The diagrams at right represent piles of pennies. Which pile has more pennies? Explain your reasoning. a. Write two different numerical expressions to represent the number of pennies in each pile. b. Write a number comparison statement (using >, <, or = ) to show if the number of pennies in one pile is greater than the other or if they are the same.
6 161. Aria and 19 of her friends plan to go to a baseball game. They all want to sit together. Aria wants to order the seats in the shape of a rectangle, but she cannot decide on the best arrangement. She starts by considering one row of 20 seats. Draw a diagram showing Aria s idea for a seat arrangement. Then draw all of the other possible rectangular arrangements for 20 seats. Label each arrangement with its number of rows and the number of seats in each row. Are all arrangements practical? Explain 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 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. a. 10 pennies b. 15 pennies c. 25 pennies How many pennies are represented by each expression below? a. 3 + (4 5) b. (4 3) + 7 c. (2 3) (4 2)
7 Comparisons Mathematical symbols are used to compare quantities. The most commonly used symbols are the two inequality signs (< and >) and the equal sign (=). You can see how these symbols are used below. greater than: > 7 > 5 less than: < 3 < 5 equal to: = = 3 greater than or equal to: 4 4 less than or equal to: 8 9 Natural, Whole, and Prime 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 another without remainder, the first one is called a factor of the second. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. 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, 29, 31, and 37.
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