Answers Investigation 1

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1 Answers Investigation Applications. ; Possible explanation: 9 9 more than. is, so 0 9 is. ; Possible explanation: Since 8 6 =, 9 6 is close to.. ; Possible explanation: 6 is only 6 less than whole. 4. ; Possible explanation: of 00 is 0, so is closer to.. ; Possible explanation: is 0.0 and 4 is 0.7, so 0.67 is close to. 6. 0; Possible explanation: is a very small amount. It does not have any tenths in it, and is equivalent to tenths. 7. ; Possible explanation: 7 8 is a little less than and 4 9 is a little less than. Together, a little less than and a little less than is a little less than or closer to than to. 8. ; Possible explanation: 4 is closest to, and 8 is a small number that will make the sum closer to, but not greater than,. 9. ; Possible explanation: is a little more than... and. is.6, which is greater than, and closest to. 0. 0; Possible explanation: It would take two 4 s to equal exactly. Since 8 is less than 4, 4 and 8 is less than and closer to 0.. ; three tenths and eight tenths have a sum greater than, so the total sum here is greater than.. ; 0 4 is equivalent to.4 which, when added to 0.7, is.77, which is close to.. 0.; Possible explanation: is a bit more than half, and 0 is a small fraction, not big enough to push close to ; Possible explanation: 4 is exactly in the middle of 0 and. 0 is a small fraction so the sum is closer to, but not greater than.. 0; Possible explanation: 4 = 8. Also, 4 is exactly halfway between 0 and , so Therefore the sum is closer to 0 than to. 6. a. This is incorrect , so the sum here is less than 4. b. Correct. Each fraction is equal to, so the sum is. c. Correct , so the sum of and 6 is also greater. d. Correct. 0 = and 8 7 4, so the sum is greater than Possible answer: 4 and 7 8. Possible answer: 8 and Possible answer: 8 and 0. Possible answer: 8 and 7 8. a. The greatest possible sum is b. The least possible sum is a. No; In the price list, the whole numbers add up to 9: = 9. The cheese is +.9, which brings the total to There s clearly enough in the cost of other items to put the total over +0. (The exact cost of the groceries is +.4.) b. Possible answers: Milk, cheese, avocado (+4.9); eggs, cheese, honey, bread ( +4.9) c. cereal, honey, and avocado ( +4.94). 8 is closest to. Possible explanation: 4 8 =. The other 8 makes 8 just a little more than, but not close to 8 8, or whole. Let s Be Rational

2 Answers Investigation 4. Soo has enough molding. 7 8 is 8 less than whole and 8 7 is 7 7 more than whole or 7. Since 7 7 8, when the 7 is added to the 7 8 it will be greater than yard, and the total will be greater than yards.. He had about 8 quarts. 6. No; If you add 4 to 4, you will get. But 8 is a little less than 4, so there is not enough. 7. a. Marigolds: 0 ; Lantana: 0 ; Impatiens: 0 ; Petunias: 0 ; Lilies: ; Begonias: 0 ; Tulips: 0 ; Daisies: 0 ; Irises: 0 b = 0 c = 6 0, or 0 d. Incorrect. Possible explanation: The number sentence for the situation is 0-0 = If you work out the subtraction problem on the left of the equal sign and the addition problem on the right, the answers are not the same. e. Possible combinations that total 0, the fraction planted with impatiens: Marigolds + Petunias + Lantana: = = 6 0 = 0 Lilies + Petunias: = 6 0, or 0 Marigolds + Begonias + Tulips + Daisies: = 6 0, or 0 8. a = 6 of the page is used for ads. b. - 6 = 6 of the page remains (three 4 -page ads, or * 4 ) plus 4 8 (four 8 -page ads, or 4 * 8 ) plus 0 6 (ten 6 -page ads, or 0 * 6 ) = 7 8 pages = 8 pages = 8 = 6 of the lasagna is eaten, leaving 6 of the lasagna uneaten = = 7 8 of a small bag of chips or is greater: + 6 = = 9 6 = = 0 0 ; = = is greater: = = 6 = = 0 ; - 0 = = = = = = = = or In all of the problems, you add unit fractions where one fraction is half the size of the other. The fraction in each part with the lesser denominator is twice the value of the unit fraction with the greater denominator. You can think of the unit fraction with the lesser denominator as two unit fractions with the greater denominator. This gives a sum with a in the numerator over the greater denominator.. No. If 4 6 of all the pizza were eaten, this would be less than one whole pizza. If there are eight sections in each pizza, then people are eating eighths. And all together they ate 4 8 or 6 8 pizzas. Let s Be Rational

3 Answers Investigation Connections. Divide the rectangle into thirds. Each third of the rectangle represents 0, of the original whole. So, 00, is represented by the blue area.. beans. There are 9 beans shown in the picture representing three fifths. So each fifth must contain beans. Since the whole is five fifths, there must be beans on the counter. 4. a. is 0. and 0, is about 0. and, 4 is 0. and, is about 0.66 or 66, (Some students will argue for 0.67 and 67,, if the convention of rounding up is used.) 4 is 0.7 and 7, 6 is about 0.6 and 6, (Some students will argue for 0.7 and 7,, if the convention of rounding up is used.) is 0. or 0, 8 is 0. or., b. (See Figure.). D 6. a and d a. This set of fractions can be renamed as hundredths because each denominator,, 4, and, are factors of 00. b. This set of fractions cannot be renamed as hundredths. The denominator 0 is a factor of 00, but and are not. c. This set of fractions cannot be renamed as hundredths. The denominators 6 and 8 are not factors of 00. d. This set of fractions can be renamed as hundredths because all the denominators,, 0, and 0, are factors of a. The least possible sum is 4. This is the sum of the least numbers in each of these intervals. b. The greatest possible sum is. That is the sum of the greatest numbers in each of these intervals. 8. N = 9. N = 60. N = 8 6. N = 4 Figure Let s Be Rational

4 Answers Investigation 6. 6 = = = = 6 = G 68. Answers will vary. Possible answers include: Foley, Burg, Walker, Theule, and Lapp = a. of a whole; squares = of a whole; 9 squares = whole; so squares = of a whole. b. 7 9 of a whole; squares = or 9 of a whole; 9 squares = or 9 9 of a whole; so 7 squares = 7 9 of a whole. 70. a. Note: The purple shaded region represents. The green shaded region represents the portion of. The blue shaded region represents 6. Or, b. Note: The Xs indicate the regions subtracted. The purple and green shaded regions each represent. The blue shaded region represents. The remaining portion is 4. Or, 7. a. Since 70 0 = 7 0 and 0 = 0, the sum of these two numbers is the same no matter which forms we use. b. Answers will vary. Possible answers include: and c. Any problem with common denominators should be easy to solve. This includes Maribel s problem as well as this one: Let s Be Rational 4

5 Answers Investigation Extensions 7. Answers will vary. Possible answers include: 7. Answers will vary. Possible answers include: It is always possible to find a fraction between any two fractions on the number line. One way to know this is that we can rewrite each given fraction as an equivalent fraction, using common denominators, and then choose a new fraction with a numerator between the two new numerators. For example, given and we can rewrite as 4 and 6, and then we know is between these. Another method would be to take the number that is halfway between the two fractions, which is the average of the two fractions. For example, to find the fraction between 4 and 4 we can take the average. ( ) = ( ) = * 0 = 40 Now 4 = = 4, so 40 is between 4 and a. page costs +, 6 page costs +0, 8 page costs +0, 4 page costs +40, page costs +80, full page costs +60. b. 40 * + 0 * * = +0 c. Yes. 8 page costs +40 and 6 4 page costs +40, for a total of +80. d. Answers will vary. Possible answers include: Any combination adding up to page. This includes two 4 -page ads, or one 4 -page ad and two 8 -page ads. 76. a. acres; 6 people can clear twice as much in the same amount of time. b. 4 acre; one fourth of the people can clear one fourth as much in the same amount of time. c. 4 people; three times as many people are needed to clear three times the area in the same amount of time. d. 48 people; twice as many people are needed to complete part (c) in half of the time. 77. a. Answers will vary. Possible answers include: - 4 =. b. Answers will vary. Possible answers include: - 6 = 0. Let s Be Rational