(2.5) , , Because 5 2 means 5? 5 and 5 4 means 5? 5? 5? 5, 5 4 also equals 5 2? 5 2 = 25? 25 = 625.

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Clementine Atkinson
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Answers Investigation ACE Assignment Choices Problem. Core Other Applications ; Connections Problem. Core,, Other Applications,,, Connections,, ; unassigned choices from previous problems Problem.

1 Answers Investigation ACE Assignment Choices Problem. Core Other Applications ; Connections Problem. Core,, Other Applications,,, Connections,, ; unassigned choices from previous problems Problem. Core,, Other Connections ; Extensions, ; unassigned choices from previous problems Problem. Core Other Applications, Connections ; Extensions ; unassigned choices from previous problems Adapted For suggestions about adapting Exercise and other ACE exercises, see the CMP Special Needs Handbook Connecting to Prior Units : Moving Straight Ahead Applications. a. Number of Cuts b. =,; n Ballots c.. After cuts, there would be,, = ballots, which is over million ballots, but is less than million. Note to the Teacher Students should be familiar with exponential notation from the grade unit Prime Time.... (.).,..,. Because means? and means???, also equals? =? =.. Because has one more factor of than has, it equals? =,,? =,,.. A.. is less than million; possible explanation: The product of six s must be less than the product of six s, which is, or million.. is less than million; possible explanation: =????????? =???? =. So =, which is less than ( million).. is greater than or million, because to find, you multiply by itself six times. The result must be greater than if you multiply by itself times.. =. or... or. or ACE ANSWERS Investigation Exponential Growth

2 . a. Angie s Ancestors Generation Ancestors,,,. a. If you go back generations, the number of ancestors exceeds the number of people who have ever lived! Amoeba Reproduction Time (hr) Amoebas,,,,, Angie s Ancestors b. a = t c. hours. At hours, there will be,, amoebas present. Ancestors,,, Generation b. a = n, where a is the number of ancestors and n is the generation number. c.,. You can find this by adding , =,. Note to the Teacher: See the page for a description of how to use a calculator to find the sum of a sequence. The ancestor pattern is identical to the pattern in the paper-cutting activity of Problem. and is very similar to the chessboard pattern in Problem.. The only noticeable difference is that, in Problem., n = doesn t make sense because there is not a square. (The graph for the rubas problem in Problem. is a translation of the graph that applies to both Problem. and the ancestor problem.) d. Amoebas,,,,,, Amoeba Reproduction Time (hr) e. The pattern of change in the number of amoebas is similar to the pattern of change in the number of ancestors because it is exponential. The difference between the two patterns is that the number of amoebas increases more rapidly than the number of ancestors because the number of amoebas quadruples at each stage, while the number of ancestors only doubles. Growing, Growing, Growing

3 . a. (Figures,, and ) b. Plan : d = n Plan : d = n Plan : d = n c. Plan. a. Grandmother s and Aunt Lori s plans are linear and Uncle Jack s and Mother s plans are exponential. b. Grandmother: a = n - Mother: a = n Aunt Lori: a =.n +. Uncle Jack: a = n c. Grandmother: $; Mother: $,,,; Aunt Lori: $.; Uncle Jack: $,. a. Graph represents y = x because it is a curve. Graph represents y = x + because it is a straight line. b. In both graphs, the horizontal change is constant. In the graph of y = x +, the vertical change is also constant. In the graph of y = x, the vertical change increases.. a. Linear b. y =.x +. a. Exponential b. y = x. Neither. a. Exponential b. y = ( x ). Neither (In fact, this is quadratic. Students will study quadratic relationships in the unit Frogs, Fleas, and Painted Cubes.) ACE ANSWERS Figure Plan $ $ $ $ $ $ $ $ $ $ $, $, $, Figure Plan $ $ $ $ $ $ $ $, $, $, $, Figure Plan $ $ $ $ $ $, $, $, Investigation Exponential Growth

4 Connections. a. After cuts:,, <, in.; about. ft.; after cuts:,,, <,, in.; about. mi. b. cuts. A foot-high stack has =, ballots. Because cuts gives, ballots and cuts gives, ballots, Chen would have to make at least cuts to get, ballots.. Square : $.; square : $,.; square : $,,.; square : $,,,.; square : about $. ; square : about $.. a. When n =, the number of rubas is =.. The stack would have been about (. ). =. in., or. ft, or. mi high. b. It is, mi?, ft/mi? in./ft <.? in. to the moon, so it would take.?. <.? rubas to reach the moon.the stacks on squares (.? rubas) and above will reach the moon.. a. r = + (n ) or r = n + b. The graph will look linear. You might want to ask students to draw the graph. Number of Rubas Another Reward Plan Square Number c., rubas;, rubas. Slope: ; y-intercept: -. Slope: -.; y-intercept:.. Slope: ; y-intercept:. Answers will vary. Any equation with a coefficient of less than. Possible answers: y = x + ; y = x +. a. Square : =,,, Square : =,,, Square : =,,, b. Square. This is because it is the number of rubas on square times nine more factors of. c. The display probably reads. E. There is an E in the middle of the number. d.. (There are occasions when the calculator display will not give the last few digits exactly.) e. =. ; =. ; =. ; =. ; =. f. Possible answer: To write a number in scientific notation, place a decimal in the original number to get a number greater than or equal to but less than. To compensate for placing the decimal in the original number, multiply the new number by a power of ten that will give you back your original number Values given for Exercises are for the standard screen of the TI-. Different calculators will give different results..... Extensions. a. Eq. : r = - = - = Eq. : r = = = b. Eq. : r = - =, Eq. : r = = =, c. The equations give different values of r because the value of n is used differently. In one equation, is subtracted from n and the result becomes the exponent of ; in the other, n is used as the exponent of, and is subtracted from the result. Growing, Growing, Growing

5 Note to the Teacher For n $, the result of Equation will always be greater than that of Equation because the exponent is greater. Subtracting from the greater exponential contribution is almost insignificant.. a. b = n, where b is the number of ballots made and n is the number of cuts b. From the table, must equal. When you evaluate with a calculator, the answer is. c. The value of any number b raised to a power of is. Note to the Teacher Talk with students about why this makes sense. Because b = b and because exponents tell us how many factors of the base to use, b b = b + must be equal to + factors of b, which is just b; so b should be to make all of the other ideas about exponents work out. Some students might say that for the pattern to continue backwards, must equal. Explain that mathematicians decided the world of mathematics would make more sense if b were defined to be for b.. a. (Figure ) b. Each entry in the total column is less than the entry in the next individual square column. Another pattern is that we can double the total rubas at a square and add to get the total rubas at the next square. c. t = n d. The total t will exceed,, when squares have been covered: t = =,, rubas. e. With all squares covered, the total would be t = <. rubas.. a c. Answers will vary. Possible Answers to Mathematical Reflections. One key property of exponential growth patterns is that each time you increase the value of the independent variable by, you multiply the value of the dependent variable by the same constant number, the growth factor. Another key property of exponential growth patterns is that the numbers begin to grow very quickly. The graphs of exponential growth patterns are increasing curves.. With exponential growth patterns, you multiply the y-value by a constant each time the x-value increases by. With linear growth patterns, you add or subtract a constant to the y-value each time the x-value increases by. Also, the graphs of linear patterns are straight lines, while graphs of exponential growth patterns are curves. ACE ANSWERS Figure Square Reward Plan Number of Rubas on Square Total Number of Rubas, Investigation Exponential Growth

Answers. Investigation 4. ACE Assignment Choices. Applications. Latisha s Licorice. Licorice Remaining (in.) Problem 4.1. Problem 4.2. Problem 4.

Answers. Investigation 4. ACE Assignment Choices. Applications. Latisha s Licorice. Licorice Remaining (in.) Problem 4.1. Problem 4.2. Problem 4. Answers Investigation ACE Assignment Choices Problem. Core,, Other Unassigned choices from previous problems Problem. Core Other Connections ; Extension ; unassigned choices from previous problems Problem.