# problems palette of David Rock and Mary K. Porter 6. A local musician comes to your school to give a performance

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1 palette of problems Davd Rock and Mary K. Porter 1. If n represents an nteger, whch of the followng expressons yelds the greatest value? n,, n, n, n n. A 60-watt lghtbulb s used for 95 hours before t burns out. What s the lfe of the lghtbulb n klowatt-hours? 3. How many multples of 4 from 1 through 1000 do not contan any of the dgts 6, 7, 8, 9, or 0? 4. Fnd the dfference between the sum of the frst 1,000,000 postve even numbers and the sum of the frst 1,000,000 postve odd numbers. 5. Whle fshng, you and a frend dscover that your boat has sprung a leak. You estmate that the leak lets n water at a rate of 8 gallons per mnute. Whle you row, your frend can dump water out of the boat at a rate of 3 gallons per mnute. The boat wll float as long as there are fewer than 40 gallons of water n t. If you are 600 feet from the dock, how fast, n mles per hour, do you need to row the boat (on average) to make t back to shore before the boat snks? 6. A local muscan comes to your school to gve a performance n the gym. The gym s arranged wth chars so that the 150 members of your grade are all seated n rows. Each row contans the same number of chars, and every char s occuped. If 10 more chars were added to each row, everyone could be seated n 4 fewer rows, allowng the back-row students to be closer to the performance. How many chars are n each row n the orgnal confguraton? 7. A mathematcs teacher collects old books. One day, a student asked hm how many old math books he has. He repled, If I dvde the books nto two unequal whole numbers, then 64 tmes the dfference between the two numbers equals the dfference between the squares of the two numbers. How many old math books does the teacher have? 8. Usng an analog clock, fll n the blank to complete the pattern of tmes: p.m., 1:37 a.m., 11:05 a.m., 8:33 p.m. 3 Mathematcs Teachng n the Mddle School Vol. 14, No. 1, August 008 Copyrght 008 The Natonal Councl of Teachers of Mathematcs, Inc. All rghts reserved. Ths materal may not be coped or dstrbuted electroncally or n any other format wthout wrtten permsson from NCTM.

2 Prepared by Davd Rock, Columbus State Unversty, 45 Unversty Ave., Columbus, GA 31907, and Mary K. Porter, Sant Mary s College, Notre Dame, IN MTMS readers are encouraged to submt sngle problems or groups of problems by ndvduals, student groups, or mathematcs clubs to be consdered for publcaton. Send to the Palette edtor, Davd Rock, at rock_davd@colstate.edu. MTMS s also nterested n students creatve solutons to these problems. Send to The Thnkng of Students edtor, Edward S. Mooney, at mooney@lstu.edu. Both problems and solutons wll be credted. For addtonal problems, see the NCTM publcaton, Menu Collecton: Problems Adapted from Mathematcs Teachng n the Mddle School (stock number 76). 9. Fnd the tmes that fall exactly between the other two tmes: 10:49 a.m.,, 1:57 p.m. 10:49 p.m.,, 1:57 p.m. 10. In a quz game, a player earns 5 ponts for every correct answer and loses 3 ponts for every wrong answer. If Steve s score s 0 after answerng 40 questons, how many questons dd he answer correctly? 13. What are the next three letters n ths sequence? c, a, x, f, b, v,, c, t, l, d, r, o, e,,, 14. How many squares (of any sze) are n the fgure below? 11. What percentage of the fgure shown here s shaded? 1. What are the next three letters n ths sequence? 15. How many nonsquare rectangles (of any sze) are n the fgure shown n queston 14? c, a, f, b,, c, l, d, o, e,,, (Solutons on pages 34 36) Vol. 14, No. 1, August 008 Mathematcs Teachng n the Mddle School 33

3 solutons to palette (Alternatve approaches to those suggested here are encouraged.) (Contnued from pages 3 33) 1. If n 0, then n yelds the greatest value. If n = 1, then n and /n both yeld the greatest (same) value. If n >, then n yelds the greatest value. Makng a table helps to determne the greatest value n each case. (See table 1 at rght.). 5.7 klowatt-hours; 60 watts 95 hours reveals the number of watt hours used, whch s 5700 watt hours. To convert watts to klowatts, multply by 1/1000 snce a watt s 1/1000 of a klowatt. Therefore, 5700 watt hours yelds /1000 = 5.7 klowatt hours All multples of 4 are even. The only sngle-dgt multples of 4 are 4 and 8. Therefore, 4 would be the only one-dgt value that does not contan a 6, 7, 8, 9, or 0. Next, examne the two-dgt numbers. Snce we cannot use the dgts 6, 7, 8, 9, we only look for even numbers less than 60 wth unts dgts of and 4. The two-dgt multples of 4 would be restrcted to 1, 4, 3, 44, and 5. Next, examne three-dgt numbers. They would have to end n 1, 4, 3, 44, and 5 but could only start (hundreds dgt) wth 1,, 3, 4, and 5. Therefore, there are 5 5 = 5 threedgt possbltes. There s 1 one-dgt number, 5 two-dgt numbers, and 5 three-dgt numbers, whch sum to = 31 possbltes. Extend the problem to more than three dgts. Snce the same 5 two-dgt numbers must be present n all the numbers wth the condtons mposed, a pattern s created for the total number of possbltes. For example, wth the numbers 1 through 10,000, there are = possbltes. Table 1 A tabular soluton to problem 1 n n / /n n n n undefned ,000,000. One strategy would be to try a smpler problem. The sum of the frst 10 postve odd numbers (1, 3, 5, 7, 9, 11, 13, 15, 17, 19) s 100, and the sum of the frst 10 postve even numbers (, 4, 6, 8, 10, 1, 14, 16, 18, 0) s 110. The dfference s 10, whch s the number of odd and even numbers. We can generalze as follows: The frst odd number s 1, and the frst even number s. The dfference between the sums s 1. If we look at the sums of the frst three odd numbers ( = 9) and the frst three even numbers ( = 1), we notce that the dfference s 3, the number of even and odd numbers. Therefore, the even numbers wll be 1 greater for each number. Snce we are workng wth the frst 1,000,000 even and the frst 1,000,000 odd numbers, the dfference s 1,000, Approxmately 3.7 mph. Snce per mnute the leak lets n 8 gallons and your frend can remove 3 gallons, the water comes n at a rate of 5 gallons per mnute. Therefore, you have 8 mnutes before you reach the 40-gallon lmt, because 40/5 = 8. You must travel 600 feet n 8 mnutes; 600/8 = 35 feet per mnute. Convert 35 feet per 1 mnute to mles per hour. Multply 35 ft. 1 mn. 60 mn. 1 hr. = 19,500 ft. per hr. Next, multply 19,500 ft./1 hr. by 1 m./580 ft. (or dvde 19,500 by 580, snce there are 580 feet n a mle) to yeld approxmately 3.7 mles per hour. 600 ft. 8 mn. 60 mn. 1 hr. 1 m. 3.7 mph 580 ft. 34 Mathematcs Teachng n the Mddle School Vol. 14, No. 1, August 008

5 example, f s found three letters after c n the alphabet. Thus, the sequence formed by the underlned terms looks lke ths: c, f,, l, o, r, u. Therefore, the orgnal sequence contnues n ths way: c, a, f, b,, c, l, d, o, e, r, f, u, g,..., so the next three terms n the orgnal sequence are r, f, u. 13. p, r, f. If we examne ths sequence (c, a, x, f, b, v,, c, t, l, d, r, o, e), we notce that t s actually made up of two sequences: The underlned terms (x, v, t, r) and the ones that are not underlned (c, a, f, b,, c, l, d, o, e). The terms that are not underlned comprse the sequence n the prevous problem, so the next terms of that sequence, as determned n the soluton to the prevous problem, are r, f. The underlned terms (x, v, t, r) also follow a pattern: Each term s the letter n the alphabet that s found two letters before the prevous term. For example, v s found two letters before x n the alphabet. Thus, the sequence formed by the underlned terms looks lke ths: x, v, t, r, p. Therefore, the orgnal sequence contnues n ths way: c, a, x, f, b, v,, c, t, l, d, r, o, e, p, r, f,..., so the next three terms n the orgnal sequence are p, r, f squares. There are 4 squares of sze 1 1 (1 unt by 1 unt). There are 3 squares of sze. There s only 1 square of sze 3 3 and just 1 square of sze 4 4. Thus, there are 9 squares n ths fgure rectangles. As shown n the prevous problem, 9 squares are n ths fgure. Count the number of nonsquare rectangles n ths fgure: 7 rectangles are 1, 3 rectangles are 1 3, 1 rectangle s 1 4, 4 rectangles are 3, rectangles are 4, and only 1 rectangle s 3 4. Ths gves us a total of 1 nonsquare rectangles. Thus, we have = 7 rectangles n the fgure. l Soluton to August s Solve It (Contnued from page 31) The number of cows n pen C s four tmes the number of cows n pen B. There are multple ways to solve ths problem, ncludng by example and algebracally, as shown below. The frst condton ndcates that the number of anmals n each pen was the same before and after the anmals escaped. Let varables represent the number of anmals n each pen. The varable wll be the same value before and after the anmals escaped. Let A represent the number of anmals n each pen. The second condton ndcates that the number of pgs n pen A s the same as the number of pgs n pen B. Snce these numbers are equal, let one varable represent both values. Let x represent both the number of pgs n pen A and n pen B. Smlarly, snce the number of sheep n pen A s the same as the number of sheep n pen C, use the same expresson to represent each value. From the fourth condton, the number of sheep n pen A s /3 the number of pgs n pen A; the number of sheep n pen A and pen C can each be expressed by (/3)x. Table 1 lsts Table 1 The anmals after they escaped Number of cows Table The updated table Number of cows the anmals n each pen after they escaped. There are (/3)x sheep n pen A and (/3)x sheep n pen C. Orgnally, there were A sheep n pen B. Ths means that there are A (/3)x (/3)x, or A (4/3)x, sheep stll n pen B. There are x pgs n pen A and x pgs n pen B. Orgnally, there were A pgs n pen C. Ths means that there are A x x, or A x, pgs n pen C. See table for these updates. We know how many sheep and pgs are n each pen, and we know the total anmals n each pen. We can fnd the number of cows n each pen by subtractng the number of both sheep and pgs from the orgnal number of anmals n the pen. By examnng pen B, we know that the number of anmals s A. The number of sheep n the pen s A (4/3)x, and the number of pgs s x. So the number of cows n pen B s A (A (4/3)x) x, or (1/3)x. In pen C, the number of anmals s A; the number of pgs s A x, and the number of sheep s (/3)x. So the number of cows n pen C s A (A x) (/3)x, or (4/3)x. Ths means that the number of cows n pen C, (4/3)x, s four tmes the number of cows n pen B, (1/3)x. l Pen A Pen B Pen C Number of sheep (/3)x (/3)x Number of pgs x x Pen A Pen B Pen C Number of sheep (/3)x A (4/3)x (/3)x Number of pgs x x A x 36 Mathematcs Teachng n the Mddle School Vol. 14, No. 1, August 008

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