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,`~..~..~.r.. I A. MARCY'S DOTS Participant Handout A pattern of clots is shown below. At each step, more clots are aclclecl to the pattern. The number of clots aclclecl at each step is more than the number aclclecl in the previous step. The pattern continues infinitely. ( 1 St step) (2n~ step) (try step) 2 Dots 6 Dots 1 2 Dots Marcy has to determine the number of clots in the 20th step, but she does not want to clraw all 20 pictures and then count them. Explain or show how she could clo this and give the answer that Marcy should get for the number of clots. Dicl you use the calculator on this question? ~ Yes ~ No SOURCE: National Assessment of Educational Progress (NAEP), 1992 Mathematics Assessment

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Sample student responses - - te-n Unit ~ of et ~ 5 , - 4~ I' I~ 5~. - , L. =~ - er ~ ~4 t~ - th~ Ah ~ ~ &M pve r~ ~ t~' ~ ~ ~ 5G Ail 30 16 ~0 - 930d~b ~4 `~ -1~^ MY ,~. . _~O Ad. , ~- - ~ ~ ~ - - ~ ~J ~- = \~ -15 - ~ -- ~ - - - t ~ a_ _~r ~ ~ ~ ~ ~ ~ ~ ~ Aft 411;, ~ ~q ~ ~O ~ ~ ~t t~+ q~ +~ - )~? ~ Iowa 11~ yet Q0 ~ ~S m~Y ~ ~ 6 of ~ S-.F~ ~` : Am - ' ~` ~,~ ~ ~ ~ ~ ~: ~ $ I__ f ~ ~ ac~ {~: fop ~: ~ plot ]: ~/ /~l topic 93 4~4 ~ ~ Bed ~` BETA ~, ~50 ~ , Ct ~ ~ 4 4~t~ L it- ~ Ins ~ ~ ,~ x ~ ~ ~1 to ,: - At. - ~. APPE N DIX 4 O~ cocci - arc cat on 0 ~0` ]0 4 ~ ~)0 ~:-~3 $= 1 ~ - 1.~ 2~........ ~ z.~.

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~LT~Pu~ ~ ~ - ~ `~ h t ~" h 0~' ~-0 SOFT \t Fly +~t~ ~^ -Gil dh, it_ ~ ~Q ~AFT hats ~ =$ t~ ~ ~ ~ ~ - ~9 _ ~Q,1 ~:~ : `4~ - aft ' l ~Rho ~ OF ~ 1 r ~ ~5 a. ~ I; ~ ~- -~4 ~ ~ ~ ~' ~if ~ ~4 ~ _~ ~ ~ _> (~r 2) 76 . ~ Of _ e, ~ - ~5 ~- ~ - -,_ ~__[/~ ~ ~ h ~ ~ t ~ ~ ~ ~ ~ ~ ~ ~ ~ 3 ~ ~ it. - Ivy -~h~ Bet I,_ ~ -~ C~3 . ~ ~ ~ ~ =~ ~ h ~ ~ ~ ~] ~ ~ ~Ash oh ch ~ T' ?~ I, ~-~ ~ ~ ~& -A - Aim' MARCY'S DOT ACTIVITY ~r 4~ ~ t ~ }6 ~ $ ~ - t ~i bard ~ Ah ~I oh ~ ~ ~ ~ The ~;l ~ ~ ~ ~ 6~ ~ ~ ~ ~ ~ ~ 5 ~ ~ _ ~ ~ ~ ~i ~ ~ p ~ 7h ~ r~w ~ ~ ;< 2 ~ 9, ~I LIZ L I Hi-` ~ 3 ~ d 1 %., {] ~ ~T g ~h ~ ~ I ~ 1 3 hxr I ~ ~ ~ I ~ ~ ~ 3-i5 i Ah4`tO 4~ 2t0r7~'F ~lv 4 x l ~ l xl I ~ 16 1 (7.~X; l ~l l'r;~1, ~ld ~e =~$( ~h i~ 11 II5lT th~ C~ ~ 1 ~ h Oh 4n ch~ ~ 4um ~ i 0 n i ~ ~1 ~ ~ ~ ~ Ol5, ~~O ;~ 5~$~` $~' ~ ~ ~ ~ ~ ~ T'~ ~ ~ i ~ 39 3~ ~ t`~ ~ ,5~ ~ ~ ~ t d~ ~

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FaciIitator's guicIe for the Marcy's Dots activity Marcy's Dots 1. To get participants involved in thinking about content and learning, have them solve Marcy's Dots, a prob- lem in the content area of algebra and function from the 1992 National Assess- ment of Educational Progress (NAEP) Grade ~ Test. About 20% of the test for grade ~ assessed algebra and function, and this is Apical of an extended con- structed-response question. The following information might be shared after or at an appropriate time during the discussion. Do not start off with this but use it after people have become engaged and come to some conclusions about the problem and about the ways students found their solutions. Only 6% of the students provided a satisfactory or better re- sponse; 6% provided a generalization; 10% made some attempt at a pattern; 63% provided inaccurate or irrelevant information; and 16% did not respond. (Remember that this test is a no stakes test for students.) 2. Investigate Student responses. Points that might be made in the .. . cl~scuss~on: Wrong answers usually occurred because they had the wrong notion of the pattern. APPE N AX 4 Student ~ alternated between adding 4 and adding 6. Student 5 multiplied by 3 then by 2 as the pattern. Student 7 multiplied by 3 then by 2. Student 12 increased by multiplying by2. No recognition of pattern: Student 9 added the numbers in the problems. Student 3 added 6, the last set of dots. Right Answers: Student 2 used the pattern within the pattern and the picture; wrote out all steps (recursion). Student 4 used relation between rows and columns and was able to general- ize to a rule. . Student 6 used relation between rows and columns from picture and was able to generalize to a rule. Student ~ used the pattern within the pattern: recursively adding two more each time (4, 6, 8, ...) Student 10 focused on relationship between rows and columns, wrote out all steps (recursion). Student 11 used relationship numbers and was able to generalize to a rule. .

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3. Discuss the problem from the perspective of its role in the middle grades mathematics curriculum. How does the plenary session on content and learning relate to the problem? How does the problem fit into the larger picture of algebra and algebraic reason- ing? Why is it important for students to recognize and be able to work with patterns? 4. Hand out the excerpts from the draft section of the middle grades algebra section from "Principles and Standards for School Mathematics: Discussion Draft," the Standards 2000 MARCY'S DOT ACTIVITY draft being prepared for dissemination, comment and input this fall. Provide a few minutes for people to read the excerpts. How does the mathematics in this task relate to the discussion of algebra in the document? Resources for Marcy's Dots Dossey, I.A, Mullis, I.V.S., & [ones, C.O. (1993~. Can students do mathematical problem solving? resavltsirom constra~cted-response questions in NAEP's 1992 Mathematics Assessment. Washington, DC: National Center for Educa- tion Statistics. Kenny, P.A, Zawojewski, J.S., & Silver, E.A (1998~. Marcy's Dot Problem. Mathematics Teaching in the Middle School, 367}, 474-477.

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TABLE 2. ~ ~ National Results for Demographic Subgroups for the Extenclecl-Response Task, Marcy's Dot Pattern, Gracle 8 No Satisfactory Response Incorrect Minimal Partial Satisfactory Extended or Better Nation 16(1.0) 63(1.3) 10(0.7) 6(0.7) 1 (0.2) 5(0.6) 6(0.7) Northeast 18(3.2) 61 (3.2) 10(1.9) 4(0.7) 2(0.5) 6(1.8) 8(1.6) Southeast 20(2.0) 6A (2.2) 9(1.5) 3 (07) 1 (oh) ~ (1.1) ~ (1.3) Central 10(1.5) 65(2.1) 10(1.~) 8(1.~) 1 (oh) 6(1.1) 7(1.~) West 16(2.0) 62(2.8) 10(1.1) 7(1.8) 0(0.2) ~ (1.1) ~ (1.1) White 12(1.1) 63(1.5) 11 (0.8) 7(0.8) 1 (0.2) 6(0.8) 8(0.9) Black 2A (2.9) 67(2 9) 6 (1.6) 2 (0.9) 0 (0.0) 1 (o.5) 1 (o.5) Hispanic 28(2.8) 61 (3.1) 7(2.0) 3(1.2) 0(00) 1 (o.5) 1 (o.5) Male 19(1.5) 63(2.2) 8(1.0) 5(0.9) 1 (0.2) 5(0.9) 5(09) Female 13(1.2) 63(1.6) 12(1.1) 6(1.0) 1 (o.3) 5(0.8) 6(0.9) Advantaged Urban 8(2.9) 62(5.1) 10(1.9) 6(1.6) 1 (0.6) 11 (2.5) 13(2.6) Disadvantaged Urban 32 (3.9) 59 (A 7) ~ (1.3) ~ (1.9) 1 (0.6) 1 (o.5) 1 (o.7) Extreme Rural 16(2.9) 69(3.6) 8(2.3) 2(1.1) 1 (o.7) A(2.0) 5(2.3) Other 15 (1.3) 62 (1.5) 11 (0.9) 6 (0.9) 1 (0.2) ~ (0.7) 5 (0.7) Public 16(1.2) 64(1.4) 9(0.8) 6(0~7) ~ (0.2) 4(0.6) 5(0.6) Catholic and Other Private 11 (1.7) 56(2.7) 12(1.6) 7(1.2) 2(0.9) 10(2.2) 13(2.0) The standard errors of the estimated percentages appear in parentheses. It can be said with about 95 percent certainty that for each population of interest, the value for the whole population is within plus or minus two standard errors of the estimate for the sample. In comparing two estimates, one must use the standard error of the difference (see Appendix for details). When the proportion of students is either 0 percent or 100 percent, the standard error is inestimable. However, percentages 99.5 percent and greater were rounded to 100 percent and percentages 0.5 percent or less were rounded to 0 percent. Percentages may not total 100 percent due to rounding error. SOURCE: National Assessment of Educational Progress (NAEP), 1992 Mathematics Assessment APPE N DIX 4