make it 36 - university of idaho...edited by barbara britton,[email protected], and carla...

T he goal of the “Problem Solvers” departmentis to foster improved communication amongteachers by posing a problem for K–6 teach-

ers to try with their students. Every teacher canbecome an author: pose the problem, reflect onyour students’ work, analyze the classroom dia-logue, and submit the resulting insights to thisdepartment. Every teacher can help us all betterunderstand children’s capabilities and thinkingabout mathematics with their contributions to thejournal. Remember that even student misconcep-tions provide valuable information.

Classroom SetupSpend a few minutes discussing the problem withthe class. You may give the students centimetergraph paper, scissors, square tiles, or geoboards.Begin by asking the class to cut out a square thatmeasures 6 centimeters by 6 centimeters using cen-timeter graph paper. Give your students time to thinkthrough the problem. Have them explain why thearea is 36 square centimeters. Challenge the class to

find another rectangle that has the same area. When the students successfully find a rectangle

with an area of 36 square centimeters, challengethem to manipulate the dimensions and find all therectangles with this area. Agree as a class aboutwhether you will consider a 4 × 9 rectangle to bethe same as a 9 × 4 rectangle. Watch to see if anyof your students consider rectangles whose sideshave lengths that are not whole numbers. Encour-age students to record the rectangles that they findalong with their measurements.

The problem can be modified to make it easierfor younger children by using a smaller number forthe area. Working with larger units and tiles wouldalso be helpful. For example, find five shapes withan area of 12 square inches. Allow students to usea geoboard or square tiles and consider rectangleswith only whole-number lengths.

As students begin to investigate other shapesincluding triangles, trapezoids, and parallelograms,the problem becomes more open-ended and chal-lenging. Encourage students to look for patterns tohelp them generate more examples. Again, theyshould find a way to record the shapes they dis-cover along with their measures. An extension ofthis problem is to ask students to determine whichof their shapes has the largest perimeter. Can theyfind the shape with the largest perimeter that has anarea of 36 square centimeters?

We are interested in how your studentsresponded to this month’s problem (or the modi-fied version for younger children) and how theyexplained or justified their reasoning. Please sendus your thoughts and reflections. Include informa-tion about how you posed the problem and samplesof students’ work, or even photographs showingyour problem solvers in action. Send your resultswith your name, grade level, and school by April1, 2005, to Carla Tayeh, Mathematics Department,Eastern Michigan University, Ypsilanti, MI 48197.Selected submissions will be published in a subse-quent issue of Teaching Children Mathematics andacknowledged by name, grade level, and schoolunless otherwise indicated. ▲

(Solutions to a previous problem begin on page 332.)

Make It 36

330 Teaching Children Mathematics / February 2005

Edited by Barbara Britton, [email protected], and Carla Tayeh, [email protected], Eastern Michigan University

PROBLEM SOLVERS Car la Tayeh and Barbara Br i t ton

Problem

Begin with a square that measures 6centimeters by 6 centimeters. The areaof the square is 36 square centimeters.How many different shapes can youfind that have the same area? Forexample, in addition to the square,how many other rectangles can youfind that have an area of 36 squarecentimeters? How many triangles canyou find that have an area of 36square centimeters? Can you make atrapezoid that has an area of 36square centimeters? What about aparallelogram? What other shapescan you construct that will have anarea of 36 square centimeters?

This material may not be copied or distributed electronically or in any other format without written permission from NCTM. Copyright © 2005 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.

Teaching Children Mathematics / February 2005 331

This problem has the potential to help students build connections among a number of related concepts in geometry. Students iden-tify a variety of shapes with the same area and come to recognize that regions can be rearranged and the area will remain constant.With more exploration, students can begin to build a solid foundation on which to derive the formulas for area. For example, the areaof a triangle is half the area of a given rectangle. Hence, students might begin with a rectangle whose area is 72 square centimetersand then cut it in half to make a triangle with an area of 36 square centimeters. Some students may generate entire families of trian-gles with an area of 36 square centimeters by drawing triangles with the same base and height as in figure 1. Other students maybegin to explore the connections between the formulas for the area of a rectangle and the area of a parallelogram by transforming arectangle into a parallelogram (see fig. 2). Using this strategy, students can generate a family of parallelograms with the same area.

In the elementary mathematics classroom, encouraging students to focus on one topic in more depth and allowing them the timeto explore helps students build connections and make sense of important mathematical concepts. In this problem, while investigat-ing shapes with the same area, students begin to develop strategies for finding areas of figures without relying solely on memorizedformulas.

Where’s the Math?

Figure 1Family of triangles with an area of 36 square centimeters

Figure 2Square and parallelogram with an area of 36 square centimeters

T he Muffin Mania problem appeared in theFebruary 2004 issue and generated manyhigh-quality responses and ideas. The origi-

nal problem and its extensions appear below.

Goldilocks’s grandmother hands a basket offreshly made peanut-butter-and-chocolatemuffins to Goldilocks and says, “These are foryou, darling, but they just came out of the ovenand need to cool before you can eat them.Please set them outside on the porch. You canenjoy them later.”

Goldilocks thanks her grandmother and care-

fully places the basket on the front porch oftheir forest home. She then goes inside to take anap while the muffins cool.

Three bears stroll by the cottage and smell thewonderful aroma of the sweet muffins. They fol-low the smell right to Goldilocks’s front porchand smile when they see both the nameplate onthe door and the basket of marvelous muffins.Papa Bear approaches the basket first and eatsexactly 1/4 of the muffins. “Mmm,” groans PapaBear, “these are yummy.” Next, Mama Bear eatsexactly 1/3 of the remaining muffins. “You areright, Papa,” she declares. “These are yummy.”Finally, Baby Bear goes to the basket and eatsexactly 1/2 of the muffins left by Mama Bear. Helicks his lips and says, “Mmm, much better thanporridge.” The three bears pat their full bellies,smile contentedly at one another, and continuetheir walk through the forest.

When Goldilocks awakens from her nap, sheimmediately runs to the porch to grab the basketof muffins. She is startled to discover that only3 muffins remain in the basket. “What happenedto all the muffins?” she exclaims. “I know therewere more than 3 in this basket when I put ithere. I wonder how many muffins were in thebasket to begin with.”

Use the information from this story to helpGoldilocks determine how many muffins werein the basket. If Papa Bear took 1/4 of the orig-inal muffins, Mama Bear took 1/3 of what heleft, Baby Bear took 1/2 of what remained, andonly 3 muffins are left in the basket, how manymuffins were in the basket when Goldilocksfirst put them on the porch? Once you have asolution, explain to Goldilocks how you cameup with your answer.

Extensions • How many muffins did each bear eat? How do

you know? Are these results surprising to you?

Solutions to the MuffinMania Problem: Cooking

Up Some Thinking


Robert Mann, [email protected], teaches mathematics and mathematics education at West-ern Illinois University. His work focuses on problem solving, functions, and activity-basedmathematics.

Edited by Robert Mann, [email protected], and Kim Hartweg, [email protected],Department of Mathematics, Western Illinois University, Macomb, IL 61455-1390

PROBLEM SOLVERS Robert Mann

Figure 1Jacob’s first attempt at solving the Muffin Mania problem

• What if the bears ate 1/4, 1/3, and then 1/2 ofthe muffins and left 4 muffins for Goldilocks?How many muffins would have been in the bas-ket originally and how many would each bearhave eaten? What if only 1 muffin remained?

• If you know the number of muffins left forGoldilocks at the end of the story—3, 4, 5, 1,and so on—can you explain how to find the totalnumber of muffins that the grandmother madeand the number that each bear ate?

• Suppose the bears eat the muffins in reverseorder. First, Baby Bear takes 1/2 of the originalmuffins; next, Mama Bear takes 1/3 of what isleft; finally, Papa Bear takes 1/4 of the remain-ing muffins, again leaving just 3 in the basketfor Goldilocks. How many muffins did eachbear eat and how many muffins were in the bas-ket originally?

The Muffin Mania problem proved to be bothaccessible and challenging to students at differentgrade levels. The problem and its extensions “cookedup” intriguing responses as well as a lot of insightfulthinking in classrooms across North America. Theseresponses provided much food for thought as I pre-pared this article. I have decided to present a smor-gasbord of the many enticing responses.

Sandra Kelly is a school-wide enrichmentteacher at Spruce Run School in Clinton, New Jer-sey. She read the Muffin Mania problem to a smallgroup of first and second graders and had them inde-

pendently work toward a solution. She also gavethem a copy of the problem and a note addressed toGoldilocks to use as a recording sheet for their finalexplanation. Kelly’s students were not at all intimi-dated by the fractions in this problem and were ableto provide some very thoughtful solutions.

Roman, a first-grade student at Spruce Run,recorded the following on his note to Goldilocks:

I used Unifix cubes to solve the problem. I said,let’s say there’s twelve muffins in the basket.Papa Bear ate one-fourth of all the muffins.Mama Bear ate one-third of the rest of themuffins. Baby Bear ate one-half of the rest ofthe muffins. There are three muffins left in thebasket. The answer is twelve.

Roman apparently used a guess-and-checkapproach to solve the problem and tested hisanswers with the help of Unifix cubes. Although hewas unclear in explaining what he did in each step,it is clear that this first grader had a solid under-standing of 1/2, 1/3, and 1/4. Roman’s use of “allthe muffins” and “the rest of the muffins” indicatesthat he understood the changing “whole” aspect ofthis fraction problem.

The aspect of a changing “whole” did create dif-ficulties for some students. Figure 1 shows oneattempt by second grader Jacob. Jacob used aworking-backward strategy to determine that therewere six muffins in the basket when Baby Bear ate


Figure 2

Jacob's second attempt at the problem

half of them. Jacob then used that same total of sixmuffins when computing Mama Bear’s share.Although he correctly determined that 1/3 of 6 is 2,his answer for Mama Bear’s amount was inaccu-rate because he took 1/3 of Baby Bear’s wholerather than 1/3 of what was in the basket whenMama Bear first took her share.

Jacob apparently realized that his initial answers

did not work, however, because he did not finish thework in figure 1 and instead submitted the addi-tional solution in figure 2. In this explanation, Jacobcreatively used a variety of mathematical operationsto determine that there were indeed twelve muffinsin the basket originally. Although his work andanswers for Baby Bear’s and Mama Bear’s amountsindicate that he still has some misconceptions with


Figure 3Second-grade student Angela’s solution

this problem, Jacob should be commended for hisperseverance, his creative thinking, and his excellentcomputational and writing skills.

Caleb, a classmate of Jacob and Roman, alsoused a work-backward strategy to obtain his solu-tion. He wrote the following explanation toGoldilocks:

First I added 3 and 3 and got 6. Then I added 6and 3 and got 9. After that I added 9 and 3 andgot 12. So I think the number is 12.

Your friend,Caleb

P.S. I added 3 and 3 because Baby Bear ate 1/2of what Mama Bear left.

Caleb’s solution demonstrates that working back-ward is an effective method for solving this prob-lem if one focuses on the constant number addedback each time rather than on the changing wholeof the fractions involved.

Angela, another second grader from SpruceRun, combined the working-backward strategy, theUnifix-cubes manipulative, and a keen understand-ing of fractions to determine the solution to theMuffin Mania problem. Figure 3 shows her intu-itive response.

Like Roman, Angela used Unifix cubes to helpher with this problem. Instead of using them tocheck numerical guesses, however, Angela usedthe cubes as muffins and separated them into fourgroups (Papa’s, Mama’s, Baby’s, and basket’s).Because three cubes were left in the basket group,she knew there had to be three muffins in each ofthe other “fourths” that she had created. Angela didan outstanding job in explaining and diagrammingher insightful technique and solution. She alsosigned the letter to Goldilocks with “love” and sub-mitted several wonderful pictures of the MuffinMania story. Figure 4 shows one of these pictures.

Angela’s classmate Andrew also provided anoutstanding solution and explanation. His superbwork is in figure 5. Andrew offers an excellent wayof presenting this solution to students of any age.


Figure 4Angela’s depiction of Goldilocks, Grandma, and the basket of muffins

By beginning with a representation of fourths, hewas able to show the portion of muffins removedeach time while at the same time avoiding the chal-lenges of the changing whole. This representationclearly shows that although the fraction removeddiffers for each bear, the actual number of muffinsremoved remains the same. Thus, because threemuffins were left in the basket in this case, three

must have also been removed by each of the hun-gry bears. Andrew’s explanation has not only pro-vided validation for his classmates’ solutions andreasoning but also established a tangible connec-tion to the extension question of what happens ifthere are 4, 5, and so on left. Notice that Andrewrefers to the three muffins remaining in the basketonly in the final stages of his solution. This fact and


Figure 5Andrew’s generalizable solution


Andrew’s diagram-supported explanation demon-strate that the same mathematical technique couldbe used to determine the original number ofmuffins regardless of how many remain in the bas-ket. This young second grader not only solved thefraction-based original problem but also clearlyexplained his reasoning and, in doing so, provideda solution strategy that easily generalizes to anynumber of remaining muffins. Andrew and hisclassmates clearly used the Muffin Mania problemto cook up some profound and flavorful thinking!

Appetizing solutions were not limited to theyoung students from New Jersey. Robert Buyea’sand Lauren McGiveny’s fourth graders fromBethany Community School in Bethany, Connecti-cut, also provided many enticing responses. Onequestionable morsel offered by students from thisclass was that 1/4 + 1/3 + 1/2 = 1 whole. Buyeaexplained, “Somehow they knew this wasn’t a truestatement, but they felt it had to be based on howthey were interpreting the problem.” Jennifer’swork in figure 6 depicts how students struggledwith this idea of fractional parts and wholes.Although this work was still inconclusive for Jen-nifer, it is promising that she attempted to use somany different fraction representations whilewrestling with this problem.

The changing whole in the Muffin Mania prob-lem perplexed many of the fourth-grade students.Even if they realized that the whole was not con-stant, they would still confuse the number of muffinseaten with the number of muffins left. Figure 7shows an example of this. Zach and Seun carefullyrecorded the steps to their guess-and-check strategy.They correctly determined that if they started with48 muffins, Papa Bear would have eaten 12 of them.Then, however, they proceeded to take 1/3 of those12 rather than 1/3 of the 36 remaining in the basket.Although they did correct their reasoning in theirsecond set of work with the guess of 48, theyreverted back to their mistaken whole approachwhen testing the guesses of 72 and 36.

McGivney, the student teacher, explained howshe helped the students think through this changing-whole predicament:

I had Jennifer work backward using what sheknew from the problem to figure out what shedid not know. Once she saw that the threemuffins that were left were half of the muffinsthat were left after Mama Bear ate her share, shebegan to realize that each bear ate from a newwhole. This was a concept many students strug-

gled with. I guided Jennifer and other studentsthrough this misconception by having themthink about how many muffins were in the bas-ket when each bear arrived and began eatinginstead of just focusing on how many were left.

This guidance appears to have been very successfulbecause Zach, Seun, Jennifer, and many of the otherfourth-grade students in Buyea’s class submittedwell-explained solutions to the original MuffinMania problem and its extensions. Zach and Seun’scolorful explanation to the original problem andextension question 1 can be seen in figure 8, andtheir solutions to extension question 2 (How manymuffins were there originally if 4 or 1 or 5 muffinswere left?) appear in figure 9. Although these stu-

Figure 6Jennifer tries to make sense of the given fractions.

dents initially struggled with this problem, theyshowed immense growth by not only solving theoriginal problem but also using what they hadlearned in that process to quickly determine solu-tions to similar problems. Like the first and secondgraders before them, these fourth graders were alsocooking up some meaningful connections while

working on the Muffin Mania problem.Linda and Anupriya, fourth graders in Buyea’s

class, were also cooking up some thinking whileworking on the Muffin Mania problem. Their solu-tion, which appears in figure 10, reveals a clearunderstanding of the fractional “whole” as well asa tremendous sense for fraction operations in gen-


Figure 7Misconception of the “whole”

Answer to “How many did each bear eat and did that surprise you?”

Figure 8

eral. Their explanations indicate that these youngstudents have a solid conceptual understanding offractions, parts, and wholes!

Once the fourth graders in Buyea’s classroomhad successfully solved the original question and itsfirst two extensions, they were quick to realize that“taking the remaining amount and multiplying it by4 would give the original amount,” no matter howmany muffins were left in the basket. Jason and Rayexplained this phenomenon by using the diagram infigure 11. Similar to Andrew’s solution in figure 5,this diagram shows that the solution method itself isindependent of the number of remaining muffins.The key is to begin with 4 equal portions and con-secutively remove 1/4, 1/3, and 1/2 of what is lefteach time. The students showed that, in general, theoriginal number of muffins can be determined sim-ply by multiplying the number remaining in thebasket by 4. Like Andrew, these fourth graders haveextended a convenient mathematical procedure to ageneral mathematical principle.

But why stop there? Like true mathematicians,Buyea’s students began to question the limits oftheir generalization. First, Jason and Ray restatedtheir principle as the following: “Since Papa Bearwas first and he ate 1/4, you can get the whole bymultiplying what is left by the denominator.” Glenfurther conjectured, “It doesn’t matter how manyfractions, but just as long as you use the denomi-nator of the smallest fraction.” To prove his con-jecture, Glen created an example in which PapaBear ate 1/5, Mama Bear ate 1/4, “Teen” Bear ate1/3, and Baby Bear ate 1/2 of the remainingmuffins, leaving 6 muffins in the basket. He thendemonstrated in figure 12 that the total number ofmuffins was simply 6 (muffins left) times 5(denominator of smallest fraction) for a total of 30muffins. Buyea and his class hypothesized that thismethod would always work if every fraction with anumerator of 1 was used from the original down to1/2. Whether that conjecture is true remains to beexplored and defined by these young mathemati-cians. What is true, though, is that these childrentook one originally challenging problem andextended it to create many other interesting ideasand possibilities. As McGivney commented, “Youcould almost see smoke coming from their earsbecause they were thinking so hard.”

Thinking was also on the menu in Jeri Delongand Lori Stewart’s fourth-grade classroom atLeona Middle School in Shadyside, Ohio. Delongand Stewart explained that their students had only“limited exposure” to fractions when the teachers

presented the Muffin Mania problem to them. Afterreading the problem to their students and givingthem centimeter cubes to use as muffins, the teach-ers were surprised when several students solved themystery in a matter of minutes. Although many ofthe students came up with a correct answer of 12muffins, many of them baked up interesting waysof determining this result.

Some of the students successfully used a guess-and-check strategy to come up with 12 muffins.Others, like Shanelle, discovered that if you addthe three denominators (4, 3, 2) and the leftovermuffins (3) you end up with 12 muffins. Similarly,Aleska and Lydia (see fig. 13) added 1/4 + 1/3 toget 2/7, then added 2/7 + 1/2 to get 3/9, then added3 + 9 to get 12. Although these techniques do notappear to be mathematically sound, they certainlyinvite many intriguing questions and investigations


Figure 9Solutions for extension question 2 (“What if

there were 4, 1, or 5 muffins left?”)

Figure 10Linda’s and Anupriya’s solutions

Figure 11General representation of the Muffin Mania problem


and provide an excellent opportunity to addressand test the related misconceptions.

Other groups from Leona Middle School intu-itively perceived the problem as four equal groups(including the amount left in the basket) andquickly determined the original amount. Loganexplained, “I put them in groups and I knew thegroups had to be even. There were three left andthen they all had to have three muffins.” Logan’saccompanying illustration is in figure 14. Noticethat the four labeled groups have three items ineach group. This diagram is similar to that ofAndrew’s in figure 5 and Jason and Ray’s in figure11. In each case, the diagram can easily be adjustedfor varying amounts of remaining muffins. Infact, Logan’s classmate Brody quickly general-ized this notion by explaining to his teachers,“It’s easy; just multiply it [the leftover amount]by 4.” Once again, a procedure has become aprinciple and a problem has become a catalyst formathematical thinking and communication. Iwonder if Logan’s diagram could be extended to“prove” the conjecture from Buyeas’s classroom.Smells like something is cooking again!

Kathleen Chadwick from Holy Cross Schoolin Runson, New Jersey, provided a clear exampleof how the Muffin Mania problem producedmathematical thinking and discussion in herfifth-grade classroom. The following conversa-

tion occurred as one group worked toward a solu-tion.

Shelby. Let’s start by working backwards. Ifthere were 3 left and Baby Bear took half, theremust have been 6 to start with. If Mama bear took1/3, there must have been 18. Papa Bear took 1/4so there must have been . . . Wait, this isn’t right.I thought we could work backwards but I think Iam not doing it right.

Kevin. Shelby, let’s try something different.What if we guess a number and use the counters?

Shelby. What if we start with 10? First wewould have to take a fourth of 10. This is notgoing to work. We have to pick a multiple of 4.


Figure 12Glen testing his conjecture

Figure 13Aleska and Lydia cooking up ideas

Photograph by Jeri Delong and Lori Stewart; all rights reserved

Kevin. What if we pick 16?Andrew. I don’t think that will work. We will

have too many muffins left.Kevin. Well, if we take 1/4 of 16 we get 4. So

then Papa Bear would get 4 and there are 12 left.Then Mama Bear would take 1/3 of the muffins,which would be 4, and there would be 8 left.Baby Bear then would take half, which would be4. We do have too many muffins. What if we pick12? I don’t think it will work. We might still havetoo many muffins.

Andrew. Let’s try. If we start with 12 (taking12 counters and putting them in equal piles) andI took 1/4 of them, I would have 3 and therewould be 9 left. Then I took 1/3 and I have 3 with6 left, and then take 1/2 and I have 3 left. See, itworks.

This conversation captures many of the com-mon elements students were likely to encounterwhen working on the Muffin Mania problem.This group used both working-backward andguess-and-check strategies to obtain a solution.The students had difficulty with the changing-whole aspect of the problem but used the manip-ulatives to better understand the situation. Theycorrectly deduced that the solution number must

be a multiple of 4 and used that knowledge tonarrow their guess-and-check choices. Finally,while working together to solve the problem, thestudents in this group—like many others, Iexpect—exhibited sound fraction skills,employed mathematical reasoning, and clearlycommunicated and represented their thoughts,ideas, and solution.

Chadwick’s fifth graders also worked on theextensions to the Muffin Mania problem and dida great job of determining and explaining thesolutions. Conor’s solution page in figure 15 is aclear, concise summary of the Muffin Maniaproblem and its extensions. Chadwick com-mented, “The students were able to find theanswer to these questions faster than the firstquestion. The students were able to apply whatthey had previously discovered.” As anotherextension to this problem, Chadwick’s classmade the tasty muffins from the Goldilocks story.This gave them the opportunity to see what 1/4,1/3, and 1/2 of a tablespoon looks like and pro-vided them with more fraction experience. It alsoprovided them with a yummy treat for their hardwork! In Rome, Georgia, Tiffany Troxell alsoused food to enhance the investigation of theMuffin Mania problem. A student teacher inMichelle Major’s fifth-grade class at Berry Ele-mentary School, Troxell presented the problemand boxes of mini-muffins to her eager students.She asked them to “show me with the muffinsand on paper how they were working on theirproblems.” One of Troxell’s muffin groups isshown at work in figure 16.

Many of the students in this class used aguess-and-check approach to solve the problemand, like other classes, some students becameconfused with the changing-whole element of theproblem. Many groups still came up with ananswer of 12, however, and many also used aworking-backward strategy for at least part oftheir solution. Other groups came up with ananswer of 12 using unconventional methods. Forexample, one group noticed that 12 was the leastcommon denominator for 2, 3, and 4, and thusguessed 12 as a solution. Another student, Emily,explained, “What we did was 1/4 ÷ 1/3 was 3/4.Then we did 3 × 4 = 12.” Both of these methodsexhibited common fraction procedures (leastcommon denominator and fraction division) butlacked meaningful support for using those opera-tions. Both of these groups correctly checkedtheir answer of 12, however, and should be


Figure 14Logan’s illustration

applauded for their creative thinking and strongarithmetic skills.

Another fifth-grade student, Jordan, was ableto quickly come up with the correct answer of 12.He wrote the following response: “I thought ofwhat 3 is 1/2 of and got 6. Then I knew that 1/4+ 1/3 is close to 1/2 and I guessed 12, which isright.” This student displayed some keen numbersense and sharp intuition by quickly determiningand checking a value of 12 for the original num-ber of muffins. As Troxell stated, “His mind wasreally turning gears quickly!” Troxell and herband of merry muffin makers are seen displayingtheir fine work in figure 17.

The following statement from teacher Kath-leen Chadwick should serve as a concise sum-mary for this investigation:

This was a fun as well as educational project.The students obtained a better understandingof multiplying whole numbers and fractions.They visualized what a half, third, and fourthof a group would look like. The studentsapplied what they discovered in previousproblems and incorporated this knowledgeinto the discovery of a generalization for allthe problems. The students were able toapproach a problem together and solve it. Thegroup members had some disagreements, butproved to one another that their guess wascorrect.

Although Chadwick’s statements are a commen-tary on the work in her fifth-grade class, they caneasily be applied to the problem-solving work bythe other first- through fifth-grade classes in thisarticle. While solving the Muffin Mania problem


Figure 15Conor’s solution page

Figure 17Berry Elementary Muffin Masters

Figure 16Muffins galore!

Photograph by Tiffany Troxell and Michelle Major; all rightsreserved

Photograph by Tiffany Troxell and Michelle Major; all rightsreserved

and its extensions, students of all ages workedsuccessfully with fraction operations and repre-sentations, effectively explained their reasoningand solutions, discovered and applied new math-ematical procedures, and created and tested con-jectures and generalizations. These young chefscreated a buffet of enticing mathematical ideasand savory academic insights.

A special thanks to the head chefs and well-managed kitchen classrooms of:

Kathleen Chadwick’s fifth graders, Holy CrossSchool, Rumson, New Jersey

Jeri Delong and Lori Stewart’s fourth graders,Leona Middle School, Shadyside, Ohio

Sandra Kelly’s first and second graders, SpruceRun School, Clinton, New Jersey

Lauren McGivney and Robert Buyea’s fourthgraders, Bethany Community School,Bethany, Connecticut

Tiffany Troxell and Michelle Major’s fifthgraders, with Dr. Carla Moldavan, Berry Ele-mentary School, Rome, Georgia ▲


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