Fruit Baskets and GCF

8/2/2019 Fruit Baskets and GCF

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pPosing mathematics problems indierent ways will raise studentslevel o cognitive demand because itwill push them to think more deeplyabout mathematics (Stallings 2007).By engaging students in a task thatrequires them to determine their ownsolution strategies, students will gaina deeper understanding o the math-ematical concept explored through thetask. One activity, in particular, ledseventh-grade students to develop anunderstanding o the concept o great-est common actor (GCF) throughtheir solutions to dierent situationsposed in a problem.

The problems posed to the studentscan be read in the activity sheet. The

As students fill baskets with

differing numbers of fruit,

they develop the concept of

GCFoften before learning

the formal definition.

Carol J. Bell, Heather J. Leisner,

and Kristina Shelley

a Ff

222 MatheMatics teaching in the Middle school Vol. 17, No. 4, November 2011

MARTTISALMELA/ISTOCKPHOTO.C

OM

Copyright 2011 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.This material may not be copied or distributed electronically or in any other format without written permission from NCTM.


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Vol. 17, No. 4, November 2011 MatheMatics teaching in the Middle school 223

F

gcmm

Fgoal o the activity was or students toexplore the concept o greatest com-mon actor beore the term was intro-duced. When developing the activity,smaller numbers were chosen or thequestions so that manipulatives couldeasily be used. The activity sheet con-tains eighteen questions. Six questionseach involving two, three, and ourpieces o ruit allowed plenty o oppor-tunity or students to explore sortingdierent quantities o ruit. As morenumbers were included, the dicultylevel increased; however, students couldstill see that the pattern they oundworking with ewer numbers held trueregardless o the number o dierenttypes o ruit being sorted.

diViding uP Fruit Baskets

Students worked in groups o twoor three to solve the Fruit BasketChallenge problem, using coloredPopsicle sticks to represent dierentruits. The students then used coloredpencils to draw a visual representationo their solutions to each question.Visual learners beneted rom thisapproach because they were able tounderstand both the number o ruitper basket as well as the number obaskets that could be made.

Most students had no knowledgeo the concept o greatest commonactor beore investigating the FruitBasket Challenge. The students had,however, explored prime numbers up

avy f


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to 100 using the sieve o Eratosthe-nes, so they were amiliar with primeand composite numbers. They werealso amiliar with the concepts odivisibility and prime actorization

as well as the rules o divisibility. Onthe classroom wall an 8 t. 10 t.grid listed the prime actorizationso the counting numbers rom 1 to60 (see fg. 1). This chart was devel-oped in a previous class period usingthe Building Numbers rom Primesactivity described by Burkhart (2009).The grid was used to identiy patternsand relationships between the primeactorizations o dierent numbersand was a good reerence or the Fruit

Basket Challenge problem.Ater the students worked through

an example problem with the teacher,they individually explored possiblesolutions to each o the questions. Asthe questions began to involve moretypes o ruit, it was necessary or eachgroup o students to create a key toindicate which Popsicle-stick colorrepresented each ruit.

Students employed a variety o

solution strategies. Almost all groupsused the Popsicle sticks to help nda solution. A ew o the advancedstudents understood that they wereusing the greatest common actor othe given numbers to determine thenumber o baskets. Since they werealready amiliar with the vocabulary,the students easily solved the prob-lems using their prior knowledge oGCF. Other students had to sort thePopsicle sticks to solve each problem.

Students separated the Popsiclesticks into stacks o ruit, with thesmallest number o ruit in one stack.The stacks that did not contain thesame number o Popsicle sticks asthe others were redistributed. Forinstance, in solving question 3 onthe activity sheet (20 apples and15 oranges), teen stacks were initiallycreated, with each stack containing1 orange (see fg. 2a). The 20 apples

a B ayThe students and the teacher discuss their work thus ar on flling baskets andfnding the greatest common actor o 16 and 8:

Teacher: I noticed that you answered our baskets or the frst question. Letstake a look at that. Heres some manipulatives rom yesterday. Can you setup the problem again?

Students: Sure. We got our baskets with 4 apples and 2 oranges in each.Teacher: Look at one o your baskets. You have 4 red sticks and 2 yellow

sticks. Could you divide them into any more identical groups?Students: I could divide this basket to make two baskets with 2 apples and

1 orange.Teacher: Yes! Lets do that with all o them. What would that look like?Students: I I had 2 apples and 1 orange in each basket, Id have eight

baskets altogether.Teacher: Using the manipulatives, show me what that looks like.

[The student places the manipulatives into 8 stacks with 2 red sticks and1 yellow stick in each stack.]

Teacher: Great. Lets look at the chart. What do you notice about the actorso 16 and the actors o 8?

Students: They each have three 2s.Teacher: What do we do with the prime actorization to get the original

number back: add, subtract, or what should we do?Students: We multiply!Teacher: Whats 2 2 2?Students: Eight.Teacher: How many ruit baskets did you fnd or question 1?Students: Eight.

F. 1 A prime actorization grid helped students identiy common actors in

subsequent activities.


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were then distributed among the -teen stacks. Since ve stacks had1 orange and 2 apples and ten stackshad 1 orange and 1 apple, the tenstacks o 1 orange and 1 apple were

redistributed among the ve stacks,which resulted in each stack having4 apples and 3 oranges (see fg. 2b).Thus, a total o 5 baskets were created,with each basket having 4 apples and3 oranges. O the ve stacks o Pop-sicle sticks, each stack contained 4 redPopsicle sticks (apples) and 3 purplePopsicle sticks (oranges).

Another solution strategy usedby groups o students was to breakup the same color o Popsicle sticks

into equal groups and then comparethe number o dierent-color stacksto determine how to orm the bas-kets o ruit. For instance, in solvingquestion 2 (12 apples and 9 oranges),one group created our stacks with3 apples in each stack and another

(a)At let: Ten stacks o 1 orange and 1 apple;at right: fve stacks o 1 orange and 2 apples

(b)Five stacks o 3 oranges and 4 apples

F. 2 Some students grouped and then regrouped their Popsicle sticks to determine

the GCF o 20 apples (red) and 15 oranges (purple).

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three stacks with 3 oranges in each(see fg. 3a). Since there were at leastthree stacks o each color (apples andoranges), three stacks were createdthat used all the stacks o orangesand three o the stacks o apples. Theremaining stack o 3 apples was then

distributed to those newly ormedstacks (see fg. 3b). Thus, three ruitbaskets were ormed; each basketcontained 4 apples and 3 oranges.

The most dicult question orstudents to solve was probably ques-tion 4 (7 apples and 13 oranges), sinceit required that only one basket bemade. Many o the groups questionedwhether the problem could be solved.However, with some questioning by

the teacher, the students eventually

recognized that 7 and 13 are primenumbers so they could not be dividedinto groups with equal numbers.In stating a solution, one studentresponded, Because we cant split upthe apples and oranges equally, wehave to put them all into one basket.

Although this question consistedo two prime numbers, the studentscould have been given a question withrelatively prime numbers, such as 7apples and 4 oranges or 9 apples and 4oranges, to get a solution o one bas-ket. In question 12, students observedthat all three numbers (4, 8, and 17)have no common actor even though 4and 8 have a greatest common actoro 4. The solution is thus one basket

or the three dierent types o ruit.

Occasionally, students separatedthe ruit into stacks using a commonactor, but not thegreatestcommonactor. For example, in question 1(16 apples and 8 oranges), a groupo students produced our baskets;each basket contained 4 apples and

2 oranges. In this case, the studentsidentied a common actor but notthegreatestcommon actor.

Although the students might havemisunderstood the instructions ornding the most baskets, it provideda good teachable moment because 4and 2 have a common actor. A di-erent arrangement could be made tocreate more baskets. It was also a goodopportunity to help students under-

stand the distinction between com-mon actors and the greatest commonactor without having introduced thisvocabulary.

To distinguish between thesetwo concepts, the prime actoriza-tion chart on the classroom wallwas used to compare actors o thegiven numbers. For instance, whenstudents compared the actors o 16and 8 rom question 1, they noticedthat both numbers have three 2s.Through questioning by the teacher(see the sidebaron p. 224) and usingthe prime actorization chart on thewall (see fg. 1), students were able tounderstand the relationship betweenthe number o ruit baskets and theprime actors that were common tothe given numbers.

As the dialogue continued, the stu-dents in this group were better able tounderstand how the common actors

(a)Four stacks o 3 apples and three stacks o 3 bananas

(b)Three stacks o 3 bananas and 3 apples and an additional apple

on each stack

F. 3 Other students grouped their Popsicle sticks and distributed the extras to

determine the GCF o 12 apples (red) and 9 oranges (purple).

By referencing the prime factorization chart,

students were able to see how all the common

factors of a set of numbers were used to form

the GCF even though they had not beenintroduced to the term.


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in the chart were used to determinethe number o baskets needed. Thisconnection helped students when theywere asked to identiy any patternsthat they noticed. Class discussion al-

lowed students to relate the maximumnumber o baskets ormed to theGCF o the given numbers. The stu-dents concluded that when each givennumber was divided by the GCF,that amount told them the number opieces o that type o ruit to put ineach basket.

Building a deFinition oF

greatest coMMon Factor

Since this activity was done over a

two-day period, the rst day was de-voted to exploration and investigationso that students had a chance to solveeach question and write their nalanswer. One groups written solutionsare shown in fgure 4. As homework

F. 4 One group presented a colorul visual representation o their fnal solutions.


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no common prime actor remaining.Students could see that the maximumnumber o baskets that could be as-sembled was ound by multiplying thecommon prime numbers 2 2 2 = 8and that each basket contained 2

apples and 1 orange, which were thenumbers remaining ater division bythe GCF.

Bundling the Work

together

By using a real-lie example, such asthe Fruit Basket Challenge, studentsmay be more likely to remember howto calculate the GCF and why it isimportant. I we are going to help

students gain a deeper understand-ing o mathematical concepts and theprocedures used with these concepts,we need to provide more opportuni-ties or students to explore mathemat-ics through specic tasks and engagestudents in problem-solving activities.Giving students the Fruit BasketChallenge allowed them to explorethe concept o greatest common actorbeore the term was introduced. It alsoprovided a visual context and allowedthem to develop their own solutionstrategies or solving problems involv-ing the GCF. Through this activityexploration, students were able tobetter understand the importance onding the GCF and how it related toa real-lie example.

As educators, we are constantlystriving to engage students, encouragelearning, and answer that all-too-amiliar question, When are we ever

rom the rst day, students refectedon the patterns they discovered sothat they could discuss their ideaswith their group members and otherclassmates the next day. During the

second day, students used their workto develop a method or nding thegreatest common actor.

When asked to share their tech-niques or solving the rst ew ques-tions in the activity, some studentsexplained that they were amiliarwith the term factorand that llingthe baskets was related to nding acommon actor. Other students hadnot yet been able to move away romthe manipulatives, so a document

camera was used to visually explainthe solutions. All students then sawthe connection. By reerencing theprime actorization chart on the wallduring the class discussion, somewere able to see how all the commonactors o a set o numbers were usedto orm the greatest common ac-tor even though they had not beenintroduced to the term. Studentswere asked questions involving larger

numbers on the grid, such as 40 and60, so that they could use the patternsthey ound to develop a procedure orcalculating the GCF.

Since students were amiliar withthe rules o divisibility, it was naturalor them to use the divide-by-a-com-mon-prime method when developinga procedure or calculating the great-est common actor. For instance, usingthis prime method in question 1 (16apples and 8 oranges) yields the resultin fgure 5. In this method, since 16and 8 have a common prime actoro 2, we divide each number by 2,resulting in respective quotients o 8and 4, which are written below 16 and8. Again, since 8 and 4 have a com-mon prime actor o 2, dividing eachnumber by 2 results in respective quo-tients o 4 and 2, which are writtendirectly below 8 and 4. The procedurecontinues until the two numbers have

2 16 8

2 8 4

2 4 2

2 1

F. 5 This table demonstrated the

divide-by-a-common-prime method to

fnd the GCF o 16 and 8.

going to use this? This activity isdesigned to encourage student inquiry.Students must work with a partnerto problem solve and determine away to nd how many o each ruit

should be in each basket and howmany baskets can be made. Mostimportant, we have given studentsa reason to learn the concept o thegreatest common actor since helpingood pantries is something that manypeople are amiliar with or have done.This activity not only caters to severallearning styles, including visual andkinesthetic, but also opens up discus-sions among students and allows themto orm their own conclusions beore

the teacher mentions the termgreatestcommon factor.

reFerences

Burkhart, Jerry. 2009. Building Numbers

rom Primes.Mathematics Teaching in

the Middle School15 (3): 15667.

Stallings, L. Lynn. 2007. See a Dierent

Mathematics.Mathematics Teaching in

the Middle School13 (4): 21217.

c J. B, [email protected], teaches math-

ematics education courses

at Northern Michigan

University in Marquette.

She is interested in how

uture teachers commu-

nicate and make sense o

the mathematics they will

someday teach.

h J. l,

[email protected]

.edu, teaching in Madison,

Wisconsin, developed this

lesson while student teaching at Father

Marquette Middle School in Marquette.

k sy, krissy.shelley@gmail

.com, teaches seventh-grade and eighth-

grade mathematics at Father Marquette

Middle School. She is interested in mak-

ing mathematics meaningul or middle

school students through the use o real-

world applications.


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activity sheet

from the November 2011 issue of

Name ______________________________

the Fruit Basket challenge

Local ood pantries are assembling ood baskets or donation. Determine how many baskets volunteers can make using thegiven ruit, so that each basket contains the same number o each kind o ruit. The goal is to make as many identical bas-

kets as possible, with no ruit letover. Use colored markers or pencils to model your work, and show what should appear

in a basket.

exmp: 12 apples and 8 oranges

aw: 4 baskets total. Each basket contains 3 apples and 2 oranges. Draw 3 red circles or the apples and draw 2

orange circles or the oranges.

1. 16 apples and 8 oranges2. 12 apples and 9 oranges

3. 20 apples and 15 oranges4. 7 apples and 13 oranges

5. 7 apples and 14 oranges

6. 8 apples and 22 oranges

7. 6 apples, 8 oranges, and 10 bananas

8. 3 apples, 6 oranges, and 12 bananas

9. 10 apples, 15 oranges, and 25 bananas10. 10 apples, 10 oranges, and 16 bananas11. 9 apples, 15 oranges, and 6 bananas12. 4 apples, 8 oranges, and 17 bananas13. 4 apples, 6 oranges, 8 bananas, and 12 pears14. 6 apples, 12 oranges, 15 bananas, and 9 pears15. 3 apples, 3 oranges, 3 bananas, and 3 pears

16. 8 apples, 14 oranges, 4 bananas, and 8 pears17. 9 apples, 6 oranges, 12 bananas, and 18 pears18. 5 apples, 10 oranges, 15 bananas, and 10 pears19. What patterns did you notice?20. Did you have an answer in which just 1 ruit basket could be made? Why?21. How did you decide how to group your items?

Fruit Baskets and GCF

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Transcript of Fruit Baskets and GCF