ideas

IDEASAuthor(s): Sharon L. YoungSource: The Arithmetic Teacher, Vol. 38, No. 7 (MARCH 1991), pp. 24-33Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41194811 .

Accessed: 17/06/2014 11:47

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extendaccess to The Arithmetic Teacher.

http://www.jstor.org

This content downloaded from 185.44.78.190 on Tue, 17 Jun 2014 11:47:50 AMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=nctm

http://www.jstor.org/stable/41194811?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


IDEAS

iji^ilè^^^^l

ПТ£ПгТШН|

IDEAS section for this month focuses on gathering, using, and interpreting data

about fingerprints as a basis for integrating mathematics and science. Patterns, classification, logical reasoning, and mathematical rela- tionships are explored by making graphs, clas- sifying fingerprints, and matching identical fingerprints, and in a parent-involvement activity sheet This month the reproducible sheets for the IDEAS section are designed to be used by multiple grade levels. Included are an activity sheet for parents to use with their children, three class-activity sheets, and a data sheet. Conse- quently, a teacher may want to reproduce and use several sheets.

IDEAS

LEVELS 1-8

FINGERPRINT DATA SHEET

Directions

This data sheet should be duplicated and used when indicated for the class-activity sheets and for the take-home parent-child activity sheet. The following information will help you and your students interpret the fingerprint data:

1 . The classification of fingerprint-pattern types currently used by the Federal Bureau of Inves- tigation (FBI) is based on the Henry system. This system recognizes three major categories of arches, loops, and whorls named according to the patterns made by the ridges of the fingers. • Arches. The ridges in a plain arch (A) enter from one side of the finger, flow smoothly to

Edited and prepared by Sharon L. Young Mathematics Education Consultant Palm Harbor, FL 34683

This section is designed for teachers who wish to give students new insights into familiar topics in grades K-8. This material can be reproduced by classroom teachers

for use in their own classes without requesting permission from the National Council of Teachers of Mathematics.

04

form a curve resembling a hill in the center, and then exit the opposite side of the finger. The ridge pattern in a tented arch (T) is similar to a plain arch but forms a sharp upthrust that resembles a tent. • Loops. Two pattern types are classified as loops: the ulnarloop (U) and the radial loop (R). m these two loop patterns, ridges enter from one side of the finger, curve to form a loop, and exit from the same side. These two loop patterns can cause some confusion because they are identified according to the hand from which the fingerprint is taken - left or right. With hands outstretched and palms down, as in taking fingerprints, one can identify two wrist bones in each arm. The ulna bone is at the outside edge of each wrist, nearest the little finger. The radius bone is at the inside edge of each wrist (see fig. 1). The loop fingerprints on each hand are identified according to whether they slope to- ward the ulna or the radius of that hand. • Whorls. Four types of patterns are labeled as whorls. The ridges in the center of the plain whorl form a sort of whirlpool. Although the next two patterns have the word loop in their names, they are classified as whorls because the ridge lines do not enter and exit on the same side of the finger. Tbe central-pocket loop resembles a combination of an ulnar or radial loop and a plain whorl. The center of the double-loop pattern appears to form the letter S. Finally, the accidental pattern may bear some resemblance to the other patterns but not great enough to be classified as any of the others. Its ridges tend to enter from one side and exit on the opposite side,

so it is classified as a whorl. When individual fingerprints are classified by symbols, all whorls are given the symbol W.

2. Seven basic ridge characteristics are used to identify or match fingerprints. It is unlikely that any one fingerprint has all seven characteristics; however, it is very likely that a fingerprint has multiples of one characteristic. The ending ridge is, as it sounds, a ridge line that just suddenly ends. A bifurcation, or two-pronged fork, is a ridge that divides into two continuing ridges. Similarly, a trifurcation, or three-pronged fork, is a ridge that divides into three continuing ridges. A dot, or island, is a small ridge section that stands by itself. An enclosure is a ridge that briefly divides into two then returns to being one ridge, resembling the eye of a needle. A bridge is a short ridge that connects two ridges running

ARITHMETIC TEACHER



^^^^^^^^^1

^^^^^^^^н

roughly parallel to each other. A hook, or spur, is a ridge that divides into two, as in a birfucation or fork, but one of the ridges is very short and ends suddenly. These seven characteristics are used in court as evidence of the matching of fingerprints. 3. Loops are the most frequently occurring patterns of fingerprints, followed by whorls and finally by arches.

Extensions

1 . Students could conduct their own library research to find out more about fingerprints. The following resources are suggested: Knill, George. "Applications: Fingerprints and

Fractions." Mathematics Teacher 13 (No- vember 1980):608-9.

Millimaki, Robert H. Fingerprint Detective. Philadelphia: J. B. Lippincott Co., 1973. It is important to note that many encyclope-

dias have erroneous or at least misleading information in their articles on fingerprints. When de- scribing the ulnar-loop and the radial-loop patterns, many encyclopedias fail to mention that these patterns change depending on from which hand the fingerprint has been taken. Students could make errors in labeling if they accept the information uncritically. 2. Send home a copy of "Fingerprints at Home" along with a copy of the "Fingerprint Data Sheet." Ask students to record the results of this home activity and then to share their results with the class.

Suggestions for taking fingerprints Two of the activity sheets, "Thumbprint Graph" and "Fingerprint Classifications," require that students have their fingerprints taken. Here are some suggestions that will aid in taking fingerprints: 1 . Police departments and sheriffs' offices typi- cally are happy to send an "Officer Friendly" to classrooms to talk about their jobs. During such a visit, the officers could take fingerprints of students that would be kept by the students. 2. One method of taking fingerprints in the classroom uses a stamp pad inked with a dark color. Carefully press one finger at a time on the ink pad and then slide it over the surface of the pad. Press the finger onto white paper. Note that the "sliding" of the finger over the pad gives a much clearer print than just pressing the fingers on the pad. The latter approach tends to clump the ink. 3. Another method of taking fingerprints is to use black watercolor and a small paintbrush. Using brush, carefully paint the surface of each finger and then press it onto white paper.

MARCH 1991

4. Teachers may find the following suggestions helpful in assisting students in taking their fingerprints. • Practice taking your own prints first so that y ou can get a feel for what technique will work best with your own supplies. • A clearer print can often be obtained by making more than one print from the same finger without reinking or repainting. For example, in figure 2, the print on the left was pressed on the paper first. ТЪе ink was too heavy to get a clear print, so a second print was made, resulting in a very clear image. • Although the official method for taking fingerprints includes "rolling" prints as well as "pressing" them (see fig. 3), the pressed print is more appropriate for younger students, since a clear pressed print is more easily obtained. • Students in grades 4-8 are capable of taking each other's prints or possibly their own.

IDEAS

LEVELS 1-3 THUMBPRINT GRAPH

Objective With assistance from an adult, students classify their thumbprints by type of pattern. They then determine if the two thumbprints are of the same type or different types, and with other students in their group, they graph the thumbprint data and write about their results.

Materials 1 . A copy of the "Thumbprint Graph" activity sheet for each student 2. A copy of the "Fingerprint Data Sheet" for each student

25



^^^^^^^^н 3. A stamp pad inked with a dark color or black watercolor and a brush

4. A magnifying glass 5. Graphing supplies: graph paper or large sheets of paper or posterboard and marking pens or crayons

Directions

1 . Ask students questions like these, discussing their responses as a class: • What do you know about fingerprints? • Have you ever had a copy of your fingerprints made? • What are some uses for fingerprints? • Do you think your fingerprints are the same as mine or different? Why? • Do you think the fingerprints from all your fingers are the same or different? Why? 2. Distribute copies of the "Fingerprint Data Sheet" and discuss the pictures of the various patterns of fingerprints without going into great detail. Ask students such general questions as "Which fingerprint looks like it has a hill in the middle?" Discuss with students the idea that fingerprints are unique for every finger, which is why they are so important in identifying people. 3. Set up a fingerprint center in the classroom for the purpose of taking thumbprints of each student. The thumbprints should be placed directly onto a copy of the "Thumbprint Graph" worksheet. The teacher or an adult helper should take the prints during one or more class sessions. As the prints are taken, assist students in identifying with the use of a magnifying glass each print to determine its pattern. Have students record the type of pattern in item 1 .

4. Place students in groups of four to six students. Discuss group behaviors: help one another, disagree in an agreeable way, listen to others in your group, take turns.

5. In item 2 have students determine whether their thumbprints are of the same type or of different types. 6. Distribute graphing materials so that each group can make a large group graph to complete item 3. Group members could make their own copy on their individual sheet to take home.

7. After the graphs have been completed, each group should study its graph and make some observations about the displayed data.

8. The data from the graphs can be grouped together to make a class graph.

Extensions

1 . Take a full set of fingerprints for each student. Label each print with its pattern. Then have

26

students make individual graphs to show how many arches, loops, and whorls appear in their ten fingerprints. See the sample in fig. 4.

2. Combine the data from extension 1 to make a class graph.

IDEAS

LEVELS 4-8 FINGERPRINT CLASSIFICATIONS

Objective Students take a full set of fingerprints and then classify the set according to an established for- mula. Finally, students explore discrete mathematics by determining the number of possible combinations for sorting sets of fingerprints into groups for a retrieval system.

Materials

1 . A copy of the "Fingerprint Data Sheet" for each student

2. A copy of the "Fingerprint Classifications" activity sheet for each student

3. A magnifying glass for each group of students

4. A stamp pad inked with a dark color or black watercolor and a brush for each group of students

5. A calculator

Directions

1. Ask students questions like these, discussing their responses as a class: • What do you know about fingerprinte? • Have you ever had a copy of your fingerprints made? • What are some uses for fingerprints? • Do you think your fingerprints are the same as mine or different? Why? • Do you think the fingerprints from all your fingers are the same or different? Why? • How do you think the FBI (Federal Bureau of Investigation) sorts and stores all the millions of fingerprints it has on file? (It uses a classification system.) 2. Distribute copies of the "Fingerprint Data Sheet" and discuss the pictures for the various types of fingerprint patterns. Ask students such general questions as "Which fingerprint looks like it has a hill in the middle?" Discuss with students the idea that every finger has a unique fingerprint, which makes possible identification of prints. 3. Distribute copies of the "Fingerprint Classifi- cations" worksheet to each student and discuss the classification procedure presented and used in item 1 . Have the students study the sample carefully, noting that each fingerprint is numbered 1-10 in the upper-lèft comer of each box. Note also that below each print is a number value, either 1 , 2, 4, 8, or 16, and a place to record the label for that print's pattern type. As indicated in the directions, the primary classification is based only on whorls that appear in the

ARITHMETIC TEACHER



^^^^^^^^^н fingerprint set. It is recorded as a ratio. For example, if no whorls appear, the ratio is

1 1*

If all ten fingers have whorls, then the primary classification is recorded as

32 32*

The secondary classification uses letters to rep- resent the patterns of the index fingers. The letter for the right index finger is placed to the right of the number above the line in the primary classification, and the letter for the left index finger is placed immediately below the line to the right of that number. 4. Have students work in groups of three to five students. Direct them to take a full set of their fingerprints. Because of the risk of smudging a fingerprint, it is helpful to have students actually take the prints on a separate piece of paper, cut them out, and then glue or paste them to the activity sheet in the appropriate spaces. In this way, students can feel free to take multiple prints of a finger until they get one that is clear. Students may also find it helpful to use the magnifying glass as they take each print to make sure that it is clear enough to classify. 5. After the full set of fingerprints has been taken, students can classify them according to their primary and secondary classifications, as directed. 6. Teachers may find it helpful to discuss item 2 with students, asking them for suggestions on how to determine the number of different groups into which prints can be sorted. Encourage students to think about all the different combinations of numerical values that could be recorded for the primary classification. Ask such questions as these: • What combination uses the smallest values?

( γ for 0 whorls)

• What combination uses the largest values?

( || for 10 whorls)

• What combinations can be recorded with a "1" above the line?

/111111 1 " ' J_ ' ' V 2' 3' 4' 5' 6' Г

" ' ' 32 /

Have students work in groups to develop a method for making sure that they have every possible combination. One way is to have one student make a list of all those with a "1" above the line, another student make a list of those with a "2" above the Hne, and so on, until they have listed all combinations. Students may also real- ize after making three or four lists that since each

MARCH 1991

list contains thirty-two different combinations and thirty-two lists are possible, the answer can be determined by multiplying 32 χ 32 or by using the counting constant on the calculator to add the number 32 thirty-two times: [+] 32 [=] Ξ Ξ · · -El· For this calculator-key code, press [Ξ] thirty-two times. 7. Students can use a similar listing method to find the number of possible combinations of the secondary classification. For example, those with a plain arch on the right index finger have these five possible combinations:

A A A A A, Α Τ U Ε W

8. To find the number of combined primary and secondary classification groups, students could again use a listing method to get started. How- ever, this method would be too tedious and time- consuming to continue for all possible combinations. For

A A

alone, 1024 combinations with the primary classification are possible. Only a few of these combinations are shown here:

1 A 2 A 3 A 4 A 5 A 6 A 7 A 1A1A1A1A1A1A1A

One method for obtaining all possible combinations is to multiply the number of primary classification groups by the number of secondary classification groups. 9. It is important to note that a complete classification of a set of fingerprints includes more than just the primary and secondary classifications. A complete classification includes the small-letter group and the subsecondary classification, which is extremely technical. Milli-

maki's book (see extension 1 under "Fingerprint Data Sheet") describes these other procedures in great detail appropriate for older students.

Extensions 1 . Students can work in groups to Compile data on the frequency of patterns for their individual groups or for the entire class. An actual count can be made or percentages calculated and then compared to the frequency data on the "Finger- print Data Sheet." A double-bar graph can be used to display the data, as shown in figure 5. 2. Give students a recorded primary classification for a set of f ingerprints and ask them to indicate without having seen the fingerprints which of the ten fingerprints would have to be whorls. For example, given the primary classification

j_ 18'

ask students to determine which fingers would have whorl prints. To solve this problem, students will have to work backward Studying the procedure for finding the primary classification given on the "Fingerprint Classifications" activity sheet, they can first subtract 1 from each number to find the total value for the whorls on the even-numbered fingerprints and on the odd- numbered fingerprints. This calculation yields

A 17*

Next, they find how 6 and then 17 can be sums of 1, 2, 4, 8, and 16. The number 6 can be written only as 2 + 4; the number 1 7 can be written as 1 + 16. Since the number 6 represents whorls for the even-numbered fingerprints, the next step is to find which even-numbered fingerprints have values of 2 and 4. The fingerprint boxes on the

27



^^^^^^^^в

^^^^^^^^н

activity sheet reveal that the left middle finger (fingerprint 8 is even and has a value of 2) and the left thumb (fingerprint 6 is even and has a value of 4) have these values. Similarly, students need to find which odd-numbered fingerprints have values of 1 and 1 6. The left ring finger and the right thumb have these values. Thus, on the basis of just the primary classification, students can deduce which fingers in the set of fingerprints have whorls, as shown in figure 6.

Answers 1. Answers will vary, depending on individual fingerprints. 2. Fingerprint sets can be sorted into the following numbers of groups:

Numbers of Basis different groups

Primary classification only 1024(32x32)

Secondary classification only 25(5x5)

Combinations of primary and secondary classifications 25 600 ( 1 024 χ 25)

IDEAS

LEVELS 4-8 FINGERPRINT DETECTIVE

Objective Students use fingerprint characteristics to prove that two fingerprints match.

Materials 1 . A copy of the "Fingerprint Data Sheet" for each student 2. A copy of the "Fingerprint Detective" activity sheet for each student 3. A magnifying glass for each pair of students

Directions

1 . Distribute copies of the "Fingerprint Data Sheet" and discuss the seven fingerprint characteristics with the students. See the teaching notes under "Fingerprint Data Sheet" for a description of these characteristics. 2. Distribute copies of the "Fingerprint Detec- tive" activity sheet. Study the three labeled characteristics of the fingerprint at the top of the sheet. Discuss with students that similarity of patterns is not the only match used to identify fingerprints. They have to match on many different characteristics. ТЪе comparison must show

2Ш

the location of the characteristics, as shown in figure 7. For example, the information that two fingerprints each had eight bifurcations would not be sufficient to determine that the fingerprints matched, as all eight of those bifurcations would have to be in the same position on each fingerprint for the prints to be identical. To use fingerprint matches as evidence in court, some states require as few as eight separate matching characteristics on the two prints, whereas other states require twelve and even fifteen or eight- een matching characteristics as proof that the prints match. 3. Each pair of students should have a magnifying glass. If not enough magnifying glasses are available, the teacher may have to alternate this activity with another and extend it over several days.

Extension Have each smdent make two copies of her or his

right thumbprint on different slips of white paper. Place students in groups. Have each student place one of his or her right thumbprints in a paper bag. Next, have a student pick one of the thumbprints from the bag and compare it to the fingerprints that the students are still holding. Prove whose thumbprint was drawn by finding twelve matching characteristics. Answers 1. Maria took a piece of her own birthday cake before the party started! 2. Answers will vary, as students may identify characteristics different from those given here. A sample answer is given in figure 8. The corresponding matching cha

1. Bifurcation 2. Ending ridge 3. Enclosure 4. Bifurcation 5. Bifurcation 6. Bifurcation 7. Bifurcation 8. Ending ridge 9. Bifurcation 10. Dot 11. Bifurcation 12. Hook

ARITHMETIC TEACHER



IDEAS JK'»

Fingerprints at Home Dear Parents,

Three Methods for Obtaining Prints Your child has been using fingerprints to investigate such mathematical ideas as patterns in nature. This Follow one of these methods to obtain prints: sheet shows several different activities that use fin- -, Vjsjt yQur |ocd sheriff or роПсе station and have gerprmts. You and your child could select one or

your fingerprints taken. You will be given the more of the activities to do together. c You will need- 2

c s|jde fingertips over an ink pad/ J ̂

' • a copy of the "Fingerprint Data Sheet/' then press on white paper. jil

^ VL

• a magnifying glass or a pair of reading glasses, 3. Use a small paintbrush to paint fin- ННкш and gertips with dark watercolor paint, ^^^1

• an ink pad or dark water colors. then press on white paper. | ' /

Note: When taking your own prints, "^? / you may need to practice to obtain T^s clear prints. >VN^^>

ж f ^V^' Go to the library. Find ЕЖ ^W ' books about fingerprints.

if Take thumbprints of all family members, and '' Some science books and

Щ label them with the name, left or right thumb, '' such books

information. about the body have

I and the type of pattern (see the data sheet). M such information. 1 In your family, whose thumbprints are of the ■ I A same pattern as yours? Look at all the thumb- I ^ L· ^=^ 1' prints you took. Which type of pattern occurs M J^T^^

- ^ t_^

"* ^=^ ̂ ^"^ VV most often - arches, loops, or whorls? ^^^^^

Two of the thumbprints below are identical. Which are they? and

Thumbprint A Thumbprint В Thumbprint С Thumbprint D Thumbprint Ε

From the Arithmetic Teacher, March 1991



ideas -^γ·;

Thumbprint Graph Work in a group. 1. Take your thumbprints in the boxes below. Write the type of pattern below each print.

Use the "Fingerprint Data Sheet" to help you.

Left thumb Right thumb

Pattern:

2. Are your thumbprints both of the same type or are they of different types? Circle one answer.

Same type Different types

3. Graph the answers for your group.

Our Two Thumbprints

Same Type

Different Types

0 12 3 4 5 6 Number of Students

4. Write about your results




IDEAS -^y^ ^яИ Name

Fingerprint Classifications 1 . Find the fingerprint classification for your set Sample

of fingerprints. Use the "Fingerprint Data Ι Ί, Right thumb I 2. Right index I 3. Right middle I 4. Right ring 1 5. Right little I oheet to help. _ ____ъ _шь -

■ Roll a set of your fingerprints in the boxes ^Щр ^Щт- ISP^l Шшк Щщ

■ Label each print with its pattern type: ^ть* W^· <^^m* -

A: Plain arch T: Tented arch £ , ,UL L16 _, , **

, 16

o , fU .,„ 8

Q VV 8 1ЛТи|, Left little ,4

M --,.,. 6· £ , Left thumb 7. _, , Left index , 8. o , Left middle .,„ Q 9. Left ring 10. Left little U: Ulnar M

loop --,.,. R: Radial loop ^^^ ^м - ■- ^^^ ~

■ Circle the values in the bottom right-hand ^BjHf ^№. *· Sü Ш&-.. т: д^др J^Hk

corner of each box for prints that are ?&шш ШШ&А Ш&-.. ^^ ^^^^ whorls. |U4|R2lWgjUi|Wfl

- Record the primary and secondary classi- The c|assification for the samp|e prints is 4U

fications. ' K

The primary classification uses values based on whorls:

Total value for whorls on even-numbered fingers + 1 _ Left middle + left little + Τ = 2 + 1+1 = 4 Total value for whorls on odd-numbered fingers + 1 No whorls + 1 0+1 1

The secondary classification uses letters for index-finger labels:

Right index label = U Left index label R

1. Right thumb 2. Right index 3. Right middle 4. Right ring 5. Right little

~To~" 16 8 8" 4

6. Left thumb 7. Left index 8. Left middle 9. Left ring 10. Left little

4 1 2 I 2 1 1 1 1

The primary and secondary classifications for my set of fingerprints are and .

2. Into how many different groups could sets of fingerprints be sorted on the basis of primary classification only? Of secondary classification only? Of combined primary and secondary classifications?

Tell how you know.




IDEAS ^ yj Паи Name

Fingerprint Detective 1 . Work with a partner to find out who

took a piece of Maria's cake before the birthday party! abb

Prove your suspicion by finding and J^ Ы^ l

marking at least twelve characteristics ^/l·***. Γ#'·ν ̂̂ Irshs of the suspect's thumbprint that match ^^^$5~гЛ^|£Е5^3 those of the thumbprint found on ^ 4 . f*J Г"*88"^^ the cake knife. Three matching ^

_ *

characteristics have already been ^ Л- _ ^j

identified. Use the "Fingerprint Data Sheet" to help you.

Enlarged right thumbprint lifted from cake knife

Bifurcation

Ending ridge Enclosure

ЩМЕ Mom Dad

Maria David

2. List twelve matching characteristics.

1. "Si^LArgQcVvOn 5. 9.

2. ^i'di('C' plOlOig 6. 10. 3. T^Aclo^rg. 7. 1 1 .

4 8. 12.




IDEAS ^у5

Fingerprint Data Sheet Types of Fingerprint Patterns

Arches 'v^ r^_ τ *■

Plain arch (A) Tented arch (T)

Loops 4>r<r* - ^^.y^.,

Right-hand ulnar loop (U) or

Left-hand radial loop (R)

Right-hand radial loop (R) or

Left-hand ulnar loop (U)

Whorls (W) Ι //////χ/τ^=^^^$&' Γ - ~

ΛΝ 4v, Ι Γ 7] r~f

_ - 71

Plain whorl Central-pocket loop Double loop Accidental

Source: Federal Bureau of Investigation, 1990

Characteristics of Fingerprints Frequency of Occurrence of Patterns

Ending ridge ^^^ Dot or island Qy

Bifurcation /^? Enclosure á^Ž' Arches 5% (Two-pronged fork) ^_.^^> ^-^2 Loops 65%

Bridge ^^^ ^-^2

Whorls 30% Trifurcation /^^2) ^^^ (Three-pronged fork) Ο^^ΖΖ? Hook or spur /^^

^^^ Source: Federal Bureau of Investigation,

1990




ideas

Documents