DOCUMENT R F S U ME SE 004 520 ED 024 5b5 By-Foley, Jack L. Action with is Contained inDivision. Pub Date Aug 67 Note- 1Sp. EDRS Price MF -$0.25 HC-10.85 Fractions, *Inst-JctionalMateriats, Low Ability Descriptors- Arithmetic, *ElementarySchool Mathematics, Students, *Mathematics Education Act Title III Identifiers-Elementary and Secondary developed for the projecA Program for This booklet, one of aseries has been teachers, is Pupils. A prolect team,including inservice Mathematically Underdeveloped The materials being used to writeand develop thematerials for this program. include the ideaof reciprocals,and countingfractional developed in this booklet above principles mathematical problemsinvolving theapplication of the parts. Several activity.Accompanying these areincluded inthe sectionsdesigned for student of "Teaching StrategyBooklet" which willinclude a description booklets will be a academic games,and suggested teacher techniques,methods, suggested sequences, visual materials. (RP) ..11101.

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FROM THE THIS DOCUMENT HAS BEENREPRODUCED EXACTLY AS RECEIVED IT.POINTS OF VIEW OR OPINIONS ,PERSON OR ORGANIZATION ORIGINATING EDUCATION STATED DO NOT NECESSARILYREPRESENT OFFICIAL OFFICE OF POSITION OR POLICY.

CONTAINED ESEATole. EC

PROJECT MATHEMATICS

Proiect Team

Dr. Jack L. Foley, Director Elizabeth Basten, AdministrativeAssistant Ruth Bower, Assistant Coordinator Wayne Jacobs, AssistantCoordinator Gerald Burke, AssistantCoordinator Leroy B. Smith, MathematicsCoordinator for Palm BeachCounty

Graduate and StudentAssistants Secretaries

Jean Cruise Novis Kay Smith Kathleen Whittier Dianah Hills Jeanne Hullihan Juanita Wyne Barbara Miller Larry Hood

Donnie Anderson Connie Speaker Ambie Vought Dale McClung

TEACHERS

Sister Margaret Arthur Mrs. Christine Maynor Mr. John Atkins, Jr. Mr. Ladell Morgan Mr. Lawrence Bernier Mr. Charles G. Owen Mr. Harry Berryman Mrs. Margaret Patterson Mr. Ricke Brown Sister Ann Richard Hrs. Nicola Corbin Mr. Carl Sandifer Mrs. Gertrude Dixon Mrs. Elizabeth Staley Mrs. Dellah Evans Mr. James Stone Mrs. Marilyn Floyd Mrs. Linda G. Teer Mrs. Katherine Graves Mr. James Wadlington Mrs. Aleen Harris Mrs. Marie Wells Mr. Earl I. Hawk Mr. Ronald Whitehead Mr. Arthur Herd Mrs. Mattie Whitfield Mrs. Alice Houlihan Mr. James Williams Mr. Harold Kerttula Mr. Kelly Williams Mrs. Mary Kisko Mr. Lloyd Williams

August, 1967

?or information write: Dr. Jack L. Foley,Director Bldg. S-503, School Annex 6th Street North West Palm Beach, Florida Reciprocals In theunit on multiplication offractions, the idea of reciprocalswas examined. A more "dashing" name forreciprocals was given. Do you recall the othername? It was multiplicative inverses. A pair of iscalled "reciprocals" or"multiplicative inverses" if their product is one. First, review the idea by completing the followingactiVIT7. A model, given a count or measureof one, is drawn. A is given, and theproblem is to "count" how manytimes the fraction "is contained in" the one unit. Do this to supply the answerto the problem. The first two are examples. (Check each division bymultiplying.)

Fraction In One Unit Division Problem

1 2

4 3 l or 5 3

1

1 3 2

1 rm.

Check

1

2 2 5. or 1 O -

Check

1

4

Check

Ow. 1

Counting Fractional Parts For each activity below, adivision model is drawn and the division problem is given besidethe model. "Count" the of times the fraction"is contained" in the model to obtain the answer to each divisionproblem. The first two are examples.

Model Fraction Division

a* lio I 444, 41' 1 t I 1 2

1 How many times is contained in 2? Do you count4 ? 1 Then is contained in 2 four times. 3

Model Fraction Division

2 6

3 3.

1 I (Haw many7s are there in3?)

3 3 =

2 3 _

2 6

Remember Now is the time toapply the idea of areciprocal. number tells"haw many times" the numberis that a reciprocal of a Consider the following contained in one unit. (The product is one.) problem: First, count thenumber of times thefraction appears in one the number of units. unit (itsreciprocal.) Then multiply by 4

. 2 Example: 3 7 = 3

First, count the number of 3?- ,s in one unit.

1of another two-thirds 2

1 two-thirds 1 Answer: 1 or 3 2

. 2 Then, in three units, the answer is: 3 7 3-

3 X =

3 X 73 = 9 = 4 1

2 Show this by counting 3- ,s in three units.

= r

Try this for: 6 = 2

1 First, the number of .2. ,s in one unit?

Answer: 2

2 Then, in 6 units: 6 X - = 12 (One-half is in 6 itnits 1 12 times.)

Notice that in the division problems you simply multiply by the reciprocal of the . The reciprocal tells how many times the divisor is in one unit.

LI 16x

21 and1 times.) (Three-fourths is in 16 units 3

Activities is true for the Complete each problembelow. Show your answer first two problemsby drawing a modeland counting.

3 2. 8 1. 6 .11011

. 3 4. 4 3. 3 5

2 5. 9 5 6. 12 .1. 5 0

4 = 8. . = 7. 15

SION 7 10. 5 9. 7 12 15 6 A second way of looking at dividing by a fraction is to use the idea of multiplying the divisor and the dividend by the same number (except zero). The will remain unchanged.

Examine this idea below. Remember that division can be written as: 10 = 10 2 2 (We know the answer is 5.)

Now multiply both the numerator and the denominator by the .same number.

lov 5 7 11.

lo 6 6o

10 4 ko 2 V

Can you see why the quotient is unchanged? Is it the same as multiplying by one? 5 6 4 Are T./ .472 etc. all names for one?

You are right. It is equivalent to multiplying by one. Now let us rewrite the division problem:

16 -1,. = as 16=

Since we know how to divide by a whole number, why not multiply the above divisor (two-thirds) so that the product is a whole number. Then multiply the numerator (16) by the same thing. What can you multiply times two-thirds to get a whole number? X 52 = a whole number

Did you pick any of these answers?

(31 61 9, 12, 15,...) 7

Try some of these:

16 = 2 3

16 X 2 = F-1

= ^ 2 3 =11

All the above can berewritten as: 2 16 -,-

(16 X 3) ( X 3) =

48 2

Actually, the simplestdivisor to use wouldbe ones What can you multiplythe divisor by toget a product of one? You are right if you sayits reciprocal. Then: 3 48 16 is equal to: 16 v 2 = 24 2 2 " 1 2

OR:

to:(16 X2-)4(2-X 48 1- 16 4. 2is equal 2' 3 2 8 Actually all this is not necessary if you want the fastest way to write down the divisor times itsreciprocal, since the answer will always be one.

. 2 Then: 167 -

16x =

24

This is the fastest way.

Try a fraction divided by a fraction:

3 (How many one-halves are in three-fourths?)

First count:

By counting do you see one one-halfandiltof another one-half?

Then: 3 17.

By the faster method:

3. 1 MM. 7 7 MM.

3

6 1 9

Activities

Complete as many problems as you can without paper or pencil.

3 . 1 = 2 =1/

3 =r1

1 . 3

5 . 7 1 . 7 .6. 7 .8. 10. 5. = 7

The next activity involves dividing a fraction by a whole number. The ideas used in solving other problems will work. For example: 1 . 3 7 7T

1 One way to picture this problem is to take and cut it into 3 equal parts. One of these three parts is the answer. gth

I Notice the answer isE1of the whole or just plain

Now look at the two problems below:

and

1 . 3 T

, 10 Examine these problems:

1 v 1 1 . 7 2

Do you notice the relationbetween multiplication offractions and division?They are inverse operations.

Activities

See how many you cancomplete mentally.

6. 2 1. 4 2 = 3 7

5 . 2. IT 7 5=

+ 12 = 3' 2

= 4. 2. 4.. 8= 9. .6. 7 17

5 . = 10. 2- 6 = 5. .6. 720 7

In division, where mixednumbers are involved, it is easier to hrenamen and thendivide. For example: 1 1 I: 6 4 2 = II: 33- 27 =

6 7 10 v 2 18 6 ,37 5 x 3 20 = 7 15 11

Activities look again Work the followingproblems. You might want to at examples Iand 11 on thepreceding page.

. 1 1 2 5 -g 1. 2 l3 2. 3 3-

8 1.1Pa

2 n1 1^1 .. 12 6lg....,3-1 _ _ 5. -2. 7 1V 3 =

2 0111 IRMO 7. 18 9 = 8.112 -1. 2

3 . 10.15 : 6v3 = 9. 3720 = 3 12 Activities

This exercise includesall the differentkinds of fraction Solve some as division problemsyou've worked on inthis unit. work. a classactivity and completethe others as individual

3 47 WON. 2. 1. 7 1110

- 1 4. 12 43 . 1 3 2

6. + 5. +15

4 1 7 . 1 8. 9 7. 5u 7 "2"

6 1 . 11 10. + 12 = 9. 7 If 7

12. 3 1=11111 7 2. +5 11. 74

2 11 3 . 14. 13. .4. 7 10= 5' 15