BOSTON UNI'/ERSITY SCHOOL OF EDUCATION ANALYSIS OF DENOMINATE NUMBERS IN TWO SERIES OF ARITHMETIC TEXTBOOKS Submitted by Henry Arnold Schulz In oartial fulfilment of requirements for the degree of Master of Education 1933

Bosion University School of Education Library

I

Table of Contents

Introduction Pa, The Study- Part I Strayer-Upton Arithmetics

Relations among units of measure Informational matter with reference

to units of measure Units of measure merely as names Table facts in examoles and problems Reductions in denominate numbers Computation with denominate numbers Summary

Part II Hoyt-Peet Arithmetics

Relations among units of measure Informational matter with reference

to units of measure Units of measure merely as names Table facts in examples and nroblems Reductions in denominate numbers Computation with denominate numbers Summary

Final Summary and Concluding Observations

Bibliography

INTRODUCTION

This study reports the analysis of two arithmetic

text-book series with respect to the units of weight

and measure and more especially the denominate numbers,

that is, numbers which refer to units of measure, as,

for example, 3 inches, 4 pounds. Miss Vernette B.

Moore constructed an analysis of denominate numbers

found in four text-books for the purpose of determining

the concepts and skills needed in teaching denominate 1

numbers. This study assumed that the elements and

processes found in these text-books were proper teaching

material. The studies by Wilson with respect to the

social usage of arithmetic and the more recent

2 3 4

investigations of Louth, Sala, and Traniello, into the

uses of weights and measures in business and industry,

present evidence which leads us to be critical of the

subject matter found in text-books in connection with

denominate numbers. The first step in a critical

1. Moore, Vernette B» An Analysis of Denominate Numbers with Data on Their Teaching. State University of Iowa, 1929,

2. Louth, Mary. Units of Measurement in Industry. B.U. Graduate School thesis, 1931.

3. Sala, V. Denominate Numbers used in the Factories of New Britain* B.U. School of Education thesis, 1931.

4» Traniello, A. Use of Tables of Measurement in

Business. B.U. School of Education thesis, 1930.

2

examination of a text-book in respect to denominate

numbers must necessarily be an analysis which shows

the content of the text-book*

The present study offers an analysis of each of

the following two text-book series:

Strayer-Upton

Strayer-Upton Arithmetics

American Book Company

Three Book Series^ 1928 edition

Hoyt and Peet

The New Everyday Arithmetic Houghton Mifflin Company

Three Book Series; first book, 1930 edition; second and third book, 1927 edition.

Units of weight and measure have been classified

under six main headings, namely,

1» Relations among units of measure.

2. - Informational matter with reference to

weights and measures.

3. Denominate numbers merely mentioned in written problems and involving no computation with denominate numbers as such.

4. Table facts in examples and written problems.

5. Reduction of denominate numbers*

6. Computation with denominate numbers. The units of weight and measure assembled under

each of these main headings have been further sub-divided into groups according to the usual •^table" classification*

3

In other words, all units of measure dealing with length have been grouped under "Linear Measure"; those dealing with square units of measure, under "Square Measure", and so on.

This analysis does not include tables of weights and measures found in the rear of the text'-books. These tables are intended for reference and pupils are not required to memorize them as they are in the case of those tables found in the text proper. All computation with United States money (including changing of money), computation with metric measures, as well as computations by the 100, or by the 1000, have been excluded from the analysis, since they are decimal. Examples and written problems in mensuration have also been excluded, since they properly form a part of intuitive geometry, rather than of denominate numbers. It also appears proper to exclude work in connection with telling time, making schedules for buses, trolley cars and trains, the use of the calendar, drawing to scale, graphs, work interest, banking, commission, discount, installment buying, stocks, bonds, insurance, equations, taxes, duty, changing Fahrenheit readings to Centigrade readings and visa versa, food problems involving the use of the terra calories^

#

( ( I I <

#

PART ONE

The following analysis covers the units of weight and measure, with special reference to denominate numbers, as found in the three book series of the Strayer-Upton Arithmetics.

r

4

RELATIONS AMONG UNITS OF MEASURE

Tables I to XIV which follow show the various tables of measures found in the Strayer-Upton series of arithmetics, each Table representing one table of measure, except that counting articles and counting paper have been grouped together under Table VIII and the metric tables of length, of capacity, and of weight, have been grouped together in Table XII* In each case the equalities found in the table of measure has been stated to the left of the figure and the frequency of occurrence of these equalities is indicated in the upper left corner of each square.. All other relations found in the texts of the Strayer-Upton Series are listed to the right and the frequency of occurrence of these equalities, either in written problems or in some other isolated form, is recorded in the lower right of the cell.

r

5

TABLE I- RELATIONS A..10NG UNITS OF UNITED STATES MONEY

FOUND IN THE STRAYER*UPTON SERIES.

Equalities to the left are found in tables; frequency of occurrence is shown in upper left of each cell. All other equalities found in the texts are listed to the right; frequency of occurrence is shown in lower right of cell. Blank indicates no occurrence.

10 mills s 1 cent

5 cents 1 nickel

lo cents St 1 dime

25 cents = 1 quarter

50 cents = § dollar

100 cents = 1 dollar

10 dimes = 1 dollar

Total

1 mill r 1/10 cent

1 nickel = 5 cents

1 dime = 10 cents

1 quarter = 25 cents

h dollar s 50 cents

1 dollar = 100 cents

1 dollar = 10 dimes

c

6

TABLE II. RELATIONS AMONG UNITS IN LINEAR MEASURE FOUND IN THE 3TRAYER-UPT0N SERIES.

Equalities to the left are found in tables of measures; frequency of occurrence is shown in upper left of each cell* All other equalities found in the texts are listed to the right; frequency of occurrence is shown in lower right of cell. Blank indicates no occurrence,.

12 inches

= 1 foot

4

1

f oot

5 12 inches

36 inches

= 1 yard

1

1

1

yard

- 36 inches

3 feet r 1

yard

3

1

yard

- 3 feet

16| feet =

1 rod

2

1

1

rod =

16| feet

5280 feet

5 1 mile

2

I

1

mile

= 5280 feet

5f yards =

1 rod

2

1

1

rod =

5f yards

5

1

mile

= 1760 yards

320 rods -

1 mile

2

1

mile

- 320 rods

Total

16

7

c

t

r

TABLE III. RELATIONS AMONG UNITS IN SQUARE MEASURE

FOUND IN THE STRAYER-UPTON SERIES.

Equalities to the left are found in tables of measures; frequency of occurrence is shown in upper left of each cell. All other equalities found in the texts are listed to the right frequency of occurrence is shown in lower right of cell* Blank indicates no occurrence*.

144 sq»in* « 1 sq»rt^

9 sq,.ft» r 1 sq..yd*

30f sq.yd» - 1 sq,.rd*

160 sq^rd^ - 1 acre

43,560 sq»ft^ - 1 acre

640 acres ^ 1 sq..mi

Total

20

1 sq# ft.. — 144 sq#in.

1 sq» yd» = 9' sq» ft*

1 sq» rd^ = 30t sq» yd,

1 acre = 160 sq» rd,

1 acre s 43,560 sq* ft,

1 sq.. mi • X 640 acres

8

TABLE IV.

RELATIONS AMONG UNITS IN CUBIC MEASURE

FOUND IN THE 3TRAYER-UPT0N SERIES^

Equalities to the left are found in tables of measures; frequency of occurrence is shown in upper left of each cell. All other equalities found in the texts are listed to the right; frequency of occurrence is shown in lower right of cell. Blank indicates no occurrence.

3

1728 cu,in, K 1 cu,ft^

1 cu,. ft* s. 1728 cu, in.

3

27 cu,ft, = 1 cu,yd.

1 cu. yd, - 27 cu, ft.

6

Total

i

9

TABLE V. RELATIONS AMONG UNITS IN DRY MEASURE FOUND IN THE STRAYER-UPTON SERIES.

Equalities to the left are found in tables of measures; frequency of occurrence is shown in upper left of each cell. All other equalities found in the texts are listed to the right frequency of occurrence is shown in lower right of cell. Blank indicates no occurrence.

2 pt. 3 1 qt. 8 qt» = 1 pk. 4 pk. = 1 bu.^ Total..

1 qt. r 2 pt. 1 pk. t 8 qt. 1 bu. - 4 pk^

c

10

TABLE VI.

RELATIONS AMONG UNITS IN LIQUID MEASURE

FOUND IN THE STRAYER-UPTON SERIES.

Equalities to the left are found in tables of measures; frequency of occurrence is shown in upper left of each cell. All other equalities found in the texts are listed to the right; frequency of occurrence is shown in lower ri^ht of cell. Blank indicates no occurrence.

3

2 pt^

1 qt»

1 qt.

2 pt.

3

4 qt,.

1 Kal»

1 gal.

4 qt,.

6

Total

II

TABLE VII. RELATIONS AL^ONG UNITS IN MEASURE OF WEIGHT

FOUND IN THE STRAYER-UPTON SERIES.

Equalities to the left are found in tables of measures; frequency of occurrence is shown in upper left of each cell. All other equalities found in the texts are listed to the right; frequency of occurrence is shown in lower rip;ht of cell* Blank indicates no occurrence*

16 oz. = 1 lb*

3

1

1 lb, - 16 oz.

100 lb, - 1 cwt.

8

1 cwt, ^ 100 lb.

8000 lb, « 1 T.

3

2000 lb, =: 1 T,

Total

8

1

18

TABLE VIII. RELATIONS AMONG UNITS IN TABLES OF COUNTING

FOUND IN THE STRAYER-UPTON SERIES ►

Equalities to the left are found in tables of measures; frequency of occurrence is shown in upper left of each cell. All other equalities found in the texts are listed to the right; frequency of occurrence is shown in lower right of cell. Blank indicates no occurrence.

12 units 9» 1 doz.

12 dozen = 1 gross

144 units T. 1 gross

20 units s 1 score

Total

24 sheets -s. 1 quire

480 (500) sheets * 1 ream

20 quires =. 1 ream

Total

1 doz. = 12 units

1 gross s 12 dozen

1 gross s 144 units

1 score s 20 units

1 quire = 24 sheets

1 ream =. 480 (500) sheets

1 ream - 20 quires

c

13

TABLE IX ► RELATIONS AMONG UNITS IN MEASURE OF TIME

FOUND IN THE STRAYER-UPTON SERIES*

Equalities to the left are found in tables of measures; frequency of occurrence is shown in upper left of each cell. All other equalities found in the texts are listed to the right; frequency of occurrence is shown in lower right of cell. Blank indicates no occurrence..

60 seconds r 1 minute

60 minutes ^ 1 hour

24 hours =. 1 day

7 days r 1 week

365 days - 1 year

366 days = 1 leap year

4 weeks = 1 month

52 weeks s. 1 year

12 months - 1 year

100 years r 1 century

Total

28

1 minute = 60 seconds

1 hour = 60 minutes

1 day s 24 hours

1 week = 7 days

1 year =365 days

1 leap year 366 days

1 month = 4 weeks

1 year = 52 weeks

1 year = 12 months

1 century =. 100 years

c

14

TABLE X. RELATIONS AI/iCNG UNITS IN MEASURE OF CAPACITY FOUND IN THE STRAYSR-UPTON SERIES •

Equalities to the left are found in tables of measures; frequency of occurrence is shown in upper left of each cell. All other equalities found in the texts are listed to the right; frequency of occurrence is shown in lower right of cell. Blank indicates no occurrence.

1 gal, - 231 cu.in, 31| gal. = 1 bbl. 7| gal. s 1 cu.ft. 1 cu. ft. ^ 7^ gal. Total

3

1

1

1

3

3

10

S

231 cu. in. c 1 gal. 1 bbl. X 3lf gal.

i

15

TABLE XI * RELATIONS AMONG UNITS IN MEASURE OF CONTENT

FOUND IN THE STRAYER-UPTCN SERIES »

Equalities to the left are found in tables of measures; frequency of occurrence is shown in upper left of each cell. All other equalities found in the texts are listed to the right frequency of occurrence is shown in lower right of cell. Blank indicates no occurrence.

1 bu, ^ 2150 cu. in.

8 bu,. » 1 cu, ft.

1 bu, s li cu, ft.

1 T, hard coal s 35 cu,f t

1 cu, ft. = 8 bu.

Total

12

2150 cu. in, r 1 bu.

It cu. ft, = 1 bu.

#

16

TABLE XI I • RELATIONS AMONG UNITS OF METRIC MEASURES

FOUND IN THE STRAY SR-UPTON SERIES.

Equalities to the left are found in tables. Frequency of occurrence of these table facts is noted in upper left of cell.

1 mm. t 1/1000 m,

1 cm, s 1/100 m.

1 dm. - 1/10 ttt,

1 Dm, - 10 m.

1 Hm, = 100 m.

1 Km, - 1000 m. Total

1 ml. = 1/1000 1,

1 HI. = 100 1,

Total

1 Kg, - 1000 g.

1 M. T. = 1000 Kg,

Total

c

17

TABLE XIII. RELATIONS BETWEEN UNITS OF METRIC MEASURES AMD ENGLISH MEASURES FOUND IN STRAYER-UPTON

Equalities to the left are found in tables of measures*. Frequency of occurrence of these table facts is noted in upper left of cell. All other equalities found in texts are listed to the right; frequency of occurrence is noted in lower right of cell. Blank indicates no occurrence.

1 m. = 39,37 in. 1 tn, - 1.1 yd. 1 Km* tt ,6 mi.

Total

Total

1 liter = about 1 qt.

Total

1 Kg, r 2»2 pounds

18

TABLE XIV. RELATIONS IN MISCELLANEOUS MEASURES

FOUND IN THE STRAYER-UPTON SERIES ►

This table shows relations in miscellaneous measures and the frequency of their occurrence in the Strayer-Upton Arithmetics*

Money

1 English pound sterling = $4»8665 1 Linear Measure

1 knot 5 1.15 mi. I

Capacity and Content

1 cu.ft. ice s 56.4 lb, 1

1 cu.ft, water r 62.5 lb. 4

1 cu.ft. marble or granite - 168 lb» 2

1 cu.ft. pine = 25 1 lb. t

1 cu.ft. hard coal •= 57 lb. 1

1 bu.. potatoes^ 60 lb, 1

I bu. cherries r 50 lb» 1

I bu. wheat <=. 60 lb. I

1 bu. wheat :s 60 lb. 1

1 gal. waters 8.3 Ib^ I

Weight

1 lb. sugar = 2 cups 2

1 lb. flour s 4 cups I

1 pt. milk s 2 cups 1

1 lb. butter s 32 tbsp. 1

1 long ton = 2240 lb. I

1 ton of hay = 500 cu» ft. 1

Liquid

1 pt. = 2 glasses 1

1 qt. = 4 glasses 1

1 gal. s 16 glasses I

1

19

INFORMATIONAL MATTER WITH REFERENCE TO UNITS OF MEASURES.

Since actual use of units of weight and measure is always in connection with articles and commodities, an understanding of the meaning and use of units of weight and measure can only come through commodities. As soon as a commodity is thoroughly un- derstood, the appropriate unit of measure to be used in con- nection with it has been learned incidentally.

Units of weight and measure are not learned as such, but in connection with commodities. It is this idea of the con- nection or inter-relation between commodities and units of weight and measure that the expression "informational matter", used here in connection with denominate numbers, is meant to convey. No manipulation or computation with denominate numbers is involved, but simply an understanding that a given unit of measure is used in connection with a given commodity.

The instances in which nothing more is involved than an understanding of the meaning of a unit of measure through its use with a commodity, have been gathered and grouped under two headings, namely, those appearing in statement form and those appearing in question form. For instance, the statement is made that "When we want to know how hot or how cold it is, we look at a thermometer and read the number of degrees of temperature."^ The information is imparted that

!• Strayer-Upton Arithmetics, Lower Grades, p. 238

(

20

temperature is measured by degrees and is listed under "Informational Matter in Statement Form", Again, the question is asked, "By what measures of weight are these thing sold: Butter, spices, coffee, meat, iron, large quantities of ice, small quantities of ice, pepper, hay?"^^ This is placed under the heading of "Informational Matter in Question Form",-

Inf ormational Matter in Statement Form found in

Strayer-Upton..

Dry Measure:

Fruits, grains, vegetables are measured by the pint, the quart, the peck, and the bushel. Also sold by the pound..

Liquid Measurer

Milk is bought by the pint and quart; Milk is bought by the gallon in large quantities; Gasoline is bought by the gallon..

Measures of weight;

Ice is bought by the pound;

Ice is bought by the ton (large quantities);

Spices and pepper by the ounce;

Butter and meat by the pound;

Freight and express by the hundredweight;

Hundredweight is used in baying grain and cotton;

Hundredweight is used in selling milk to creamery;

Ton is used to weigh such things as coal and ice.

1- Strayer-Upton Arithmetics, Lower Grades, p# 237 #2

21

Miscellaneous Measuresr

Electricity

Gas

Lumber

Finishing lumber, molding

Temperature

Angles

Freezing point of water Boiling point of water Blood heat

kilowatt hour (watt hour)

cubic feet

board feet

lineal foot

Degree

degree

32 degrees F^ 212 degrees F. 98i degrees F..

Informational Matter in Question Form found in Strayer-Upton*

Telling ages of children, finding heights and weights*

What groceries are bought by the pound?

How are the following sold: butter, spices, coffee,

meat, iron, large quantities of ice, small

quantities of ice, pepper, hay? Tell what unit of measure you use to findr

The height of a door;

The length of a carpet;

The length of a house;

The thickness of a board;

The weight of a ship;

The length of a road;

The weight of a man;

The length of a candle;

The amount of coal you burn each winter;

The amount of ice cream needed for 8 people;

The distance from Chicago to San Francisco;

The amount of gasoline used on an automobile trip.

o

22

UNITS OF MEASURE MERELY A3 NAMES

Many of the written problems contain units of measure which,

however, involve no computation or manipulation with denominate

numbers or such» The units of measure are mentioned merely as

words. The following problems mi,a;ht be cited to show the type

of written problem involved: "Express the following lengths to

the nearest tenth of a foot: 5.29 ft., 7»21 ft*, 4.38 ft.,

6»42 ft»"^ "How much must Mrs. King pay at the fish store for

some codfish weighing 2 5/3 lb. and costing 19 a pound."

"The Woolworth Building in the city of New York is 792 ft* high.

Jack's house is 72 ft. high. How many times as high as Jack's

3

house is the Woolworth Building?" "Mrs. Lee bought 3t yd. of curtain goods. How many curtains, each 7/8 yd. long, can she make from this?"^

1 Strayer-Upton, Middle Grades, p.216,j^4

2 Ibid* 88 #Q

3 " 32 #2

4 " 212/^5

23

TABLE XV. UNITS OF MEASURE AS '.VORDS IN STRAYER-UPTON

The table shows the frequency of occurrence of units of measure involving no computation with denominate numbers as such*

Linear Measure

Inch Feet Yard Mile

7 16 63 22

Square Measure

Square yard Acre

L 16

Cubic Measure

Cubic foot

1

Dry Measure Quart Bushel

3 5

Liquid Measure

Pint

Quart

Gallon

3 8 8

Wei^?ht Measure

Ounce Pound Ton

4

75 12

Counting Pair Dozen

24 37

24

Time Measure

Seconds I

Minutes 1

Hour 39

Day 32

Week 42

Month 25

Year 44

Miscellaneous Measures

Kilowatt hour Cubic foot (gas)

5 1

25

TABLE FACTS IN EXAMPLES AND V/RITTSN PROBLEMS

Tables XVI to XXI show the number of times a table fact or the reverse statement is called for in either examples or in written problems. For example, a table fact may be called for in an example of the following

1

type: "How many square inches equal 1 square foot?"

Or a written problem may involve the knowledge of a

table fact so that it may be solved, as for instance

in the following written problem: "Mr. Archer finds

that the distance around the deck of a steamship is 264

ft. How many times must he go around the deck to walk 2

a mile?" To solve this problem involves a knowledge of the fact that 52SO feet equal i mile.

1. Strayer-Upton , Lower Grades, p. 305 #1, 8. Strayer-Upton, Middle Grades, p. 309 j^4.

Q

26

TABLE XVI. TABLE FACTS IN UNITED STATH:S MONEY ASKED IN

EXAMPLES AND WRITTEN PROBLEMS- STRAYER-UPTON .

In each instance, the fact in parenthesis indicates the fact asked for. Read to the left of the figure thus: What are 10 mills equal to? Frequency of occurrence is shown in the column of cells to the left. Read to the right of the figure thus: What is one nickel equal to? Frequency of occurrence is shown in the column of cells to the right. The frequency of occurrence in examples is shown in the upper left of the cell; the frequency of occurrence in written problems is shown in the lower right of each cell. Blank indicates no occurrence.

10 millssCl cent)

5 cents=(l nickel)

10 centsnCl dime)

25 centssCl quarter)

50 centssCI dollar)

100 cents^Cl dollar)

10 dimeszCl dollar)

1 mill=(l/10 cent)

1 nickels (5 cents)

1 dimesClO cents)

1 quarter=(25 cents)

|- dollars(50 cents)

1 dollarsClOO cents)

1 dollarrClO dimes)

0

27

TABLE XVII TABLE FACTS IN LINEAR MEASURE ASKED IN EXAMPLES

AND WRITTEN PROBLEMS- STRAYER-UPTON

In each instance the fact in parenthesis indicates the fact asked for. Read to the left of the figure thus: What are 12 inches equal to? Frequency of occurrence is shown in the column of cells to the left. Read to the right of the figure thus: What is one foot equal to? Frequency of occurrence is shown in the column of cells to the right. The frequency of occurrence in examples is shown in the upner left of the cell; the frequency of occurrence in written problems is shown in the lower right of each cell. Blank indicates no occurrence.

1 foot=(l2 inches)

1 yards (76 inches)

1 yard=(3 feet)

1 rod=(16| feet)

1 mile=(5280 feet)

1 rode (51 yards)

1 mile=(1760 yards)

1 milea(320 rods)

12 inchesaCl foot) 36 inchesrCl yard)

3 feet-(l yard)

16| feet=.(l rod)

5280 feetrCl mile)

5|- yards^Cl rod)

320 rodsrCl mile)

5

6

28

TABLE XVIII. TABLE FACTS IN SQUARE MEASURE ASKED IN EXAMPLES

AND '.VRITTEN PROBLEMS- STRAYER-UPTON

In each instance the fact in parenthesis indicates the fact asked for. Read to the left of the fip^ure thus: What are 144 square inches equal to? Frequency of occurrence is shown in the column of cells to the left. Read to the right of the figure thus: What is 1 square foot equal to? Frequency of occurrence is shown in the column of cells to the right. The frequency os occurrence in examples is shown in the upper left of the cell; the frequency of occurrence in written problems is shown in the lower right of each cell.

144 sq. in.sd sq.ft.)

9 sq.ft.-sCl sq.yd.) 307 sq.yd.rd sq.rd.)

160 sq.rd. r(l A. )

640 A.-(l sq.mi.)

36 sq . mi . = (1 tp. )

1 sq.ft.s(144 square in.)

1 sq.yd. s(9 sq.ft.) 1 sq.rd. r(30t sq.yd.)

1 A.- (i60 sq.rd.)

1 sq.mi.T;(640 A. )

1 tp. =( 36 sq. mi . )

5

0

29

TABLE XIX. TABLE FACTS IN DRY MEASURE ASKED IN EXAMPLES

AND 'vVRITTEN PROBLEMS- STRAYER -UPTON

In each instance the fact in parenthesis indicates the fact asked for. Read to the left of the figure thus: What are S pints equal to? Frequency of occurrence is shown in the column of cells to the left. Read to the right of the figure thus: How many pints are there in one quart? Frequency of occurrence is shown in the column of cells to the right. The frequency of occurrence in examples is shown in the upper left of the cell; the frequency of occurrence in written problems is shown in the lower right of each cell. Blank indicates no occurrence •

2 pintsaCl quart)

8 quarts^d peck)

4 pecks-id bushel)

1 quart=.(2 pints)

1 peck?:(8 quarts)

1 bushelTi(4 pecks)

30

TABLE XX. TABLE FACTS IN LIQUID MEASURE ASKED IN EXAMPLES

AND WRITTEN PROBLEMS — STRAYER-UPTON

In each instance the fact in parenthesis indicates the fact asked for. Read to the left of the figure thus: What are 2 pints equal to? Frequency of occurrence is shown in the column of cells to the left. Read to the right of the figure thus; One quart is equal to how many pints? Frequency of occurrence is shown in the column of cells to the right. The frequency of occurrence in examples is shown in the upper left of the cell; the frequency of occurrence in written problems is shown in the lower right of each cell, ^lank indicates no occurrence •

2 pints:t(l quart)

4 quarts^Cl gallon)

1 quart=(2 pints) 1 gallon=(4 quarts)

31

TABLS XXI. TABLE FACTS IN TIME ASKED IN EXAMPLES AND

'JVRITTEN PROBLEMS - STRAY^R-UPTON

In each instance the fact in parenthesis indicates the fact asked for. Read to the left of the figure thus:What are 60 seconds equal to? Frequency of occurrence is shown in the column of cells to the left. Read to the right of the figure thus: How many seconds in 1 minute? Frequency of occurrence is shown in the column of cells to the right. The frequency of occurrence in examples is shown in the upper left of the cell; the frequency of occurrence in written problems is shown in the lower right of each cell. Blank indicates no occurrence*

60 secondssCl minute)

60 minutessd hour)

24 hourssCl day)

7 days::(l week)

365 dayssCl year)

366 dayszCl Ip.year)

52 weeks-id year) 12 monthssd year)

100 yearsrCl century)

1 minute:5(60 seconds)

1 hour=(60 minutes)

1 day-(24 hours)

1 weeks:.(7 days)

1 yearr(365 days)

1 year=(366 days)

1 years(52 weeks)

1 yearrd2 months)

1 century=.(lOO years)

32

REDUCTION OF DENOMINATE NUMBERS

Reduction of denominate numbers is the process of changing their denomination without changing their value* Denominate numbers may be changed from a higher to a lower denomination, as for example, when 8 miles are reduced to feet; denominate numbers may also be changed from a lower to a higher denomination, as when 21 pints are reduced to gallons. The former process, is known as descending reduction; the latter, as ascending reduction.

Meaning and Reading of Tables XXII to XXXIII. The upper figure shows descending reductions. The unit of measure at the head of the column of vertical squares in these figures read in connection with each of the cells in these columns indicate the denomination from which the reduction is made; when these terms are read in connection with the row of horizontal cells to the left, they indicate the denomination to which the reduction is made,- The lower figures show ascending reductions. The unit of measure read in connection with the horizontal rows of cells to the right indicate the denomi nation from which the reduction is made; read in

0

connection with the vertical column of cells it refers to the unit to which the ascending reduction is made. Frequency of occurrence of reductions appearing in examples is shown in the upper left of each cell; those appearing in written problems are shown in the lower right of each cell.

34

TABLE XXII. REDUCTIONS IN UNITED STATES MONEY

STRAYEH-UPTON

The upper figure shows descending reductions: the lower figure ascenaing reductions. The frequency of reductions ap-earing in examples is shown in the upper left of cell; those appearing in written problems in lower right. For further details as to meaning and reading of table see page 32.

Dollars

Half dollars

Quarters

1

Dimes

1

1

Nickels

1

2

2

7

1

^ Cents 3

Dollars

Half dollars

Quarters

1

Dimes

1

Nickels

1

1

Cents 2

5

1

4

f

€

35

TABLE XXIII. REDUCTIONS IN LINEAR MEASURE-«-3TRAYER-UPT0N

The upper figure shows descending reductions; the lower figure ascending reductions. The frequency of reductions appearing in examples is shown in the upper left of each square; those appearing in written problems are shown in the lower right of each square. For further details as to meaning and reading of tables see page 32^

Miles

2

Rods

2

1

Yards

6

1

9

3

Feet

1.

3

6

2

inches

Miles

Rods

Yards

8

Feet

Inches 1

;4

36

TABLS XXIV. REDUCTIONS IN SQUARE MEASURE^-tSTRAYSR-UPTON

The upper figure shows descending reductions; the lower figure ascending reductions. The frequency of reductions appearing in examples is shown in the upper left of each square; those appearing in written problems are shown in the lower right of each square. For further details as to meaning and reading of table see page 32,

Square *"iles 1

Acres

il

I t

Square Rods

Square Yards

Square Feet

Square inches

Square Miles

Acres

1

Square Rd.

Square Yards

Square Feet

2

1

Square inches

2

3

37

TABLE XXV. REDUCTIONS IN CUBIC MEASURE- STRAYER-UPTON

The upper fip;ure shows descending reductions; the lower figure ascending reductions. The frequency of reductions appearing in examples is shown in the upper left of each cell; those appearing in written problems are shown in the lower right of each cell.

Cords

Cubic Yards

2

Cubic Feet

2

Cubic Inches

Cords

Cjabic Yards

3

Cubic Feet

6

38

TABLE XXVI. REDUCTIONS IN DRY MEASURE; STRAYER-UPTON

The upper figure shows descending reductions; the lower figure af?cending reductions. The frequency of reductions appearing in ex-mples is shown in the upper left of each square; those appearing in written problems are shown in the lower right of each square. For further details as to reading and meaning of table see page 32.

Bushels

1

2

Pecks

6

4

5

2

Quarts

1

1

Pints

Bushels

Pecks

1 '

Quarts

1

4

2

1

Pints

3

1

39

TABLE XXVII. REDUCTIONS IN LIQUID MEASURE- STRAYER-UPTON

The upper figure shows descending reductions; the lower figure ascending reductions. The frequency of reductions appearing in examples is snown in the upper left of each cell; those appearing in written problems are shown in the lower right of each cell. For further details as to reading and meaning of table see page 32.

Barrels

Gallons

1

9

2

Quarts

9

1

4

1

Pints

2

2

Cups or Glasses

Barrels

Gallons

Pints

Quarts |2~~

t

2 :

Cups or Glasses

4

40

TABLE XXVIII. REDUCTIONS IN WEIGHT MEASURES- STRAYER-UPTON.

The upper figure shows descending reductions; the lower figure ascending reductions. The frequency of reductions apnearing in examples is shown in the upper left of each square; Those appearing in written problems are shown in the lower right of each square. For further details as to reading and meaning of table see page; 32,

Tons

Cwt.

4

1

2

Pounds

27

1

Ounces

Tons

Cwt.

1

Pounds

8

2

9

Ounces

20

2

€

0

TABLE XXIX. REDUCTIONS IN COUNTING- STRAYER-UPTON

The upper figures show descending reductions; the lower figures descending ref^uctions. The frequency of reductions ap'-'earing in examples is shown in the upper left of each cell; those appearing in written problems are shown in the lower right of each cell.

Gross

Score

Ream

Dozen

Quire

3

1

2

6

Unit

1

Sheet

Gross

Ream

Score

Quire

Dozen

Shee

Unit

1

5

i

42

TABLE XXX. REDUCTIONS IN TIME- STRAYER-UPTON

Century

Year

Month

The upner figure shows descending reductions; the lower figure ascending reductions. The frequency of reductions appearing in examples is shown in the upper left of each cell; those appearing in written problems are shown in the lower risht of each cell. Further details as to meaning and reading of table may be found on page 32.

Week

Day

Hour

14

Minute

Second

Year

Month

Week

Day

Hour

^inute

8

10

Second

#

#

43

TABLE XXXI . REDUCTIONS IN MEASURES OF CONTENT AND CAPACITY'

3TRAYER-UPT0N

The upper figure shows reductions from pints, gallons, etc. to cubic feet and cubic inches and the frequency of their occurrence* The lower figure shows the reverse, that is, the changes of cubic feet and cubic inches to pints, gallons etc. Those occurring in examples are shown in the upper left of the cell; those occurring in written problems in the lower right of each ceil.

00

0)

d

I-H

0)

0)

o

0)

t3

CO

-p

rH

XI

C

C

C

i-H

0)

=J

o

cd

3

O

O

PQ

CU

S

1

1

Cubic Feet Cubic Inches

2

00

CO

c

I-H

0)

o

0)

tJ

■p

I-H

-C

C

ra

c

r-i

OT

C

•H

as

O

o

O

PU

Cubic Feet

1

16

6

5

21

2

Cubic inches

1

2

44

TABLE XXXII. REDUCTION'S IN METRIC MEASURES- STRAYER-UPTON

Reductions from metric measures to Enp.lish equivalents and visa versa. The fortner are shown in the upper fif^ures, namely in fig. 1, fig. 5 and fig, 6; the latter in the lower figures, fig. 4, fig. 7 and fig. 8.

Km. Dm. m, dm.

Miles Yards

Feet

Inches

Fig. 1. Length

Fig. 2

Km. Dm. m. dm.

Miles Yards

Feet

Inches

Fig. 3.

Fig. 4.

c

TABLE XXXII. (Continued)

Liter HI.

Quart

M.T. Kg,

Gallon

Pound

Bushel

Ton

Fig, 5. Capacity

Fig, 6. Weight

Liter HI.

M.T. Kg.

Pound

Ton

Quart Gallon Bu shel

Fig. 7, Weight

Fig, 8. Capacity

46

TABLE XXXIII. REDUCTIONS IN MISCEL- ANE0U3 MEA3URE3-

STRAYER-UPTON

Fig, 1 and 2 show chanp;es from knots, leagues to miles; and bushels, pecks to pounds and ounces; fig. 3 and 4 show the reverse process, namely changes from pounds, ounces to bushels, pecks etc.

CO

(0

r-i

C

i-H

<D

w

0)

o

0)

CO

x:

-P

0)

0)

o

c

r-i

i-H

•r-l

(d

cd

PQ

(X.

CQ

Pounds Ounces

m o

0) D

0)

ii-iiles

Fig. 1

Fig. 2

Pounds

(0

0)

0)

iH

C

l-l

0)

m

to

o

,— 1

U

CD

(0

o

c

l-l

u

rH

3

•H

cd

CO

CQ

{I.

OQ

CQ

Ounces

Fig. 3

Fig. 4

47

COMPUTATION WITH DENOMINATE NUMBERS

Generally speaking, denominate numbers are of two

kinds, namely, simple denominate numbers and compound

denominate numbers. vSimple denominate numbers consist

of units of measure of but one denomination, as, 1 inch.

Compound denominate numbers consist of two or more units

of measure of the same kind, such as, 2 yards 3 feet

5 inches. Lennes calls numbers involving two or more

units of different kinds, as for example those expressed

in "miles oer hour", "ton-miles" etc. "quasi-denominate" 1

numbers.

Computation in the fundamental processes involving these three types of denominate numbers and the frequency of their occurrence are shown in Tables XXXIV to XXXVI,

Written problems in connection with the clock and the calendar which involve computation, are grouped separately under Time,

1. Lennes, N» J. The Teaching of Arithmetic, page 281.

48

TABLE XXXIV. COMPUTATION WITH SIMPLE DENOMINATE NUMBERS

STRAYER- UPTON

The first column marked "Addition" shows the number of

occurrences of computation in adding the denominate numbers

involving the units of measure listed to the left of the

figure; the second column shows the number of occurrences

in subtraction, etc. Examples snown in upper left of cell;

written problems in lower right of each cell.

c

Inches Feet

Yards

Miles Acres

Square *"iles Quart (Dry)

Peck

Bushel

Pint (Liquid) Gallon

^arrel

C o

o u

CO

I

•H C rH O Q-, rH •H -P P

c o

•r-l

•rH >

Q

16

6

4

8

S

1

13

18

12

1

9

1

I

•5

2

11_

35

9

2

4

1

1

!

»

i

3

1 1

L

J

1

. 2

r —

1

i

1

I

f

)

TABLE XXXIV (Continued)

d

o

c

+>

o

(d

•fH

o

c

•P

•H

o

o

o

(d

cu

+>

'H

(D

•H

•p

•P

•H

n

iH

>

•i-t

<

03

3

P

Ounce

2

Pound

1

1

37

18

i

14

Dozen

1

Seconds

1

4.

3

ii'iinutes

2

3

3

2

Hours

i

...15

S

5

3

Days

L ■ -

i ^

3

Years

Clock-Calendar iwinutes-"ec .

L

.

I 1

r

Hours-Min.

4

9

2

!

i

Month-Day

1

« f

Yr.Mo.Day

il

i

c

TABLE XXXIV (Continued)

c

o

<

Degree (Temp) Cu.Ft. (gas)

Kilowatt

c

O

1-1

c

■P

o

o

+>

C

o

r-4

o

L.

•r-l

m

+3

1-1

^

i-l

>

iH

(0

5

10

1

31

!

1

51

TABLE XXXV. COMPUTATION WITH "QUASI-DENOMINATE" NUMBERS-

STRAYER-UPTON

The first column marked "Addition" shows the number of occurrences of computation in addinp^ the denominate numbers involving the units of measure listed to the left of the figure; the second column shows the number of subtraction computations, etc. Examples are shown in upper left of the cell; written problems in lov.er right of each cell.

Feet per second Yards per second Miles per second

Miles per minute Miles per hour Miles per day Knots per hour Km. per hour Miles per gallon Bushels per acre

c

i C

o

f-i

o o

c

■p

c

o

o

r-* -P

o

ft

«D

a.

+>

U

w

f-i

+>

x>

r-l

>

•d

3

3

<

03

P

1

1

2

10

1

1

1

6

i

1

2

18

36

4

3

14

1

1

1

1 — ^

L .,

i .

1 1

|_.._.

__J

j

!

21

1

L

Bosion University

t

TABLE XXXVI. COMPUTATION WITH COMPOUND DENOMINATE NUMBER3-

STRAYER-UPTON

The first column marked "Addition" shows the number of occurrences of computation in adding the denominate numbers involving the units of measure listed to the left of the figure; the second column shows the number of subtraction computations, etc. Examples are shown in upper left of the cell; written problems in lower right of each cell.

Feet inches Yards inches Yards feet Mi .rd. yd. ft. in.

Bushels pecks Pecks quarts Gallons quarts Pounds ounces Days hours ■ Hours minutes

C

O

c

4^

o

cd

o

c:

•p

t-i

c

O

o

o

t-i

cS

a.

•i-i

+^

u

•H

0)

•i-i

+>

+>

•r-l

•a

X>

r-l

>

3

3

<

to

a

Q

22

I. A...

! 1 i 1

I 2_ 1

L...

■2

1 "iT

I

2 »

c

I

53

SUMMARY OF THE STRAYER-UPTON ARITHMETIC SERIES

In Table xxxvii we have a summary of the data gathered with reference to the denominate numbers found in the Strayer-Upton Arithmetic Series. The data has been gathered under five headings. One group of facts has been omitted from this summary, namely, "Informational Matter With Reference To Denominate Numbers".. The reader may refer to pages 19 to 21 for this data.

Reading and Meaning of Table XXXVII. The tables of measures involved in the summary are listed to the left of the figure. Column 1 shows the number of equalities found in each of the tables; column 2 shows the number of units of measure used as mere words, while column 3 shows the number of written problems involved; in column 4 we find the number of table facts asked in examples and written problems and column 5 shows the number of examples and written problems involved; column 6 shows the total number of reductions, while column 7 shows the number of examples and written problems involved in these reductions; column 8 shows the number of units of measure used in computation and column 9 the number of examples and written problems involved in computation with simple denominate numbers; column 10 shows the number of compound denominate numbers used and the final column the number of examples and written problems involved in computation with compound denominate numbers.

( (

54

SUMMARY OF STRAYER-UPTON. TABLE XXXVII.

For reading and meaning oi table, see page 53.

U. S. Money Linear

Square

Cubic Dry

Liquid Weight Counting

Time

Clock Calendar

Capacity Content

Metric

English Equivalents

QuasiTDen.^No,

Misc*lileasures

m

0)

tH

rH

cu

a

B

U

(fi

cr

• o

X

(xj

rH

to.

0}

[

l5

c

T

•P

W

o

*rl

0)

cd

CO

o

CO

-P

m

c

(D

(D

•rH

<0

<D

rH

«-H

-P

rH

■p

i-H

o

a.

0)

a.

O

a.

cu

E

rH

e

C3

D.

B

s

«J

(tj

S

c6

o

X

00

X

X

o

X

o

W

w

w

o

w

o

rH

in

«>-

CO

rH

rH

6

6

1?

14

27

\

7

4

LOB

6

14

p

15

50 16

4

4

l45

4

15

30

6

2

17

2

2

11

14

4

2

6

2

1

1

4

8

3

2

8

> ' 1

1

9

21 18

3

A

2

8

2

3

21

2

p

10

31

3

3

1

1

3

3

91

5

61

2

2

1

3

lo

7

2

61

5

4

1

7

10

7

184

3

16

43

5

vjVJ

2

4

7

4

20

10

8

59

10

3

5

10

20

2

6

3

5,Q

(

c

. J. . _

» _ . • . - . ... f .Ik-.-..,. » . . H.

1

c

55

TABLE XXXVIIA. DETAILED SUMMARY OF REDUCTIONS- Strayer-Upton

Column one shows the total number of descending reductions with reference to each table of measure listed to the left of the figure. The second column shows the total number of examples (upper left of cell) and the total number of written problems (lower right of cell). Column 3 and 4 show similar data for ascending reductions.

+j •

o -P

3 o

xi 3

Q) to 'C5 0)

<D <1> 0)

r-J K r-l

m Ci. CL.

OS'S

CO flj o <d

(D X 0) X

Q cd << (a

United States "^oney

7

14

a

7

13

3

Linear Measure

9

30

6

6

20

10

Square Measure

6

8

6

6

4

Cubic Measure

2

4

2

4

Dry Measure

4

13

9

5

8

9

Liquid Measure

5

24

7

5

7

13

Weight Measure

> - - .

31

i_ ..

2

3

30

12

Counting Time

.1..

10

25

_ .-8..

1 6

1

1

iT^

Engl»Eq»ofMetric Weights

4

1

9

' 1

(

--A

Capacity and Content

4

1

12.

6

7 !

i I

Misc. Measures

7

6

4i

1

47 j

c

PART TWO

The following analysis covers the units of weight and measure, with special reference to denominate numbers, as found in the three book series of the New Everyday Arithmetic, or as they will be called in this study Hoyt-Peet.

i

< (

56

RELATIONS AMONG UNITS OF MEASURE

Tables XXXVIII - XLIX which follow show the various tables of measures found in the New Everyday Arithmetic or Hoyt-Peet, each Table representing; one table of measure, except that counting articles and counting paper have been grouped together in one table, as have also the various measures of the Metric system. In each case the equalities found in the table of measure has been stated to the left of the figure and the frequency of occurrence of these equalities is indicated in the upper left corner of the corresponding square • All other relations found in the texts of the Hoyt-Peet Series are listed to the right and the frequency of occurrence of these equalities, either in written problems or in some other isolated form, is recorded in the lower right of the cell#

The Hoyt-Peet Series uses the puzzle element to introduce tables of weight and measure* One of the numbers in each relation of the table is omitted, for instance, — inches — 1 foot",.^ The pupil is asked to "complete the table".. This is merely a variation in the method of introducing children to the relations in the tables of weights and measures..

1.. Hoyt-Peet, Second Book, p. 144

i

< (

57

TABLE XXXVI II • RELATIONS AwlONG UNITS OF UNITED STATES MONEY

FOUND IN THE HOYT-PEET SERIES.

Equalities to the left are found in tables; frequency of occurrence is shown in upper left of each cell. All other equalities found in the texts are listed to the rip;ht; frequency of occurrence is shown in lower right of cell. Blank indicates no occurrence.

10 mills 7L 1 cent

5 cents st 1 nickel

10 cents = 1 dime

25 cents s 1 quarter

50 cents ^ § dollar

100 cents ss 1 dollar

10 dimes a 1 dollar Total

1 mill ^ 1/10 cent

1 nickel = 5 cents

1 dime s 10 cents

1 quarter a 25 cents

5 dollar 50 cents

1 dollar = 100 cents

1 dollar s 10 dimes

58

TABLE XXXIX. RELATIONS Aiv"ONG UNITS OF LINEAR MEASURE

FOUND IN THE HCYT-PEET SERIES*

Equalities to the left are found in tables of measures; frequency of occurrence is shown in upper left of each cell. All other equalities found in the texts are listed to the right frequency of occurrence is shown in lower right of cell. Blank indicates no occurrence*

12 inches =. 1 foot

36 inches » 1 yard

3 feet =t 1 yard

16w feet - 1 rod

5280 feet s. 1 mile

5i yards s 1 rod

1760 yards s. 1 mils

320 rods = 1 mile

Total

1 foot - 12 inches

1 yard r 36 inches

1 yard ^ 3 feet

1 rod = 16|- feet

1 mile s 5280 feet

1 rod = 5|- yards

1 mile - 1760 yards

1 mile r 320 rods

59

TABLE XL. RELATIONS AwiONG UNITS IN SQUARE MEASURE

FOUND IN THE HOYT-PEET SERIES.

Equalities to the left are found in tables of measures; frequency of occurrence is shown in upper left of each cell. All other equalities found in the texts are listed to the right; frequency of occurrence is shown in lower right of cell. Blank indicates no occurrence.

144 sq»in,. = 1 sq»ft.

9 sq^ft* s 1 sq»yd*

30# sq^yd. s 1 sq.rd.

160 sq»rd,. =- 1 acre

43,560 sq»ft. s 1 acre

209 ft» square - 1 acre

640 acres = 1 sq»mi. Total

1 3q»ft» =t 144 sq.in.

1 sq»yd. =t 9 sq»ft»

1 sq.rd* 30|- sq»yd,

1 acre s 160 sq»rd,

1 acre = 43,560 sq.ft,

1 acre - 809 ft. square

1 sq.mi. -5 640 acres

60

TABLE XLI.

RELATIONS AMONG UNITS IN CUBIC MEASURE

FOUND IN THE HOYT-PEST SERIES.

Equalities to the left are found in tables of measures; frequency of occurrence is shown in upper left of each cell. All other equalities found in the texts are listed to the right frequency of occurrence is shown in lower right of cell. Blank indicates no occurrence.

I

1728 cu.in* z 1 cu.ft.

1 cu.ft. - 1728 cu.in.

1

27 cu.ft. =■ 1 cu.yd.

1 cu.yd. = 27 cu.ft.

1

128 cu.ft. = 1 cord

2

Total

1

61

TABLE XLII. RELATIONS AiTiONG UNITS IN DRY MEASURE

FOUND IN THE HOYT-PEET SERIES •

Equalities to the left are found in tables of measures; frequency of occurrence is shown in upper left of each cell. All other equalities found in the texts are listed to the right; frequency of occurrence is shown in lower right of cell. Blank indicates no occurrence.

2 pints s 1 qt, 8 qt» = 1 pk, 4 pk. 3 1 bu.

Total

1 qt, - 2 pt, 1 pk. r 8 qt, 1 bu, z 4 pk.

#

62

TABLE XLIII. RELATIONS AMONG UNITS IN LIQUID MEASURE

FOUND IN THE HOYT-PEET SERIES •

Equalities to the left are found in tables of measures; frequency of occurrence is shown in upper left of each cell. All other equalities found in the texts are listed to the right; frequency of occurrence is shown in lower right of cell. Blank indicates no occurrence.

2 pt* s 1 qt. 4 qt» » 1 gal. Total

1 qt. = 2 pt. 1 gal. r 4 qt,

63

TABLE XLIV. RELATIONS AMONG UNITS IN MEASURES OF WEIGHT

FOUND IN THE HOYT-PEET SERIES.

Equalities to the left are found in tables of measures; frequency of occurrence is shown in upper left of each cell. All other equalities found in the texts are listed to the ri^^ht; frequency of occurrence is shown in lower right of cell. Blank indicates no occurrence.

16 oz» s I lb.

100 lb» X 1 cwt»

2000 lb. = 1 T.

Total

1 lb» 16 02.

1 cwt. s. 100 lb.

1 T» -s 2000 lb.

64

TABLE XLV. RELATIONS AMONG UNITS IN TABLES OF COUNTING

FOUND IN THE HOYT-PEET SERIES.

Equalities to the left are found in tables of measures; frequency of occurrence is shown in upper left of each cell* All other equalities found in the texts are listed to the right; frequency of occurrences is shown in lower ri,a;ht of cell* Blank indicates no occurrence*

12 units - 1 doz*

144 units - 1 gross

12 doz* - 1 gross

20 units - 1 score

Total

1 doz* - 12 units

1 gross - 144 units

1 gross - 12 doz*

1 score - 20 units

24 sheets - 1 quire

480 C500) sheets - 1 ream

20 quires - 1 ream

1 quire - 24 sheets

1 ream - 480 (500) sheets

1 ream - 20 quires

t

■J

65

TABLE XL VI,. RELATIONS AMONG UNITS IN MEASURES OF TIME

FOUND IN THE HOYT-PEST SERIES.

Equalities to the left are found in tables of measures; frequency of occurrence is shown in upper left of each cell. All other equalities found in the texts are listed to the right frequency of occurrence is shown in lower right of cell. Blank indicates no occurrence.

60 seconds - 1 minute

60 minutes - 1 hour

24 hours - 1 day

7 days - 1 week

365 days - 1 year

366 days - 1 leap year

52 weeks - 1 year

12 months - 1 year

100 years - 1 century

Total

1 minute - 60 seconds

1 hour - 60 minutes

1 day - 24 hours

1 week - 7 days

1 year - 365 days

1 leap year - 366 days 4 weeks - 1 month

1 year - 52 weeks

1 year - 12 months

1 century - 100 years

(#

66

TABLE XL VII. RELATIONS AMONG UNITS OF METRIC MEASURES

FOUND IN THE HOYT-PEET SERIES.

Equalities to the left are found in tables of measures. Frequency of occurrence of these table facts is noted in upper left of cell.

10 mm. s 1 cm* 10 cm» :t 1 dm, 10 dm* s. 1 m* 10 m, =1 Dm. 10 Dm. - 1 Hm. 10 Hm. - 1 Km. 10 Km. ^ 1 Mm. Total

1

1

1

1

1

7

Fig* 1. Length

(•

TABLE XL VI I (CONTINUATION)

Equalities to the left are found in tables of measures Frequency of occurrence of these table facts is noted upper left of cell.

100 sq» mm,. = 1 sq» cm,

100 sq, cm, - 1 sq» dm,

100 sq, dm, ^1 sq, m,

100 sq, m. r 1 sq. Dm,

100 sq. Dm, = 1 sq, Hm,

100 sq, Hm. - 1 Km, Total

Fig, 2, Square Measure

TABLE XL VI I (CONTINUATION)

Equalities to the left are found in tables of measures Frequency of occurrence of these table facts is noted upper left of cell.

L

1000 cu. rnrn* -t 1 cu, cm.

1

1000 cu. cm» » 1 cu.. dm.

1

1000 cu. dm. - 1 cu. m.

3

Total

Fig. 3. Cubic Measure

TABLE XL VI I (CONTINUATION)

Equalities to the left are found in tables of measures Frequency of occurrence of these table facts is noted upper left of cell.

10 ml, 51 1 cl. 10 cl, = 1 dl. 10 dl. =11. 10 1. = 1 Dl. 10 Dl. = 1 HI. Total

1 1

Fig. 4. Capacity

TABLE XL VI I (CONTINUATION)

Equalities to the left are found in tables of measures Frequency of occurrence of these table facts is noted upper left of cell.

10 cg» s 1 dg,. 10 dg» rig. 10 g» = 1 Dg. 10 Dg. s 1 Hg. 10 Hg» t;. 1 Kg, 10 Kg. - 1 Mg.. 10 Mg. r 1 10 Q.. r 1 M.T. Total

1

1

1

1

1

1

1

8

Fig. 5. Weight

1

71

TABLE XL VII I • RELATIONS BETWEEN UNITS OF METRIC MEASURES

AND ENGLISH MEASURES FOUND IN HOYT-PEET.

Equalities to the left are found in table of measure. Frequency of occurrence of these table facts is noted in upper left of cell.

1 m» •= 39»37 in, 1 Km, - ,62 mi, 5/8 mi, 1 Ha, - 2,47 acres 1 1, :r ,91 qt, 9Dry) 1 1, 5t 1.06 qt, (Liquid) 1 HI, =. 2,84 bu, 1 Kg, = 2,2 lb, 1 M,T, - 1.1 T Total

8

72

TABLE XLIX^ RELATIONS IN MISCELLANEOUS MEASURES

FOUND IN THE HOYT-PEET SERIES •

This table shows relations in miscellaneous measures and the frequency of their occurrence in the Hoyt-Peet Ari thmetics*

Linear Measure

1 Knot = 1»1516 statute miles 1

Capacity and Content

1 cu» ft» water = 62|- lb» 1

12 cu» ft» air = 1 lb. 1

5 cu, ft, wheat s 4 bu. 1

5 cu. ft, ear corn s 2 bu, 1

1 bu. = 2150.42 cu, in. 3

1 bu. jz 1 i cu. ft. I

1 bu. corn (shelled) ^ 56 lb. 8

I bu. potatoes ^ 60 lb. L

1 gal. water s 231 cu, in. 3

1 bbl. flour r: 196 lb. 1

1 glass s k pt* 1

Weight

1 long ton :t 2240 lb. 1

1 short ton = 2000 lb. 1

360 degrees = 1 circle

2

73

INFORMATIONAL MATTER WITH REFERENCE TO UNITS OF MEASURES.

Since the actual use of units of weight and measure is always in connection with articles and commodities, an under- standing of the meaning and use of units of weight and measure can only come through commodities* As soon as a commodity is thoroughly understood, the appropriate unit of measure to be used in connection with it has been learned incidentally,.

Units of weight and measure are not learned as such, but in connection with commodities* It is this idea of the con- nection or inter-relation between commodities and units of weight and measure that the expression "informational matter", used here in connection with denominate numbers, is meant to convey. No manipulation or computation with denominate numbers is involved > but simply an understanding that a given unit of measure is used in connection with a given commodity.

The instances in which nothing more is involved than an understanding of the meaning of a unit of measure through its use with a commodity, have been gathered and grouped under two headings, namely, those appearing in statement form and those appearing in question form*

74

For instance, the statement is made that "In measuring land and other surfaces, the square inch, the square foot, and the square yard are used^"'^ The information is imparted that square measures are used to measure surfaces and is, therefore, listed under "Informational Matter in Statement Form**,. Again, the question is asked, "What are the measures that are commonly used for apples, potatoes, wheat, and other fruits, vegetables, and grains?"^ This is placed under the heading of "Informa- tional Matter in Question Form,

Informational Matter in Statement Form found in Hoyt-Pee t. Money:

The mill, the thousandths part of a dollar, is often used in the calculations of business men and public officials^

The standard foreign coins are the following: British pound sterling British shilling German mark French franc Italian lira Russian ruble Spanish peseta Mexican peso

Square Measure:

In measuring surfaces the square units used are the square inch, the square foot, the square yard, the square rod, the acre, and the square mile»

1. Hoyt-Peet, Second Book, page 148. 2^ H It II •• n 3^51^

75

Cubic Measure:

In measuring the contents of a box, a grain bin, a wagon, or a freight car, cubic units are some- times used*

Weight Measurer

Freight is shipped by the hundredweight; also by the net ton (20001b») and the gross ton (2240)lb,

Liquid Measure:

Milk is bought in pint or quart bottles. Counting:

Shoes are bought in pairs*

Time r

A month counting 30 or 31 days (28 or 29 in Feb.) is called a calendar month; one counting 4 weeks is called a lunar month.

Miscellaneous Measures:

Angles are measured by degrees; Electrical energy is measured by the watt or kilowatt. Origin of the thermometer and the reason for dividing it as it is.

Metric measures:

In measuring land, the square Dm. is called an are; the square Hm. is called hektare.

In measuring wood the cubic meter is called a stere*

How the length of the meter was arrived at.

Informational Matter in Question Form found in Hoyt-Peet.

Name the coins used in buying and selling; What unit of measure do you use in measuring height? What measure is used in selling cloth? Name measure used in finding the length of cloth; the width?

What measure is used for the depth of a building lot? What measure is used for the width of a field? What measure is used for finding distances between cities?

c

76

What unit is used for railway distances? Pace and measure a distance one rod long,. Name a place 1 mile from school.

Which units would one use in measuring the surface of

a small sheet of silver? In measuring the area of a room; of a sidewalk; of a

field, of a building lot, of a farm? Which

unit is used for the area of a continent or

an ocean?

What unit of measure is used for apples, potatoes, wheat, oats, fruits, vegetables, and grains?

What unit of measure is used in computing quantity of earth excavated?

By what weight is butter sold?

What unit of measure is used in measuring one's weight?

What units of me-^sures are used in mixing a cake?

in canning fruit? What unit of weight is used for first call mail; for parcel post; for hay?

By what unit of measure are eggs sold? name fruits that are sold by the same measure.

In selling screws and other small articles, what name is given to 12 dozen?

What units of measure are used for giving one's age?

Name units of measure used for each of the following electricity, area of a farm, area of a room, coal, wheat, potatoes, oranges, sugar, hay, strawberries, peaches, time, gas, rainfall, temperature, land, distance at sea, speed of a ship, height above sea level?

77

UNITS OF MEASURE MERELY AS NAMES

Many of the written problems contain units of

measure which, however, involve no computation or

manipulation with denominate numbers or such.

The units of measure are mentioned merely as words.

The following problems might be cited to show the type

of written problem involved: "Some of the fast vessels

in the United States Navy travel at the rate of 22.5

1

knots or 25.911 miles, per hour." "Two pounds of

candy are to be put into boxes holding half pound each.

2

How many boxes are needed?" "A railway was built

at the cost of |24,000 per mile. At that rate,

how many miles of railway can be built for $2,400,000?"

1. Hovt-^Peet. Second Book, page 203, §\&

2. Ibid. page 135, ^\

3. " "83. #18

0-

78

TABLE L. UNITS OF MEASURE A3 WORDS IN HOYT-PEET

The table shows the frequency of occurrence of units of measure involving no computation with denominate numbers as such.

Linear Measure

Inches Feet Yards Miles

Square Measure

Acres Sq, iJ'iiles

Cubic Measure

Cord

Dry ii^easure

Quarts

Pecks

Bushels

Liquid Measure

Pints

Quarts

Gallons

38 29 112 35

17 5

3

2 34

3 16

12

Weip:ht "^easure

Pounds Ounces Ton

Counting

Pair Dozen

Time

121 1 13

35 62

Seconds 1

"^inutes 40

Hours 61

Days 44

n'eek 50

""onth 21

Year 24

Miscellaneous Measures

Knots 1

Cu, ft. (gas) 3

Kwt. ' 8

Barrel 17

Bales 3

TABLE FACTS IN EXAMPLES AND WRITTEN PROBLEMS

Tables LI to LIX show the number of times a table fact or the reverse statement is called for either in examples or in written problems. For example, a table fact may be called for in an example of the following type, "How many inches in one foot?" Or a written problem may involve the knowledsze of a table fact so that it may be solved, as for instance in the following written problem: "Mr. Archer finds that the distance around the deck of a steamsnip os 264 ft. How many times must he ro around the deck to walk a mile? To solve this problem involves a knowledge of the fact that there are 5280 feet in one mile.

80

TABLE LI. TABLE FACTS IN UNITED STATES MONEY ASKED IN EXAMPLES AND WRITTEN PROBLEMS - HOYT-PEET

In each instance, the fact in parenthesis indicates the fact asked for. Read to the left of the figure thus: What are 10 mills equal to? Frequency of occurrence is shown in the column of cells to the left. Read to the right of figure thus: What is one nickel equal to? Frequency of occurrence is shown in the column of cells to the right. The frequency of occurrence in examples is shown in the upper left of the cell; the frequency of occurrence in written problems is shown in the lower right of each cell. Blank indicates no occurrence.

10 mills =(1 cent) 5 cents =. (1 nickel) 10 cents 5 (1 dime) 25 cents s (1 quarter) 50 cents r (I dollar) 100 cents *(1 dollar) 10 dimes =.(1 dollar)

13

I

18

1

21

PI

7

1 mill r (1/10 cent) 1 nickel =. (5 cents) 1 dime ^ (10 cents) 1 quarter - (25 cents) I dollar = (50 cents) 1 dollar (100 cents) 1 dollar - (10 dimes)

81

TABLE LI I. TABLE FACTS IN LINEAR MEASURE ASKED IN

EXAMPLES AND WRITTEN PROBLEMS- HOYT-PEET

In each inst-^ince the fact in parenthesis indicates the fact asked for. Read to the left of the figure thus : What are 12 inches equal to? Frequency of occurrence is shown in the column of cells to the left. Read to the tight of the figure thus: What is one foot equal to or how many inches are there in one foot? Frequency of occurrence is shown in the column of cells to the right. The frequency of occurrence in examples is shown in the upper left of the cell; the frequency of occurrence in written problems is shown in the lower right of cell, ^lank indicates no occurrence.

12 inches* (l foot)

3 feetr (l yard) 16^ Feetr (l rod)

5280 feetr (1 mile) 5| yards- (1 rod)

1760 yards-(l mile)

320 rodsT. (1 mile)

1 foot:: (12 inches)

1 yardz (3 feet)

1 rod^ (16| feet)

1 miler (5280 feet)

1 rod's (5| yards) 1 miles (1760 yards)

1 mile-s (320 rods)

t

82

TABLE LIII. TABLE FACTS IN SQUARE MEASURE ASKED IN

EXAMPLES AND WRITTEN PROBLEMS- HOYT-PEET

In each instance the fact in parenttjesis indicates the fact asked for. Read to the left of the figure thus: What are 144 square inches equal to? Frequency of occurrence is shown in the column of cells to the left. Read to the right of the figure thus: What is 1 square foot equal to? Frequency of occurrence is shown in the column of cells to the right. The frequency of occurrence in examples is shown in the upper left of the cell; the frequency of occurrence in written problems is shown in the lower right of each cell.

144 sq. in.=(l sq.ft.)

9 sq. ft. r (1 sq. yd.)

ZO^ sq. yd. s (1 sq. rd.)

160 sq. yd. -r (1 A. )

640 A. = (1 sq. mi . )

36 sq. mi. - (1 tp.)

1 sq. ft. ^(144 sq. in.)

1 sq. yd, c(9 sq. ft.)

1 sq. rd. - (30i sq. yd.)

1 A. i (160 sq. yd. )

1 sq. mi. - (640 A. )

1 tp. r (36 sq. mi . )

83

TABLE LIV, TABLE FACTS IN CUBIC MEASURE ASKED IN

EXAMPLES AND WRITTEN PROBLEMS - HOYT-PEET

In each instance, the fact in parenthesis indicates the fact asked for. Read to the left of the figure thus: What are 1728 cu. in. equal to? Frequency of occurrence is shown in the column of cells to the left. Read to the ri)!7ht of the fif?ure thus: How many cu. in. in one cu. ft.? Frequency of occurrence is shown in the column of cells to the rif?ht. The frequency of occurrence in examples is shown in the upper left of the cell; the frequency of occurrence in written problems is shown in the lower right of each cell. Blank indicates no occurrence.

1728 cu. in = ( 1 cu.f t. ) 27 cu. ft. -z. (1 cu. yd. )

1

1

1 cu. ft. T (1728 cu. in.) 1 cu. yd. r (27 cu. ft. )

84

TABLE LV. TABLE FACTS IN DRY MEASURE ASKED IN EXAI»/IPLES AND WRITTEN PROBLEMS - HOYT-PEET

In each instance the fact in parenthesis indicates the fact asked for. Read to the left of the figure thus: What are 2 pints equal to? Frequency of occurrence is shown in the column of cells to the left. Read to the ri^^ht of the figure thus: How many pints are there in one quart? Frequency of occurrence is shown in the column of cells to the right. The frequency of occurrence in examples is shown in the upper left of the cell; the frequency of occurrence in written problems is shown in the lower rio'ht of esch cell. Blank indicates no occurrence .

2 pins =. (1 quart) 8 quarts = (1 peck) 4 Decks =. (1 bushel)

2

5

5

4

n

1 quart ^ (2 pints) 1 peck r (8 quarts) 1 bushel = (4 pecks)

85

TABLE LVI. TABLE FACTS IN LIQUID MEASURE ASKED IN EXAMPLES

AND \VRITTEN PROBLEMS - HOYT-PEET

In each instance the fact in parenthesis indicates the fact asked for. Read to the left of the fiG;ure thus: What are 2 pints equal to? Frequency of occurrence is shown in the column of cells to the left. Read to the right of the figure thus: One quart is equal to how many pints? Frequency of occurrence is shown in the column of cells to the right. The frequency of occurrence in examples is shown in the upper left of the cell;) the frequency of occurrence in written problems is shown in the lower ris^ht of each cell; Blank indicates no occurrence ,

2 pints :5 (1 quart)

4 quarts - (l gallon)

1 quart = (2 pints)

1 gallon c (4 quarts)

0^

86

TABLE LVII. TABLE FACTS IN MEASURE OF WEIGHT ASKED IN EXAIvlPLES

AND WRITTEN PROBLEMS - HOYT-PEST

In each instance, the fact in oarenthesis indicates the fact asked for. Read to the left of the fip;ure thus: What are 16 oz. equal to? Frequency of occurrence is shown in the column of cells to the left. Read to the right of the figure thus: How many oz. in 1 pound? Frequency of occurrence is shown in the column of cells to the right. The frequency of occurrence in examples is shown in the upper left of the cell; the frequency of occurrence in written problems is shown in the lower right of each cell. Blank indicates no occurrence.

16 ounces s (1 pound)

100 lb. s(l cwt. )

2000 pounds = (1 ton)

1 pound - (16 ounces)

1 cwt. (100 lb.)

2000 pounds = (l ton)

87

TABLE LVIII. TABLE r-'^ACTS IN COUNTING ASKED IN EXAMPLES

AND WRITTEN PROBLEMS - HOYT-PEET

In each instance, the fact in parenthesis indicates the fact asked for. Read to the left of the figure thus: What are 12 equal to? Frequency of occurrence is shown in the column of cells to the left. Read to the right of the figure thus: How many equal one dozen? Frequency of occurrence is shown in the column of cells to the right. The frequency of occurrence in examples is shown in the upper left of the cell; the frequency of occurrence in written problems is shown in the lower right of each cell. Blank indicates no occurrence.

3

12 = (1 dozen)

16

144 :r (1 gross)

12 dozen r (1 <2ross)

1 dozen - (12)

1 gross - (144)

1 gross = (12 dozen)

88

TABLE LIX. TABLE FACTS IN TIMS ASKED IN EXAMPLES AND

WRITTEN PROBLEMS - HOYT-PEET

In each instance the fact in parenthesis indicates the fact asked for. Read to tne left of the figure thus: What are 60 seconds equal to? Frequency of occurrence is shown in the colu;nn of cells to the left. Read to the ri/ht of the fi'^ure thus: How many seconds in 1 minute? Frequency of occurrence is shown in the column of cells to the right. The frequency of occurrence in examples is shown in the upper left of the cell; the frequency of occurrence in written problems is shown in the lower right of each cell. Blank indicates no occurrence.

60 seconds - (1 minute)

60 minutes =. (l hour) 24 hours = (1 day)

7 days - (1 week)

365 days - (1 year)

366 days -(l Ip. year)

52 weeks - (1 year) 12 months - (I year)

100 years - (1 century)

1

1

3

5

1

1

2

3

i ■

11

1 minute (60 seconds)

1 hour r (60 minutes)

1 day - (24 hours)

1 week z (7 days)

1 year z (365 days)

1 Ip. year =. (366 days)

1 year - (52 weeks)

1 year - (12 months)

1 century - (100 years)

0*

89

REDUCTION OF DENOMINATE NUMBERS

Reduction of denominate numbers is the process of changing their denomination without changing their value. Denominate numbers may be changed from a higher to a lower denomination, as for example, when Z miles are reduced to feet; denominate numbers may also be changed from a lower to a higher denomination, as when 21 pints are reduced to gallons. The former process is known as descending reduction; the latter, as ascending reduction.

Meaning and Reading of Tables LXII to LXXIII.

The upper figure shows descending reductions. The unit of measure at the head of the column of vertical squares in these fi.^ures read in connection with each of the cells in these columns indicate the denomination from which the reduction is made; when these terms are read in connection with the row of horizontal cells to the left, they indicate the denomination to which the reduction is made.- The lower figures show ascending reductions. The unit of measure read in connection with the horizontal rov.s of cells to the right indicate the denomination from which the reduction is made; read in

90

connection with the vertical column of cells it refers to the unit to which the ascending reduction is made. Frequency of occurrence of reductions appearing in examples is shown in the upper left of each cell; those appearing? in written problems are shown in the lower risht of each cell.

91

TABLE LX. REDUCTIONS IN UNITED STATES MONEY-HOYT-PEET

The upper figure shows descending reductions; the lower figure ascending reductions. The frequency of reductions appearing in examples is shewn in the upper left of cell; those appearing in written problems in lower right. For further details as to meaning and reading of table see page 89.

Dollars

Half dollars

Quarters

Dimes

28

11

Nickels

21

Cents

Mills

Dollars

Half dollars

Quarters

Dimes

Nickels

Cents

5

5

2

Mills

92

TABLE LXI. REDUCTIONS IN LINEAR MEASURE - HOYT-PEET

The upper figure shows descending reductions; the lower figure ascending reductions. The frequency of reductions appearing in examples is shown in the upper left of each square; those appearing in written problems are snown in the lower right of each square. For further details as to meaning and reading of tables see page 89,

Miles

TT

Rods

Yards

46

49

Feet

43

Inches

Vli les

Rods

1

Yards

1

Feet

24 i 4

!

L 311

8

Inches

20

10

19

1

93

TABLE LXII. REDUCTIONS IN SQUARE MEASURE - HOYT-PEET

The upper fiejure shows descending redactions; the lower figure ascending reductions. The frequency of reductions appearing in examples is shown in the upper left of each square; those appearing in written oroblems are shown in the lower right of each square. For further details as to meanin.? and reading of table see page 89.

Square Miles

Acres

5

2

Square Rods

1

1

Square Yards

. I,...

2

3

1

6

2

Square Feet

I

Square

Square Miles

Square

Acres

1

Square

Rods

2

6

Square Yards

1

1

Square Feet

4

7

1

1

1

5

Inches

2

TABLE LXIII. REDUCTION?? IN CUBIC MEASURE - HOYT-PEET

The upper fip;ure shows descending reductions; the lower figure ascendinp reductions. The frequency of reductions appearing? in examples is shown in the upper left of each cell; those appearin,-? in written problems are shown in the lovi/er right of each cell.

Cords

Cubic Yards

4

Cubic Feet

2

1-

Cubic Inches

Cords

Cubic Yards

Cubic

Feet

5

8

1

Cubic Inches

2

1

95

TABLE LXIV. REDUCTIONS IN DRY MEASURE- HOYT-PEET

The upper fifj-ure shows def^cendinp; redactions; the lower figure ascending reductions. The frequency of reductions appearing in examples is shown in the upper left of each square; those appearing in written problems are shown in the lower right of each square. For further details as to reading and meaning of table see page 89,

Bushels

s

2

Pecks

1

2

Quarts

Pints

Bushels

Pecks

3 '

Quarts

1

2

1

■

i i

Pints

€

8

96

TABLE LXV. REDUCTIONS IN LIQUID MEASURE - HOYT-PEET

The upper figure shows descending redactions; the lower fi/^ure ascending reductions. The frequency of reductions appearing in examples is shown in the upper left of each cell; those annearin-^ in written oroblems are shown in the lower right of each cell. For further details as to reading and meanin.? of table see page 89.

Barrels

■' (rallons

Quarts

Pints

Cups or Glasses

Barrels

Gallons

Quarts

Pints

97

TABLE LXVI. REDUCTIONS IN WEIGHT MEASURES - HOYT-PSET

The UDDer fierure show?? descendin«y reductions; the lower figure ascendirif^ redactions. The frequency of reductions appearins- in examples is shown in the upper left of each square; those appearin-? in written problems are shown in the lower right of each square. For further details as to reading and meaning'- of table see page 89,

T:ons

8

Cwt .

Pounds

15

Ounces

Tons

r

Cwt.

Pounds

3

3

Ounces

10

6

98

TABLE LXVII. REDUCTIONS IN COUNTINO - HOYT-PEET

The upner figures show descendine reductions; the lower figures descending reductions. The frequency of reductions appearing in examples is shown in the upper left of each cell; those appearin.-'- in written problems are shown in the lower rip;ht of each cell.

Gross

Score

Ream

Dozen

1

13

17

Unit

Quire

Sheet

Gross

Ream

Score

-r

Quire

Dozen

•

Sheet

Unit

1

5

99

TABLE LXVIII. REDUCTIONS IN TIMS - HOYT-PEST

Century

Year

lonth

The upper figure shows descending reductions the lower fip;ure ascendinp; reductions. The frequency of reductions appearing in examples is shown in the upper left of each cell; those appearing in written problems are shown in the lower right of each cell.

Further details as to meaning and reading of table may be found on page 89.

Week

Day

Hour

13

Minute

Second

Century

Year

Month

Week

Day

Hour

Minute

Second

100

TABLE LXIX. REDUCTIONS IN MSASUR'"':S OF CONTENT AND CAPACITY-

HOYT-PEET

The upper figure shovvs reductions from pints and gallons etc. to cubic feet and cubic inches and the frequency of their occurrence. The lower figure snows the reverse, that is, the changes of cubic feet and cubic inches to pints, gallons, etc. Those occurring in examples are shown in the upper left of the cell; those occurring in written problems in the lower right of each cell.

Gallons

Bushels

Pounds

Tons

1

Cubic feet

Cubic inches

m

C

r^

m

o

a>

•d

Xi

C

m

iH

m

c

CS

r±

o

O

a.

Eh

Cubic Feet Cubic Inches

101

TABLE LXX REDUCTIONS I'^ METRIC MEASURES- Hoyt-Peet

Reductions from metric measures to English equivalents and visa versa. The former are shown in fig, 1,2,5 and 6; the latter in fig. 3,4,7 and 8. For further details as to the reading of the table refer to page 89,

Km. Dm. m. dm.

2

Miles

Y

ards Feet Inches

H

a. sQ.m.

Sq, mi.

Acres

Sq. Rods

Fig. 1, Length

Fig. 2. Square ^"^easure

Km. . Dm. m. dm.

m Ha,

Sq. Mi. Acres Sq. Rd.

whiles

ards

Feet

Inches

Fig. 3

fig. 4

TABLE LXX (Continued)

Liter HI.

1

1 1

Quart

M. t.

Kg.

1

Gal.

1

1

Pound

Bushels

1

Ton

Fig. 5 Fig. 6

Mt. Kg.

Pound Ton

Fig. 7

Liter HI.

Quart Gallon

Bushel

Fig. 3

c

103

TABLE LXXI REDUCTIONS IN FOREIGN MONEY- HOYT-PEET

The upper figure shows the number of examples and problems involving changes from United States money to foreign money; the lower figure the number involving changes from foreign money to its equivalent in U.S. money.

United States Money to

Pounds (English)

Francs 1 Marks

Lira

U.S. Money

Pounds to

Francs to Marks to

t

104

COMPUTATION '.VITH DENOMINATE NUMBERS

Gener ally speaking, denominate numbers are of t'Ao kinds, namely, simple denominate numbers and compound denominate numbers. Simple denominate numbers consist of units of measure of but one denomination, as, 1 inch. Compound denominate numbers consist of two or more units of measure of the same kind, such as, 8 yards 3 feet 5 inches. Lennes calls numbers involving two or more units of different kinds, as for example those expressed in •♦miles per hour", or "ton-miles" "quasi-denominate" numbers.^

Computation in the fundamental processes involving these three types of denominate numbers and the frequency of their occurrence are snown in Tables LXXII to LXXIV.

L. Lennes, N.J. The Teaching of Arithmetic, p. 281

( (

i t

105

TABLE LXXII. COMPUTATION WITH SIMPLE DENOMINATE NUMBERS-

HOYT-PEET

Unit of measure involved noted on the left; the computation involved noted at head of each column; frequency of occurrence of examoles of each noted in upner left; written problem lower right.

Inches Feet

Yards

Rods

Miles

Square Feet Square Rods Acres

Square Miles Cubic Inches

Cubic Feet

Bushels

c o

•H -P tH

•d <

c o ft +> o «J u

CO

I

o C f-t o

rH -H

CU-P

•rH +J rH

c

O

•H CO

■r-i >

Q

9

55

7

27

5

21

21

9

9

22

6

16

4

12

4

7

5

43

3

40

3

24

5

3

1

2

L -23.

1

23

i

1

1

3

1

1"!

»- . . .

2

9

5

„,.,

> 1

1

L.. ^

1

1

i

TABLE LXXII (Continued)

106

Pints

Gallons

Barrels

Ounces Pounds

Hundredweight

Ton

Dozen

Seconds

Minutes

Hours

Days

Degrees

Cu. ft. (gas)

c o

<

c

o

c

-p

Q

•H

o

-P

•H

C

o

e-H

0

<d

O4

•H

u

•^^

OT

-p

-p

JO

1-1

>

3

CO

Q

1

1

1

1

2

1

1

1

1

1

1

3

L -14..

3

8

4

16

2

1

jl

1

1

•

. A

2

... 5

2

1

1

i

1 1 ' 1

i

1

.V

■ 1

1 1

, J

1 !

1

4

1

1

1

3

>

1

2

1 1

1

5

6*-

4

107

TABLE LXXII (Continued)

Same as previous, but units of time in connection with the clock or the calendar.

c

*

o

ft

ta •

d

•p

c a.

o

o

•rH 1

•rH

cri

> •

-P

U

•H

+J

o •

n

X)

> <<

d

<

CO

n

Hour

7

2

nour "iinut.es

1

13

23 19

Month day

1

Yr.Mo. Day

8

Years

4

Mo. Day Hr. "^in.

1

>

1

i

1 1

/

▲

108

TABLE LXXIIt.

COMPUTATION WITH "QUASI-DENOMINATE" NUMBERS- HOYT-PEET

The first column marked "Addition" shows the number of occurrences of computation in adding?; the denominate numbers involving the units of measure listed to the left of the figure; the second column shows the number of subtraction computations , etc. Examples are shown in upper left of the cell; written problems in lower right.

c o

•fH -P •iH

n <

c o

+> o

(0

u

CO

c

o

■p

CO

o

D. ft +^ <-l

3

c

o

to

>

Q

Miles per minute Miles per hour

Miles per day

Miles per day,hr,min

Nautical mi. per day

Miles per gallon Bushels per acre

Tons oer acre

2

2

6

9

11

19

45

2

16

1

t

1

1

1

t 1

1

f i

i . . .i

3

10

2

5

109

TABLE LXXIY. COMPUTATION WITH COMPOUND DENOMINATE NUMBERS-

HOYT-PEET

The first column marked "Addition" shows the number of occurrences of computation in addin^j; the denominate numbers involving the units of measure listed to the left of the figure; the second column shows the humber of subtraction computations, etc. Examples are shown in upper left of each cell; written problems in lower right.

c

o

c

■p

o

ClJ

o

-P

•H

C

o

o

O

•H

ex.

•H

+>

u

(0

l-l

>

3

•i-l

<

CO

Q

Feet inches

3

6

3

8

6

Yard inches

4

6

5

3

Yard feet

1

Rods feet

1

2

Miles Rods

1

2

Hours ^inutes

1

Sq, yd* Sq. ft.

1

Sq. ft. Sq.in.

1

Bushels pecks

1

110

SUMMARY OF THE HOYT-PEST ARITHMETIC SERIES In Table LXXV we have a summary of the data gathered with reference to the denominate numbers found in the Hoyt-f' eet arithmetics. The data has been grouped under five headings. One group of facts has been omitted from this summary, namely, "Informational Matter with Reference to Denominate Numbers". The reader may refer to pages 73-76 for this data.

Reading and Meaning of Table LXX.V: • The tables of measures involved in the suminary are listed to the left of the figure. Column 1 shows the number of equalities found in each of the tables; column 2 shows the number of units of measure used as mere wo^ds, while column 3 shows the number of written problems involved; in column 4 we find the number of table facts asked in examples and written problems and column 5 shows the number of examples and WEitten problems involved; column 6 shows the total number of reductions, while column 7 shows the number of examples and written problems involved in these reductions; column 8 shows the number of units of measure used in computation and column 9 the number of examples and written problems involved in computation with simple denominate numbers; column 10 shows the number of compound denominate numbers used and the final column shows the number of examplr=s and written problems involved in computation with compound denominate numbers.

Ill

TABLE LXXV. SUMMARY OF HOYT-PEET ARITHMETICS

FOR reading and meaning of table see page 110,

U.S. Money Linear Measure Square Pleasure Cubic Measure

Dry Measure Liquid ^^easure Weight Measure Counting

Time

Clock Calendar

Metric Measures

English Equi valents

Foreign Exchange Quasi-Den.No. Misc. Measures

0)

CO

■p

01

o

f\ %j

■iH

CO

03

O

+>

(D

Ul,

<U

•H

1-1

rH

•H

-P

rH

C.

0)

Qu

O

a

T>

s

iH

E

3

U

03

X3

ClJ

O

X

CC

X

0)

W

(r:

rH

to

IT)

(0 0) rH

Du E

cc X

w

c o

iH

■p

■P 3

CL

E O P

(0 Q) iH &.

E CO X

c

3

o

E o o

CO cr>

10 (D iH

a. E <d X W

4

59 IC

10

91

4

4

214

5

16

2

16

231 7"^

5

35

32S

5

48

3

2

22

14

18 29

4

3

16

2

2

2

1

1

2

2

5

20 11

2

2

3

3

39

r

3

11 16

9

16

4

1

1

1

1

1

2

3

31

2

6

1

5

18 13

3

4

4

3

3

135

3

6

8

40 23

4

17

52

3

2

97

2

3

17

5

18 29

1

1

6

7

246

7

9

20

7

26 11

4

7

13

6

37

85

29

8

17

21 6

7

13

5

8

2

134

15

5

32

2

11 6

s

112

TABLE LXXVa. DETAILED SUMMARY OF REDUCTION- HOYT -Peet,

Column one shows the total number of descending reductions with reference to each table of measure listed to the left of the figure. The second column shows the total number of ascending reductions. Column 3 and 4 show similar data for a'^.cending reductions.

United States ^oney Linear Measure Square Measure Cubic Measure Dry Measure Liquid iVieasure Weight Measure

Counting Time

Engl. Eq. of Metr We. Foreign Exchange

•

•

o

o

3

TJ

to

t3

CO

0)

0)

0)

<D

l-l

r-t

•

a.

a.

o

E

•

E

(0

o

c6

0)

x:

X

o

W

79

12

X 'J

3

?3

— H

7

159

7

72

23

50

16

12

7

7

8

21

■

6

14

2

3

1

10

10

6

3

•2,

t

3

1

1

9

9

i 3

2

1

i

9

4

1 ]

PA

16

4

♦

. . .

12

13

5

1 2

3

18

11

1 ,

20

! 4

3

10

1

^3

8

» 9

8

i

5

I

4

9

3

i ^

1

4

1

—

\

(5

nal Summary and Concluding Observations

113

TABLE LXXVI. FINAL SUMMARY

Tables ^

Number of tables in texts 16 13

Number of relations in tables 68 60

Number of Miscellaneous relations 20 15

Total number of relations taught 88 75

Units of measure merely mentioned 26 30

Table facts in e>:am|;^les and written problems

Number of table facts 20 28

Number of examples and problems involved 61 168

Reductions

Number of reductions 109 109

Examples and problems involved 482 639

Comoutation

Simple units of mea=;ure 23 27

Examples and problems involved 399 673

"Quasi-Denominate" numbers 10 8

Examples and problems involved 136 136

Compound denominate numbers 10 8

Examples and problems involved 73 54

114

Concluding Observations.

1. One notes the absence of many of the tables of measurement found in older arithmetic text-books but now generally considered obsolete. However, the metric system is still retained in both series,

a doubtful procedure. The Upton-Strayer series also retains the tables of so-called "useful equivalents in liquid measure," called in this study "measure of capacity," as well as the table of "useful equivalents in dry measure," called in this study, "measure of content."

2. Each series insists on the memorization of the entire

table of measure as such. The Fourth Yearbook of the

Department of Superintsndence stated that "tables

1

as such are unnecessary." Louth also found in her study that tables of measurement, as such, are seldom used in their entirety. Why then, should a table, as such, be memorized?

3. The Hoyt-Peet Arithmetics retain foreign exchange. It will be recalled that the Fourth Yearbook of the Department of Superintendence recommended that no provision for formal drill be made for work

in foreip-n exchange.

c

115

4. The greatest emphasis in each series of texts is placad on reductions. The Fourth Yearbook of the Department of Superintendence asserted that "no reductions, either ascsndinj? or descending, occur in business situations,"^ This has been confirmed by the studies of Louth, Sala, and Traniello.

5. There is a modern tendency to eliminate computation with compound denominate numbers entirely. But a large number of examoles and problems with compound denominate numbers have still been retained in these texts.

6. Tho much of the work with denominate numbers is done incident?.lly in connection with other phases of arithmetic, such work must be justified on the basis of its social usefulness.

7. More emphasis needs to be placed on teaching the

use and meaning of units of wei,g;ht through commodities. But to say that "fruits, grains, vegetables are measured by the nint, the quart, the peck, and the bushel" has little value. Each commodity must be joined with the specific unit of measure with which it is used.

1, Fourth Yearboo'.-., Department of Superintendence, p. 178,

cc

... t '

116

Selected bibliography

1. Fourth Yearbook, Department of Superintendence, ^'^ational Education Association, 1926. pp.l73-220»

2. Lennes, N.J. The Teaching of 'Arithmetic. New York: Macmillan Company, 1929, pp.278-288»

3. Louth, ^iary. iJnits of Measurement in Industry. B.U. Graduate school thesis, 1931. ms.94p.

4. Moore, Vernette B, An Analysis of Denominate Numbers with Data on Their Teaching. State ^niversity of Iowa, Master's thesis of the Graduate College, 1929.

5. Sala, V. Denominate ^^umbers used in the Factories of New Britain. B.U. School of Education thesis, 1931. ms» 80p»

6. Traniello, A. Use of Tables of ^'Measurement in Business.

B.U. School of Education thesis, 1930. ms.70p.

7. Wilson, G.M. "Arithmetic", Department of Superintendence, Third Yearbook, Washington, D.C. National Education Association, 1925, p. 35-109,

8. Wilson, G.M. A Survey of the Social and business Usage of Arithmetic. Contributions to Education, No. 100, Teachers College, Columbia University, 1919,

9. Wilson, G.M. What Arithmetic Shall We Teach Houghton Mifflin Compansi, 1926.

c

I