Mathematics Manual IV
Elementary II
Dr. Pamela Zell Rigg
© Copyright 2006
Montessori Research and Development
16492 Foothill Boulevard, San Leandro, CA 94578-2105
(510) 278-1115 FAX (510) 278-1577
Mathematics Volume IV
TABLE OF CONTENTS
CHAPTER 1:
RATIO AND PROPORTION
1. A. INTRODUCTION .............................................................................................. 2
1. B. ANALYSIS OF LESSONS BY LEVEL ....................................................................... 3
2. THE CONCEPTS OF RATIO AND PROPORTION ....................................................... 4
3. MEANS AND EXTREMES ......................................................................................... 7
4. SOLVING PROPORTIONS ....................................................................................... 9
5. SOLVING WORD PROBLEMS WITH THE PROPORTION CHART............................... 11
CHAPTER 2:
PERCENTAGE
1. A. INTRODUCTION .............................................................................................17
1. B. ANALYSIS OF LESSONS BY LEVEL ..................................................................... 18
2. THE CONCEPT OF PERCENTAGE............................................................................19
3. CHANGING FRACTIONS TO DECIMALS WITH THE MONTESSORI PROTRACTOR ... 22
4. CHANGING DECIMALS TO PERCENTAGES ............................................................ 23
5. CHANGING PERCENTAGES TO DECIMALS ............................................................ 25
6. CHANGING FRACTIONS TO PERCENTAGES ...........................................................28
7. AN ANALYSIS OF THE PERCENTAGE PROBLEM: FIRST CASE ....................................31
8. AN ANALYSIS OF THE PERCENTAGE PROBLEM: SECOND CASE ...............................34
9. AN ANALYSIS OF THE PERCENTAGE PROBLEM: THIRD CASE ..................................36
10. WORKING PERCENTAGE PROBLEMS WITH RATIO............................................... 39
11. WORD PROBLEMS FOR PERCENTAGE STUDY ......................................................40
12. THE STUDY OF SIMPLE INTEREST ........................................................................42
13. CALCULATION OF INTEREST .............................................................................. 44
14. CALCULATION OF RATE OF INTEREST ................................................................ 47
15. CALCULATION OF PRINCIPAL .............................................................................49
16. CALCULATION OF TIME ......................................................................................51
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CHAPTER 3:
A STUDY OF INTEGERS
1. A. INTRODUCTION ....................................................................................54
1. B. ANALYSIS OF LESSONS BY LEVEL ............................................................ 55
2. THE NUMBER LINE ......................................................................................56
3. VERTICAL NUMBER LINE ..............................................................................57
4. GREATER THAN AND LESS THAN ................................................................ 58
5. ADDITION/SUBTRACTION OF INTEGERS ON THE NUMBER LINE ..................60
6. MULTIPLICATION/DIVISION OF INTEGERS ON THE NUMBER LINE ................61
7. THE ADDITIVE INVERSE .............................................................................. 62
8. DERIVING RULES .........................................................................................63
CHAPTER 4:
OTHER BASE SYSTEMS
1. INTRODUCTION .........................................................................................66
2. USE OF SPINDLES TO SHOW OTHER BASES..................................................67
3. USE OF NUMBER RODS TO SHOW OTHER BASES ........................................68
4. PLACE VALUE IN OTHER BASE SYSTEMS .......................................................70
5. NUMERATION IN OTHER BASE SYSTEMS ......................................................73
6. ADDITION AND SUBTRACTION IN OTHER BASE SYSTEMS ............................75
CHAPTER 5:
A STUDY OF CUBING
1. A. INTRODUCTION ....................................................................................78
1. B. ANALYSIS OF LESSONS BY LEVEL ............................................................ 79
2. REVIEW OF SQUARING .............................................................................. 80
3. FROM THE SQUARE TO THE CUBE OF A NUMBER ........................................82
4. FROM A CUBE TO A SUCCEEDING CUBE ......................................................84
5. FROM A CUBE TO A NON-SUCCEEDING CUBE .............................................86
6. CUBING THE SUM OF A BINOMIAL..............................................................88
7. CUBING THE SUM OF A TRINOMIAL ............................................................91
8. INTRODUCTION TO THE HIERARCHICAL BINOMIAL CUBE ...........................95
9. GIVING AN ALGEBRAIC VALUE TO THE CUBE OF THE BINOMIAL ................ 97
10. INTRODUCTION TO THE HIERARCHICAL TRINOMIAL CUBE .......................98
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11. USING THE HIERARCHICAL TRINOMIAL CUBE TO CALCULATE THE CUBE OF
A TRINOMIAL ....................................................................................... 100
12. GIVING AN ALGEBRAIC VALUE TO THE CUBE OF THE BINOMIAL ............. 102
CHAPTER 6:
A STUDY OF CUBE ROOT
1. INTRODUCTION TO CUBE ROOT .............................................................. 104
2. CUBE ROOT OF ONE DIGIT ....................................................................... 106
3. CUBE ROOT OF A BINOMIAL ..................................................................... 107
4. CUBE ROOT OF A TRINOMIAL ................................................................... 115
5. CUBE ROOT OF A TRINOMIAL FOUND ABSTRACTLY .................................. 120
6. PARTICULAR CASES OF THE CUBE ROOT ................................................... 121
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CHAPTER 1
RATIO AND PROPORTION
1.
A. Introduction
B. Analysis of Lessons by Level
2.
The Concepts of Ratio and Proportion
3.
Means and Extremes
4.
Solving Proportions
5.
Solving Word Problems with the Proportion Chart
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1. A. INTRODUCTION
These lessons follow Study of Fractions and Study of Decimals. They may precede Percentage,
or they may be presented as an introduction to percentage. A study of Ratio and Proportion
prepares the child for the practical use of percentage; every case of percentage can be fit into a
workable ratio and solved through the use of a proportion equation.
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1. B. ANALYSIS OF LESSONS
BY LEVEL
Level 1:
Introduction to Ratio and Proportion when studying Fractions
Level 2:
Complete study of Ratio and Proportion
Level 3:
Review of Ratio and Proportion, if needed
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2. THE CONCEPTS OF RATIO
AND PROPORTION
Material:
Grammar Symbols
Ratio Charts 1, 2, 3, 4, 5
Paper and pencil for labels
Presentation:
1. “Today we are going to study a new way to look at numbers as well as a ‘sure-fire’ way to solve
problems. We call this new work ‘ratio’.” The adult takes from the symbols box 2 noun triangles
and 3 verb circles.
2. “I have decided to set up a ratio of triangles to circles; I can express this as:”
2:3
or
or
two is to three
or
2/3
Δ Δ OOO
(Chart #1)
3. “If I take another set of (2) triangles, I shall need to take another set of (3) circles. Now my new
ratio is expressed as:”
4:6
or
four is to six
or
4/6
Lay out triangles and circles.
or
Δ Δ OOO
Δ Δ OOO
(Chart #2)
4. “I can reverse the ratio and begin with the circles:”
3:2
or
or
three is to two
or
3/2
OOO Δ Δ
(Chart #3)
5. “Let’s set up another ratio: 2 circles to 1 triangle.
Suppose we extend this to 6 circles. Then I shall have 3 triangles.”
or
6:3
or
OO Δ
OO Δ
six is to three
or
6/3
OO Δ
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6. “We can show that two ratios are equal. If I set up a relationship such as: 1 triangle to 3
circles, then twice this would be 2 triangles to 6 circles:”
1:3
or
1/3
=
Δ OOO
2:6
or
2/6
Δ OOO
Δ OOO
(Chart #5)
“These equal ratios are called a proportion. 1/3 = 2/6 is a proportion.”
7. Show the Ratio Charts 1, 2, 3, 4, 5.
8. The child makes charts of his/her own in a notebook.
9. The child practices writing ratios from familiar objects. (See “Problems”, next page.)
Δ Δ OOO
Δ Δ OOO Δ Δ OOO
triangles : circles
2:3
triangles : circles
4:6
triangles
circles
2
3
triangles
circles
4
6
Chart 1
Chart 2
OOO Δ Δ
OO Δ
OO Δ
OO Δ
circles : triangles
3:2
circles : triangles
6:3
circles
triangles
3
2
circles
triangles
6
3
Chart 3
Chart 4
Δ OOO
Δ OOO Δ OOO
1:3
=
2:6
1
3
=
2
6
Chart 5
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PROBLEMS
Express the following as ratios:
1.
The ratio of one nickel to one dime.
2.
The ratio of one nickel to one quarter.
3.
The ratio of one ounce to one pound.
4.
The ratio of 3 pints to one gallon.
5.
The ratio of 3 inches to one foot.
6.
The ratio of one nickel to one dollar.
7.
The ratio of 60 pounds to 50 pounds (lbs).
8.
The ratio of 2 days to one week.
9.
The ratio of 3 months to one year.
10.
The ratio of girls to boys in your class.
11.
The ratio of boys to girls in your class.
12.
The ratio of 25 centimeters to one meter.
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3.
MEANS AND EXTREMES
Material:
Chart 6
Paper and pencil
2 colored pencils (green and purple)
Prepared problems
Aim:
To learn application of ratio in word problems.
Presentation:
1. Now that we understand what a ratio is, we shall learn how to use this to solve problems. There
are two ways to set up a proportion. The adult writes this on paper or on a chalkboard first
before showing the chart.
2. Write the first and last number in green, the second and third in purple.
Say while writing: “One is to three as two is to six.”
1:3 = 2:6
3. We can also write it this way, again using two colors:
1/3 = 2/6
4. We call the first and last numbers (the green digits, 1 and 6) the extremes. The second and
third numbers (purple digits, 3 and 2) are called the means.
5. The rule for proving the equality of a proportion is: The product of the means (3 x 2) equals the
product of the extremes (1 x 6).
6. Show Chart 6. The child places this chart in a notebook.
7. The child practices identifying a proportion.
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1:3
=
1
3
=
2:6
2
6
The first and last numbers are called the Extremes.
The second and third numbers are called the Means.
Chart 6
PROBLEMS
Directions:
1. Decide whether the 2 ratios form a proportion.
2. Write the ratios with either = or ≠ between them.
1.
1/4
2/5
2.
3/10
21/70
3.
14/16
5/6
4.
24/16
27/18
5.
12/15
16/20
6.
4/120
5/200
7.
8/18
10/27
8.
6/16
9/24
9.
5/10
4/8
10.
11/27
4/9
11.
10/.85
16/1.35
12.
3/.45
8/1.00
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4.
SOLVING PROPORTIONS
Material:
Ratio Chart 7
Paper and pencil or chalkboard
Balance scale, preferably with two plates
Presentation:
1. We are now ready to learn how to compute with the proportion. One of the numbers, either a
mean or an extreme, will be unknown. We shall learn how to discover this unknown through
solving an Equation.
2. An equation always solves an equality. The equal sign acts as the central beam of the scale.
3. Demonstrate with a balance scale. The material in the one tray must equal the weights in the
other tray.
4. The numbers on the one side of the equation must equal the numbers on the opposite side of
the = sign.
5. Let us use a sample equation: 3/4 = n/12
6. The “n” stands for the unknown. (We could use any letter.)
7. Remember the rule: The product of the Means equals the product of the Extremes.
8. Do the multiplication: 36 = 4n.
9. Multiply each side of the equation by the reciprocal of 4 which is 1/4.
10. Factor out the 4’s and divide 36 by 4.
9=n
11. Show Chart 7.
12. The child does many problems after placing Chart 7 in a notebook.
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Solving a Proportion
2
3
= 6
n
Write the Proportion.
2xn =3x6
Multiply Means and Extremes.
2 x n = 18
Multiply each side by the
reciprocal of 2 which is 1/2.
2 x n = 18
2
2
1
9
2 x n = 18
2
2
1
1
Factor out.
Chart 7
PROBLEMS
Write out each step of the problem according to Chart 7. Remember to keep the equal signs
under each other, as in Chart 7.
1.
2/3 = 6/n
2.
4/5 = 24/n
3.
16/24 = 18/n
4.
n/7 = 21/49
5.
1/3 = 25/n
6.
4/5 = 24/n
7.
3/10 = n/70
8.
21/56 = 3/n
9.
5/6 = n/432
10.
5/8 = 105/n
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5. SOLVING WORD PROBLEMS
WITH THE PROPORTION CHART
Introduction:
The child must be competent in solving the previous number problems before any attempt
is made to approach word problems. The difficulty in this process is the setting up of the
equation. After that is achieved, the actual computation is simple.
Material:
Blank Proportion Chart. A rectangular card on which are drawn the following figures. The
long rectangles are colored.
:
_____ = _____
:
Blank slips of paper
Pencil
Prepared problems
Presentation:
1. Present the chart.
2. Read a sample problem. “For two hours of work I get 90 ¢. For one hour I would get ____?
(Since it is the process that is important here, ignore the child who gives the answer
immediately.)
3. “In order to solve this problem (and problems like this) we shall use the chart.”
4. “We have two kinds of quantity in the problem. One that refers to time - that is the word
‘hours’.” Write this word and place it on the colored rectangle.
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“The other quantity refers to money: ‘cents’.” Write this word and place it on the other
colored rectangle. (It does not matter which word is at the top and which is at the bottom. The
important thing is that the numbers placed in the proportion correspond to the words.)
5. “Now we’ll write the numbers that correspond to our words in the blank squares. For two
hours, I get 90 ¢...” Write the ‘2’ on a slip of paper and the ‘90 ¢’ on another; then place them
on the chart.
cents
90
_____ = _____
2
hours
6. “Now for the second part of the problem: For one hour I would get _____? The number 1 refers
to hour.” Write “1” on a slip of paper and place it in the ‘hours’ row.
7. “I don’t know how many cents I should get, so I shall use the letter ‘n’ for the unknown quantity
and place it in the cents column.”
cents
90
n
____ = ____
2
hours
1
8. The child solves the equation: 45¢
90 x 1 = 2 x n
90/2 = 2n/2
45
=1n
Note:
It is very important that the child practice setting out the concepts or words on this chart. It is
the setting up of a proportion in this way that helps to clarify what is given in the problem and
what is asked.
The same problem can be done by placing the problem thus:
hours
cents
2
90
=
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n
2 x n = 90 x 1
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The child can see that the actual computation is the same. Stress that the unknown quantity is
best placed on the left side of the equation.
Group Activity:
Before allowing the children to work on these problems individually, have the children each
take a problem, read it aloud, write out the terms, and set them on the chart. Computation is
done later.
PROBLEMS I
1.
For two hours, I get 90 ¢. For one hour, I get _____.
2.
Eight out of 10 is the same as _____ out of 30. (Consider means and extremes.)
3.
One hundred miles in 2 hours is the same as _____ miles in 4 hours.
4.
Two deliveries in five minutes is the same as _____ in 20 minutes.
5.
If I can answer 5 problems in 6 minutes, then I can answer 10 problems in _____ minutes.
6.
If I can buy 10 gallons of gas for $12.00, then I can buy 12 gallons of gas for _____.
7.
Joan was paid $2.80 for babysitting for 4 hours. At the same rate, what would she be paid
for 9 hours?
8.
Jon got the same grade for his first two math tests. The first had 20 problems and he did 17
correctly. The second test had 60 problems. How many of these did he do correctly?
9.
On a trip, Mr. Santoro drove 156 miles in 3 hours. At that rate, how far could he drive in
12 hours?
10.
Angela read 17 pages of her book in 45 minutes. At that rate, how long would it take her
to read 68 pages?
11.
When doing subtraction problems, Larry can answer 15 in 10 minutes. At that rate, how
long would it take him to answer 75 problems.
12.
Sol delivered 9 newspapers in 20 minutes. At this rate, how many newspapers can he
deliver in 60 minutes?
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PROBLEMS II
Continued practice is needed in learning how to set up word problems. It is helpful to continue
to use the chart and to set up the equation in many different ways.
1.
Jim has a picture 3 inches long and 2 inches wide. He wants to enlarge it on a sheet of paper.
The paper is 18 inches long. How wide would the enlargement be?
2.
Tennis balls are selling 3 for $1.65. How much did you pay for 9 balls?
3.
The Boy Scouts went on an all-day hike. They hiked 8 miles in 3 hours. How far could they
hike in 6 hours if they kept the same pace?
4.
Candy sold at the rate of 6 pieces for 25¢. At this rate, how many pieces can be bought for
one dollar?
5.
At three pounds for 45¢, how much would 15 pounds of potatoes cost?
6.
Marilyn saved $4.50 from her allowance in 9 weeks. At that rate, how long would it take her
to save $22.50.
7.
Terry was paid $2.25 for a job it took him 3 hours to do. At that same rate, how much should
he be paid for a job taking 5 hours?
8.
A candy recipe calls for 2 cups of brown sugar for every cup of milk. Jane had only 1 1/2
cups of sugar. How much milk should she use?
9.
David, who is 48 inches tall, stood next to a tree in his front yard. If he cast a shadow 32
inches long and the tree cast a shadow 40 feet long (480 inches), how tall is the tree?
10.
A motorist traveled 312 miles in 6 hours. At the same rate, how long would it take him to
travel 416 miles?
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Extension:
Similar figures in geometry are based on the same proportion between sides. Provide command
cards for children to construct similar figures to some of the geometry polygons.
Example: If these two triangles are similar, how tall should the larger triangle be?
base
=
6
21
_____ _____
height
10
n
10
6
n
21
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CHAPTER 2
PERCENTAGE
1.
A. Introduction
B. Analysis of Lessons by Level
2.
The Concept of Percentage
3.
Changing Fractions to Decimals with the Montessori Protractor
4.
Changing Decimals to Percentages
5.
Changing Percentages to Decimals
6.
Changing Fractions to Percentages
7.
An Analysis of the Percentage Problem: First Case
8.
An Analysis of the Percentage Problem: Second Case
9.
An Analysis of the Percentage Problem: Third Case
10.
Working Percentage Problems with Ratio
11.
Word Problems for Percentage Study
12.
The Study of Simple Interest
13.
Calculation of Interest
14.
Calculation of Rate of Interest
15.
Calculation of Principal
16.
Calculation of Time
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1.
A. INTRODUCTION
The study of percentages is the culmination of the study of decimals, ratio, and proportion.
The understanding of percentages has practical application in daily life. Percentage study
culminates in its application to banking calculations involving interest.
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1.
B. ANALYSIS OF LESSONS
BY LEVEL
Level 1:
Level 2:
All lessons.
Level 3:
Review, if necessay.
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2.
THE CONCEPT OF PERCENTAGE
Material:
Large red circle divided into 100 parts.
Fraction insets divided into tenths.
Felt decimal board
Slips of paper or cards
Colored felt-tipped pens (3 colors)
Piece of red construction paper
Blank chart for: Fraction - Decimal - Percentage
Presentation:
1. “Please show me the fractional piece for one-tenth.” Invite the child to write one tenth on a
card, selecting a color.
1
10
2. “Please show me a decimal quantity for one-tenth.” Invite the child to write one tenth in
decimal form on a card, using a different colored pen. Place on the decimal board.
0.1
3. Invite the child to trace the fraction tenth on the red paper - and cut it into ten piecesapproximately. Each of these would be one hundredth. Invite the child to write one hundredth
as a fraction.
1
100
4. Show a decimal quantity for one hundredth. Invite the child to write one hundredth as a
decimal. Place on the decimal board.
0.01
5. Show the large circle divided into one hundred parts. “This circle contains 100 sectors - the
whole circle contains one hundred hundredths - or one whole.” Write this (in the fraction color)
as a fraction.
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100
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6. We have a new way of writing parts of numbers if we consider the whole to be made of 100
hundredths. We call it “PERCENT”.
7. Write CENT on a paper in a third color.
Cent
This comes from the Latin root meaning ‘hundred’. We have many words in English that come
from this Latin root: century, centennial, centurion, cent = penny. There are one hundred cents
in a dollar.
8. Write the word PER. This is also Latin, meaning ‘parts of’.
Per
9. If we take just one sector of the large circle, we have one percent. There is a special way to write
the percent. Write this on a card; it contains the / (write it) and the 2 circles (add these to form
the symbol for percent.)
%
When we write quickly, we connect the first zero and the /.
Thus, one percent is written:
1%
10. Take the blank chart:
Fraction
Decimal
Percent
Take the cards that were written for the presentation and ask the child to place them on the
chart.
11. Ask a child to choose a certain number of sectors on the percent circle: e.g. 15.
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12. The child writes this on a card, places it on the chart, then writes the same as a fraction over
hundred:
15
100
and as a decimal:
0.15
and places these on the chart.
13. Each child in turn makes a set of Fractions, Decimals, and Percents. Don’t worry about lowest
terms in fractions yet.
14. Continue forming sets of terms.
Percent Card Game:
A set of cards can be made on durable cardstock containing sets of Fractions, Decimals, and
Percents, as above. The children may devise their own rules for the game. A set of cards would
include the Fraction, Decimal, and Percent for the same quantity. Cards may be dealt to the
players with a stack of cards in the center (depending on the number of players). The object of
the game is to form as many sets as possible.
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3.
CHANGING FRACTIONS TO DECIMALS
WITH THE MONTESSORI PROTRACTOR
Material:
Montessori Protractor (divided into 100 sections)
Blank Percent Chart (as shown in step #10 of the previous presentation)
Small pieces of paper or cards
Fraction insets (circles)
Presentation:
1. Take the half from the insets and place it in the protractor. Read the numbers on the protractor.
2. Write cards for 1 and 0.50.
2
3. Place these cards on the chart.
4. Continue with the thirds. Take one third and place it in the protractor. Read the protractor to the
nearest number.
5. Write the cards 1 and 0.33.
3
6. Continue for each inset.
7. The child may place this chart in his/her notebook.
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4.
CHANGING DECIMALS TO
PERCENTAGES
Material:
Set of numeral cards such as those from the Bank Game
Card with a percent sign: %
Red bead or small disc for the decimal point
Presentation:
1. Construct a decimal number with the cards, placing the bead as the decimal point.
2. Read the number: Seventy-five hundredths.
3. Another way to say this is seventy-five percent.
4. Add the percent sign and move the bead two places to the right and remove the zero card.
5. When changing a decimal to a percent, move the decimal point two places to the right and drop
the initial zero. These two mean exactly the same quantity.
6. Build a second decimal number and ask the child to make it a percent. (Do not use whole
numbers with the decimal as yet.)
7. Build a third number such as : 0.6 with the cards and bead. We cannot move two places to the
right since there is only one number: 6. But 6 tenths is the same as 60 hundredths (recall work
with decimals.)
8. Add a zero to the 6 and move the bead. Now we can add the percent sign. Our quantity is 60%.
9. Repeat steps 7 and 8 with another number such as 0.3. Make sure the child understands the
process.
10. Now try a more complicated decimal: 2.3456.
Read it: “Two and three thousand, four hundred fifty-six ten-thousandths.”
Recall the process and make the necessary changes for a percent.
The answer: 234.56%.
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Note:
We can have a percent larger than 100%.
We can have a percent with numbers to the right of the decimal point.
Since decimals and fractions are interchangeable, this last number could also be written:
56
234 100 %
It is not necessary to point out this last to the child, however. He/she will meet percents with
fractions later on.
PROBLEMS
The child should work through many conversions from decimals to percents, using the cards
and the bead if necessary.
1. Set out the cards and the ‘decimal bead.’
2. Move the bead and place the percent card.
3. Write out what you have done. E.g. 0.32 = 32%
1.
0.42
0.07
0.3
2.
0.98
1.234
0.55
3.
0.06
0.11
1.23
4.
2.675
0.67
0.01
5.
0.2
2.55
3.313
6.
0.94
0.8
0.6
7.
1.11
4.498
23.67
8.
1.225
0.004
0.0005
9.
1.001
1.0001 1.011
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5.
CHANGING PERCENTAGES
TO DECIMALS
Materials:
Set of numeral cards such as those from the Bank Game
Card with a percent sign: %
Red bead or small disc for the decimal point
Presentation:
1. Remember how we changed a decimal to a percent:
Move the decimal point two places to the right and add the % sign.
2. Demonstrate this with a number. E.g.: 0.536. Set out with cards and bead. Move the decimal
point; add the %.
3. We have shown that 53.6% is the same as 0.536. Reform the original number: Remove the %
sign, move the bead back to its original position, and add the zero.
We moved the decimal point 2 places to the LEFT and dropped the percent sign (%). This is
opposite to the way in which we changed a decimal to a percent.
4. Build 35% with the cards. We have no decimal point. We can presume that it is after the 5.
Place a bead.
5. Now we can change it to a decimal: 0.35.
6. Build 9% with the cards. We have no decimal point, but we can put it after the 9.
7. We do not have 2 places to the LEFT of the decimal point. But we know that 9% equals
9 hundredths (since percents are always parts of 100).
As a decimal, 9 hundredths can be written: 0 09. We must add a zero before the 9 in order to
move the decimal point 2 places.
8. Place the zero, move the decimal bead, and remove the % sign.
9. Practice with other decimals if necessary.
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Mathematics Volume IV
PROBLEMS
Use the cards and bead or write directly on paper. E.g. 53% = 0.53
1.
29%
46%
62%
81%
2.
6%
9%
4%
39%
3.
15%
3%
90%
99%
4.
72.1%
5.6%
12.2% 145%
5.
167%
212%
349%
0.4%
6.
0.033%
0.03% 0.2%
0.4%
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Mathematics Volume IV
Extensions of Percent Study:
Writing the Complete Table of Equivalences.
This is an extensive exercise, begun with the adult and carried out as a student project. This
work is placed in the child’s notebook for reference as well as for completeness of the work.
Material:
Children’s notebooks
Ruler
Pencil
Presentation:
1. Show the child how to measure the page in his/her notebook, dividing the page into 4 equal
parts. For most filler paper sheets, this will be 4.5 cm apart.
TABLE OF EQUIVALENCES
Percent %
Fraction
Fraction in
Lowest Terms
Decimal
100
1%
1
100
1
100
0.01
2%
2
100
1
50
0.02
2. We’’ll begin with one percent and write how this would be written under each heading.
3. Go on to 2% - this will involve changing
2
100
to
1
50
4. Continue all the way to 100%.
5. A large wall chart may be made by the adult as a Control for the exercise and as a research
chart.
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Mathematics Volume IV
6.
CHANGING FRACTIONS
TO PERCENTAGES
Material:
Completed Chart from the previous lesson ( 1% through 100% as fractions and decimals)
Proportion Chart (from Chapter 1)
Fraction insets and circles
Paper and pencil
Presentation:
1. Place the circle insets on the table. “We want to see if we have included all the fractions in
our Percentage Chart.”
2. Begin with the whole unit: Is it on the chart?
Yes, at the end: 100/100 which equals 1/1 - 1.
3. What about the halves?
Yes, 50% = 50/100 = 1/2 = 0.50.
4. Do we have the thirds?
No, this fraction does not occur on the chart.
5. Set the thirds aside and continue with the fourths.
6. Continue through all the fractions.
7. We discover that the following fractions are not on the chsrt:
thirds, sixths (except for 3/6 which equals 1/2), all sevenths, some eighths, all ninths.
8. Write fraction cards for all that do not appear on the chart:
1
3
2
3
1
6
2
6
3
6
4
6
5
6
1
7
2
7
3
7
4
7
5
7
6
7
1
8
2
8
3
8
4
8
5
8
6
8
7
8
1
9
2
9
3
9
4
9
5
9
6
9
7
9
8
9
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Mathematics Volume IV
9. We can eliminate some of these as equivalent fractions. E.g.: 2/6 = 1/3 as above.
10. Leave on the table only those which are not equivalent to others. We have circled those which
can be removed.
11. Take the Proportion Chart. Our unknown in each case is the actual percent. Percent is always
a number over 100, so we can set up the card as follows:
“One is to three as some number is to 100.”
12. If necessary, work out the proportion.
1/3
= n/100
3xn
= 1 x 100
3n
= 100
3n/3
= 100/3
n
= 33 1/3
1
n
________ = ______
3
100
13. 1/3 equals 33 1/3%
This is a repeating decimal and cannot be more accurate.
14. Try the same process with 2/3.
15. We are now ready to make a Chart from these Special Fractions. Why are they Special? They
are made up of denominators which are not factors of 100 (or 10).
16. Mark off the notebook page as before: 4.5 cm, 4 columns.
SPECIAL FRACTIONS
Percent %
Fraction
100
Fraction in
Lowest Terms
33.33 %
33 1/3
100
1
3
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Decimal
0.333
29
Mathematics Volume IV
17. Insert the answers as found in the above proportion. This time we begin with the fraction in
lowest terms: 1/3. Then we write in the equivalences as we have discovered them to be.
18. Continue with all the Special Fractions. Upon completing this chart, we shall have all the
percent, decimal, and fraction equivalences less than one whole.
Note:
The work necessary in computing these equivalences in invaluable to the child in his/her future
work with percentage.
Control:
The child may see through the work in proportion that a ‘short cut’ method of finding the
percent is to:
Divide the Numerator of the Fraction by the Denominator.
1
3
=
0.33 1/3
3 ) 1.00
2
3
0.66 2/3
3 ) 2.00
Also: If 1/3 equals 0.33 1/3, then 2/3 is twice that amount or 0.66 2/3.
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Mathematics Volume IV
7. AN ANALYSIS OF THE
PERCENTAGE PROBLEM:
FIRST CASE
Introduction:
1. First we know that we cannnot use the expression 25 % of any other percent to calculate a
quantity. The percent must first be changed into a fraction or a decimal before any computation
can be done.
2. The child has worked a great deal on these equivalences in order to be able to concentrate on the
difficulties involved in ‘setting up’ a percentage problem.
3. Although there are other ways of solving percentage problems, we shall use Ratio and Proportion
in this study since the same process can be used for every type of problem.
4. There are 3 parts to every percentage problem:
e.g. 50 %
Percent
of 8
equals 4
Whole
Part
Therefore, there are three types of problems:
a) When the percent is unknown:
n % of 8 equals 4.
b) When the whole is unknown:
50 % of n equals 4.
c) When the part is unknown:
50 % of 8 equals n.
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Mathematics Volume IV
5. In percentage problems, the percent sets up a ratio. We can set up a percentage chart which
is based on our proportion chart:
Proportion Chart
:
_____
= _____
:
Percentage Chart
Part
:
____
Whole
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:
= ____
100
32
Mathematics Volume IV
6. In the problem: 50 % of 8 equals _______, we can put the values we know in the percentage
chart.
Part
:
50
_____
Whole
:
100
n
= _____
8
7. Now we can multiply the means and the extremes and find the number.
50
100
=
n
8
1
2
=
n
8
2n
=
8
n
=
4
simplified:
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Mathematics Volume IV
8. AN ANALYSIS OF THE
PERCENTAGE PROBLEM:
SECOND CASE
Material:
The Percentage Chart
Slips of paper and pencil
Prepared problems of the three types of percentage
Large red circle divided into 100 parts
Presentation:
1. Now we take a second sample problem:
50 % of some number equals 4. Write the problem on a long strip of paper.
2. Invite the child to identify each part of the problem: The child cuts apart the problem and
places the parts on the percentage chart.
Part
:
____
Whole
:
= ____
100
50 % is the percentage.
Turn over the word ‘of’ and write ‘X’.
Some number is the whole; turn over the paper and write ‘n’.
The number ‘4’ is the part.
3. Children continue to study samples of this second type of problem.
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Mathematics Volume IV
PROBLEMS:
SECOND CASE
1.
32% of what is 16?
2.
80% of what is 12?
3.
75% of what is 45?
4.
18 is 15% of what?
5.
144 is 96% of what?
6.
42 is 12% of what?
7.
28% of a is 49.
8.
55% of z is 77.
9.
20% of k is 15.
10.
36 is 90% of what?
11.
21 is 7% of what?
12.
160 is 32% of what?
13.
88% of what is 660?
14.
75% of what is 45?
15.
60% of what is 42?
16.
25% of r is 13.
17.
90% of h is 72.
18.
64% of t is 144.
19.
90 is 72% of what?
20.
27 is 36% of b.
21.
16 is 64% of what?
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Mathematics Volume IV
9. AN ANALYSIS OF THE
PERCENTAGE PROBLEM:
THIRD CASE
Material:
The Percentage Chart (as in the previous lessons)
Slips of paper and pencil
Prepared problems of the three types of percentage
Large red circle divided into 100 parts
Presentation:
1. Our third problem is similar: What percent of 8 is 4?
2. The child writes the problem on a long strip of paper.
3. The child cuts apart the various parts of the problem and places them on the Percentage Chart,
describing them in the process.
Part
:
____
Whole
:
= ____
100
‘What percent’ is the unknown. Turn the paper over and write n%.
‘Of’ - Turn over and write ‘X’.
The number ‘8’ is the whole.
The number ‘4’ is the part.
4. The child continues to study more samples of this type of percentage problem.
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Mathematics Volume IV
TABLE OF EQUIVALENCES
Percent %
Fraction
100
Fraction in
Lowest Terms
Decimal
1%
2%
1/100
2/100
1/100
1/50
0.01
0.02
3%
4%
3/100
4/100
3/100
1/25
0.03
0.04
5%
6%
5/100
6/100
1/20
3/50
0.05
0.06
7%
8%
7/100
8/100
7/100
2/25
0.07
0.08
9%
10%
9/100
10/100
9/100
1/10
0.09
0.10
11%
12%
11/100
12/100
11/100
3/25
0.11
0.12
13%
14%
13/100
14/100
13/100
7/50
0.13
0.14
15%
16%
15/100
16/100
3/20
4/25
0.15
0.16
17%
18%
17/100
18/100
17/100
9/50
0.17
0.18
19%
20%
19/100
20/100
19/100
1/5
0.19
0.20
21%
22%
21/100
22/100
21/100
11/50
0.21
0.22
23%
24%
23/100
24/100
23/100
6/25
0.23
0.24
25%
26%
25/100
26/100
1/4
13/50
0.25
0.26
27%
28%
27/100
28/100
27/100
7/25
0.27
0.28
29%
30%
29/100
30/100
29/100
3/10
0.29
0.30
100%
100/100
1/1
1.00
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Mathematics Volume IV
SPECIAL FRACTIONS
Percent %
33 1/3 %
66 2/3 %
Fraction
100
331/3/100
662/3/100
Fraction in
Lowest Terms
1/3
1/6
Decimal
0.331/3
0.662/3
2/6
5/6
1/7
2/7
3/7
4/7
5/7
6/7
1/8
3/8
5/8
7/8
1/9
2/9
4/9
8/9
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Mathematics Volume IV
10. WORKING PERCENTAGE PROBLEMS
WITH RATIO
1.
6 is what % of 40?
2.
3 is what % of 6?
3.
72 is what % of 80?
4.
40 is what % of 50?
5.
32 is what % of 80?
6.
42 is what % of 42?
7.
What % of 160 is 72?
8.
What % of 400 is 125?
9.
What % of 120 is 78?
10.
What % of 150 is 144?
11.
What % of 100 is 5?
12.
What % of 32 is 24?
13.
81 equals n% of 324.
14.
68 equals n% of 80.
15.
85 equals n% of 150.
16.
20 equals n% of 50.
17.
3 = n% of 30.
18.
26 = n% of 50.
19.
36 is what % of 200?
20.
11 is n% of 25.
21.
17 is n% of 68.
Note:
Each type of problem is written in a variety of ways to enable the child to conquer this challenge.
These same problems may be used in the following study of Percentage through Ratio and
Proportion.
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Mathematics Volume IV
11. WORD PROBLEMS FOR
PERCENTAGE STUDY
1. In 1972, about 90% of the households in
New Brunswick had telephones. What
fraction of households was this?
2. The total length of the Saturn V is about
280 feet. The length of the first stage is
about 140 feet. The length of the first stage
is what percent of the total length?
3. At our school, 150 of the students play
tennis. This is 30% of all the students.
How many students are in the school?
4. A rock brought back from the moon
weighs 3 pounds when it is 8,000 miles
from Earth. This is 25% of its weight on
Earth. What would the rock weigh on
Earth?
5. An astronaut weighing 200 pounds on
Earth would weigh 76 pounds on Mars.
The weight on Mars is what percent of
the weight on Earth?
6. A basketball team won 70% of the
games they played in one season. They
won 14 games. How many games did
they play?
7. The strings on a tennis racket weigh 0.5
of an ounce. This is 4% of the total
weight of the racket. What is the weight
of the racket?
8. The Apollo spacecraft weighs about 48
tons when fully fueled. Its lunar module
weighs about 16 tons. What percent of
the total weight of Apollo is the weight of
the lunar module?
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Mathematics Volume IV
9. A new saxophone is worth $320.00. The
sales tax is 5% of the price. What is the
tax on the saxophone?
10. It costs $5.00 to make a pair of pants.
At a store, the pants are sold for $12.00.
What is the percent of increase from the
manufacturing price to the selling price?
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Mathematics Volume IV
12. THE STUDY OF SIMPLE INTEREST
This section came about as the result of a Montessori class in which the children were responsible
for buying their own school supplies from the school store (open only every 15 days) and in
accordance with a specific sum of money deposited by their parents. They kept a checkbook to
keep their accounts and in general were responsible for the management of it.
The four cases to be studied are:
I.
Calculation of Interest
II.
Research to find the Rate of Interest
III.
Research to find the Principal
IV.
Research to find the Period of Time
Material:
Golden Bead material in quantity, particularly squares and unit beads
Green skittles from Division materials
Envelope containing 4 nomenclature cards in red:
1. Principal
2. Interest
+
÷
3. Time
4. Rate of Interest
Signs:
x
=
Envelope of symbols:
100
P
p
Montessori Research and Development © 2006
r
$
%
T
t
I
x
?
42
Mathematics Volume IV
Presentation of Nomenclature Cards:
This is given to a group of children who have a general idea of the function of interest but not an
exact definition.
Show cards:
Principal
At the bank, the money we deposit is called Principal
Principal.
Time
The period we leave it there is called Time
Time.
Rate of Interest
This is how much the bank gives you for every dollar you
leave in the bank each year.
Interest
The amount the bank gives you on all of your money is called
Interest.
Interest
Presentation of Symbols:
Place the cards next to the symbols (first using the smaller letter). Explain that symbols referring
directly to money are capitalized. Remove the cards and do a Three Period Lesson on the
symbols.
Explain that the r refers to percent (per hundred), so we place the card of 100 next to it as well
as the symbol for the dollar $ .
Principal
p
P
Time
t
T
Interest
I
Rate of Interest
r
Montessori Research and Development © 2006
100
%
$
43
Mathematics Volume IV
13. CALCULATION OF INTEREST
Material:
Golden Bead squares and unit beads
Interest symbols and signs
Paper and pencil
Small bead tray
The Case:
Begin with the calculation of simple interest, using easy quantities.
My Principal deposited in the bank is $26.00. The bank will give me 2 cents for every dollar
Rate
(100 cents) left in the bank (Rate
Rate).
Time
If I leave the money in the bank for 3 years (Time
Time), what will my interest be at the end of this time?
Presentation: First Level
1. Line up the symbols on the table.
P
r
t
2. Write out labels showing the amounts.
$26.00
2
3
I
Interest
3. Pair these with the symbols. The question mark is placed unter the I (Interest
Interest), which we don’t
know.
P
r
t
I
?
4. Consider one hundred square to represent one dollar (100 cents). Set out 26 squares representing
the Principal
Principal.
oo
oo
oo
oo
oo
oo
oo
oo
oo
oo
oo
oo
oo
oo
oo
oo
oo
oo
oo
oo
oo
oo
oo
oo
oo
oo
5. For the rate, put 2 beads under each square.
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Mathematics Volume IV
6. Collect all 52 of these beads and exchange for 5 ten bars and 2 loose beads:
oo
52
7. Remove the hundred squares.
8. Explain that the 52 represents what the bank gives you after one year, but we must find the
amount for 3 years.
9. Line up 3 green skittles and position 52 in beads under each one.
10. Collect all the bead material for the three years and total to get the interest. 156
11. Write a label for 156 and place under the symbol instead of the ‘?’
I
156 cents
Second Level: Replacing Unit Beads with Labels
1. The procedure is similar to the first presentation.
2. Lay out the symbol cards and the problem.
3. After laying out the amount in Golden Bead squares, do not use beads for the rate, but write the
number ‘2’ on little slips of paper and place these on each square.
4. Gather the 26 slips of paper. “If we have repeated 26 x 2, we have 52.”
5. Take the skittles. Write 52 on 3 slips and give one to each skittle. “52 taken 3 times is 156.”
Write a slip with ‘156’ and place it under the ‘?’ and remove the ‘?’.
Third Level: Mathematical without Materials
1. State the problem. The principal is $26.00, the rate is 2%, the time is 3 years; find the interest.
2. Lay out the symbols:
%
P
r
t
I
Place the % sign first, then take it away because the child should understand that the r
represents a percent.
3. I must divide the Principal. How many groups of 100 are in 2600?
Write: 2600 ÷ 100 = 26.
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Mathematics Volume IV
4. Multiply the number of hundreds by the rate. Write: 2 x 26 = 52¢ (annual interest).
5. Multiply the interest for one year by the number of years.
Write: 52¢ x 3 = $1.56 (Interest for 3 years).
Fourth Level: Abstraction, Building the Formula
1. Ask: “For what are we looking?” (Interest)
I =
2. We took the Principal and divided it by 100.
I = P ÷ 100
3. We multiplied the result by the rate.
I = P ÷ 100 x r
4. We then multiplied by the time.
I = P ÷ 100 x r x t
5. This can also be written:
I =
6. We can also write P x r x t over 100.
I = P x r x t
100
7. As a final step, we can take away the x signs.
I = Prt
100
Montessori Research and Development © 2006
P x r x t
100
46
Mathematics Volume IV
14. CALCULATION OF
RATE OF INTEREST
Material:
Golden Bead squares and unit beads
Interest symbols and signs
Paper and pencil
Small bead tray
Presentation:
1. State the problem. I left the Principal of $26.00 in the bank for 3 years. I received $1.56 for the
interest, but I don’t remember what the rate of interest was.
First Level:
1. Line up the symbols and the slips of paper for each amount.
P
2600
I
156
t
3
r
?
2. Set out the Principal in Golden Beads. (2600) Exchange the thousands for hundreds and line
them up.
3. We know the interest was 156¢. Get Golden Beads and distribute them one at a time below the
squares. Each will receive 6.
4. Put away all but one square and its 6 beads.
oooooo
“For each 100, I received 6 units, but that was after 3 years.”
5. Take 3 skittles and distribute the 6 beads. Each skittle receives 2 beads. The rate of interest is 2.
Place this on a slip of paper under the ‘r’.
r
2
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Mathematics Volume IV
Second Level: Replacing Beads with Slips of Paper
1. Repeat as above, setting out the golden squares.
2. Instead of distributing golden beads for the 156, have 156 pieces of paper with 1 written on
them. Distribute these to each square. There will be 6 for each.
3. Take one square and its slips of paper and replace the others.
4. Distribute these slips to 3 skittles.
5. One skittle received 2 slips. Write ‘2’ and put it in the place of the ‘?’.
Third Level: Without Material
1. 2600 ÷ 100
= 26 hundred squares
2. 156 ÷ 26
= 0.06, the rate for 3 years
3. 0.06 ÷ 3
= 0.02, the rate of interest for 1 year.
Fourth Level: Building the Formula
1. r =
2. r = P ÷ 100
3. We have to divide the interest we received by the capital so we move over the P ÷ 100 and
the formula looks like:
r = I ÷ P ÷ 100
4. Then we divide it all by the time:
r = I ÷ P ÷ 100 ÷ t
5. We can change the formula to:
r = I ÷
6. And change it again to:
r = I x 100 ÷ t
P
7. And again to:
r = I x 100 x 1
P
t
8. Or:
r = I 100
Pt
9. And we get the equation:
r = I 100
Pt
Montessori Research and Development © 2006
P ÷ t
100
48
Mathematics Volume IV
15. CALCULATION OF PRINCIPAL
Material:
Golden Bead squares and unit beads
Interest symbols and signs
Paper and pencil
Small bead tray
Green skittles
First Level:
1. State the problem. We left some money in the bank for 3 years and have earned $1.56 interest.
The rate was 2% per year. What was our Principal?
2. Place the symbols and labels.
P
?
t
3
I
156
r
2%
3. Take the amount of interest in Golden Bead material (156):
15 ten bars
6 units
This was my interest for 3 years. I want to find the interest for one year. Take 3 skittles.
Exchange the beads and distribute them. Each receives 52 beads. This is the interest for one
year.
4. Take what one skittle received. With this I must make groups of 2 to represent the rate. Exchange
the quantity for units. Divide into groups of 26. This represents the number of hundreds in the
Principal. Put one hundred square with each group of 2.
5. Collect the 26 hundred squares and exchange them for 2 thousand cubes and 6 hundred squares.
This is the Principal. Write it on a slip of paper and place the answer in the place of the ‘?’.
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Mathematics Volume IV
Second Level: Without Materials
1. State the problem: Knowing the rate (2%), the time (3), and the interest (156), determine the
capital.
2. Think about what was done:
a) Divide the interest by the time:
156 ÷ 3 = 52 (annual interest)
b) The rate was 2, so we did a group
division. (How many groups of 2 in 52?)
52 ÷ 2 = 26 (number of 100’s)
c) Then we gave 100 to each group.
26 x 100 = 2600 ($26.00) Principal
Third Level: The Formula
1. We want to find the formula for P (Principal).
2. P = I ÷ t
3. P = I ÷ t ÷ r
4. P = I ÷ t ÷ r x 100
5. I ÷ t ÷ r = I ÷ r
t
6. P = I ÷ r x 100
t
7. I ÷ r = I x 1
t
t
r
8. I x 1 x 100
t
r
9. I x 1 =
I
t r txr
10. Therefore, the equation is :
P =
I x 100
txr
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Mathematics Volume IV
16. CALCULATION OF TIME
Material:
Golden Bead squares and unit beads
Interest symbols and signs
Paper and pencil
Small bead tray
Presentation:
1. State the problem. I left the Principal of $26.00 in the bank and gained $1.56. The interest rate
was 2%, but I don’t remember how long it was in the bank.
First Level:
1. Line up the symbols:
P
2600
I
156
r
2
t
?
2. Take the Principal in Golden Beads. Exchange for hundreds and place them on the table.
3. Get the 156 Interest in loose beads. Distribute them to the hundreds. Each receives 6.
4. Take these beads and see how many groups of 2 can be made. These groups are the number
of years I left the money in the bank. Write a 3 and place it under the symbol.
Second Level: Writing Analysis of Previous Work
1. State the problem.
2. Lay out the symbols.
3. Write: 2600 ÷ 100
= 26
(number of hundreds)
156 ÷ 26
=6
(Interest per hundred)
6 ÷ 2
=3
(Time)
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Mathematics Volume IV
Third Level: Writing the Formula
1. We are looking for the formula for Time: t =
2. P ÷ 100
3. I ÷ P ÷ 100
4. I ÷ P ÷ 100 ÷ r
5. I ÷
P ÷ r
100
6. I x 100 ÷ r
P
7. I x 100 x 1
P
r
8. I x 100
P x r
9. The equation is:
t = 100 I
Pr
Activities:
1. The children use prepared problems on Command Cards which are easy to calculate.
2. Practice using bank and savings booklets.
Point of Consciousness:
Give the child a comprehension of the function - the way in which the problems are solved
rather than the exact result.
Note:
This study refers only to simple interest, but it is a key which will permit the adult and the child
to progress on to other types of interest, i.e. compound interest. For compound interest, it is not
necessary to use a special method. At a certain point, we can’t use materials but must go to
abstraction.
The numbers should always be the same for each operation so there are no difficulties in
calculation.
Always take the Time in years, not months or days. Don’t use fractions of money, i.e. 50¢.
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Mathematics Volume IV
CHAPTER 3
A STUDY OF INTEGERS
1.
A. Introduction
B. Analysis of Lessons by Level
2.
The Number Line
3.
Vertical Number Line
4.
Greater Than and Less Than
5.
Addition/Subtraction of Integers on the Number Line
6.
Multiplication/Division of Integers on the Number Line
7.
The Additive Inverse
8.
Deriving Rules
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