math 2 album

8/14/2019 Math 2 Album

1/95

Copyright 2009 | Jonathan Feaglewww.FreeMontessori.org

General Outline

I Powers of Numbers 1

II Negative Numbers 15

III Non-Decimal Bases 27

IV Word Problems 45

V Ratio & Proportion 65

VI Algebra 83

Math II


2/95

i

Copyright 2009 | Jonathan Feagle

www.FreeMontessori.org

Contents

I Powers of Numbers 1A. Powers of 2 2

Presentation:

Passage One: Introduction p.2Extension I: erminology

Extension II: Exploration with Bases Other than wo

Passage wo: Different Unit Size p.6

Passage Tree: Hierarchical material p.8

B. Exponential Notation 10

Presentation:

Passage One: Behavior of Exponents when Multiplying Number of the Same Base p.10

Passage wo: Behavior of Exponents when Dividing Numbers p.12

II Negative Numbers 15A. Addition Using Negative Numbers 16

Presentation:

Passage One: Te Snake Game with Negative Numbers and Negative Changing p.16

Passage wo: Writing p.18

Passage Tree: Introduction to the en Bar p.18

B. Subtraction of Sign Numbers 20

Presentation:

Deriving the Rule for Subtracting Sign Numbers p.22C. Multiplication of Sign Numbers 24

Presentation:

D. Division of Sign Numbers 25

Presentation:

III Non-Decimal Bases 27A. Numeration 29

Passage One: Numeration

Part A: Counting on a strip p.29

Part B: Bases larger than 10 p.29

B. Operations in Bases 31

Part A: Addition p.31

Part B: Subtraction p.33

Part C: Multiplication p.35

Part D: Division p.36


3/95

ii



C. Conversion from One Base to Another 37

Part A: o convert a number from any base to base 10 p.37

Part B: o convert from base 10 to another base p.39

Part C: Changing to bases larger than 10 p.41

Part D: Changing bases using the base chart p.42

Example I:

Example II:Example III:

IV Word Problems 45A. Introduction to Word Problems 46

B. Distance, Velocity and ime 47

Presentation:

Passage One: Introduction p.47

Passage wo: Solving for Distance p.47

Level One

Level wo

Level Tree

Passage Tree: Solving for Velocity p.49

Level One

Level wo

Level Tree

Passage Four: Solving for ime p.52Level One

Level wo

Level Tree

C. Principal, Interest, Rate and ime 53

Presentation:


Passage wo: Solving for Interest p.54

Level One

Level wo

Level Tree

Passage Tree: Solving for Rate p.56

Level One

Level wo

Level Tree

Passage Four: Solving for Principal

Level One


4/95

iii



Level wo

Passage Five: Solving for ime p.62

Level One

Level wo

Level Tree

V Ratio & Proportion 65A. Ratio 66

Presentation: Introduction


Passage wo: Introduction to the Language p.66

Passage Tree p.68

Passage Four p.68

Passage Five: Exploring the Idea Arithmetically p.68

Passage Tree:

Passage Four:

Passage Six: Ratios Written as Fractions p.70

Passage Seven: Stating the Ratio Algebraically p.71

Passage Eight: Word Problems p.72

Example A

Example B

Example B Algebraically

Example CExample C Arithmetically

Example C Algebraically

B. Proportion 77


Exercise One: Determining if Something is in Proportion p.78

Exercise wo: Proportion Between Geometric Figures p.78

Exercise Tree: With 3 Dimensional Figures p.79

Exercise Four: p.80

C. Calculations with Proportion 81

Exercise One: Arithmetically p.81

Exercise wo: Algebraically (for older children) p.81

Exercise Tree: Applications p.82

Example I

Example II


5/95

iv



VI Algebra 83A. Introduction to Algebra 84

Exercise One: Balancing an Equation p.84

Exercise wo: Balancing an Equation When Something is aken Away p.85

Exercise Tree: Balancing an Equation When Something is Multiplied p.85

Exercise Four: Balancing an Equation When it is Divided p.85

B. Operations With Equations 86Exercise One: Addition p.86

Exercise wo: Subtraction p.86

Exercise Tree: Multiplication p.86

Exercise Four: Division p.86

C. Algebraic Word Problems 87

Example I: p.87

Example II: p.87

Example III: p.87

Example IV: p.88

Example V: p.88

Example VI: p.88

Example VI: p.89

Example IX: p.89

Example X: p.90

Example XI: p.90

Example XII: p.90


6/95

5



I Powers of Numbers

Contents

A. Powers of 2 2

Presentation:


Extension I: erminology


Passage wo: Different Unit Size p.6

Passage Tree: Hierarchical material p.8

B. Exponential Notation 10

Presentation:

Passage One: Behavior of Exponents when Multiplying Number of the Same Base p.10

Passage wo: Behavior of Exponents when Dividing Numbers p.12


7/95

6



A. Powers of 2

Introduction:

Tis material is not presented to the child until after the exercises with squares and cubes, including notation and

operations have been completed. Tis material is designed to present to the child the powers of numbers beyond

three, and to present the hierarchical material in another way to reinforce in the childs consciousness what is meantby the powers of ten. Tis lesson may be presented several times, perhaps a month or so apart, particularly if the

child is having difficulties with it.

Materials:

Te Power of wos Box, the cubing material, small tickets of paper and pencils

Presentation:

Passage One: Introduction

1. Hold up a small red cube. State that it is one, a unit, and that it is powerless. Set it on the mat.

2. Now we will make a group of two. Move the first cube over, setting a second beside it.

3. Recognize this as your first group of two, call it two to the first power, and write and place a ticket under

it.

4. Now we will take 2 to the power of 1, two times. Move the group of two over and add two to it, form-

ing a square.5. It makes a square. Write a ticket stating 22, place it under the square, stating that this is what it is.

6. Now we will take 2 to the power of 2, two times. Move the square over and add four cubes to it, form-

ing a cube.

7. Replace the built cube with the cube of the same size from the cubing material. State that it is two to the

power of three and label it as such.

8. Continue in the same way, each time taking to the previous power two times, double the number of

cubes, exchange if possible, state its name and label it.

9. When two to the power of nine has been completed, reverse the procedure, dismantling the cube, ex-

changing as necessary, laying the pieces out at each level, and reading their names.

10. When you return to the cube, ask the child what you called it (unit). Remind her that you said it had no

power. Label it as 20, and state its value as one.

11. Te child may wish to write and place tickets stating the value (23, 2x2x2, 8) or the number of the previ-

ous power times two (23x2, 24x2, etc.).


8/95

7




9/95

8



Extension I: erminology

1. After the child has had some practice, tell her that the first number with which we make these powers of

numbers, the larger one, is called the base.

2. Continue, stating that the second number, the smaller one written above the first, is called the exponent.

3. Ask the child what the exponent is (what does it tell you?)

4. Note that when you have multiplied the base by itself the number of times directed by the exponent, youhave reached that power of the base.


1. Present the unit; state that this time you would like to work with groups of three. Assemble one group of

three cubes.

2. Ask the child how this should be written (31), have the child write and place a ticket.

3. How do we get to the next power? Determine that you would take three 3 times. Replace the three red

cubes with a three square. Have the child write and place a label (32).

4. Continue, taking 32three times, replace the square with the cube and label it accordingly (33).

5. Note that you do not have two more cubes of three. Add two stacks of three squares to the cube and

label (34).

6. Note that for 35, you lack squares with which to construct the shape, ask the child what it would look

like.

7. Lead the child to see that it would be the nine square. Lay it out.

8. You may continue to the power of nine, or the child may wish to reverse the procedure as you did for thepower of two.

9. You may repeat with other bases. Also you may compare bases to each other, laying one base behind the

other. Compare their sizes, shapes and other relationships.


10/95

9



23base

exponent

what does the exponent tell you?

2 x 2 x 2 = power of the base

Extension I:

= 31 = 32

then 3

1

three times then 3

2

three times

= 33

= 34

then 33three times

=

=34taken three times = 35

Extension II:


11/95

10



Passage wo: Different Unit Size

1. ake out the Power of wos Box, stating that you are going to look at it again.

2. We will let the small cube (small yellow) be the unit, and we are in the base of two, what will we label

this cube?

3. Have the child write and place a label reading 20

below the unit cube.4. Place the other cube next to it, label it 21, and replace the two cube with the prism, putting the cube

back over its original label.

5. Lets take 21two times to build to the next power. Put the two cubes beside the prism, then exchange it

for the square, replacing the cubes in their original locations. Have the child write a label.

6. Repeat, building the two square to a cube with the prism, and two cubes. Exchange, replace the other

pieces and label the cube.

7. Repeat in the above fashion with 24, constructing the prism, then dismantling the other pieces back in

their places.

8. Continue to 26, the limit of the material.

9. Summarize that the unit can be of any size.


12/95

11



21 22

23 24

25

26

20

exchange

21

exchange

22

exchange

23

exchange

24

exchange

25

exchange

26


13/95

12



Passage Tree: Hierarchical material

Note: For this passage, you will need the hierarchical material.

1. Have the child lay out the hierarchical material in their families.

2. Have the child name each family. State that we know this is the decimal system, since each family is

composed of ten of the previous (deci = ten).3. Note that in the decimal system, the base is ten. Label the unit 100, ask the child what it means when

you have a base (how much it takes to go from one power to the next).

4. Is that what has been done to the ten? Is it ten to the first power? Te child may verify by counting.

Ask the child what it is called (101). Have him write and place a label.

5. Lets go to the next power, what do we have to do? (multiply the ten by ten) have the child write and

place a label.

6. For the next level, we take ten to what power? (3rd) Have the child write and place the label by the

cube.

7. Continue to the sixth power (106). Te child may wish to place the powers of two alongside. Note which

powers have the same shape.

8. Te child may also wish to lay out the numerical values of the powers of ten. Note that the number of

zeros equals the exponent number.


14/95

13



101

102

103

21

22

23

10 100 1000


15/95

14



B. Exponential Notation

Introduction:

Tis work allows the child to investigate and explore exponential numbers. It should not be presented until the

child has completed the powers of two and ten.

Materials:

Te cubing material, pencils and paper

Presentation:

Passage One: Behavior of Exponents when Multiplying Number of the Same Base

1. Propose the problem: 33x 32=. Note that the child can probably do it, but that you are going to try a

new way.

2. We are going to take the three cube32times. Lay out the three cube.

3. Ask what 32is (9). State that you will take the three cube nine times.

4. Layout nine three cubes (assembled with three squares), forming a square.

5. We can write this as a power of three, as three to the fifth power.

6. How did I know that? Demonstrate by reconstructing the shape. Start with a three square (32), times 3

makes the three cube (33), times three yields a line of 3 three cubes (34), times three is 3 rows of 3 three

cubes (35).

33

x 32

= 35

7. Propose another problem: 32x 33=. Note that the quantity is 32, and it is to be taken 33times (27).

8. Set out 27 three squares. ake one, stating that it is 32. Add two more to form 33. Add six more to form a

line of three cubes (34). Place the remaining 18 to form a square of 27 three squares (35).

9. Note that you will express the answer as a power of the base. In this case, the base is three, and youve

taken it to the power of five (35is the answer).

10. After some work, ask the child if he can make an observation about the answers. Note the following rule

When multiplying exponential numbers of the same base, add the exponents and express the

answer as the base with the sum of the exponents.

11. Later, propose the problem: 53x 5 =. State that the problem says to take a five cube five times.

12. Add four groups of 5 five squares to the cube. Note that you get 54.

13. What can we say about the five in this problem? (It is five to the power of one and it gets added to the

cube to make 54.


16/95

15



33x 32=

33

= 9So,x

() x 9 =

32

= 35


17/95

16



Passage wo: Behavior of Exponents when Dividing Numbers

1. Propose the problem: 35 33=. Write in fraction form. Lay out three to the fifth, forming a large square.

2. Ask how many three cubes are contained in the square. [Lay the cubes in a line.]

3. Have the child count the cubes (9). Ask her if it is a number that can be written in the same base as the

others (32).

4. Note the rule:

When dividing exponential numbers of the same base, subtract the exponents and express the

answer as the base with the difference of the exponents.

5. Propose the problem: 32 32=. Note that the answer would be 1, or 30 (expressing the unit as the base

to the zero power).


18/95

17



35

= 35

How many three cubes in the square?

9 cubes

9 = 32

33=


19/95

18




20/95

19



II Negative Numbers

Contents

A. Addition Using Negative Numbers 16

Presentation:

Passage One: Te Snake Game with Negative Numbers and Negative Changing p.16

Passage wo: Writing p.18

Passage Tree: Introduction to the en Bar p.18

B. Subtraction of Sign Numbers 20

Presentation:

Deriving the Rule for Subtracting Sign Numbers p.22

C. Multiplication of Sign Numbers 24

Presentation:

D. Division of Sign Numbers 25

Presentation:


21/95

20



A. Addition Using Negative Numbers

Note: It helps to give kids examples of negative numbers before starting: If I have $7 and I owe John $8, I really

have -$1.

Materials:

A mat, boxes containing the colored bead bars from 1 - 10, the negative bead bars 1 - 10, the black and white

bead stairs and the red and white bead stairs.

Presentation:

Passage One: Te Snake Game with Negative Numbers and Negative Changing

1. Lets call the colored bead bars positive, and the gray ones negative.

2. Have the child form a long snake, dictating to him positive and negative numbers to be placed:

+8 + +9 + -8 + +8 + -9 + +6 + +5 + -8 + +4 + +6

3. Have the child lay out the black and white stair. Bring the first two bars down from the snake (+8 + +9).

4. Exchange them for a ten bar and the seven from the black and white stair. Attach this to the snake.

5. Place the used bars in a pile at the top. Bring down the black and white 7 and the - 8.

6. When you add -8 and +7, you get a negative 1 (-1) | How are we going to do that without a -1 bar?

7. Replace the bars with the-

1 bar. Place the black and white 7 back into the b/w bead stair, and start anegative pile at the top.

8. Bring down the next two bars (-1 and +8). Continue as described above until the end of the snake.

9. Count to see whats left on the snake. Ask the child how you are going to check it. Point out that positive

and negative beads at the top represent the whole snake.

10. We can add all the positives, and all the negatives, then find the difference between them. Place bars in

each pile in like groups and add their values on paper.

11. Compare the two answers, place a check if correct.


22/95

21




23/95

22



Passage wo: Writing

1. Dictate a number (+8) and have the child take the bar out. Show him how to say and write it.

2. Continue dictating, saying and writing the rest of the numbers in the problem below, noting that the +/-

signs are placed close to the numbers in the problem.

3. Point out that the problem is a sum. Have the child work it out and place the answer at the end of the

problem.

+8 + -9 + +6 + -5 =

Passage Tree: Introduction to the en Bar

Note: Te child will most likely run across the following on her own, and you will therefore show her what to do

then. It need not be a separate lesson.

1. Lay out and record the problem:

+7 + +2 + +3 + -9 + -8 + +2 + -9 + -3 + +7 + +9 + +4 + +9 + -7 + -3 + -6 + +2 + +6 + +8 =

2. Have the child work out the problem as usual. When she encounters a negative sum over nine, intro-

duce her to the negative ten bar, and have her place it in the snake, followed by the five from the red and

white bead stair.

3. Continue in the manner described in Passage One through the snake. When the snake is complete, de-

termine its value by canceling out the positive and negative ten bars.

4. Have the child check the answer. She may do so in a similar way, canceling the positive and negative barsto get the answer. Have her record the answer

+7 + +2 + +3 + -9 + -8 + +2 + -9 + -3 + +7 + +9 + +4 + +9 + -7 + -3 + -6 + +2 + +6 + +8 =14

5. Lead the child to the rule by having her complete the following statements:

When you add numbers of different signs (you subtract and the answer has the sign of the larger

number in the problem).

When adding numbers of the same sign.(you add them and the sign remains the same).

6. Te child may discover that she can add all the same sign numbers and subtract the different sign totals

and assign the sign of the larger number.

7. And/or, she may discover canceling. When doing this, she should cross out the canceled numbers, then

add the same signs and subtract as described above.


24/95

23



Passage Tree:


25/95

24



B. Subtraction of Sign Numbers

Materials:

A mat, boxes containing the colored bead bars from 1 to 10, the negative bead bars from 1 to 10, the black and

white bead stairs and the red and white bead stairs.

Presentation:

1. Propose the problem: +7 + +3 + +4 + +3 + -4 =. Have the child build the snake, record the values, solve the

problem and write the answer at the end (+13).

2. Recompose the snake and the problem by removing the +4 bar and hiding it in your hand. Have the

child work the new problem and state the answer.

3. State that you will record the problem in a special way. Note that you took the +4 away and record this

problem under the original:

(+7 + +3 + +4 + +3 + -4) - +4 = +9.

4. Recompose the problem again by replacing the +4 and removing the -4. Have the child work the problem

as above.

5. Write the problem, asking what you took away, and subtract that at the end:

(+7 + +3 + +4 + +3 + -4) - -4 = +17.

6. Do a series of similar problems, recording them as you go, so that you may make observations at the end7. Ask the child what she notices. Lead her to understand thatwhen a positive number was subtracted,

the answer got smaller, while when a negative number was subtracted, the answer got larger .

8. Te child may check her observations on other problems.


26/95

25



+7 + +3 + +4 + +3 + -4 =

Hide the four bar in hand.

Recorded as: (+7 + +3 + +4 + +3 + -4) - +4= +9

Replace +4 bar and remove the -4 bar

then write the problem showing

what you took away (-4).

Recorded as: (+7 + +3 + +4 + +3 + -4) - -4= +9


27/95

26



Deriving the Rule for Subtracting Sign Numbers

1. Restate the rule arrived at above. Hold up the nine bar, and say that it is positive nine. Write it down.

2. Ill take +4 away from that. Record this: +9 - +4, and cover four beads on the bar.

3. Ask the child whats left (+5), record this.

+9 - +4 = +5

4. Its also possible to have -9 - -4. Help the child to show you this on the negative nine bead bar. Deter-

mine that the answer is negative five and write:

-9 - -4 = -5

5. You can also have -9 - +4 =. But there isnt a positive four on the negative nine bar. Place both a positive

and a negative four bar beside the negative nine bar. Affirm with the child that you have now added zero

to the negative nine bar.

6. ake the positive four bar away, determining that -13 is the answer. Record this:

-9 - +4 = -13

7. Propose the problem:+

9 --

4 =. Have the child take the positive nine bar and add zero to it by placing apositive and a negative four bar beside it. Remove the negative four to arrive at +13.

+9 - -4 = +13

8. Have the child look at the first problem (+9 - +4 = +5). Ask if there is another way to get a +5. Determine

that you can change the sign of the subtrahend and add.

9. Write out the rule and try it on other problems:

When subtracting sign numbers, change the sign of the subtrahend and add it to the minuend.


28/95

27



-9 - -4 =

cover four

-9 - +4 =

add zero

take away the +4

= -13 +9 - -4 =

take away the -4

= 13

add zero

cover four


29/95

28



C. Multiplication of Sign Numbers

Materials:

A mat, boxes containing the colored bead bars from 1 - 10, the negative bead bars 1 - 10, the black and white

bead stairs and the red and white bead stairs.

Presentation:

1. Roll out the mat, and ask the child what is on it (nothing). I would like you to give me +5 three times.

2. Place the bars on the mat and ask the child what is there now (+15). We took +5, +3 times and we got+15. Have him write:

+5 x +3 = +15

3. Now, I would like you to give me -5 three times. Lay them out after the child hands them to you, and

have him restate the problem with the answer:

-5 x +3 = -15

4. Lay out three positive and three negative five bars, and ask the child what is there (zero).

5. ake away the three positive five bars and ask what is left (-15). (ake +5 a -3 times.) Record:

+5 x -3 = -15

What is here? Zero!

What is here now? -15.

Take +5 negative three times.

+5 x -3 =


30/95

29



6. Again, lay out three positive and three negative five bars, and state that you want to take negative five

negative three times.

7. Remove three negative five bars, and ask the child whats left (+15). Record:

-5 x -3 = +15

8. After some practice, help the child to observe the rule:

When you multiply numbers of the same sign, the answer is positive. When you multiply numbers

of different signs, the answer is negative.

D. Division of Sign Numbers

Materials:

In addition to the materials mentioned above, division cups and skittles will be necessary.

Presentation:

1. Place two skittles on the mat. Gather 4 positive seven bars into your hand. Show them to the child, ask

her what they are (four +7 bars), and what their value is (+28).

2. Im sharing these bars between the skittles. Have the child record:

+28 +2 = +14

3. Note that this matches the multiplication rule (+ x + = +).

4. Return the positive sevens and gather four negative

seven bars into your hand. Ask the child what they are

(four -7 bars), and what their value is (-28).

5. Distribute the bars to the skittles, ask the child what

each skittle got (-14), and have her write the problem:

-28 +2 = -14

6. Gather the negative seven bars again. Ask the child

how many groups of -14 you could make (+2).

7. Ask how to write this as a division problem and

record:

+28

+2 = 14


31/95

30



-28 -14 = +2

8. Propose that 2 people are debtors, they owe money,

and each owes +14. Place two +7 bars each in two cups.

Ask the child what is there (+28).

9. Ask her how to write this as a division problem. Tereare +28 beads in the cups (write +28), and two debtors.

Te debtors are negative because they took the +28

away (write -2).

10. State that the debtors took the positive 28 away, do

so, and ask the child what each skittle now gets (-14).

Record the problem and answer:

+28 -2 = -14

11. Determine that the rule is the same as that for multiplication: When you divide numbers of the same

sign, the answer is positive. When you divide numbers of different signs, the answer is negative.

-28

+2 =

-14


32/95

31



III Non-Decimal Bases

Contents

A. Numeration 29


Part A: Counting on a strip p.29

Part B: Bases larger than 10 p.29

B. Operations in Bases 31

Part A: Addition p.31

Part B: Subtraction p.33

Part C: Multiplication p.35

Part D: Division p.36

C. Conversion from One Base to Another 37

Part A: o convert a number from any base to base 10 p.37

Part B: o convert from base 10 to another base p.39Part C: Changing to bases larger than 10 p.41

Part D: Changing bases using the base chart p.42

Example I:

Example II:

Example III:


33/95

32



Introduction:

Children should have worked extensively with the decimal system. Tey must understand that in our system there

are only nine symbols (well, okay, ten (i.e. 0)) to represent digital quantity. As soon as ten is reached, it is necessary

to move to the next place and employ the zero as a marker in the units category. At ten, we no longer have a group

of units, we have a single unit of the next higher order: we have a ten. Te child must recognize that ours is a place

value system in which zero is a necessity.Te child must have worked with powers of number between 2 and 10 and exponential notation. Tey are also

aware of the geometric shapes of the various powers, i.e.: the power of zero is a point, and the powers of 2 and 5

create lines. Te child also must have studied the history of numbers and understand that other cultures have used

different-base number systems than ours, and had entirely distinct concepts of number (the Egyptians used no place

value) from ours.

Materials:

Use as appropriate: the colored bead bars, the cubing material, the golden unit beads, a roll of adding machine

tape, number bases board, made of felt and marked into four categories as below, a chart of numeration in four or

more bases with base 10 in red, blank paper tickets, and pencils

Sixteen

Fifteen

Fourteen

Thirteen

Twelve

Eleven

Ten

Nine

Eight

Seven

Six

Five

Four

Three

Two

0 1 2 3 4 5 6 7 8 9 A B C D E F

0 1 2 3 4 5 6 7 8 9 A B C D E

0 1 2 3 4 5 6 7 8 9 A B C D

0 1 2 3 4 5 6 7 8 9 A B C

0 1 2 3 4 5 6 7 8 9 A B

0 1 2 3 4 5 6 7 8 9 A

10

10

10

10

10

11

11

11

11

12

12

12

13

1413

0 1 2 3 4 5 6 7 8 9 10 11 12 1413 15

0 1 2 3 4 5 6 7 8

0 1 2 3 4 5 6 7

0 1 2 3 4 5 6

0 1 2 3 4 5

0 1 2 3 4

0 1 2 3

0 1 2

0 1

10

10

10

10

10

10

10

10

11 12 1413 15 16

11 12 1413 15 16 17

11 12 1413 15 16 20 21

11 12 1413 15 20 21 22 23

11 12 1413 20 21 22 23 24 30

11 12 13 20 21 22 23

11 12

30 31 32 33

20 21 22

11

100 101 110 111 112102 120

100 101 110 111 1000 1001 1010 10111100 1101 1110 1111

TeR

elationsamongafewbasesystems.


34/95

33



A. Numeration


Part A: Counting on a strip

1. Introduce the number bases board, perhaps relating it to a question that the child may have had regard-ing numbers in other bases.

2. Lets work in the base of five and see how to count and write in it. Write 5 on a ticket and place it

into the box at the top of the board.

3. Cut a long strip from the roll of adding machine tape and write base 5 at the top. Place a unit bead in

the units column of the board and write 1 on the strip.

4. Place another bead on the board and write 2 under the 1. Continue until five beads are laid in the

unit column.

5. Exchange the beads for a five bar and record 10(read one, zero) on the strip.

6. Place a gold bead in the unit column and record 11. Place a second and record 12.

7. Continue until there are five beads in the unit column. Exchange for a five bar and record 20 on the

strip.

8. Continue further until there are four unit beads and four five bars on the board. Exchange the beads for

a bar, and exchange the bars for a square, recording 100 on the strip. Continue in the same manner,

writing 101 for the next unit.

9. Encourage the child to work in different bases. You may wish to give two children different bases and

run a race to see who can get to 1000 first.

Part B: Bases larger than 10

1. Note that you can only write a single digit in a particular place.

2. Place a ticket reading 12into the box at the top of the base board.

3. Place beads to 9 on the board, recording them as in Part A. At ten, note that a single digit symbol is

necessary for this quantity, since 10 has a different meaning in this base. Suggest t for ten and e for

eleven. Ask what twelve will be (10).

4. Note that the last bead invites an exchange to the next level. Ask the child what we would normally

exchange for (a bar). Acknowledge that there is no bar for twelve, invite the child to find an appropriate

solution.


35/95

34



Cubes Squares Bars Units5

use strip

of paper

Cubes Squares Bars Units5

base 5

1

2

3

4

5

10

1

2

3

4

5

20

10

20

3040

50

100

200

300

400

500

1000


36/95

35



B. Operations in Bases

Part A: Addition

1. Review that when the child was in the primary classroom, learning to add, he used an addition chart.

Show the child the base ten addition chart.

2. State that we can do the same thing in other bases. Suggest that they make their own so they can refer toit when adding in other bases.

3. Discuss with the child how to set the chart up, referring to the base ten chart.

4. First lay out the top and left sides of the chart. Ten have the child fill in the chart by adding the num-

bers on the axes. Start with 1 + 1, write the answer (2) in the correct space.

5. Continue as above, picking numbers to add at random.

6. Te complete board should look like the example on the page to the right:

7. Note that now you can add with any number in this base.

8. Point out that when working in a non-decimal base, first you must know what base you are working in.

9. You may either write the base at the top of the page if all the work is in the same base. Or you can

write it to the lower right of the number - 12five

10. Propose the problem:

12five

13five

4five

+3five

11. Begin by adding 4five

and

3five

to get 12five

(from

the chart). Add 3five

to

this for 20five

. Add 2five

to

complete the column and

get 22five

. Have the child

record 2five

as the first digit

of the answer and carry in

their heads.

12. Add the mentally-car-

ried-2 to the two 1s in

the second column for an

answer of 42five

.

10

2 3 4 10

3 4 10 11

4 10 11 12

10 11

11

11

12

12

12

13

14

20

13

13 14

1 2 3 4

1

2

3

4

10

Base 5 Addition Chart


37/95

36



1

2

3

2 3

3

3

4

base 10 addition chart

4

5

6

7

9

8

10

4 5 6 7 8 9 101

2

4

5

6

7

8

9

10

11

4 5 6 7 8 9 10 11

5 6 7 8 9 10 11 12

12

12

12

12

12

12

12

12

11

11

11

11

11

11

11

10

10

10

10

10

10

9

9

9

9

9

8

8

8

8

7

7

76

6

5 13

13

13

13

13

13

13

13

14

14

14

14

14

14

14

15

15

15

15

15

15

16

16

16

16

16

17

17

17

17

18

18

18

19

19

20


38/95

37



Part B: Subtraction

1. Display the base ten subtraction chart, and note how the chart is set up. Set the numbers up in a similar

fashion for a base five chart.

2. Have the child choose the numbers she wants to subtract. Te child may want to lay the beads on the

base board to solve the problem. Make sure she exchanges as necessary.

3. Continue working until the chart is completed.4. Note that they are now prepared to do any subtraction problem in base five. Suggest the following:

24five

21five

123134five

-13five

-3five

-23412five

11five

13five

44222five

101 2 3 4

1

2

3

4

10

Base 5 Subtraction Chart

S

u

b

t

r

a

h

e

n

d

Minuend

0 1 2 3 4

0 1 2 3

0 1 2

0 1

0


39/95

38



1 2 3 4 5 6 7 8 9 10

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9

1 2 3 4 5 6 7 8

1 2 3 4 5 6 7

1 2 3 4 5 6

1 2 3 4 5

1 2 3 4

1 2 3

1 2

1

0-1-2-3-4-5-6-7-8-9

0-1-2-3-4-5-6-7-8

0-1-2-3-4-5-6-7

0-1-2-3-4-5-6

0-1-2-3-4-5

0-1-2-3-4

0-1-2-3

0-1-2

0-1

minuendBase 10 Subtraction Chart

S

u

b

t

r

a

h

e

n

d


40/95

39



Part C: Multiplication

1. Set up a chart for base five multiplication in a manner similar to the preceding. Randomly fill in the

chart by working out the problems on the base board.

2. Have the child work out problems using the chart. She should carry in her head, not on paper. Suggest:

123five

x 3five

424five

Multiplicand

Multiplier

101 2 3 4

1

2

3

4

10

Base 5 Multiplication Chart

1 2 3 4 10

2 4 11 13 20

3 11 14 22 30

4 13 22 31 40

10 20 30 40 100

0


41/95

40



Part D: Division

1. Look at the base ten division chart as a guide to making a base five division chart.

2. Note that you will need the products from the multiplication chart to start the division chart.

3. Point out that the product of 9 x 9 (81) is the largest digit on the base ten chart, and that the corre-

sponding number in base 5 would be the product of 4five

x 4five

(31five

). Start with this at the top of the

chart.

4. Note that the products (quotients?) on the base ten chart descend to zero in order and write them assuch on the base five chart.

5. Also, note that not all the squares are filled in - Only whole quotients are used.

6. Fill the chart randomly, checking your work against the multiplication chart.

7. Use the chart to solve problems in the same way as in base ten. Check by multiplying and adding the

remainder. Suggest:

(20344five

3five

)

3244r2five

3244five

3five

20344five

x 3five

-14 20342five

13 + 2five

-11 20344five

24

-22

2

base 5 division c hart

1

2

3

4

10

31 30 22 20 14 13 11 10 4

30 2 1

4 3 2 1

10 4 3 2

10 4 3 2

31 30 22 20 14 13 11 10 4


42/95

41



C. Conversion from One Base to Another

Note: Children must understand that in different bases, the categories represent different powers.

Part A: o convert a number from any base to base 10

1. Suppose I wanted to know what a number in base five equaled in base ten. How could I figure thatout?

2. Propose the problem: 1432five

= ____ten

. Have the child lay the beads, bars, squares and cubes onto the

base chart in the appropriate places. Note that unit beads are the same in either base.

3. Direct the child to the cube (53). State that it is 1 times 53, or 125. Record:

1 x 53= 125

4. Direct the child to the squares. Ask what one is worth (52or 25), and how many there are (4).

1 x 53= 125

4 x 52= 100

5. Direct the child to the bars. Determine that each bar represents 51and that there are 3. Record 3 x 51

below the others.

6. Direct the child to the units. Ask how they may be expressed as a power of five. Record 2x50=2 below the

others and add them together, putting the answer in the original equation:

1 x 53

= 125 1432five= 242ten4 x 52= 100

3 x 51= 15

2 x 50= + 2

242

7. Note that you wrote the number in expanded notation in order to make the transition from base five to

base ten.


43/95

42



C

ubes

Squares

Bars

Units

5

use

strip

ofpaper

10

20

30

12

53

4

x52

3

x51

2

x50

125

+

100

+

15

+

2

=

242

ten

base

5

1432

five=_______ten

1000

100

200

300

400


44/95

43



Part B: o convert from base 10 to another base

1. Suppose I wanted to know what a number in base ten equalled in base four. How could I figure that

out?

2. Propose the problem: 54ten

= ____four

. Have the child place 5 ten bars and 4 unit beads on the base

board.

3. Have the child exchange as many of the beads as he can into 4-bars (dont have him exchange to squaresor cubes yet).

4. Lets record what we did. Write: 54 4 = 13r2.

5. Have the child exchange the bars for squares and record what he did: 13 4 = 3r1.

6. Lets see if we can exchange any more. OOPS, 3 4 doesnt work.; state that youll take it one step

further to show the fact that you cant change any more. Write: 3 4 = 0r3 Note that what you did was

take out multiples of four, the remainders stayed behind.

7. Point out that you continue because whats left over becomes the quotient (work to zero); the remainders

are the digits of the preceding columns. Te first remainder (2) is the number in the digits column, and

the second remainder is in the second (41) column.

54ten

= 312four

54 4 = 13 r 2 (40)

13 4 = 3 r 1 (41)

3 4 = 0 r 3 (42)


45/95

44



Cubes Squares Bars Units

Exchange for 4 bars (54 4 = 13r2)

Exchange for 4-squares (13 4 = 3r1)



54ten

= _______four

Can we exchange any more? What happens if we divide 3 by 4? = 0r3.

54ten

= _______312four


46/95

45



8. Have the child repeat the same procedure for:

6821ten

= 100100122three

6821 3 = 2273 r 2 (30)

2273 3 = 757 r 2 (31)

757 3 = 252 r 1 (32)

252 3 = 84 r 0 (33

) 84 3 = 28 r 0 (34)

8 3 = 9 r 1 (35)

9 3 = 3 r 0 (36)

3 3 = 1 r 0 (37)

1 3 = 0 r 1 (38)

9. Te child may check his answer by expanding it, as was done when converting from a given base to base

ten: 1 x 38= 6561

0 x 37= 0

0 x 36= 0 etc.

Part C: Changing to bases larger than 10

1. Propose the problem: 6821ten

= ____twelve

.

2. Ask the child if he expects the base 12 number to be larger or smaller than the base 10 (smaller).

3. Work out the problem in the manner described above, substituting t and e for 10 and 11 as

necessary.

6821ten

= 3b45twelve

6821 12 = 568 r 5 (120)

568 12 = 47 r 4 (121)

47 12 = 3 r b (122)

3 12 = 0 r 3 (123)


47/95

46



Part D: Changing bases using the base chart

Example I:


= ____ten

.

2. Lets do this one using the rule. Its in base five right now, lets change it to base ten.

3. We need to know how many groups of 10 there are in this base five number. Well find that out by di-

viding by ten. Tis chart will tell us what number ten is in base five. Find the number youre converting

to in the base ten column and slide across to the base five column to see what its equivalent is (20five).4. Divide 1432

fiveby 20

five, noting that the answer is in groups of ten.

44 r 2

20five

1432five

-130

132

- 130

2

5. Continue to divide out the answers as demonstrated above:

1432five

= 242ten

1432five

20five

= 44 r 2 (100)

44five

20five

= 2 r 4 (101)

2five

20five

= 0 r 2 (102)

Example II:


= ____four

. Complete in the same manner described above.

1424five

= 3233four

1424 4 = 214 r 3

214 4 = 24 r 3

24 4 = 3 r 2

3 4 = 0 r 3


48/95

47



Example III:


= ____seven

. Complete in the same manner described above, noting that

sometimes when changing to a larger base, the remainder may need to be changed. (7 = 12 in base

five[10 (5) + 2])

1424five= 461seven

424five

12five

= 114 r 1

114five

12five

= 4 r 3

4five

12five

= 0 r 4


49/95

48



notes:


50/95

49



IV Word Problems

Contents

A. Introduction to Word Problems 46

B. Distance, Velocity and ime 47

Presentation:Passage One: Introduction p.47

Passage wo: Solving for Distance p.47

Level One

Level wo

Level Tree

Passage Tree: Solving for Velocity p.49

Level One

Level wo

Level Tree

Passage Four: Solving for ime p.52

Level One

Level wo

Level Tree

C. Principal, Interest, Rate and ime 53

Presentation:


Passage wo: Solving for Interest p.54Level One

Level wo

Level Tree

Passage Tree: Solving for Rate p.56

Level One

Level wo

Level Tree


Level One

Level wo

Passage Five: Solving for ime p.62

Level One

Level wo

Level Tree


51/95

50



A. Introduction to Word Problems

Word problems are an important aspect of the cosmic approach to mathematics. Just as the other mathematical

areas are considered to have abstract and practical applications, so do word problems. It is best to introduce word

problems through situations that arise in class. If, however, problems do not arise, some may be made up. Te scope

and variety of vocabulary used to describe operations in our language should be employed when doing word prob-

lems in class. It is in this way that the child comes to understand how all of these words describe the operations.

Word problems should not be a constant feature in the classroom. Tey should be brought in on occasion. Te

first set of word problems should be coded as to what operation is required to complete it. After the children have

explored these and are comfortable, these coded problems may be exchanged for uncoded problems. Later, problems

that are uncoded and which involve mixed operations may be introduced. Further, when these problems are mas-

tered, it is time to introduce more complex problems involving decimals, fractions and mixed operations.

When teaching word problems, remember to solve them in a step-by-step fashion, ensuring that the child under-

stands the method. First, read the entire problem, making sure the child knows all the words in it. Ten, help the

child to determine what you know by listing to the facts presented in the problem, what do you want to know

by evaluating the request of the problem, and how to solve it (what operations on what numbers, etc.). Te child

should then carry out the work of the problem and check her work by asking herself if the answer seems right, given

what is known, and what is to be discovered.

Te following section covers two types of word problems, those involving distance, time and velocity, and those

involving interest rate, principal and time, leading to endeavors with formulas. Each type is presented in three levels.

Te first levelis introductory, sensorial and may be presented around seven years of age. Te second levelleads toabstraction, and more precise identification of the problems request. Level two is presented between seven and eight

years. Te third levelis abstract and presents the rule for the type of problems solution. It is presented around eight

to nine years of age.


52/95

51



B. Distance, Velocity and ime

Materials:

Te golden bead material, a box of tickets containing one each for velocity, distance, time and their abbreviations

(v, d, t), what is known?, what is wanted?, operation signs, blanks, two fraction bars, and pencils

Presentation:


1. Set up a race: measure a straight course, set out start and finish lines, and have the children run. Record

each childs time and the distance of the race.

2. Make a chart of this information. As you do, mention that the straight course is a distance and that the

childrens different foot speed affected the times.

3. Ensure the children understand the relationships between time, distance and speed, then introduce the

terms, using velocity for speed.

4. Before beginning the problems, ask the children to remind you what the name of what they ran was

(distance). Introduce the distance card.

5. Ask the children what the stopwatch recorded (how long, time). Introduce the time card.

6. Ask the children why some finished before others (they were faster). You could introduce velocity as the

measurement of fastness. Introduce the velocity card, and all the abbreviation cards. Lay the abbrevia-

tions alongside the terms.

Note:Sometimes problems will arise from this discussion. Solve them first.

Passage wo: Solving for Distance

Level One


If a plane travels 500 miles per hour, how far will it travel in 3 hours?

2. What information does the problem give us? Lay out the ticket reading What is known?

3. We know the plane was going 500 miles per hour; what is that (velocity)? Place the velocity card, then

its abbreviation to the right of the What is known? card.

4. Write 500 on a blank ticket and place it to the right of the cards.

5. Determine what else is known (time) and place that below the velocity material. Finish the layout by

placing the What is wanted? card below What is known? and place the distance card to the right

with a question mark at the end:


53/95

52



6. Get out 5 hundred squares, noting that this is the velocity. State that the plane went that fast for 3 hours

and put out 3 skittles.

6. Place 5 hundred squares beneath each skittle. Add them together for the answer (1500 miles).

7. Write 1500 on a ticket, replacing the ? with it.

8. Continue with other similar problems before advancing to Level wo.

Level wo

1. Have the child reconstruct the final layout from Level One.

2. Ask the child what operations were used on what numbers (multiplication). Lay these tickets out with

operation cards.

3. Propose other problems and

solve them in the same manner.

Soon, the child wont need the

material at all.

Layout the cards thus:


54/95

53



Level Tree

1. Propose the problem from Level One again. Tis time, ask the child how we might express the problem

using the abbreviations.

2. We want to know what the distance was. Place out the d abbreviation card with an equal sign written

on a ticket

3. Ask what you did (multiplied velocity by time). Lay out these abbreviations:

4. Have the child solve the problem using the formula. Continue practicing on other problems

Passage Tree: Solving for Velocity

Level One


A plane travels

1500 miles in 3hours. At what

speed is it travel-

ing?

2. alk through lay-

ing out the cards for

what is known and

what is wanted.

3. Place out 15 hundred-squares, stating that they are the distance. Placing out three skittles, state that this

is the time.

4. Distribute the squares to the skittles. Note that the plane travels 500 miles per hour. Write a ticket to

reflect this, and replace the ? with it.

5. Note that youve solved for velocity.

D =



55/95

54



Write ticket for the answer -

500


56/95

55



Level wo

1. Have the child lay the cards, skittles and squares out in the manner above.

2. Ask the child what operation he used to perform the lay out (division).

3. Determine that you divided 1500 miles by 3 hours lay this out with the tickets:

Level Tree

1. Propose the problem from Level One again. Ask the child how we might express the problem using the

abbreviations.

2. We want to know what the velocity was. Place out the v abbreviation card with an equal sign written

on a ticket

3. Ask what you did (divided distance by time). Lay out these abbreviations:

4. Work out the problem on paper, plugging the facts into the formula.


57/95

56



Passage Four: Solving for ime

Level One


A plane traveled 1500 miles at 500 miles per hour. How many hours did the plane fly?

2. Lay out the cards and information as below:

3. Place out a stack of 15 hundred-squares. Count from this stack groups of five for each hour traveled.

4. Place a skittle on top of each stack. Count the skittles for the answer.

Level wo

1. Ask the child what you did. Note that you found out how many groups of 500 there were in 1500.

2. Note that you divided, and lay out the tickets as follows:

Lay out the cards thus -


58/95

57



Level Tree

1. Determine the formula, noting that you were searching for

time. Lay out

Note that you divided the distance by the velocity:

2. Plug in the number and solve using the formula.

C. Principal, Interest, Rate and ime

Materials:

A box labeled Interest containing cards marked What is known?, What is wanted?, principal, rate, interest,

time, P, , R, I, 100, years, ?, several marked $, and division bars.

Presentation:

Note:Tis should be presented after the distance, time and velocity problems, and after the child understands

fractions.


1. Give an oral introduction stemming from an article read or a savings passbook.

2. Note that the money in the account is called the principle, and the bank uses it to loan to other people.

Tey are using your money, so they pay you a certain amount. Tis is called interest. Tey charge the

borrower interest on the loan they made as well.

3. o make the calculations easier, the bank pays you a certain amount based on every hundred dollars in

the account for a specific period of time. Tis is called the rate.

4. Because the amount is paid for every hundred dollars, it is called apercent(per= for, and cent= hun-

dred).

5. One way to pay interest is every year. Te interest is paid at a certain rate for every year the bank has

your money. Te amount of interest you money earns depends on the number of years it is left in the

account.

6. Te term for which the money is left is called the time.


59/95

58



Passage wo: Solving for Interest

Level One

1. Propose the Problem:

I left $1600 in the bank for 3 years. Each year the bank paid me $2 for every $100. How much did

they pay me?

2. Lay out what is known and wanted as follows:

3. Place 16 hundred-squaresin a square on the rug to represent the principle.

4. Place 3 skittlesbeside them to represent the years.

5. Ask the child how much was paid for each $100 ($2). Place two beads on each of the hundred squares.

6. Remind the child that $2 per $100 was paid for each year, but the money was in the bank for three

years. State that what is laid out represents one years earnings.

7. Collect the beads from the hundred squares, exchanging as necessary. State that this is what you got in

one year, but the money was there for three years. Place the 32 beads under one skittle.

8. As you place 32 beads in front of each of the other skittles, note that now you are putting out the total

interest earnings.

9. Collect the beads and bars together and count them for the answer.

10. Replace the ?card with a ticket reading $96.



60/95

59



Level wo

1. Ask the child what you did. Note that you took the principal times the rate. Lay it out:

2. Tis told us how much I earned per year. Now we have to multiply by the number of years. Lay it out:

3. Do the arithmetic for an answer of $96.

Level Tree1. Work out the formula, noting that you were solving for interest, and place out:

2. Ask the child what you did (took the principal times the rate then times time). Lay it out, noting that

you dont write multiplication symbols between the letter in a formula:

Solve similar problems on paper using the formula.


61/95

60



Passage Tree: Solving for Rate

Level One


I left $1600 in the bank for three years. Te total interest I earned was $96. What was the rate of inter-est?

2. Lay out the cards, talking through what is known and wanted:

3. Lay 16 hundred-squaresinto a large square. Set three skittlesbeside them:

4. Bring out 96 beads. State that for the rate, you must find out how much was earned each year. Distrib-

ute the beads evenly to each (32).

5. ake the beads from one skittle, exchange them, and distribute them one at a time to each of the 16

hundred squares. When finished, note that the rate is 2 for each 100, and that this can also be read as

2%. Replace the ? card with a ticket reading 2.



62/95

61



Passage Tree


63/95

62



Level wo

1. Ask the child what you did. You found how much you got for each year by dividing $96 by 3 years.

2. Ten, you divided that by the amount of money (96/3 1600).

3. Lets do the arithmetic. 96 divided by 3 is 32. Now 32 divided by 1600; lets make this a fraction.

4. Set it up as a fraction and reduce terms to something over 100.

32/1600 = 2/100

Level Tree

1. State that you needed to find the rate, and place:

2. Ask the child what you did in the problem (first, I t, then I/t p). Lay this out in cards:

3. Note that in formulas, you dont use division signs. You can invert the divisor and multiply:

4. Ten, multiply the fractions to get:


64/95

63




Level One


I received $96 for money I left in the bank for 3 years at a rate of $2 per hundred. How much did I putin the bank?


3. First, lets figure out how much we got each year. ake out 96 beadsand distribute them to 3 skittles.

Note that each skittle got 32 beads.

4. Saying We only need to worry about one years interest to find the principal. Put two skittles and their

beads away.

5. Te rate is 2 per hundred or 2%.Lets see how many groups of two we can make. Make 16 pairs of

beads, exchanging as necessary.

6. Match the pairs of beads to hundred squares, since each two beads represents a hundred deposited.

7. Count the hundred squares to get the answer (1600); Replace the ? card with a ticket reading 1600.



65/95

64



Level wo

1. Ask the child what you did. Note that you took the interest ($96) and divided it by the number of years

(3) to discover how much interest was earned each year. Lay this out in tickets:

2. Note that you took the yearly interest and divided by the rate (2/100) to learn the principal. Lay this out

in tickets:

Do the arithmetic, invert and multiply, then divide for the principal (1600):


66/95

65



Level Tree

1. Lets work out a formula for solving for principal. Lay out the cards and signs, continue, o get the

principal, we first divided the interest by time, then we divided by the rate.

2. In a formula, we cant have a division sign, so well make the rate a fraction, invert it and multiply.:

3. Multiply and alphabetize the terms to get the following. Use this formula to solve other problems for

principal.


67/95

66



Passage Five: Solving for ime

Level One


I left $1600 in the bank at a rate of $2 per hundred for 3 years, and the bank paid me $96. How longwas my money in the bank?


3. Lay out 16 hundred-squaresin a square to represent the principal. Noting that it is the interest - place

out 96 beads.

4. State that for each $100 in the bank, you were paid a certain amount. Share out the interest beads

until all are out. Tere should be 6 on each hundred square.

5. We know that each $100 received $2 per year. Put out a skittle, lay 2 beads beside it.

6. Continue with a second then a third skittle, noting that each represents a year of time.

7. Determine that there are 3 skittles representing 3 years. Replace the ? card with a ticket reading 3.



68/95

67



Level wo

1. Ask the child what you did. Determine that you divided 96 by 1600. Lay this out in tickets:

2. Note that you got an answer, then divided that by the rate (2/100 annually). Lay out:

3. Lets invert and multiply to find out how long the money was in the bank.


69/95

68



Level Tree

1. We were solving for time. Lay out:

2. We divided the interest by the principal, then by the rate. Lay out:

3. Remove the by inverting and multiplying:

4. Multiply and alphabetize for:

5. Have the child use this formula to work out some problems on paper.


70/95

69



V Ratio & Proportion

Contents

A. Ratio 66



Passage wo: Introduction to the Language p.66Passage Tree p.68

Passage Four p.68

Passage Five: Exploring the Idea Arithmetically p.68

Passage Tree:

Passage Four:

Passage Six: Ratios Written as Fractions p.70

Passage Seven: Stating the Ratio Algebraically p.71

Passage Eight: Word Problems p.72

Example A

Example B


Example C

Example C Arithmetically


B. Proportion 77


Exercise One: Determining if Something is in Proportion p.78Exercise wo: Proportion Between Geometric Figures p.78

Exercise Tree: With 3 Dimensional Figures p.79

Exercise Four: p.80

C. Calculations with Proportion 81

Exercise One: Arithmetically p.81

Exercise wo: Algebraically (for older children) p.81

Exercise Tree: Applications p.82

Example I

Example II


71/95

70



A. Ratio

Introduction:

Ratios compare objects. Tis comparison is one of division. Because of this, it is vital that the divisor and divi-

dend be identified. For example, the ratio of length to width of a 3x5-card is 5 to 3, while its length to width ratio

is3 to 5. One yields an answer greater than one, the other, an answer less than one.

Te child will have worked with ratios before this, comparing the unknown length of an object to a fixed length,

and investigating the relationship of pi to the radius and circumference of a circle.

Materials:

Te geography stamps, various objects from the environment, the peg board and pegs, paper and pencils



1. Place 2 green pegsacross from 3 red pegson the pegboard (bead bars may also be employed here).

2. We have some pegs here. Lets compare them. One way I can do this is to say that there are 2 pegs here

to 3 pegs there.

3. Tere is another way of saying this. We could say the ratio of green pegs to red pegs is 2 to 3.

4. Using the geography stamps, make illustrations of ratios. For example, a ratio of corn to wheat of 5 to 3.

5. Show the child how to write the ratio as such: 5 : 3. Te child may wish to find and express ratios ofobjects in the environment.

Passage wo: Introduction to the Language

1. After the child has had some experience, point out that the order in which the objects are stated is im-

portant.

2. Introduce the term antecedent(ante-meaning before) to describe the first term in the ratio.

3. Also introduce consequenceto describe the second term in the ratio.

4. Note that the antecedent is the number to which the consequence is compared. Ratios are always stated

antecedent to consequence.

5. Write 2 : 3. State that the green pegs are two thirds of the red ones, and this is what is meant when say-

ing that the antecedent is compared to the consequence.

6. Switch the terms, and read the ratios of red to green as 3 : 2. Note that the red pegs are 1 times the

green ones.

length = 5 inches

width = 3 inches


72/95

71




73/95

72



Passage Tree

1. Lets look at some other ratios. Place two red pegsacross from three green pegs.

2. State that you are going to do something else. Add two rows of each below the first.

3. Have the child examine the second group. Lead him to see that the ratio is still 2 : 3 even though there

are more beads.

4. Have the child make another ratio using the whole quantity of pegs (6 : 9).

5. Te child may notice that these are multiples and can be reduced to the first quantities.6. Ask how the child would figure out if 23 : 36 had the same ratio as the ones youd been working with.

Lead the Note that if you multiply or divide both terms by the same quantity, the ratio will remain the

same.

Passage Four

1. Lets say that we have two numbers in a ratio of 2 : 4, and the smaller of the terms is 8, how can we find

out what the other term is? Write tickets for 2 and 4.

2. Have the child lay out 2 red pegsto 4 green pegs.

3. We know the smaller number will be eight, which of these will be the smaller number (the red)?

4. Lay out red pegs in groups of 2 beneath the first, as the child lays out corresponding groups of four green

pegs.

5. Note that each line of the red pegs contains 2 while each line of green is four. Stop laying out pegs when

8 red is reached.

6. Count the pegs in each group, noting that 8: 16is the same ratio as 2: 4.

7. Work through other examples, giving the ratio and one term and having the child find the second.

8. Later, the children may pose problems to each other.

Passage Five: Exploring the Idea Arithmetically

1. Lay out 2 groups of pegs in the ratio 2 : 4 (perhaps 8 : 16).

2. o reach 8 what did I do? Lead the child to understand the smaller number was multiplied by some-

thing.

3. What number did we multiply 2 by to get 8 (4)?

4. o keep the same ratio, we have to multiply the other side by 4 as well. Write

down the work youve done so far:

5. Repeat with other examples, working them out on the board and recording them.

2 : 4

2x4 : 4x4

8 : 16


74/95

73



Passage Tree:

Passage Four:


75/95


76/95

75



Passage Seven: Stating the Ratio Algebraically

Note:Present this lesson after the child has been introduced to

algebra.

1. Propose the same problem: two numbers are in a ratio of

2 : 3and the smaller number is 6.

2. Set the problem up and solve it as follows:

3. Check as follows:

4. What if the ratio was 3 : 2? Set up 9 : 6 pegson the board.

5. Ask the child what this ratio would be in a

fraction (9/6). Ask what the 9 is compared to

(6).

6. If we were given this ratio, and we knew only

that the larger number is 9, we could set up

the problem in the same way. Set it up:

7. Work it out algebraically:


77/95

76



Passage Eight: Word Problems

Example A


A man left some money when he died. It was to be distributed to his wife and only son

in a ratio of 2:3. Te wife received $2400, how much did the son get?

2. Te child can work it out with pegs and calculate:

3. Or she may attempt it algebraically:

4. Continue with other problems. Work up to more complicated ones.


78/95

77



Example B


Im giving you 2 numbers in a ratio of 4 : 7. Te sum of the numbers is 55. What are the numbers?

2. Lay out the ratio with pegs. Count them up to get 11, and place a ticket stating this to the right of thepegs.

3. Continue with a second row and a ticket reading 22.

4. Repeat until there are five rows which total fifty-five.

5. Count each group of pegs to

arrive at 20 and 35.

6. Lets analyze arithmetically

what we did. Record the

ratio as a fraction:

7. Note that you added 4 and

7 to get 11, and each time

you placed a row, you used

up 11 pegs. You continueduntil you reached fifty-five

(5 rows).

8. Is there another way we

could have gotten to the

answer? Lead the child to

see that you could divide 55

by 11 to get 5 and then multiply the terms

of the fraction by 5 to get the answer:

Repeat with other problems. Tink them

through arithmetically, then check with the

peg board.

11

22

33

44

55


79/95

78




1. State that if the ratio is 4 : 7, we know that 4/7thsof the larger number is the smaller number.

2. Record:

3. We also know thaty + x = 55. Work the problem out as follows:


80/95

79



Example C

1. Propose the problem below and lay it out in pegs:

Here are 2 numbers in a ratio of 3 : 5. If I say the difference of the two terms is 10, what are the

terms?

2. Determine that the difference between 3 and 5 is 2. Continue to 6 : 10, noting that the difference is 4,

closer, but not quite what were looking for.

3. Continue until 15 : 25is laid out. Note that the difference between them is 10, and these are the num-

bers you are looking for.

4. Repeat with other problems.


81/95

80



Example C Arithmetically

1. Discuss what was done arithmetically as you write the following:

2. Note that the difference between the ratio was 2, while the difference between the terms was 10. o dis-

cover what to multiply by, you divide 10 by 2.


1. Set the problem up and work it out

as follows:

2. o solve: (one way to do it)


82/95

81



B. Proportion

Introduction:

Proportion is a statement in which 2 ratios are equal. If there are more than 2 ratios, the proportion is continued.

Te children have worked with both ratios and proportion; now it is brought to their consciousness.

Prerequisite:

Children should have studied ratioand be able to balance equations.

Materials:Objects of the environment.


1. Have children make a ratio with the geography stamps e.g. 2 black sheep to 1 white sheep.

2. Ask children to do another which is in the same ratioe.g. 4 black sheep to 2 white.

3. Note that writing these ratios as fractions, we can say they are equivalent to each other; ask why.

4. alk through why pointing out when 2 ratios are equal this statement is said to be proportion and the

numbers are said to be proportional or in proportion.

5. Ask what this means; note that it means the relationship is the same. Point out how this worked with the

sheep - for every 2 black sheep there is one white sheep.

6. Children can investigate other proportions.

State, Lets write this.


83/95


84/95

83



Exercise Tree: With 3 Dimensional Figures

1. Lay out the pieces of the power of 2 cube; state that you want to find which ones are in proportion.

2. Note that the small yellow cubes are because they are the same; note how many dimensions it has(length, width and height).

3. Compare the cube to the next larger cube; write out the ratio of the length, width and height:

L = 1:2

W = 1:2

H = 1:1

4. Note that the height is not the same ratio as the length and width so it is not in proportion.

5. Continue comparing the other pieces; build the whole cube and ask if it is in proportion to anything;

note that all the cubes are proportional.

Excercise wo


85/95

84



Exercise Four:

Materials:Peg board and pegs, white decimal cards 0 to 9.

1. State that you have 2 ratios; 2:3 and 4:6; lay out the cards for these at the top of the peg board.

2. Note that you have 5 pegs in a ratio of 2 to 3; place these under the 2:3 number cards in ratio.

3. State that you have 10 pegs in a ratio of 4 to 6; lay these under the 4 to 6 number cards in that ratio.4. Ask if these are proportional; determine that they are.

5. Ask if they can express both groups as the same ratio; show this by changing the 10 pegs laid out in a line

of 4 and 6 to 2 lines of 2 and 3:

6. Note that if all the pegs look like theyre in the same ratio, theyre proportional.


86/95

85



C. Calculations with Proportion

Introduction:Tis section works with proportion arithmetically and algebraically.

Exercise One: Arithmetically

1. State when you say 1/2and 3/6are in proportion, what it means is that 1/2 is equal to 3/6.2. Record this; ask if it is true and if they can prove it to you.

3. Note that you can also determine if they are in proportion if the fractions are divided out and their quo-

tients equal:

1 2 = 0.5 3 6 = 0.5 0.5 = 0.5

4. State that you want to look at these 2 numbers in a different way; to do so youll multiply the first side

by 2.

5. Note if you do one side you have to do the other:

1/2 x 2/1 = 3/6 x 2/1 Work out:

1x2 = 3x2

2x1 6x1

6. State that you want to look at something; point out the 3x2/6x1, have the children look at the original

problem; note where the numbers are (diagonal).

7. Note that the opposite numerators and denominators are multiplied.

8. Ask why we care if this happens; state that suppose we wrote this down: 6/9 = x/129. Note that to solve for x we can cross multiply: work out:

9x = 72, x = 8

10. Show how to check this: 6/9 = 8/12 6 x 12 = 9 x 8

72=72 ck

11. Note that because this is a cross pattern, its called cross multiplication.

Exercise wo: Algebraically (for older children)

1. State that the pattern of cross multiplication works for those numbers, but what about others.

2. Set up ratios: a/c = c/d

3. Multiply this out: a/b x b = c/d x b, a = bc/d, multiply each side by d to get: ad = bc

4. Note that you have the same pattern.


87/95

86



Exercise Tree: Applications

Example I

1. Give the problem: Teres a painting 4 feet by 6 feet high. Tis painting appears in a photograph 2

inches by 3 inches. Te eye in the picture is 5 mm long; how long is the eye in the painting?

2. Set up the ratio; one of the width of the picture to the width of the painting (2/48) as equal to the widthof the eye in the picture to the width of the eye in the painting (5/x):

2/48 = 5/x (48 inches = 4 feet)

3. Work out as follows: 2x = 240, x = 120

4. Children can work on their own to make scale drawings

Example II

1. Give the problem: Tese is a recipe which calls for 9 teaspoons of pineapple juice mixed with 6 tea-

spoons of cranberry. Itll taste like a drink made with 12 cups of pineapple and 8 cups of cranberry. Is

this true?

2. Determine that it is by cross multiplying and arriving at 72 for each.

3. Continue that you only have 10 cups of pineapple juice; how much cranberry juice will be needed for

the same taste?

4. Set up as: 9/6 = 10/x, 60 = 9x, x = 6 2/3 c.

5. Suppose you only needed 10 cups? Set up and work out:

x/y = 3/2 x + y = 10

3/2y + y = 10

3y + 2y = 20

5y = 20

y = 4 x = 6


88/95

87



VI Algebra

Contents

A. Introduction to Algebra 84

Exercise One: Balancing an Equation p.84

Exercise wo: Balancing an Equation When Something is aken Away p.85Exercise Tree: Balancing an Equation When Something is Multiplied p.85

Exercise Four: Balancing an Equation When it is Divided p.85

B. Operations With Equations 86

Exercise One: Addition p.86

Exercise wo: Subtraction p.86

Exercise Tree: Multiplication p.86

Exercise Four: Division p.86

C. Algebraic Word Problems 87

Example I: p.87

Example II: p.87

Example III: p.87

Example IV: p.88

Example V: p.88

Example VI: p.88

Example VI: p.89

Example IX: p.89

Example X: p.90Example XI: p.90

Example XII: p.90


89/95


90/95

89



Exercise wo: Balancing an Equation When Something is aken Away

1. Lay out the original equation, 5 + 3 = 8; state lets look at this again; note that it is equal and is an equation.

2. Have the children watch, take the +3 away; note that it says 5 = 8, ask if this is true.

3. Ask what you could do to balance it noting you could take away from the other side as well.

4. Do the same on the right by writing a ticket for 3 and placing it with a minus sign to the right of the 8.

= - 3

5. State, Lets do it; replace the 8 bar with a 5 bar; note that it is again true, the equation is balanced.

=

6. Children can practice with this; have pick an equation, so something to it to change it then restore it.

Exercise Tree: Balancing an Equation When Something is Multiplied

1. Note that there are other things that can happen to change the equation; can also multiply,

2. Set up the equation in the same way as the original adding a multiplication sign and ticket with a 2 on it after

the 5 plus 3 which can be put in brackets:

( + ) x 2 =

3. Ask if it is an equation (no); ask what can be done to balance it; child knows to do the same to the quantity.4. Lay out a x2 in tickets after the 8 bar; have look again to see if it is really true; note it is equivalent, the equa-

tion is balanced.

5. Let children each have a turn and practice in the same way as the other exercises.

Exercise Four: Balancing an Equation When it is Divided

1. Ask what else can be done (divide it); do so by placing a division bar under the 5+3 bead bars and a 2 under it.

2. Note these are not equal any more; ask what to do; children should know to divide the 8 bar in the same way.

3. Check to see if it is balanced.

4. wo things should come to the childrens awareness:

1. what is to the left of the equal sign must be the same value as what is to the right of the equal

sign to be an equation

2. if some operation is performed on one side to change it, it can be read as true by changing the

other side in the same way.

+ /2 = ( /2)

math 2 album

Documents