##### Document Text Contents

Page 1

Copyright 2009 | Jonathan Feagle

www.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

Page 2

i

Copyright 2009 | Jonathan Feagle

www.FreeMontessori.org

Contents

I Powers of Numbers 1

A. Powers of 2 2

Presentation:

Passage One: Introduction p.2

Extension I: Terminology

Extension II: Exploration with Bases Other than Two

Passage Two: Di�erent Unit Size p.6

Passage �ree: Hierarchical material p.8

B. Exponential Notation 10

Presentation:

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

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

II Negative Numbers 15

A. Addition Using Negative Numbers 16

Presentation:

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

Passage Two: Writing p.18

Passage �ree: Introduction to the Ten 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:

III Non-Decimal Bases 27

A. 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

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Copyright 2009 | Jonathan Feagle

www.FreeMontessori.org

Part D: Changing bases using the base chart

Example I:

1. Propose the problem: 1432five = ____ten.

2. “Let’s do this one using the rule. It’s in base five right now, let’s change it to base ten.”

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

viding by ten. This chart will tell us what number ten is in base five.” Find the number you’re converting

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

4. Divide 1432five by 20five, 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 (10

0)

44five ÷ 20five = 2 r 4 (10

1)

2five ÷ 20five = 0 r 2 (10

2)

Example II:

1. Propose the problem: 1424five = ____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

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Copyright 2009 | Jonathan Feagle

www.FreeMontessori.org

Example III:

1. Propose the problem: 1424five = ____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

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Copyright 2009 | Jonathan Feagle

www.FreeMontessori.org

Example VI:

1. Give a similar problem: “�e same man and boy work together for 2 days then the man leaves the

boy to do the remaining part. How long does it take?”

2. Talk through that the boy does one section a day and the man does 3 so 4 are completed in a day.

3. Note that they work together for 2 days so 8 sections are completed; subtract this from how many all

together to find 4 more need to be done.

4. Because there are 4 more to be done and the boy can only do one section a day, it will take 4 more days

to complete.

Example IX:

1. State the problem: “A garden is divided into 12 parts. A boy and a man dig it and the man works

twice as fast as the boy. If each works for 2 days, the work is �nished. See how many each digs in a

day.”

2. Write out the know information, let:

b = # of parts boy digs

m = # of parts man digs

m = 2b

3. Note that the man digs twice as much as the boy and they dig 12 sections in all; set up and work out:

2m + 2b = 12

2(2b) + 2b = 12

6b = 12

b = 2 parts m = 4 parts

4. Plug in to check:

(2 x 4) = 2(2) = 12

8 + 4 = 12

12 = 12ck.

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Copyright 2009 | Jonathan Feagle

www.FreeMontessori.org

Example X:

1. Relate the problem: “�ere is a water tank and it’s �lled by 2 pipes from the top (a + b). When it is

needed, there is a drainage pipe c. If the tank starts o� empty, if “a” opens, it takes 8 hours to �ll,

“b” takes 12 hours to �ll, “c” can drain the tank in 6 hours. If a,b, and c are open, how long does it

take to �ll?”

2. Set the problem up so that x = the number of hours to fill the tank; set up and work out as follows:

x/8 + x/12 - x/6 = 1

3x/24 + 2x/24 - 4x/24 = 1

x/24 = 1

3. Determine that x = 24 hours to fill the tank.

Example XI:

1. Give the problem: “A laborer does a job in 15 days, another does it in 10 days. How long does it

take if they work together?”

2. Figure as follows:

x/15 + x/10 = 1 5x/30 = 1

x = 6 days

2x/30 + 3x/30 = 1 5x = 30

Example XII:

1. Give this problem: “A women bought some prized live turkeys and ducks. She spent $31 and each

turkey cost $5, each duck $2. How many of each did she buy?”

2. Let t = # of turkeys, d = # of ducks; set up so that 5t + 2d = 31.

3. Try different values of t to find three possible solutions:

t = 1 t = 3 t = 5

5 + 2d = 31 15 +2d = 31 25 + 2d = 31

2d = 26 2d = 16 2d = 6

d = 13 d = 8 d = 3

4. Note the three possibilities:

1 turkey, 13 ducks

3 turkeys, 8 ducks

5 turkeys, 3 ducks

Copyright 2009 | Jonathan Feagle

www.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

Page 2

i

Copyright 2009 | Jonathan Feagle

www.FreeMontessori.org

Contents

I Powers of Numbers 1

A. Powers of 2 2

Presentation:

Passage One: Introduction p.2

Extension I: Terminology

Extension II: Exploration with Bases Other than Two

Passage Two: Di�erent Unit Size p.6

Passage �ree: Hierarchical material p.8

B. Exponential Notation 10

Presentation:

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

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

II Negative Numbers 15

A. Addition Using Negative Numbers 16

Presentation:

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

Passage Two: Writing p.18

Passage �ree: Introduction to the Ten 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:

III Non-Decimal Bases 27

A. 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

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Copyright 2009 | Jonathan Feagle

www.FreeMontessori.org

Part D: Changing bases using the base chart

Example I:

1. Propose the problem: 1432five = ____ten.

2. “Let’s do this one using the rule. It’s in base five right now, let’s change it to base ten.”

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

viding by ten. This chart will tell us what number ten is in base five.” Find the number you’re converting

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

4. Divide 1432five by 20five, 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 (10

0)

44five ÷ 20five = 2 r 4 (10

1)

2five ÷ 20five = 0 r 2 (10

2)

Example II:

1. Propose the problem: 1424five = ____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

Page 48

47

Copyright 2009 | Jonathan Feagle

www.FreeMontessori.org

Example III:

1. Propose the problem: 1424five = ____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

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Copyright 2009 | Jonathan Feagle

www.FreeMontessori.org

Example VI:

1. Give a similar problem: “�e same man and boy work together for 2 days then the man leaves the

boy to do the remaining part. How long does it take?”

2. Talk through that the boy does one section a day and the man does 3 so 4 are completed in a day.

3. Note that they work together for 2 days so 8 sections are completed; subtract this from how many all

together to find 4 more need to be done.

4. Because there are 4 more to be done and the boy can only do one section a day, it will take 4 more days

to complete.

Example IX:

1. State the problem: “A garden is divided into 12 parts. A boy and a man dig it and the man works

twice as fast as the boy. If each works for 2 days, the work is �nished. See how many each digs in a

day.”

2. Write out the know information, let:

b = # of parts boy digs

m = # of parts man digs

m = 2b

3. Note that the man digs twice as much as the boy and they dig 12 sections in all; set up and work out:

2m + 2b = 12

2(2b) + 2b = 12

6b = 12

b = 2 parts m = 4 parts

4. Plug in to check:

(2 x 4) = 2(2) = 12

8 + 4 = 12

12 = 12ck.

Page 95

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Copyright 2009 | Jonathan Feagle

www.FreeMontessori.org

Example X:

1. Relate the problem: “�ere is a water tank and it’s �lled by 2 pipes from the top (a + b). When it is

needed, there is a drainage pipe c. If the tank starts o� empty, if “a” opens, it takes 8 hours to �ll,

“b” takes 12 hours to �ll, “c” can drain the tank in 6 hours. If a,b, and c are open, how long does it

take to �ll?”

2. Set the problem up so that x = the number of hours to fill the tank; set up and work out as follows:

x/8 + x/12 - x/6 = 1

3x/24 + 2x/24 - 4x/24 = 1

x/24 = 1

3. Determine that x = 24 hours to fill the tank.

Example XI:

1. Give the problem: “A laborer does a job in 15 days, another does it in 10 days. How long does it

take if they work together?”

2. Figure as follows:

x/15 + x/10 = 1 5x/30 = 1

x = 6 days

2x/30 + 3x/30 = 1 5x = 30

Example XII:

1. Give this problem: “A women bought some prized live turkeys and ducks. She spent $31 and each

turkey cost $5, each duck $2. How many of each did she buy?”

2. Let t = # of turkeys, d = # of ducks; set up so that 5t + 2d = 31.

3. Try different values of t to find three possible solutions:

t = 1 t = 3 t = 5

5 + 2d = 31 15 +2d = 31 25 + 2d = 31

2d = 26 2d = 16 2d = 6

d = 13 d = 8 d = 3

4. Note the three possibilities:

1 turkey, 13 ducks

3 turkeys, 8 ducks

5 turkeys, 3 ducks