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Presented to the 2012 Annual Conference American Mathematical Association of Two-

Year Colleges Jacksonville, Florida November 8, 2012

By Joyce Lindstrom, Ed.D.

Third-world economy

No natural resources

Large population for

space available

Shipping –competes with

Rotterdam as busiest port in the

world

Education – develop the best

education system in the world so

the people could become the

natural resource

Preserve ethnic identities

While I am happy that

Singapore has been able to

develop a first-world

economy so quickly, what

does that have to do with

teaching math in the US?

Fourth grade Eighth grade

Country

Average

score Country

Average

score

International average 495 International average 466

Singapore 594 Singapore 605

Hong Kong SAR1,2 575 Korea, Republic of 589

Japan 565 Hong Kong SAR1,2 586

Chinese Taipei 564 Chinese Taipei 585

Belgium-Flemish 551 Japan 570

Netherlands2 540 Belgium-Flemish 537

Latvia 536 Netherlands1 536

Lithuania3 534 Estonia 531

Russian Federation 532 Hungary 529

England2 531 Malaysia 508

Hungary 529 Latvia 508

United States2 518 Russian Federation 508

Cyprus 510 Slovak Republic 508

Moldova, Republic of 504 Australia 505

Italy 503 United States 504

Australia2 499 Lithuania3 502

New Zealand 493 Sweden 499

Scotland2 490 Scotland1 498

Slovenia 479 Israel 496

Armenia 456 New Zealand 494

Norway 451 Slovenia 493

1) Curriculum Design based on

research

Jerome Bruner from Poland

Richard Skemp from England

Zoltan Dienes from Denmark

To instruct someone... is not a matter of getting him to commit results to mind. Rather, it is to teach him to participate in the process that makes possible the establishment of knowledge. We teach a subject not to produce little living libraries on that subject, but rather to get a student to think mathematically for himself... Knowing is a process not a product.

(1966: 72)

Concrete

Pictorial

Abstract

To understand something is to relate

it to what is already known…

Without the prerequisite knowledge…

the result is rote learning, or no

learning at all.

SAIL: Structured Activities in Intelligent Learning:

SAIL Design Strengths

Conceptual understanding from Kindergarten

Algebra in the 8th grade

Geometry with proofs in the 9th grade

2) Professional Development of

Teachers

100 hours every year

Events provided by or funded by

the Ministry of Education

Lesson Studies

Shared teacher-created resources

Missouri law requires individuals with Initial Professional Certificates (IPC) to complete 30-contact hours of professional development during the first four years of teaching.

Individuals with Career Continuous Professional Certificates (CCPC) must complete 15-contact hours each year.

Missouri Department of Elementary and Secondary Web Site 2011

Number bonds

Place Value

Bar modeling

Mental math

Primary Math 1a, page 18

Copyright Marshall Cavendish 2003

Used by permission of SingaporeMath.com

Primary Math 2b, page 14



Primary Math 2b, page 15



Michael and Joris baked 64 chocolate chip

cookies. The ratio of the number of cookies

Joris baked to the number of cookies Michael

baked was 5:3. How many fewer cookies did

Michael bake than Joris?

Mathematics PSLE Revision Guide, 2nd Edition, Michelle Choo

Copyright 2009 p. 181

Michael and Joris baked 64 chocolate chip cookies. The ratio of the number of cookies Joris baked to the number of cookies Michael baked was 5:3. How many fewer cookies did Michael bake than Joris?

Michael’s cookies

Joris’ cookies

? 8 units = 64

1 unit = 8

2 units = 16; Michael had 16 cookies more than Joris

Of the people in attendance at a recent

ball game, one-third had grandstand

tickets, one-fourth had bleacher tickets,

and the remaining 11,250 people in

attendance had other kinds of tickets.

What was the total number of people in

attendance at the game?

Richard Bisk Copyright 2007 Used by permission

Of the people in attendance at a recent ball game, one-third had

grandstand tickets, one-fourth had bleacher tickets, and the

remaining 11,250 people in attendance had other kinds of tickets.

What was the total number of people in attendance at the game?

To solve by bar modeling:

1/3 1/4

1/3 1/4

Grandstand

tickets

11,250

tickets

Bleacher

Tickets

Bleacher

tickets

Grandstand

tickets

11,250

tickets

? 5 units = 11,250

1 unit = 11,250/5 = 2,250

12 units = 27,000

27,000 people attended the game

In 1983, prior to implementing this

curriculum, Singapore ranked 17 out of 26

countries tested in eighth grade

Mathematics.

Just twelve years later, in 1995, Singapore

ranked number one out of 41 countries

tested at that level and remained number

one in both 1999 and 2003.

Dr. Richard Bisk Copyright 2007 Used by Permission

In 1960 of the 30,615 students who sat for

the Primary School Leaving Exam, 45%

passed.

In 2010, of the 45,049 students who sat for

the Primary School Leaving Exam, 97.3%

assessed as suitable to proceed to secondary

school.

Because kids are visual, model drawing helps

students “see” relationships

Model drawing makes more complex word

problems possible

Number bonds are essential in early grades

Most important training is on-the-job

New/experienced teachers observe each

other’s classrooms

Ministry Of Education supports education

with professional development and classroom

resources

752.3

748.3

750.2

759.3

761.2

771

735

740

745

750

755

760

765

770

775

2006 2007 2008 2009 2010 2011

M

A

P

I

N

D

E

X

YEAR

3RD GRADE MATHEMATICS 6 YEAR TREND SINGAPORE MATH ANALYSIS

State

Wentzville

Timeline of implementation: 2008 = 160 students participated in an after school program 2009 = majority of 1st, 2nd and 3rd teachers were using Singapore Math in the classroom.

We saw higher student

achievement out of the

classrooms that used Singapore

Math. All grade levels improved…

We are very pleased with the

results and we hope to continue

to see the scores improve. Gregg Klinginsmith, Math Curriculum Coordinator, personal email, 8/15/2010

Exercise with number cards 1-10

6÷1/2 = ? And why?

Compute mentally: 12 x 97

John spent 2/7 of his money on a storybook.

The storybook cost $16. How much money

did he have at first?

Let’s try 6÷2. How many 2’s are there in 6?

There are three 2’s in 6, so 6÷2 = 3

What about 6÷3? How many 3’s are there in 6?

There are two 3’s in 6, so 6÷3 = 2

2 2 2

3 3

How many (1/2)’s are there in 6?

There are twelve (1/2)’s in 6, so 6÷(1/2) = 12

1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2

John spent 2/7 of his money on a storybook. The storybook cost $16. How much money did he have at first?

?

John’s money

$16

2 units = $16

1 unit = $8

7 units = $56

John had $56 at first.

Keane bought some marbles and gave half of them to Leon. Leon bought some stamps and gave half of them to Keane.

Keane used 5 stamps and Leon gave away 11 marbles. The ratio of the number of stamps to the number of marbles Keane had left then became 1 : 7 and the ratio of the number of stamps to the number of marbles Leon had left became 1 : 5.

How many stamps did Leon buy?

Kentucky Town and Merrily Town are 234 km apart.

Dawn left Kentucky Town for Merrily Town at 8.42

a.m. travelling at an average speed of 85 km/h. At

the same time, Jennifer left Merrily Town for

Kentucky Town. They met each other at 10.30 a.m.

(a) What was Jennifer’s average speed when she

met Dawn?

(b) If Jennifer were to increase her speed by 26

km/h before meeting Dawn, how much lesser time

would she take before meeting her?

1. Ali has $8 more than Sid. Trina has

$6 less than Ali. The three of them

have $76 in all. Find the amount of

money each of them has. Yeap Ban Har, Bar Modeling, A Problem-Solving Tool, p. 37

2. Chris started saving some money

on Monday. Each day she saved $2

more than the day before. By

Friday of the same week Chris had

saved $35. Find the amount Chris

saved on Wednesday. Yeap Ban Har, Bar Modeling, A Problem-Solving Tool, p. 40

3. There were 3 times as many

girls as boys on a bus. There were

twice as many children as adults.

There were 36 persons altogether

on the bus. How many girls were

on the bus? Yeap Ban Har, Bar Modeling, A Problem-Solving Tool, p. 56

4. Kumar took 5 days to read a book. He read 1/9

of the book on Saturday, 1/4 of the remainder on

Sunday, and the remaining 48 pages during the last

three days. How many pages were there in the

book? Cynthia Seto, Teaching Fraction, Ratio and Percentage Effectively,

p. 49

5. The ratio of the number of boys to the number of

girls in a school hall was 5:7. After ½ of the boys

and 63 girls left the hall, there was an equal

number of boys and girls remaining in the hall. How

many girls were there in the hall at first? Cynthia Seto, Teaching Fraction, Ratio and Percentage Effectively, p.

80

6. There were 126 children at a

concert. The number of boys was ¾

as many as the number of girls. When

an equal number of boys and girls left

the concert, the number of boys and

girls remaining at the concert was 5:7.

How many boys left the concert? Cynthia Seto, Teaching Fraction, Ratio and Percentage Effectively,

p. 79

9. The number of balls in Box A is ½

of the number of balls in Box B. 10%

of the balls in Box A and 10% of the

balls in Box B were moved to Box C.

As a result, the number of balls in

Box C increased by 20%. There are 72

balls in Box C now. How many balls

were there in Box B at first? [RGPS 2010 SA1 Q16]

10. Dolly had 80 more stickers than

Jenny. Dolly gave 25% of her

stickers to Jenny.

Jenny in return gave 60% of her

stickers to Dolly. In the end, Dolly

had 100 stickers more than Jenny.

How many stickers did Dolly have

at first?

[NHPS 2010 SA1 Q18]

11. The tickets for a show were priced at

$10 and $5. The number of ten-dollar

tickets available is 1½ times the number

of five-dollar tickets. 5 out of 6 ten-

dollar tickets and all the five-dollar

tickets were sold. The amount of money

collected from the sale on the tickets

was $5600. How much more would have

been collected if all the tickets were

sold? [Ai Tong 2010 CA1 Q10]

12. How many gallons of 50%

antifreeze must be mixed with 80

gallons of 20% antifreeze to obtain a

mixture that is 40% antifreeze? Lial, Hornsby, McGinnis Beginning and Intermediate Algebra, 4th

Edition, p. 158

13. Airplanes usually fly faster from west to

east than from east to west because the

prevailing winds go from west to east. The

air distance between Chicago and London is

about 4000 miles, while the air distance

between New York and London is about 3500

miles. If a jet can fly eastbound from

Chicago to London in the same time it can fly

westbound from London to New York in a 35-

mph wind, what is the speed of the plane in

still air? Encyclopaedia Britannica, quoted in Lial, Hornsby, McGinnis Beginning

and Intermediate Algebra, 4th Edition, p. 451

14. Ms. Teng, a high school mathematics

teacher, gave a test on perimeter, area,

and volume to her geometry class. Working

alone, it would take her 4 hours to grade

the tests. Her student teacher, Jonah

Schmidt, would take 6 hours to grade the

same tests. How long would it take them

to grade these tests if they work together? Lial, Hornsby, McGinnis Beginning and Intermediate Algebra, 4th

Edition, p. 452

Keane bought some marbles and gave half of them to Leon. Leon bought

some stamps and gave half of them to Keane.

Keane used 5 stamps and Leon gave away 11 marbles. The ratio of the

number of stamps to the number of marbles Keane had left then became 1 : 7

and the ratio of the number of stamps to the number of marbles Leon had

left became 1 : 5.


Leon’s stamps

Keane’s stamps

Leon’s marbles 11

Keane’s marbles

5






left became 1 : 5.


Leon’s stamps 1 unit

?

Keane’s stamps 1 unit-5

Leon’s marbles 11

Keane’s marbles

5






left became 1 : 5.


Leon’s stamps 1 unit 1 unit

Keane’s stamps 1 unit

-5 5

Leon’s marbles 1 unit 1 unit 1 unit 1 unit 1 unit 11

Keane’s marbles 1 unit 1 unit 1 unit 1 unit 1 unit 1 unit 1 unit

- 5 -5 -5 -5 -5 -5 -5

From the last two lines:

5 units + 11 = 1unit -5 + 1 unit –5 + 1 unit – 5 + 1 unit – 5 + 1 unit – 5 + 1 unit -5 + 1 unit – 5

5 units + 11 = 7 units – 35

11 = 2 units – 35

46 = 2 units, which is the number of stamps Leon bought

Kentucky Town and Merrily Town are 234 km apart. Dawn left Kentucky Town for Merrily Town at 8.42 a.m. travelling at an average speed of 85 km/h. At the same time, Jennifer left Merrily Town for Kentucky Town. They met each other at 10.30 a.m.

(a) What was Jennifer’s average speed when she met Dawn?

(b) If Jennifer were to increase her speed by 26 km/h before meeting Dawn, how much lesser time would she take before meeting her?

Kentucky 234 km Merrily

Town Town

85 km/h ------- ------? km/h

8:42 am 10:30 am 8:42 am

Dawn ----------------------------------------------------------------------Jennifer

8:42 to 9:00 is 18 minutes; 9:00 – 10:30 is 90 minutes, so the two cars travel for 108 minutes before they meet.

108 minutes = 108/60 hours = 9/5 hours

Dawn traveled 85 km/hr x 9/5 hr = 153 km

Jennifer traveled 234 km – 153 km = 81 km

To travel 81 km in 9/5 hr, Jennifer traveled 81/(9/5) = 45 km/hr

Kentucky Town and Merrily Town are 234 km apart. Dawn left Kentucky Town for Merrily Town at 8.42 a.m. travelling at an average speed of 85 km/h. At the same time, Jennifer left Merrily Town for Kentucky Town. They met each other at 10.30 a.m.

(a) What was Jennifer’s average speed when she met Dawn?

(b) If Jennifer were to increase her speed by 26 km/h before meeting Dawn, how much lesser time would she take before meeting her?

Kentucky 234 km Merrily

Town Town

85 km/h ------- ------ 71 km/h

8:42 am ? 8:42 am

Dawn -------------------------------------------------------------------------Jennifer

Jennifer is now traveling 45 + 26 = 71 km/hr, so

Dawn and Jennifer are heading toward each other at 71+85 = 156 km/hr

They will meet in 234 km ÷ 156 km/hr = 3/2 hr = 90 minutes

The time this change would save is 108 minutes – 90 minutes = 18 minutes

t

10t

10t - 1

Divisible by

9?

Divisible

by 11?

10t + 1

Divisible

by 11?

0

1

2

3

4

5

6

Let n = at x 10t +… + a2 x 102 + a1 x 101+ a0 x 100

Complete this table:

t

10t

10t - 1

Divisible

by 9?

Divisible

by 11?

10t + 1

Divisible

by 11?

0 1 0 Y

1 10 9 Y

2 100 99 Y

3 1,000 9,99 Y

4 10,000 9,999 Y

5 100,000 99,999 Y

6 1,000,000 999,999 Y

Let n = at x 10t +… + a2 x 102 + a1 x 101+ a0 x 100


For example: 53,812 can be written as:

(5 x 9999) + (3 x 999) + (8 x 99) + (1x 9) +

Divisible by 9

+ (5 x 1) + (3 x 1) + (8 x 1) + (1 x 1) + (2 x 1)

Divisible by 9?

Conclusion: A number is divisible by 9 if and only if the sum of the digits is a multiple of 9.

t

10t

10t - 1

Divisible

by 9?

Divisible

by 11?

10t + 1

Divisible

by 11?

0 1 0 Y N 1

1 10 9 Y N 11

2 100 99 Y Y 101

3 1,000 9,99 Y N 1,001

4 10,000 9,999 Y Y 10,001

5 100,000 99,999 Y N 100,001

6 1,000,000 999,999 Y Y 1,000,001

Let n = at x 10t +… + a2 x 102 + a1 x 101+ a0 x 100


t

10t

10t - 1

Divisible

by 9?

Divisible

by 11?

10t + 1

Divisible

by 11?

0 1 0 Y N 1 N

1 10 9 Y N 11 Y

2 100 99 Y Y 101 N

3 1,000 9,99 Y N 1,001 Y

4 10,000 9,999 Y Y 10,001 N

5 100,000 99,999 Y N 100,001 Y

6 1,000,000 999,999 Y Y 1,000,001 N

Let n = at x 10t +… + a2 x 102 + a1 x 101+ a0 x 100


Also, 53,812 can be written as:

5(9999) + 3(1001) + 8(99) + 1(11) +

Divisible by 11

+ 5(1) – 3(1) + 8(1) – 1(1) + 2(1)

Divisible by 11?

Conclusion: A number is divisible by 11 if and only if the number formed by (sum of odd-position digits) – (sum of even-position digits) is a multiple of 11.

Presented by

St. Charles Community College

Cottleville, Missouri 63376

[email protected]

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