Anthony B. Wight
Figure 1: Sample Solution using a 5part “multiple solutions” template
Pattern /Sketch
X
XX
XXX
XXXX
XXXXX
XXXXXX
Table of Values
# of Rows Blocks/row Total blocls
1 1 1
2 2 3
3 3 6
4 4 10
5 5 15
6 6 21
7 7 28
8 8 36
9 9 45
10 10 55
Verbal Description
Amir is arranging blocks. He puts one in the top row, 2 in the 2nd, 3 in the 3rd, etc. How many total blocks does he need to build 5 rows?
Equation
Let R = number of Rows
Let T = Total number of blocks
T = R + (R1)R/2
or
T = 1/2 R(R+1)
Graph
(Graph is available in print form)
Figure 2. Multiple Solutions 5part Problem Solving Template
Pattern /Sketch
Table of Values
Verbal Description
Equation
Graph
Figure 3. Singapore Approach Sample #1 Solution:
Ted 1 unit
Jose   
After spending money:
Ted   $20 
Jose   
/ ($50)  /
1 unit = $50  $20 = $30
Ted started with $30
Jose started with 2 units = 2 x $30 = $ 60
Figure 4. Singapore Approach Sample #2 Solution.
After the grandmother's gift:
Sam   
Kim  
Before the gift:
/  $140 >/
Sam    (gift) 
Kim   (gift) 
/$60>/

\/
/ $140 >/
    
/$60>/ /$60>/
$140  $60  $60 = $20
The grandmother gave each of them $20.
Figure 5. A sample algebra problem form www.japaneseonline.com/math
(All problems start with a sketch, but these do not download)
Hose “A” takes 45 minutes to fill the bucket with water. Hose “B” can do the same in 30 minutes.
If you use both hoses, how long will it take to fill the bucket?
You would have 1 minute to finish this problem if you were a Japanese 12 year old.

ANSWERS (Click the letter choices below to check your answer.)
12 minutes 18 minutes 16 minutes 13 minutes 21 minutes
HINT EXPLANATION NEXT QUESTION BACK TO INDEX
Press here to listen to the question.
Figure 6. Diagonals, Lines, Handshakes and Sums table
n Diagonals Lines Regions Handshakes 1st #s sum 1st Odd #s sum Ordered sums of 2 Unordered sums of 2
0





1











1



0

2

0

1

1

0



2



1

4

1

3

4

1

1

3

0

3

7

3

6

9

2

1

4

2

6

11

6

10

16

3

2

5

5

10

16

10

15

25

4

2

6

9

15

22

15

21

36

5

3

7

14

21

29

21

28

49

6

3

8

20

28

37

28

36

64

7

4

9

27

36

46

36

45

81

8

4

10

35

45

56

45

55

100

9

5

.
.
.
n n(n3) /2

n(n1) /2

n(n+1) +1/2

n(n1) /2

n(n+1) /2

(n x n)

(n2)(n1)/2

Figure 7. Addition table and grid for probability problem solving, Example 1.
1 2 3 4 5 6
1 2 3 4 5 6 7
2 3 4 5 6 7 8
3 4 5 6 7 8 9
4 5 6 7 8 9
10
5 6 7 8 9
10 11
6 7 8 9
10 11 12
Figure 8. Addition table and grid for probability problem solving, Example 2.
1 2 3 4 5 6 7
1
2 3 4 5 6
7 8
2
3 4 5 6
7 8 9
3
4 5 6
7 8 9 10
4
5 6
7 8 9 10 11
Figure 9. Graphic illustration for a “political and economic” problem.
(figure is available in print form)
Shaded = 10% of population
Shaded = 60% of Total Income
Endnotes
1. “Fewer noses Stuck In books In America, Survey Finds,” Times [New York] 8 July 2004. Reporting on the analysis by The National Endowment for the Arts of data from The Survey of Public Participation in the Arts conducted by the Census Bureau in 2002.
2. Vertical TeamTM is a term trademarked by The College Board PreAP program
3. Michelle Ralston handout on “Vertical Teams Strategies and Implications.” New Haven Public Schools Workshop, 4 November 2003.
4. Ibid.
5. Ibid.
6. Professor Roger Howe, “Comments during a meeting of The Craft of Word Problems seminar”, Spring, 2004.
7. Thomas Carpenter, Megan Loef Franke and Linda Levi, Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary School (Portsmouth: Heinemann,
8. 2003) xi.
9. Thomas Carpenter, Megan Loef Franke and Linda Levi, Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary School (Portsmouth: Heinemann, 2003) 1.
10. Thomas Carpenter, Megan Loef Franke and Linda Levi, Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary School (Portsmouth: Heinemann, 2003) xii. Adapted from a vertical team “4part problem page” presented by Michelle Ralston.
11. Primary Mathematics Texts 6A
, U.S. Edition. Curriculum Planning and Development Division, Singapore Ministry of Education. (Singapore: Federal Publications, 2003) 3.
12. Thomas Carpenter, Megan Loef Franke and Linda Levi, Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary School (Portsmouth: Heinemann, 2003) 134.
13. Ibid.
14. Thomas Carpenter, Megan Loef Franke and Linda Levi, Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary School (Portsmouth: Heinemann, 2003) 137.
15. www.japaneseonline.com/math