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# Singapore Mathematics: Problem Solving by OJXgOy

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```									        Singapore Mathematics:
Problem Solving
Linda King
Singapore Math Coach
lking@windsor.k12.mo.us

By Linda King, Windsor C-1
Singapore Math &
Solving Word Problems

Singapore Math moves students through 3
stages of learning:
– Concrete
– Pictorial
– Abstract

By Linda King, Windsor C-1
Singapore Math &
Solving Word Problems

8-Step Model Drawing is a pictorial way to
represent word problems. This method helps
students to SEE the problem and can
dramatically increase student understanding.
8-Step Model Drawing provides students with a
framework that can be used to attack MOST
word problems.

By Linda King, Windsor C-1
8 Steps for Model-Drawing

2. Decide who is involved in the problem.
3. Decide what is involved in the problem.
4. Draw unit bars of equal length.
5. Read each sentence, one at a time.
6. Put the question mark in place.
7. Work computation to the side or underneath.
8. Answer the question in a complete sentence.

By Linda King, Windsor C-1
Problem 1:

How many books did they read altogether?

By Linda King, Windsor C-1
Step 1:

How many books did they read altogether?

By Linda King, Windsor C-1
Step 2:

How many books did they read altogether?

By Linda King, Windsor C-1
Step 2:

How many books did they read
altogether?

By Linda King, Windsor C-1
Step 2:
Who else is the problem about?

How many books did they read altogether?

By Linda King, Windsor C-1
Step 2:
Who else is the problem about?

How many books did they read altogether?

Haley

By Linda King, Windsor C-1
Step 3:

How many books did they read altogether?

Haley

By Linda King, Windsor C-1
Step 3: What is the problem about?

How many books did they read altogether?

books

Haley’s
books

By Linda King, Windsor C-1
Step 4:
Draw unit bars of equal length.

books

Haley’s
books

By Linda King, Windsor C-1
Step 4:
Draw unit bars of equal length.

books

Haley’s
books

By Linda King, Windsor C-1
Step 5: Read each sentence, one at a time.

books

Haley’s
books

By Linda King, Windsor C-1

books

Haley’s
books

By Linda King, Windsor C-1

1         1       1       1       1   1   1
books

Haley’s                      At the beginning of
books                       1st grade, one unit bar
represents 1 unit.

By Linda King, Windsor C-1
Step 5:
Read each sentence, one at a time.

1        1       1       1       1   1   1
books

Haley’s
1
books

By Linda King, Windsor C-1

1         1       1       1       1   1   1
books

Haley’s
1
books

By Linda King, Windsor C-1

1         1       1       1       1   1   1
books

Haley’s
1    1    1       1       1       1
books

By Linda King, Windsor C-1
Step 5:
Read each sentence, one at a time.

How many books did they read
altogether?
1        1       1       1       1   1   1
books

Haley’s
1   1    1       1       1       1
books

By Linda King, Windsor C-1
No new information. Move to step 6.

1        1       1       1       1   1   1
books

Haley’s
1   1    1       1       1       1
books

By Linda King, Windsor C-1
Step 6:
Put the question mark in place.

How many books did they read
altogether?
1        1       1       1       1   1   1
books

Haley’s
1   1    1       1       1       1
books

By Linda King, Windsor C-1
Step 6:
Put the question mark in your drawing.

1        1       1       1       1   1   1
books

Haley’s   1   1    1       1       1       1           ?
books

By Linda King, Windsor C-1
information to the drawing as needed.

1        1       1       1       1   1   1
books

Haley’s   1   1    1       1       1       1           ?
books

By Linda King, Windsor C-1
information to the drawing as needed.

1        1       1       1       1   1   1   8
books

6
Haley’s   1   1    1       1       1       1           ?
books

By Linda King, Windsor C-1
Step 7:
Work the computation.

1        1       1       1       1   1   1   8
books

6
Haley’s   1   1    1       1       1       1           ?
books

By Linda King, Windsor C-1
Step 7:
Work the computation.

8
1          1       1       1       1   1   1
books

Haley’s   1   1      1       1       1       1   6           ?
books
8 + 6 = 14 or count the unit bars

By Linda King, Windsor C-1
Place the answer next to the question

8
1          1       1       1       1   1   1
books
14

Haley’s   1   1      1       1       1       1   6           ?
books
8 + 6 = 14

By Linda King, Windsor C-1
Step 8:

8
1            1       1       1       1   1   1
books
14

Haley’s    1    1       1       1       1       1   6             ?
books
8 + 6 = 14                1st graders fill in the
blank in a sentence.

By Linda King, Windsor C-1
Step 8:

8
1        1       1       1       1   1   1
books
14

Haley’s                                        6           ?
1   1    1       1       1       1
books
8 + 6 = 14

By Linda King, Windsor C-1
Problem 2:

40¢ each
2. Sara bought 2 pears.
She paid _____¢ .

By Linda King, Windsor C-1
Step 1:

40¢ each
2. Sara bought 2 pears.
She paid _____¢ .

By Linda King, Windsor C-1
Step 2: Who is the problem about?

40¢ each
2. Sara bought 2 pears.
She paid _____¢ .

By Linda King, Windsor C-1
Step 2: Who is the problem about?

40¢ each
2. Sara bought 2 pears.
She paid _____¢ .

Sara

By Linda King, Windsor C-1
Step 3: What is the problem about?

40¢ each
2. Sara bought 2 pears.
She paid _____¢ .

By Linda King, Windsor C-1
Step 3: What is the problem about?

2. Sara bought 2 pears.                        40¢ each
She paid _____¢ .

Sara’s
pears

By Linda King, Windsor C-1
Step 4: Draw unit bars of equal length.

2. Sara bought 2 pears.                        40¢ each
She paid _____¢ .

Sara’s
pears

By Linda King, Windsor C-1
Step 4: Draw unit bars of equal length.

2. Sara bought 2 pears.                        40¢ each
She paid _____¢ .

Sara’s
pears

By Linda King, Windsor C-1
Step 5:
Read each sentence, one at a time.

2. Sara bought 2 pears.                        40¢ each
She paid _____¢ .

Sara’s
pears

By Linda King, Windsor C-1

2. Sara bought 2 pears.
She paid _____¢ .                              40¢ each

Sara’s
pears

By Linda King, Windsor C-1

2. Sara bought 2 pears.
She paid _____¢ .                              40¢ each

Sara’s
pears     1       1

By Linda King, Windsor C-1
Step 5:
Read each sentence, one at a time.

2. Sara bought 2 pears.
She paid _____¢ .                              40¢ each

Sara’s
pears
1      1

By Linda King, Windsor C-1
drawing.

2. Sara bought 2 pears.
40¢ each
She paid _____¢ .

Sara’s
pears     1         1

By Linda King, Windsor C-1
No new information in this sentence.
2. Sara bought 2 pears.
40¢ each
She paid _____¢ .

Sara’s
pears     1         1

By Linda King, Windsor C-1
Is there any other information that I
2. Sara bought 2 pears.
She paid _____¢ .                              40¢ each

Sara’s
pears
1       1

By Linda King, Windsor C-1

2. Sara bought 2 pears.
She paid _____¢ .                              40¢ each

Sara’s
pears
1       1

By Linda King, Windsor C-1

2. Sara bought 2 pears.
She paid _____¢ .                              40¢ each

Sara’s   40¢
pears
1       1

By Linda King, Windsor C-1
Step 6: Put the question mark in place.

2. Sara bought 2 pears.
She paid _____¢ .                              40¢ each

Sara’s   40¢
pears
1       1

By Linda King, Windsor C-1
Step 6: Put the question mark in place.

2. Sara bought 2 pears.
She paid _____¢ .                              40¢ each

Sara’s   40¢
pears
1       1          ?¢

By Linda King, Windsor C-1

2. Sara bought 2 pears.
She paid _____¢ .                              40¢ each

40¢
Sara’s
pears
1       1         ?¢

By Linda King, Windsor C-1

2. Sara bought 2 pears.
She paid _____¢ .                              40¢ each

Sara’s   40¢    (40¢)
pears
1       1          ?¢

By Linda King, Windsor C-1
Step 7: Work computation.
2. Sara bought 2 pears.
She paid _____¢ .

Sara’s   40¢     (40¢)
pears                       ?¢
1        1

By Linda King, Windsor C-1
Step 7: Work computation.
2. Sara bought 2 pears.
She paid _____¢ .

Sara’s   40¢     (40¢)
pears                       ?¢
1        1

40 + 40 = 80

By Linda King, Windsor C-1
Place the answer in the drawing next to
the question mark.
2. Sara bought 2 pears.
She paid _____¢ .

Sara’s   40¢     (40¢)
pears                       ?¢
1        1

40 + 40 = 80

By Linda King, Windsor C-1
Place the answer in the drawing next to
the question mark.
2. Sara bought 2 pears.
She paid _____¢ .

Sara’s   40¢     (40¢)
pears                       ?¢         80
1        1

40 + 40 = 80

By Linda King, Windsor C-1
Step 8: Answer in a complete sentence.
2. Sara bought 2 pears.
She paid _____¢ .

Sara’s   40¢     (40¢)
pears                       ?¢         80
1        1

40 + 40 = 80

By Linda King, Windsor C-1
Step 8: Answer in a complete sentence.
2. Sara bought 2 pears.
She paid _____¢ .

Sara’s   40¢     (40¢)
pears                       ?¢         80
1        1

40 + 40 = 80
She paid 80¢.

By Linda King, Windsor C-1
Problem 3:

70¢
3. Devi has 55¢. She wants to buy the pen.

She needs ________¢ more.

By Linda King, Windsor C-1
Step 1: Read the entire problem.

70¢
3. Devi has 55¢. She wants to buy the pen.

She needs ________¢ more.

By Linda King, Windsor C-1
Step 2: Who is the problem about?

70¢
3. Devi has 55¢. She wants to buy the pen.

She needs ________¢ more.

By Linda King, Windsor C-1
Step 2: Who is the problem about?

70¢
3. Devi has 55¢. She wants to buy the pen.
She needs ________¢ more.

Devi

By Linda King, Windsor C-1
Step 3: What is the problem about?

70¢
3. Devi has 55¢. She wants to buy the pen.
She needs ________¢ more.

Devi

By Linda King, Windsor C-1
Step 3: What is the problem about?

70¢
3. Devi has 55¢. She wants to buy the pen.
She needs ________¢ more.

Devi’s
money

By Linda King, Windsor C-1
Step 4: Draw unit bars of equal length.

70¢
3. Devi has 55¢. She wants to buy the pen.
She needs ________¢ more.

Devi’s
money

By Linda King, Windsor C-1
Step 4: Draw unit bars of equal length.

70¢
3. Devi has 55¢. She wants to buy the pen.
She needs ________¢ more.

Devi’s
money

By Linda King, Windsor C-1
Step 5: Read each sentence, one at a time.

3. Devi has 55¢. She wants to buy the pen.
She needs ________¢ more.                        70¢

Devi’s
money

By Linda King, Windsor C-1
Add the information to the problem.

3. Devi has 55¢. She wants to buy the pen.
She needs ________¢ more.                        70¢

Devi’s
money

By Linda King, Windsor C-1
Add the information to the problem.

3. Devi has 55¢. She wants to buy the pen.
She needs ________¢ more.                        70¢

Devi’s
55¢
money

This unit bar represents Devi’s
money. Transition at end of 1st

By Linda King, Windsor C-1
Step 5: Read each sentence, one at a time.

3. Devi has 55¢. She wants to buy the pen.
She needs ________¢ more.                        70¢

Devi’s
55¢
money

By Linda King, Windsor C-1
Add the information to the problem.

3. Devi has 55¢. She wants to buy the pen.
She needs ________¢ more.                        70¢

Devi’s
55¢
money

By Linda King, Windsor C-1
There is no new information to add to the
drawing in this sentence.

3. Devi has 55¢. She wants to buy the pen.
She needs ________¢ more.                        70¢

Devi’s
55¢
money

By Linda King, Windsor C-1
Step 5: Read each sentence, one at a time.

3. Devi has 55¢. She wants to buy the pen.
She needs ________¢ more.                        70¢

Devi’s
55¢
money

By Linda King, Windsor C-1

3. Devi has 55¢. She wants to buy the pen.
She needs ________¢ more.                        70¢

Devi’s
55¢
money

By Linda King, Windsor C-1
3. Devi has 55¢. She wants to buy the pen.
She needs ________¢ more.                        70¢

Devi’s
55¢       more
money

By Linda King, Windsor C-1
How much money does she need in all?
Add the information to the problem.

3. Devi has 55¢. She wants to buy the pen.
She needs ________¢ more.                        70¢

Devi’s
55¢       more
money

By Linda King, Windsor C-1

3. Devi has 55¢. She wants to buy the pen.
She needs ________¢ more.                        70¢

70¢
Devi’s
55¢       more
money

By Linda King, Windsor C-1
Step 6: Put the question mark in place.

3. Devi has 55¢. She wants to buy the pen.
She needs ________¢ more.                        70¢

70¢
Devi’s
55¢       more
money

By Linda King, Windsor C-1
Step 6: Put the question mark in place.

3. Devi has 55¢. She wants to buy the pen.
She needs ________¢ more.                        70¢

70¢
Devi’s
55¢       more
money

?

By Linda King, Windsor C-1
Is there any other information that I could

3. Devi has 55¢. She wants to buy the pen.
She needs ________¢ more.                        70¢

70¢
Devi’s
55¢       more
money

?

By Linda King, Windsor C-1
No other information.

3. Devi has 55¢. She wants to buy the pen.
She needs ________¢ more.                        70¢

70¢
Devi’s
55¢       more
money

?

By Linda King, Windsor C-1
Step 7: Computation.

3. Devi has 55¢. She wants to buy the pen.
She needs ________¢ more.                        70¢

70¢
Devi’s
55¢       more
money

?

By Linda King, Windsor C-1
Step 7: Computation.

3. Devi has 55¢. She wants to buy the pen.
She needs ________¢ more.

70¢
Devi’s
55¢       more                     70 – 55 = 15
money

?

By Linda King, Windsor C-1
Place the answer by the question mark.

3. Devi has 55¢. She wants to buy the pen.
She needs ________¢ more.

70¢
Devi’s
55¢       more                     70 – 55 = 15
money

?      15

By Linda King, Windsor C-1
Step 8: Answer in a complete sentence.

3. Devi has 55¢. She wants to buy the pen.
She needs ________¢ more.

70¢
Devi’s
55¢       more                     70 – 55 = 15
money

?        15

By Linda King, Windsor C-1
Step 8: Answer in a complete sentence.

3. Devi has 55¢. She wants to buy the pen.
She needs 15¢ more.

70¢
Devi’s        55¢       more                     70 – 55 = 15
money
?        15

By Linda King, Windsor C-1
Problem 4:

4. There were 287 people in a hall.
52 of them were children.

Begin with      Steps 1 through 4
of the 8-Steps for Model-Drawing.

By Linda King, Windsor C-1
Problem 4:                       After Step 4.

4. There were 287 people in a hall.
52 of them were children.

people

Now do Step 5, adding information,
one sentence at a time.

By Linda King, Windsor C-1
Problem 4:          Step 5: After 1st sentence.

4. There were 287 people in a hall.
52 of them were children.

287

people

By Linda King, Windsor C-1
Problem 4:          Step 5: After 2nd sentence.

4. There were 287 people in a hall.
52 of them were children.

287
children
people
52

By Linda King, Windsor C-1
Problem 4:          Steps 5 & 6: Place the question
mark, using the information in
the third sentence.

4. There were 287 people in a hall.
52 of them were children.

287

52

By Linda King, Windsor C-1
Problem 4:           Step 7: Computation.

4. There were 287 people in a hall.
52 of them were children.

287

52

By Linda King, Windsor C-1
Problem 4:              Step 7: Completed.

4. There were 287 people in a hall.
52 of them were children.

287
52                     235
287 – 52 = 235

By Linda King, Windsor C-1
Problem 4 completed:
4. There were 287 people in a hall.
52 of them were children.

287
52                     235
287 – 52 = 235

By Linda King, Windsor C-1
Problem 5:

5. Tracy bought 6 bags of oranges.
There were 10 oranges in each bag.
How many oranges did she buy altogether?

Try this one on your own using the
8 Steps for Model Drawing.

By Linda King, Windsor C-1
Problem 5 Drawing:
5. Tracy bought 6 bags of oranges.
There were 10 oranges in each bag.
How many oranges did she buy altogether?

1    2          3          4          5    6

Tracy’s   10   10        10         10           10   10
oranges
?

By Linda King, Windsor C-1
Problem 5 Solution:
5. Tracy bought 6 bags of oranges.
There were 10 oranges in each bag.
How many oranges did she buy altogether?

1      2          3          4          5    6

Tracy’s     10    10         10         10           10   10
oranges
?         60
6 x 10 = 60 or
10 + 10 + 10 + 10 + 10 + 10 = 60

Tracy bought 60 oranges.

By Linda King, Windsor C-1
Problem 6:

6. There are 304 girls in a school.
There are 46 fewer boys than girls.
a. How many boys are there in the school?
b. How many children are there in the school?

Try this one on your own using the
8 Steps for Model Drawing.
By Linda King, Windsor C-1
Problem 6 Drawing:
6. There are 304 girls in a school.
There are 46 fewer boys than girls.
a.   How many boys are there in the school?
b.   How many children are there in the school?

304
girls

b)?
boys          a)?                 fewer
- 46

By Linda King, Windsor C-1
Problem 6 Solution:
6. There are 304 girls in a school.
There are 46 fewer boys than girls.
a.   How many boys are there in the school?
b.   How many children are there in the school?

304
girls

b)?   562
boys          a)?    258          fewer
- 46
304 – 46 = 258
258 + 304 = 562          There are 258 boys in the school.
There are 562 children in the school.

By Linda King, Windsor C-1
Problem 7:

7. Melissa has 6 postcards.
Sally has 3 times as many postcards as Melissa.
How many more postcards does Sally have than
Melissa?

Try this one on your own using the 8 Steps
for Model Drawing.

By Linda King, Windsor C-1
Problem 7 Drawing:
7. Melissa has 6 postcards.
Sally has 3 times as many postcards as Melissa.
How many more postcards does Sally have than Melissa?

Melissa’s
6
postcards

1x    2x        3x
Sally’s
postcards   (6)   (6)       (6)

?

By Linda King, Windsor C-1
Problem 7 Solution:
7. Melissa has 6 postcards.
Sally has 3 times as many postcards as Melissa.
How many more postcards does Sally have than Melissa?

Melissa’s
6
postcards
6 + 6 = 12
1x    2x        3x
Sally’s
postcards      (6)   (6)       (6)

?    12

Sally has 12 more postcards than Melissa.

By Linda King, Windsor C-1
Problem 8
8. Brian has 6 goldfish.                     Work through
He has 5 times as many                          Step 5:
guppies as goldfish.                        sentence 2.
STOP                                            THEN,
If he puts his guppies
equally into 3 tanks, how
many guppies are in each
tank?

By Linda King, Windsor C-1
Problem 8: Step 5, sentence 3
8. Brian has 6 goldfish.
He has 5 times as many guppies as goldfish.
If he puts his guppies equally into 3 tanks, how many guppies
are in each tank?

Brian’s
goldfish     6

x1      x2      x3         x4         x5
Brian’s
guppies

By Linda King, Windsor C-1
Problem 8: Step 5, sentence 3
8. Brian has 6 goldfish.
He has 5 times as many guppies as goldfish.
If he puts his guppies equally into 3 tanks, how many guppies
are in each tank?
Pull down the bar.
Brian’s                               Too confusing to divide the
6                           guppie bar again.
goldfish

Brian’s      x1      x2      x3         x4         x5
guppies

By Linda King, Windsor C-1
Problem 8: Step 5, sentence 3
8. Brian has 6 goldfish.
He has 5 times as many guppies as goldfish.
If he puts his guppies equally into 3 tanks, how many guppies
are in each tank?
NOW, divide the “pull-down”
Brian’s
6                      bar into 3 parts and label.
goldfish

Brian’s      x1     x2      x3         x4         x5
guppies

tank 1        tank 2           tank 3

By Linda King, Windsor C-1
Problem 8: Step 5, sentence 3
8. Brian has 6 goldfish.
He has 5 times as many guppies as goldfish.
If he puts his guppies equally into 3 tanks, how many guppies
are in each tank?

Brian’s                                   Complete the problem.
6
goldfish

Brian’s     x1      x2      x3         x4         x5
guppies

tank 1        tank 2           tank 3

By Linda King, Windsor C-1
Problem 8 Drawing:
8. Brian has 6 goldfish.
He has 5 times as many guppies as goldfish.
If he puts his guppies equally into 3 tanks, how many guppies
are in each tank?

Brian’s
6
goldfish

Brian’s     x1      x2      x3         x4         x5
guppies     (6)     (6)     (6)        (6)        (6)   (30)

(30)

tank 1 ?       tank 2            tank 3

By Linda King, Windsor C-1
Problem 8 Solution:
8. Brian has 6 goldfish.
He has 5 times as many guppies as goldfish.
If he puts his guppies equally into 3 tanks, how many guppies
are in each tank?

Brian’s
6
goldfish

Brian’s     x1        x2      x3         x4         x5
guppies     (6)       (6)     (6)        (6)        (6)   (30)

(30)
5 x 6 = 30
tank 1 ?        tank 2             tank 3       30 ÷ 3 = 10
10
Brian has 10 guppies in each tank.
By Linda King, Windsor C-1
Problem 9:

9. The capacity of a bucket is 6 liters.
5 buckets of water are needed to fill up a tank.
What is the capacity of the tank?

Try this one on your own using the 8 Steps
for Model Drawing.

By Linda King, Windsor C-1
Problem 9 Drawing:
9. The capacity of a bucket is 6 liters.
5 buckets of water are needed to fill up a tank.
What is the capacity of the tank?

Bkt 1   Bkt 2   Bkt 3      Bkt 4        Bkt 5
Tank’s
capacity     6     (6)      (6)        (6)         (6)

?

By Linda King, Windsor C-1
Problem 9 Solution:
9. The capacity of a bucket is 6 liters.
5 buckets of water are needed to fill up a tank.
What is the capacity of the tank?

Bkt 1    Bkt 2   Bkt 3      Bkt 4        Bkt 5
Tank’s
capacity     6      (6)      (6)        (6)         (6)

? 30
5 x 6 = 30

The capacity of the tank is 30 liters.

By Linda King, Windsor C-1
Contact Linda King: lking@windsor.k12.mo.us
(636) 464 – 4400, extension 4399

By Linda King, Windsor C-1

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