Introduction to Singapore Math
INTRODUCTION TO SINGAPORE MATH
Welcome to Singapore Math! The math curriculum in Singapore has
been recognized worldwide for its excellence in producing students
highly skilled in mathematics. Students in Singapore have ranked at
the top in the world in mathematics on the Trends in International
Mathematics and Science Study (TIMSS) in 1993, 1995, 2003, and
2008. Because of this, Singapore Math has gained in interest and
popularity in the United States.
Singapore Math curriculum aims to help students develop the
necessary math concepts and process skills for everyday life and to
provide students with the ability to formulate, apply, and solve
problems. Mathematics in the Singapore Primary (Elementary)
Curriculum cover fewer topics but in greater depth. Key math
concepts are introduced and built-on to reinforce various
mathematical ideas and thinking. Students in Singapore are
typically one grade level ahead of students in the United
The following pages provide examples of the various math problem
types and skill sets taught in Singapore.
At an elementary level, some simple mathematical skills can help
students understand mathematical principles. These skills are the
counting-on, counting- back, and crossing-out methods. Note that
these methods are most useful when the numbers are small.
The Counting-On Method
Used for addition of two numbers. Count on in 1s with the help
of a picture or number line.
7 + 4 = 11
+1 +1 +1 +1
7 8 9 10 11
The Counting-Back Method
Used for subtraction of two numbers. Count back in 1s with the
help of a picture or number line.
16 - 3 = 13
-1 -1 -1
13 14 15 16
The Crossing-Out Method
Used for subtraction of two numbers. Cross out the number of
items to be taken away. Count the remaining ones to find the
20 - 12 = 8
A number bond shows the relationship in a simple addition or
subtraction problem. The number bond is based on the concept
"part-part-whole." This concept is useful in teaching simple
addition and subtraction to young children.
3. Addition Number Bond (double and single digits)
2 + 15
To find a whole, students must add the two parts.
To find a part, students must subtract the other part from the
The different types of number bonds are illustrated below.
Number Bond (single digits)
3 (part) + 6 (part) = 9 (whole) 9 (whole) - 3 (part) = 6 (part)
9 (whole) - 6 (part) = 3 (part)
Addition Number Bond (single digits)
The Adding-Without-Regrouping Method
= 9 + 1 + 4 = 10 + 4
Make a ten first.
+1 5 3
6 4 5
70 Must-Know Word Problems Level 3
Students should understand that multiplication is repeated
addition and that division is the grouping of all items into equal
Repeated Addition (Multiplication)
Mackenzie eats 2 rolls a day. How many rolls does she eat in 5
days? 2 + 2 + 2 + 2 + 2 = 10
5 2 = 10
She eats 10 rolls in 5 days.
The Grouping Method (Division)
Mrs. Lee makes 14 sandwiches. She gives all the sandwiches
equally to 7 friends. How many sandwiches does each friend
14 7 = 2 Each friend receives 2 sandwiches.
One of the basic but essential math skills students should
acquire is to perform the 4 operations of whole numbers and
fractions. Each of these methods is illustrated below.
14 HTO 14 9 2
= 2 + 5 + 10 = 7 + 10
Regroup15 into 5 and 10.
4. Subtraction Number Bond (double and single digits)
12 - 7
2 10 10 - 7 = 3
3+ 2= 5
5. Subtraction Number Bond (double digits)
3 + 5 8
T O 2 1 6 8 8 9
20 - 15
10 10 5
10 - 5 = 5 10 - 10 = 0 5+ 0= 5
Since no regrouping is required, add the digits in each place
The Adding-by-Regrouping Method
In this example, regroup 14 tens into 1 hundred 4 tens.
3. The Adding-by-Regrouping-Twice Method
11. The Addition-of-Fractions Method
T O 18 6 6 5 5 1
_1 × 2
_1 × 3 2 3 _5_ × 3 = ___ + ___ =
H T O
73 9 -32 5 41 4
× 2 +
4 12 12 12
Always remember to make the denominators common before adding
12. The Subtraction-of-Fractions Method
_1_ × 5 _1_ × 2 _5__ _2__ _3_ 2 × 5 - 5
× 2 = 10 - 10 = 10
Always remembers to make the denominators common before
subtracting the fractions.
13. The Multiplication-of-Fractions Method
1 _3_ _1_ _1_ 5 ×39 = 15
When the numerator and the denominator have a common multiple,
reduce them to their lowest fractions.
14. The Division-of-Fractions Method
Regroup twice in this example.
First, regroup 11 ones into 1 ten 1 one. Second, regroup 15 tens
into 1 hundred 5 tens.
4. The Subtracting-Without-Regrouping Method
Since no regrouping is required, subtract the digits in each
place value accordingly.
5. The Subtracting-by-Regrouping Method
5 78111 - 2 4 7 334
_7_ ÷ _1_ = _7_ × _6_ = _1_4_ = 4 _2
In this example, students cannot subtract 7 ones from 1 one. So,
regroup the tens and ones. Regroup 8 tens 1 one into 7 tens 11
6. The Subtracting-by-Regrouping-Twice Method
When dividing fractions, first change the division sign
(÷) to the multiplication sign (×). Then, switch the
numerator and denominator of the fraction on the right hand side.
Multiply the fractions in the usual way.
Model drawing is an effective strategy used to solve math word
problems. It is a visual representation of the information in word
problems using bar units. By drawing the models, students will know
of the variables given in the problem, the variables to find, and
even the methods used to solve the problem.
Drawing models is also a versatile strategy. It can be applied
to simple word problems involving addition, subtraction,
multiplication, and division. It can also be applied to word
problems related to fractions, decimals, percentage, and ratio.
The use of models also trains students to think in an algebraic
manner, which uses symbols for representation.
The different types of bar models used to solve word problems
are illustrated below.
HTO 78 90100
9 6 39 1 3 3
In this example, students cannot subtract 3 ones from 0 ones and
9 tens from 0 tens. So, regroup the hundreds, tens, and ones.
Regroup 8 hundreds into 7 hundreds 9 tens 10 ones.
7. The Multiplying-Without-Regrouping Method
TO 2 4
The model that involves addition
Melissa has 50 blue beads and 20 red beads. How many beads does
O: Ones 2 T: Tens
Since no regrouping is required, multiply the digit in each
place value by the multiplier accordingly.
8. The Multiplying-With-Regrouping Method
50 + 20 = 70 The model that involves subtraction
Ben and Andy have 90 toy cars. Andy has 60 toy cars. How many
H T O
13 24 ×
9 3 7
does Ben have?
In this example, regroup 27 ones into 2 tens 7 ones, and 14 tens
into 1 hundred 4 tens.
9. The Dividing-Without-Regrouping Method
2 -2 0
Since no regrouping is required, divide the digit in each place
value by the divisor accordingly.
10. The Dividing-With-Regrouping Method
30 -30 0
In this example, regroup 3 hundreds into 30 tens and add 3 tens
to make 33 tens. Regroup 3 tens into 30 ones.
90 - 60 = 30 The model that involves comparison
Mr. Simons has 150 magazines and 110 books in his study. How
many more magazines than books does he have?
150 - 110 = 40
The model that involves two items with a difference
A pair of shoes costs $109. A leather bag costs $241 more than
the pair of shoes. How much is the leather bag?
$109 + $241 = $350
70 Must-Know Word Problems Level 3
The model that involves multiples
Mrs. Drew buys 12 apples. She buys 3 times as many oranges as
apples. She also buys 3 times as many cherries as oranges. How many
pieces of fruit does she buy altogether?
Oranges ? Cherries
13 × 12 = 156
The model that involves multiples and difference
There are 15 students in Class A. There are 5 more students in
Class B than in Class A. There are 3 times as many students in
Class C than in Class A. How many students are there altogether in
the three classes?
Jack's height is _2_ of Leslie's height. Leslie's height is _3_
of Lindsay's height. If 34
Lindsay is 160 cm tall, find Jack's height and Leslie's height.
? $539 ÷ 7 = $77
Tie (2 units) → 2 x $77 = $154 Belt (5 units) → 5 x
$77 = $385
11. The model that involves comparison of fractions
Class A Class B Class C
(5 × 15) + 5 = 80
7. The model that involves creating a whole
Ellen, Giselle, and Brenda bake 111 muffins. Giselle bakes twice
as many muffins as Brenda. Ellen bakes 9 fewer muffins than
Giselle. How many muffins does Ellen bake?
1 unit → 160 ÷ 4 = 40 cm
Jack's height (2 units) → 2 × 40 = 80 cm
Thinking skills and strategies are important in mathematical
problem solving. These skills are applied when students think
through the math problems to solve them. Below are some commonly
used thinking skills and strategies applied in mathematical problem
Comparing is a form of thinking skill that students can apply to
identify similarities and differences.
When comparing numbers, look carefully at each digit before
deciding if a number is greater or less than the other. Students
might also use a number line for comparison when there are more
3 is greater than 2 but smaller than 7.
A sequence shows the order of a series of numbers. Sequencing is
a form of thinking skill that requires students to place numbers in
a particular order. There are many terms in a sequence. The terms
refer to the numbers in a sequence.
To place numbers in a correct order, students must first find a
rule that generates the sequence. In a simple math sequence,
students can either add or subtract to find the unknown terms in
Example: Find the 7th term in the sequence below.
1, 4, 7, 10, 13, 16 ?
160 cm Leslie's height (3 units) → 3 × 40 = 120
Ellen Giselle Brenda
111 + 9
8. The model that involves sharing
(111 + 9) ÷ 5 = 24 (2 × 24) - 9 = 39
There are 183 tennis balls in Basket A and 97 tennis balls in
Basket B. How many tennis balls must be transferred from Basket A
to Basket B so that both baskets contain the same number of tennis
Basket A Basket B
183 - 97 = 86 86 ÷ 2 = 43
9. The model that involves fractions
George had 355 marbles. He lost _1_ of the marbles and gave _1_
of the remaining 54
marbles to his brother. How many marbles did he have left?
? 5 parts → 355 marbles
1 part → 355 ÷ 5 = 71 marbles 3 parts → 3
× 71 = 213 marbles
10. The model that involves ratio
Step 1: Step 2:
2nd 3rd 4th 5th 6th 7th term term term term term term
This sequence is in an increasing order.
4 - 1 = 3 7 - 4 = 3
The difference between two consecutive terms is 3.
16 + 3 = 19
The 7th term is 19.
Aaron buys a tie and a belt. The prices of the tie and belt are
in the ratio 2 : 5. If both items cost $539,
(a) what is the price of the tie?
(b) what is the price of the belt?
Visualization is a problem solving strategy that can help
students visualize a problem through the use of physical objects.
Students will play a more active role in solving the problem by
manipulating these objects.
The main advantage of using this strategy is the mobility of
information in the process of solving the problem. When students
make a wrong step in the process, they can retrace the step without
erasing or canceling it.
The other advantage is that this strategy helps develop a better
understanding of the problem or solution through visual objects or
images. In this way, students will be better able to remember how
to solve these types of problems.
70 Must-Know Word Problems Level 3
Some of the commonly used objects for this strategy are
toothpicks, straws, cards, strings, water, sand, pencils, paper,
4. Look for a Pattern
This strategy requires the use of observational and analytical
skills. Students have to observe the given data to find a pattern
in order to solve the problem. Math word problems that involve the
use of this strategy usually have repeated numbers or patterns.
Example: Find the sum of all the numbers from 1 to 100.
assumptions will eliminate some possibilities and simplifies the
word problems by providing a boundary of values to work within.
Example: Mrs. Jackson bought 100 pieces of candy for all the
students in her class. How many pieces of candy would each student
receive if there were 25 students in her class?
In the above word problem, assume that each student received the
same number of pieces. This eliminates the possibilities that some
students would receive more than others due to good behaviour,
better results, or any other reason.
Representation of Problem
In problem solving, students often use representations in the
solutions to show their understanding of the problems. Using
representations also allow students to understand the mathematical
concepts and relationships as well as to manipulate the information
presented in the problems. Examples of representations are diagrams
and lists or tables.
Diagrams allow students to consolidate or organize the
information given in the problems. By drawing a diagram, students
can see the problem clearly and solve it effectively.
A list or table can help students organize information that is
useful for analysis. After analyzing, students can then see a
pattern, which can be used to solve the problem.
Guess and Check
One of the most important and effective problem-solving
techniques is Guess and Check. It is also known as Trial and Error.
As the name suggests, students have to guess the answer to a
problem and check if that guess is correct. If the guess is wrong,
students will make another guess. This will continue until the
guess is correct.
It is beneficial to keep a record of all the guesses and checks
in a table. In addition, a Comments column can be included. This
will enable students to analyze their guess (if it is too high or
too low) and improve on the next guess. Be careful; this
problem-solving technique can be tiresome without systematic or
Example: Jessica had 15 coins. Some of them were 10-cent coins
and the rest were 5-cent coins. The total amount added up to $1.25.
How many coins of each kind were there?
Use the guess-and-check method.
There were ten 10-cent coins and five 5-cent coins.
10. Restate the Problem
When solving challenging math problems, conventional methods may
not be workable. Instead, restating the problem will enable
students to see some challenging problems in a different light so
that they can better understand them.
The strategy of restating the problem is to "say" the problem in
a different and clearer way. However, students have to ensure that
the main idea of the problem is not altered.
How do students restate a math problem?
First, read and understand the problem. Gather the given facts
and unknowns. Note any condition(s) that have to be satisfied.
Next, restate the problem. Imagine narrating this problem to a
friend. Present the given facts, unknown(s), and condition(s).
Students may want to write the "revised" problem. Once the
"revised" problem is analyzed, students should be able to think of
an appropriate strategy to solve it.
11. Simplify the Problem
One of the commonly used strategies in mathematical problem
solving is simplification of the problem. When a problem is
simplified, it can be "broken down" into two or more smaller parts.
Students can then solve the parts systematically to get to the
Step 1: Step 2:
Simplify the problem.
Find the sum of 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10.
Look for a pattern.
1 + 10 = 11 4 + 7 = 11
2 + 9 = 11 5 + 6 = 11
3 + 8 = 11
Describe the pattern.
When finding the sum of 1 to 10, add the first and last numbers to
get a result of 11. Then, add the second and second last numbers to
get the same result. The pattern continues until all the numbers
from 1 to 10 are added. There will be 5 pairs of such results.
Since each addition equals 11, the answer is then 5 × 11 =
Use the pattern to find the answer.
pairs) in the sum of 1 to 100.
Note that the addition for each pair is not equal to 11 now. The
addition for each pair is now (1 + 100 = 101).
50 × 101 = 5050
The sum of all the numbers from 1 to 100 is 5,050.
The strategy of working backward applies only to a specific type
of math word problem. These word problems state the end result, and
students are required to find the total number. In order to solve
these word problems, students have to work backward by thinking
through the correct sequence of events. The strategy of working
backward allows students to use their logical reasoning and
sequencing to find the answers.
Example: Sarah has a piece of ribbon. She cuts the ribbon into 4
equal parts. Each part is then cut into 3 smaller equal parts. If
the length of each small part is 35 cm, how long is the piece of
3 × 35 = 105 cm 4 × 105 = 420 cm
The piece of ribbon is 420 cm.
The Before-After Concept
The Before-After concept lists all the relevant data before and
after an event. Students can then compare the differences and
eventually solve the problems. Usually, the Before-After concept
and the mathematical model go hand in hand to solve math word
problems. Note that the Before-After concept can be applied only to
a certain type of math word problem, which trains students to think
Example: Kelly has 4 times as much money as Joey. After Kelly
uses some money to buy a tennis racquet, and Joey uses $30 to buy a
pair of pants, Kelly has twice as much money as Joey. If Joey has
$98 in the beginning,
(a) how much money does Kelly have in the end?
(b) how much money does Kelly spend on the tennis
Number of 10¢ Coins
Number of 5¢ Coins
Total Number of Coins
7 × 10¢ = 70¢
8 × 5¢ = 40¢
7 + 8 = 15
70¢ + 40¢ = 110¢ = $1.10
8 × 10¢ = 80¢
7 × 5¢ = 35¢
8 + 7 = 15
80¢ + 35¢ = 115¢ = $1.15
10 × 10¢ = 100¢
5 × 5¢ = 25¢
10 + 5 = 15
100¢ + 25¢ = 125¢ = $1.25
(a) $98 - $30 = $68 2 × $68 = $136
Kelly has $136 in the end.
(b) 4 × $98 = $392
$392 - $136 = $256
Kelly spends $256 on the tennis racquet.
7. Making Supposition
Making supposition is commonly known as "making an assumption."
Students can use this strategy to solve certain types of math word
70 Must-Know Word Problems Level 3