What is Singapore Math?
What Is Singapore Math?
You may be wondering what Singapore Math is all about, and with
good reason. This is a totally new kind of math for you and your
child. What you may not know is that Singapore has led the world in
math mastery for over a decade; its students become competent and
proficient mathematicians at very early ages. Even better, they
grow to be capable problem solvers who think mathematically with
ease. Wouldn't it be nice if your child could enjoy the same
success with math?
Well, there's good news: We're teaching Singapore Math to your
child this year. So let's discover what it's all about and how you
can help your child succeed. It all begins with understanding the
curriculum and seeing some examples-just what we'll do today.
First, you need to know that Singapore Math takes a slightly
different mathematical approach than what you may be used to. It
revolves around several key number‐ sense strategies: (1)
building number sense through part‐whole thinking, (2)
understanding place value, and (3) breaking numbers into decomposed
parts or friendlier numbers, ones that are easier to work with in
the four operations (addition, subtraction, multiplication and
Second, Singapore Math does something dramatically different
when it comes to word problems. It relies on model drawing, which
uses units to visually represent a word problem. Students learn to
visualize what a word problem is saying so they can understand the
meaning and thus how to solve the problem.
Third, we have mental math, which teaches students to calculate
in their heads without using paper and pencil. Sure, your child
will still need to commit some facts to memory, but mental math
will teach him or her to do calculations using proven strategies
that don't require pencil and paper.
Fourth, the strategies taught in Singapore are layered upon one
another. One strategy is the foundation for another one. You'll
notice this as you read through this letter. For example, students
need prior knowledge of bonding in order to be successful at
strategies they will learn later on (like vertical addition).
Last, Singapore Math teaches students to understand math in
stages, beginning with concrete (using manipulatives such as
counters, number disks, dice, and so on), then moving to pictorial
(solving problems where pictures are involved), and finally working
in the abstract (where numbers represent symbolic values). Through
the process, students learn numerous strategies to work with
numbers and build conceptual understanding.
With time and practice, they eventually master the traditional
methods and algorithms. Let's take a closer look at all the layers
of Singapore Math.
Singapore Math is a base‐10 system. A number's place value
is determined from right to left, starting with the ones and moving
through the tens, hundreds, thousands, ten thousands, one hundred
thousands, to a million and beyond. In fourth grade, we add the
study of decimals into the mix. In class, we use tools such as
place value boards with disks and cards to help us organize,
visualize, and understand what these numbers actually mean and how
they relate to one another.
As we perform mathematical operations, we can move place value
disks from column to column on the place value board to demonstrate
regrouping (making a ten out of 10 ones).
An algorithm is a systematic, step‐by step procedure to
solve a problem using a mathematical operation. For example, with
subtraction, we have learned to line our numbers up vertically so
that the digits are in the correct place value columns. We've
learned to subtract the digits moving from right to left, using
regrouping or borrowing, in order to get the correct answer to the
In Singapore, traditional algorithms are taught and mastered
with the help of the place value mat. However, we also teach
alternative algorithms or strategies to solving equations often
before we teach the traditional ones. This helps us build and
reinforce our understanding of number sense and place value. This
also allows students to use a strategy that they are competent at
using for any problem. Rather than having one strategy, they may
have several to choose from, and they can use the one that's most
intuitive for them.
In the rest of this letter, I'll be explaining some of the
strategies that your child may be learning to build number sense
and an understanding of place value. Please don't hesitate to call
me or come and visit the classroom to see firsthand what Singapore
Math is all about!
Number Bonds: PartWhole Thinking
The beginning of number sense is viewing each digit as a part of
a whole. This is very similar to fact families, where a number has
specific "relatives" in its family. Let's take the number 6 as an
example. 6 is 6 and 0, 5 and 1, 4 and 2, and last, 3 and 3. This
understanding becomes very important when students start doing
operations with the number 6.
After students learn digits 1-9, they master what combinations
make 10. 10 is an anchor number in Singapore. So, in K and grade 1,
students will spend a significant amount of time learning their
bonds through 10. Number bonds can also be created
with multiplication and division fact families, using two
factors and a product. For example, 16 is 16 and 1, 2 and 8, and 4
Once students know the parts that make up each number, they can
add any numbers together by making 10s. For example, when students
add 7 + 5, they find how many 10s they can make, and label the
leftover parts as 1s. In this example, there is one 10 and two 1s
remaining, so the answer is 12. Instead of memorizing just the
fact, the student has a strategy to work with addition.
Decomposing and Branching
Students spend time learning how to break numbers into place
value groupings on the place value board. This is called
decomposing numbers or using expanded notation. After students
practice breaking numbers apart into place value groupings, we
teach them to add and subtract by place value. This is branching.
The goal with branching is for students to break numbers into place
value groupings and then do the operation with those place value
For example, 23 + 42 would be branched into tens and ones. Then
students will add each place value grouping separately, and then
add the groupings together.
Take a look at how branching works. 23 + 42
20 + 40 = 50 3+2=5 50+ 5 =55
The goal for branching is for students to eventually be able to
look at the problem and work it out mentally. Remember that mental
This algorithm uses expanded notation or decomposed place value
groupings to add each place value separately and then together. In
expanded notation, you write a number horizontally and expand or
stretch it to reflect all its place value parts.
For example, the number 8,735 would be written as (8,000 + 700 +
30 + 5).
With left‐to‐right addition, we take our expanded
parts and add them together, starting at the left and moving to the
right. Let's take the equation 45 + 33. First, we decompose numbers
into place value groupings.
45 + 33 = 40 + 30 = 70 5 + 3= 8
With students in fourth grade and up, we would group the numbers
using parenthesis, and it may look like this: (40 + 30) + (5 + 3) =
70 + 8 or 78.
Another strategy for addition is to use a vertical strategy and
work with numbers that are lined up vertically but added much
differently than you and I did when we were in school. The
difference is that instead of adding columns, moving from right to
left, and regrouping, we add each column separately and write it
down below. Then we add our partial sums together at the end.
Let's take a look at how this strategy works.
124 +152 6 70
It doesn't matter at all where we start: in the ones, tens or
hundreds column. This reinforces the commutative property of
addition in that the order in which we add doesn't matter.
Let's take a look at one more example:
567 +489 16
Multiplication and Division Strategies
In Singapore Math, multiplication and division concepts
(repeated addition, equal groups of, and so on) are taught starting
in first grade. We build upon students' prior
knowledge with addition and subtraction with disks on a place
value mat to help students visualize the groups. We also use other
manipulatives, such as counters, to make groups and arrays (equal
group rows and columns). This helps us conceptualize
By third grade, we're memorizing times tables in order to
automatically recall facts. Once we get to multidigit
multiplication, we learn some visually helpful methods to organize
numbers in order to understand how each place value grouping
relates to the others.
Distributive Property for Multiplication
The distributive property allows students to calculate once
complicated multiplication problems easily by breaking the
multidigit number into smaller factors. We distribute the larger
number into smaller, more manageable parts. Let's take a look at
how this method works.
3 x 15 =
First, we look to the multidigit factor, which is 15. We break
it into place value groupings, 10 and 5. Once they are broken into
parts, we take each part and multiply it separately by the factor
3. Then, we add the parts together.
(3x10) + (3x5) 30 + 15 = 45
This allows students to work this problem mentally rather than
write it out on paper.
Area Model or Partial Product Method
Using the area or partial product method is a fast and effective
way to multiply by place value. This model uses a box design, where
factors are broken into place value groupings and written outside
the box. Then each grouping is multiplied separately and added
together to get a final product.
For example: 45 x 26 40 5
1100 + 70 = 1170
Or 45 x 26
30 6 x 5 240 6x40 100 20 x 5 800 20 x 40
This is applying the partial product method.
You multiply each partial product and then add to get your final
Bonding and branching come in handy when we're learning
division. Branching numbers into bonds to decompose them into
friendlier, more manageable numbers allows students to work with
parts and simpler numbers. Using place value disks and charts helps
students to see division as repeated subtraction. It also allows
students to trade disks (distribute them) from place value column
to place value column, taking the dividend and breaking it apart
into an equal quantity in order to get a quotient. Any leftover
parts become our remainder.
Distributive Property for Division
You can use the distributive property with a multidigit division
problem when you have a single‐digit divisor. You distribute
your dividend into two or more friendlier parts. The catch here is
that both parts need to be a multiple of the divisor or equally
divisible by the divisor.
So, when your dividend is divided up, it may not be into place
value groupings. It is worth taking some time to practice breaking
numbers into other parts besides place value groupings. You can use
the mat to do this. For example, 45 can be broken into 40 and 5; it
can also be broken into 30 and 15 or 20 and 25. Also, your child
needs to have prior knowledge of divisibility rules.
Let's look at a few examples of how to use the distributive
property to divide.
52 ÷ 4 =
52 needs to be broken into two or more parts with both parts being
divisible by 4. 50 and 2 doesn't work, 30 and 22 doesn't work, but
40 and 12 will work.
(40 ÷ 4) + (12 ÷ 4) 10 + 3 = 13
75 ÷ 5 =
(50 ÷ 5) + (25 ÷ 5) 10 + 5 = 15
42 ÷ 3 =
(30 ÷ 3) + (12 ÷ 3) 10 + 4 = 14
Partial Quotient Division
Partial quotient division is similar to long division, but
instead of having to be exact on the top with a quotient, we make
partial quotients and then add them to get our final quotient. You
can think of it as a successive approximations or estimates. One
strategy would be to use multiples of 10s, 100s or 1,000s as your
estimates because they are easy to multiply and divide by. The
quotient is built through vertical steps (which resemble the game
Hangman), and we don't have to get the exact quotient.
We can ask ourselves, how many times do we know for sure that
our divisor can go into our dividend, or how many times can we pull
it out of our dividend? We continue to find partial quotients until
we have no remainder or a remainder is less than the divisor. We
then add our partial quotients to arrive at the final quotient.
Let's look at an example:
19 R 1 3 58
30 10 28
12 4 16
15 5 R1 19
This strategy gives each student the freedom to work with the
facts that they are competent at. Students can choose the multiple
that makes sense to them, so each equation may look different when
they turn in the work. However, the final answer should look the
Mental math is one of the cornerstones of Singapore Math as its
emphasis is on helping students to calculate mathematically in
their heads, thus developing number sense and place value. It
encourages flexibility and speed when working with numbers. We
practice mental math strategies and do lots of fun activities that
support the skill. Please come to class and participate with us one
Model Drawing is the key strategy we use to solve word problems.
Starting in second grade, we use units ( ) to help us create a
visual representation of a word problem. We learn a set of steps to
help us internalize the process.
Read the problem to get a sense of what it is asking.
Decide who and what the problem is about.
Draw units for each who and what.
Reread the problem and adjust our units to match the word problem.
Decide what the question is asking of us and place a question mark
in place. Work our computation.
Write our answer in a complete sentence.
Well, that's Singapore Math in a nutshell. What we're finding as
we teach this program is that math is math, no matter how you
package it. Number sense, place value, calculations, operations,
puzzles, word problems, problem solving, visuals,
relationships-these are all familiar parts of math. Singapore Math
just provides us with some new vocabulary and numerous strategies
that enable all students to learn mathematics to mastery.
Thank you for your continued interest and support as your child
discovers the curriculum of the world's math leaders.