1. mathplus www.banhar.blogspot.com
Dr. Yeap Ban Har
Making Connections through Multiplication and Division | Grades 3 – 5
SYNOPSIS |Using the CPA Approach which is fundamental to Singapore schools, we will explore
the learning of multiplication facts as well as multi-digit multiplication and division. The extension
between multiplication and division in whole numbers and in factions will be discussed. We will also
look at anchor tasks for lessons where the emphasis is on practicing a previously-learnt skill.
The CPA (Concrete  Pictorial  Abstract) Approach is based on Bruner’s ideas of
enactive, iconic and symbolic representations. New concepts are learnt through
concrete experiences.
Stages in Learning Multiplication Facts
1. Counting
2. Figuring Out
3. Remembering










3 x 4 =
6 x 4 =
7 x 4 =
9 x 4 =
Let’s figure out!

2. 










3. Anchor Tasks for Multi-Digit Multiplication and Division
Source | Maths – No Problem! Year 4

4. A farmer sells his crops to 3 restaurants.
He divides 525 heads of lettuce equally among the 3 restaurants.
How many heads of lettuce does each restaurant receive?
Source | Math in Focus 4
Hundreds tens Ones
  
Note | In Partitive Division, you know the number of groups. Example – Four children
shared 12 cookies equally. In Quotitive Division, you know the number of items in
each group. Example – Twelve cookies are put in packs of 4 cookies.

5. Anchor Tasks for Practice Lessons
Source | Every Child Can Do Math
Fraction Connections
1. How many halves are there in
2
1
2 ?
Source | Math in Focus 4
2. Find the value of
4
3
of 20.
Source | Primary Mathematics 4

6. mathplus www.banhar.blogspot.com
Dr. Yeap Ban Har
Developing Number Sense through Counting, Adding and Subtracting |
Grades K – 2
SYNOPSIS |Using the CPA Approach which is fundamental in Singapore schools, we will explore
what students need to be able to do to be proficient in counting, adding and subtracting. How can
teachers of younger children develop deep understanding using small numbers? The use of ten
frames, number bonds and bar models will be emphasized. We will also learn several mental
computation strategies.
The CPA (Concrete  Pictorial  Abstract) Approach is based on Bruner’s ideas of
enactive, iconic and symbolic representations. New concepts are learnt through
concrete experiences.
Teaching kids to count is fine, but teaching them what counts is best.
Bob Talbert
A fundamental idea in counting is that you can only count things in the same set i.e.
having the same ‘noun’. It is obvious that four oranges and an apples does not make
five apples. You don’t count different nouns!
In the same sense, 1 ten and 4 ones do not make 5 tens. Or, 1 half and 4 fifths do
not make 5 fifths. Or, worse, 5 sevenths!
Implications – integrate sorting with counting rather than teaching those as isolated
skills.
1. Sort the fruits.
2. Count how many in each basket.

7. Source | Earlybird Kindergarten Mathematics Reader
Let’s look at the use of ten frames for counting numbers less than 10 and numbers
greater than ten.
Implication – Help students progress from counting in ones to counting in tens and
ones.
Count, “1, 2, 3, 4, 5, 6, 7, 8, 9, 10. 1, 2. 10 and 2 is twelve.”
Instead of “1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.”

8. Anchor Tasks for Counting, Adding and Subtracting
1. Count 4 square tiles. Make a shape using 4 square tiles.
Source | Earlybird Kindergarten Mathematics

10. 2. 8 + 6
Source | Maths in Focus 1
3. Subtract 28 from 300.
Source | Primary Mathematics 2 Third Edition
Number Bonds
Two children starts with a small pile of beans. They take turn to remove either one
or two beans until all the beans have been removed from the pile. The winner is the
child who takes the last bean(s).

11. Bar Model
Source | Primary Mathematics 2 Third Edition

12. mathplus www.banhar.blogspot.com
Dr. Yeap Ban Har
Key Note | The Big Fives in Mathematics Classrooms
SYNOPSIS | What are five things that students who are good in math do? What are five key
competencies that students need to have by the time they start middle school? Who are the five
wise guys who can answer most of the questions we have in mathematics education? What did they
say about mathematics learning?
In his engaging and thought-provoking keynote, Ban Har will offer a simple framework that guides
good mathematics instruction. You will learn how students develop ability in visualization and
generalization, number sense, metacognition and communication. We will see that "it is fine to teach
children to count, but it is best to teach them what counts!" (Bob Talbert)

14.  visualization
 generalization
 number sense
 metacognition
 communication
 I can show.
 I can draw.
 I can explain.
 I can write about it.
 I can find ways to challenge myself.
 Piaget
 Dienes
 Vygotsky
 Skemp
 Bruner
 explore
 structure
 journal
 reflect
 practice
core
competencies
good
at math
elements of
effective lessons
learning
theories

15.  model
 scaffold
 let go
 explain
 counsel
Yulin wants to make 14 small identical stars and 20 large identical stars using wire.
She has made 12 small stars and 7 large ones using 960 cm of wire. The length of
the wire she used for 3 large stars is the same as that for 4 small stars.
What is the length of the wire she needs to make the remaining stars?
Source | Primary School Leaving Examination
tools
pedagogical

16. mathplus www.banhar.blogspot.com
Dr. Yeap Ban Har
Developing Foundations for Multiplication and Division | Grades K – 2
SYNOPSIS | Using the Problem-Solving Approach which is fundamental in Singapore schools, we
will explore the acquisition of multiplication facts. What are some things that young children do that
prepare them for formal learning of multiplication and division. The role of visuals such as dot
diagrams and language will be emphasized.
Source | Maths - No Problem!

18. Stages in Learning Multiplication Facts
1. Counting
2. Figuring Out
3. Remembering










Stories for Multiplication
 Equal Groups
 Array
 Comparison
 Rate
 Combination
Stories for Division
 Sharing
 Grouping
Note | In Partitive Division, you know the number of groups. Example – Four children
shared 12 cookies equally. In Quotitive Division, you know the number of items in
each group. Example – Twelve cookies are put in packs of 4 cookies.
3 x 3 =
6 x 3=
7 x 3 =
9 x 3 =
Let’s figure out!

19. Anchor Tasks
1. Find the value of 18 x 3.
2. Find the value of 54  3.

20. mathplus www.banhar.blogspot.com
Dr. Yeap Ban Har
Fractions | Grades 3 – 5
SYNOPSIS | In this session we will look at two key ideas in learning fractions and consider strategies
to teach addition, subtraction, multiplication and division of fractions.
Example 1
Share among 4 friends. They all get the same amount.

21. Example 2
Example 3
Grace arranged
2
1
,
3
2
,
4
1
in order, from the greatest to the smallest.
Is she correct?

22. 
4
1
2
1


9
8
3
1
3 
 6
3
2


4
3
3
1

 2
3
1

3
1
3

23. 3
1
Example 9
Find the value of 12
5
4
 .
Give your answer as a fraction in the simplest form.
Source | PSLE Singapore 2013
Example 10
The total cost of a pen and a book is $32. The cost of the pen is
5
3
the cost of the
book. What is the cost of the book?
Source | PSLE Singapore 2013
Example 11
Kai Li spent
3
1
of her money on 5 magnets and 11 postcards.
The cost of each magnet is 3 times the cost of each postcard.
She bought some more magnets with
4
3
of her remaining money.
How many magnets did Kai Li buy altogether?
Source | PSLE Singapore 2013

24. Example 12
Mr Lim had a total of 540 long and short rulers. After selling an equal number of
both types, he had
3
1
of the long rulers left and
6
1
of the short ones left.
What was the total number of rulers left?
Source | PSLE Singapore 2014

25. mathplus www.banhar.blogspot.com
Dr. Yeap Ban Har
Math Journals| Grades K – 2
SYNOPSIS | In this session we look at a framework for teaching to mastery and examine the role
of journaling in the process. How does journal entries look like? What do we do with them? How
does it help in assessment and differentiation?
Teaching to Mastery
To master counting, Kindergarteners should be able to tell the number of a set of
objects. Fluent at counting means able to do it with accuracy, confidence and some
degree of flexibility. It does not mean count fast!
To master number facts, First Graders should be able to recall 1 + 4 = 5 and, in
cases, they cannot recall (8 + 9), they should be able to figure out (8 + 9 = 10 + 7).
Fluency is not mere speed. When one is fluent in a language, it does not mean one
speaks fast, does it?
 Anchor Task
o Explore
o Structure
o Journal
o Reflect
 Guided Practice
 Independent Practice
The framework for teaching to master involves several stages and journaling is one
of them.

26. In an open lesson for students who had just completed Kindergarten, the anchor task
was to use unifix cubes to explore the number bonds of 5. Later, they extend their
investigation to the cases of 3 and 7.
The journal writing was to look for and write down two numbers that add up to a
given number. Owen was given 6, for instance.
Source | Owen’s Journal
Source | Alex’s Journal
Owen looked for two numbers that add up to 6.
He tried to record his results in his journal.
Owen was able to make number bonds for 6.
He was also able to see possibilities – 5 and 1,
3 and 3 as well as 4 and 2. Is recording was
less successful as his ability to form numerals was
still not developed. He also attempted to use
symbols (not used in this lesson) that he had little
understanding of.
Although not required, Alex had chosen to record the results of the
investigations in his journal. It can be seen that he went from random
recording to making a systematic list. It is interesting to observe the
case of 0 continued to elude many students.
Journal allows advanced learner to experience differentiated
lesson. In this case, Alex went to explore cases involving negative
numbers while waiting for the class to complete their investigation..

27. In an open lesson for students who had just completed Grade 1, the anchor task was
to add three single digit numbers such that the vertical sum equals the horizontal sum.
After working on it with their friends and discussing it, students document how it is
done.
__________________________________________________________________
__________________________________________________________________

28. __________________________________________________________________