Carlo Magno, PhD
De La Salle University, Manila
read and write large whole numbers and
round off whole numbers to the nearest
thousands and millions.
find the greatest common factor and the
least common multiple of given numbers.
apply divisibility rules for 2, 3, 4, 5, 6, 8, 9, 10,
11 and 12 on different contexts.
simplify a series of operations on whole
numbers and solve problems involving these.
perform the four fundamental operations on
fractions and mixed numbers and solve related
investigate the relationship between fractions
and decimal numbers.
explore, know and understand the concept and
value of a decimal number.
add and subtract decimal numbers with values
through thousandths and solve problems
multiply decimal numbers of values up to the
hundredths and solve problems involving these
divide decimal numbers of values up to the
hundredths and solve problems involving these
manipulate ratios and solve problems involving
ratios and proportions
know and understand the concept of percent
and to solve problems involving percents.
explore the concept of integers and compare
integers with whole numbers, fractions and
use the notation and computation of positive
whole number powers to find values of numbers.
perform the four basic operations on integers
perform a series of operations following the
GEMDAS rule and solve problems involving a
series of operations on whole numbers.
add and subtract fractions and decimal
numbers and solve problems involving these
multiply fractions and decimal numbers and
solve problems involving multiplication of
fractions and decimal numbers.
divide fractions and decimal numbers and
solve problems involving division of fractions
and decimal numbers.
use ratio and proportion in a variety of ways
and solve problems in different contexts.
use percent in a variety of ways and solve
problems in different contexts.
What do you think is the problem in the
child’s concept in subtraction?
Number sense is the ability to understand
and use numbers and operations flexibly.
Number sense results in a view of numbers as
meaningful entities and an expectation that
mathematical manipulations and outcomes
should make sense.
Number sense involves:
knowing how to write and represent numbers in different
recognizing the quantity represented by numerals and
other number forms
discovering how a number relates to another number or
group of numbers
Number sense develops gradually and varies as a
visualizing them in a variety of contexts
relating to them in different ways
In the primary and intermediate grades, number
sense includes skills such as:
representing numbers with manipulatives and models
understanding place value in the context of our base 10
writing and recognizing numbers in different forms such as
expanded, word, and standard
expressing a number different ways—5 is "4 + 1" as well as
"7 - 2," and 100 is 10 tens as well as 1 hundred
ability to compare and order numbers—whole
numbers, fractions, decimals, and integers
the ability to identify a number by an attribute—such as
odd or even, prime or composite-or as a multiple or factor
of another number
Whole numbers describe the number of students in a
class or the number of days until a special event.
Decimal quantities relate to money or metric
Fractional amounts describing ingredient measures or
Negative quantities conveying temperatures below
zero or depths below sea level.
Percent amounts describing test scores or sale prices.
Able to estimate and make a meaningful
interpretation of its result.
Number sense develops as students:
understand the size of numbers,
develop multiple ways of thinking about and
use numbers as referents,
develop accurate perceptions about the effects of
operations on numbers (Sowder 1992).
Using numbers flexibly when mentally
Judging magnitude of numbers and
reasonableness of results
Moving between number representations
Relating numbers and symbols and
Making sense of numerical situations
Looking for links between new information
and previously acquired knowledge
There should be a balance in teaching
mathematical algorithms with mental
computations, problem solving and the use
Problem-centered projects show how children
who engage in meaningful arithmetic
problem solving also develop autonomy and
adaptability in the learning situation.
Partitioning numbers using tens and ones.
"First I added the 20 and 10 and got 30.Then I
added the 9 and 4 and got 13.Then I added
the 10 from 13 to 30 and added 3 more and
Counting on or back from a number. "First I
counted on from 29 by tens and went 29, 39.
Then I counted on 4 more — 40, 41, 42, 43."
Using "nice numbers." Nice numbers are
multiples of 10 or other numbers that are
easy to work with. "I know that 30 plus 15 is
45, but 29 plus 14 is 2 less than that, so it’s
Translating to a new problem. "I took one
away from the 14 and gave it to the 29 to
make 30.Then I had 30 plus 13, which is 30
plus 10 plus 3, which is 43.
Mentally estimates and calculates addition
and subtraction to 100 using strategies based
on ones and knowledge of number facts.
Use alternative strategies to decompose
(break up) two-digit numbers to add and
subtract numbers, ex. “to subtract 43 I took
off the 40 and then 3 more’, or ’26 and 43 is
43, 53, 63 and 6 more is 69’.
Adds a list of one-digit numbers or amounts
in an efficient way, e. g. 76c + 49c +14c … 76c
+ 14c makes 70, 80, 90 plus 50c is Php 1.40
less 1c makes Php 1.39.
Multiplies and divides by one-digit numbers
involving multiples of 10.
Uses a variety of strategies to add and
subtract whole numbers and money only
when mental strategies are inadequate.
Uses regrouping in number exploration and in
computation such as in showing
decomposition (346 is 34 tens and 6 ones or 3
hundreds and 46 ones ) or when needing to
subtract (46 ones becomes 3 tens and 16
Uses estimation and approximation to round
off and justify whether sums and differences
Uses multiplication and division as inverse
operations and recognize the commutativity
or multiplication (36 9 = 4, 4 x 9 =36.
Shows understanding of multiplication and
division based on equal groups and uses
written methods for calculating products and
quotients such as expanded
multiplication, ex. 345 x 3 is 300 x 3 40 x 3
5 x 3, and division, such as 345 3 as 3 x ?
makes 300 3 X ? makes 45.
Devise shorter methods for written
calculation based on understanding of
strategies based on groupings.
Uses a calculator appropriately by entering
digits correctly and in the right order for all
Makes estimates and checks reasonableness
Checks whether a prediction about numerical
operations or patterns holds true when
computational demands are beyond the
child’s levels of ability, ex. 15 + 15 = 130, so 115
+ 115 = 230, so 1115 + 1115 = 2230.
Tests understanding of place value such as
increasing digits by ten or pretending that a
numerical key is inoperable.
Finds relationships between operations such
as 45 X 6 is the same as 45 + 45 + 45 +45 + 45
Finds related patterns in the number system
by exploring the effect of multiplying and
dividing by 10 including decimals.
Explores rounding-off options.
Product of 12 and 6 is _____
12 X 6 =
6 X 12 =
120 X 60 =
12 X 0.6 =
Complete the pattern: 6, __, 18, __, 36
List factors 0f 72 ______
1. Make up a number story that matches 12 x
6 =. Explain how you got the answer.Think
about how your answers would change if you
changed the problem to 120 x 6.Would the
answer be smaller or larger if it was changed
to 120 x 0.6? Explain your thinking by
recording your solutions.
2. Is 12 x 6 the same as 6 x 12? Explain your
thinking by recording your solution.
3.Why is 12 6 different from 12 x 6? Explain
or thinking. Make up a problem to show how
the two number facts are different.
Teaching with manipulatives and models
place-value and hundreds charts
Use of frames
How many counters are there?
How many spaces without counters are
Hundreds chart (121)
•Help children recognize number
•Work with large numbers
•visual support for problem solving
Many teachers use the calendar as a source of
counting, patterns, number sequence, odd and even
numbers, and multiples of a number
create word problems related to the calendar
A hundreds chart can help them count the
number of days in school, and the current day’s
number can be the "number of the day.
On the 37th day of school, children may describe
that number as 30 plus 7, 40 minus 3, an odd
number, 15 plus 15 plus 7, my mother’s age, or 1
more than 3 dozen.
The calculator can be a fun tool for quick
mental computation practice.
Have the children predict the sum or
difference and the calculator is used to
Press 0 + 2 = on the calculator.
Then press any other number, such as 7, and
hold your finger over the = key.
Have the children predict the sum then press the
= key to confirm it.
As long as they continue to press only a number
and =, the "machine" will continue to add 2.
The use of the calculator and the immediate
feedback reinforce computation and encourage
children to keep playing
Guess My Number
Choose a secret number and tell children a range that your
number falls within.
You can start small, with 1 to 10, or use a larger range
(such as 1 to 100, 25 to 75, or 150 to 250).
Have children guess your number and tell them whether
their guess is larger or smaller than your number.
Children will quickly develop strategies that help them
zero in on your secret number.
To extend this game, choose a secret number from a wide
range such as 1 to 500 and give one clue, such as that it is
even, it ends in 4, or the sum of the digits is 9; then ask
students to start guessing.
Stand Up and Be Counted
Ask children to describe the number 25 in as many
ways as they can ("number of the day,“) and record
their ideas as an example.
Each child draw from a bag of squares numbered 1
through 100 and write down as many ways as they
can to make the number they drew.
Ask a volunteer to stand up and read one statement at
a time about his or her number.
If that statement is true for other children’s numbers,
they stand up.
If it is not true, they remain seated.
Through discussion, the children can begin to focus on
the characteristics of the numbers and their
Help students identify whole-number
relationships that are different from
decimals, fractions, and integers.
Students may unsuccessfully try to apply
these relationships to decimals, fractions, or
34.5 > 3.456,
1.11 < 1.111,
68.2 = 68.20
The whole number 6 is greater than the
whole number 5, but when unit fractions have
these numerals as denominators, the
relationship is reversed, and 1/5 > 1/6.
Similarly, -5 > -6
Read and discuss any of the many available counting
books that illustrate numbers up to 10, 20, and so on.
Identify ways that numbers are represented in print.
▪ "When are numbers shown in standard form?
▪ Word form?
▪ Short-word form?
▪ When are actual numbers used?
▪ When are rounded numbers used?"
Ask students to write about number
representations by defining and giving
examples of different forms of numbers-
standard, expanded, word, and short-word.
For students in primary grades, find and discuss
different ways numbers are used in the
environment (e.g., addresses, time,
temperatures, grades, speed limits, phone
numbers, on recyclable plastics).
Have students in intermediate grades find
population figures for town or city, state, and
country.Then ask them to compare and order
the populations they found with those other
represent numbers with concrete objects.
Choose linear measurements relating to
science, such as the sizes of dinosaurs, and
represent lengths using pieces of string or yarn.
Label the strings, and then compare and order
Make comparisons between string
lengths, string length and classroom
dimensions, string length and students'
height, and so on.
Give examples of games to enhance number
Discuss and practice writing numbers using
scientific notation. Have them find examples
of measures written with scientific notation
and identify the situations in which they are
used and why.