This document discusses addition and subtraction of integers using two-colored counters. It provides examples of representing integer addition and subtraction problems using red and yellow counters to model positive and negative numbers. Rules for adding and subtracting integers with the same sign or different signs are explained. An example problem about the distance between a submarine and airplane is worked out to demonstrate subtracting a negative number by adding its additive inverse. Finally, an evaluation activity called FACEing MATH is described that has students answer math questions to complete drawings of faces.

1. INTEGERS:
ADDITION & SUBTRACTION
Presented by: CARLO JUSTINO J. LUNA
Conrado P. Macapinlac, Sr. High School
SST-1 / Math Leader

2. EXPLORE!
JOSE’s SAVINGS
Jose saves Php 8 from his day’s
allowance. However, he has to
pay Php 5 to the class treasurer.
After paying the class treasurer,
how much is left of his savings for
the day?

3. COLORED COUNTERS can be used
to model this problem. The idea of
using colored counters or rods to
represent signed numbers goes a
long way back.

4. HISTORICAL TRIVIA
In 200 BC, Chinese accountants used
black (negative) rods for debts and red
(positive) rods for credits. However,
it was Girolamo Cardano (1501-1576)
who introduced negative numbers
and their properties formally.
Nowadays, if there are more credits, the
account is positive or “in the black.” If
there are more debits than credits, the
account is in debt or is “in the red.”

5. Operations with integers can be
modeled using two-colored
counters.
Positive
+1
Negative
-1

6. ZERO PAIRS
They have a value of zero.

7. Now, we’re ready to solve the
problem above using counters. Let
us represent – 5 + ( 8 ) using red
and yellow counters.

8. When using two-colored counters
to model addition, build each
addend then find the value of the
collection.
(– 5) + ( 8 ) = 3
Jose has Php 3
left.
zero pairs

9. Modeling addition of integers using
counters:
7 + −6 = 1

10. Here is another example:
−4 + −3 = −7
(Notice that there are no zero pairs.)

11. Build the following addition
problems:
1. 5 + (–3)
2. –5 + (9)
3. –4 + (–2)
4. 1 + (8)
5. –9 + (6)

12. While using counters helps us
visualize addition of integers, these
are inconvenience to use with
large numbers. And so, we have a
rule on adding integers with the
same sign and adding integers
with unlike signs.

13. Adding Integers with the Same Sign
To add integers with the same sign,
add their absolute values and
common their common sign.
Adding Integers with Different Signs
To add integers with different signs,
get the difference between their
absolute values and use the sign of
the integer with the greater absolute
value in the answer.

14. Try this!
1. 12 + (15)
2. –3 + (7)
3. –9 + (-4)
4. 4 + (–12)
5. –2 + (–3) + (–4)
6. 5 + (–7) + (–2)

15. EXPLORE!
Distance Between
A submarine cruises at a depth of
40 meters. Directly above it, an
airplane flies at an altitude of 192
meters. How far apart are they?

16. To find the distance between the
submarine and the airplane, we
represent a depth of 40m as –40
and a height of 192m as +192. To
obtain the difference between the
two, we write 192 – (–40).

17. Based on the illustration, it is clear that
the distance from the plane to the
submarine is 232 meters, that is 192 + 40
= 232. Thus, we say that these two
expressions are equal.
192 – (–40) = 192 + 40
232 = 232

18. Subtracting –40 has the same
effect as adding +40. We state this
finding as a rule on subtracting
signed numbers.
Subtracting Signed Numbers
To subtract a signed number, add
its additive inverse.
For any real numbers x and y:
𝑥 − (−𝑦) = 𝑥 + (+𝑦)
𝑥 − +𝑦 = 𝑥 + (−𝑦)

19. Examples: Subtract.
1. 5 − −3 = 5 + 3 = 8
2. 2. −3 − 2 = −3 + −2 = −5
3. 8 − −9 = 8 + 9 = 17
4. −5 − −12 = −5 + 12 = 7
5. 6 − −3 = 6 + 3 = 9
6. 10 − 4 = 10 + −4 = 6

20. EVALUATION
ACTIVITY: FACEing MATH by Kristin DeWit
Materials: crayons
copy of the blank face
copy of the 27 questions
The learners will answer 27 questions. They will do this by creating
wonderful faces while answering the questions. In this activity, the
learners who do not typically enjoy lecture/discussion days will find
themselves motivated to complete the lesson in order to draw and
color the desired face. Although all the student’s drawings should
have the same facial features, each face will look unique because of
the distracters in the choices of answers. FACEing MATH has never
been this fun! 