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Introduction
Eva is going camping and is thinking
about what clothes to take. It is in the
mountains and the temperature
drops 6ºc for every kilometer you
ascend. Your house is at sea level and
the temperature is 14ºC. The camp is 3
kilometers high and you need to know
how many degrees the temperature
will drop to choose clothes
1 2 3 km
1
ºC
2
3
4
5
6
7

Activity 1
What operation do you have to perform to know how much the temperature
will drop?
(-6) x (-3) (-6) x 3
6x3 6 x (-3)

Activity 1
Exactly! If the temperature drops 6ºc (-6) every kilometer and there are
3 kilometers, we will have to multiply 3 by -6
(-6) x (-3) (-6) x 3
6x3 6 x (-3)

Explanation
We still don't know how to
multiply a positive number by a
negative number. But we can
see multiplication as repeated
addition 6 x 3 = 6 + 6 + 6
(-6) x 3

Activity 2
How can you write the above multiplication as a repeated addition?
6 + 6 + 6 (-3) + (-3) + (-3)
3 + 3 + 3 (-6) + (-6) + (-6)

Activity 2
Indeed!
6 + 6 + 6 (-3) + (-3) + (-3)
3 + 3 + 3 (-6) + (-6) + (-6)

Activity 2
And since we already know how to add negative numbers, we can perform this
operation. What is the result of this sum?
(-6) x 3 = (-6) + (-6) + (-6) =
2 3 4 5
6 7 9 0 +
1
8
-

Activity 2
(-6) x 3 = (-6) + (-6) + (-6) =
2 3 4 5
6 7 9 0 +
1
8
- 1

Activity 2
(-6) x 3 = (-6) + (-6) + (-6) = Very good!
1
2 3 4 5
6 7
8
9 0
-
+

Activity 2
By multiplying a negative number by a positive number, what type of
number have we obtained?
Positive Negative

Activity 2
Exactly!
Positive Negative

Explanation
When multiplying a negative
number by a positive number
we always obtain a negative
number. What result will we
obtain when multiplying a
positive number by a negative
number?
Distributive
property

Activity 3
Use the distributive property to rewrite this operation:
a x (b - c) =
a x b - c a x b - a x c
a x b + a x c (a x b) - c

Activity 3
Correct!
a x (b - c) =
a x b - c a x b - a x c
a x b + a x c (a x b) - c

Explanation
The first thing we have to do is
write the integer as the
difference of two positive
numbers. Write any pair of
positive numbers whose
difference is equal to -3
6x ( ) = ?
-
6 x (-3) = ?

Explanation
Very good! 1 and 4, since 1
minus 4 equals -3
6x ( - ) = ?
6 x (-3) = ?
1 4

Activity 3
Use the distributive property to rewrite this product:
6 x (1 - 4) =
6 x 1 - 6 x 4 = 6 x 1 - 4 =
6 x 1 + 6 x 4 = 6 x (-4 + 1) =

Activity 3
Correct!
6 x (1 - 4) =
6 x 1 - 6 x 4 = 6 x 1 - 4 =
6 x 1 + 6 x 4 = 6 x (-4 + 1) =

Activity 3
What is the result of the first multiplication?
6 x (1 - 4) =
6 x 1 - 6 x 4 =
6 x (-3) =
?

Activity 3
Very good! And the result of the second multiplication?
6 x (1 - 4) =
6 x 1 - 6 x 4 =
6 x (-3) =
6 ?
-

Activity 3
Indeed!
6 x (1 - 4) =
6 x 1 - 6 x 4 =
6 x (-3) =
6 24
-

Activity 3
Now do the subtraction
6 x (1 - 4) =
6 x 1 - 6 x 4 =
6 x (-3) =
6 24 = ?
-

Activity 3
Very good!
6 x (1 - 4) =
6 x 1 - 6 x 4 =
6 x (-3) =
6 24 -18
=
-

Explanation
If we multiply a positive number by a negative number, we always get a
negative number
(+6) x (-3) = (+6) x (1 - 4) = (+6) x 1 - (+6) x 4 = 6 - 24 = -18

Explanation
What result will we have then
when multiplying a negative
number by another negative
number? What we have to do is
write the second multiplicand
as the difference of two
positive numbers. Write any
pair of positive numbers whose
difference is equal to -3
(-6) x ( - ) = ?
(-6) x (-3) = ?

Explanation
Very good! 2 and 5
(-6) x ( - ) = ?
(-6) x (-3) = ?
2 5

Activity 4
Use the distributive property to rewrite this product:
(-6) x (2 - 5) =
(-6) x 2 + (-6) x 5 = (-6) x 2 - 5 =
(-6) x 2 - (-6) x 5 = (-6) x (-5 + 2) =

Activity 4
Correct!
(-6) x (2 - 5) =
(-6) x 2 + (-6) x 5 = (-6) x 2 - 5 =
(-6) x 2 - (-6) x 5 = (-6) x (-5 + 2) =

Activity 4
What is the result of the first multiplication?
(-6) x (2 - 5) =
(-6) x 2 - (-6) x 5 =
(-6) x (-3) =
?

Activity 4
Very good! And the result of the second multiplication?
(-6) x (2 - 5) =
(-6) x 2 - (-6) x 5 =
(-6) x (-3) =
(-12) ?
-

Activity 4
Indeed! Now do the subtraction
(-6) x (2 - 5) =
(-6) x 2 - (-6) x 5 =
(-6) x (-3) =
(-12) (-30)
- = ?

Activity 4
Indeed! Now do the subtraction
(-6) x (2 - 5) =
(-6) x 2 - (-6) x 5 =
(-6) x (-3) =
(-12) (-30)
- = 18

Explanation
If we multiply a negative number by a negative number, we always get
a positive number
(+6) x (-3) = (+6) x (1 - 4) = (+6) x 1 - (+6) x 4 = 6 - 24 = -18

Summary
To multiply two integers, we multiply
the two numbers ignoring their sign
and then we add it. If the two factors
have the same sign, we add a positive
sign, otherwise we add a negative
sign
+x+=+
+ x - = -
- x - = +
- x + = -
6 x 3 = 18
6 x (-3) = 18
(-6) x (-3) = 18
(-6) x 3 = -18

Activity 5
What is the result of multiplying -7 by -5?
(-7) x (-5) =
2 3 4 5
6 7 9 0 +
1
8
-

Activity 5
(-7) x (-5) =
2 3 4 5
6 7 9 0
+
1
8
-

Activity 5
(-7) x (-5) =
2
3
4 5
6 7 9 0
+
1
8
-

Activity 5
(-7) x (-5) = Very good!
2
3
4
5
6 7 9 0
+
1
8
-

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