3) Complete a decimal fraction sum for
each number.
1) Calculate:
A) −2 + 6
4
B) −2 − 6
−8
C) −2 + −5
−7
Starter
2) Calculate:
A)
–2 + 5 + –8 = −5
B)
3 – –6 + 4 = 13
C)
–7 + 2 – –3 = −2
D) –12 – –6 + –9 = −15
A) 0.56
5
6
+
10 100
5
6
+
10 100
B) 0.287
2
8
7
+
+
10 100 1000
4) Round each number to 𝑥 decimal
places ( 𝑥 dp ):
1)
32.1981
( 2 dp )
4)
32.20
2)
18.973
26.90
0.07854
0.079
( 2 dp )
( 1 dp )
19.0
3)
26.89951
5)
( 3 dp )
0.00873
0.0
( 1 dp )
21/10/2023
Multiplying and dividing with
directed numbers
Learning Objectives:
Be able to multiply and divide negative numbers, extending to
calculations involving both operations
Negative numbers: Multiplying
Multiplication is repeated addition.
How can we describe these calculations?
3 × 4 = ‘the sum of 3 lots of 4’
4 + 4 + 4 = 12
8 × 5 = ‘the sum of 8 lots of 5’
5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 = 40
4 × −5 = ‘the sum of 4 lots of −5’
(−5) + (−5) + (−5) + (−5) = −20
Would we
describe
−5 × 4
differently?
Negative numbers: Multiplying
Negative numbers: Multiplying
−3 × 4 = −12
−4 × 3 = −12
−1
−1
−1
−1
−1
−1
−1
−1
−1
−1
−1
−1
-4 + -4 + -4 = -12
4 × −3 = −12
4×2=8
4×1=4
4×0=0
4 × −1 = −4
4 × −2 = −8
4 × −3 = −12
−4
−4
−4
−4
−4
-3 + -3 + -3 + -3 = 12
Negative numbers: Multiplying
What’s the same?
What is the value of the
What’s different?
missing number?
18
6
6
6 × 3 = 18
6
?
−18
−6
−6
−6 × 3 = −18
−6
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
Negative numbers: Multiplying
Negative numbers: Multiplying
3 × 4 = 12
−3 × −4 = 12
−3 × 2 = −6
−3 × 1 = −3
−3 × 0 = 0
−3 × −1 = 3
−3 × −2 = 6
−3 × −3 = 9
−3 × −4 = 12
×
+3
+3
+3
+3
+3
+3
=
−3 × 4 = −12
×
=
−3 × −4 = 12
×
=
3 × −4 = −12
×
=
Negative numbers: Multiplying
×
=
×
=
×
=
×
=
Negative numbers: Multiplying
Negative numbers: Multiplying - Try
Multiplying Negative Numbers
positive × positive = positive
4 × 5 = 20
positive × negative = negative
4 × −5 = −20
−4 × 5 = −20
negative × negative = positive
−4 × −5 = 20
8 × −3 = −24
−6 × −2 = 12
−7 × −4 = 28
−6 × 4 = −24
−15 × −3 = 45
Negative numbers: Multiplying
Negative numbers: Multiplying
Negative numbers: Multiplying
−3 × −5 × 4 =
60
I think the product will be a negative
number because there are some
positive numbers and some negative
numbers in the calculation
Do you agree with Teddy?
−3 × −5 × 4
= 15 × 4 = 60
−3 × −5 × 4
= −3 × −20 = 60
Challenge
a
b
c
d
e
f
Find two numbers which have the specified sum
and product.
? -4
Sum = −7 Product = 12  -3,
Sum = 1 Product = −12  4,
? -3
Sum = −3 Product = −10
 2,
? -5
? -3
Sum = 5 Product = −24  8,
? -6
Sum = −11 Product = 30  -5,
Sum = −1 Product = −30
 5,
? -6
Negative numbers: Dividing
−6 ÷ 2 = −3
−6 ÷ 3 = −2
−1
−1
−1
−1
−1
−1
−1
−1
−1
−1
−1
−1
−1
−1
−1
−1
−1
−1
−1
−1
−1
−5 × 3 = −15
−3 × 5 = −15
−15 ÷ 3 = −5
−15 ÷ 5 =−3
Negative numbers: Dividing
−24
−6
−6
−6 × 4 = −24
−24 ÷ 4 = −6
−6
−6
4 × −6 = −24
−24 ÷ −6 = 4
Division is the inverse
(opposite) operation of
multiplication, so the same
rules apply as before.
Negative numbers: Dividing
What result do we expect from this calculation?
Dividing Negative Numbers
positive ÷ positive = positive
20 ÷ 5 = 4
negative ÷ positive = negative
−20 ÷ 5 = −4
positive ÷ negative = negative
20 ÷ −5 = −4
negative ÷ negative = positive
−20 ÷ −5 = 4
−20 ÷ −4 = 5
Why?
−20 ÷ 4 = −5
−20 ÷ 2 = −10
−20 ÷ 1 = −20
−20 ÷ −1 = 20
−20 ÷ −2 = 10
−20 ÷ −4 = 5
Negative numbers: Dividing
?
−42 ÷ 6 = −7
−42 ÷ 7 = −6?
−42 ÷ −6 = 7?
30 ÷ −5 = −6
−50 ÷ 2 = −25
?
42 ÷ −7 = −6
÷
=
÷
=
÷
=
÷
=
−35 ÷ −7 = 5
12 ÷ −3 = −4
Negative numbers: Dividing
Negative numbers: Dividing
Negative numbers: Dividing
Negative numbers: Dividing
Negative numbers: Multiplying and Dividing
Multiplying Negative Numbers
positive × positive = positive
4 × 5 = 20
positive × negative = negative
4 × −5 = −20
−4 × 5 = −20
negative × negative = positive
−4 × −5 = 20
Dividing Negative Numbers
positive ÷ positive = positive
20 ÷ 5 = 4
negative ÷ positive = negative
−20 ÷ 5 = −4
positive ÷ negative = negative
20 ÷ −5 = −4
negative ÷ negative = positive
−20 ÷ −5 = 4
Calculate:
4 × −1 = −𝟒?
?
−3 × 7 = −𝟐𝟏
−5 × −4 = 𝟐𝟎?
−8 ÷ 2 = −𝟒?
−21 ÷ −7 = 𝟑 ?
9 ÷ −3 = −𝟑?
2
−5 = 𝟐𝟓?
−2 3 = −𝟖?
?
−10 − 2 2 = 𝟏𝟒𝟒
Negative numbers: Multiplying and Dividing
Multiplying Negative Numbers
positive × positive = positive
4 × 5 = 20
positive × negative = negative
4 × −5 = −20
−4 × 5 = −20
negative × negative = positive
−4 × −5 = 20
Dividing Negative Numbers
positive ÷ positive = positive
20 ÷ 5 = 4
negative ÷ positive = negative
−20 ÷ 5 = −4
positive ÷ negative = negative
20 ÷ −5 = −4
negative ÷ negative = positive
−20 ÷ −5 = 4
−5 × 6 ÷ −3 =
10
Negative numbers: Multiplying and Dividing
Multiplying Negative Numbers
positive × positive = positive
4 × 5 = 20
positive × negative = negative
4 × −5 = −20
−4 × 5 = −20
negative × negative = positive
−4 × −5 = 20
Dividing Negative Numbers
positive ÷ positive = positive
20 ÷ 5 = 4
negative ÷ positive = negative
−20 ÷ 5 = −4
positive ÷ negative = negative
20 ÷ −5 = −4
negative ÷ negative = positive
−20 ÷ −5 = 4
7 × −4 ÷ −2 = 14
Negative numbers: Multiplying and Dividing
Multiplying Negative Numbers
positive × positive = positive
4 × 5 = 20
positive × negative = negative
4 × −5 = −20
−4 × 5 = −20
negative × negative = positive
−4 × −5 = 20
Dividing Negative Numbers
positive ÷ positive = positive
20 ÷ 5 = 4
negative ÷ positive = negative
−20 ÷ 5 = −4
positive ÷ negative = negative
20 ÷ −5 = −4
negative ÷ negative = positive
−20 ÷ −5 = 4
−12 ÷ −3 ÷ 2 = 2
Negative numbers: Multiplying and Dividing
1
a
b
c
d
e
f
g
Calculate the following:
?
−3 × −2 = 𝟔
−12 ÷ 4 = −𝟑
?
−9 2 = 𝟖𝟏
?
4 × −13 = −𝟓𝟐
?
?
16 ÷ −2 = −𝟖
−30 ÷ −6 = 𝟓?
?
−8 × 3 = −𝟐𝟒
2
a
b
c
d
3
a
b
c
d
e
Determine the missing number:
8 × ∎ = −32
∎ =? −𝟒
−6 × ∎ = −18 ∎ =? 𝟑
−8 ÷ ∎ = −2
∎ =? 𝟒
−2 2 × ∎ = 32 ∎ =? 𝟖
Calculate the following:
−4 2 ÷ −2 2 = 𝟒?
?
−3 2 × −7 = −𝟔𝟑
50 ÷ −5 2 = 𝟐?
1 − −1 × 2 − −2 = 𝟖?
−3 3 ÷ −3 2 = −𝟑
?