Page 29 - vectormsint_catalogue

5 5 5 5 5 5 Alge br aic expr e s sion s 5 5
Algebraic expressions
7 7 Use known properties to simplify the algebraic expressions.
4 4 Do the calculations.
(a) 2(7 p ) + 3( p
) + 3(
(a) 2(7p − q) + 3(p + 2q) = __________________________________________________________________
(a) 2(7
Example
(b) −6x + 5[−2x + 3(4 − x)] = ________________________________________________________________
(a) 3a + 9a = 12a
k
− 17
k
k
(b) 21k − 17k = 4k (c) 3[x(2 − 3y) + 5(4 − xy)] = ________________________________________________________________
k
= 4
y
(c) 4x · 5y = 20xy
· 5
· 5
x
+ 1 + 2(
+ 1 + 2(
: 6
(d) 36t : 6s = 36t 6s = 6t s (d) 4x + [3x − x + 1 + 2(x − 3)]= ______________________________________________________________ Lower Secondary
t
t
8 8 Factorise the algebraic expressions. Explorers! Maths 1, e-Workbook
(a) 11x − 9x − 5x =
(b) 6ab + ab − 8ab = Example Factorisation
Factorisation
= 3(
Factorisation is the process of changing an algebraic
(a) 3x + 3xy + 3z = 3(x + xy + z) Factorisation is the process of changing an algebraic
= 3(
= 3(
z
x
z
(c) 25xy · 15ab =
expression to an equivalent in the form of a product of two
(
(
x
(b) xy − ay + y = y(x − a + 1) expression to an equivalent in the form of a product of two
or more factors.
st
=
(d) 63kl : 9st = or more factors.
kl
: 9
st
kl
Tip
Tip
5 5 Use known properties to simplify the algebraic expressions. We can use the distributive property to factorise an
op
e
he di
s
i
s
ac
t
or
o f
r
t
y t
e an
t
i
ibu
t
W
r
v
s
e pr
e t
e can u
alg e br aic expr e s sion .
algebraic expression.
(a) 16x − 8 − 5x =
(b) 9a − 12 + 3a + 6 =
z
− 18
y
z
=
− 18
(a) 18x − 18y + 18z = ______________________________________________________________________
+ 7
(c) 5p − 6q + 7p − 2 =
(c) 5
(c) 5 p + 7 p (b) 26a + 13b − 26c = _____________________________________________________________________
(d) y + x − 3y + 4x =
(c) abc − ab − ca = ________________________________________________________________________
(e) −17x − 4y + 32x + 3y = (d) 3xz − 6xy − x = ________________________________________________________________________
− 4
− 4
y
xz
− 6
xz
(f) −10m + 37n − 13m − 31n =
(e) kl − lx + yl + la = ______________________________________________________________________
z
− 4
− 4
z
− 9
z
z
y
(g) 3x − 4y + 8z − 9x + 3y − 6z = =
(f) 3a − 6ax + 12ay = ______________________________________________________________________
(g) 4mn − 16n + 12m = ____________________________________________________________________
6 6 Use the distributive property to expand the algebraic expressions.
9 9 Use factorisation and calculate.
Distributive property
operty
Distributive pr
b
a
·
a a · (b + c) = (a · b) + (a · c) c ) (a) 189 · 74 + 189 · 26 = ___________________________________________________________________
+
b
) = (
·
) + (
a
· (
c
) − (
·
a
b
−
a a · (b − c) = (a · b) − (a · c) c )
· (
c
·
b
) = (
a
(b) 255 · 995 + 255 · 5 = ___________________________________________________________________
(a) 8(a + 3) = (b) a(b − c) = (c) 104 · 3 + 104 · 7 = _____________________________________________________________________
k
k
− 7
l
(c) c(ab − b) = (d) −4(5k − 7l) = (d) 252 · 34 + 252 · 66 = ___________________________________________________________________
) =
l
(
(
y
−
z
z
+ 2
+ 2
p
(e) −6m(11n − k + 2p) = (f) 3x(y + 3z − w) =
k
+ 2
k
30 31
3 Decimals 3
3.1 Decimals 3.2 Decimals and
De
im
De
Decimals
al
al
• Draw Ss’ attention to the 3 De c c c im al s s s 3 fractions
im
theory section Decimals. Changing decimals into
• Explain to Ss that decimals Apply your knowledge decimal fractions
are numbers that have a All about maths t h s 3.1 Decimals
u
A
a
ll ab
t m
o
The number system
whole number part and a The numb e r sys t e m Decimals are numbers that have a whole number part and a fractional part. The 1 Write the numerals. • Draw Ss’ attention to the
we use in modern
(a) 38 hundredths
fractional part and that the times was born in i z mī fractional part of a decimal number comes after the whole part and the decimal (b) 121 hundredths theory section Changing
K
a
. Al-
ār
w
I
ndi
decimals into decimal
India. Al-Kwārizmī
0 C
0 C
ar
ar
E) w
E) w
fractional part of a decimal ( ( (around 820 CE) was ic ic a a s s an i i an point and always has a value smaller than one. (c) 243 hundredths fractions.
ound 82
ound 82
The numbers 14,75, 1,041 and 0,00175 are decimals.
a
he
a
t
t
t m
he
t m
ea
ea
a gr
a gr
a great mathematician
t
t
a
m
m
a
number comes after the and astronomer. He . H e We can write a decimal number in the form of a decimal fraction. (d) 4 tens 3 ones 5 tenths 7 hundredths
onome
s
r
t
and a
r
o
ok t
ot
w
e a b
r
whole part and the decimal wrote a book that h a t e r 215,105 = 215 105 (e) 11 tens 4 ones 9 tenths 5 hundredths • Explain to Ss that changing
included this number
inclu
s numb
d t
hi
de
(f) 3 hundreds 8 ones 7 tenths 5 hundredths
m ba
s
e
t
sys
e
d on
point and always has a value system based on a a t t ic ic s s . . 4,8 = 48 10 14,78 = 1478 1000 a decimal into a decimal
100
Indian mathematics.
I I
m
he
m
a
ndi
ndi
an m
an m
a
t
he
t
r
r
smaller than one. Later, through Arabic ie ie s s 2 Complete the sentences. fraction depends on
abic
t
ough A
hr
ough A
r
hr
r
e
t
a
e
a
L L
abic
, t
, t
i
commercial activities
t
r
(a) In 5890,73:
i
omme
omme
vit
vit
c c
i
c
t
r
al ac
c
i
al ac
indu
r
• Explain to Ss that the t t this Hindu-Arabic Place value of decimals (i) the digit 5 is in the place. It stands for . the number of decimal
A
A
s H
s H
abic
abic
hi
indu
-
-
hi
r
numbers 14,75, 1,041 and number system also We can use a place value table to show the value of each digit in a decimal. (ii) the digit 8 is in the place. Its value is . places it has and that the Explorers! Maths 1, Τeacher’s Book
spread to the western
0,00175 are decimals and world. In this number (iii) the digit 7 is in the place. It stands for . denominators of decimal
system we use zero (0)
(iv) the digit 3 is in the place. Its value is .
s a pl
r and
ac
e holde
that we can write a decimal a as a place holder and , 2 , (b) In 85,43, the digit 3 is in the place. fractions are indices of 10, so
, 1
e
nine more digits, 1, 2,
digits
nine mor
number in the form of a 3, 4, 5, 6, 7, 8, and 9, to , t o Hundredths Thousandths (c) In 329,41, the digit is in the hundredths place. we write as many zeros as
, 6
, 8
, 4, 5
3
, 7
, and 9
r
pr
n
s
e
e
e
t numb
decimal fraction. represent numbers. e r rs e n . t Thousands Hundreds Tens Ones Tenths (d) The value of the digit 8 in 812,47 is . the number of decimal places
f
e call t
he di
e
W
f
We call the different
s o
t
ion
• Have Ss study how we can c combinations of these s s e 4 7 8 5 , 6 8 4 (e) The digit 6 in 73,46 stands for . each decimal number has.
a
f t
he
ombin
t
ten digits, numerals.
,
e
n digits
nume
al
r
write 4,8, 14,78 and 215,105 in 3 Write the numerals. • Have Ss study how we can
the form of decimal fractions. 4000 or 4 thousandths or 1000 4 (a) 8 thousandths (b) 24 thousandths write 0,9, 0,11, 0,451 and
4 thousands
700 or decimal comma 8 hundredths or 100 8 (c) 109 thousandths (d) 230 thousandths 0,087 in the form of decimal
7 hundreds (e) 2419 thousandths (f) 4718 thousandths
All about maths Note 80 or 8 tens 6 tenths or 10 6 fractions.
The decimal comma
• Focus Ss’ attention on the is the symbol that 5 or 5 ones
separates the whole
section All about maths. from the fractional 3.2 Decimals and fractions
• Point out to Ss that the part. Changing decimals into decimal fractions
number system we use Reading decimal numbers How we change a decimal into a decimal fraction depends on the number of
in modern times was When reading decimals, we first read the whole part, then the decimal comma, decimal places it has. The denominators of decimal fractions are indices of
10, so we write as many zeros as the number of decimal places each decimal
born in India. Al-Kwārizmī and finally we read each digit of the fractional part separately. number has.
(around 820 CE) was a • How do we read 19,482? 0,9 = 9 0,11 = 11 0,451 = 451 0,087 = 87
great mathematician and Tens Ones Tenths Hundredths Thousandths (1 decimal 10 (2 decimal 100 (3 decimal 1000 (3 decimal 1000
astronomer. He wrote a book 1 9 , 4 8 2 place) places) places) places)
that included this number nineteen comma four eight two
system based on Indian 44 45
mathematics. Later, through
Arabic commercial activities
this Hindu-Arabic number
system was also spread to • Focus Ss’ attention on the place value table, and explain to Ss how
the western world. In this we read a number in each place value. • Have Ss do the activities in the Apply your knowledge section.
number system we use zero
(0) as a place holder and nine Apply your knowledge
more digits 1, 2, 3, 4, 5, 6, 7, 8, Note
and 9 to represent numbers. Point out to Ss that the decimal comma is the symbol that separates 1 (a) 0,38 (b) 1,21 (c) 2,43
• Explain that the di erent the whole from the fractional part. (d) 43,57 (e) 114,95 (f) 308,75

combinations of these ten 2 (a) (i) thousands, 5000 (ii) hundreds, 800
digits are called numerals.
Reading decimal numbers (iii) tenths, 0,7 (iv) hundredths, 0,03
Place value of decimals • Draw Ss’ attention to the theory section Reading decimal numbers. (b) hundredths
(c) 1
• Explain to Ss that when reading decimals, we first read the whole
• Draw Ss’ attention to the part, then the decimal comma and finally we read each digit of the (d) 800
theory section Place value of fractional part separately. (e) 0,06
decimals. • Ask Ss How do you read 19,482?
• Explain to Ss that we can use • Focus Ss’ attention on the place value table, and explain to Ss that 3 (a) 0,008 (b) 0,024 (c) 0,109 (d) 0,230 (e) 2,419 (f) 4,718
a place value table to show the we read the number 19,482 as nineteen comma four eight two.
value of each digit in a decimal.
54 55
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