Transcript 3 - Dynamic Learning from Hodder Education

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Collecting like terms
Multiplying with variables
Understanding expressions
Substituting into expressions
Substituting negative numbers
Mastering Mathematics © Hodder and Stoughton 2014
Combining variables – Worked Examples
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6b means 6 x b.
An expression is a collection of terms.
A term is a single number or a variable. It can
also be a product of numbers or variables.
Examples:
t
5 3n ab m 7pq
4a + 6b + a – 2b + 5
Write 4a, not a4.
You usually write
this as a, not 1a.
Like terms are terms that use exactly the same variable.
5T and 3T are like terms as they use the same letter. 3a2 and 6a2 are also like terms.
6w and 3v are unlike each other. They are not like terms.
5a and 6a2 are not like terms as they use different powers of a.
Like terms can be combined into a single term. This is called collecting or gathering terms,
and to do it to an expression is simplifying the expression.
For example: 2a + 3b + 4a + 2b = 6a + 5b.
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Mastering Mathematics © Hodder and Stoughton 2014
Combining variables – Worked Examples
Collecting like terms
1. Simplify these expressions:
a) 5e + 2e
b) 6v – 2v + 4v
c) 7t – 12t
d) 3p + p – 4p
2. All of these expressions simplify to 5m.
What is underneath the sticky notes?
a) 8m –
b) 3m – 4m +
c) m – 3m
+4m.
a) 8m – 3m = 5m.
a) 5e + 2e = 7e
b) 6v – 2v + 4v = 8v
b) 3m – 4m
= -m. This isproblems
6m less thanthose
5m.
c) 7t – 12tMake
= -5t up some
d) 3p + p – 4p like
=0
So the in
answer
is 3m
– 4m
6m = 5m.
question
2 for
your+ partner
to
Remember
to
write
p
instead
of
1p.
c) The m solve.
terms that can be seen add to 2m.
If the
cancelled
better
to
Anvariables
extra 3m have
is needed:
m – out
3m then
+ 3mit’s
+ 4m
= 5m.
write 0 instead of 0p.
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Answer 1 Answer 2
Combining variables – Worked Examples
Collecting like terms
1. Simplify these expressions:
a) e2 + 2e2
b) 7t3 – 2t3
c) 7w +12w2 – 2w
2. Simplify these expressions:
a) 2f + 3 + 4f b) 8 + 6y – 9y c) 3g + 7 – 8g
a)
a)
b)
c)
b)
c)
Make a set of domino cards with
e2 +expressions
2e2 = 3e2 that have to be
2f +matched.
3 + 4f = 6f + 3
3
7t
2t3example,
= 5t33 can’t
The–For
number
combined
twobecards
might with
fit the 6f.
The
terms
w8 are
like terms. They can be
8 + together
6y
– 9yin=like
+this:
-3y
combined
It is easierto
togive
write5w.
thisThe
as 8w2– term
3y. must be kept
separate. 4a – 3a 3a + 5a 9a – a 7a + 2a
3g + 7 – 8g = -5g + 7
2 – 2w = 5w + 12w2
The
answer
is
7w +12w
ThisPlay
can the
be written
as 7a
–partner.
5g.
game
with
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Answer 1 Answer 2
Combining variables – Worked Examples
Multiplying with variables
1. Simplify these expressions:
a) 2v × 3
b) m × 3a
c) 4n × 2n
2. How many different expressions can you find
that simplify to 6e2f2?
Example: 2e × 3f × e × f
Investigate
what
a)
2v×3
ismany
the same
ashappens
3×2v. when you
There
are
combinations.
simplify other powers.
Themultiply
answer and
is 6v.
The numbers must be either 1 and 6 or 2 and 3.
2×p
3 ×a.
b) m×3a
means m
×3
This is the same as
The e2Example:
can comepfrom
e x e=or from a number×e2.
3×m
×a. So the answer is 3ma.
p×p×p×p×p
2×p3
Always
writeanswers
the
before the letter.
Some
possible
include:
= pnumber
5
= p×n
c)
4n ×2n = 4 ×2
×n.
2e×3e×f×f
e2×2f×3f
6e×e×f2
= 8 ×n ×n
Can you
a rule?
2.
= 8ndiscover
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Combining variables – Worked Examples
Understanding expressions
1. Match the expressions in words to the
equivalent algebraic expressions.
Double n and then add m
2n – m
Double n and then subtract from m
2n + m
Double n and then take away m
m – 2n
2. Write an algebraic expression for these word
expressions.
What
How many different
does expressions can
you write using
2(m +any
n) of these
symbols: mean?
Alan
Meena
+ – ×You
( )add
c m
d and
4 n
together first, then
multiplytobysay
2. what
Challenge your partner
each one means
a) Add p and q together and then multiply by 3.
b) Add p to three times q.
c) Add 3 to p and then multiply by q.
a) 3(p + q)
Remember the number is written before the
Double
n andwhen
then multiplying.
add m
2n – m
variables
b) 3q + np and then subtract from m
Double
2n + m
This is the same as p + 3q.
Double
n and then take away m
m – 2n
c) (p + 3)q
It is neater to write q(p + 3) or q(3 + p).
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Combining variables – Worked Examples
Substituting into expressions
Work out the value of these expressions.
easier
keep
track a
Use e = 5, f =It’s
2 and
g to
=3
to make
of your
working
you
set of expressions
that
take ifevery
write the numbers you
value from 1 to
26.
have substituted below
Use the values e = 5, f = 2 and g = 3.
a) 3ef + 2
b) -2f + g
c) (4 + 3e)
d) 3f + g2
e) 12 – 4e
f) 4(e – 4g)
their place in the
Use your expressions
to write a
expression.
coded message for a friend to
decode.
g) 4fg – 2ef + 3eg
a)
3 e f +2
= 3×5×2 + 2
= 30
+2
= 32
b)
-2 f + g
= -2×2 + 3
= -4 + 3
= -1
c)
(4 + 3e)
= (4 + 3×5)
= (4 + 15)
=  ×19 = 9
e)
12 – 4 e
= 12 – 4×5
= 12 – 2
= -8
f)
4 (e – 4 g)
= 4×(5 – 4×3)
= 4×(5 – 12)
= 4×-7
= -28
h)
4 f g–2 e f +3 e g
= 4×2×3 – 2×5×2 + 3×5×3
= 24
– 20
+ 45
= 49
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d)
3 f + g2
= 3×2 + 3×3
= 6 + 9
= 15
Answer
Combining variables – Worked Examples
Substituting negative numbers
1. Work out the value of these expressions when
j = -2 and k = 3 and n = -4
a) 3k + 2j + 4
b) n(j + 5)
c) jk – n
Take extra
care with
negative
numbers.
2. Substitute v = -2 into these expressions.
a) v2 – 2
b) (2v + 7)
v2 –+2 2j +b)4 (2b) v +n7) (j + 5)
3k
== 3×3
-22 +–2×-2
2
+ 7) –2 + 5)
+ 4 = (2×-2
= -4×(
== -2×-2
9 + – 2-4 + 4 = (-4 +=7)
-4×3
== 49 – 2
= ×3 = -12
=2
= 1
c)
j k – n
= –2×3 – -4
= -6
+4
= -2
a)
a)
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Combining variables – Worked Examples
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Combining variables – Worked Examples