Transcript 1 - Dynamic Learning

Steps
Collecting like terms
Multiplying numbers and letters
Finding the value of expressions
Mastering Mathematics © Hodder and Stoughton 2014
Combining variables – Developing Understanding
Collecting like terms
Bruce knows that 2r + r is the same as 3r.
He writes down some other sets of terms that
add up to 3r.
2 r + r = 3r
r + r + r = 3r
r + 2r = 3r
1. How many sets of terms can you find that
add to 4r?
2. Which expression is the odd one out?
a) 3m + 5n + 2n + m + 2m
b) 2n – 3m + 4m + 2n + 3n + 5m
c) 10m + 8n – 4m – 6n + 2n
There
3+ways
of
3rwithout
into
like
3mwere
+ 5n
2npossible
+
m splitting
+ 2m
= 6m
+ 7n
There
are
seven
ways
Term
2nterms.
3msubtraction.
+is4m
+2n + number
3n + 5mor= a6m
+ 7n
using
A– term
a single
variable.
There
were
7–ways
splitting
4r into
like
10m
+can
8n
4m
– of
6n
2n = together
6m
IfAthe
like
terms
are
collected
term
also
be
the+product
of+ 4n
terms.
they
are allorequivalent
numbers
variables. to 4r.
thetonumber
ofsoways
for 5r.
c) Investigate
does
not add
6m + 7n
it could
also
Examples:
be the odd one
t 5out.
3n ab m 7pq
What do you notice?
Like terms are terms that use exactly
Can you use a formula to describe your
the same variable.
rule?
5T and 3T are like terms as they use
Test your formula with other values.
the same letter.
6w and 3v are not like terms.
You can also have different orders.
+b)r is= the
4r odd one
r + r + r + r = 4r
=out
4r as the
r + r + 2r = 4r r + 2r + r = 4r 2r + r + r = 4r
subtracted
expression has six
r + 3r = 4r
3r + r = 4r
There
are
4
different
ways.
terms.
terms.
2r + 2r = 4r
So there are 7 different ways.
r + a)
r +is rthe
+ rodd
= 4rone
r + 2r
2r out
+ 2ras= it
4rhas no r + 3r
Menu
Back Forward Cont/d
More
Vocabulary
Mastering Mathematics © Hodder and Stoughton 2014
Q1
Opinion 1 Opinion 2 Answer
Q2
Opinion 1 Opinion 2 Answer
Combining variables – Developing Understanding
Collecting like terms
2a + 7b is an expression in a and b.
3a – 4b is also an expression in a and b.
They can be added together and the like
terms can be collected.
2a + 7b + 3a – 4b = 5a + 3b
1. Find three different expressions that can
be added together to give 5a + 3b.
2. Rearrange these cards to make an
algebraic magic square. Every row and
every column must add to the same
amount.
Make
algebraic
square.
Thereyour
are own
lots of
possiblemagic
answers.
Like
9a
–together
2b while
2a
+
+terms
4b
5ato–you
2b
Begin
with
a3b2 by
2a square
The ‘a’s
have
to add
Like terms use exactly the same letter
work
makeout
5a.an efficient method.
or combination
of
letters
7a
-2a
a to
++4b
3a
– 3b
b 3 to
+together
3b . 7a
3b
The make
‘b’s +have
add
Now
a
by
3
square.
They can be combined into a single
make 3b.
term.-4a
is called
4b
3a –
+collecting
3b -4a
3aor
b
3aThis
++ 3b
9a
2b
+– 4b
Can
you make
one with more than two
gathering
terms.
It
could
be:
4a
+ b, 3a adds
+ b and -2a++5b.
b.
Each
row and
variables?
Example:
2a +column
3b + 4a + 2bto= 8a
6a + 5b
Did you find a different answer?
How many more answers can you find?
There are 8‘a’s and ‘b’s altogether in 5a +
3b soHow
that’s
the same
as variable
8ab.
many
of each
are
there
altogether?
So the
answer
could be:How
ab +many
3ab + 4ab.
rows are they spread between?
So what must each row total be?
You have to keep the ‘a’s and the ‘b’s
separate.
It could be: 4a + b, 3a + b and -2a + b.
Menu
Back Forward Cont/d
More
Vocabulary
Mastering Mathematics © Hodder and Stoughton 2014
Q1
Opinion 1 Opinion 2 Answer
Q2
Clue
Answer
Combining variables – Developing Understanding
Collecting like terms
Liz is organising an evening at
the Spotted Dragon for people at
work. She knows how many
people are going. She doesn’t
know how much the tickets will
cost yet.
She writes £f for the cost of a
full ticket.
She writes £d for the cost of a
disco ticket.
Spotted
Dragon
says
that up
thethe
cost
will be
for
Cutting
room:
2fhow
+ 4d
ItThe
doesn’t
matter
you
add
terms,
but£15
they
Expression
the
full
and
£5 forThe
thef disco.
Sewing
room:
3f
+ 5d
must
beevening
kept separately.
and d terms can’t be
How
much does the
group have to pay?
Despatch:
+ 3d
combined.
A collection of 2f
numbers
and letters combined together
How many different ways can you find to work out the
using arithmetic signs.
total correct
cost? answer
question
asks for
the +
cost
The
is 13f
13d.of the tickets, not the
Which
easiest
way?
numberisofthe
people
who
are going.
Why is it easier to gather like terms first?
1. The total cost in pounds of all the tickets for
people in the office is 6f + d.
What are the cost in pounds of tickets for
each of the other departments?
2. Write an expression for the cost of tickets for
the whole group.
Cutting
2 + 4Total
= 6 tickets
Total
= Cutting
+ Sewing
room room:
+
Cutting
room: room
2f + 4d
= cost of full tickets + cost of disco
Sewing room: 3 + 5 = 8 tickets
Sewing Despatch
room: 3f+ +Office
5d
tickets
= 2f + 4d +
f + 3d + 6f + d2 + 3 = 5 tickets
Despatch:
2f3+f +
3d5d + 2Despatch:
= 2f + 3f + 2f + 6f + 4d + 5d + 3d + d
= 13fd
= 13f + 13d
Menu
Back Forward
More
Vocabulary
Mastering Mathematics © Hodder and Stoughton 2014
Q1
Opinion 1 Opinion 2 Answer
Q2
Opinion 1 Opinion 2 Answer
Combining variables – Developing Understanding
Multiplying numbers and letters
The length of this rectangle has
been covered up.
Its area is 3 ×
3
This can be written as a
formula: A = 3l
m
1. The length and width are
both unknown in this
rectangle.
3
Write a formula for its area
using l for the length and w for the width.
m
The length and width are different
Variables
are squared
inshapes
the same
way
What
other
formust
do
you
5 cm formulae
measurements
so
we
use
two
as numbers. Variables
know?
different variables.
The letters used in a formula.
2 means s x s.
sThey
Can
you
find
some
formulae
thatcan
use
a
Area =
width
×
length
represent
numbers
which
l3 and
w.
l
2 Call
cm them
=2×5
cube
like (or
r instead
change
vary). of a square?
2
= 10 cm2
So A = s .
A = lw
w
Remember ‘no sign’ means multiply.
2. The sides of this square are
hidden.
Write a formula for its area
using s for the length of the side.
Area
= s x sw
. is under one of the
The width
This
is the same as Area = s2.
notes.
The length l is hidden under the
Area = s x s.
other note.
So the Area = lw.But there’s no need to
A = ss.
Menu
Back Forward Cont/d
More
Vocabulary
Mastering Mathematics © Hodder and Stoughton 2014
The same number
could be under both.
Call it n for the
number.
write
So A =the
n xsign.
n.
Q1
Opinion 1 Opinion 2 Answer
Q2
Opinion 1 Opinion 2 Answer
Combining variables – Developing Understanding
Multiplying numbers and letters
Each
rectangle
has an area B.
Bothsmall
opinions
are correct.
Three
of them
up the large rectangle.
The total
areamake
is 6de.
The
shows that
+ B + 3e
B =and
3B.width
Thediagram
large rectangle
hasBheight
2d.
1.SoHow
many= ‘C’s
are
its area
3e ×
2d. C
there
in this
This
shows
that:
diagram?
3e × 2d Write
= 6dean
C
orexpression
2d × 3e =that
6de
they
2dshows
× 3e how
=2×
dx3×e
C
can be =counted
in d × e
2×3×
groups.= 6de
2. Each small rectangle
has width d and height e.
Each small area is de.
2d
C
C+C=2C
de
C
C+C=2C 3e
C
C+C=2C
Back Forward Cont/d
Mastering Mathematics © Hodder and Stoughton 2014
Bde = 6de
2x3
They are set out in 2 columns. Each
column adds to 3CB.
There are 6 ‘C’s altogether.
Each row has 2 small rectangles.
So 3C + 3C = 6C
The area in each row adds to 2de.
The total area is 2de + 2de + 2de = 6de.
Both opinions are correct.
x 2de
= 6de
The diagrams 3show
that:
d
3 × 2C = 6C or 2 × 3C = 6C
d
de
e
e
Write an expression for
the area of the large
rectangle.
Menu
There
are 3 rows.
row
adds to 2C.
Each column
has 3Each
small
rectangles.
The area
column adds to 3de.
There
are in
6 each
‘C’s altogether.
B
The2C
total
area
So
+ 2C
+ 2is
C3
=de
6C+ 3de = 6de.
When terms are multiplied together the
numbers can be multiplied separately.
e
Q1
Opinion 1 Opinion 2 Answer
Q2
Opinion 1 Opinion 2 Answer
Combining variables – Developing Understanding
Multiplying numbers and letters
The
small
cuboid
is d cm
wide
Opinion
is correct
for high,
parts ea)cm
and
b), and
but
3
f not
cm long.
Its
volume
is
d
×
e
×
f
cm
.
c).
This is written as def cm3.
Opinion  is fully correct.
multiplied
in anyisorder:
1.The
Thelengths
area ofcan
thebe
front
of the stack
The area of the top is 2f x 4e.

f face are
The
is made
from on
8 rectangles.
a)
There
8 cuboids
the top layer.
Product
The area of each is fe.
There are 3 layers, so that’s 3 x 8 =
24e cuboids. Each one has volume def.
Two
numbers
are
This
shows
that
2or
f=
xvariables
4
e =def
8fe
.
3dstack
b) Volume
of
24
. that
c) 3d x 4e multiplied
x 2f = 3 together.
x 4 x 2 x d x e x 2f
f
= 24 def
4e
d
3d × 4e.
3dThe
× 4e
× is
2fmade
=3×4×2×d×e×f
face
de
3d
from 12 small= 24 def
3drectangles.
× 2f × 4eThe
=3×2×4×d×f×e
4e
area of each =is24
de.def
××
4e3=×12de.
2f This
× 4eshows
× 3d that
= 2 3d
×4
f×e×d
Write a similar
expression
for the area of
= 24
def
top and side faces.
The area of the side is 2f x 3d.
The face is made from 6 rectangles.

The area of each is fd.
a) If you slice the stack along the top
there
arethat
two2f
piles
of =
126fd
cuboids. So
This
shows
x 3d
that’s 24 altogether. Each one has
volume def.
b) Volume of stack = 24 def.
c) To get
the volume
you add
upthe
all the
Always
multiply
the numbers
and
edges. separately.
variables
That
to 3before
d + 4ethe
+ 2variables
f.
Write
thecomes
numbers
2. a) How many small cuboids are there in
the stack? What is the volume of each
one?
b) What is the volume of the whole stack?
c) Do you get the same volume if you
multiply the length, width and height of
the stack?
Menu
Back Forward
Vocabulary
Mastering Mathematics © Hodder and Stoughton 2014
Q1
in a product.
2f x 4e = 8fe
2f x 3d = 6fd
Top
Side
Answer
Q2
Opinion 1 Opinion 2 Answer
Combining variables – Developing Understanding
Finding the value of expressions
The statement compares two expressions
involving the variable m.
1. Is the statement always true, sometimes
true or never true?
If you think ‘sometimes’, when is it true
and when is it not true?
It depends on the value of m.
8m – 2 is larger than 4m + 18 when m > 5.
When m = 3
8m – 2
=8×3–2
= 24 – 2
= 22
When m = 7
8m – 2
=8×7–2
= 56 – 2
= 54
When m = 5
8m – 2
=8×5–2
= 40 – 2
= 38
4m+18 = 30
4m+18 = 46
4m+18 = 38
Smaller
Menu
Larger
Back Forward Cont/d
More
Expression
Think
of
any
number.
A collection of numbers and letters
The value of
combined together using arithmetic
– 2 expressions that
Can
you make 8m
up two
signs.
more
thevalue
valuewhen
of your
will bothisgive
thethan
same
4m + 18into them?
number is substituted
Variables
The letters used in a formula.
Example:
They represent numbers which can
My
number
5.
change
(or is
vary).
8m –
is always
larger as there
arethe
I need
to 2find
two expressions
that are
eightwhen
‘m’s 5
compared
with into
onlythem.
four in
same
is substituted
4m + 18.
They could be 3n + 2 and 22 – n
These both have the value 17 when n = 5
is substituted.
When
m=3
4m + 18 = 30
But 8m – 2 = 22
So 8m – 2 is smaller.
The same
Vocabulary
Mastering Mathematics © Hodder and Stoughton 2014
Q1
Opinion 1 Opinion 2 Answer
Combining variables – Developing Understanding
Finding the value of expressions
Here are three expressions in k.
Remember that k2 means k × k.
Also k means  × k.
To work this out find half of k, or k ÷ 2.
1. Find values of k for which each of these
expressions is negative.
2. The three expressions are to be put into
order. Find a value of k for each one that
gives it the middle value.
When k = 1,
k + 6 =  × 1in+ 6
Expressions
There
are many
values
k for paper
which to
Use
graphing
software
orofgraph
=  that
+ 6k2the
A
quick
way
of
saying
k + graphs
6 and 8
3k three
are negative.
draw
of–the
expressions
= 6
expression
uses
this
variableLargest
k
+
6
above. Superimpose all three on the
When k =21, 8 – 3k = 8 – 3 × 1
But kgraph.
will always be positive whatever
same
8= –8 3k
–3
An
expression
the value of k. in terms of k would use
5
Middle
only the variable k =combined
with
Look at where the2graphs cross. Can you
When
k = 1, and arithmetic
k = 1 ×signs.
1
numbers
Remember
two negative
numbers
see
why eachthat
expression
can have
the
=1
Smallest
Example:
3k
+
7
multiply
to give
a positive number.
middle
value?
4k2 – 3k + 5
What are the critical values of k when the
order of the expressions changes?
When k
k == 4,
-14,k
k++66 =
=  ×
x -14
When
4 + 6+ 6
When k = 2,
k + 6 =  × 2 + 6
I agree with the first two in
=
-7
+
6
=2+6
=1+6
= -1 [k + 6 is negative]
Opinion , but the value
=8
Middle
=7
Largest
When k = 5, 8 – 3k = 8 – 3 x 5
for k2 is wrong.
When k = 4, 8 – 3k == 88 ––315
×4
When k = 2, 8 – 3k = 8 – 3 × 2
== 8-7
– 12
8 –negative.
6
[8 – 3k is negative]
k2 can’t ever= be

–
2
4 x Smallest
=2
Smallest
When k = -4,
k =
= -4
-4
–
2
2
2
When k = 4,
k == 4 16
× 4[k is negative] When k = 2,
k =2×2

= 16 Largest
=4
Middle
Menu
Back Forward Cont/d
More
Vocabulary
Mastering Mathematics © Hodder and Stoughton 2014
Q1
Opinion 1 Opinion 2 Answer
Q2
Answer 1 Answer 2 Answer 3
Combining variables – Developing Understanding
Finding the value of expressions
Avonford Academy awards house points for
achievement and voluntary service. House
points are taken away if homework is not
finished. The expression 3a + 5v – 2h is used to
calculate the number of points gained
1. What do the variables a, v and h represent?
2. Jo has volunteered 7 times this term. She
thinks there’s been a mistake because her
term total is zero points. What do you think?
aJois has
the gained
points 5
awarded
v points for
for achievement.
volunteering.
vThis
is the
points5awarded
for
volunteering.
is worth
x 7 = 35
points
hThat’s
is thean
points
taken off for no homework.
odd number.
If 2 points are taken off for every missing
homework it’ll never get to zero. The lowest
ait iscould
the number
awards given.
get to isof1 achievement
point. 
v is the number of volunteering awards given.
h is the number of times homework is missing.
Menu
Back
More
Mastering Mathematics © Hodder and Stoughton 2014
Q1
Some possible answers are:
Make
a = 5,avspreadsheet
= 7, h = 15 to help the House
Leader
the
points.
Points keep
= 3 ×track
5 + 5of×
7 total
–2×
25
House points
= 15 + 35 – 50
Mike goes
Avonford Academy.
3 points
= 0toAchievement:
He
finishes
the
term
with
345 points.
service:
points
a = 25, v Voluntary
= 7, h = 55
He
has =
volunteered
times.
Points
3No
× homework:
25 + 53×
7–2– points
2 × 55
= 75 + 35 – 110
How many
= 0 times could he have missed
his
homework?
A student
is given 3 points for every
achievement. So the total points for
achievement is 3 × the number of
awards.
 is wrong
because sheawards
could
aOpinion
is the number
of achievement
have gained some extra points for
given.
vachievement.
is the number of volunteering awards
That could bring her total up to an
given.
even number before any homework
h is the number of times homework is
points are taken off. 
missing.
Opinion 1 Opinion 2 Answer
Q2
Opinion 1 Opinion 2 Answer
Combining variables – Developing Understanding
Editable Teacher Template
Information
Vocabulary
1. Task – fixed
More
2. Task – appears
Q1
Opinion 1
Q1
Opinion 2
Q1
Answer
Q2
Opinion 1
Q2
Opinion 2
Q2
Answer
Menu
Back Forward
More
Vocabulary
Mastering Mathematics © Hodder and Stoughton 2014
Q1
Opinion 1 Opinion 2 Answer
Q2
Opinion 1 Opinion 2 Answer
Combining variables – Developing Understanding