E1, E2, E3
L1, L2
GCSE

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ENTRY LEVEL MATHS

Multiplication and division

General stuff

When you see some groups of same items then multiplying is the fastest way to calculate the total.How many white pads are there?
One way you could say:there are two rows by 4 pads
that's 4+4=8
or 2×4=8
Other way you could say:there are four columns by 2 pads
that's 2+2+2+2=8
or 4×2=8
Remember: you can swap the numbers either way round when multiply them
2×4=4×2=8
Practice tip: download 2 and 3 times table, print it, stick it on the wall and practise each day.
When you feel confident with it, then move on with next tables:
download 4 and 5 times table;
download 6 and 7 times table;
download 8 and 9 times table.

Practice tasks:

  • How many cubes are there?
    4 rows by 5 cubes makes
    5+5+5+5 = 4×5 = ?
    OR
    5 columns by 4 cubes makes
    4+4+4+4+4 = 5×4 = ?
    4×5=20
    5×4=20
  • How many cubes are there?
    How many dots are there?
    × = cubes
    × = dots
    2×3 = 6 cubes
    6×3 = 18 dots
  • How many 5p coins are there?
    How much money is there?

    There are 8 coins of 5p
    That makes 8×5p = 40p
  • Write these as multiplication sums:
    7+7+7+7+7 = 35
    8+8+8 = 24
    4+4+4+4 = 16
    two tens makes twenty
    three fives are fifteen
    5×7 = 35
    3×8 = 24
    4×4 = 16
    2×10 = 20
    3×5 = 15
Find missing numbers:
  • 4×2=
    9×2=
    3×2=
    7×2=
    2×=10
    ×2=16
    2×=12
    ×2=2
    4×2=8 9×2=18
    3×2=6 7×2=14
    2×5=10 8×2=16
    2×6=12 1×2=2
  • double five
    two lots of two
    twice eight
    three multiplied by two
    double nine
    six times two
    twice ten
2×5=10
2×2=4
2×8=16
3×2=6
2×9=18
6×2=12
2×10=20

Opposite operations

Practice tasks:

Can you see the four sums of the multiplication and division:

  • 2×4=8
    4×2=8
    8÷2=4
    8÷4=2

  • 3×6=18
    6×3=18
    18÷3=6
    18÷6=3

  • 4×6=24
    6×4=24
    24÷4=6
    24÷6=4
Find missing numbers:
2×5= 16÷8=
3×=21 45÷=9
×4=16 ÷7=2
2×5=10 16÷8=2
3×7=21 45÷5=9
4×4=16 14÷7=2

Multiplication words

Division words

Practice tasks:

Find missing words:
  • The number 35 is p t of 5 and 7
  • The numbers 8 and 9 are f s of 72
  • The number 48 is m e of either 6 or 8
  • Since 5×7=35, the number 35 is product of 5 and 7
  • Since 8×9=72, the numbers 8 and 9 are factors of 72
  • Since 6×8=48, the number 48 is multiple of either 6 or 8

Find missing words:
  • The number 6 d s 36
  • The number 25 is d e by 5
  • 24 o r 8 makes 3
  • Since 36÷6=6, the number 6 divides 36
  • Since 25÷5=5, the number 25 is divisible by 5
  • Since 24÷8=3, then we can say 24 over 8 makes 3

Find missing words:
  • In the "bus stop" division we write the d d under the "bus stop"
  • Q t may include the whole part and the remainder
  • 5.27 is a d l number
  • In the "bus stop" division we write the dividend under the "bus stop"
  • Quotient may include the whole part and the remainder
  • 5.27 is a decimal number

The use of times table

You can download your times table here.

Practice tasks:

Find product:
  • 3×6=?
  • 4×5=?
  • 7×8=?
  • 3×6=18
  • 4×5=20
  • 7×8=56

Find quotient:
  • 30÷6=?
  • 81÷9=?
  • 24÷8=?
  • 30÷6=5
  • 81÷9=9
  • 24÷8=3

Find factor:
  • 8×=32
  • ×9=27
  • 8×4=32
  • 3×9=27

Find divisor:
  • 63÷=9
  • 63÷7=9

Find dividend:
  • ÷7=6
  • 42÷7=6

Multiplication

Multiplication facts

0 × any number = 0 e.g. 0×0=0, 0×1=0, 0×2=0, 0×3=0, etc.

1 × any number = same number e.g. 1×0=0, 1×1=1, 1×2=2, 1×3=3, etc.

10 × any integer = same number with 0 at the end e.g. 10×1=10, 10×5=50, 10×20=200, 10×37=370, etc.

100 × any integer = same number with 00 at the end e.g. 100×4=400, 100×16=1600, 100×23=2300, 100×60=6000, etc.

1000 × any integer = same number with 000 at the end and so on...

Short multiplication - by a single digit number

Practice tasks:

Long multiplication - by a few digit number

Practice tasks:

Division

Division facts

any number ÷ 0 is undefinedand therefore not allowed
so NEVER divide by zero

0 ÷ any number = 0 e.g. 0÷1=0, 0÷7=0, 0÷24=0, 0÷158=0, etc.

any number ÷ 1 = same number e.g. 1÷1=1, 8÷1=8, 49÷1=49, 515÷1=515, etc.

any number ÷ same number = 1 e.g. 1÷1=1, 9÷9=1, 36÷36=1, 284÷284=1, etc.

integer that ends with zeros ÷ 10 = cross out last zero e.g. 40÷10=4, 100÷10=10, 170÷10=17, 1050÷10=105, etc.

integer that ends with zeros ÷ 100 = cross out last two zeros e.g. 300÷100=3, 800÷100=8, 1300÷100=13, 21000÷100=210, etc.
and so on...

Short division - by a single digit number

Practice tasks:

basic medium harder
6322÷2= 1316÷5= 656÷7=
636÷3= 306÷8= 728÷9=
6342÷7= 1367÷4= 7790÷9=
basic medium harder
6322÷2
=3161		 1316÷5
=263 rem 1
=263.2		 656÷7
=93 rem 5
=93.714...
636÷3
=212		 306÷8
=38 rem 2
=38.25		 728÷9
=80 rem 8
=80.888...
6342÷7
=906		 1367÷4
=341 rem 3
=341.75		 7790÷9
=865 rem 5
=865.555...

Long division - by a few digit number

Practice tasks:

basic medium harder
3465÷11= 4863÷15= 8063÷26=
8463÷21= 7189÷14= 36254÷44=
basic medium harder
3465÷11
=315		 4863÷15
=324 rem 3
=324.2		 8063÷26
=310 rem 3
=310.115...
8463÷21
=403		 7189÷14
=513 rem 7
=513.5		 36254÷44
=823 rem 42
=823.954...

Level 1 and Level 2

GCSE

A STRAIGHT LINE

A straight line we can represent in coordinate plane by an equation y=mx+c

  • x and y are two coordinates of each particular point that belongs to the line
  • m is gradient that describes the slope of the line
  • c is y-intercept where the line crosses y-axis
We can create table of values from equation of the line where for each chosen x coordinate we can calculate corresponding y coordinate. And other way round: from the table of values with coordinates from at least two points we can work out equation of the straight line

From the equation to the graph

Practice tasks:

Create table of values and draw the graph of the straight line:
y = -x + 2

From the points to the equation

Practice tasks:

Work out the equation and draw the graph of the straight line
x y
8 -1
-4 5
Two points
M(-2,-1) and N(2,3)
belong to the same line.
Work out the equation of the straight line.
y = x+1

Intercepts from the table

Practice tasks:

Work out the intercepts:
x y
-3 -12
-2 -9
-1 -6
(0,)
(,0)
(0,-3)
(1,0)

Simultaneous equations

Task 1: one solution
Tasks 2 and 3: other cases

Correction of the mistakes:
Mistake1 read "different" instead of "diferent";
Mistake2 read "=-2-3/2" instead of "=-2+3/2" and following read "=-7/2" instead of "=-1/2"

Practice tasks:

How many solutions are there in following sets of the simultaneous equations? Can you name them?
  • 2x+y=9
    4x+2y=20
    Equations in standard form
    y = -2x+9
    y = -2x+10
    Lines are parallel. There are no solutions.
  • 3x-y=12
    6y=18x-72
    Equations in standard form
    y = 3x-12
    y = 3x-12
    Lines match each other. There are infinite amount of solutions - all along the line.
  • 4x+2y=10
    2x-2y=14
    Equations in standard form
    y = -2x+5
    y = x-7
    There is one solution
    x = 4 and y = -3
    OR
    point (4,-3)
  • What are essential requirements to determine the amount of solutions for linear simultaneous equations?
    a) exactly one solution;
    b) no solutions;
    c) infinitely many solutions?
    In standard form y=mx+c there should be:
    a) different coefficient m next to x;
    b) same coefficient m but different number c;
    c) same coefficient m and same number c.
  • If one of the lines is y= -75x+57 then what is possible equation for another line to have:
    a) exactly one solution;
    b) no solutions;
    c) infinitely many solutions?
    The other line have to come in standard form y=mx+c to:
    a) different coefficient m next to x and no matter of c,
    e.g. y= -5x+57 or y= -5x+5 or y= 75x+57;
    b) same coefficient m but different number c,
    e.g. y= -75x+50 or y= -75x-57;
    c) same coefficient m and same number c, so the only option is y= -75x+57